HYPERSPACES
PURE
AND APPLIED
A Program of Monographs,
MATHEMATICS Textbooks,
EXECUTIVE
and Lecture Notes
EDITORS
Earl J. Taft Rutgers University New Brunswick, New Jersey
EDITORIAL M S. Baouendi Universi~ of California, San Diego Jane Cronin Rutgers Universi~ Jack K. Hale Georgia Institute of Technology
Zuhair Nashed University of Delaware Newark, Delaware
BOARD Anil Nerode Cornell Universify Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University
S. Kobayashi University of California, Berkeley
Gian-Carlo Rota Massachusetts Institute of Technology
Marvin Marcus University of Calrlfornia, Santa Barbara
David L. Russell Virginia Polytechnic Institute and State University
W. S. Massey Yale University
Walter Schempp Universitat Siegen
Mark Teply University of Wisconsin, Milwaukee
MONOGRAPHS AND TEXTBOOKS PURE AND APPLIED MATHEMATICS
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. z:. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.
IN
K. Yano, Integral Formulas in Riemannian Geometry (I 970) S. Kobayashi, Hyperbolic Manifolds and Holomorphrc Mappings (I 970) V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; Littlewood, trans.) (1970) B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translatron ed.; K. Makowskr, trans.) (1971) L. harici er a/., Functional Analysis and Valuation Theory (I 971) S. S. Passman, Infinite Group Rings (1971) L. Domhoff, Group Representation Theory. Part A: Ordinary Representatron Theory. Part B: Modular Representation Theory (1971, 1972) W. Boofhby and G. L. Weiss, eds., Symmetric Spaces (1972) Y. Mafsushima, Differentiable Manifolds (E. T. Kobayashr, trans.) (1972) L. E. Ward, Jr., Topology (1972) A. Babakhanian, Cohomological Methods in Group Theory (1972) R. Gilmer, Multiplicative Ideal Theory (I 972) J. Yeh, Stochastic Processes and the Wiener Integral (I 973) J. Barros-Nero, Introduction to the Theory of Distributions (I 973) R. Larsen, Functional Analysis (1973) K. Yano and S. lshihara, Tangent and Cotangent Bundles (1973) C. Procesi, Rings with Polynomial Identities (1973) R. Hermann, Geometry, Physics, and Systems (I 973) N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) J. Dieudonnt!, Introduction to the Theory of Formal Groups (1973) /. Vaisman, Cohomologv and Differential Forms (I 973) B.-Y. Chen, Geometryof Submanifolds (1973) M. Marcus, Finite Dimensronal Multilinear Algebra (in two parts) (1973, 1975) R. Larsen, Banach Algebras (I 973) R. 0. Kujala and A. L. Vitter, eds., Value Distribution Theory: Part A; Part Deficit and Bezout Estimates by Wilhelm Stall (1973) K. 8. Stolarsky, Algebrarc Numbers and Diophantine Approxrmation (1974) A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) B. R. McDonald, Finite Rings with Identity (1974) J. Satake, Linear Algebra (S. Koh et al., trans.) (I 975) J. S. Go/an, Localization of Noncommutative Rings (1975) G. Klambauer, Mathematical Analysis (1975) M. K. Agosfon, Algebraic Topology (I 976) K. R. Goodearl, Ring Theory (1976) L. E. Mansfield, Linear Algebra with Geometric Applications (1976) N. J. Pullman, Matrix Theory and Its Applicatrons (1976) B. R. McDonald, Geometric Algebra Over Local Rings (I 976) C. W. Groetsch, Generalized Inverses of Linear Operators (1977) J. E. Kuczkowski and J. L. Gersfing, Abstract Algebra (1977) C. 0. Christenson and W. L. Voxman, Aspects of Topology (1977) M. Nagata, Field Theory (1977) R. L. Long, Algebraic Number Theory (1977) W. F. Pfeffer, Integrals and Measures (1977) R. L. Wheeden and A. Zygmund, Measure and Integral (I 977) J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) K. Hrbacek and T. Jech, Introduction to Set Theory (I 978) W. S. Massey, Homology and Cohomology Theory (I 978) M. Marcus, Introduction to Modern Algebra (1978) E. C. Young, Vector and Tensor Analysis (I 978) S. 8. Nadler, Jr., Hvperspaces of Sets (I 978) S. K. Segal, Topics.in Group Kings (1978) A. C. M. van Roofi, Non-Archimedean Functional Analysis (1978) L. Corwin and R..Szczarba, Calculus in Vector Spaces (I 979) C. Sadosky, Interpolation of Operators and Singular Integrals (I 979)
A.
8:
54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109.
J. Cronin, Differential Equations (1980) C. W. Groefsch, Elements of Applicable Functional Analysis (1980) 1. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. /. Freedan, Deterministic Mathematical Models rn Population Ecoloav (1980) S. B. Chae, Lebesgue Integration (1980) C. S. Rees el al., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Drzech, Plane Algebraic Curves (1981) R. Johnsonbaugh and W. E. F’faffenbefger, Foundations of Mathematical Analysrs (1981) W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Corwin and R. H. Szczarba, Multivanable Calculus (? 982) V. /. Isfr~fescu, Introduction to Linear Operator Theory (1981) R. D. Jiirvinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem andP. E. Ehrlich, Global Lorentzian Geometry (1981) D. L. Armacost, The Structure of Locally Compact Abelian Groups (I 981) J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981) K. H. Kim, Boolean Matrix Theory and Applications (I 982) T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) D. B. Gauld, Differential Topology (I 982) R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) M. Carmeli, Statistical Theory and Random Matrices (1983) J. H. Carruth et a/., The Theory of Topological Semigroups (I 983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Earner?, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (I 983) F. Van Oystakyen and A. Vefschor&, Relative Invariants of Rings (1983) 1. Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (I 984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. NiQt&escu, Relative Finiteness in Module Theory (I 984) K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (I 984) F. Van Oystaeyen and A. Verschoren, Relative lnvanants of Rings (1984) B. R. McDonald, Lrnear Algebra Over Commutative Rings (I 984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lre Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (I 985) A. J. Jerri, Introduction to Integral Equations with Applications (1985) G. Karpilovsky, Projective Representations of Finite Groups (I 985) L. Nariciand E. Beckenstein, Topological Vector Spaces (1985) J. Weeks, The Shape of Space (1985) P. R. Gnbik and K. 0. Kortanek, Extrernal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) G. D. Crown et a/., Abstract Algebra (1986) J. H. Carruth et a/., The Theory of Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (I 986) M. W. Jeter, Mathematical Programmrng (1986) M. Alfman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (I 986) A. Verschoren, Relative Invariants of Sheaves (1987) R. A. Usmani, Applied Lrnear Algebra (1987) P. B/ass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) J. A. Reneke et a/., Structured Hereditary Systems (1987) H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) R. Harte, lnvertibility and Singularity for Bounded Linear Operators (1988)
110. 1 11. 112. 1 13. 1 14. 115. 116. 1 17.
1 18. 1 19. 120. 121.
122. 123.
124. 125. 126.
127. 128.
129. 130.
131. 132. 133.
134. 135. 136. 137. 138. 139.
140. 141. 142. 143. 144.
145. 146. 147.
148. 149. 150.
151 152. 153. 154. 155. 156.
157. 158. 159. 160. 161. 162. 163. 164.
G. S. Ladde ef al., Oscillation Theory of Differential Equations with Deviating Arguments (I 987) L. Dudkin et a/,, Iterative Aggregation Theory (1987) T. Okubo, Differential Geometry (1987) D. 1. Stancl and M. L. Stancl, Real Analysis with Point-Set Topology (1987) T. C. Gard, Introduction to Stochastic Differential Equations (19881 S. S. Abhyankar, Enumerative Combmatorics of Young Tableaux (19881 H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Walks, Combinatorial Designs (19881 W. Wi@aw, Topological Fields (1988) G. Karpilovsky, Field Theory (I 988) S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) W. Kozlowski, Modular Function Spaces (1988) f. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (I 989) M. Pave/, Fundamentals of Pattern Recognition (I 989) V. Lakshmikantham ef a/., Stability Analysis of Nonlinear Systems (1989) R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (19891 N. A. Wafson, Parabolic Equations on an Infinite Strip (1989) K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) B. Fine, Algebraic Theory of the Bianchi Groups (1989) D. N. Dikranian era/., Topological Groups (1989) J. C. Morgan II, Point Set Theory (1990) P. Eiler and A. Witkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (I 990) J.-P. Florens e? a/., Elements of Bayesian Statistics (1990) N. She//, Topological Fields and Near Valuations (1990) B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engrneers (1990) S. S. Ho//and, Jr., Applied Analysis by the Hilbert Space Method (1990) J. Okninski, Semigroup Algebras (I 9901 K. Zhu, Operator Theory in Function Spaces (I 990) G. 6. Price, An Introduction to Multicomplex Spaces and Functions (I 991) R. 8. Darsr, Introduction to Linear Programming (I 991) P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (I 991) W. C. Brown, Matrices and Vector Spaces (1991 i M. M. Rao andZ. D. Ren, Theory of Orlicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yano. Comolex Alaebraic Geometrv (1991) D. G. Hoffman et a/.,-Coding Theory (1991) . M. 0. Gonzdlez, Classical Complex Analvsis (I 992) M. 0. Gondlez; Complex Analysis (1992) L. W. Baggeff, Functional Analysis (I 992) M. Sniedovich, Dynamic Programming (1992) R. P. Agarwal, Difference Equations and Inequalities (I 992) C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) C. Swarfz, An Introduction to Functional Analysis (1992) S. B. Nadlef, Jr., Continuum Theory (1992) M. A. Al-Gwaiz, Theory of Distributions (19921 E. Pen-y, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modellinq in Science and Engineering (1992) A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analvsis (1992) A. Charlier et a/., Tensors and the Clifford Algebra (1992) P. Bile/ and T. Nadzieja, Problems and Examples in Differential Equations (1992)
165. 166. 167. 168. 169. 170. 171.
E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-De/abri&re, Classical Sequences in Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkarni and B. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Lousfau and M. Dillon. Linear Geometry with Computer Graphics (1993) W. V. Petryshyn. Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T. A. Eick, Elementary Boundary Value Problems (I 993) 174. M. Pave/, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio er a/., Noncommutative Distributions (1993) 175. 176. W. I%/&, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) 177. 178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (I 994) P. Neirraanmgki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems 179. (1994) Differential Equations: Introduction and Qualitative Theory, Second 180. J. Cronin, Edition (I 994) 181. S. Heikkil.+’ and V. Lakshmikanrham. Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao, Exponential Stability of Stochastic Differential Equations (1994) B. S. Thomson, Symmetric Properties of Real Functions (1994) 783. 184. J. E. Rubio, Optimization and Nonstandard Analysis (1994) 185. J. L. Bueso et a/., Compatibility, Stability, and Sheaves (1995) 186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) 187. M. R. Dame/, Theory of Lattice-Ordered Groups (1995) 188. Z. Naniewicz and P. D. Panagioropoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) 189. L. ./. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (I 995) 190. L. H. Erbe er a/., Oscillation Theory for Functional Differential Equations (1995) 191. S. Agaian et a/., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) 192. M. /. Gil’, Norm Estimations for Operation-Valued Functions and Applications (1995) 193. P. A. Griller, Semigroups: An Introduction to the Structure Theory (1995) 194. S. Kichenassamy, Nonlinear Wave Equations (1996) 195. V. F. Krorov, Global Methods in Optimal Control Theory (I 996) 196. K. /. Beidar er a/. , Rings with Generalized Identities (1996) 197. V. 1. Arnaurov et a/, Introduction to the Theory of Topological Rings and Modules (1996) 198. G. Sierksma, Linear and Integer Programming (1996) 199. R. Lasser, Introduction to Fourier Series (1996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1996) 201. D. Redmond, Number Theory (1996) 202. ./. K. Beem et a/., Global Lorentzian Geometry: Second Edition (1996) 203. M. Fonrana et a/., Prijfer Domains (1997) 204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) 205. C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (I 997) 206. E. Spiegel and C. J. O’Donnell. Incidence Algebras (1997) 207. 8. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (I 998) 208. T. W. Haynes et a/., Fundamentals of Domination in Graphs (1998) 209 T. W. Haynes er a/., Domination in Graphs: Advanced Topics (1998) 210. L. A. D’Alorro et a/., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) 21 1. F. Halrer-Koch, Ideal Systems (1998) 212. N. K. Govil et a/., Approximation Theory (I 998) 213. R. Cross, Multivalued Linear Operators (1998) 214. A. A. Marrynyuk, Stability by Liapunov’s Matrix Function Method with Applications (I 998)
215. 216. 217.
A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) A. Manes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999) G. Kafo and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) Additional
Volumes
in Preparation
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HYPERSPACES Fundamentals
Alejandro
and Recent Advances
Wanes
Universidad National Ciudad Universitaria,
Autdnoma Mkxico
de Mbxico
Sam B. Nadler, Jr. West Virginia University Morgan town, West Virginia
MARCEL
DEKKER, INC.
NEW YORK - BASEL
ISBN: O-8247-1982-4 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 2 12-696-9000; fax: 2 12-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 8 12, CH-400 I Basel, Switzerland tel: 44-6 I-26 I-8482; fax: 44-6 l-26 l-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright0
1999 by Marcel Dekker, Inc. All Rights Reserved.
Neither this book nor any part may be reproduced or transmitted in any form or by any means. electronic or mechanical. including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit) 10987654321 PRINTED
IN THE UNITED
STATES
OF AMERICA
To Margarita
and Elsa
This Page Intentionally Left Blank
Preface We wrote the book with two purposes in mind: to present the fundamentals of hyperspaces in a pedagogically appropriate way and to survey the research aspects of the field. The first book on hyperspaces was written in 1978 and is referred to here as HS (Hyperspaces of Sets, by Sam B. Nadler, Jr., Marcel Dekker, Inc.). HS was written more as a research monograph than as an introductory text. In contrast, we have written this book as a text, as well as a reference. The book emphasizes the hyperspaces 2” and C(X), where X is a continuum. We also include material about symmetric products, containment hyperspaces, selections, spaces of segments and spaces of Whitney levels. We define some concepts differently than in the literature so as to obtain the most appropriate formulations for theorems (e.g., the definition in 13.1 and the theorem in 17.7 compared with the original version of the theorem in HS, p. 79). The book contains historical comments and references to original sources, either incorporated into the main body of the text or at the ends of sections. For a brief history of hyperspaces up until 1978, see HS (pp. xii-xiv). The first six chapters are solely concerned with the fundamentals of hyperspaces, providing a basic overview of the subject and a foundation for further study. The remaining chapters include more specialized material. The book is replete with exercises that we hope will enable the reader to gain a deeper understanding of material and more facility with techniques. A number of exercises contain new ideas, interesting sidelights, applications, and comments; the exercises are therefore an integral part of the book. Examples abound throughout the book. Moreover, we devote all of Chapter II to basic examples; the detailed verifications of the properties of the examples in Chapter II put the techniques used on a solid basis. The book is reasonably self-contained. We provide proofs for almost all results in the first six chapters and for many results in the remaining chapters. When we do not prove a result, we sometimes discuss the ideas V
vi
PREFACE
involved in the proof; we always refer to the literature where a proof can be found. Furthermore, we include basic material (usually with proofs) that is not specifically about hyperspaces: absolute retracts and Z-sets (section 9)) Peano continua (section lo), boundary bumping (section 12), and general theorems concerning the fixed point property (section 21). Of course, we limit our treatment of these topics to ideas and results used in hyperspaces. A tremendous amount of research has been done on hyperspaces in the past twenty years. New developments have begun, and many of the almost two hundred research questions in HS have been partially or completely answered. Therefore, we felt it was time to survey the research in hyperspaces that has occurred since HS appeared. We do this beginning with Chapter VII, covering recent research thoroughly. For example, we include complete details for the especially important solution to the dimension problem (section 731, the characterizations of Class(W) (section 671, the n-od problem (section 70), and the product problem (section 79). The most active research topic in hyperspaces has been Whitney properties and Whitney-reversible properties. We cover the results in Chapter VIII. The table at the end of Chapter VIII gives the reader a bird’s-eye view of the resultsand their relations to eachother. or Problem refers to Starting with Chapter VII, the term Question a question or problem whose answer is not known (at least by us). The last chapter is wholly concerned with questions. The first two sections of the last chapter quote the questions from HS and discuss their current status. The final section discussesmore questions, a number of which are original with this book. After the references for the last chapter is a list of the papers concerned with hyperspaces that have appeared since 1978 (we apologize to those whom we overlooked in our search of the literature, and we ask them to let us know so that we can include their papers in a subsequentprinting of the book). Our general notation is standard. Typographical considerations lead us to use X as well as cl(A) (or cl-~(A)) to denote the closure of A; cl is usually used when the expression over which we are taking the closure consists of several letters and/or symbols (however, we do not mix the two ways of denoting closure in the sameproof, definition, etc.). Other notation is explained as it comes up. We express our gratitude to several people for their help: Janusz J. Charatonik and Sergio Ma&s, who read the manuscript and gave us many beneficial suggestions; Gerard0 Acosta and Fernando Orozco, who helped us search the literature; Joann Mayhew, who patiently and diligently did a splendid job typing a large portion of the book; the people at Marcel Dekker, Inc., especially Maria Allegra, who were always patient and helpful; and the students Daniel Arkvalo, Fklix Capulin, Benjamin Es-
PREFACE
vii
pinoza, Fanny Jasso, Maria de J. L6pez, Jorge Martinez, Ver6nica Martinez de la Vega, Albert0 C. Mercado, Fernando Orozco, Patricia Pellicer and Likin C. Sim6n, who took a one year course based on the manuscript for the book. We also thank the Instituto de Matematicas of the Universidad National Autbnoma de Mexico, and the Mathematics Department of West Virginia University, for the use of resources during the preparation of the book. We express our gratitude to Gabriela SanginCs for formatting the final version of the book. It is our hope that the book will serve well as a text to attract people to the field and that researchers will find the book valuable.
Alejandro Illanes Sam B. Nadler, Jr.
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Contents Preface
V
Part
One
I.
The
Topology
for
3
Hyperspaces
1. The General Notion of a Hyperspace ..............
3
Topological Invariance ...................... Specified Hyperspaces ......................
5 6
Exercises ............................. 2. The Hausdorff Metric H,.J .................... Proof That Hd Is a Metric ....................
7
A Result about Metrizability
9 11 12
of CL(X) .............
Exercises ............................. 3. Metrizability of Hyperspaces .................. Metrizability of 2x ........................ Metrizability and Compactness of CL(X) Exercises .............................
14 16 16 ...........
18 19
4. Convergence in Hyperspaces ................... L-convergence, TV-convergence ................. Relationships between L-convergence and Tv-convergenece When X Is Compact Hausdorff ................. Countable Compactness Is Necessary .............. Exercises ............................. References ............................... ix
20 20 .
22 25 26 26 28
X II.
TABLE OF CONTENTS Examples:
Geometric
Models
for
Hyperspaces
5. C(X) for Certain Finite Graphs X ............... XanArc ............................. X a Simple Closed Curve .................... XaNoose ............................. X a Simple n-od ......................... Historical Comments ....................... Exercises ............................. 6. C(X) When X Is the Hairy Point ................ Exercises ............................. 7. C(X) When X Is the Circle-with-a-Spiral. ........... Cones, Geometric Cones ..................... The Model for C(X) ....................... Knaster’s Question ........................ When C(Y) x Cone(Y) ..................... Exercises ............................. 8. 2x When X Is Any Countably Infinite Compacturn ...... Cantor Sets ............................ Preliminary Results ....................... Structure Theorem ........................ Uniqueness of Compactifications ................ The Model for 2’ ........................ Exercises ............................. References ............................... III.
2’
and
C(X)
for
Peano
Continua
X
9. Preliminaries: Absolute Retracts, Z-sets, Torunczyk’s Theorem Exercises ............................. 10. Preliminaries: General Results about Peano Continua .... Exercises ............................. 11. The Curtis-Schori Theorem for 2x and C(X) ......... When 2; and CK(X) Are Z-sets ................ The Curtis-Schori Theorem ................... Further Uses of Torunczyk’s Theorem ............. Exercises ............................. References ...............................
31
33 33 35 36 39 44 44 46 50 51 51 53 59 59 62 64 65 65 67 67 70 72 72 75
76 79 80 83 85 85 89 90 91 94
TABLE OF CONTENTS IV.
Arcs
xi
in Hyperspaces
97
12. Preliminaries: Separation, Quasicomponents, Boundary Bumping ............................
97
Exercises .............................
103
13. A Brief Introduction to Whitney Maps
............
105
..................
Definition of a Whitney Map
105
Existence of Whitney Maps ...................
106
Exercises .............................
108
14. Order Arcs and Arcwise Connectedness of 2x and C(X) Definition of Order Arc
.....................
Arcwise Connectednessof 2x and C(X)
...
110 110
............
110
Application: 2’ > I”O ......................
114
Original Sources .........................
116
Exercises .............................
117
15. Existence of an Order Arc from As to Ai
...........
119
Necessary and Sufficient Condition ...............
119
Application: Homogeneous Hyperspaces ............
122
Original Sources .........................
124
Exercises .............................
124
16. Kelley’s Segments ........................
127
Kelley’s Notion of a Segment ..................
127
Results about Segments .....................
128
Addendum: Extending Whitney Maps .............
131
Original Sources .........................
132
Exercises .............................
132
17. Spacesof Segments, S,(E)
...................
134
Compactness ........................... S,(R) M 8(B) S,(Z’),S,(C(X)) Application:
134
.......................... When X Is a Peano Continuum
136 ......
Mapping the Cantor Fan Onto 2x and C(X)
138 . . 140
Original Sources .........................
141
Exercises .............................
141
TABLE OF CONTENTS
xii
18. When C(X) Is Uniquely
Connected
. . . .
143 145
Uniqueness of Arcs in C(X) When X Is Hereditarily Indecomposable . . . . . . . . . . . . . . . . . . .
146
Original Exercises References Shape
Sources
Theorem
. . . . . . . . . . .
. .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
148 148
. .
148
. . . . . . . . . . . . . . . . . . . . . . . .
149
and Contractibility
153
19. 2’ and C(X)
of Hyperspaces
as Nested Intersections
of ARs
.........
153
2’, C(X) Are Acyclic
......................
155
2’, C(X) Are crANR.
......................
155
2’, C(X)
Are Unicoherent
....................
Whitney
Levels in C(X)
2’, C(X)
Have Trivial Sources .........................
Original Exercises
157 .............
Are Continua Shape ..................
.............................
.................... 20. Contractible Hyperspaces The Fundamental Theorem ...................
159 160 161 161 164 164
X Contractible, X Hereditarily Indecomposable ........ Property (K) (Kelley’s Property) ................
166
Theorem about Property (K) .................. X Peano, X Homogeneous ....................
168
Original Sources ......................... Exercises .............................
175
References ............................... VI.
. .
Structure of Arcs in C(X) When X Is Hereditarily Indecomposable . . . . . . . . . . . . . . . . . . .
The Characterization
V.
Arcwise
Hyperspaces
and the Fixed Point Property
167 173 176 177
181
21. Preliminaries: Brouwer’s Theorem, Universal Maps, Lokuciewski’s Theorem . . . _ . . . . . . . . . . . . . . . . . 181 Original Sources . . . . . . . . _ . . . . . . . . . . . . . . . . 186 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
.. . x111
TABLE OF CONTENTS 22. Hyperspaces with the Fixed Point Property .......... Peano Continua. ......................... Arc-like Continua. ........................ Circle-like Continua ....................... A General Theorem ....................... Dendroids ............................. Hereditarily Indecomposable Continua ............. Addendum: Dim[C(X)] > 2 ................... Original Sources ......................... Exercises ............................. References ...............................
Part VII.
187 187 190 192 193 196 196 197 197 199
Two 205
Whitney Maps 23. Existence and Extensions .................... Exercises ............................. 24. Open and Monotone Whitney Maps for 2’ Exercises ............................. 25. Admissible Whitney Maps ................... Exercises .............................
..........
26. A Metric on Hyperspaces Defined by Whitney Maps ..... Exercises ............................. References ............................... VIII.
187
Whitney Properties and Whitney-Reversible Properties 27. Definitions ............................ Exercises ............................. 28.ANR ............................... Exercises ............................. 29. Aposyndesis ........................... Exercises ............................. 30. AR ................................ Exercises .............................
205 206 207 215 216 225 227 227 228
231 231 233 234 236 238 239 239 245
xiv
TABLE OF CONTENTS 245 246 247 247
31. Being an Arc ........................... ............................. Exercises ......................... 32. Arc-Smoothness ..................... 33. Arcwise Connectedness ............................. Exercises .......................... 34. Being Atriodic
Exercises
.............................
35. C*-Smoothness, Class(W) and Covering Property ............................. Exercises ............. 36. Tech Cohomology Groups, Acyclicity ................... 37. Chainability (Arc-Likeness) ............................. Exercises 38. Being a Circle .......................... ............................. Exercises .......................... 39. Circle-Likeness ............................. Exercises 40. Cone = Hyperspace Property .................. Exercise .............................. .......................... 41. Contractibility ............................. Exercises 42. Convex Metric .......................... ............................. Exercises ............................ 43. Cut Points
............................. Exercises ......................... 44. Decomposability Exercises
.............................
............................ 45. Dimension ............................. Exercises ...................... 46. Fixed Point Property ............................. Exercises 47. Fundamental Group ....................... ............................. Exercises ........................... 48. Homogeneity Exercise .............................. ........................... 49. Irreducibility
Exercises
251
.............................
......
251 253 253 256 257 257 259 259 259 259 260 261 262 262 264 265 265 265
267 267
268 268 269 270 270 271 271 271 272 273
276
xv
TABLE OF CONTENTS 50. Kelley’s Property ......................... ............................. Exercises ......................... 51. X-Connectedness
276 278 279 280
............................. 52. Local Connectedness ....................... ............................. Exercises ......................... 53. n-Connectedness ............................. Exercises Exercises
54. Planarity Exercises
281 281 281 283
............................. .............................
............................ ............................. ............................ 56. Pseudo-Arc .............................. Exercise 57. Pseudo-Solenoids and the Pseudo-Circle ........................... 58. R3-Continua Exercise ..............................
283 284 284
55. P-Likeness Exercises
285 286 286 286 286
............
287
59. Rational
Continua ........................ ............................. Exercises 60. Shape of Continua ........................ 61. Solenoids ............................. 62. Span. ............................... ........................... 63. Tree-Likeness
287 287 287 290
64. Unicoherence ........................... ............................. Exercises
292 293
Table Summarizing Chapter References ............................... IX.
VIII
291 292
................
Whitney Levels .......................... 65. Finite Graphs Exercises ............................. 66. Spaces of the Form &(X, t) Are ARs Exercises 67. Absolutely Exercises
............................. C*-Smooth, Class(W) .............................
294 299 305 305 313 314
.............
and Covering
Property
318 . . 319 325
xvi
TABLE OF CONTENTS 68. Holes in Whitney Levels ............... References .........................
X.
.
XII.
............
347
of C(X)
72. Previous Results about Dimension of Hyperspaces ............................. Exercises 73. Dimension of C(X) for a-Dimensional Continua X ............................. Exercises
......
74. Dimension of C(X) for l-Dimensional References ...............................
......
between
76. Retractions between Hyperspaces ....................... Exercises 77. Induced Maps .................... ....................... Exercises References ......................... XIII.
Continua
X
358 359
. . . . 363 . . . . 368 , . . . 371 . . . . 379
..........
. . 1 . 381 . . . . 387 . . . . 390 395 395 ........
395 399 400
.....................
Spaces of Finite Subsets ........................... Admissibility Hyperspace
......
347 348 349 357
Hyperspaces
More on Contractibility of Hyperspaces ............... 78. More on Contractible Hyperspaces Contractibility vs. Smoothness in Hyperspaces R3-Sets ..............................
Maps Preserving
326
342 344 345
...............................
Special Types of Maps 75. Selections ....................... ....................... Exercises
.
337 341
Exercises ............................. 71. Neighborhoods of X in the Hyperspaces Exercises .............................
Dimension
.
333 333 336
Exercises ............................. 70. Cells in Hyperspaces .......................
XI.
9
. . . . . 329
General Properties of Hyperspaces 69. Semi-Boundaries .........................
References
.
402 Contractibility
..........
403
xvii
TABLE OF CONTENTS More on Kelley’s Property .................... Exercises ............................. References ...............................
405 406 408
XIV. Products, Cones and Hyperspaces ............... 79. Hyperspaces Which Are Products ............................. Wrinkles. Folds. ............................... Proof of the Main Theorem ................... ............................. Exercises 80. More on Hyperspaces and Cones ................ ............................. Exercises References ...............................
413 413 414 415 421 423 424 431 434
XV.
437
Questions 81. Unsolved and Partially Solved Questions of [56] ........ 82. Solved Questions of [56] ..................... 83. More Questions .......................... General Spaces .......................... Geometric Models ........................ Z-Sets ............................... ....................... Symmetric Products Size Maps ............................. The Space of Whitney Levels for 2x .............. Aposyndesis ............................ Universal Maps .......................... References ............................... Literature
Special Index
Related to Hyperspaces
Symbols
of Continua
Since 1978 ....
437 463 470 470 471 471 471 472 473 473 473 474 478 497 499
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HYPERSPACES
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Part
One
This Page Intentionally Left Blank
I.
The Topology Hyperspaces
1.
The General
Notion
for
of a Hyperspace
Let X be a topological space with topology T. A hyperspace of X is a specified collection of subsets of X with the Vietoris topology (which we will define in 1.1). For convenience, we exclude the empty set 0 from being a point of a hyperspace. Also, to avoid pathology, we restrict our attention to hyperspaces whose points are closed subsets of X (cf. Exercise 1.13). Thus, the largest hyperspace of X is CL(X)
= {A c X : A is nonempty and closed in X}
with the Vietoris topology, which we now define. 1.1 Definition. Let (X, T) be a topological space. The Vietoris topology for CL(X) is the smallest topology, TV, for CL(X) having the following natural property: (-4 E CL(X) : A C U} E TV whenever U E T, and {A E CL(X) : A c B} is TV-closed whenever B is T-closed. We may exhibit a base, Bv, for TV as follows. First, note the following general notation: For any finitely many subsets Sr , . . . , S, of X, let (Sl,...,
S,) = {A E CL(X)
1.2 Theorem.
: A c lJ~==,Si and A II Si # 0 for each i}.
Let (X, T) be a topological space, and let
Bv = {(VI,...
, U,,) : Vi E T for each i and n < oo}.
Then Bv is a base for TV. 3
4
I. THE TOPOLOGY FOR HYPERSPACES
Proof. The sets that are used to define TV in 1.1 can be expressed using the notation for the members of BV as follows: for any U E T and any T-closed set B, {A E CL(X) : A c U} = (U); {A E CL(X) : A c B} = CL(X) - (X,X - B). Hence, the definition in 1.1 says that TV is the smallest topology for CL(X) containing all the sets of the forms (U) and (X,U) for U E T. In other words, letting S = {(U) : U E T} U {(X,U)
: U E T},
the collection S is a subbase for TV, which means that the family, S*, of all finite intersections of members of S is a base for TV. Hence, it suffices to prove that S* = Bv. To seethat 13~ c S*, let VI, . . . , U,, E T(n < oo) and simply observe that , Un) = (ULW
Wl,...
f-l (q-=1(X,
Vi)).
Finally, we prove that S’ c Bv. First, we prove that if Z.4,W E Bv, then U n W E Bv. To prove this, assume that U = (VI,. . . , Uk) and W = (WI,, . . , W,), where Vi, Wi E T and k, m < 00. Let U = Ut=,Ui and let W = Uz”=, Wi. Then, it is easy to check that U n w = (VI n W, . . . , Uk n W, WI
n U, . . . , W, n U).
Therefore, U n W E Bv. Thus, to prove that S’ c Bv, it suffices to prove that the intersection of any two members of S is a member of Bv. (Note: If & denotes the empty subcollection of S, then, clearly, nE E S*; however, since & is a collection of subsetsof CL(X), we see that nE = CL(X), which implies that n& E BV since CL(X) = (X).) We show that the intersection of any two members of S is a member of BV in (l)-(3) below (where UI, U2 E T): (I)
(u,)
n
(u2)
(2)
(ud
n
Ku2)
=
(uln =
u2); Wdmu2);
(3) (x, W n (x, U2) =
(x,
VI,
u2).
This completes the proof that S’ c Bv.
n
The theorem in 1.2 leads us to think of the Vietoris topology in a geometric way, as in Figure 1 (top of next page), where A and B are points in the same member (VI, . . . , U,,) of the base 0~.
TOPOLOGICAL
A,B E
INVARIANCE
5
(&,...,u5)
Figure 1
Topological
Invariance
We prove the fundamental theorem about the topological CL(X). The symbol M means “is homeomorphic to.” 1.3 Theorem.
If X x Y, then CL(X)
invariance
of
x CL(Y).
Proof. Let h be a homeomorphism from X onto Y. Define a function, h’, on CL(X) as follows: h*(A) = h(A) for each A E CL(X). Since h is a closed map, h’ maps CL(X) into CL(Y); moreover, h’ maps CL(X) onto E CL(X) and h*(h-l(B)) = B. CL(Y) since for any B E CL(Y), h-‘(B) Since h is one-to-one, we seethat h* is one-to-one.
Now, note the two equalities below (the proof of the first one usesonly that h is a closed map; the proof of the second one usesthat h is one-to-one and maps X onto Y and that h is continuous): (1) (h*)-‘((Wr,...,W,)) open in Y; (2) h*(W,...
= (h-‘(W~),...,h-‘(W,)),
where each IV, is
,Um)) = (h(U,), . . . , h(Um)), where each U, is open in X.
By (l), the fact that h is continuous, and 1.2, we seethat h’ is continuous. By (2), the fact that h is an open map, and 1.2, we seethat h” is an open map and, hence, that (h*)-’ is continuous.
I. THE TOPOLOGY
6
FOR HYPERSPACES
Therefore, we have proved that h* is a homeomorphism from CL(X) onto CL(Y). I The converse of 1.3 is false. This is easily seen by letting X and Y be nonhomeomorphic, indiscrete spaces. However, much more satisfying examples come from 8.10 and 11.3 (regarding 8.10, seethe secondparagraph of section 8). We comment about the proof of 1.3. The idea of forming h* from h when h is continuous is an important idea in hyperspace theory. For example, results about the fixed point property for hyperspaces can be proved using this idea (see section 22). For more information, seesection 77 (where h* is denoted by 2h and h’ /C(X) is denoted by C(h)).
Specified
Hyperspaces
By considering simple topological properties - connectedness,compactness, etc. - we are led to consider hyperspaces that consist of sets having one or more of these properties. We introduce some of these hyperspaces in 1.4-1.8. Of course, each of the hyperspaces has the subspace topology obtained from the Vietoris topology on CL(X). = {A E CL(X)
1.4 Definition.
CLC(X)
1.5 Definition.
2x = {A E CL(X)
: A is connected}.
: A is compact}.
We note that 2x = CL(X) when X is compact. We also note that when X is Hausdorff, 2x = {A c X : A is nonempty and compact}. 1.6 Definition.
C(X)
= {A E 2dY : A is connected}; in other words,
C(X) = 2x n CLC(X). Starting with Chapter II, the book is almost exclusively about the two hyperspaces 2x and C(X) when X is a compact metric space. For the next definition, (Al denotes the cardinality of a set A. Also, recall that a Tl-space is a topological space, X, such that (2) is closed in A’ for each 2 E X. 1.7 Definition. Assume that X is a Ti-space. For each n = 1,2,. . . , let F,,(X) = {A c X : 1 5 IAJ 5 n}, which we consider as a subspace of CL(X) or 2”. The space F,(X) is called the n-fold symmetric product of X; the l-fold symmetric product of X is also called the space of singletons (note Exercise 1.15).
Symmetric products provide a simple way to obtain interesting spaces. In particular, the n-fold symmetric product of X and the Cartesian product
7
EXERCISES
of n copies of X are usually quite different. This is true even when X is a simple metric continuum and n = 2. For example, see Exercise 1.26. Basic open questions about symmetric products are in 83.11-83.14; also, see 78.19 and 78.20. 1.8 Definition. For a Ti-space, X, let F(X) F(X) the space of finite subsetsof X.
= U~i,,F,(X).
We call
An open problem and related results about F(X) are in 78.19 and the comment following 78.20. We conclude this section by noting that the topological invariance of CL(X) in 1.3 is also true of each of the hyperspaces in 1.4-1.8: 1.9 Theorem. If X = Y, then ‘tli(X) z ‘&(Y), where ‘&(.X) and xi(Y) are the hyperspaces of X and Y defined in 1.i for each i = 4,. . . ,8 (we assumethat X and Y are Ti-spaces when i = 7 or 8). Proof. For each i, the homeomorphism h’ in the proof of 1.3 maps the hyperspace ?&(X) onto the hyperspace ?&(Y). n
Exercises 1.10 Exercise. Let (X,T) be the Sierpinski two-point space: X = (0, 1) with the topology T = {X,0, (0)). What familiar space is (C-W), TV)? Describe the space (CL(X), TV) when X = (0, 1,2} with the topology T consisting of the sets X, 0, (0, I}, { 1,2}, and { 1). 1.11 Exercise. For any topological space X, CL(X) is a To-space. (Recall that a To-space is a topological space, (Y,T), such that for any ~1, ~2 E Y with yi # ~2, there exists U E T such that yi E U and ?/j $ U for some choice of i and j.) 1.12 Exercise. converse is false. 1.13 Exercise.
If X is a Ti-space, then CL(X)
is a Ti-space. The
Let (X, T) be a topological space, and let N(X)
= {A c X : A #
0).
The Vietoris topology for CL(X) extends naturally to what we would call the Vietoris topology for N(X): simply replace CL(X) with N(X) throughout 1.1. However, we did not consider the hyperspace N(X) because it almost always fails to have decent separation properties. In fact, prove that
8
I. THE
N(X) with its Vietoris space.
TOPOLOGY
FOR
HYPERSPACES
topology is not a Ti-space unless (X, T) is a discrete
1.14 Exercise. If X is a regular space, then CL(X) is a Hausdorff space. If CL(X) is a Hausdorff space and X is a Ti-space, then X is a regular space. 1.15 Exercise.
If X is a Tr-space, then X M Fi(X).
1.16 Exercise. If X is a Hausdorff space, then F,(X) is closed in CL(X) for each n = 1,2,. . .. Give an example of a Ti-space, X, such that Fl (X) is not closed in CL(X). 1.17 Exercise. If X is a connected Tr-space, then F,(X) for each n; hence, CL(X) is connected.
is connected
1.18 Exercise. If X is a Ti-space, then X is separable if and only if CL(X) is separable. (Note: the proof of the “if” part usesthe Axiom of Choice.) 1.19 Exercise. X. Let
Let X be a Ti-space, and let K be a closed subset of CLK(X)
Then, CLK(X)
= {A E CL(X)
: A > K}.
is closed in CL(X).
1.20 Exercise. CL(X). 1.21 Exercise.
If X is a normal space, then CLC(X)
is closed in
Let (X, T) be a Ti-space. For any Ui , . . . , lJ,, E T (n <
m), Cl((Ul,
. . ., UT&))
=
(~l,...,cL)
where cl denotes the closure operator for (CL(X), TV) and ??, denotes the T-closure of Vi for each i. (Note: The assumption that (X, T) is a Ti-space is only used to prove containment in one direction.) 1.22 Exercise. Let (X, T) be a topological space, and let Y be a closed subset of X. Then, the Vietoris topology for CL(Y) obtained from the subspace topology for Y is the same as the subspacetopology for CL(Y) obtained from the Vietoris topology TV for CL(X). In other words, using juxtaposition with vertical lines to denote subspace topologies, (TIY)v = Ty]CL(Y).
2. THE HAUSDORFF
METRIC
Hd
9
1.23 Exercise. The union map u : CL(X) x CL(X) + CL(X), which is defined by u(A,B) = A U B for each A,B E CL(X), is continuous. (x denotes Cartesian product). 1.24 Exercise. WR’).
Find a subset, 2, of the plane R2 such that Z M
1.25 Exercise. Let X be the subspace of the plane R2 consisting of the interval [-1, l] on the s-axis and the interval [0, l] on the y-axis. Find a subset, 2, of Euclidean S-space R3 such that 2 M Fz(X). 1.26 Exercise. Let S’ denote the unit circle in the plane R2 (i.e., S’ = {(z,y) E R2 : x2 +y2 = 1)). Then F2(S1) is a Moebius band; that is, Fz(S’) is homeomorphic to the quotient space obtained from [0, l] x [0, l] by identifying the point (0,~) with the point (1,l - y) for each 9 E [0, 11. Remark. Some of the results in the exercises above are in [6]. If you read [6], you should be aware of two errors: (c) on p. 156 and 4.5.3 on p. 162 (cf. 0.66.5 of [7, pp. 40-411 and [8 or 91, respectively). You should also be careful to remember the standing assumption on p. 153 of [6] that the spacesX are almost always assumedto be Ti-spaces; this assumption was not taken into account when the comment was made about 2.4.1 of [6] in 0.66.6 of [7, p. 411. 2.
The Hausdorff
Metric
Hd
In Figure 1, p. 5 we illustrated what it means for two points, A and B, of CL(X) to be in the same basic open set, (Vi,. . . , V,), of the Vietoris topology. Now, let us assume that X has a metric, d, that induces the topology on X. Then, Figure 1 suggests that we consider the “Vietoris distance” between A and B to be lessthan a real number, r, provided that A and B satisfy the following two similar conditions: (a) for each a E A, there exists b E B such that d(a, b) < r; (b) for each b E B, there exists a E A such that d(b,a) < T. It would then be natural to define the “Vietoris distance” between A and B to be equal to the infimum over all real numbers T satisfying (a) and (b). However, there is an apparent technical problem with the definition just proposed: If, for example, A is not bounded and B is bounded, then there is no real number r satisfying (a). We can solve this problem by assuming that d is a bounded metric for X. This assumption would not cause any loss of generality since every metric can be replaced by a bounded metric
I. THE TOPOLOGY FOR HYPERSPACES
10
that induces the same topology for the space (e.g., replace the metric d with the bounded metric D = d/(1 + d) [4, p. 2111). Of course, even on assuming that d is a bounded metric for X, we still have to check that the “Vietoris distance” is actually a metric. But first let us provide convenient notation and repeat the definition of the “Vietoris distance” in terms of the new notation. We will call the “Vietoris distance” the Hausdorff metric since it was first considered by F. Hausdorff [3, p. 1451.
Let (X, d) be a metric space. For any 3: E X and any A E CL(X), d(z,A)
= inf {d(z,a)
For any r > 0 and any A E CL(X),
let
: a E A}.
let
Nd(r, A) = {x E X : d(z, A) < T}
(Figure 2). We call Nd(r, A) the generalized open d-ball radius
in X about A of
r.
Note that (a) and (b) above say that A c Nd(r,B) and B c Nd(r,A), respectively. Hence, the following definition is simply a reformulation of what we referred to as the “Vietoris distance” from A to B.
Nd(r, A) Figure 2
PROOF
THAT
Hd Is A METRIC
11
2.1 Definition. Let (X, d) be a bounded metric space. The Huusdorff metric for CL(X) induced by d, which is denoted by Hd, is defined as follows: for any A,B E CL(X), Hd(A, B) = inf {r > 0 : A c Nd(r, B) and B c Nd(r, A)}.
Proof
That
Hd Is a Metric
We prove that Hd is, indeed, a metric. 2.2 Theorem.
If (X, d) is a bounded metric space,then Hd is a metric.
Proof. We begin by noting that, since d is a bounded metric, Hd is a real-valued (nonnegative) function. We next observe, as is evident from 2.1, that Hd is a symmetric function; in other words, Hd(A, B) = Hd(B, A) for all A, B E CL(X). Now, assume that A,B E CL(X) such that Hd(A,B) = 0. Then, by 2.1, A c N~(E,B) for all E > 0. Hence, fixing p E A, there exists b, E B for each n = 1,2,. . . such that d(p,M < l/n. Thus, since {b,}~zl converges to p and B is closed in X, p E B. Therefore, we have proved that A c B. A similar argument shows that B c A. Hence, A = B. The converse - Hd(A, A) = 0 for all A E CL(X) - follows immediately from 2.1. It only remains to prove the triangle inequality for Hd. For this purpose, it is useful to have the following fact (which is an elementary consequence of definitions): (#) for any K, L E CL(X) Now, let A, B, C E CL(X). &(A,
and E > 0, K C Nd(Hd(K, L) + E, L).
We prove that c) 5 &(A,
B) + Hd(B, c).
Let E > 0. Let a E A. Then, by (#), there exists b E B such that (1) d(a, b) < Hd(A, B) + E. Since b E B, we seeby using (#) again that there exists c E C such that (2) d(b, c) < Hd(B, C) + E. By (l), (2), and the triangle inequality for d, we have that (3) d(a, c) < Hd(A, B) + Hd(B, C) + 2~.
12
I. THE TOPOLOGY
FOR HYPERSPACES
Thus, since a was an arbitrary point of A, we have proved that (4) A C Nd(&(A,
B) + H@,
c) + 25, c).
A similar argument (which starts with a point of C) shows that (5) c c Nd(Hd(c, B) + Hd(B, A) + 2~, A). Recall from the second paragraph of the proof that Hd is a symmetric function; hence, we may rewrite (5) as follows: (6) c C Nd(Hd(A, B) f Hd(B, c) + 2E, A). By (4), (6), and 2.1, Hd(A,C) 5 Hd(A,B) since E > 0 was arbitrary, Hd(A, C) 5 &(A,
A Result
about
+ Hd(B,C) + 2s. Therefore, B) + Hd(B, C). n
Metrizability
of CL(X)
The discussion at the beginning of the section shows how the idea for the Hausdorff metric arises naturally from examining the basic open sets (VI,... , Un) for the Vietoris topology. In particular, when the metric d for X is bounded, the conditions in (a) and (b) at the beginning of the section seem to describe the interpretation of the Vietoris topology in Figure 1, p. 5. Thus, when X has a bounded metric, we would expect that the Vietoris topology for CL(X) and the Hausdorff metric topology for CL(X) would be the same. However, this is not so. In fact, for Ti-spaces X, the Vietoris topology for CL(X) is not even metrizable (by any metric) unless X is compact. The following lemma gives the principal reason (as you will seefrom the proof of the succeeding theorem). 2.3 Lemma. If Y is an infinite, discrete space, then the Vietoris topology for CL(Y) does not have a countable base. Let ,0 be a base for the Vietoris topology for CL(Y). Note that Proof. each A E CL(Y) is open in Y. Hence, for each A E CL(Y), there exists a,4 E /I such that A E f?A C (A). Thus, it follows easily that (1) A = USA for each A E CL(Y). Now, it follows immediately from (1) that if A, A’ E CL(Y) such that A # A’, then DA # a,&. In other words, the function A + f?jA is a one-toone function from CL(Y) into p. Hence,
A RESULT ABOUT METRIZABILITY c-4
ICW)l
OF CL(X)
13
I IPI-
Since Y is a discrete space, CL(Y) = {A c Y : A # 0}; thus, since Y is an infinite set, CL(Y) is uncountable. Therefore, by (2), the base p is uncountable. w 2.4 Theorem. Let (X, T) be a Ti-space. then (X, T) is a compact, metrizable space.
If (CL(X),
TV) is metrizable,
Proof. Since (CL(X),Tv) is metrizable and X M Fi(X) (by 1.15), it is clear that (X,T) must be metrizable. We prove that (X,T) is compact. Suppose that (X, T) is not compact. Then, since (X, T) is metrizable, there is a countably infinite, closed (in X), and discrete subspace, (Y, TIY), of (X,T). Since Y is closed in X, we have by 1.22 that CL(Y) with its own Vietoris topology (TIY) v is a subspace of (CL(X), TV). Thus, since we are assuming that (CL(X), 2’“) is metrizable, we have that (1) (CL(Y),
(TIY)“)
is metrizable.
Now, note that (Y, TIY) is a Z’i-space (since (Y, T)Y) is discrete); also, note that (Y,TIY) is separable (since Y is countable). Hence, by 1.18, (2) (CL(Y),
(TIY)v)
is separable.
By (1) and (2), there is a countable base for (TIY)“. However, infinite, discrete space, this contradicts 2.3. Therefore, (X,T)
Y being an is compact.
n
The theorem in 2.4 is somewhat disappointing since it limits the generality in which (CL(X),T v ) is metrizable to compact spaces X. However, we will be consoled by the important theorems in the next section: They show that for any metric space (X, d), the hyperspace 2x with its Vietoris topology is metrizable by the Hausdorff metric; they also show that the converse of 2.4 is true and, moreover, that it is the Hausdorff metric that does the metrization. 2.5 Remark. Let (X, d) be a metric space. Let A, B E 2x (i.e., A and B are nonempty, compact subsets of X). Then there is a real number T > 0 such that A C Nd(r, B) and B C Nd(r, A) (e.g., r = l+ diameterd(AUB)). Thus, whether d is bounded or not, the formula for Hd(A, B) in 2.1 produces a real number when A, B E 2x. Now, recall that the assumption in 2.2 that d is bounded was only used at the beginning of the proof of 2.2 to know that Hd is real valued. Therefore, the proof of 2.2 shows that Hd is a metric for 2x whether d is bounded or not. We call this metric the Hausdorff metric
14
I. THE TOPOLOGY
FOR HYPERSPACES
for 2 x induced by d. We use the same symbol, Hd, for this metric that we
used in 2.1 (since we know that doing so will never cause any confusion).
Exercises 2.6 Exercise. Let (X, d) be a bounded metric space. Let A, B, A’, B’ E CL(X) such that A’ C A and B’ c B. Then &(A
2.7 Exercise.
U B’, B U A’) 5 Hd(A, B)
Let (X,d)
be a bounded metric space. Then, for any
A, B E CL(X),
Hd(A, B) = ma ‘1:~ d(a,B), ;F; d(b,A)}. 2.8 Exercise. Let (X, d) be a metric space. If A, B E 2x, then there are points, p E A and q E B, such that Hd(A, B) = d(p, q). Also, when (X, d) is a bounded metric space, is the analogous result for A, B E CL(X) true? 2.9 Exercise. Let (X, d) be a metric space. If A, B E 2” and if A c Nd(r, B) and B C Nd(r, A), then &(A, B) < r. Is the analogous result for A, B E CL(X) true (assuming, of course, that (X, d) is a bounded metric space)? 2.10 Exercise. Use modifications of the proof of 2.4 to prove the following result. Let (X,T) be a Tr-space. If (CL(X),Tv) has a countable base, then (X, T) is compact. Can you also conclude, as in 2.4, that (X,5”) is metrizable? 2.11 Exercise. Even though Hd # Hdt whenever d # d’, there is a much more important reason to involve the given metric on X in our notation for the Hausdorff metric for CL(X): two bounded metrics, d and d’, may induce the sametopology on X, yet the Hausdorff metrics Hd and ff& may induce nonhomeomorphic topologies on CL(X). Verify that this can happen by letting X = { 1,2,. . .} and by using the metrics d and d’ defined as follows: 1, ifz#y d(s,y) = 0, ifz=y. { and d’(s,y)=/i-iIforallx,yEX.
EXERCISES
15
2.12 Exercise. If (X, d) is a totally bounded metric space, then THE C TV, where THY is the topology for CL(X) induced by Hd. (A metric space, (X, d), is said to be totally bounded provided that for each E > 0, X is the union of finitely many sets each of which has diameter less than E.) 2.13 Exercise. Let (X,d) be a bounded metric space. Then (l), (2), and (3) are equivalent: (1) (X,d) is totally bounded; (2) (CL(X), Hd) is totally bounded; (3) (CL(X), Hd) is separable. Note that the equivalence of (1) and (3) makes 2.11 trivial. 2.14 Exercise. Let (X, d) be a bounded metric space. If (X,d) is complete, then (CL(X), H d ) is complete. (A metric space is said to be complete provided that every Cauchy sequence with respect to the given metric converges.) For each n = [Hint: Let {Ai}zl b e a Cauchy sequence in CL(X). 1,2,. . ., let Y, = d(UgO,,Ai). Consider Y = IIF&Y,.] Remark.
An important
consequence of Exercises
2.13 and 2.14 is in
3.5. 2.15 Exercise. If (X,d) is a complete metric space, then (2”,Hd) is complete. [Hint: Use Y in the hint for 2.14. Here, A, E 2x for each i; prove that E;, hence each Y,, is totally bounded and complete, therefore compact [5, P. WI 2.16 (i.e., d(s, for Fz(I) First, R2. Next, that
Exercise. Let I = [0, 11, and let d denote the usual metric for I t) = Js - tl for all s, t E I). Let Hd denote the Hausdorff metric induced by d. d ) is topologically embeddable in the plane prove that (Fz(l),H find A~,Az,BI,Bz Hd(Ai,
E Fz(I),
Bj) = iHd(Al,
with
AI # A:! and BI # BP, such
AZ) whenever
i # j.
Therefore, even though (Fz(I), Hd) is topologically embeddable in R2, we see that (Fz(l),H d ) is not isometrically embeddable in any Euclidean space R* (with the metric given by JC~=“=,(G - yiJ2 for (G)L, (yi)L E R”). (An isometry is a distance preserving map between two metric spaces.)
16
3.
I. THE TOPOLOGY
FOR HYPERSPACES
Metrizability
of Hyperspaces
We determine when (CL(X),Tv) and (2x, T~l2~) are metrizable. We show that when they are metrizable, they are metrizable by the Hausdorff metric. We first set down some convenient notation. We will be concerned with 2x and (later on) with hyperspaces, 31, contained in 2*. We could expand on previous notation to fit this situation as follows: (U,, . . . ,Un) nX, HdlR x ‘H. TV\‘% However, the expanded notation is cumbersome. Thus, we often simply use the previous notation and rely on the context for clarification. We also rely on the context for clarification regarding the following notation. Let (X,d) be a metric space, and let 3t c 2x. Then, THY denotes the topology for % induced by the Hausdorff metric Hd for ?t (cf. 2.5). Also, for any A E ?l and T > 0, BH, (T, A) denotes the open Hd-ball in 31 with radius r and center A; in other words, aH,(T,A)
= {B E ‘?i : Hd(A,B)
Metrizability
of Zx
We begin with the following important theorem. Note that the theorem implies that the converse of 2.4 is true (see 3.4). 3.1 Theorem. If (X,d) is a metric space, then (2x,Tv) In fact, TV = THY,
is metrizable.
Proof. We prove first that THY > TV. Taking into account our agreement about notation (above), we seefrom section 1 that TV is the topology for 2x generated by the sets (17) and (X, U) for all U E T (where T is the topology on X). Therefore, to prove that THY > TV, it suffices to show that (U) E THY and (X, U) E THY for all U E T. SO, let U E T. We show that (U) E THY as follows. Note that if U = X, then (U) = 2* and, hence, (U) E THY. Therefore, we assume that U # X. NOW, let A E (U). Let E = d(A,X
- U)(= inf{d(a,z)
: a E A and z E X - U}).
Note that E > 0 since A is a compact subset of the open set U (E exists since A and X - U are nonempty). Furthermore, we seethat BHd(&, A) C (U) as follows: if B E BHd(E, A), then Hd(A, B) < & and, thus, it follows from 2.1 that B C Nd(&,
4;
METRIZABILITY
OF 2’
17
hence, by the way E was defined, we see that B c U, i.e., B E (U). Thus, starting with A E (U), we showed that a,(~, A) C (U) for some E > 0. Therefore, we have proved that (1)
(u)
ETH~.
We show that (X, U) E THY as follows. Let A E (X, U). Then, Afl U # 8. Let p E An U. Since p E U and U E T, there exists 6 > 0 such that
We prove that BH~(~,A) 6. Thus, by 2.1,
C (X,U).
Let B E BHd(6,A).
Then, &(A,B)
<
A C Nd(6, B). Thus, since p E A, there exists b E B such that d(b,p) < 6. Hence, from the way we chose 6, we seethat b E U. Therefore B II U # 0, which shows that B E (X, U). This proves that B&(6, A) C (X, U). Thus, starting with A E (X, U), we showed that aHd (6, A) C (X, U) for some b > 0. Therefore, we have proved that
Since we have proved (1) and (2) for any U E T, it follows (as noted above) that THY > TV. We now prove that THY c TV, By 1.2, it suffices to show that for each open Hd-ball, 23~~(T, A), there are finitely many, open subsets, UI, . . . , U,,, of X such that
AE (VI,...
,un)
C aH,(T,A).
To prove this, let A E 2x and let r > 0. Then, since A is compact and nonempty, there are finitely many, open subsets, Ui , . . , , U,, of X satisfying (3)-(5) below: (3) A c u&Vi; (4) A II Vi # 0 for each i; (5) diameter (Ut) < T for each i. It is evident from (3) and (4) that A E (VI,. . . , Un). Therefore, we will be done with the proof once we show that
WI,...
,un)
C BH~(~,A).
I. THE TOPOLOGY
18
FOR HYPERSPACES
To show this, let K E (Vi,. . . , Un). Then, since from (4) and (5) that K c Nc+(r, A);
K c
Uy==, U,, it follows
also, since K fl Vi # 0 for each i, it follows from (3) and (5) that
A c N~(T, K). Hence, Hd(A, K) < r by Exercise 2.9; thus, K E f?~,(r, Therefore, we have proved that THY c TV. n
A).
Let us note that we can state 3.1 in the following way. 3.2 Theorem. If (X, T) is a metrizable topological space, then (2x, TV) is metrizable; moreover, if d is any metric for X that induces T, then
T\J = THY. The converse of the first part of 3.2 is true for Ti-spaces: 3.3 Theorem. Let (X, T) be a Tr-space. Then, (2”) if and only if (X,7’) is metrizable.
T~I) is metrizable
Proof. Assume that (2x, TV) is metrizable; then, since Fi(X) c 2~~ and X z Fi (X) (by Exercise 1.15), we seethat (X, T) is metrizable. The converse is in 3.2. n
Metrizability
and Compactness
of CL(X)
Compare the result about 2x in 3.3 with the following result about
CL(X). 3.4 Theorem. Let (X, T) be a Ti-space. Then, (CL(X), able if and only if (X, T) is a compact, metrizable space.
TV) is metriz-
Proof. Assume that (X,T) is a compact, metrizable space. Then, is (2”, TI/) is metrizable by 3.2 and 2x = CL(X). Therefore, (CL(X),Tv) metrizable. The converse is in 2.4. n The next theorem is fundamental to the development of the theory of hyperspaces. The most general version of the theorem is in Exercise 3.12. 3.5 Theorem. If (CL(X),Tv) is compact.
(X,T)
is a compact, metrizable space, then
EXERCISES
19
Proof. Let d be a metric for X that induces T. Note that (X,d) is totally bounded and complete. Hence, (CL(X), Hd) is totally bounded and complete by Exercises 2.13 and 2.14. Thus, (CL(X), Wd) is compact [5, p. 201. Now, note that CL(X) = 2x (since X is compact); hence, by 3.1, THY = TV. Therefore, (CL(X), TV) is compact. n 3.6 Corollary. Let (X, T) be a Ti-space. If (CL(X), then (CL(X), TV) is compact. Proof.
The corollary follows at once from 3.4 and 3.5. n
3.7 Corollary. If (C(X), TV) is compact. Proof.
TV) is metrizable,
(X,T)
is a compact, metrizable space, then
The corollary follows immediately from 3.5 and Exercise 1.20.
n
3.8 Remark. Assume that (X,T) is a metrizable topological space. We will often want to think of the Vietoris topology on 2x in terms of the Hausdorff metric. The theorem in 3.2 says that we may do this without taking into account any specific metric d that induces T. Thus, we will often omit d from our notation; in other words, we will denote the Hausdorff metric for 2-x by H and the associated topology by TH. In particular, when we say that H is the Hausdorff metric for 2-x, we mean that H is the metric Hd obtained from some metric d for X that induces T. Finally, for z c 2”, we conform to our agreement about notation at the beginning of the section; namely, we use H and TH to denote the Hausdorff metric for ‘?f and the topology for 7-i, respectively. Regarding the comments in 3.8, we must mention that, on occasion, a specific metric (or type of metric) for X that induces T is important when considering properties of 2”. On such occasions, we revert to the notation Hd and THY.
Exercises 3.9 Exercise. Let (X,T) be a compact Hausdorff space. Then, (CL(X),T”) is metrizable if and only if B(X) is a G&-set in CL(X). (A subset, 2, of a topological space, (Y, T), is said to be a Ga-set in Y provided that Z = f$ZiUt, where U, E T for each i.) 3.10 Exercise. Give an example of a topological space, (X,T), that (CL(X),Tv) is metrizable but (X,T) is not metrizable.
such
20
I. THE TOPOLOGY
FOR HYPERSPACES
3.11 Exercise. Let (X, T) be a met&able space. Then, (CL(X), is metrizable if and only if (CL(X), TV) is compact.
TV)
3.12 Exercise. We know that the following result is true when (X, T) is metrizable (combine 3.5, 3.11, and 2.4): If (X, T) is a topological space, then (CL(X), TV) is compact if and only if (X,T) is compact. Prove this general result by using the Alexander Subbase Lemma, which we state as follows: Let (X, T) be a topological space, and let S be a subbase for T; if every cover of X by membersof S has a finite subcover, then (X, T) is compact [5, p. 41. [Hint:For the “if” part, assumethat CL(X)=(UiE~(Uz))U(U+y(X, V,)), where Vi, Vj E T for all i and j (recall S in the proof of 1.2); then consider when u~~JV~ = X and when u~~JV~ # X. For the “only if” part, be careful - {z} may not be a point of CL(X) for 5 E X.1
4.
Convergence
in Hyperspaces
We describe convergence of sequencesin CL(X) directly in terms of the topology on X. Our description enables us to picture convergence as if it were taking place in X. By viewing convergence in CL(X) as occurring in X, we will often enhance our understanding of various aspects of the theory of hyperspaces. Besides, it is a natural tendency to want to think of hyperspace phenomena in terms of the base space X as much as possible: the space X seemssimpler, lessformidable, than the hyperspace CL(X). In 4.2 we define a notion of convergence for sequencesof subsets of X; we call the notion L-convergence. We will see that convergence (of sequences)in CL(X) implies L-convergence when X is regular; we will also see that the converse implication is true when X is countably compact. Hence, convergence in CL(X) and L-convergence are equivalent when X is a compact Hausdorff space. These results are sufficient for our purpose since the rest of the book is almost entirely about hyperspaces of compact metric spaces;moreover, the results can not be much more general (as you will see, notabIy by 4.9).
L-convergence,
TV-convergence
We will define L-convergence using the two companion notions in the following definition. be 4.1 Definition. Let (X, T) be a topological space, and let {Ai}zl a sequenceof subsetsof X. We define the limit inferior of {Ai}z=, , denoted
L-CONVERGENCE, by lim inf Ai, and the limit superior follows (illustrated in Figure 3):
TV-CONVERGENCE
21
of {Ai}z”=, , denoted by lim sup A,, as
(1) lim inf Ai = {x E X: for any U E T such that z E U, U all but finitely many i};
n Ai # 0 for
(2) limsupA,={x~X:foranyU~TsuchthatxEU,UnAi#Ofor infinitely many i}. 4.2 Definition. Let (X,T) be a topological space, let {Ai}z, be a is Lsequence of subsets of X, and let A C X. We say that {Ai}zl convergent in X to A, which we denote by writing Lim Ai = A, provided that lim inf -4, = A = lim sup A,.
For example, let {Ai}& be the sequencedepicted in Figure 3; we see are each that {Ai}& is not L-convergent whereas {Asi}E”=, and {A~-l}z~ L-convergent (with different limits). At this time, you may find it beneficial to work some (or all) of the exercisesin 4.10-4.17. These exercises are directly concerned with the concepts that we just introduced, and their solutions do not depend on any material in the rest of the section.
liminf,limsup Figure 3
(4.1)
22
I. THE TOPOLOGY
FOR HYPERSPACES
We discuss some terminology and phraseology that we use in the theorems. Convergence in CL(X) means, of course, convergence with respect to the Vietoris topology TV. We call this convergence TV-convergence. We want to always remind ourselves of the inherent difference in perspective between TV-convergence and L-convergence: TV-convergence takes place in CL(X) whereas L-convergence takes place in X. We emphasize the and “L-convergence in difference by saying “TV-convergence in CL(X)” X.”
Relationships
between L-convergence TV-convergence
and
Our theorems are concerned with implications between TV-convergence and L-convergence of sequences.It would be awkward, if not distracting, to continually refer directly to sequencesand their limits in the statements of the theorems. Thus, we simply say that one type of convergence implies the converges to other, by which we mean the following: If a sequence,{Ai}El, A with respect to the first type of convergence, then the sequence {Ai}zl converges to A with respect to the second type of convergence. Note the following simple but useful lemma. 4.3 Lemma. Let (X, T) be a topological space. Let {A,}z, be a sequence of subsets of X, and let .4 c X. Then, (1) and (2) below are equivalent:
(1) Lim -4, = A; (2) A
c
lim inf Ai and lim sup Ai c A.
The lemma follows immediately from the definitions in 4.1 and 4.2, it being evident from 4.1 that lim inf Ai c iim sup A,. H Proof.
4.4 Theorem. Let (X,T) be a regular topological space. Then, TVconvergence in CL(X) implies L-convergence in X. Proof. Assume that {A,)zO=, is a sequencein CL(X) such that {Ai}p”l TV-converges to A in CL(X). We will show that Lim ‘4i = A by using 4.3. First, we show that A C lim inf Ai. Let a E A, and let U E T such that a E U. Note that A E (X,U). Thus, since {Ai}zO=, TL+converges to A, there exists N such that
Ai E (X, U) for all i 2 N;
RELATIONSHIPS
BETWEEN
L-CONVERGENCE
AND..
.
23
in other words, Ai n U # 8 for all i 2 N. This proves that a E lim inf Ai. Therefore, we have proved that (1) A c lim inf Ai. Next, we show that lim sup A, c A. Let CCE X - A. Since (X,T) is regular, there exist U,W E T such that z E U, A c W, and U n W = 0. Then, since A E (IV) and {Ai}:“=, T v - converges to A, there exists A4 such that Ai E (IV) for all i 2 M. Thus, since U n W = 0, we see that A, n U = 0 for all i 2 M. This shows that x $! lim sup A,. Therefore, since we started with any point x # A, we have proved that (2) lim sup Ai c A By (l), (2), and 4.3, Lim Ai = A. H We turn our attention to the converse of 4.4, that is, to determining when L-convergence implies TV-convergence. Of course, we must only consider L-convergence for sequences whose terms are in CL(X). Under this trivial but necessary restriction, we show that L-convergence implies TVconvergence when X is countably compact (4.6); we also show that this implication between convergences is sufficient for a Ti-space X to be countably compact (4.9). Recall that a topological space, X, is said to be countably compact provided that every countable, open cover of X has a finite subcover [5, p. 11. Let (X,T) be a countably compact topological space. If is a sequenceof nonempty subsets of X, then
4.5 Lemma.
{Ai}fZl
lim sup Ai # 0. Proof. First, let us note the following fact (whose proof we leave as an exercise in 4.11): (#) lim sup Ai = n:zp,l[cl(u~nA,)]. Next, let U, = X - cl(Ug=,A,) for each n = 1,2,. . .. Now, supposethat lim sup Ai = 0. Then, by (#), Ur=iU,, = X. Hence, (X,T) being countably compact, finitely many of the sets U,, cover X. Of these finitely many sets that cover X, let U,,, be the one having the largest
24
I. THE TOPOLOGY FOR HYPERSPACES
index m. Then, since U, c U,,, for each n < m, obviously means that +&A,) = 0. Therefore, lemma. #
in particular,
A, = 0. This contradicts
U,,, = X. This
an assumption
in the
4.6 Theorem. Let (X,T) be a countably compact topological space. Then, L-convergence in X for sequences in CL(X) implies TV-convergence in CL(S). be a sequence in CL(X) such that {Ai}zl is LProof. Let {Ai}gl convergent in X to a subset, A, of X (i.e., Lim Ai = A). First, we show that A E CL(X). By 4.2, A = lim sup Ai; hence, by 4.5, A # 0. The exercise in 4.12 shows that A is closed in X. Therefore, A E CL(X). is TV-convergent to A. Let Ur , , . . , U,, E NOW, we show that {Ai}gl T(n < co) such that
AE (&,...,Un). We must show that A, E (U,, . . . , U,,) for all sufficiently large i. We do this by finding integers, M and N, such that (1) and (2) below hold: (1) A, n U, # 0 for all i 2 M and each j = 1, . . . , n; (2) Ai c lJjn=rUj for all i 2 N. We find M so that (1) holds as follows. Consider any one of the sets U, by fixing j 5 n. Since A E (Ul, . . , U,), there is a point p E A fl U,. Since p E A and A = Lim Ai, we have by 4.2 that p E lim inf A,. Thus, since p E Uj and U, is open in X, we have by 4.1 that Uj fl A, # 0 for all but finitely many i. In other words, there is an integer, Mj, such that
Now, having obtained such an integer Mj for each j 5 n, we let
M = max {M1,...,Mn}. It is then evident that (1) holds for this integer M. Next, we show that there exists N such that (2) holds. For this purpose, let Y = x - uy&. Suppose, by way of obtaining a contradiction, that there is no integer N such that (2) holds. Then there is a subsequence, {Ai(k)}rzO=l, of {Ai}zl such that
WHEN X Is COMPACT HAUSDORFF (a) Ai
25
n Y # 0 for each k = 1,2,. . ..
We will use lim supy in denoting limit superiors with respect to the space (Y,TIY). Note that (Y,TIY) is countably compact (since Y is closed in X). Therefore, by (a), we may apply 4.5 to the space (Y,TIY) and to the sequence{Aick) n Y}p=i to conclude that (b) lim su~~(Ai(k) n Y) # 0 Note the following fact (which is evident from the definition in 4.1): (c) lim su~,(A~(~l
n Y) c lim sup Ai.
Now, since A = Lim Ai, we have by 4.2 that A = lim sup Ai. Hence, we see from (b) and (c) above that A n Y # 0; however, since A E (VI, . . . , U,J, clearly A c u$L~U~ and, hence, A n Y = 0. Thus, we have obtained a contradiction. Therefore, there must be an integer N such that (2) holds. By (1) and (2), Ai E (VI,. . . , U,J for all i 2 max {M, N}. Therefore, TV-converges to A. n we have proved that {Ai}zl
When
X Is Compact
Hausdorff
The following theorem is a consequenceof the two preceding theorems. 4.7 Theorem. Let (X,2’) be a compact Hausdorff space. Then, Lconvergence in X for sequencesin CL(X) .is equivalent to TV-convergence in CL(X). Proof. Since compact Hausdorff spacesare regular and countably compact, the theorem follows from 4.4 and 4.6. n 4.8 Corollary. Let (X,T) be a compact, metrizable space. Then, Lconvergence in X for sequencesin CL(X) is equivalent to convergence in CL(X) with respect to the Hausdorff metric. Proof.
The corollary follows from 4.7 and 3.2. I
An analogue of 4.8 for 2x when (X,T) is only metrizable is in Exercise 4.19. One incidental consequenceof 4.8 is worth noting: When (X,T) is a compact, metrizable space, then the Lim operator is the sequential closure operator for the Vietoris topology for CL(X). In contrast, when (X,T) is only metrizable, Lim may not be a sequential closure operator for any topology for CL(X) or, even, Fs(X) - seeExercise 4.20.
26
I. THE TOPOLOGY
Countable
FOR HYPERSPACES
Compactness
Is Necessary
We prove that the condition of countable compactness in 4.6 can not be weakened when (X, T) is a Tr-space. In fact, we prove even more: We restrict ourselves to F2(X) and we restrict ourselves to sequencesthat have a nonempty Lim. In the interest of conciseness,let us agree for the moment to call a sequence, {Ai}cl, of subsetsof X a nontrivial sequenceprovided that Lim A, # 0. 4.9 Theorem. Let (X,T) be a Ti-space. If L-convergence in X for nontrivial sequencesin FZ(X) implies TV-convergence in Fz(X), then (X, T) is countably compact. Proof. Suppose that (X,T) is not countably compact. Then, since (X, T) is a Ti-space, there is a countably infinite subset, Y, of X such that Y has no limit point in X [l, p. 2291. We assumethat Y # X (as we may, by removing a point from Y if necessary), and we let p E X - Y. Now, let yi for i = 1,2, . . . be a one-to-one indexing of the points of Y. For each i=l,2,...,let A
= {~,a).
We seeeasily that Lim A, = {p} (if z # p then, since z is not a limit point of Y and (X,T) is a Ti-space, there exists U E T with x E U such that is a nontrivial sequence U n A, # 0 for at most one i). Hence, {Ai}zl in Fs(X) with Lim Ai = (p}. However, the sequence { Ai}gi does not TV-converge in Fz(X) to {p} since {p} E (X - Y) but Ai 4 (X - Y) for any i (note that (X - Y) E TV because Y is closed in X). H Some research questions related to material in this section and the preceding section are in 83.1-83.5.
Exercises 4.10 Exercise. Let (X,T) be a topological space, and let What is Lim Ai when A, = A for each i = 1,2,. . .? 4.11 Exercise. i = 1,2,. . ., then
If (X,T)
A
C
X.
is a topological space and Ai C X for each
lim sup Ai = nr=i(cl(U&Ai)]. Remark. Regarding the formula for lim sup Ai in 4.11, there can not be a similar type of formula for lim inf Ai [2]!
EXERCISES
27
4.12 Exercise. Let (X,T) be a topological space, and let Ai C X for each i = 1,2,. . .. Then, lim sup Ai and lim inf A, are each closed in X; hence, assuming that Lim Ai exists, Lim Ai is closed in X. 4.13 Exercise. Let (X,T) be a topological each i = 1,2,. . .. Then . . lim mf Ai = hm mf A,, lim sup Ai = lim sup x2, and, assuming that Lim Ai exists, Lim A, = Lim xi. 4.14 Exercise. i = 1,2,. . ., then
space, and let A, c X for
If (X, T) is a topological spaceand A,, Bi C X for each
lim sup (Ai U &) = (lim sup Ai) U (lim sup Bi). Is the analogous formula for limit inferiors also valid? (Compare with the result in the next exercise.) 4.15 Exercise. foreachi=1,2,.... exists and, in fact,
Let (X,T) be a topological space, and let Ai, B, c X If Lim Ai and Lim Bi each exist, then Lim (A, U B2)
Lim (A, u &) = (Lim Ai) U (Lim Bi). 4.16 each i = If A1 If AI
Exercise. Let (X,T) be a topological space, and let Ai C X for 1,2, . . .. c Ax c ..., then Lim Ai = cZ(UEiAi). > AZ > ..., then Lim Ai = ~I~~cZ(A~).
4.17 Exercise. If {Ai}g”=, is a sequenceof connected subsetsof R’ and Lim A, exists, then Lim Ai is connected. Give an example of a sequence, {Ai}fZO=,, of compact, connected subsets of a metric space, X, such that Lim Ai exists, Lim Ai is compact, but Lim Ai is not connected. 4.18 Exercise. Let (X,T) be a compact Hausdorff space. Then, Lconvergence in X for sequencesin C(X) is equivalent to TV-convergence in C(X). Hence, when (X, T) is compact and metrizable, L-convergence in X for sequencesin C(X) is equivalent to convergence in C(X) with respect to the Hausdorff metric.
28
I. THE
TOPOLOGY
FOR HYPERSPACES
4.19 Exercise. Let (X, T) be a metrizable topological space. If { Ai}zI is a sequence in 2x such that Lim Ai E 2x, then {Ai}gI converges with respect to the Hausdorff metric to Lim Ai. Conversely, convergence in 2X with respect to the Hausdorff metric implies L-convergence in X. 4.20 Exercise. The purpose of this exercise is to show that the Lim operator is not necessarily a sequential closure operator for any topology for Fs(X) when the space (X, T) is metrizable (recall the discussion in the second paragraph after the proof of 4.8). First, let us define what it means for Lim to be a sequential closure operator. Let (X,T) be a topological space. Let ?/ C CL(X). For any A C ?t, define L(d) as follows: L(d) = {B E 3-1: B = Lim Ai for some sequence{Ai}c”=, in A}. Then, saying “Lim is a sequential closure operator for some topology for 3-1” simply means that the function L is a closure operator. (It is easy to see that L always satisfies the first three axioms for closure operators in [4, p. 381; hence, L is a closure operator when L is idempotent, i.e., when L 0 L = L.) Now, for our example. Let X be the subspace of R’ consisting of the following numbers: 0; 1; p, = 1+ (l/n) for n = 1,2,. . .; & = (l/n) + (l/k) for n = 2,3,. . ., k = 1,2,. . ., and for which qk < l/(n - 1). Prove that Lim do this by is not a sequential closure operator for any topology for F2(X); considering the subset A of F2(X) given by d={{p,,qL}:n=2,3
,...
andk=l,2
,... }.
References 1.
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1967 (third printing).
2.
R. Engelking, Sur l’impossibilite’ de dt?j%ir la limite topologique infkrieure ci l’aide des ope’rations dt!nombrables de l’algtbre de Boole et de l’ope’ration de fermeture, Bull. Acad. Polon. Sci. Cl. III 4 (1956), 659-662; M.R. 19 (1958), 668.
3. 4.
F. Hausdorff, Mengenlehre, Walter de Gruyter & Co., Berlin, 1927. K. Kuratowski, Topology, Vol. I, Acad. Press, New York, N.Y., 1966.
5. 6.
K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. Sot. 71 (1951), 152-182.
REFERENCES 7.
8. 9.
29
Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. R.E. Smithson, First countable hyperspaces, Proc. Amer. Math. Sot. 56 (1976), 325-328. Daniel E. Wulbert, Subsets of first countable spaces, Proc. Amer. Math. Sot. 19 (1968), 1273-1277.
This Page Intentionally Left Blank
II. Examples: Geometric Models Hyperspaces
for
A geometric model for a hyperspace is a picture that shows what the hyperspace looks like. We obtain many geometric models for hyperspaces. When natural limitations prevent us from drawing a picture of the entire hyperspace, the geometric model will contain enough information to give us a clear mental image of the hyperspace. All of our spaces X are nonempty, compact, and metric. We call a nonempty, compact, metrizable space a compactum; we call a connected compactum a continuum [24]. We use the words subcompactum and subcontinuum when referring to a compactum and a continuum (respectively) as being a subset of a space. According to the terminology just introduced, 2x is the hyperspace of all subcompacta of the compactum X, and C(X) is the hyperspace of all subcontinua of X. All of our geometric models are for these two hyperspaces. In fact, most of our geometric models are for C(X) (the exception being in section 8). Thus, 2” plays a minor role in this chapter; however, 2x plays a major role in Chapter III, which is in reality a continuation of the study of geometric models. Let us note some pertinent information from Chapter I. Let X be a compactum. Then, 2” and C(X) are also compacta (by 3.1, 3.5, and 3.7). We will be concerned with proving that various functions defined on (subspaces of) 2” or C(X) are continuous. Since 2x and C(X) are metrizable (by 3.1), we can use sequences in proving that the functions are continuous. With this in mind, we recall that for sequences in 2x or C(X), TV-convergence, Hausdorff metric convergence, and L-convergence in X are equivalent to each other (by 4.7, 4.8, and Exercise 4.18). 31
32
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
We take this opportunity to give somegeneral terminology and notation that we use throughout the chapter. We use the term nondegenerate when referring to a space t,o mean that the space has at least two points. We use II or x in denoting Cartesian products. We tacitly assumethat all Cartesian products have the Tychonoff (product) topology [18, p. 1471. We will frequently be concerned with Cartesian products of intervals, most often when all the intervals are [O,l]. We use [0, 11% to denote the ith coordinate factor for such Cartesian products. Now, for n = 1,2,. . ., an n-cell is a space that is homeomorphic to I” = IIy==,[O, l]i. A l-cell is called an arc; an end point of an arc, A, is either one of the two points of A that are the image of the end points of [O,l] under any homeomorphism of [O,l] onto A. For each n = 1,2,. . ., aP denotes the manifold boundary of In; in other words, dI” = ((~&
E I” : 5, = 0 or 1 for some i).
A space that is homeomorphic to aP is called an (n- l)-sphere. A l-sphere is called a simple closed curve. Any n-cell is an n-manifold with boundary; we denote the manifold boundary of any n-cell, 2, by dZ. A Hilbert cube is a space that is homeomorphic to I* = llfZo=,[0, l]i. We note that the standard metric, d,, for I” is defined as follows [18, p. 2121: dm((s,)~l,
(yi&)
= 2
2-‘1xi -
yil forall (zi)El, (yi)El E P.
i=l
The dimension of a spacemeans the topological dimension of the space [9]. We mostly use dimension in describing geometric objects - usually polyhedra - for which dimension has a clear geometric meaning. The only time that we use dimension extensively in a technical way is in section 8, where we are concerned with zero-dimensional spaces (we define this notion near the beginning of section 8). We often use the term “finitedimensional”, which the reader can interpret as meaning that the space being referred to is embeddable in R” for some n. The term “infinitedimensional” means not finite-dimensional. We denote the dimension of a space, Y, by dim(Y). Finally, it will be helpful in building many of our geometric models to consider the following subspacesof 2x and C(X). Let X be a compactum, and let K be a subcompacturn of X. We define 25 and CK(X) as follows: 2; = {A E 2x : A > K},
CK(X)
= {A E C(X) : A 3 K}.
We refer to 2: and CK(X) as containment hyperspaces;more precisely, 2; is the containment hyperspacefor K in 2x, and CK(X) is the containment
5. C(X)
FOR CERTAIN FINITE GRAPHS
X
33
&per-space for K in C(X). For a point p E X, we write 2: and C,(X) for the containment hyperspaces for (p}. We note that since 2x and C(X) are compacta, containment hyperspaces are compacta (by Exercise 1.19).
5.
C(X)
for Certain
Finite
Graphs
X
A finite graph is a continuum that can be written as the union of finitely many arcs, any two of which can intersect in at most one or both of their end points. Alternatively, a finite graph is a compact, connected polyhedron of dimension zero or one (the zero-dimensional casebeing the polyhedron that consists of only one point). We construct geometric models for C(X) when X is an arc, a simple closed curve, a noose, and a simple n-od. Regarding 2x when X is a finite graph, seeChapter III.
X an Arc 5.1 Example. We construct a geometric model for C(X) when X is any arc. Specifically, we show that C(X) is a 2-cell. In addition, we determine which subcontinua of X form the manifold boundary of C(X). We first consider the case when X = [0, 11. Observe that the points of C([O,l]) are the closed intervals [a, b] withO
is a 2-cell
34
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
C([O, 11) for 5.1 Figure 4
Proof of 5.1.1. Having just shown that C([O, 11)is a 2-cell, we seethat C(X) is a 2-cell immediately from 1.9 (for i = 6). To prove that K’(X) is as we claim in 5.1.1, it suffices to exhibit one homeomorphism of C(X) onto C([O, 11) that takes Fl (X) u C,(X) U C,(X) onto aC([O, 11) (see 1.3.3 of [33, p. 31). To do this, we start with a homeomorphism, Ic, from X onto [0, I]. We then define k* : C(X) + C([O, 11) as follows:
k*(A) = k(A) for each A E C(X). We already know that k’ is a homeomorphism of C(X) onto C([O, 11) (since k* is defined in the same way in which h’ was defined in the proofs of 1.3 and 1.9). Also, since any homeomorphism between arcs must take end points to end points, we have that
k(h nl) = {O,l). Hence, it follows easily using the formula for k’ that k*[Ji(X)
UC,(X)
u C,(X)]
= Fl([O, 11)u Co([O,l]) U cl([O, 11).
X A SIMPLE CLOSED CURVE
35
Therefore, in view of the formula for aC([O, 11) in (2) above 5.1.1, lc* is a homeomorphism that does what we wanted. n
X a Simple
Closed
Curve
5.2 Example. We construct a geometric model for C(X) when X is any simple closed curve. We show that C(X) is a 2-cell and that aC(X) = E(X). Let us first consider the case when X is the unit circle, S’, in the plane (S’ = {(x1,x2) E R2 : xy +s; = 1)). We define a homeomorphism, h, from C(S’) onto the unit disk D = {(51,z2) E R2 : xf + xz 5 1). We begin by defining h(A) when A is any arc in S1. Let C(A) denote the length of any given arc A in S’ ; let m(A) denote the point of A that divides A into two subarcs of equal length. Then, let h(A) be the point that lies on the straight line segment from (0,O) to m(A) and that is of (Euclidean) distance l(A)/27r fr om m(A) (left-hand side of Figure 5). In other words, considering m(A) as a vector,
h(A) = (1 - [l(A)/274
C(S’) for 5.2 Figure 5
. m(A).
36
II.
EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
Thus, h maps all the arcs in S’ of a given length, t, onto the circle with center (0,O) and radius 1 - (t/2n). NOW, there is only one way that we can define h on the rest of C(S’) so that the resulting function is continuous: we let h({s}) = z for each {z} E Fi(Sl), and we let h(S’) = (0,O). Thus, we have defined h(A) for every A E C(P). It follows rather easily that h is a homeomorphism of C(S’) onto the disk D; also, clearly, h maps Fi(S’) onto dD. Therefore, C(S’) is a 2-cell and dC(S’) = F,(P). The result for simple closed curves in general follows from what we have just shown by adapting the proof of 5.1.1 to the present situation. Let X be any simple closed curve, let k be a homeomorphism from X onto S’ , and let k* : C(X) + C(S) be defined in the same way h” was defined in the proof of 1.3. Then, k’ is a homeomorphism of C(X) onto C(Si); also, by the formula for k’, k* maps Fi(X) onto Fi (S’). Therefore, from what we showed about C(P), we seethat C(X) is a 2-cell and that K’(X) = Fi (X). n
It is appropriate at this time to remark that geometric models for hyperspacescan be used to prove theorems that are not ostensibly about the models themselves. We illustrate this with an application of 5.2 to descriptive set theory in Exercise 5.13.
X a Noose A nooseis a finite graph consisting of a simple closed curve and an arc whose intersection is one of the end points of the arc. 5.3 Example. We show that when X is a noose, the 3-dimensional polyhedron in Figure 6 (top of the next page) is a geometric model for C(X). By 1.9, it suffices to show the result when X = S1 U J, where S’ is the unit circle in R2 and J is the interval [l, 21in the z-axis. We will usually not distinguish between t and (t, 0) for 1 5 t < 2. Also, we will often consider R3 as being R2 x R’. We build a homeomorphism on C(X) in three stages. We start by obtaining a homeomorphism, cp, on C(Sl) such that the images of C(S’) and the containment hyperspace Ci(S’) are suitable. We then extend cp in a special way to a homeomorphism, g, on C(S’) u C,(X). Finally, we extend g to a homeomorphism, h, that maps C(X) onto the polyhedron in Figure 6.
X A NOOSE
C(X),X
37
a noose (5.3) Figure 6
Step 1: A special homeomorphism, cp, of C(S’) into R2. By 5.2, we can consider C(S’) as being the unit disk D in R2. By using the homeomorphism h in the proof of 5.2, we see that Cr(S’) is a 2-cell and that
cl (9) n 6C(S1) = { (1)). Now, let D’ be the disk in D with center at (l/2,0) to l/2. Then there is a homeomorphism, cp, of C(Sl) (p[Cr(S’)] = D’ (by the Schoenflies Theorem [33, p. methods that are based on knowing precisely what the under the homeomorphism h in the proof of 5.2). For use in the next step, we note that (1) cp({ll) = U>O)> which follows from the conditions that cp satisfies and D’naD
and radius equal onto D such that 471, or by ad hoc image of Cr(S’) is
from the fact that
= ((1,O)).
Step 2: Extending cpin a special way to a homeomorphism, g, of C(S’)U Cr (X) into R3. We do this with the help of a homeomorphism, f, of Cr (X) onto D’ x [0, l] that we define as follows. Note that each A E Cl(X)
38
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
can be written uniquely in the form BA U [I, TV], where BA E Ci(S’) and tA E [1,2]. Furthermore, as is easy to see, the function given by il + (BA , tA) is a homeomorphism of Ci (X) onto Ci (S’) x [l, 21. Define f: C,(X) + D’ x [O,l] by f(A) = (y(BA),tA
- 1) for all A E Cl(*Y).
It follows easily that f is a homeomorphism of Ci (X) onto D’ x [0, l]. Now, we use f and ‘p to define the homeomorphism g that we want for Step 2:
(cp(A),O), if A E C(S’) g(A) =
f(A),
if A E C,(X).
The fact that g is well defined is easy to verify: if A E C(Sl) n C,(X), then A E Cr(S’) and, thus, BA = A and tA = 1; hence, by the formula for f, f(A) = (cp(A),O). The fact that g is one-to-one is also easy to verify: since cp and f are one-to-one, we need only show that A = K under the assumption that for some A E C(S’) and I< E Cl(X);
(cp(A)>O) = f(K)
under this assumption, we have from the formula for f that (v(A),@
=
(v’(BK),tK
-
1);
hence, tK = 1, which means that BK = K; thus, (cp(A),O) = (v(K), Oh therefore, finally, A = K (since cpis one-to-one). Now, since g is one-to-one and since ‘p and f are homeomorphisms on the compact spacesC(S’) and Ci (X) (respectively), we seethat g is a homeomorphism of C(S’) U Cl(X) into R3. We need two specific facts about g - (2) and (3) below - for use in = Step 3. Recall from Step 1 that (p[C(S’)] = D; thus, since f[Cl(X)] D’ x [0, 11, we have that (2) g[C(S’) u C,(X)] = (D x (0)) u CD’ x [O, 11). Next, by the formula for f and by (1) near the end of Step 1, observe that f([l,t])
= ((l,O),t
- 1) for all [l,t]
E Cl(J);
thus, since g(Ci(J) = f]Ci(J), we have that (3) g([l,t]) = ((l,O),t - 1) for all [l,t] E Cl(J). Step 3: Extending g to a homeomorphism, h, of C(X) onto the polyhedron in Figure 6, p. 37. Denote the polyhedron in Figure 6 by P. Let
X A SIMPLE
n-oD
39
T denote the triangular 2-cell in P with vertices (( 1, 0), 0), ((l,O), l), and ((2,0), 1). Let K = cl(P - T). By (2) of Step 2, g maps C(S’) U Ci (X) homeomorphically onto K. We also note the following three relevant facts: C(X) = C(S’) u Cl(X) u C(J), [CW)
u
Cl(X)]
n
C(J)
=
Cl(J),
and, by (3) at the end of Step 2, T n K = g[C, (cl)]. Thus, to obtain the homeomorphism h of C(X) onto P that we want, it follows that we only need to find a homeomorphism, j, of C(J) onto T such that j]Ci(J) = g]Ci(J). Th is is easy to do. Define j : C(J) + T by letting j([s, t]) = ((s, 0), t - 1) for all [s, t] E C(J). We seethat j satisfiesour requirements: j is a homeomorphism of C(J) onto T (J’ being a simple modification of the homeomorphism that we defined in 5.1), and j]Ci(J) = g(Ci(J) by (3) of Step 2. Therefore, letting h(A) =
g(A),
if A E C(S’) U Cl(X)
j(A),
if A E C(J)
we seethat h is a homeomorphism of C(X) onto the polyhedron in Figure 6, p. 37. n
X a Simple
n-od
In 5.4 we construct a geometric model for C(X) when X is any simple n-od. Note the following terminology. A simple n-od (n > 3) is a finite graph that is the union of n arcs emanating from a single point, V, and otherwise disjoint from one another (Figure 7, top of the next page). The point 2, is called the vertex of the simple n-od, and each of the n arcs is called a spoke of the simple n-od. A simple 3-od is called a simple triod. A standard simple n-od is a simple n-od lying in some metric linear space such that each spoke of the simple n-od is a straight line segment. An n-fin (n > 3) is a continuum that is the union of n 2-cells (called fins) all of which intersect in a single point and any two of which intersect only in that point (Figure 8, bottom of the next page). Any n-fin, F, evidently contains a simple n-od any two of whose spokeslie in the manifold boundaries of different fins of F; we call a simple n-od that is situated this way in F a base of the n-fin F. (The use of the word “base” comes from thinking of an n-fin in the following way: If Y is a simple n-od with vertex U,
40
II. EXAMPLES:
GEOMETRIC
MODELS
Simple n-od Figure 7
n-fin Figure 8
FOR HYPERSPACES
X A SIMPLE
n-OD
41
then the decomposition spaceof Y x [0, l] obtained by shrinking {v} x [0, 11 to a point is an n-fin; and Y x (0) is a base of this n-fin.) 5.4 Example. We show that when X is a simple n-o& a geometric model for C(X) is the n-dimensional polyhedron that is represented in Figure 9 (top of the next page). The polyhedron may be described as follows: It is the result of attaching an n-fin to the n-cell I” by identifying a base of the n-fin with a standard simple n-od lying in dI”. To be more precise, let F,, denote an n-fin, and let Y be a standard simple n-od in dI”; then, the polyhedron that Figure 9 is intended to suggest is a quotient space - it is the attaching space F, Uf I” that is obtained from the free (disjoint) union of F, and I” by means of a homeomorphism f of a base of F,, onto Y. Regarding our requirement that Y be a standard simple n-od, see 5.5. (For basic information about attaching spaces,see, e.g., [7, p. 1271 or [34, p. 651.) For the proof that C(X) M F, Uf I” when X is a simple n-od, we denote the vertex of X by v and the spokes of X by S1, . . . , S,. The proof focuses on two dominant parts of C(X), namely, the containment hyperspace &(X) and 3 = U~==,C(S~,). Note that C(X) = C,(X) u 3. Also, note that C,(X) n 3 = Ur==, C,(Si), which we denote by f?. We prove first that CV(X) is an n-cell; in fact, as is more pertinent, we define a homeomorphism, (p, of C,(X) onto I” such that (p(B) is a standard simple n-od in dI”. We define ‘p in terms of its coordinate functions . , (Pi. For each i < n, let & be a homeomorphism of Si onto [O,l] such ~~~‘[i(v) = 0; then de&e (Pi : Cv(X) + [0, l] as follows: pi(A) = sup[&(A n Si)] for each A E C,,(X). It follows readily that each (pi is continuous (verify first that the function that assignsA n Si to each A E C,(X) is continuous). Clearly, each pi maps CV(X) onto [O,l]. Also, the family of functions (PI,. . . , I++ separates points: if A, B E C,,(X) such that A # B, then clearly A n Sj # B n Sj for somej; hence, it follows easily that cpj(A) # ‘pj (B). Therefore, we now seethat cp= (cpl,... , cp,) is a homeomorphism of C,,(X) onto I”. Finally, cp(L?)is a standard simple n-od in 81” since v(B) = Y, where Y = {(ti)F=l E I” : ti = 0 for all but at most one i}. Next, we prove that 3 is an n-fin and that B is a base of 3. To seethat 3 is an n-fin, simply note that each C(Si) is a 2-cell (by 5.1 .l) and that C(Si) n C(S,) = {{v}>
whenever i # j.
42
II. EXAMPLES:
GEOMETRIC
C(X),X
MODELS
FOR HYPERSPACES
a simple n-od (5.4) Figure 9
To seethat f? is a base of the n-fin .T, we make three observations: (1) C,,(S) is an arc for each i (cf. 5.1.1); (2) C,(Si) n C,(Sj) = {{u}}
whenever i # j;
(3) C,(Si) C LW(Si) for each i (cf. 5.1.1). BY (1) and (21, B is a simple n-od with vertex {w} and spokes C,,(S); by (3), the spokes of B lie in the manifold boundaries of different fins of T. Therefore, f3 is a base of FT. On the basis of what we have shown, it is easy to see why C(X) M F, Uf In. Nevertheless, we include a proof. For convenience, we choose F, to be F’; we assumethat 3 rl In = 0 (so that we may form the attaching space F,, Uf I” with F,, = 3). We then let f = cp[L?,which maps B onto Y C I”. Before proceeding directly with the proof, we make a few preparatory comments about the nature of the points of F, Uf I”. The points of F,, Uf I” are equivalence classesof points of the disjoint
X
A SIMPLE
43
n-OD
union F, U I”. The relevant equivalence relation, -, on F, U In is the one generated by declaring that B N f(B) for each B E B (recall that a c F,, since F, = F). Therefore, since f = (~]a is one-to-one, the only nondegenerate equivalence classesare {B, p(B)} for the points B E L?. We adopt the following notation for the points of F, Uf I”: for any z E F, UP, [z]- denotes the equivalence classwith respect to - that contains z. Thus, (#)
[B], = {B,p(B)}
= [p(B)],
whenever B E L3,
and [z]~ = { z } whenever z E F, U In - (B U q(D)). In light of the comments just made (especially (#)), we see that the following formula gives a well-defined function, h, from C(X) to F, Uf 1” (recall that f = ‘p]L?): h(A) =
M-1 b(A)l-
ifAEF 7 if A E C,,(X)
Therefore, since ‘p is a homeomorphism of C,,(X) onto In, it follows easily that h is a homeomorphism of C(X) onto F,, Uf I”. Furthermore, LJ is a base of the n-fin 3 = F,,, the attaching map f is a homeomorphism of I3 onto Y, and Y is a standard simple n-od in d1”. n 5.5 Remark. We used the standard simple n-od Y in df” to construct a geometric model for C(X) in 5.4. This resulted in an especially clear geometric model - and, after all, that is our goal. Nevertheless, we make three comments about our choice of Y in 5.4 (reasons that justify the comments will follow) : (1) we could not have let Y be any simple n-od in dI” when n > 4; (2) we could have only required that Y be tame in 81” (definition follows); (3) when X is a simple triod and Y is any simple triod in 813, then C(X) M F3 Uf I3 (where F3 and f are as in 5.4). Before we justify the comments, we give a definition for the terminology in (2). A simple n-od, 2, in i?In is said to be tame in df” provided that there is a homeomorphism, h, of 81” onto OI” such that h(Z) is a standard simple n-od; otherwise, Z is said to be wild in dI” [33]. Now, concerning the comment in (l), there are wild simple n-ods in 81” for any n 2 4 (since there are wild arcs in dP for n 2 4 [33, p. 841). On the other hand, Y in 5.4 is tame in 81”; thus, since the homeomorphism cp in 5.4 from Cv(X) onto I” takes Z?onto Y, we seethat D is tame in &7,(X). Therefore, C(X) could not possibly be homeomorphic to F, Uf I” if f were a homeomorphism of D onto a wild simple n-od in 81”. This verifies (1). The comment in (2) is evident from the definition of tame and from the
44
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
fact that the simple n-od B in 5.4 is tame in G’Cv(X). Finally, (3) follows from (2) since any simple triod in 131~is tame in d13 (which can be proved using the Schoenflies Theorem [33, p. 471). More about finite graphs is in section 65 and in the exercises at the end of section 72.
Historical
Comments
Duda did an in-depth study of C(X) when X is a finite graph ([4]-[6]). He provided a lot of specific information that can be used in constructing geometric models for C(X), and he constructed several such models (see, especially, [6, pp. 248-2551). He also obtained general results, two of which are particularly relevant to what we have done: A continuum, X, is a finite graph if and only if C(X) is a polyhedron [4, p. 2761; if P is a polyhedron and dim(P) 2 3, then P NN C(X) for at most one finite graph X [4, p. 2831. Earlier, Kelley had proved that if X is a Peano continuum, then dim [C(X)] < 00 if and only if X is a finite graph [16, p. 301. Furthermore, Kelley provided a formula for calculating dim [C(X)] when X is a finite graph [16, P. 301 (seethe Remarks in [4, p. 2781). We mention that Kelley stated (without proof) the “only if” part of the first result of Duda quoted above (see [16, p. 311). Recent developments concerning C(X) when X is a finite graph are mostly concerned with Whitney levels (e.g., see [ll] and [13]-[15]; we discusssome of these results in section 65).
Exercises 5.6 Exercise. Construct a geometric model for C(X) when X is a figure eight. (A figure eight is a finite graph consisting of two simple closed curves that intersect in a single point.) 5.7 Exercise. Construct a geometric model for C(X) when X = S’ U 2, where S1 is the unit circle in R2 and Z is the straight line segment from Kw) to GAO). 5.8 Exercise. Construct a geometric model for C(X) when X = S’ U Y U Z, where S1 is the unit circle in R2, Y is the straight line segment from (1,0) to (2,0), and 2 is the straight line segment from (-1, 0) to (-2, 0). 5.9 Exercise. What is dim [C(X)] when X is the finite graph drawn in Figure 10 (top of the next page)? In other words, what is the largest n such that C(X) contains an n-cell? 5.10 Exercise.
Let X be a finite graph.
45
EXERCISES
x for 5.9 Figure IO
If C(X) is a 2-cell, then X is an arc or a simple closed curve (and conversely). Can C(X) be an n-cell for n > 2? If C(X) is the polyhedron in Figure 6, p. 37, then X is a noose. (Prove this without using Duda’s theorem [4, p. 2831, which we stated in the last paragraph of the section.) 5.11 Exercise. Let X be an arc with end points p and q. In the proof of 5.1.1, we used k’ to show that
K(X)
= Fl (X) u C,(X)
u C,(X).
Avoid the use of k* by proving the following fact directly: If A E C(X), then A has an open neighborhood that is homeomorphic to R2 if and only if A 4 Fi(X), p $! A, and q 4 A. 5.12 Exercise. The purpose of this exercise is to gain some insight into a new concept; we will use the concept in connection with geometric models in the next exercise.
46
II. EXAMPLES: GEOMETRIC MODELS
FOR HYPERSPACES
Let A be a collection of nonempty sets. A choice function for A is a function f : A 4 ud such that f(A) E A for each A E A. Choice functions are often called selections when they are continuous on a hyperspace. The Axiom of Choice says that choice functions for A always exist [12]. Thus, there is a choice function for any hyperspace. However, the Axiom of Choice says nothing about the continuity of a choice function for a hyperspace: (1) Let I = [0, 11. Find two selections for 2’. Prove that these are the only two selections for 2’. (2) Find a selection for C(X) when X is any n-od. Regarding (1) and the change to C(X) in (2), we remark that the arc is the only continuum X for which there is a selection for 2x. In fact, the arc is the only continuum X for which there is a selection for Fz(X). These results are in [19] and can be deduced from 1.9 and 7.6 of [22]; see Exercise 75.19. 5.13 Exercise. We give the application of geometric models of hyperspacesthat we mentioned after the proof of 5.2. Prove that there is no continuous choice function for C(S’) by using the geometric model for C(S’) in 5.2. [Hint: Use a form of the Brouwer fixed point theorem.] Remark. For applications of the result in 5.13 and for more about selections, seesection 75 and [25, pp. 253-2671. 5.14 Exercise. Let h : C(S’) + D be the homeomorphism that is defined in the proof of 5.2. Let p = (l,O). What is the precise shape of the image of the containment hyperspace C,(S1) under h? What medicalmathematical phraseology might appropriately describe removing the interior of h[C,(Sl)] from D?
6.
C(X)
When
X Is the Hairy
Point
Our previous geometric models are for finite-dimensional hyperspaces C(X). We now construct a geometric model for C(X) when C(X) is infinitedimensional. Specifically, we construct a geometric model for C(X) when X is the hairy point. We depict the hairy point in Figure 11 (top of the next page). It is a natural, infinite extension of simple n-ods that can be defined as follows. The hairy point is a continuum that is the union of countably infinitely many arcs HI, Hz,. . . satisfying the following conditions: All the arcs Hi
6. C(X)
WHEN X Is THE HAIRY POINT
Hairy point Figure 11
co-fin Figure 12
47
48
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
emanate from a single point, v, and are otherwise disjoint from one another, and limi+, diameter (Hi) = 0. Each of the arcs Hi is called a hair, and the point v is called the follicle of the hairy point. We use the following terminology, which is an extension of terminology preceding 5.4. An co-fin (Figure 12, bottom of the previous page) is a continuum that is the union of countably infinitely many P-cells Fl , F2, . . . all of which intersect in a single point and any two of which intersect only in that point, and such that limi+, diameter (Fi) = 0. Each of the 2-cells Fi is called a fin. Any m-fin, F, obviously contains a hairy point any two of whosehairs lie in the manifold boundaries of different fins of F; we call a hairy point that is situated this way in F a base of the oo-fin F. We now give the example for this section. 6.1 Example. Let X be the hairy point. We obtain a geometric model for C(X) as follows (the model is drawn in Figure 13, top of the next page): We assumethat the Hilbert cube I” has its standard metric d, (defined near the beginning of the chapter). Let Y = {(t%)zr E I”
: ti = 0 for all but at most one i}.
Note that Y is a hairy point, its J‘th hair being the points of Y such that t, = 0 for all i # j (the da-diameter of the jth hair is 2-j). Next, let F, be an oa-fin such that F, n IO0 = 0. Finally, let f be a homeomorphism of a base of Foe onto Y. We show that C(X) x F, uj I=‘=, where F, Uf Iw is the attaching space obtained from F, U I” by means of the attaching map f ([7, p. 1271or [34, p. 651). The proof that C(X) M F, Uf P is similar to the proof in 5.4. We sketch the proof, omitting the details of the verifications (since the details are straightforward modifications of the details in 5.4). Let v denote the follicle of X, and let H1, Hz,. . . denote the hairs of X. Let T = UgiC(Hi), and let B = C,(X) n FT. Note that C(X) = C,(X)
u FT.
6. C(X)
WHEN
X Is THE HAIRY
POINT
49
C(X), X the hairy point (6.1) Figure 13
We show first that C,(X) is a Hilbert cube; in fact, we define a homeomorphism, (p, of C,(X) onto I” such that cp(B) = Y (where Y is as defined above). We define the coordinate functions, vi, of ‘p as follows: for each i= 1,2,..., let & be a homeomorphism of Hi onto [O,l] such that
=
sup
[<:(A n Hi)] for each A E CL(X).
It follows that cp = (cpi, cp2, . .) is a homeomorphism of C,(X) such that q(B) = Y (cf. details in 5.4).
onto P
We seethat F is an oo-fin from the following facts: each C(Hi) is a 2-cell (by 5.1.1); C(H,) n C(Hj) = {{u}} w henever i # j; and, letting d denote the metric on X, the diameter of C(H,) with respect to the Hausdorff metric Hd is equal to the diameter of Hi with respect to d.
50
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
Next, noting that B = uE=,C~(H~), we seethat t? is a base of the oo-fin F (cf. details in 5.4). Now, proceeding as we did in the last three paragraphs of 5.4 (with F = F, and In replaced by I@‘), we see that C(X) z F, Uf P. n The example in 6.1 is from [26, p. 2441. Near the beginning of the section, we described the hairy point as a natural, infinite extension of simple n-ods. Another such extension of simple n-ods is the harmonic fan (Figure 23, p. 92). Steps outlining the construction of a geometric model for C(X) when X is the harmonic fan are in 11.9. In spite of the similarity between the harmonic fan and the hairy point, the techniques employed in 11.9 are quite different than the techniques used in this section.
Exercises 6.2 Exercise. Construct a geometric model for C(X) when X is the following continuum: X = Yr U A U Ys, where Yr and Ys are mutually disjoint hairy points, A is an arc, and the only point of A n Y, is the follicle of Y, for each i. 6.3 Exercise. Construct a geometric model for C(X) when X is the following continuum: X = Yr USUY,, where Yr and Y2 are mutually disjoint hairy points, S is a simple closed curve, and the only point of S n Yi is the follicle of Yi for each i. 6.4 Exercise. Let X be the continuum in R2 consisting of the straight line segment from (0,O) to (1,0) and the straight line segmentsfrom (l/i, 0) to (l/i, l/i) for each i = 1,2,. . .. Let p = (O,O), and prove that C,(X) is a Hilbert cube. Remark. The continuum in 6.4 is called the null comb. Now, let X be a Peano continuum that is not a finite graph. Then it can be shown that X contains a hairy point or a null comb ([lS, p. 301 or [26, pp. 2452461). Thus, by 6.1 and 6.4, C(X) contains a Hilbert cube. This proves the essential half of the following result of Kelley: If X is a Peano continuum, then dim[C(X)] < 00 if and only if X is a finite graph [16, p. 301. For related material including applications and stronger results, see [lo], [ll], and [26]. Results about dim[C(X)] for any continuum X are in Chapter XI.
7. C(X)
7.
C(X)
X Is THE CIRCLE-WITH-A-SPIRAL
WHEN
When
51
X Is the Circle-with-a-Spiral
The circle-r&h-a-spiral is the continuum X in Figure 14; X = S1 u S, where S’ is the unit circle in R” and S is the spiral given in polar coordinates bY S={(r,19):T=l+&j and B 2 0). In 7.1 we show that C(X) is homeomorphic to the cone over X when X is the circle-with-a-spiral. Let us recall the definition of the cone over a space and its geometric interpretation.
Cones,
Geometric
Cones
Let Y be a topological space. The cone over Y, which we denote by Cone(Y), is the quotient space obtained from Y x [0, l] by shrinking Y x { 1) to a point; in other words, Cone(Y) is the quotient space Y x [0, l]/ N, where N is the equivalence relation on Y x [0, I] given by (yr , ti) N (~2, t2) if and
The circle-with-a-spiral Figure 14
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II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
only if (yi, ti) = (~2, t2) or ti = t2 = 1. The point Y x (1) of Cone(Y) is called the vertex of Cone(Y); the subset Y x (0) of Cone(Y) is called the baseof Cone(Y). If Y is a compactum, then Cone(Y) is topologically the same as the space G(Y) that results from the following geometric construction (we illustrate G(Y) in Figure 15): We may assumethat Y c E, where E = R” for some n or E = IO0 [18, p. 2411. Fix a point p E E. Consider the space E x [0, 11. Let 2, = (p, 1). For each y E Y, let yw denote the straight line segment in EX [0, I] from (y,O) to v (i.e., yv = {t.v+[l-t].(y,O) : 0 5 t 5 1)). Let G(Y) = u{yv: y E Y}. We now prove that G(Y) and Cone(Y) are homeomorphic. Let x : Y x [0, l] + Cone(Y) denote the quotient map, and let f : Y x [0, l] + G(Y) be given by f(y, t) = t. 21+ [l - t] (y, 0) for all (y, t) E Y x [0, 11.
Geometric cone G(Y) Figure 15
THE MODEL FOR C(X)
53
Note that f o 7r-l is a function (since f(y, 1) = v for all points (y, 1) in Y x [0, 11); hence, f o r -’ is continuous (since f is continuous and rr is a quotient map - see 3.2 of [7, p. 1231 or 3.22 of [24, p. 451). Clearly, for-1 is one-to-one. Also, Cone(Y) is compact (since Y x [0, l] is compact and rr is continuous). Therefore, f o 7r-’ is a homeomorphism of Cone(Y) onto G(Y). Referring to the construction above, we call G(Y) the geometric cone over Y; we call the point v of G(Y) the vertex ofG(Y) (note that f o 7r-l defined above takes the vertex of Cone(Y) to v), and we call the subset Y x (0) of G(Y) the base ofG(Y). We note that the construction of G(Y) can be carried out as above for any separable metric space Y. However, for example, if Y is the subspace of R’ consisting of the integers, then Cone(Y) ic: G(Y).
The Model
for C(X)
7.1 Example. Let X be the circle-with-a-spiral, X = S1 U S (Figure 14, p. 51). We prove that C(X) M Cone(X); thus, G(X) is a geometric model for C(X). We use the following notation. For any arc, A, in X, we let t(A) denote the arc length of A. Let z,y E X such that x # y. If z,y E S, then xy denotes the arc in S from x to y; if x, y E S’, then xy denotes the arc in S’ from x to y in the counterclockwise direction. For any z E X, xx = {z}. For any z E S, xS’ denotes the smallest subcontinuum of X containing {x} US’ (hence, zS1 x X for each z E S). We define a subset, Y, of R3 as follows (Y is depicted in Figure 16, top of the next page). We let T denote the radial projection of X onto S’; more precisely, for each x E X, r(x) is the point of S’ that lies on the straight line segment in R2 from the origin to z. For each x E S, let x2, be the closest point in S to x such that ~(xxg,) = S’; let V, be the vertical line segment in R3(= R2 x R’) from (x, 0) to (x, [(zxsx)). Note, as in Figure 16, that [(x52,) > 27rfor each 2 E S. Let Y = (U{V2 : 2 E S}) u (S’ x [0,2n]). (We note that Y M X x [0,27r].) We will define a homeomorphism, f, of C(X) - {S’} onto Y - (Sr x (27r)). We will then “extend” f to a homeomorphism, h, of C(X) onto
54
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
Y in 7.1 Figure 16
the quotient space, Y/(S’ x {Zr}), that is obtained from Y by shrinking S1 x (27r) to a point. Finally, we will seethat Y/(Sl x (27r)) M Cone (X). The map f is made up of two maps, cr and /3, which we define in the next two paragraphs. We define o, which will be a homeomorphism of C(S’) - {S’} onto S’ x [0,27r), as follows. For each 2 E 5”) let
A, = {xy : y E S'}, and let cyz : A, + {x} x [0,2~) be given by Q, (zy) = (z, l(zy)) Then let Q:: C(9)
for each sy E A,.
- {S’} + S’ x [0,2x) be given by
cu(zy) = az(xy)
for each zy E C(S’) - {S’}.
THE MODEL FOR C(X)
55
We define p, which will be a homeomorphism of C(X) - C(S’) onto U{Vz : 5 E S}, as follows. For each z E S, let f?; = {xy : y E xx2r} and let aq = {xy : y E (x&l) I-IS} u {xs’}; note that f?: and Bq are arcs in C(X) such that f?: fl Bz = (zc~~}. For each x E S, define subarcs, V,’ and V,“, of V, as follows: v,’ = {x} x [U,ZK], v,” = {x} x [27T,tyXX2~)]. Now, for any z E S, let /?i be the homeomorphism off?: onto Vi given by . [(xy))
P5(XY) = (XT & Then let 8’ = u{Bi : x E S}, vl = u{V,’ : 2 E S}, and let p’ be the homeomorphism
for each zy E Bk.
of B1 onto V1 given by
p’ (xy) = &(xy)
for each xy E B’.
Next, let 82 = u{BZ : x E S} and let v2 = u{V,’ : x E S}. It is geometrically evident that O2 and V2 are homeomorphic; nevertheless, we need an especially well-behaved homeomorphism, p2, of B2 onto V2. We define /I2 in terms of homeomorphisms (pi : B2 + S x [l, 21 and 92 : V2 + S x [l, 21. Let cpi be the homeomorphism of a2 onto S x [l, 21 defined as follows: for each XY E u2 cpl(XY) = b , t(zz~‘~+Y&,j) and cpl (x,9’) = (x, 2) for each zS1 E f?2. Let ‘p2 be the homeomorphism of V2 onto S x [l, 2] defined as follows: p~(z, t) = (x, &$& + 1) for each (z, t) E V2. Now, let ,B2 = (p;’ o (pl. Clearly, p2 is a homeomorphism Also, ,02 agrees with ,B’ on L?l n U2 since for any x E S, P2 (xX2,)
and
=
cp;l
[cpl (xX2,)1
=
(Pi1
[(XT
I)]
=
b,2r)
of B2 onto V2.
56
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
Therefore, finally, we can define the homeomorphism p of C(X) - C(9) onto U{Vz : 5 E S} by letting
P(A) =
P’(A), P2(A),
if A E I?’ if A E f?2.
Now, we define f : C(X) - {S’} + Y - (S’ x (27r)) by the following formula: a(A), if A E C(S’) - {Sl} f(A) = if A E C(X) - C(S’) 1 P(A), It follows easily that f is a homeomorphism of C(X) - {S1} onto Y - (5” x {2x1). We can not extend f to a continuous function of C(X) into Y - each point of S’ x (2~) would have to be the value of such an extension at the point S’ of C(X)! Note that this is the only problem in extending f to a homeomorphism on all of C(X). Thus, the next step is natural: we change S’ x (2~) to a single point. Let 2 be the quotient space obtained from Y by shrinking S’ x (27r) to a point, which we denote by v. Let q : Y + Z be the quotient map; in other words (remembering that the points of 2 are equivalence classes), Cl(Y) =
{y}, { V,
if y E Y - (S’ x (2n)) if 1JE S’ x (27r).
We have drawn 2 in Figure 17 (top of the next page) - Figure 17 shows how 2 would look if 2 were obtained from Y in the most natural geometric way, namely, as the end result of a deformation of R3 that pulls S’ x (27r) in along concentric circles to the point v = ((0, 0), 2n). Define h : C(X) + Z as follows:
h(A) =
q o f(A),
if A # S’
V,
if A = S1.
It follows easily that h is a homeomorphism of C(X) onto Z. Finally, we show that Z M G(X). For convenience, we let G(X) be the geometric cone over X with vertex v = ((O,O), 2x); seeFigure 18, p. 58. We use the notation and auxiliary maps defined in the next two paragraphs. Concerning G(X), let y be a homeomorphism of S onto [r, 2n) (as we will see, the choice of ?r as the first point of the range of y will not be significant - any point t > 0 such that t < 23~would do as well). For each straight line segment 22) in G(X) such that z E S, let p, be the point of xv at height y(z). Thus, the points ps spiral around in G(X) in a continuous, increasing path that is headed toward v as the points x advance around S.
THE
MODEL FOR C(X)
57
2 = Y/S’ x (2~) in 7.1 Figure 17
Concerning 2, recall that q is the quotient map of Y onto Z. It is implicitly indicated by Figures 16 and 17 that for each z = (x,0) E S, q(V%) is the straight line segmentfrom z(= q(z)) to q(z, &(n2,)). Let e = (2,O) E S, and let t, denote the third coordinate (height) of q(e,l(eez=)). We consider the point q(e, Qeez*)) to be the point at the top of the right-hand side of 2 in Figure 17; in particular, t, > 27r. Let X be a homeomorphism of S onto (27r,te] such that for each z E S - {e}, X(s) is strictly less than the third coordinate of q(s,l!(zzz,)) (which is the point at the top of q(Vz)). For each z E S, let 2%be the point of q(V,) at height X(z). Now, we describe a homeomorphism, g, of G(X) onto Z. For each 2 E S, g maps zv into Z as follows: g(x) = z; g maps the straight line segment sp, linearly onto the straight line segment zzz in q(Vz); g maps the straight line segment p,v onto a spiral, Sz., that begins at zz(= g(p%)), ends at w, and is asymptotic to the spiral Szc at the top of Z in Figure 17
58
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
(when 2 = e, g maps p,v onto the spiral S,J. It is easy to see how to choose the spirals S,, for all z E S so that g is a homeomorphism from U{zv : z E S} onto 2 - q(S’ x [0,2~)). Defining g on the rest of G(X) is simple: Note from Figures 17 and 18, that G(S’) = q(S1 x [0,27r]), and let g : G(S’) t q(S’ x [0,2x]) be the identity map. Now, having defined g on all of G(X), it follows easily that g is a homeomorphism of G(X) onto Z. We have shown that C(X) z Z (by the map h) and that Z M G(X) (by the map g). Also, as we showed for any compacturn near the beginning of
G(X), X the circle-with-a-spiral (7.1) Figure 18
KNASTER’S QUESTION the section, G(X) z Cone(X). Therefore, C(X) Figure 18 is a geometric model for C(X). W
59 M Cone(X)
and G(X)
in
The example in 7.1 is due to James T. Rogers, Jr. [32, p. 2831.
Knaster’s
Question
The example in 7.1 led to the answer to a question asked by B. Knaster in 1952. Knaster’s question, which appeared in the (unpublished) New Scottish Book, is as follows: If Y is a continuum with the fixed point property, then must C(Y) have the fixed point property? We briefly discuss how 7.1 led to the answer to Knaster’s question. Let X be the circle-with-a-spiral. Ronald Knill showed that Cone(X) does not have the fixed point property ([17]; a more directly accessible proof is in [3, p. 1291). Using Knill’s result and 7.1, Rogers observed that C(X) does not have the fixed point property [32, p. 2821. In fact, this was the first example of a continuum whose hyperspace does not have the fixed point property; however, this example does not answer Knaster’s question since the continuum X also does not have the fixed point property. Nevertheless, consider the natural extension of X obtained by adding the unit disk, D, to X (D = ((2, y) E R” : 2’ + y2 5 1)) - X U D has the fixed point property and C(X U D) does not have the fixed point property. The proof that C(X U D) does not have the fixed point property is in [27] and is done by showing that C(X) is a retract of C(X U D); a proof that X U D has the fixed point property is in [3, p. 1231. For results about hyperspaces that have the fixed point property, see section 22.
When
C(Y)
M Cone(Y)
We discuss the following natural question: Which continua, Y, have the property that C(Y) z Cone (Y)? We do not suggest that the original motivation for the question should come from 7.1. Instead, it comes primarily from the following two general similarities between C(Y) and Cone(Y). First, there are arcs in Cone(Y) that go from the base of Cone(Y) up to the vertex of Cone(Y) (namely, the arcs yv in G(Y)); similarly, there are arcs in C(Y), called order arcs, that go from Fi (Y) up to Y (14.6). Second, there is a natural projection of Cone(Y) onto [O,l] that measures the height of points in Cone(Y); similarly, there is a continuous function from C(Y) into [0, oo), called a Whitney map, that measures the height (with respect to the partial order of containment)
60
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
of points in C(Y) (13.4). We are therefore led to represent C(Y) as in Figure 19, which resembles the way we represented Cone(Y) as G(Y) in Figure 15, p. 52. Lest our discussion of similarities be misleading on one point, let us make an observation. We know from 7.1 that C(X) M Cone(X) when X is the circle-with-a-spiral; however, there is no homeomorphism of C(X) onto Cone(X) that takes the point X of C(X) to the vertex of Cone(X). See Exercise 7.5. In spite of the general similarities between C(Y) and Cone(Y) that we mentioned above, these spacesare usually quite different. We can seethis from geometric models in the two preceding sections (5.1 and 5.2 are exceptions since they show that C(Y) M Cone(Y) when Y is an arc or a simple closed curve). In fact, if Y is a finite-dimensional Peano continuum such that C(Y) M Cone(Y), then Y is a finite graph; hence, Y is an arc
C(Y), Y a continuum Figure 19
WHEN
C(Y)
rz CONE(Y)
61
or a simple closed curve. The first part of this result is a consequence of comments in the remark following 6.4; the second part is left as an exercise in 7.9. One specific difference between C(Y) and Cone(Y) that is worth mentioning concerns dimension. On the one hand, dim[Cone(Y)] < M whenever dim(Y) < oo (cf. [25, p. 3011); on the other hand, dim[C(Y)] = 00 whenever dim(Y) 2 2 ([20]; see Chapter XI). Therefore, we have the following result (which was known before [20] - see [25, p. 3111):
7.2 Theorem. If Y is a finite-dimensional continuum such that C(Y) x Cone(Y), then dim(Y) = 1.
We note that the assumption that Y is finite-dimensional is necessary in 7.2. For example, according to (2) of 11.3, C(P) M I” and, by 9.7, I” x Cone(P). Indecomposable continua play a significant role in trying to determine when C(Y) x Cone(Y). We define indecomposable continua and make somecomments about them. Then we return to our discussion about when C(Y) z Cone(Y). A continuum is said to be decomposableprovided that it is the union of two proper subcontinua (proper means not equal to the whole space). A continuum that is not decomposable is said to be indecomposable. A continuum is said to be hereditarily decomposableprovided that all of its nondegenerate subcontinua are decomposable. A continuum is said to be hereditarily indecomposableprovided that all of its subcontinua are indecomposable. Although it may seem that all nondegenerate continua are decomposable, this is far from being the case. Most continua in R”(n 2 2) are hereditarily indecomposable, where most means that they form a denseGJ set of points in the hyperspace C(P) ([21]; also proved in 19.27 of [25, p. 6131). The following results of Bing [2, p. 2701point out the abundance as well as the diversity of hereditarily indecomposable continua: Any n-dimensional continuum (n < oo) contains an (n - 1)-dimensional, hereditarily indecomposable continuum; in addition, there are infinite-dimensional, hereditarily indecomposable continua. Some specific examples of indecomposable continua are constructed in [24, pp. 7, 13, 21, and 221. We return to our discussion of when C(Y) M Cone(Y). The remainder of the discussion centers around the next theorem, which is due to Rogers [30, p. 2861. We remark that the theorem is significantly stronger than 7.2 (it implies 7.2 by the result in [2, p. 2701that we stated above).
62
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
7.3 Theorem. If Y is a finite-dimensional continuum such that C(Y) M Cone (Y), then Y contains at most one nondegenerate indecomposable continuum. In Concerning the continuum Y in 7.3, Y itself can be indecomposable. fact, aside from the arc and the simple closed curve, the first continua Y for which it WLXS shown that C(Y) M Cone(Y) were indecomposable. They were the nonplanar solenoids; the fact that their hyperspace and cone are homeomorphic was of interest in connection with an embedding theorem. Specifically, Rogers proved that if Y is any planar circle-like continuum, then C(Y) is embeddable in R3 [31, p. 1661. In order to show the necessity of being planar, Rogers then proved that C(Z) M Cone(Z) whenever 2 is a nonplanar solenoid [31, p. 1671(also proved in [8]); this showed that C(Z) is not embeddable in R3 by the theorem about cones in [l]. In view of 7.3, finite-dimensional continua,Y, such that C(Y) fi: Cone(Y) are closeto being hereditarily decomposable. Thus, it is natural to consider the following question: Which hereditarily decomposablecontinua, Y, have the property that C(Y) z Cone(Y)? This question has been completely answered [23]:
7.4 Theorem. Let Y be a hereditarily decomposable continuum. Then, C(Y) z Cone(Y) if and only if Y is one of the eight continua in Figure 20 (next page). More about hyperspaces and cones is in section 80.
Exercises 7.5 Exercise. Let X be the circle-with-a-spiral (Figure 14, p. 51). Recall from 7.1 that C(X) M Cone(X). Prove that if h is any homeomorphism of C(X) onto Cone(X), then h(S’) = v, where v is the vertex of Cone(X). [Hint: Cone(X) - {v} is not arcwise connected.] 7.6 Exercise. Let Y be the continuum in (4) of Figure 20; Y is often called the Warsaw circle or the &(1/z)-circle. Determine the subcontinuum of Y that would have to correspond to the vertex of Cone(Y) under all homeomorphisms of C(Y) onto Cone(Y). 7.7 Exercise, Let Y be the continuum in (3) of Figure 20; Y is called the sin(l/x)-continuum. Prove that C(Y) x Cone(Y). 7.8 Exercise. Let Y be the Warsaw circle ((4) of Figure 20). Prove that C(Y) M Cone(Y) by making use of 7.7.
EXERCISES
63
(1)
(3)
(5)
w
(4)
(6)
Continua Y for 7.4 Figure 20
64
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
7.9 Exercise. We mentioned the result stated below when we were discussing differences between C(Y) and Cone(Y) (in the two paragraphs preceding 7.2). Prove the result in an ad hoc manner (i.e., without using 7.4). The arc and the simple closed curve are the only finite graphs, Y, for which C(Y) M Cone(Y).
8.
2x When
X Is Any
Countably
Infinite
Compactum
We obtain a geometric model for 2x when X is any countably infinite compacturn. More generally, we obtain a geometric model for 2x when X is any zero-dimensional, infinite compactum with a dense set of isolated points. In fact, we show that all such compacta have the same geometric model for their hyperspace. The model is in Figure 21, p. 71, and the results are 8.9 and 8.10. As a simple illustration of the results, consider the following two compacta: Y
=
{O&
)... },
2
=
Yu{l+L=1,2,...}. n
By 8.10, 2’ and 2z are homeomorphic to the compactum P in Figure 21; hence, 2” z 2’ even though Y $ 2. This is the simplest example showing that the converse of 1.3 is false for compact metric spaces (recall the comments following the proof of 1.3). Concerning the example above, Ponomariov was apparently the first person to ask if the converse of 1.3 is true for compacta [29, p. 1951; Pelczyriski [28] obtained the results stated here in 8.9 and 8.10 as a negative answer to Ponomariov’s question. Most of the results in this section are due to Pelczynski (the exceptions being 8.1 and 8.6). We use the following terminology. There are several notions of dimension [9, p. 1531;they agree with one another for separable metric spacesbut not in general. It suits our purpose best to define zero-dimensional for topological spacesin general just as it is defined for separable metric spacesin [9, p. lo]: We say that a topological space, Y, is zero-dimensional, written dim(Y) = 0, provided that Y # 0 and there is a base for the topology for Y such that each member of the base is both closed and open in Y. For example, any nonempty, countable, metrizable space is zero-dimensional (Exercise 8.11).
CANTOR SETS
65
From now on, we say that a set is clopen in Y to mean that the set is both closed and open in Y. We say that a topological space, Y, is perfect provided that every point of Y is a limit point of Y. An isolated point of a space, Y, is a point of Y that is not a limit point of Y (i.e., a point 9 E Y such that {y} is open in Y).
Cantor
Sets
Cantor sets play a central role later in the section. The Cantor Middle-third set is the subspace, C, of [O,l] that is obtained as follows: C = IT~lCi, where Cr = [0,1/3]~[2/3, l] and, assuming inductively that we have defined Ci, we define CI+i to be the subset of C, obtained by removing from C, the middle-third open interval of each maximal subinterval of C,. Any space that is homeomorphic to C is called a Cantor set. Cantor sets can be characterized intrinsically; we state the characterization in 8.1 for use later. Proofs of 8.1 are in many texts, including [24, p. 1091 and [34, p. 2161. (The characterization in 8.1 is often stated in terms of totally disconnected compacta instead of zero-dimensional compacta see 12.11). 8.1 Theorem. A spaceis a Cantor set if and only if it is a zero-dimensional, perfect compactum.
Preliminary
Results
We use the results in this part of the section to prove the structure theorem in 8.7. 8.2 Lemma. Let (X, d) be a metric space, and let A E 2”. If some point, p, of A is a limit point of X, then there is a sequence, {Ai}zl, converging to A in 2x such that p E Ai and A, # A for each i. Proof. Since p is a limit point of X, there is a sequence, {pi}zi, X - {p} such that {pi}:1 converges to p. Let
ri = 2-‘d(p,pi)
for each i.
Now, define A, for each i as follows: ifpi #A if pi c A.
in
66
II. EXAMPLES:
GEOMETRIC
MODELS
FOR HYPERSPACES
It follows easily that the sequence {Ai}zl just defined has the required < 3r, for each i, properties (concerning convergence, note that Hd(A,Ai) and recall 3.1). n Let (X, d) be a metric space, and let A E 2x. Then, 8.3 Proposition. A is an isolated point of 2x if and only if each point of A is an isolated point of X. Proof. If A is an isolated point of 2x, then it follows immediately from 8.2 that each point of A must be an isolated point of X. Conversely, assume that each point of A is an isolated point of X. Then, since A is compact, A must be a finite set, say
,4 = {al,. . . ,a,,}. Clearly, {A} = ({al }, . . . , {a,}), which is open in 2x since each {ui} is open in X (recall 1.2). In other words, A is an isolated point of 2x. n We reformulate 8.3 in a more readily applicable form by using the following notation: For a space Y, Z(Y) denotes the set of all isolated points of Y. 8.4
If (X,d) is a metric space, then Z(2x) = 2’(“).
Proposition.
Proof. The result is simply a restatement of 8.3 using the notation just introduced. n 8.5 Corollary. Let (X,d) be a metric space. Then, 2(2-‘) is dense in 2x if and only if Z(X) is dense in X. Proof.
The corollary follows easily from 8.4. n Let (X,T)
8.6 Proposition.
be a Tl-space. Then, dim(X) = 0 if and
only if dim(2-Y) = 0. Proof. Assume that dim(X) = 0. Let W = {W, : (Y E A} be a base for T such that each W, is clopen in X. Let
fl = {(W,(I),
. . . I Wa,,,) l-l 2” : n < 00 and Waci) E W for each i}.
Since the members of W are clopen in X, it follows from 1.2 and Exercise 1.21 that the members of R are clopen in 2x. Furthermore, R is a base for the Vietoris topology for 2x (Exercise 8.12). Therefore, dim(2x) = 0. Conversely, assumethat dim(2x) = 0. Then, since Fl (X) c 2x, we have that dim[Fl(X)] = 0. Hence, by Exercise 1.15, dim(X) = 0. n
STRUCTURE THEOREM
Structure
67
Theorem
For a space Y, we let h/(Y) d enote the set of all nonisolated points of Y; in other words, N(Y) = Y - Z(Y). Under the assumptions on X, the theorem below says that 2” is a. metric compactification of the integers with a Cantor set as the remainof a space, Y, is a compact space, Q, in which der. (A compactification Y is embedded as a dense subset; Q - Y is called the remainder of the compactijkation.) 8.7 Theorem. Let X be a zero-dimensional, infinite compactum such that Z(X) is dense in X. Then, 2,’ is a compacturn such that Z(2,‘) is dense in 2,y and N(2*‘) is a Cantor set. Proof. By 3.1 and 3.5, 2x is a compact,um. By 8.5, Z(2x) is dense in 2”. Therefore, it remains to show that &‘(2dY) is a Cantor set. We will use 8.1. 111order to apply 8.1, we prove (l)-(4) below. Note that if Y is any infinite compactum, then N(Y) # 0 and n/(Y), being closed in Y, is compact. Thus, since 2.’ is an infinite compactum, we have that (1) wm # 0 and (2) JV(~~) is compact. By 8.6, dim(2x) = 0. Therefore, since it is evident that any nonempty subspace of a zero-dimensional space is zero-dimensional, we have by (1) that (3) dim[i\/(2-‘)I = 0. By 8.3, ni(2x) = {A E 2” : A n N(X) # 0). Hence, it follows from 8.2 that (4) N(2.‘) is perfect. By (l)-(4) and 8.1, n/(2x)
is a Cantor set. n
Uniqueness
of Compactifications
The proposition below says that, for a given compactum Y, there is only one metric compactification of the integers with Y as the remainder. (The analogue for metric compactifications of other spacesis not necessarily true; e.g., (7) and (8) in Figure 20, p. 63 are nonhomeomorphic compactifications of R’ with a simple closed curve as the remainder.) 8.8 Proposition. Let Y and Z be infinite compacta such that Z(Y) is dense in Y and Z(Z) is dense in Z. If N(Y) M ni(Z), then Y x Z.
68
II. EXAMPLES:
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MODELS
FOR HYPERSPACES
Proof. Let h be a homeomorphism of n/(Y) onto JV( 2). We will extend h to a homeomorphism, h,, of Y onto 2. Let d and D denote metrics for Y and 2, respectively. For any natural number, n, and any infinite subset, Yu, of Z(Y), we let Y(n,Ye) denote a given set of n distinct points of Ye with the following property: (1) d(yo>NW) L ~YJV)) for any yo E YO - Y(n,Yo) and any y E l’(n, lb). The meaning of .Z(n, Z’s), where n is any natural number and ZO is any infinite subset of Z(Z), is analogous. We use induction to define Rk, Sk, and hk for each k = 0, 1,2, . . .; Rk and Sk will be finite subsetsof Z(Y) and Z(Z), respectively, and hk will be a one-to-one function from Rk onto Sk. Let & = So = 0 and ho = Q) Assume inductively that we have defined finite sets Re C 1(Y) and Se C Z(Z) and one-to-one functions he from Re onto St for all e such that 0 < C 5 k - 1. Note that N(Y) and J/(Z) are compact; hence, there are finite (nonempty) sets, Al,
=
{ak,l,
B/c
=
{bk,lrbk,2,.
ak,2,
. . . , ak,tQk)
1 c
N(Y)
c
N(z),
. . ,bk,m(k)}
such that each point of n/(Y) (respectively, N(Z)) is within l/k of a point of Ak (respectively, Bk). Noting that z(Y) - UiiJ Re is an infinite set, let Pk = Y(n(k),Z(Y)
- u;z;R&
For each point p E 4, let a(p)
=
min{j
: d(p,
ak,j)
=
d(p,
Ak)}.
Index the points of Pk as pk,i,pkJ, . . . ,pk,n(k) so that if s < t, then either (2) or (3) holds: (2)
@k,s)
<
dpk,t),
(3) Q(Pk,s) = +k,t) and @k,s, a~&~,,)) > d(Pk,t, ‘&,a(ph,t))~ is an infinite set, we can define two sets, PL and Next,, since Z(Z) - UfliS, QL?as follows: let P;,~, ~6,~~ . . . , pk,+) be n(k) distinct points of Z(Z) Ut$Se
such that for each i, D(Pi,i, h(‘k.a(pb.r)) I d(Pk,t>‘k,a(pk,i));
UNIQUENESS
OF COMPACTIFICATIONS
69
then let and let Q’k = Z(m(k),I(Z)
- [Pi u (uf:$%)]).
For each point qf E Q’,, let p(q))
= min{j :
D(q',bk,j)
=
D(q',Bk)}.
Index the points of Qk as ., so that if s < t, then, as is analogous to (2) and (3) above, either (4) or (5) holds: qL,l,
q6,2r..
qk,mck)
(4) Pk4,,) < Pk7LJl (5) P(qIL,s)= PC&) and Wq k,87bk,L3(q;“,) 2 WA,,1 hc7;., 1). In analogy with how we defined PL in terms of Pk and Ak, we define Qk in terms of QL and Bk as follows: noting that, Z(Y) - [Pk U (UfziRe)] is be m(k) distinct points of Z(Y) an infinite set, let qk,l , qk,2, . . . , qk,m(k) such that for each i, [pk u (uf:; &)I d(qk,z> h-1(bk,/3(&)))
5
D(qh,i~bk,4(q~~,,))~
and let &k = {qk,l,??k,2,.
. ,qk,m(k)h
Finally, let Rk = Pk U Qk and Sk = Pk U Qk, and define hk : Rk + Sk as follows: hk(pk,t)
=
p;,, for all pkj E 9,
hk(qk,z)
=
9;,i
for
all
qk,i
E
Qk.
Therefore, by induction, we have defined Rk, Sk, and the one-to-one function hk from Rk onto Sk for each k = 0,1,2,. . .. We will define the homeomorphism h, of Y onto Z using Rk, Sk, hk, and h. We first prove the following two facts: (6) UFO=,& = z(Y); (7) UFO=,& = z(z). To prove (6), supposethat (6) is false. Then there is a point ys E Z(Y) UpEo Rk. Hence, by the way we defined Rk and 9, yo 51u&Pk
= u&Y(n(k),Z(Y)
- u;z;Rt).
70
II. EXAMPLES:
Thus, since Y/OE Z(Y) (#)
GEOMETRIC
FOR HYPERSPACES
for each k, we see from (1) that
U~~~Rp
4!/o,N(Y))
MODELS
L &N(Y))
for all p E uglPk.
From the definitions of Pk and &, we see that the sets PI, Pz, . . . are mutually disjoint and nonempty; hence, UpE=lPk is an infinite set. Thus, since Y is compact,, inf {d(p,N(Y))
: p E ur&Pk}
= 0.
Hence, by (#), ~YO,N(Y)) = 0; however, since yo E Z(Y), this is impossible. Therefore, (6) must be true. The proof of (7) is similar (using the sets Q’,) and is therefore omitted. Now, let h, be given by
h,(y) = h(y)1 h(Y),
if g f N(Y) ifyE&
and note that h, is well defined since the setsN(Y), RI, R2,. , . are mut,ually disjoint. Also, h, is defined on all of Y by (6). We seethat h, maps I’ onto all of 2 by (7) since hk(Rk) = Sk for each k and h[N(Y)] = N(Z). We see that h, is one-to-one since h and each hk is one-to-one and since the sets N(Z), 4, SZ,. . . are mutually disjoint. Finally, it follows that h, is continuous by using the uniform continuity of h, the properties of the sets AEL and Bk (k = 1,2,. . .), and (2)-(5); for details, see[28, p. 871. Therefore, since Y is compact and 2 is Hausdorff, h, is a homeomorphism of Y onto 2.
n
The
Model
for
2x
We now come to the main results. We will refer to a specific compacturn P, which we describe as follows (Figure 21, top of the next page): P is a metric compactification of the integers with a Cantor set as the remainder (by 8.8 there is only one such compactification). In Figure 21, the Cantor Middle-third set C is the remainder, and the isolated points of P lie above the midpoints of the maximal intervals in [0, l] - C (we could just as well have chosen the isolated points of P to be the midpoints of the maximal intervals in [0, l] - C; we did not do so only out, of considerations of clarity in Figure 21). We call P the Pelczy&ki compacturn. 8.9 Theorem. Let X be a zero-dimensional, infinite compacturn such that Z(X) is dense in X. Then 2x M P, where P is the Pelczyfiski compactum in Figure 21; in other words, P is a geometric model for 2x.
THE MODEL FOR 2’
71
l
.
. ......
. . . ....
. .... ....
.
.
.
.
. ....
. .... ....
Pelczyriski
compacturn
. ....
. .. :.
. . . ....
P
Figure 21
By 8.8, any compactum, Y, such that Z(Y) is dense in 1’ and is a Cantor set is homeomorphic to P. Therefore, by 8.7, 2” z P. W
Proof.
N(Y)
8.10
Corollary.
If X is a countably infinite compactum, then 2.’ M P.
Proof. By the exercise in 8.11, X is zero-dimensional. Hence, by 8.9, it suffices to show that Z(X) in dense in X. Let Y = X - cl[Z(X)]. Suppose that Y # 0. Since Y is open in X, Y is topologically complete [18, p. 4081; thus, we may apply the Baire theorem [18, p. 4141to seethat, since Y is nonempty and countable,
Z(Y) # 0. However, since Y is open in X, clearly Z(Y) definition of Y, IL(Y) = 0.
c Z(X)
and hence, by the
72
II. EXAMPLES:
GEOMETRIC
MODELS
FOR Hyp~~sp~c~s
Therefore, having established a contradiction, we conclude that y = 0. This proves that Z(X) is dense in X. n An open question related to 8.9 is in 83.6.
Exercises 8.11
Any nonempty, countable, metrizable space is zero-di-
Exercise.
mensional. 8.12 Exercise. then ((6,.
Let (X, T) be a topological space. If &Jis a base for T, . ’ ,Bn)n2 *’ : Bi E B for each i and n < oo}
is a base for the Vietoris topology for 2x. (We used this result in the proof of 8.6.) However, the analogous result for CL(X) is false: Give an example of a metrizable space, (X,T), and a base, f?, for T such that
is not a base for the Vietoris topology for CL(X). Construct a geometric model for 2dYwhen X = CUD, 8.13 Exercise. where C is the Cantor Middle-third set and D = (1 + k : 11= 1,2,. . .}. 8.14 Exercise. Let X be a compacturn. Then, 2x is embeddable in R’ if and only if X is zero-dimensional. 8.15
Exercise.
The converses of 8.7 and 8.9 are true (assuming that
X is a compacturn).
References Ralph Bennett and W. R. R. Transue, On embedding cones over circularly chainable continua, Proc. Amer. Math. SOC. 21, (1969), 275-276. R. H. Bing, Higher-dimensional hereditarily indecomposablecontinua, Trans. Amer. Math. Sot. 71, (1951), 267-273. R. H. Bing, The elusive jixed point property, Amer. Math. Monthly 76, (1969), 119-132. R. Duda, On the hyperspace of subcon~in~aof a finite graph, I, Fund. Math. 62, (1968), 265-286.
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R. Duda, Correction to the paper “On the hyperspace of subcontinua of a finite graph, I”, Fund. Math. 69, (1970), 207-211.
6.
R. Duda, On the hyperspace of subcontinua Math. 63, (1968), 225-255.
7.
James Dugundji, (third printing).
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Carl Eberhart, The semigroup of subcontinua of a solenoid, Proc. of the Univ. of Houston Point Set Topology Conf. (University of Houston, 1971), D.R. Traylor, Ed., 1971, 9-10.
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Witold Hurewicz and Henry Wallman, Dimension University Press, Princeton, New Jersey, 1948.
of a finite graph, II, Fund.
Topology, Allyn and Bacon, Inc., Boston, Mass., 1967
10. Alejandro Illanes, (1988), 57-65.
Cells and cubes in hyperspaces,
Theory, Princeton Fund.
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11. Alejandro Illanes and Isabel Puga, Determining finite graphs by their large Whitney levels, Top. and its Appls. 60, (1994), 173-184. 12. Thomas J. Jech, The Axiom of Choice, North-Holland sterdam, Holland, 1973. 13. H. Kato, Whitney (1987), 367-381.
continua of curves, Trans.
Amer.
Pub. Co., AmMath.
Sot. 300,
14. H. Kato, Whitney continua of graphs admit all homotopy types of compact connected ANRs, Fund. Math. 129, (1988), 161-166. 15. H. Kato, A note on fundamental dimensions of Whitney graphs, J. Math. Sot. Japan 41, (1989), 243-250.
continua
16. J. L. Kelley, Hyperspaces 52, (1942), 22-36.
Math.
17. Ronald J. Knill, (1967)) 35-46. 18. K. Kuratowski
of a continuum,
Cones, products
Trans.
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and fixed points, Fund.
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of Sot.
60,
Topology, Vol. I, Acad. Press, New York, N.Y., 1966.
19. K. Kuratowski, S. B. Nadler, Jr., and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Pol. Acad. Sci. 18, (1970), 5-11.
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MODELS
FOR HYPERSPACES
20. Michael Levin and Yaki Sternfeld, The space of s&continua of a 2dimensional continuum is infinite dimensional, Proc. Amer. Math. sot, 125, (1997), 2771-2775. 21. Stefan Mazurkiewicz, Sur les continus absolument indecomposabtes, Fund. Math. 16, (1930), 151-159. 22. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. sot. 71, (1951), 152-182. 23. Sam B. Nadler, Jr., Continua whose cone and hyperspace are homeomorphic, Trans. Amer. Math. Sot. 230, (1977), 321-345. 24. Sam B. Nadler, Jr. Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 25. Sam B. Nadler, Jr. Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 26. Sam B. Nadler, Jr., Locating cones and Hilbert cubes in hyperspaces, Fund. Math. 79, (1973), 233-250. 27. Sam B. Nadler, Jr. and J. T. Rogers, Jr., A note on hyperspaces and the fixed point property, Colloq. Math. 25, (1972), 255-257. 28. A. Pelczynski, A remark on spaces2x for zero-dimensional X, Bull. Pol. Acad. Sci. 13, (1965), 85-89. 29. V. I. Ponomariov, Novoye prostranstvo zamknutykh mnozhestv i mnogoznachnye otobrazhenya bikompaktov, (in Russian) [A new space of closed subsets and multi-valued mappings of bicompact spaces] Mat. Sb. Novaya Seriya 48, (19591, 191-212. 30. James T. Rogers, Jr., Continua with cones homeomorphic to hyperspaces,Gen. Top. and its Appls. 3, (1973), 283-289. 31. James T. Rogers, Jr., Embedding the hyperspaces of circle-like plane continua, Proc. Amer. Math. SOC. 29, (1971), 165-168. 32. James T. Rogers, Jr., The cone = hyperspace property, Can. J. Math. 24, (1972)) 2799285. 33. T. Benny Rushing Topological Embeddings, Acad. Press, New York, N.Y., 1973. 34. Stephen Willard General Topology, Addison-Wesley Pub. Co., Reading, Mass., 1970.
III. 2x and C(X) for Peano Continua X A space is said to be Eocally connected provided that each of its points has a neighborhood base of connected, open sets. A locally connected continuum is often called a Peano continuum in honor of Giuseppe Peano: in 1890, Peano gave the first example of a space-filling curve, namely, a continuous function of the closed interval [O,l] onto the square I2 [28]; later, Hahn [18] and Mazurkiewicz [23] showed that every locally connected continuum is a continuous image of \O,l] (and conversely). Simple examples of Peano continua include any finite graph, any n-cell, the Hilbert cube, and the hairy point. On the other hand, for example, the familiar continuum that is the closure of the graph of y = sin( l/z), 0 < x 5 1, is not a Peano continuum. The topic of this chapter is of fundamental importance; it is centered around a problem that took over fifty years to resolve. Therefore, it seems only fitting to begin with a historical discussion. In Poland in the early 1920’s, it was conjectured that 2’ is the Hilbert cube, where I = [0, 11. The conjecture first appeared in print in 1938 [36]. There were compelling reasonsto believe the conjecture; nevertheless, for many years a proof eluded everyone who worked on it. Finally, in the 1970’s, Schori and West proved that 2’ is, indeed, the Hilbert cube ([29] and [30]). They went on to extend their result to 2x when X is any finite graph [31]. Returning to 1938, Wojdyslawski [36] asked the following question: Is 2x the Hilbert cube whenever X is a (nondegenerate) Peano continuum? We remark that the question was restricted to Peano continua because they are the only continua X for which 2x is a Peano continuum ([33] and [34]). In 1939, Wojdyslawski provided strong evidence in support of an affirmative answer to his question; he proved that for any Peano continuum X, 2x and C(X) are absolute retracts [37]. And there the matter rested 75
76
III.
2x
AND
C(X)
FOR
PEANO CONTINUA
X
- for over thirty years - until Schori and West proved their result about 2’ (mentioned above). Then, in I974 and 1978, Curtis and Schori published their landmark papers ([9] and [lo]). They answered Wojdyslawski’s question (a&matively), and they obtained two definitive results about C(X). Their results are as follows (where X is a nondegenerate Peano continuum): 2x is the Hilbert cube; C(X) is the Hilbert cube in the only situation when it could be, namely, when every arc in X has empty interior in X; and C(X) is a Hilbert cube factor (i.e., C(X) x Q z Q, where Q denotes the Hilbert cube). These results led to similar results for containment hyperspaces ([9], [lo, p. 241). We note that the results about C(X) were proved for a special case in an earlier paper by West [35]. The Curtis-Schori-West techniques of proof involve the delicate use of inverse limits, subtle and complicated maneuvers with refinements of partitions, and what were, at the time, fairly new results about infinitedimensional topology. But, the Curtis-Schori papers are not the end of our story. Enter H. Torunczyk, who was hard at work in the mid 1970’s trying to characterize Hilbert cube manifolds. I remember the day when a preprint of Torunczyk’s results was received in the mail by a renowned topologist. The first reaction was “This can’t be right - we prove the converse the first day of class,” which was followed five hours later with “By God, it is right!” At the end of the paper, Torunczyk gave an elegant proof of the Curtis-Schori result for 2x as an application of his characterization theorem. This brings us to the purpose of this chapter - proving the first two Curtis-Schori results mentioned above using Torunczyk’s theorem. We present the necessary background for the proof in the first two sections of the chapter.
9.
Preliminaries: Absolute Retracts, Toruriczyk’s Theorem
Z-sets,
We discuss the concepts that are necessary for understanding Torunczyk’s theorem. We state Torunczyk’s theorem at the end of the section. The following notions are due to Borsuk [3]. A retraction is a continuous function, T , from a space, Y, into Y such that T is the identity on its range (i.e., ~(T(Y)) = f-(y) f or each y E Y). A subset, 2, of Y is said to be a retract of Y provided that there is a retraction of Y onto 2. A compacturn, K, is called an absolute retract (written AR) provided that whenever K is embedded in a metric space, Y, the embedded copy of K is a retract of Y.
9. PRELIMINARIES:
ABSOLUTE RETRACTS, Z-SETS,...
77
We will give some examples of absolute retracts using the corollary in 9.2. First, we prove the important characterization theorem in 9.1, which uses the following terminology. A compactum, K, is called an absolute extensor (written AE) provided that whenever B is a closed subset of a metric space, M, and f : B + K is continuous, then f can be extended to a continuous function F : M + K. (Note: F being an extension off means that FIB = f.) 9.1 Borsuk’s if K is an AE.
Theorem
[3]. A compactum,
K, is an AR if and only
Proof. Assume that K is an AR. By Urysohn’s metrization theorem Then, since K is an AR, [21, p. 2411, we can assume that K C I”. there is a retraction, r, of Ice onto K. Now, let B be a closed subset of a metric space, M, and let f : B + K be continuous, where f has coordinate functions fi : B -+ [0, l]i. By Tietze’s extension theorem [21, p. 1271, each fi can be extended to a continuous function gi : M + [0, 11%.Let g = (g& Then we see that T o g : M + K K is an AE. The converse is easy: Assume copy of K in a metric space Y, continuous function r : Y + K’. 9.2 Corollary. then K is an AR.
: M + I’=‘.
is a continuous
extension of
f.
Therefore,
that K is an AE, let K’ be an embedded and extend the identity map for K’ to a n
Let K be a compacturn in I”.
If K is a retract of Im,
Proof. The first part of the proof of 9.1 shows that any retract of 10° is an AE. Therefore, by 9.1, any retract of I” is an AR. W Now, let us use 9.2 to give some simple examples of absolute retracts. By 9.2, 10° itself is an AR (the identity map of 1” being a retraction). By 9.2, any n-cell is an AR (the map (Zi)ci + (xi)&, where X: = zi for all i 5 n and x!, = 0 for all i > n, is a retraction of I” onto a copy of In). Next, let P denote the polyhedron in Figure 6, p. 37. Evidently, P is contained in the 3-cell J = I$=,[-1,2]i. It is easy to envision a retraction of J onto P. Therefore, since J is an AR, P is an AR (it being obvious that a retract of an AR is an AR). A similar argument (using an n-cell) shows that the polyhedron in Figure 9, p. 42 is an AR. On the other hand, simple examples of compacta that are not absolute retracts come from realizing that every AR is a Peano continuum. This
78
III.
2’ AND C(X)
FOR PEANO CONTINUA
X
fact is a consequence of the following observations: Every compactum is embeddable in IO0 [21, p. 2411; I” is a Peano continuum; and every retract Of a Peano continuum is a Peano continuum. We note that an n-sphere for n > 0 is an example of a Peano continuum that is not an AR (since dI”+l is not a retract of In+l [21, p. 3141). For a systematic treatment of the fundamentals of absolute retracts, see
PI. We turn our attention to the idea of a Z-set. In [l], Anderson defined a notion about closed subsets of certain infmitedimensional, linear spaces. He called the notion Property Z (the letter Z was intended to be suggestive of the zigzag motions that Anderson, Klee, and others used in moving points around in infinite-dimensional spaces). Anderson’s definition of Property Z has been modified, resulting in what are now called Z-sets. We define Z-sets and briefly discuss them. The definition we give is from [5, p. 21, except that we restrict our attention to Z-sets in compacta. Let Y be a compacturn with metric d. A closed subset, A, of Y is said to be a Z-set in Y provided that for each E > 0, there is a continuous function, f6, from Y into Y - A such that f, is within E of the identity map on Y (i.e., d(f,(y),y) < E for all y E Y). For example, it is easy to see that 81” is a Z-set in I”. On the other hand, no point of In - dI” is a Z-set in I” (which follows using the Brouwer fixed point theorem [21, p. 3131). Therefore, a closed subset, A, of In is a Z-set in In if and only if A c aIn. We will contrast Z-sets in I” with Z-sets in the Hilbert cube I”. We use the standard metric d, for I” given by
dw((~i)~l,
(yi)&)
= 22-“/xi
- yil for all (xi&,
(yi)&
E 103.
i=l
Unlike In, any point of 103 is a Z-set in I”. Let E > 0. Fix j 2 1 such that 2-j (Pi)~“,l E I”. such that q # pj. Define fc : IO0 + 103 as follows: f<((~i)&)
To see this, let p = < E, and let q E [0, l]
= (~1,. . . , zj-1, q, x3+1,. . .) for each (zi)Ei
E I”.
Then, clearly, p 4 f6(I”) and f 6 is within E of the identity map on I”O (with the metric d,). A similar argument shows that any finite set (and any countable, closed set) in IO0 is a Z-set in I”. This implies that, unlike In, there is a sequence of Z-sets in IO” whose union is dense in I”. We will give another simple example of Z-sets in Tco. The example concerns Z-maps, which we now define.
EXERCISES
79
A continuous function, f, from a compactum, Yi, into a compactum, Yz, is called a Z-map provided that f(Yr) is a Z-set in Yz. Now, for each n = 1,2,. . . , let fn : I” + 10° be the following “projection” : fn((zi)gl)
= (xi,. . . ,znrO,O,.
. .) for all (si)p”i
E 10°.
We see that each fn is a Z-map (by using coordinate replacement, as we did above). Note that the sequence {fn}r=i converges uniformly to the identity map on I” (with respect to the metric d,). Therefore, we have determined an elementary fact about Hilbert cubes: if Q is a Hilbert cube, then the identity map on Q is a uniform limit of Z-maps. The converse, for absolute retracts, is Torunczyk’s theorem! 9.3 Toruriczyk’s Theorem [32]. Let Y be an AR. If the identity on Y is a uniform limit of Z-maps, then Y is the Hilbert cube.
map
We remark that 9.3 is only a special case of Theorem 1 of [32, p. 341; 9.3 follows from Theorem 1 of [32] by using Chapman’s contractibility theorem [S] (which says that any compact, contractible, Hilbert cube manifold is the Hilbert cube). Easy-to-state open questions about Z-sets in hyperspaces are in 83.883.10.
Exercises 9.4 Exercise. is a Z-set in Y.
Let Y be a compacturn.
The finite union of Z-sets in Y
9.5 Exercise. Determine all the Z-sets in Y when Y is the closure in R* of the graph of y = sin(l/z), 0 < z 5 1. 9.6 Exercise. Give an example of a closed subset of IO0 that has empty interior in IO3 and that is not a Z-set in P. 9.7 Exercise. Prove that I” M Cone(F’) by using 9.3. [Hint: To prove that Cone(lm) is an AR, let Y = {(yi)Er E 103 : y1 = 0}, and let G(Y) be the geometric cone over Y with vertex v = (l,O, 0,. . .). Consider the “nearest point” map r : 10° + G(Y) defined as follows: for each x E I”, T(X) is the unique point of G(Y) that is nearest to x.1 9.8 Exercise. Let X and Y be compacta. Let Yx denote the space of all continuous functions from X into Y with the uniform metric p; in other
80
III.
words,
2x AND C(X)
FOR PEANO CONTINUA
X
if d denotes the metric on Y and f, g E Y x, then df,g)
= sup {d(f(s)>g(z))
If A is a Z-set in Y, then {f E Yx : f(X) conversely).
: x E X1. nA = 0) is dense in Yx (and
9.9 Exercise. Let Y” be as in 9.7. If Y is a Hilbert YX : f is a Z-map} is dense in Y x.
10.
Preliminaries:
General Results Continua
cube, then {f E
about
Peano
We will use three classical theorems about Peano continua in the next section. We state these theorems in 10.1, 10.2, and 10.3. Proofs of 10.1 and 10.2 can be found in, for example, [22, pp. 254-2571 or Chapter VIII of [27]; proofs of 10.3 are in [2] and [25]. 10.1 Theorem. Let X be a Peano continuum. Then, for each E > 0, X is the union of finitely many Peano continua of diameter < E. A space, X, is said to be arcwise (or pathwise) connected provided any two points of X can be joined by an arc in X. 10.2 Theorem.
Every Peano continuum
is arcwise
that
connected.
A convex metric for a space X is a metric, d, for X that induces the topology on X and for which midpoints always exist: for any x, y E X, there exists m E X such that d(x,m) 10.3 Theorem.
= id(x, y) = d(m,y).
Every Peano continuum
has a convex metric.
Convex metrics were first studied by Menger [24]. The theorem in 10.3, which is due to Bing [2] and Moise [25], answered a question in [24]. The convexity of the Hausdorff metric has been studied by Duda in [12] and [13] (regarding the problems in [12, p. 331, see 4.5 and 4.6 of [13]). Let us note a few elementary properties of convex metrics. We will use 10.5 and 10.6 in the next section. It is easy to see that a compacturn with a convex metric must be a continuum. In fact, a compacturn with a convex metric must be a Peano continuum (Exercise 10.13, which will be easy by the end of the section). The following proposition shows that if a continuum has a convex metric, then the points of the continuum can be joined by metrically-straight line segments, that is, by arcs that are isometric to intervals in R1. (Recall that an isometry is a distance preserving map.)
10. PRELIMINARIES:
GENERAL
RESULTS
ABOUT
PEANO
CONTINUA
81
10.4 Proposition. Let X be a continuum with a convex metric d. Then any two points, x and y, of X can be joined by an arc, J, in X such that J is isometric to the closed interval [0, d(x, y)]. Proof. Let m(1/2) be a midpoint for x and y. Then let m(1/4) be a midpoint for x and m(1/2), and let m(3/4) be a midpoint for m(1/2) and y. Then let m(/~/8) be a midpoint for m([k - 1]/8) and m([lc+ l]/S), where Ic = 1,3,5,7 and where m(0) = z and m(l) = y. A formal induction, in accordance with the indicated pattern, gives us the following subset, M, of x: A4 = {m(/~/2~) : n = 1,2,. . . and k = 1,. . ,2n - l}, . a midpoint for m([kwhere each m( k/2n) 1s J = R. Then, letting f(z)
= d(x,z)
1]/2n) and m([k+
1]/2n). Let,
for each z E J,
it follows that f is an isometry of J onto [O,d(x, y)]. We note that J must, therefore, be an arc since isometries are homeomorphisms. Also, x,3 E J by the way we constructed J. n The next two properties of convex metrics involve generalized closed balls. Note the following definition (which is what is expected in view of how we defined generalized open balls in Chapter I). Let (X,d) be a metric space, let T > 0, and let A E CL(X). The generalized closed d-ball in X about A of radius T, which is denoted by C~(T, A), is defined as follows: &(T,A)
= {x E X : d(x,A)
5
T}.
We remark that Cd(f, A) may not be the sameas the closure of N~(T, A) even when the space (X, d) is an arc; seeExercises 10.10 and 10.11. The following proposition may not be true when the metric d for the continuum X is not convex (Exercise 10.12).
T
10.5 Proposition. Let X be a continuum with a convex metric d. Fix > 0. Then, for any A, B E 2x, Hd[Cd(T,
A),
cd(T,
WI
5
Hd(A,
B).
Proof. Let A, B E 2”. We use the formula for Hd in the exercise in 2.7; thus, it suffices (by symmetry) to show that (#)
d(x, Cd(T, B))
5 Hd(A,
B) for all x E cd(T, A).
82
III.
2’ AND C(X)
FOR PEANO CONTINUA
X
To prove (#), let 2 t C~(T, A). Since it is obvious that (#) holds if z E Crl(r, B), we assume for the proof that 2 $ Cd(r, B). Since z E Cd(r, A) and A E 2x, there exists a E A such that d(z, a) 5 T. Since B E 2”, there exists b E B such that d(n,b) = d(a, B). Note that d(z, b) > T (by our assumption that z # Cd(r, B)). Now, by 10.4, there is an arc, J, in X from 5 to b such that J is isometric to [0, d(z, b)]. S’mce d(z, b) > r, there is a point y E J such that d(y, b) = r. Thus, since 2 and b are the end points of J and J is isometric to [0, d(x, b)], the usual triangle inequality for x,y, and b (with y as the repeated point) is an equality: 4x7 Y) + d(~, b) = 4x, b). Therefore,
since d(y, 6) = T, d(z, y) = d(z, b) - T < d(z, a) + d(a, b) - r;
thus, since d(z, n) 5 r and d(n, b) = d(a, B), we have that
d(z, 9) I d(a, B). Hence, by 2.7, d(z, y) 5 Hd(A, B). Therefore, proved (#). n
since y E C~(T, B), we have
Our final property of convex metrics concerns the connectednessof generalized closed balls. 10.6 Proposition. Let X be a continuum with a convex metric d. Then, for any A E C(X) and T > 0, C,(r,A) E C(X). Proof. The fact that Cd(r, A) is connected when A E C(X) is a simple consequenceof 10.4. n
We will use the following simple result about unions of Peano continua several times in the next section. 10.7 Proposition. If Zi and 2s are Peano continua (in a given space) and if 21 f? 22 # 0, then 21 U 22 is a Peano continuum. Proof. Let Z = Zi U 22. Obviously, 2 is a continuum that is locally connected at each point of 2 - (2, n 22). Note that if z E 21 n 22 and I”, is a connected neighborhood of z in Zi, then VI u Vz is a connected neighborhood of z in 2 (however, VI u Vz may not be open in 2 even if Vi is open in Zi for each i). Thus, since Zi and Zz are Peano continua, it follows t,hat
EXERCISES
83
(*) each point of 2 has a neighborhood base of connected sets. Now, fix p E Z and let N be any neighborhood of p in Z. Let No denote the interior of N in Z. Let G denote the component of p in N” (i.e., G is the union of all the connected subsets of No containing p). Note that G is connected. We show that G is open in Z. Let z E G. By (*), z has a connected neighborhood, E, in N”. Note that G U E is a connected subset of N“ containing p; thus, since G is the union of all such sets, clearly E c G. Therefore, since E is a neighborhood of z in Z and since z was an arbitrary point of G, we have proved that G is open in Z. Hence, G is a connected, open subset of Z and p E G c N. Therefore, having shown that there is such a G for any p E Z and for any neighborhood N of p, we have proved that Z is locally connected. n The proof of 10.7 leads us to make some comments that will explain why the proof was, perhaps, harder than expected. There are two natural ways to localize connectedness: Let X be a topological space, and let p E X; X is locally connected at p provided that p has a neighborhood base of connected, open neighborhoods; X is connected im kleinen (cik) at p provided that p has a neighborhood base of connected neighborhoods (that is, connected sets that contain p in their interiors in X). It is obvious that if X is locally connected at p, then X is cik at p. However, the converse is false even for continua, as the continuum in Figure 22 (top of the next page) illustrates. Nevertheless, if a topological space is cik at every point, then it is locally connected at every point. This is what most of the proof of 10.7 was devoted to showing. Finally, we recall Wojdyslawski’s theorem for use in the next section. We mentioned his theorem in the historical discussion at the beginning of the chapter. 10.8 Wojdyslawski’s Theorem [37]. If X is a Peano continuum, then 2x and C(X) are absolute retracts. For a short proof of Wojdyslawski’s theorem based on a characterization of Lefschetz, see 4.4 of Kelley [20, p. 281.
Exercises 10.9 Exercise. Give an example of a convex metric for a simple closed curve. Do the same for a noose and a simple n-od.
84
III. 2’ AND C(X)
FOR PEANO CONTINUA
X
X cik at p, not lot. corm. at p Figure 22
10.10 Exercise. Give an example of a metric, d, for an arc, X, such that d induces the topology on X and for some p E X and some T > 0, cwd(r,
{PI)1 # Cd(T, {PI).
10.11 Exercise. Let X be a continuum with a convex metric d. Then, for any -4 E 2x and any T > 0, cl[Nd(r, A)] = Cd(r, A). (The equality here, without requiring d to be convex, is equivalent to the continuous variance of balls [26]). 10.12 Exercise. Give examples of metrics, dl and dz, for an arc, X, such that dl and d2 induce the topology on X and have the following properties: the inequality in 10.5 fails for dl, some T > 0, and some A, B E 2x; the inequality in 10.5 is true for d2 even though d2 is not a convex metric. 10.13 Exercise. If a compactum, X, has a convex metric, then X is a Peano continuum. (This is a converse of 10.3.) 10.14 Exercise. If X is a Peano continuum, then F,(X) is a Peano continuum for each n. If Fl(X) is a Peano continuum, then X is a Peano continuum.
11. THE CURTIS-SCHORI
11.
The Curtis-Schori
THEOREM FOR 2x AND C(X)
Theorem
85
for 2x and C(X)
We prove the Curtis-Schori Theorem in 11.3. First, we prove an important result about when containment hyperspaces are Z-sets.
When
2; and CK(X)
Are Z-sets
We prove the theorem in 11.2. We will use 11.2 in the proof of the Curtis-Schori Theorem in 11.3. We facilitate part of the proof of 11.2 with the following technical lemma (whose proof uses a function in [ll]). 11.1 Lemma. Let J be an arc with end points p and q. There is a continuous function cp : 2J + 2’ such that cp has the following properties: If A E 2J and S c {p, q}, then p(A) # J, cp(S) = S, and (p(AUS) = (p(A)US. Proof. For convenience, [-l,l]. If A E 2’, let
we first
prove the lemma for the arc Y =
a+ = inf A C-I[0, l] a- =supAn[-l,O]
if A f~ [0, l] # 0, ifAn[-1,0]#0,
a0 = inf (1~1 : a E A}. If A E 2y such that 0 4 A, let AU {2a+ - l}, y(A)
=
Au {2a1 A u {2a+
if A
+ l}, - 1,2a-
+ l},
Note that y is continuous on 2y - 2:. follows:
c
(0, l]
if A c [-1,0) if An [-1,O) # 0 # An (O,l]. Finally, define ‘p : 2’ -+ 2’ as
ifAn[-$,$I=0 -Y(A), ifOEA v(A) = [A - C-1, l)] u t-1, 11, { [y(A)-(2a0 - 1,1 - 2a”)] U {2a0 - 1,1 - 2a0}, if 0 < a0 5 5. We leave it for the reader to check that cphas the properties in the lemma (for the case when J = Y). The general result for any arc J now follows readily: Let h be a homeomorphism from J onto Y, let h’ : 2’ + 2’ be as in the proof of 1.3, and let ‘p : 2’ -+ 2y be as just constructed; then, (h*)-’ o cpo h* is the required map for 2J. n Let A be an arc in a space X. We say that A is a free arc in X provided that A without its end points is open in X.
86
III.
2’ AND C(X)
FOR PEANO CONTINUA
If X is a space and B C X, then int(B)
X
and B” denote the interior
of
B in X. 11.2 Theorem [ll, p. 1621. Let X be a nondegenerate Peano continuum. Let K be a closed subset of X such that K” # 0. Then, 2% is a Z-set in 2x; also, if K contains no free arc in X, CK(X) is a Z-set in C(X). Proof. In view of the definition of a Z-set, we note to begin with that 2; is closed in 2x by 1.19; also, therefore, CK(X) is closed in C(X). NOW, let d denote a metric for X. In determining how close a map is to the identity map on 2x or C(X), we will use the Hausdorff metric Hd as it is specifically defined in 2.1 (recall 3.1). We first prove the theorem for 2; assuming that K contains a free arc in X. Let E > 0. Then, since K contains a free arc in X, K contains a free arc, J, in X such that diameterd(J) < E. Let p and q denote the end points of J, and let cp : 2J -+ 2’ be as guaranteed by 11.1. Define fE : 2x + 2x as follows: ifBnJ=0
f’(B)={fk-Jo)u~(BnJ),
ifBnJf0.
The continuity of fc follows from J being a free arc in X and from the properties of cp in 11.1. We see that fc maps 2x into 2*Y - 2: since J C K and since, by 11.1, p(A) 3 J for any A E 2’. Finally, f( is within E of the identity map on 2x (with Hd) since diameterd(J) < E. Therefore, we have proved that 25 is a Z-set in 2 x in the case when K contains a free arc in x. Next, we prove the theorem for 2; assuming that K does not contain a free arc in X. In other words, we show that for each E > 0, there is a continuous function, gL, from 2x into 2x - 2: such that g( is within E of the identity map on 2x (with Hd). Let t > 0. Recall from the statement of the theorem that K’ # 0, and let p E K“. By 10.1, X = uFz2=,Xi, where n < oo, each Xi is a Peano continuum, and diameterd(Xi) < c/4 for each i. We define what is commonly called the star of p with respect to Xi,. as follows: S(p)
= U(Xi
: p E Xi).
Without loss of generality (recall that p E K’), c K and St(p) # X. Let enough so that St(p) C = {Xj
. . , X,
we assume that E is small
: p 4 Xj and X, n St(p) # S}.
WHEN
2;
AND C&(X)
ARE Z-SETS
87
is connected, it follows easily that C # 0. Since St(p) # X and X = U~==,xi For each ,Xj E C, let pj E Xj fl St(p); note that the points pj really do exist (since C # 0). Also, note by using 10.7 that St(p) is a Peano continuum. Hence, by 10.2, there is an arc, A,, in St(p) from p to each of the points pj chosen above. Let A denote the union of these arcs Aj, and let Y = Au (UC). It follows by using 10.7 again that Y is a Peano continuum. Hence, by 10.8, C(Y) is an AR. Define cy: Y -+ C(Y) as follows: a(y) = {y} for all y E Y. Then, since C(Y) is an AR, we see from 9.1 that Q can be extended to a continuous function p : St(p) U Y -+ C(Y’). Now, extend p to a function y : X -+ C(X) by the following formula:
1
p(z),
Y(Z)
=
(2>,
if 5 E St(p) U 1 if 2 E X - [St(p)
U I’].
We will prove that y is continuous; first, we show how y is used to define the final function, gc, on 2”. For each B E 2”, let g,(B) =
u{y(b)
:
b E B}.
We prove that gr has the following three properties: (4 Se maps 2x into 2x and gr is continuous; (b) g,(B) 2 K for any B E 2”; (c) ge is within Eof the identity map on 2,’ (with Hd) Proof of (a): We begin the proof of (a) by proving that y is continuous. Recall that p is continuous and that /3(y) = {y} for all y E Y; hence, the continuity of y follows easily once we prove that (#)
cl[X - &St@)]n St(p) c Y.
To prove (#), let z E cl[X - St(p)] II St(p). Let {zk}& be a sequence in and n < 03, X-St(p) such that {zk}& converges to z. Since X = Uy==,X, there exists m such that zk E X,, for infinitely many k. This implies that X, has the following three properties: (i) z E X,; (ii) p $ X, (since zk 4 St(p) for any k); (iii) X, II St(p) # 0 (by (i) since z E St(p)). By (ii) and (iii), X, E C. Therefore, by (if, z E Y. This proves (#), which implies the continuity of y.
88
III.
2x AND C(X) FOR PEANO
CONTINUA
X
Now, we use the continuity of y to complete the proof of (a). Let Since y is a continuous function from X into C(X), y(B) is a (nonempty) compact subset of C(X); hence, y(B) E 22x. Thus, since g,(B) = U-Y(B), we seeby (1) of the exercise in 11.5 that g,(B) E 2x. This proves that gEmaps 2x into 2x. The fact that ge is continuous follows from the continuity of y and from (2) of 11.5 as follows. Let 21be the union map in 11.5. Let y* : 2x + 22x be defined by
B E 2”.
y*(B) = y(B) for all B E 2x. Then we see that g( = u o r*; also, y* is continuous (by the continuity of y and by (1) in the proof of 1.3), and 21is continuous (by (2) of 11.5). Therefore, gc is continuous. This proves (a). Proof of (b): The reason that (b) is true is that St(p) $ZI’. Here are the details. We first prove that
(*I SW c y. For use in proving (*), let U = X -
u{Xi
: p $ Xi}.
Note that
u c St(p) - UC. Hence, to prove (*), it suffices to show that U-A # 0 (since Y = AU(UC)). Recall that A was defined as a finite union of arcs in St(p) and that St(p) C I(; also, recall our assumption that K does not contain a free arc in X. Therefore, A is a finite union of arcs each of which has empty interior in X. Hence, A” = 8 (as follows directly or by using the Baire theorem [21, p. 4141). Thus, since U is clearly nonempty and open in Xj we see that U 4 A; i.e., U - A # 8. Therefore, we have proved (*). Now, wecompletetheproofof(b). By(*), thereisapointq E St(p)-Y. Recall the formula for y and the fact that /3 maps into C(Y). Then we see that q $! y(z) for any 5 E X. Hence, by the formula for g6, q $! g,(B) for any B E 2x. Therefore, since q E St(p) c K, clearly g,(B) $ K for any B E 2x. This proves (b). Proof of (c): Observe that diameterd[St(p) U Y] < e. Hence, by the formula for y and the fact that 0 maps into C(Y), we seethat diameterd[{z} U r(z)]
< e for all z E X.
Thus, for any B E 2x, it follows immediately that B c Nd(c, g( (B)) and c N~(E, B). Therefore, Hd(gf(B), B) < E for all B E 2x (by Exercise 2.9). This proves (c).
SC(B)
THE CURTIS-SCHORI By (a)-(c),
THEOREM
89
2: is a Z-set in 2x.
The proof of the theorem for CK(X) is now easy. Consider the map gc]C(X), where gc is as defined above. We prove that g<]C(X) maps C(X) into C(X). Let B E C(X). Then, since y : X + C(X) is continuous, y(B) is a subcontinuum of C(X); i.e., y(B) E C[C(X)]. Thus, since g,(B) = UT(B), we see by (3) of the exercise in 11.5 that Se(B) E C(X). Hence, we have proved that gc]C(X) maps C(X) into C(X). Therefore, in view of (a)-(c) above, it follows that CK(X) is a Z-set in C(X). m
The Curtis-Schori
Theorem
There are three parts to the Curtis-Schori Theorem; the first two parts are of primary importance. We prove the first two parts and then discuss proofs for the third part. Regarding the terminology in the third part of the Curtis-Schori Theorem, a Hilbert cube factor is a space, Y, such that Y x P M I” (cf. Exercise 11.12). 11.3 Curtis-Schori Theorem ([9] and [lo]). Let X be a nondegenerate Peano continuum. Then (1) 2x is the Hilbert cube, (2) C(X) is the Hilbert cube when there is no free arc in X, and (3) C(X) is a Hilbert cube factor. Proof of (1) and (2) [32, p. 391. The proof is based on Torunczyk’s Theorem in 9.3. Regarding the first assumption in 9.3, we note right away that 2x and C(X) are absolute retracts by 10.8. We will verify the second assumption in 9.3 for 2x and then for C(X). For this purpose, we assume by 10.3 that d is a convex metric for X. Let E > 0. According to 9.3, we must show that there is a Z-map from 2x into 2x that is within E of the identity map on 2x. Define af : 2x + 2x as follows (recall that Cd is used in denoting closed d-balls): +<(A) = Cd(e,A)
for all A E 2x.
By 10.5, +c is continuous. Obviously, $, is at most E from the identity map on 2x (with Hd). Finally, we show that +c is a Z-map. Since X is compact, there are finitely many points, pl, . . _,p,, of X such that
Let Ki = Cd(c/2, {pi}) for each i = 1,. . . , n. By the first part of 11.2, 2 5, is a Z-set in 2x for each i. Hence, by Exercise 9.4, Uy!=,2$, is a Z-set in 2x.
90
III. 2’ AND C(X) FOR PEANO CONTINUA X
It is easy to seethat for each A E 2 x, there exists j such that Q,(A) E 25, ; in other words,
‘p, (2”) c uy=, 2;, . Thus, since it is evident that a closed subset of a Z-set is a Z-set, @t(2-Y) is a Z-set in 2”. Hence, we have proved that Qe is a Z-map. Therefore, having verified the assumptions in 9.3 for 2x, we have by 9.3 that 2aYis the Hilbert cube. This proves part (1) of the theorem. Next, we prove part (2) of the theorem. Assume that there is no free arc in X. Then the proof that C(X) satisfies the second assumption in 9.3 is a simple adaptation of what we just did for 2x. Namely, let ar be as above, and let pr = aelC(X). By 10.6, cpcmaps C(X) into C(X). l+om the properties of G<, we see that (pc is continuous and that (pc is at most E from the identity map on C(X). To see that cpcis a Z-map, let Ki be as aboveforeachi=l,... ,a. Then, by the second part of 11.2, CK,(X) is a Z-set in C(X) for each i. Hence, by adjusting the relevant part of what we did above to the present situation, it follows that (pt is a Z-map. Therefore, by 9.3, C(X) is the Hilbert cube. n It, is simple enough to prove part (3) of 11.3 using Wojdyslawski’s Theorem in 10.8 and Toruilczyk’s Theorem in 9.3 (Exercise 11.11). However, such a proof is really not appropriate for the following reason: When Toruticzyk proved his theorem, he used a theorem of R. D. Edwards, and a special case of Edwards’s theorem directly implies (3) of 11.3 (using 10.8). We state the special caseof Edwards’s theorem: 11.4
Edwards’s
Theorem.
Every AR is a Hilbert cube factor.
Edwards proved his theorem in the early 1970’s, but apparently he did not publish it (see [17]). A proof of the theorem is in 44.1 of [5] (combine 44.1 and 22.1 of [5] to obtain our 11.4). Thus, at the present time, there are two appropriate proofs for part (3) of 11.3 - the original Curtis-Schori proof ([9], [lo]), and the proof using 11.4 and 10.8. An open question related to 11.3 is in 83.7.
Further
Uses of Toruriczyk’s
Theorem
Toruriczyk’s Theorem in 9.3 is useful in studying hyperspaces of nonPeano continua. One illustration is in Exercise 11.9. Toruriczyk’s Theorem can be used to prove results about containment hyperspaces. Two such results are in Exercises 11.6 and 11.7 (the results are due t,o Curtis and Schori, [9] and [lo], with a different proof).
EXERCISES
91
Torunczyk’s Theorem can be used to prove results about intersection hyperspaces (see part (3) of Exercise 15.15). A result about spaces of segments whose proof uses Torunczyk’s Theorem is in 17.9. Papers about hyperspaces that use Torunczyk’s Theorem and other results from infinite-dimensional topology, aside from papers already mentioned, include 171, 181, [14]-[lSj, and 1191.
Exercises 11.5 Exercise. Let X be a compactum. function ‘u. defined by
The union map for 22x is the
u(d) = ud for each A E 22x. Verify the properties of u in (l)-(3) below (which we used in the proof of 11.2). We note that by 3.1, 22x .1s metrized by the Hausdorff metric HH induced by the Hausdorff metric H for 2x. (See Remark below). (1) u maps 22x into (in fact, onto) 2x. (2) u is continuous; moreover, H(Ud, Ua) 5 HH(d, B) for all A, B E 22x. (3) u maps C[C(X)] into (in fact, onto) C(X). Remark. Part (1) of 11.5 says that the union of a compact collection of compact sets is compact; thus, part (1) generalizes the familiar case of finite unions. 11.6 Exercise. Let X be a Peano continuum, and let K be a subcompactum of X such that K # X. Then, 2: is the Hilbert cube. [Hint: To show that 2-z is an AR, find a natural retraction of 2x onto 2; and use 10.8. To verify the second assumption in 9.3 for 2:, find a natural map from 22 onto 25d!d(C,Kjfor (small) 6 > 0, and examine the proof of 11.2 carefully.] 11.7 Exercise. Let X be a Peano continuum, and let K be a subcompactum of X such that K # X. If X - K contains no free arc in X, then CK (X) is the Hilbert cube. [Hint: The hint is similar to the hint for 11.6 except that a retraction of C(X) onto CK(X) is not as easy to find as a retraction of 2x onto 2;. The following function is helpful: Let d be a convex metric for X (10.3), and let [(A) = inf(r : Cd(r, A) > K} for each A E C(X).]
III. Remark.
2x AND C(X)
Regarding
FOR PEANO CONTINUA
11.7 (including
X
the hint), see Exercise
14.22.
11.8 Exercise. The following general result has many applications (e.g., we will use it in the next exercise). Let X be a continuum, and let P be a Peano subcontinuum of 2x or C(X). Assume that ud E P for every subcontinuum A of P. Then, P is an AR. [Hint: Use 11.5.1 11.9 Exercise. The harmonic fan is the continuum in Figure 23; it is the cone over the compacturn Y, where Y = {ei : ee = 0 and ei = l/i for i = 1,2,...}. Construct a geometric model for C(X) when X is the harmonic fan. We suggest a procedure for doing this in (l)-(5) below. The following theorem is used in (5) to give clarity to the model (for a proof of the theorem, see [l, p. 3811 or [5, p. 141):
Harmonic
fan (11.9)
Figure 23
EXERCISES 11.9.1. Anderson’s Homogeneity phism between two Z-sets in a Hilbert homeomorphism of Q onto Q.
93 Theorem [l]. Any homeomorcube, Q, can be extended to a
For what follows, recall that X = Cone(Y), where Y is as above; let v denote the vertex of X, and let eiv denote the arc in X from ci to v for each i = 0, 1,2, . . .. (1) Prove that C,,(X) is a continuum; furthermore, prove that C,,(X) a Peano continuum. (2) Prove that C,,(X) is the Hilbert cube (use (l), 11.8, and 9.3). (3) Determine a geometric model for F = U&C(eiv).
is
(4) Prove that F n C,,(X) is a Z-set in C,,(X). (5) Use (2)-(4) and 11.9.1 to obtain a geometric model for C(X). Remark. The harmonic fan is a natural generalization of a simple nod. In 5.4 we constructed a geometric model for C(X) when X is a simple n-od. It is interesting to compare the comments in 5.5 with what we did or, rather, didn’t have to do - in obtaining a geometric model for the harmonic fan. In particular, due to Anderson’s Theorem in 11.9.1, less care is required in constructing a geometric model for C(X) when X is the harmonic fan than when X is a simple n-od. One should also compare what we did with respect to the harmonic fan with what we did with respect to the hairy point in 6.1: We did not need Torunczyk’s Theorem or Anderson’s Theorem in 6.1 because we had a formula for the homeomorphism cp of Cv(X) onto 10°. Nevertheless, it is beneficial to rework 6.1 disregarding ‘p and using the procedure outlined for the harmonic fan. We also suggest working the exercises at the end of section 6 using the methods of this chapter. 11.10 Exercise. Let X be the harmonic fan with vertex v (11.9 and Figure 23, p. 92). Is 2,x the Hilbert cube? (We know from (2) of 11.9 that CV(X) is the Hilbert cube.) 11.11 Exercise. Show how part (3) of the Curtis-Schori Theorem in 11.3 follows easily from 10.8 and Torunczyk’s Theorem in 9.3. (Although such a proof is instructive, it is not appropriate for the reason mentioned preceding 11.4.) 11.12 Exercise. Prove the result stated below in an elementary way (without using Edwards’s Theorem in 11.4). See the Remark following the result.
94
III.
2x AND C(X) FOR PEANO
CONTINUA
X
A space, Y, is a Hilbert cube factor if (and only if) there is a space, 2, such that Y x 2 is the Hilbert cube. Remark. The result in 11.12 puts our definition of a Hilbert cube factor in sync with the usual notion of a factor of a Cartesian product. In contrast, [O,l] is a Cartesian factor of 1” but, using the definition of an n-cell factor that is directly analogous to our definition of a Hilbert cube factor, [O,l] is not an n-cell factor since [0, l] x In # Tn.
References 1.
R. D. Anderson, On topological infinite deficiency, Mich. Math. J. 14 (1967), 365-383. 2. R. H. Bing, Partitioning a set, Bull. Amer. Math. Sot. 55 (1949), 1101-1110. 3. K. Borsuk, Sur les re’tractes, Fund. Math. 17 (1931), 152-170. 4. K. Borsuk, Theory of Retracts, Monografie Matematyczne, Vol. 44, Polish Scientific Publishers, Warszawa, Poland, 1967. 5. T. A. Chapman, Lectures on Hilbert Cube Manifolds, Conf. Board of the Math. Sci. Regional Conf. Series in Math., Amer. Math. Sot., Vol. 28, Providence, R.I., 1976. 6. T.A. Chapman, On the structure of Hilbert cube manifolds, Compositio Math. 24 (1972), 329-353. 7. D. W. Curtis, Growth hyperspaces of Peano continua, Trans. Amer. Math. Sot. 238 (1978), 271-283. 8. D. W. Curtis, J. Quinn, and R. M. Schori, The hyperspace of compact conuex subsets of a polyhedral 2-cell, Houston J. of Math. 3 (1977), 7-15. 9. D. W. Curtis and R.M. Schori, 2X and C(X) are homeomorphic to the Hilbert cube, Bull. Amer. Math. Sot. 80 (1974), 927-931. 10. D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes,Fund. Math. 101 (1978), 19-38. 11. D. W. Curtis and R. M. Schori, Hyperspaceswhich characterize simple homotopy type, Gen. Top. and its Appls. 6 (1976), 153-165. 12. R. Duda, On convex metric spaces111,Fund. Math. 51 (1962), 23-33. 13. R. Duda, On convex metric spacesV, Fund. Math. 68 (1970), 87-106. 14. Carl Eberhart, Intervals of continua which are Hilbert cubes, Proc. Amer. Math. Sot. 68 (1978), 220-224. 15. Carl Eberhart and Sam B. Nadler, Jr., Hyperspaces of cones and fans, Proc. Amer. Math. Sot. 77 (1979), 279-288.
REFERENCES
95
16. Carl Eberhart, Sam B. Nadler, Jr., and William 0. Nowell, Spaces of order arcs in hyperspaces,Fund. Math. 112 (1981), 111-120. 17. Robert D. Edwards, Characterizing infinite dimensional manifolds topologically, Lecture Notes in Mathematics, Vol. 770, Seminaire Bourbaki (Ed. by A. Dold and B. Eckmann), Springer-Verlag, Berlin, 1980, 278-302. Characterisierung der stetige Kurven, 18. H. Hahn, Mengentheoretische Sitzungsberichte, Akad. der Wissenschaften 123 (1914), 2433. 19. Alejandro Illanes, The space of Whitney levels, Top. and its Appls. 40 (1991), 157-169. 20. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot. 52 (1942), 22-36. 21. K. Kuratowski, Topology, Vol. I, Acad. Press, New York, N.Y., 1966. 22. K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. 23. S. Mazurkiewicz, Sur les lignes de Jordan, Fund. Math. 1 (1920), 166-209. 24. K. Menger, Untersuchungen iiber allgemeine Metrik, Math. Ann. 100 (1928), 75-163. 25. E.E. Moise, Grille decomposition and convexijication theorems for compact metric locally connected continua, Bull. Amer. Math. Sot. 55 (1949), 1111-1121. 26. Sam. B. Nadler, Jr., A characterization of locally connected continua by hyperspace retractions, Proc. Amer. Math. Sot. 6’7 (1977), 167176. Theory, An Introduction, Monographs 27. Sam B. Nadler, Jr., Continuum and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, NY, 1992. 28. G. Peano, Sur une courbe qui remplit toute une aire plane, Math. Ann. 36 (1890),157-160. 29. R. M. Schori and J. E. West, 2’ is homeomorphic to the Hilbert cube, Bull. Amer. Math. Sot. 78 (1972), 402-406. 30. R. M. Schori and 3. E. West, The hyperspace of the closed unit interval is a Hilbert cube, Trans. Amer. Math. Sot. 213 (1975), 217-235. 31. R. M. Schori and J. E. West, Hyperspaces of graphs are Hilbert cubes, Pac. J. Math. 53 (1974), 239-251. 32. H. Torunczyk, On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), 31-40. 33. L. Vietoris, Kontinua zweiter Ordnung, Monatshefte fiir Math. und Physik 33 (1923), 49-62.
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2’ AND C(X)
FOR PEANO CONTINUA
X
34. T. Wazewski, Sur un continu singulier, Fund. Math. 4 (1923), 214235. 35. James E. West, The subcontinua of a dendron form a Hilbert cube factor, Proc. Amer. Math. Sot. 36 (1972), 603-608. 36. M. Wojdyslawski, SW la contractilitt des hyperespacesde continus localement connexes, Fund. Math. 30 (1938), 247-252. 37. M. Wojdyslawski, Re’tractes absolus et hyperespaces des continus, find. Math. 32 (1939), 184-192.
IV.
Arcs
in Hyperspaces
The arc structure of hyperspaces is the most fundamental and important ingredient in the general theory of hyperspaces. We present a systematic treatment of the arc structure of 2x and C(X) when X is a compactum. We also include some related topics such as spaces of segments. We begin the chapter with a section of general preliminaries - quasicomponents and boundary bumping - followed by a section about Whitney maps. We use the material in these two sections throughout the rest of the , chapter. The rest of the chapter contains our study of the arc structure of 2” and C(X). Most of the sections include applications: 2” contains a Hilbert cube for any nondegenerate continuum X (14.12); definitive results about the homogeneity of 2x and C(X) when X is a continuum (15.6 and 15.7); 2x and C(X) are continuous images of the cone over the Cantor set (17.10); and other applications that are in the exercises.
12.
Preliminaries:
Separation, Quasicomponents, Boundary Bumping
We obtain some general results concerned with connectedness for use later. The main theorems are 12.9 and 12.10. Let (X, T) be a topological space. Two subsets, E and F, of X are said to be mutually separated in X provided that
Note that this notion only depends on the subspace topology for E U F; hence, we often simply say that E and F are mutually separated (without saying in X). If (X,T) is a topological space and Y c X, then we write Y = E(F to mean that Y = E U F, where E and F are nonempty and mutually
97
98
IV. ARCS IN HYPERSPACES
separated. Thus, Y = E(F means that Y is not connected and that E and F are the “sides” of a separation of X. Let (X, T) be a topological space, and let K, L, and B be subsets of X. We say that K and L are separated in X by B provided that X - B = EIF with
K c E, L c F.
When I( and L are separated in X by B = 0, we simply say that K and L are separated in X. If any of the sets K, L, or B is known to be a one-point set, say {x}, then we use x rather than {x} in our notation and terminology. We often use the following easy-to-prove reformulation of the preceding definition. 12.1 Proposition. Let (X,T) be a topological space, and let K, L, and B be subsets of X. Then, K and L are separated in X by B if and only if there is a nonempty subset, G, of X - B such that G is clopen in X-B,KcG,andGnL=0. 12.2 Lemma. Let (X,T) be a topological space, let p E X, and let C be a nonempty, compact subset of X. If p and each point of C are separated in X, then p and C are separated in X. Proof. For each c E C, there is a clopen set, G,, in X such that c E G, and p # G, (by 12.1). Since C is compact, C is contained in a union, G, of finitely many of the sets G,. Clearly, G is clopen in X and p $ G. Therefore, since C c G, p and C are separated in X (by 12.1). n 12.3 Proposition. Let (X,T) be a topological space, and let B and C be nonempty, cotnpact subsets of X. If each point of B and each point of C are separated in X, then B and C are separated in X. Proof. By 12.2, B and each point of C are separated in X. Hence, repeating the proof of 12.2 with p replaced by B, it follows that B and C are separated in X. n
Let (X, T) be a topological space, and let 2 C X. The boundary of Z in X, which is denoted by Bd(Z), is defined as follows: Bd(Z) = cl(Z) rl cl(X - Z). EvidentIy, Bd(Z) depends on the containing space X (e.g., the boundary of I” in I” is empty whereas the boundary of I” in R’” is aP). Thus, it would be more accurate to incorporate X in our notation by writing, for
12. PRELIMINARIES: example, ary of 2 confusion Y when
SEPARATION, QUASICOMPONENTS,.
..
99
Bd,v(Z). However, we prefer to use Bd(Z) to mean the boundin the largest space under consideration; this will not cause any since we will always use Bdy (2) to denote the boundary of 2 in Y is not the largest space under the consideration.
12.4 Proposition. Let (X,T) be a topological space, let 2 c X, and let p E 2. If p and Bd(Z) are separated in X, then p and X - Z are separated in X. Proof. By 12.1, there is a clopen set, G, in X such that p E G and G n Bd(Z) = 0. W e see that G n 2 is clopen in X as follows: since G n Bd(Z) = 0, clearly
GnZ=Gncl(Z)andGnZ=GnZ”; thus, since G is clopen in X, G n Z is clopen in X. Also, p E G n Z and GnZ is disjoint from X - 2. Therefore, by 12.1, p and X - 2 are separated in X. w Let (X,7’) be a topological space, and let p E X. The component of p in X is the set of all those points, x, of X such that p and x lie together in a connected subset of X; in other words, the component of p in X is the union of all the connected subsetsof X that contain p. The quasicomponent (abbreviated qc) ofp in X is the set of all those points, x, of X such that p and x are not separated in X; in other words, the qc of p in X is the intersection of all the clopen sets in X that contain p. By a component of X, or a qc of X, we mean a component, or qc (respectively), of some point in X (the component of 0 and the qc of 0 is 0). Obviously, a component of a spaceis a maximal connected subset of the space. Thus, it may seemthat any two components of a space are separated in the space. However, this is not the case. For example, let X be the subspaceof R” consisting of the vertical lines through the points (l/n,O), n = 1,2,. . ., together with the points p = (0,O) and q = (0,l); then, {p} and {q} are components of X that are not separated in X. Nevertheless, any two components of a compact Hausdorff space are separated in the space, as we will seewhen we get to 12.8. 12.5
Lemma.
Any qc of a space, X, is closed in X.
Proof. As we noted when we defined a qc, any qc of X is an intersection of clopen sets in X. n 12.6
Lemma.
nents of X.
Any qc of a space, X, is a union of some of the compo-
100
IV. ARCS IN HYPERSPACES
Proof. Let Q be the qc of a point, p, in X. Let x E Q, and let C be the component of x in X. Let G be a clopen set in X such that p E G. Then, by the definition of a qc, G > Q. Hence, G n C # 0. Thus, since G n C is clopen in C and C is connected, we see that G II C = C; i.e., G > C. Therefore, since Q is the intersection of all the clopen sets in X that contain p, it follows that Q > C. n 12.7 Theorem. In compact Hausdorff spaces,components and quasicomponents are the same. Proof. Let X be a compact Hausdorff space. In view of 12.6, it, s&ices to prove that every qc of X is connected. Let Q be a qc of X. Suppose that Q is not connected, which is to say that Q = EIF. Note from 12.5 that E and F are closed in X. Therefore, since X is a normal space, there is an open subset, U, of X such that (1) ECUandFnU=0. Note that since U is open in X, Bd(U) = V - CT.Thus, since Q = E u F, we seefrom (1) that (2) Q n Bd(U) = 0. Now, let p E E. Then, since p E Q, we see from (2) that (3) p and each point of Bd(U) are separated in X. We will use 12.2 to show that p and Bd(U) are separated in X. For this purpose, we prove (4) and (5) below. If Bd(U) = 0, then U would be clopen in X; however, since lJ n Q # 0 and U $ Q (by (I)), we seethat U can not be clopen in X. Therefore, (4) BW) # 0. Since X is compact and Bd(U) is closed in X, (5) Bd(U) is compact. Now, by (3)-(5), we may apply 12.2 to seethat p and Bd(U) are separated in X. Thus, noting that p E U (since E c U), we have by 12.4 that p and X - U are separated in X. However, this is impossible (since p E Q and since F is a nonempty subset of Q such that F C X - U). Therefore, Q is connected. n 12.8 Corollary. Any two components of a compact Hausdorff space are separated in the space. Proof. 12.7. n
By the exercise in 12.14, the corollary is actually equivalent to
We now prove the two main theorems of the section.
12. PRELIMINARIES:
SEPARATION,
QUASICOMPONENTS,.
..
101
12.9 Cut Wire Fence Theorem. Let Y be a compact Hausdorff space, and let B and C be nonempty, closed subsets of Y. If no connected subset of Y intersects both B and C, then B and C are separated in Y. Proof. By assumption, no component of Y intersects both B and C. Hence, by 12.7, no qc of Y intersects both B and C. In other words, each point of B and each point of C are separated in Y. Therefore, by 12.3, B and C are separated in Y. H Recall that our definition of continuum in Chapter II requires that a continuum be a metrizable space. We call any nonempty, compact, connected, Hausdorff space a Hausdorff continuum. 12.10 Boundary Bumping Theorem. Let X be a Hausdorff continuum, and let U be a nonempty, open, proper subset of X. If K is a component of v, then K n Bd(U) # 0, i.e., K n (X - U) # 0. Proof. Suppose that K n Bd(U) = 0. Then, since K is a maximal connected subset of u, it follows that no .connected subset of u intersects both K and Bd(U). Also, note that K and BcZ(U) are closed subsetsof u, K # 0, and Bd(U) # 0 (E xercise 12.13). Hence, by 12.9, K and Bd(U) are separated in v; in other words, ?? = EIF with K c E, Bd(U) C F. Thus, it follows easily that X = EI(F u [X - VI). This contradicts the connectednessof X. Therefore, K n Bd(U) # 0. W We conclude the section with two applications of our main theorems. The first application concerns the relationship between zero-dimensional spaces and totally disconnected spaces. We defined zero-dimensional in section 8. A topological space, X, is said to be totally disconnectedprovided that X # 0 and no connected subset of X contains more than one point (i.e., each component of X consists of only one point). It is easy to seethat every zero-dimensional Tr-space is totally disconnected. However, a totally disconnected metric space need not be zerodimensional (an example is in Exercise 12.19). We have the following application of 12.9.
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IV. ARCS IN HYPERSPACES
12.11 Theorem. Let X be a compact Hausdorff space. dim(X) = 0 if and only if X is totally disconnected.
Then,
Proof. Assume that dim(X) = 0 (hence, X # 0). Suppose that X is not totally disconnected. Then, since X # 0, there is a connected subset, C, of X such that C contains at least two points, say p and q. Since dim(X) = 0 and X is a Ti-space, there is a clopen neighborhood, N, of p in X such that q $ N. Clearly, N n C is a nonempty, proper, clopen subset of C; hence, we have a contradiction to C being connected. Therefore, X must be totally disconnected. Conversely, assume that X is totally disconnected. To prove that dim(X) = 0, let p E X and let U be an open neighborhood of p in X. We may assume for the purpose of proof that U # X. Then, {p} and X - U are nonempty, closed subsetsof X; also, since X is totally disconnected, it is obvious that no connected subset of X intersects both {p} and X - U. Hence, by 12.9, p and X - U are separated in X. Thus, by 12.1, there is a clopen set, G, in X such that p E G and G c U. Therefore, since X # 0, we have proved that dim(X) = 0. n
The next theorem is a fundamental result about the structure of continua. In general terms, the theorem says that every nondegenerate, Hausdorff continuum contains many nondegenerate, Hausdorff subcontinua. It probably comes as no surprise that every nondegenerate continuum must contain a nondegenerate, proper subcontinuum; however, a proof of this fact would be far from obvious until now. Indeed, the following theorem is a simple application of 12.10. 12.12 Theorem. Let X be a nondegenerate Hausdorff continuum. Then, X contains a nondegenerate, proper, Hausdorff subcontinuum. Moreover, let A be a proper, Hausdorff subcontinuum of X, and let U be an open subset of X such that A c U; then there is a Hausdorff subcontinuum, B, of U such that B > A and B # A.
We prove the second part of the theorem first. Assume that A Proof. and U satisfy the assumptions in the second part of the theorem. Then, since X is a normal space, there is an open subset, V, of X such that A c V, v c U, and V # X. Let B be the component of 7 containing A (B exists since A is a connected subset of v). By 12.10, Bn(X-V)#@ thus, since A c V, we seethat B # A. Note that B c U since v C U. The other properties of B that are stated in the theorem are obviously satisfied.
EXERCISES
103
Now, we show how the first part of the theorem follows from the second part. Let p E X, and let U be an open, proper subset, of X such that p E U. Then, by the second part of the theorem (with ‘4 = {p}), there is a B, of U such that p E B and B # {p}. Clearly, Hausdorff subcontinuum, B is a nondegenerate, proper, Hausdorff subcontinuum of X. w
Exercises 12.13 Exercise, If (X,T) is a connected topological nonempty, proper subset of X, then Bd(Z) # 0.
space and 2 is a
12.14 Exercise. For any topological space, (X,T), the following two statements are equivalent: (1) any two components of X are separated in X; (2) the quasicomponents of X and the components of X are the same. (Cf. proof of 12.8.) 12.15 Exercise. Prove the following boundary bumping theorem (cf. the remark following 12.16). Let X be a Hausdorff continuum, and let 2 be a nonempty, proper subset of X. If K is a component of 2, then
cl(K) fl Bd(Z) # 0, i.e., cl(K) f~ cZ(X - Z) # 0. [Hint: Find a way to use 12.12.1 12.16 Exercise. Let X be a Hausdorff continuum, and let, 2 be a nonempty, proper subset of X. If K is a component of 2, then IC and X - 2 are not mutually separated. [Hint: Use 12.15.1 Remark. The results in 12.10, 12.12 (second part), 12.15, and 12.16 are actually one and the same result stated in different ways. We see this from how 12.10 implied the second part of 12.12, from the hints in 12.15 and 12.16, and from an easy argument that shows that 12.16 directly implies 12.10. 12.17 Exercise. We consider the following generalization of an end point of an arc. A point, p, of a topological space, (X,7’), is called an end point of X provided that p has a neighborhood base of open sets in X with one-point boundaries. For example, the point p of the continuum X in Figure 22, p. 84, is an end point of X. Prove the two results below using 12.15. If X is a Hausdorff continuum and p is an end point of X, then X - {p} is connected.
104
IV. ARCS IN HYPERSPACES
If p is an end point of a Hausdorff continuum (However, X may not be locally connected at p -
X, then X is cik at p. e.g., X in Figure 22.)
12.18 Exercise. Let X be a Hausdorff continuum, and let p E X. The composant of p in X, which is denoted by K(P), is defined as follows: &c(p)
=
{x E X : there is a proper, Hausdorff containing
both
p
subcontinuum
of X
and x}.
of X we mean a composant of some point in X. By a composant What is K(P) for various choices of p when X = [O?l]? What about when X is a simple closed curve ? What about when X is the sin(l/x)continuum in (3) of Figure 20, p. 63? (Note: A nondegenerate continuum must have either one, three, or uncountably many composants [28, p. 2041; however, this does not remain true for Hausdorff continua [I].) Prove the following result using 12.15: If X is a nondegenerate Hausdorff continuum and p E X, then n(p) is dense in X and connected.
12.19 Exercise. We give an example of a totally disconnected metric space that is not zero-dimensional (cf. 12.11). Let Q denote the set of all “rational points” in the Hilbert space & of all square-summable sequences of real numbers; explicitly, : xiER’foreachiand
e2 = u~,)~_,
~x~
with the metric, d, given by
d((xtL,
hEo=,)
=
g(zi
- yi)2
for all (xi)gr,
(yi)zi
E 12,
i=l
and
Q = {(xi):1
E e2 : xi is rational
for all i}.
Show that Q is totally disconnected but that Q is not zero-dimensional. [Hint: To prove that dim(Q) # 0, let U be any bounded, open subset of Q such that (O,O, . . .) E U. It suffices to prove that U is not closed in Q. Let al be a rational number such that, for pl = (al, O,O, . . .), pl E U and d(pl, Q - U) < 2-l. Then let a2 be a rational number such that, for ps = (ai, az, O,O, . .), p2 E U and d(p2, Q - U) < 2-‘. Consider the that is obtained by induction according to the indicated sequence {p,}p& pattern. Let p = (a,)z=,; why does p E e2?]
13. A BRIEF INTRODUCTION 12.20
Exercise.
The following
TO WHITNEY
definition
MAPS
105
is a generalization
of the no-
tion of a simple n-od: An n-od (2 5 n < 00) is a Hausdorff continuum, Y, for which there is a (Hausdorff) subcontinuum, 2, such that Y - 2 = ElIE21...1E, (i.e., Y - 2 = Ur==,Ei, ,?$ # 0 for each i, and Ei and E3 are mutually separated whenever i # j). A 3-od is called a triod. If Y is an n-od, then Y = Uy==,Y,, where Yi is a Hausdorff continuum for each i, M = f&Yi is a Hausdorff continuum, Y, - M # 0 for each i, and Yi n Y3 = M whenever i # j. Remark.
Some results about n-ods and hyperspaces are in Exercises
14.18-14.20.
13.
A Brief
Introduction
to Whitney
Maps
In the early 1930’s, Hassler Whitney constructed special types of functions on spacesof sets for the purpose of studying families of curves ([38] and [39]). In 1942, J. L. Kelley made significant use of Whitney’s functions in studying hyperspaces [16]. Beginning in the early 1970’s, connections between Whitney’s functions and the structure of hyperspaces were investigated extensively and systematically by numerous people (seeChapter XIV of [29] for what was known up until 1979). In particular, Whitney maps (as Whitney’s functions are now called) are closely connected with the arc structure of hyperspaces. For this reason it is appropriate at this time to devote a short section to constructing a Whitney map for any hyperspace, thereby proving that Whitney maps always exist. We will see later that Whitney maps lend elegance and clarity to proofs.
Definition
of a Whitney
Map
The definition of a Whitney map is essentially an axiomatic formulation of what one might mean by a general measureof the relative geometric sizes of sets: Let X be a compactum, and let ‘U C 2x. A Whitney map for fl is a continuous function ‘w : Z + [O,oo) that satisfies the following two conditions: (1) for any A, B E 31 such that A c B and A # B, w(A) < w(B); 13.1
Definition.
(2) w(A) = 0 if and only if A E ‘H n Fl(X). For example, assuming that [O,l] has its usual (absolute value) metric, the diameter map is a Whitney map for C([O, 11). On the other hand, let X be an arc in the unit circle S’ such that X has arc length greater than a, and assumethat X has the Euclidean metric; then the diameter map is
IV. ARCS IN HYPERSPACES
106
not a Whitney map for C(X). In fact, a diameter map is rarely a Whitney map for C(X), and a diameter map is never a Whitney map for Zx when X contains more than two points (Exercise 13.9). Nevertheless, diameter maps are useful in constructing Whitney maps (as we will see when we prove 13.4).
Existence
of Whitney
Maps
In 13.4 we prove that Whitney maps exist for any hyperspace of a compactum. In the proof of 13.4, we construct the Whitney map that was constructed in [39, pp. 245-2461. Let X be a compacturn with metric d; the diameter map with respect to d is the function diamd : 2x -+ [0, cc) that is defined as follows: for each A E 2”,
diamd(A) = lub{d(z, y) : z:,y E A}. 13.2 Lemma. Let X be a compactum with metric d. Then the diameter map diamd : zx + [O,00) is continuous. Proof. Let E > 0, and let A, B E 2x such that Hd(A, B) < & (where Hd is the Hausdorff metric in 2.1). Since A is compact, there exist ai,u2 E A such that diamd(A) = d(al, a~). Since al, a2 E A and A c Nd(s, B), there exist bi, bz E B such that d(al,bl) < E and d(az,bn) < E. Hence, d(al,az)
5 d(al,bl)
+ d(bl,b2)
Therefore, since diamd(A) = d(al,az),
+ d(b2,a~)
< d(bl,bz)
+ 2~.
we have that
(1) diamd(A) < d(bl, bs) + 2~ 5 diamd(B) f 2~. A similar (symmetric) argument shows that (2) diamd(B) < diam,j(A) + 2E. By (1) and (2), (diamd(A)- diamd(B)( < 2~. n 13.3 Lemma. Let X and Y be compacta, and let continuous. Let f* : 2x -+ Zy be defined as follows:
f*(A) Then, f’ is continuous.
=
f(A)
for each A E 2x.
f :
X
-+
Y be
EXISTENCE
OF WHITNEY
MAPS
Proof. Let Wi, . . . , W,, be finitely many open subsets of Y. using that f is a closed map, we see that . . .) Wn)) = (f-ywl),
tf*r’twl~
107
Then,
. . ., f-1(wl)).
Therefore, since f is continuous, we seefrom 1.2 that f * is continuous. n 13.4 Theorem. If X is a compactum, then there is a Whitney map for any hyperspace of X. Proof. It is evident from 13.1 that if w is a Whitney map for 2x and R C 2”) then the restricted map w]‘R is a Whitney map for 3t. Therefore, it suffices to show that there is a Whitney map for 2x. Let d denote a metric for X. Let 2 = {zi, 22, . . . , z,, . . .} be a countable, dense subset of X. For each n = 1,2,. . . , define fn : X + [0, l] as follows: fn(z) = & for each 2 E X. Next, for each n = 1,2,. . . , define w, : 2” + [0, l] as follows: wn(A) = diameter [fn(A)] for each A E 2x. Now, define w : 2x + (0, l] as follows: w(A) = CrEp=l2-n. wn(A) for each A E 2x. We prove that w is a Whitney map for 2”. To seethat w is continuous, simply note that each w,, is continuous by the lemmas in 13.2 and 13.3 (since wn = diam, o f,t, where p is the usual metric for [O,l]). Next, to show that w satisfies (1) of 13.1, let A, B E 2x such that A c B and A # B. Since A c B, clearly fn(A) c fn(B) for each n; hence, w=(A) 5 wn(B) for each n. Therefore, to prove that w(A) < w(B), it suffices to prove the following:
(if)
wi(A) < wi(B) for some i.
Proof of (#): Let p E B - A, and let T = $d(p,A). Note that T > 0 (since p $! A and A is closed in X), and recall that 2 is dense in X. Hence, there exists zi E 2 such that d(p,zi) < T. We show that wi(A) < wi(B) as follows. Since d(p, zi) < r, b-4 f&4 = riT;it;;;;;r > ii+7 Since d(zi, A) > T, (b) fi(a) = & < & for each a E A. By combining the inequalities in (a) and (b), we seethat (c) lubfi(A) I & < fib). Thus, since p E B, we seeimmediately from (c) that (d) lubfi(A) < lubfi(B).
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IV. ARCS IN HYPERSPACES
Now, we note the following A c B):
inequality
(e) .dbf@) L Ofi( It follows at once from (d) and (e) and, therefore, completes the proof Finally, we show that w satisfies that A E Fi (X), say A = {z}; then, w,(A)
(which
that that (2) for
is simply due to the fact that
wi(A) < 2ui(B). This proves (#) w satisfies (1) of 13.1. of 13.1. Let A E 2”. First, assume each n = 1,2,. .,
= diameter [fn({z})]
= 0;
therefore, w(A) = 0. Next, assume that .A 4 Fi(X); then, for some k = 1,2,. . . , fk(A) is nondegenerate (since 2 is dense in X); hence, wk(A) > 0; therefore, w(A) > 0. R We construct three other Whitney maps in Exercises 13.8. More about Whitney maps is in Chapters VII-IX.
13.5, 13.7, and
Exercises 13.5 Exercise. Let X be a compacturn with metric d. Prove that the function UJ defined below is a Whitney map for 2x. (The notation F,(A) stands for the n-fold symmetric product of A (1.7).) Fix A E 2x. For any given integer n 2 2, define X, : F,(A) + [0, co) index points of K using all the integers as follows: For each K E F,(A), 1,2,. . . ) n (thus requiring the indexing not to be one-to-one when (K( < n); then, if K f F,(A) and K = {ar,a2,. . . ,a,}, let X,(K)
= min{d(ai,aj)
Next, for each n 2 2, let wn(A) w(A)
= lub(X,(K) = 2
2’+.
: i # j}. : K E Fn(A)}.
Finally,
let
wn(A).
n=2
Remark. The Whitney map in 13.5 was, historically, the first Whitney map [38]. It has a very natural and useful property that Whitney maps in general do not have - see Exercise 13.6. 13.6 Exercise. Prove that the Whitney map ‘w in 13.5 has the following property: For any A, B E 2x such that A and B are isometric (as subsets of X), w(A) = w(B). Give an example of a Whitney map that does not have the property just mentioned.
EXERCISES
109
[Hint for second part: Let X = [0, l] with the usual metric d, let w be the Whitney map for 2 *’ in 13.5, and define cp on 2x by v(A) = 1ubA for each A E 2x; consider the product map ‘p . w.] 13.7 Exercise. Let X be a compacturn. Prove that the function w defined below is a Whitney map for 2x. (The construction is due to Krasinkiewicz [19].) Let B = {Bi,Bs,. . .} be a countable base for the topology for X. We assume that for each i = 1,2,. . . , Bi # 0 and ??, # X. Then, for each pair (Bi, Bj) such that Bi C Bj, there is a continuous function f,? : X + [0, l] such that Ji(Bi) = 0 and fi(X - Bj) = 1. Reindex all the functions f{ SO that they form a simple sequence fi, fi, . . . , fn, . . .. For each n = 1,2,. ., define w, : 2x + [0, l] by
w,(A) = diameter [f,%(A)] for each A E 2”. NOW, define w : 2x -+ [0, I] by
w(A) = 2
2Tn. w,(A) for each A E 2x.
13.8 Exercise. Let X be function w defined below is a due to Illanes [13].) Fix A E 2x. For any given E X such that exist zlr...,zn w(A)
a compacturn with metric d. Prove that the Whitney map for 2x. (The construction is integer n 2 1, let wn(A) = inf{e > 0: there A c U~T~N~(E, {xi})}. Then let = 5
2-n. w*(A).
n=l
13.9 Exercise. If X is a compacturn that contains more than two points, then the diameter map with respect to any metric for X is not a Whitney map for 2x. If X is a nondegenerate, arcwise connected continuum such that the diameter map with respect to some metric for X is a Whitney map for C(X), then X is an arc. (See the discussion following 13.1.) 13.10 Exercise. Let X be a compacturn with metric d, and let w be converges in 2x to any Whitney map for 2x. Then a sequence, {A,}&, A if and only if lim+,w(Ai) = w(A) and for any E > 0, Ai C Nd(c, A) for all i sufficiently large.
110
IV. ARCS IN HYPERSPACES
13.11 Exercise. Note the following definitions. Let C be a collection of sets. A maximal member of C is an F E C such that no member of C properly contains F; a minimal member of C is an E E C such that no member of C is properly contained in E. Prove the following theorem using 13.4. Maximum-Minimum Theorem. Let X be a compacturn. If C is a nonempty, closed subset of 2x, then there is a maximal member of C and there is a minimal member of C. Remark. The Maximum-Minimum Theorem has numerous applications. In particular, it can often be used in place of Zorn’s Lemma or the Brouwer Reduction Theorem. Some applications of the MaximumMinimum Theorem are in [28, pp. 68 and 130-1311;also, for example, see the proofs of 14.5 and 15.3 here.
14.
Order
Arcs and Arcwise
Connectedness C(X)
of 2x and
We are ready to begin our study of the arc structure of hyperspaces. We introduce the notion of an order arc in 14.1. We use order arcs to prove the important theorem that 2x and C(X) are arcwise connected when X is a continuum (14.9). We apply this theorem to show that 2x contains a Hilbert cube when X is any nondegenerate continuum (14.12).
Definition
of Order
Arc
Any arc has a natural total ordering; any hyperspace has a natural partial ordering (namely, containment). An order arc in a hyperspace, ‘I-l, is an arc (Y c 7-i such that the partial ordering of containment for ?i agrees on a: with the total ordering on Q. We find it convenient to state the definition of an order arc as in 14.1 below. A collection, N, of sets is called a nest provided that for any Ni, N2 E N, NiCNzorNzCNi. 14.1 Definition. Let X be a compactum, and let 31 c 2x. An order arc in 3c is an arc, Q, in 7i such that a is a nest.
Arcwise
Connectedness
of 2x and C(X)
We prove several lemmas about nests. The lemmas lead to a quick proof that 2” and C(X) are arcwise connected when X is a continuum (14.9).
ARCWISE
CONNECTEDNESS
OF 2x AND C(X)
111
14.2 Lemma. Let X be a compactum, let 31 C 2x, and let w be a Whitney map for 7-1. If N is a compact nest in ?i, then WIN is a homeomorphism. Proof.
The lemma follows easily from the definition in 13.1. n
A nest from A0 to Al is a nest, N, such that Ao, Al E N c Al for all N E N.
N and A0
c
14.3 Lemma. Let X be a compactum, and let Ao, AI E C(X) such that A0 c Al. If M is a maximal nest in C(X) from A0 to Al, then M is compact. Proof. Since C(X) is compact (by 3.7), it suffices to prove that M is closed in C(X). Let Mi E M for each i = 1,2,. . . such that {Mi}Ei converges to some A E C(X). We show that A E M. First, we show that M U {A} is a nest. Let M E M. Then, M is a nest, Mi > M for infinitely many i or Mi C M for infinitely many i. In the first case, A 3 M (cf. Exercise 1.19); in the second case, A C M since C(M) is compact by 3.7. Therefore, since M is a nest, it follows that M U {A} is a nest. Next, note that, since A0 c M, c Al for each i, A0 C A C AI. Hence, we have shown that M U {A} is a nest in C(X) from A0 to AI. Therefore, by the maximality of M, we have that A E M. n 14.4 Lemma. Let X be a compactum, and let Ao,Al E C(X) such that A0 c Al and A0 # Al. If M is a maximal nest in C(X) from AO to Al, then M is an arc from A0 to Al. Proof. By 13.4, there is a Whitney map, w, for C(X). Let to = w(Ao) and let ti = w(Al). Then, since M is a nest from A0 to A1 and A0 # Al, we see from (1) of 13.1 that to < tl and that w(M) c [to,tl]. We also know from 14.3 that M is compact. Therefore, to prove that M is an arc from A0 to Al, it suffices by 14.2 to prove that w(M) = [to, tl]. Suppose that w(M) # [to, tl]. Then, since to, tl E w(M) and since w(M) is a compact subset of [to,tl], there exist se,si E w(M) such that SO< s1 and w(M) n ( so, sI) = 0; in other words, the open interval (so, si) = SO is a component of [to, ti] -w(M). Let MO, Ml E M such that I and w(M1) = ~1. Since SO < si and M is a nest, it follows from (1) of 13.1 that MO c Ml and MO # MI. Thus, since M c C(X), MO is a proper subcontinuum of the continuum Ml. Therefore, by the second part of 12.12, there is a proper subcontinuum, B, of Ml such that B > MO and B # MO.
112
IV. ARCS IN HYPERSPACES
We show that M u {B} is a nest. Let M E M. Then, since w(M) n (so,s~) = 0, we have that w(M) 5 SOor w(M) 2 ~1. If w(M) 5 so then, since M is a nest, we see using (1) of 13.1 that M c MO; if w(M) 2 s1 then, similarly, M > Ml. Thus, since MO c B c Ml, we seethat M c B or M 3 B. It now follows easily that M U {B} is a nest. Furthermore, MU(B) is a nest in C(X) from A0 to Al (since B E C(X) and since AO C MO c B C Ml c AI). Hence, by the maximality of M, we have that B E M. However, we also have that B $ M for the following reason: w(M) n ( SO,S~)= 0 and SO< w(B) < s1 by (1) of 13.1 (since the containments MO c B c Ml are proper). The contradiction that we just obtained came from our supposition that w(M) # [to,tl]. Therefore, w(M) = [to,tl]. n 14.5 Lemma. Let X be a compactum, and let Ao, Al E C(X) such that A0 c Al. Then there is a maximal nest in C(X) from Ao to Al. Proof. The lemma is a consequenceof Zorn’s Lemma [15, p. 331. Nevertheless, we include the following proof of the lemma to illustrate the Maximum-Minimum Theorem in Exercise 13.11. Let
A = {N c C(X) : N is a compact nest from A0 to Al}. Then, A # 0 (since (Ao, Al} E A) and A is a closed subset of 2c(x) (by an elementary sequenceargument). Also, note that C(X) is a compacturn (by 3.1 and 3.7). Thus, by the Maximum-Minimum Theorem in 13.11 (with X replaced by C(X)), there is a maximal member, M, of A. Clearly, M is a maximal compact nest in C(X) from A0 to Al. Therefore, it follows from Exercise 14.14 that M is a maximal nest in C(X) from A0 to Al. n The following theorem is a straightforward but important consequence of the two preceding lemmas. 14.6 Theorem. Let X be a compactum, and let Ao, Al E C(X) such that A0 c Al and A0 # Al. Then there is a order arc in C(X) from A0 to
Al. Proof. By 14.5, there is a maximal nest, M, in C(X) from A0 to AI; by 14.4, M is an arc from A0 to Al. Therefore, according to the definition in 14.1, M is an order arc in C(X) from A0 to Al. n The theorem in 14.6 can be thought of as a continuous boundary bumping theorem: We can start at any proper subcontinuum, A, of a continuum, X, and grow continuously to X with larger and larger continua, thereby
ARCWISE
CONNECTEDNESS
OF 2x AND C(X)
113
bumping the boundaries of all neighborhoods of A (except X). Thus, 14.6 enables us to envision the boundary bumping theorems in 12.10 and 12.13 in a dynamic, hence more enlightening way. We prove our final lemma about nests: 14.7 Lemma. Let X be a compactum, and let A be a nondegenerate subcontinuum of 2x. If A is a nest, then A is an order arc (and conversely). Proof. By 13.4, there is a Whitney map, w, for 2x. By the assumptions about A and 14.2, wlA is a homeomorphism of A onto a nondegenerate, closed and bounded interval. Hence, A is an arc. Therefore, since A is a nest, A is an order arc (14.1). n
We note the following elementary, general lemma for use in the proof of 14.9. 14.8 Lemma. Let Y be a space such that for some point p E Y, there is an arc in Y from y to p for any point y E Y - {p}. Then, Y is arcwise connected.
Now, we come to our theorem about the and C(X). Note that the theorem applies themselves arcwise connected. In particular, tinua that contain no arc whatsoever! (as is 1.23 and 2.27 of [28]). 14.9
Theorem.
arcwise connectedness of 2-’ to continua X that are not the theorem applies to contrue, e.g., of the continua in
If X is a continuum, then 2” and C(X)
are arcwise
connected. Proof. Let K E 2x such that K # X. Let A0 be a component of K. Note that Ao,X E C(X) and that A0 # X. Hence, by 14.6, there is an order arc, Q, in C(X) from A0 to X. Let h be a homeomorphism of [O,l] onto (I such that h(0) = A0 and h(1) = X. Then let f be the function from [0, l] into 2-y given by
f(t) = K
u
h(t) for each t E [0, 11.
Note that f(0) = K, f(l) = X, and f is continuous (by Exercise 1.23 since h is continuous). Hence, f([O, 11) is a nondegenerate subcontinuum of 2”; furthermore, f([O, 11) is a nest from K to X (since (Y is an order arc from A0 to X and Ao c K). Therefore, by 14.7, f([O, 11) is an order arc in 2x from K to X. We have shown that for any K E 2” such that K # X, there is an arc in 2” from K to X. Therefore, 2x is arcwise connected by 14.8.
114
IV. ARCS IN HYPERSPACES
The proof that C(X) is arcwise 14.6 with Ai = X and 14.8. n
connected is simply a matter of using
The analogue of 14.9 for Hausdorff continua and generalized arcs has been obtained by McWaters [25, p. 12091. (A generalized arc is a Hausdorff continuum, X, whose topology is the order topology obtained from a total (simple) ordering on X; equivalently, a generalized arc is a Hausdorff continuum with exactly two non-cut points -. 6.16 of [28, p. 951.) We have known for some time that 2x is connected when X is a continuum --- second part of Exercise 1.17. However, 14.9 is the first result from which we know that C(X) is connected when X is a continuum. In contrast, we note that there are connected Hausdorff spacesX for which CLC(X) is not connected [21]. It is convenient for later reference to have a summary of what we know about the general structure of 2x and C(X) when X is a continuum: 14.10 Corollary. connected continua.
If X is a continuum, then 2x and C(X) are arcwise
Proof. The hyperspaces 2” and C(X) are compacta by 3.1, 3.5, and 3.7. Therefore, by 14.9, 2” and C(X) are arcwise connected continua. n
Application:
2x > I”
We prove that, for a nondegenerate continuum X, 2” contains a Hilbert cube. Ost,ensibly, this theorem has nothing to do with the arcwise connectedness of 2” ; nevertheless, the proof of the theorem is based directly on the existence of arcs in 2aY. The following general lemma sets the stage for the proof. 14.11 Lemma. Let X be a nondegenerate continuum, and let p E X. Then there is a sequence, {.4i}pOi, of nondegenerate, mutually disjoint subcontinua of S - {p} such that Lim Ai = {p}. Proof. Let {z2}g1 be a sequenceof distinct points in X-(p) such that {~i}zi converges top. A simple induction shows that, for each i = 1,2,. . ., there are open sets, Ui, in X - (p} with the following properties: Q E Vi for each i; U, n Uj = 0 whenever i # j; and diameter (Ui) < 2-j for each i. Now, by the second part of 12.12, there is a nondegenerate subcontinuum, .-&, of U, for each i. It is easy to seethat the sequence{Ai}El is asrequired. n
APPLICATION:
2x > IO0
115
14.12 Theorem. If X is a nondegenerate continuum, then 2” contains a Hilbert cube; hence, 2* is infinite-dimensional. Proof. Let p E X, and let {Ai}El be as guaranteed by 14.11. By 14.9 (and since each Ai is nondegenerate), there is an arc, oi, in 2Az for each i. Note that IIzioi is a Hilbert cube. We embed IIzo=,oi in 2dYas follows. Let (Bi)zi E I’Izioi. Note that Bi c A, for each i and that {A,}E”=, converges in 2x to {p} with respect to the Hausdorff metric H (by 14.11 and 4.8). Hence, {&}gi converges in 2x to {p}. Thus, letting
we see that h((Bi)zl) E 2x. Therefore, we have defined a function h from II,“=iai into 2”. Furthermore, h is one-to-one, which we see using that the sets Ai are mutually disjoint and that p $! Ai for any i (14.11). We prove that h is continuous. Let B” = (Bt)z”=, E IIE”=,cri, let B = (B,)z”=, E II~=,ai, and assume that the sequence {B”}rf”=, converges in IIEioi to B. Let E > 0. Since {Ai}z”=, converges in 2x to {p}, there exists N such that (1) diameter (Ai) < Efor each i 2 N. Now, since convergence in IIEi oi is coordinatewise convergence, there exists K such that (2) H(B,“, Bi) < E for each i 5 N when k 2 K. Since Bf c A, and Bi c A, for each k and i, we seefrom (1) and from the formula for H in 2.1 that (3) H(Bf,Bi) < E for each i 2 N (and every k). Combining (2) and (3), we have that (4) H(Bf,Bi) < c for each k > K. It follows easily from (4) that H(h(B”),
h(B)) < 6 for each k 2 K.
Thus, we have proved that h is continuous. Therefore, having proved that h is one-to-one, h is an embedding of the Hilbert cube IIEioi in 2”. This proves the first part of the theorem. The second part of the theorem follows from the first part by the fact that Hilbert cubes are infinitedimensional [ll, p. 491and by the subspacetheorem [ll, p. 261. n 14.13
Corollary.
are equivalent: (1) dim(X) # 0;
Let X be a compactum. Then (l), (2), and (3) below
IV. ARCS IN HYPERSPACES
116 (2) dim(2x)
= 00;
(3) 2x contains a Hilbert
cube.
Proof. Assume that dim(X) # degenerate continuum, Y. By 14.12, Therefore, we have proved that (1) (see the last paragraph of the proof 8.6. n
0. Then, by 12.11, X contains a non2y, hence 2x, contains a Hilbert cube. implies (3). Evidently, (3) implies (2) of 14.12). Finally, (2) implies (1) by
By 14.13, 2x contains a Hilbert cube whenever 2x is infinite-dimensional. The analogous statement for C(X) is false: Let X be the continuum in Figure 24; then C(X) contains an n-cell for every n, hence C(X) is infinitedimensional, but C(X) does not contain a Hilbert cube (Exercise 14.15). Some results about when C(X) contains an n-cell or a Hilbert cube are in Exercises 14.18-14.21. More results about this along with some applications are in [30]; a definitive characterization is in [12], which we present in section 70. Regarding the dimension of C(X), see Chapter XI.
Original
Sources
The notion of an order arc was first formulated in a precise way by Mazurkiewicz ([22, p. 1721and Lemma 5 of [22]). However, the specter of order arcs appeared earlier in a proof of Borsuk and Mazurkiewicz [3] (for comments about the proof, see 1.10 of [29]). Regarding specific results, 14.9
Figure 24
EXERCISES
117
for 2” is from [3]; 14.9 for C(X) and 14.6 are from the proof in [3] (see remarks in 1.10 and 1.14 of [29]); 14.12 is due to Mazurkiewicz [24] and, independently, to Kelley [lS, p. 291. The example in Figure 24, p. 116 is due to B.J. Ball and appeared in [30, pp.‘247-2481.
Exercises 14.14 Exercise. Let X be a compactum, let U c 2x, and let J\/ be a nest in Ifl. Then the closure, g, of n/ in 31 is a nest; also, if n/ is a nest from A0 to Al, g is a nest from AC,,to Al. 14.15 Exercise. Let X be the continuum in Figure 24, p. 116. Prove that C(X) contains an n-cell for every n, and prove that C(X) does not contain a Hilbert cube. (The continuum X came up in the discussion about 14.13.) 14.16 Exercise. Let X be a compacturn. then the end points of cy are no and Ua.
If (Y is an order arc in 2x,
14.17 Exercise. Let X be a compactum, and let f be a continuous Let function from [O,l] into 2x. For each t E [0, 11,let At = Uf([O,t]). a = {At : t E [O,11). If a is nondegenerate, then o is an order arc in 2x from f(0) to Uf([O, 11). 14.18 Exercise. If a continuum, X, contains an n-od for some n 1 2 (as defined in Exercise 12.20), then C(X) contains an n-cell. [Hint: Start with the result in 12.20.1 Remark. Two applications of 14.18 are in 14.19 and 14.20. The result in 14.18 is due to Rogers [31, p. 1771. The converse of 14.18 is true (see section 70). 14.19 Exercise. If a continuum, X, contains a decomposable continuum, then C(X) contains a 2-cell. [Hint: Prove that any decomposable continuum contains a 2-od, and apply 14.18.1 Remark. The converse of 14.19 is true; we will prove this in 18.8. We also note that 14.19 implies that dim[C(X)] 2 2 when X contains a decomposable continuum; we will prove in 22.18 that dim[C(X)] 1 2 for any nondegenerate continuum X; we will prove in 73.9 that dim[C(X)] = 00 when dim(X) 2 2 (see 72.5).
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IV. ARCS IN HYPERSPACES
14.20 Exercise. If a continuum, X, contains two continua, A and B, such that An B has at least n components for some n < 00, then C(X) contains an n-cell. [Hint: Prove that X contains an n-od, and apply 14.18.1 14.21 Exercise. Let X = Y x 2, where Y and Z are nondegenerate continua. Then, C(X) contains a Hilbert cube (and, thus, C(X) is infinitedimensional). 14.22 then the 11.7, and [Hint:
of X, Exercise. If X is a continuum and K is a subcompacturn containment hyperspace CK(X) is an AR. (Cf. Exercises 11.6, 14.23.) Use 14.6 to show that the result in Exercise 11.8 applies.]
14.23 Exercise. for 2; is false.
Give an example to show that the analogue of 14.22
14.24 Exercise. Let X be a continuum, and let K be a subcontinuum of X. Then the following three statements are equivalent: (1) Cl<(X) is an arc; (2) CK(X) is an order arc; (3) there is only one order arc in C(X) from K to X. Prove that the statements above are equivalent. Determine which subcontinua, K, of X satisfy the statements when X = [0, I] and when X is a noose. Remark.
In relation to 14.24, see Exercise 15.13.
14.25 Exercise. A continuous function, f, from Y into Z is called an open map provided that for any open set, U, in Y, f(U) is open in f(Y). (1) If X is a continuum and w is a Whitney map for C(X), then w is an open map. (2) A Whitney map for 2x may not be an open map: Let X = { (2, y) E R2: max{]z], Iv]} = 1 an d 1‘f a:=ltheny;}withthe Euclidean metric d, and let w be the Whitney map in Exercise 13.5. Remark.
More about open Whitney
maps is in section 24.
14.26 Exercise. Give an example of a continuum, X, and a subcontinuum, A, of X such that C(X) - {.4} is not arcwise connected. On the other hand, prove that for any continuum X and any A E C(X), C(X) - {A} is connected.
15. EXISTENCE
OF AN ORDER
ARC FROM A0 TO A1
119
Remark. A much stronger result than the second part of 14.26 is in [17]. In relation to 14.26, see Exercises 15.16, 15.17, and 18.9.
15.
Existence
of an Order
Arc from
A0 to Al
We give a useful, necessary and sufficient condition for there to be an order arc in 2,’ between two g’rven points of 2*’ (15.3). We also obtain two definitive theorems about homogeneoushyperspaces as applications. 15.1 Terminology. When we say that an order arc, a, is an order arc finm A0 to Al, we mean that A0 c ‘41 and, as for any arc, A0 and A1 are the end points of Q. (Thus, an order arc from A0 to Al is not an order arc from Al to A,; of course, an order arc from A0 to A1 is still an arc from AI to Ao.) Let -40, Al E C(X) such that A0 # Al. We already know a necessary and sufficient condition for there to be an order arc in C(X) from A0 to Al: The condition is, simply, that A0 C Al (14.6 and 15.1). However, when Ao, Al c zx, this condition is not (always) sufficient for there to be an order arc in 2.x from A0 to A,; for example, it is easy to see that there is no order arc in 21°,11from { 0) to (0, 1).
Necessary
and Sufficient
Condition
A necessary and sufficient condition for there to be an order arc in 2x from A0 to A1 can be discovered by carefully analyzing the example just mentioned. The condition is in (2) of 15.3. It is also in the following lemma, which we use in proving 15.3. 15.2 Lemma. Let X be a compactum, and let MO, MI E 2.’ such that AToc Ml, h/r, # Ml, and each component of Mr intersects Me. Then there exists C E 2” such that MO c C c Ml, MO # C # MI, and each component of C intersects MO. Proof. Since Ml c Me, there exists p E Ml - MO. Let K1 be the component of Ml containing p. Then, by an assumption in our lemma, K1 n MO # 0. Thus, since K1 is a component of MI and MI > MO, we see that K1 contains a component, Ko, of MO. Note that KO c KI - {p} and that KO is a proper subcontinuum of the continuum Kr . Therefore, by the second part of 12.12, there is a subcontinuum, B, of K1 - {p} such that B > KO and B # Ko. Now, let
C=M,,uB.
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IV. ARCS IN HYPERSPACES
We show that C has the properties in the conclusion of our lemma. Obviously, C E 2x and MO C C c MI. To see that MO # C, recall that Ke is a component of MO and that B is a continuum that properly contains Ke; hence, B $ZMO and, therefore, C # MO. To seethat C # MI, simply note that p E MI and p $! C. Finally, to prove that each component of C intersects MO, let L be a component of C. If L II B = 0, then L c MO and, hence, L O MO # 0. So, assumethat L f~ B # 0. Then, since B is a connected subset of C, we seethat L > B. Thus, since B > Ko, L > Ko. Therefore, since KO c MO and KO # 0, we again have that L ~7MO # 0. l We now prove our theorem. 15.3 Theorem. Let X be a compactum, and let Ao, AL E 2x such that A0 # Al. Then, (1) and (2) below are equivalent: (1) there is an order arc in 2x from A0 to A,; (2) A0 C Al and each component of Al intersects Ao, Proof. Assume that (1) holds. Then there is an order arc, (Y, in 2x from A0 to Al. Clearly, As C Al (by 15.1). We now prove that the second part of (2) holds. Suppose, to the contrary, that there is a component, K, of Al such that K n A0 = 0. Then, since A0 c Al, we see by 12.9 that K and A0 are separated in Ar. Hence, by definition, there are E and F such that Al = E]F with A0 c E, K C F. Let & = {A E ~1:: A c E} and let 3 = {A e a : A n F # 0). Note that & and 3 have the following properties: & # 0 (since A0 E E); F # 0 (since Al E 3); E u 3 = CY(since for all A E a, A C Al = E U F); & n 3 = 0 (since E n F = 0); and, finally, & and 3 are each closed in o (since E and F are closed in X by 1.1). Hence, o is not connected. This is a contradiction (since a: is an arc). Thus, each component of Al must intersect Ao. Therefore, we have proved that (1) of our theorem implies (2). Next, assumethat (2) of our theorem holds. Let I? =
{hi c 2x : N is a compact nest from A0 to A1 and, for any N E N, each component of N intersects Ao}.
Then, r # 0 (since {Ao, Al} E I’ by (2)); also, I’ is a closed subset of 22x (by an easy sequence argument). Therefore, there is a maximal member, M, of I’ (by the Maximum-Minimum Theorem in Exercise 13.11 with X replaced by 2x). We show that (1) of our theorem holds by showing that
NECESSARY
AND SUFFICIENT
CONDITION
121
M is an order arc in 2x from A0 to Al. Since M is a nest in 2x from A0 to AI, we only need to show that M is an arc (14.1). Let w be a Whitney map for 2x (20exists by 13.4), let to = w(Ao), and let tr = w(Al). Since M is a nest from A0 to Al, we see from (1) of 13.1 that to < tr and that w(M) c [to, tl]. Also, note that M is compact since M E r. Therefore, to prove that M is an arc, it suffices by 14.2 to prove that w(M) = [to, tl]. Suppose that w(M) # [to, tl]. Then, since to, tl E w(M) and since w(M) is a compact subset of [to, tl], there exist SO,sr E w(M) such that se < s1 and w(M) n ( se, sl) = 0. Let I&, Mr E M such that w(Mc) = se and w(Mr) = sr. We show that MO and Ml satisfy the hypotheses of 15.2. Since SO< sr, clearly MO # Ml and, by (1) of 13.1, MI $Z MO; thus, since M is a nest, MO c MI. To prove the last hypothesis of 15.2, let L be a component of Ml. Then, MI E M E r, L n A0 # 0; also, since MO E M E r, AO C MO. Hence, L f~ MO # 0. Therefore, we may apply 15.2 to obtain C E 2x such that C has the properties in (a)-(c) below: (a) MOCCC
MI; (b) MO # C # MI;
(c) each component of C intersects A4c. We will use (a), (b), and (c) to obtain a contradiction; namely, we will show thatCEMandC$M. We show that C E M. We do this by showing that M U {C} E l? (and then applying the maximality of M). Since M E r and C E 2”, the proof that M u {C} E r amounts to proving that C satisfies three conditions: (i) M c C or C c M for each M E M; (ii) Ao C C C AI; (iii) each component of C intersects Ao. To prove (i), let M E M. Then, w(M) 5 SO or w(M) 2 sr. Thus, since M is a nest, we see using (1) of 13.1 that M c MO or M > Ml. Hence, by (a), M c C or C C M. This proves (i). To prove (ii), recall that MO, Ml E M E r; hence, AO c MO and MI c Al. Thus, (ii) now follows immediately from (a). Finally, to prove (iii), let Q be a component of C. Then, by (c), QnMo # 0. Thus, since Q is a component of C and C > MO (by (a)), Q must contain a component, Qo, of MO. Since MO E M E r, Qc n A0 # 0. Thus, since Q 2 Qo, Q fl A0 # 0. This proves (iii). Now, having proved (i)-(iii), we have that M U {C} E I’. Therefore, since M is a maximal member of I?, C E M. However, C $ M since w(M) n (~0,s~) = 0 and since so < w(C) < s1 by (a), (b), and (1) of 13.1. The contradiction that we just obtained came from our supposition that w(M) # [to, tl]. Hence, w(M) = [to, tl]. This completes the proof that M
122
IV. ARCS IN HYPERSPACES
is an order arc in 2x from A0 to Al. Therefore, the theorem implies (1). n
we have proved that (2) of
We use the following descriptive terminology in the corollary below. Let CYbe an order arc in 2x from A* to Al, and let ?-f c 2x. We say that a: beginsin 3t if A0 E ?i, after which we say that LYstays in 7-i if cr c Ifl. 15.4 Corollary. Let X be a compactum, and let CYbe an order arc in 2.‘. If cy begins in C(X), then o stays in C(X). Proof. By assumption, (Y is an order arc in 2x from A0 to Ai, where >40E C(X). Let B E o such that B # Ao. Let p denote the subarc of cr from A0 to B. Note that p is an order arc and that A0 c B; thus, p is an order arc from A0 to B (15.1). Hence, by 15.3, each component of B intersects Ao. Thus, since A0 is a connected subset of B, we seethat B has only one component; in other words, B is connected. Hence, B E C(X). Therefore, we have proved that o c C(X). n
A natural generalization of 15.4 is in Exercise 15.11. We note the following simple consequenceof 15.3 and 15.4; we will use it to determine those continua, X, for which 2x and C(X) are homogeneous. 15.5
Corollary.
For any continuum X, 2x and C(X) are locally arc-
wise connected at X. Proof. Let d denote a metric for X. Let E > 0. Let A0 E 2x such that A0 # X and Hd(Ao,X) < &. By 15.3, there is an order arc, cr, in 2.’ from A0 to X; note that if A0 E C(X), then Q c C(X) by 15.4. It follows easily from the formula for Hd in 2.1 that H,j(A, X) < E for all A E CLn
Application:
Homogeneous
Hyperspaces
A topological space, Y, is said to be homogeneousprovided that for any p, q E Y, there is a homeomorphism, h, of Y onto Y such that h(p) = q. For example, any n-sphere is homogeneous(use rotations) and any topological group is homogeneous (use translations). On the other hand, no n-cell is homogeneous(by Invariance of Domain [ll, p. 951); nevertheless, the Hilbert cube is homogeneous[14] (a much stronger result is in 11.9.1). In the next two theorems, we obtain complete characterizations for the homogeneity of 2” and C(X) when X is a continuum. We then briefly discussthe situation when X is a compacturn. Our discussion leads to the corollary in 15.8.
APPLICATION:
HOMOGENEOUS
HYPERSPACES
123
15.6 Theorem. Let X be a nondegenerate continuum. Then, (l)-(3) below are equivalent: (1) 2x is homogeneous; (2) X is a Peano continuum; (3) 2x is a Hilbert cube. Proof. Assume that (1) holds. Then, by 15.5, 2x is a Peano continuum (2” being a continuum by 14.10). Hence, (2) holds by Exercise 15.10. Next, (2) implies (3) by 11.3. Finally, (3) implies (1) by [13] (or by 11.9.1). n 15.7 Theorem. Let X be a nondegenerate continuum. Then, (l)-(3) below are equivalent: (1) C(X) is homogeneous; (2) X is a Peano continuum with no free arc; (3) C(X) is a Hilbert cube. Proof. Assume that (1) holds. Then, by 15.5, C(X) is a Peano continuum (C(X) being a continuum by 14.10). Hence, X is a Peano continuum by Exercise 15.10. To prove that the second part of (2) holds, suppose to the contrary that there is a free arc, A, in X. Then, C(A) has nonempty interior in C(X) and C(A) is a 2-cell (by 5.1). Hence, by (l), every point of C(X) has a 2-cell neighborhood in C(X). Thus, X does not contain a simple triod (since if there were a simple triod in X, C(X) would contain a 3-cell, G, by 5.4 - hence, by [ll, p. 951,the points of G could not have a 2-cell neighborhood in C(X)). We have shown that X is a (nondegenerate) Peano continuum that does not contain a simple triod. Hence, X is an arc or a simple closed curve (8.40 of [28, p. 1351). Thus, by 5.1 and 5.2, C(X) is a 2-cell. However, a 2-cell is not homogeneous(as noted preceding 15.6); hence, we have a contradiction to our assumption that (1) holds. Therefore, there is no free arc in X. This completes the proof that (1) implies (2). Finally, (2) implies (3) by 11.3, and (3) implies (1) by [13] (or by 11.9.1). H We remark that 15.6 and 15.7 do not generalize to Hausdorff continua (Exercise 15.19). Let us consider for a moment the homogeneity of 2x and C(X) when X is a compacturn. The situation is quite different than when X is a continuum. In particular, there are nonlocally connected compacta, X, such that 2x and C(X) are homogeneous. For example, let X be a Cantor set. Then, 2x is a Cantor set by 8.1 (use 8.3 and 8.6); also, C(X) is a Cantor set (since C(X) = X). Therefore, since a Cantor set is a topological group [15, p. 1661,2x and C(X) are topological groups; hence, 2x and C(X) are homogeneous.
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IV. ARCS IN HYPERSPACES
We have just seen that 2x and C(X) can be topological groups when X is a nondegenerate compacturn. Our final result showsthat this can not happen when X’ is a nondegenerate continuum: 15.8 Corollary. If X is a nondegenerate continuum, then neither 2x nor C(X) is a topological group. Proof. Since topological groups are homogeneous, we see from 15.6 and 15.7 that we only need to show that the Hilbert cube is not a topological group. Note the following two facts: On the one hand, the Hilbert cube has the fixed point property (21.4); and, on the other hand, a nondegenerate topological group does not have the fixed point property (translation by an element of the group different from the identity element is a fixed point free map). Therefore, the Hilbert cube is not a topological group. n
Original
Sources
The theorem in 15.3 and the corollary in 15.4 are reformulations of results about “segments” due to Kelley [16]. We discuss Kelley’s notion of a segment in the next section (in particular, 16.7 and 16.8 are Kelley’s versions of 15.3 and 15.4). The result in 15.5 is from [7, p. 10321for C(X) (also, seethe proof of Lemma 2 of 191);the analogue of 15.5 for Hausdorff continua is in 1.136 of [29, p. 1541. The homogeneity theorems in 15.6 and 15.7 are from [29, pp. 564-5651.
Exercises 15.9 Exercise. Prove the two results below for usein the next exercise. (1) Let X be a continuum, and let Al,. . , A, be finitely many subcontinua of X. Then, (A,, . . . , A,) and (Al,. . . , A,) II C(X) are arcwise connected continua (if they are nonempty). (2) Let X be a continuum, and let A be a subcontinuum of 2x. If A II C(X) # 0, then Ufi is a subcontinuum of X. 15.10 Exercise. The following three statements are equivalent: (1) X is a Peano continuum; (2) C(X) is a Peano continuum; (3) 2x is a Peano continuum. Prove the equivalences using 15.9; do not use 10.8 or 11.3 - using them would be overkill. It is helpful to remember that continua that are cik at every point are Peano continua (seecomments following the proof of 10.7). Remark. The equivalences in 15.10 are due to Vietoris [34] and Wazewski [37]. The equivalencesare the first of several progressively strong-
EXERCISES
125
er results leading to the Curtis-Schori Theorem cussion is in the introduction to Chapter III.
(11.3) -
a historical
dis-
15.11 Exercise. For any compacturn Y, let c(Y) denote the cardinality of the set of all components of Y. Prove the following generalization of 15.4: Let X be a compactum, and let Q be an order arc in 2x. If A, B E (Y and A c B, then c(B) 5 c(A). 15.12 Exercise. Let X be a continuum. Then, for each n = 1,2,...,(A E 2x: c(A) 5 n) is an arcwise connected continuum (notation from 15.11); also, {A E 2x : c(A) < No} and {A E 2x : c(A) 5 NO} are arcwise connected. On the other hand, give an example of a continuum, X, such that {A E 2x : c(A) = n } is not arcwise connected for any n 2 2. 15.13 Exercise. Let X be a continuum, and let A E 2”. Then there are none, only one, or uncountably many order arcs in 2x from A to X. Furthermore, if there is only one order arc in 2x from A to X, then A E C(X). (See Exercise 14.24.) 15.14 Exercise. Let X be a continuum, let w be a Whitney map for 2x or for C(X), and let t E [O,w(X)]. Also, Then, w-l ([t, w(X)]) is an arcwise connected continuum. w-‘([O,t]) is a continuum when w is a Whitney map for C(X); however, w-l ([0, t]) may fail to be a continuum when w is a Whitney map for 2x (e.g., show this using X and w in (2) of Exercise 14.25). Remark. in 19.9.
An important
application
of the results for C(X)
15.15 Exercise. Let X be a compactum, 2”(K) and C(X; I<) as follows:
and let K E 2x. We define
We call 2”(K) and C(X; K) intersection hyperspaces. (1) If X is a continuum and K E 2x, then 2x(K) arcwise connected continua. (2) If X is a Peano continuum are absolute retracts.
in 15.14 is
and C(X; K) are
and K E 2x, then 2”(K)
and C(X; K)
126
IV. ARCS IN HYPERSPACES
(3) If X is a Peano continuum and K E 2”, then 2”(K) is the Hilbert cube; furthermore, if there is no free arc in X, then C(X; K) is the Hilbert cube. [Hint: For (2), use (1) and Exercise 11.8. For (3), use (2) and 9.3 (see second part of hint for Exercise 11 .S) .] Remark. Note that (3) of 15.15 immediately implies the first two parts of the Curtis- Schori Theorem in 11.3 (take K = X). The result in (3) of 15.15 is a special case of Theorem 5.2 of [5, p. 1611. 15.16 Exercise. Let Y be an arcwise connected space, and let p E Y; we say that p arcwise disconnects Y provided that Y - {p} is not arcwise connected. Let X be a continuum, and let E E 2x. If E arcwise disconnects 2x, then E E C(X). Remark. 18.9.
In relation
to 15.16, see the exercises
in 14.26, 15.17, and
15.17 Exercise. Let X be a continuum, and let E be a nondegenerate, proper subcontinuum of X; we say that E is buried in X provided that whenever A is a subcontinuum of X such that AnE#0andAn(X-E)#0, then A > E. If X is a continuum and E is buried in X, then E arcwise disconnects 2x and C(X). [Hint: Let f : [0, l] + 2x be continuous such that f(0) E C(E) - {E} and f(l) $ E. Let to = lub{t 2 1 : f(t) c E}. For any s > to, consider
{uf@o,tl) : to I t L sJ.1 Remark. The converse of the result in 15.17 is false, but the converse is true when E is decomposable (4.4 of [27]). We also note that E arcwise disconnects 2x if and only if E arcwise disconnects C(X) [27, p. 191. For more results, see [27, pp. 15-241 or [29, pp. 357-3721; one of these results is in Exercise 18.9. Our terminology buried in X has not been used before. However, the notion itself is old: Sets we call buried, except that they were not assumed to be continua, were called C-sets (in, e.g., [35]). Probably the C in C-set was intended to refer to the word composant - composants of indecomposable continua were important examples of C-sets [35].
16. KELLEY'S SEGMENTS
127
15.18 Exercise. Let Y be a space, let Z c Y, and let p E 2; we say that p is arcwise accessible from Y - Z provided that there is an arc in (Y - 2) u {P> h aving p as an end point. Let X be a continuum, and let A E C(X). Then, A is order arcwise accessiblefrom 2x - C(X) if and only if A # Fr(X). Remark. The result in 15.18 shows that each A E C(X) - Fl(X) is arcwise accessiblefrom 2.’ - C(X); nevertheless, a singleton may not be arcwise accessiblefrom 2x - C(X) (for examples, see [27, pp. 10 and 131 or [29, pp. 378 and 3831). 15.19 Exercise. Show that our theorems about homogeneous hyperspaces in 15.6 and 15.7 do not generalize to Hausdorff continua by giving examples of the following: A locally connected Hausdorff continuum, S, such that 2x is not homogeneous;a locally connected Hausdorff continuum, X, with no free generalized arc such that C(S) is not homogeneous.
16.
Kelley’s
Segments
We define Kelley’s notion of a segment [16], and we establish the basic properties of segments. This provides a foundation for the next section, where the results and ideas that we present here take on new life in a dynamic setting.
Kelley’s
Notion
of a Segment
There is a natural and especially useful way to parameterize any order arc. The method involves Whitney maps. Let us see how it is done. This will lead to Kelley’s notion of a segment in 16.1. Let X be a compactum, and let w be a Whitney map for 2”. Let cy be an order arc in 2x from A0 to Al. Let ae = w(AO) and let al = w(Al). Then, W((Yis a homeomorphism of 0: onto [aa,ar] (by 14.2). Let X denote the linear map of [0, 11onto [a~, al] given by x(t) = (1 - t) . a0 + t. al for all t E [O,l]. Finally, define the parameterization u from [O, l] onto cy by letting 0 = (wla)-’
0 A.
Now, having defined the parameterization U, let us determine some of the elementary properties of o. Our first two properties of 0 are rather evident: o is a homeomorphism of [0, l] onto cu; a(0) = A0 and o(l) = A1 (since a(O) = (~]a)-‘[X(O)] = (wlcr-‘(ao) = A 0, and similarly for a(l)).
128
IV. ARCS IN HYPERSPACES
Our next property of CJis that ‘w linearizes c - explicitly, w o 0 = A (which is obvious). We prefer to express the fact that w o c = X without mentioning XT as follows: for each t E [0, 11, w[a(t)]
= (1 - t> . w[a(O)]
+ t . w[a(l)].
The right-hand side of the equality is, indeed, A(t) since w[a(O)] as and w[o(l)] = w(A1) = al. Our final property of g is that ~7is order preserving: rr(ti)
c
o(tz) whenever
05
tl
= w(A,,)
=
<_ t2 <_ I.
This property of 0 is due to the fact that X and lu]~y are order preserving (recall (1) of 13.1). We are now in a position to define Kelley’s notion of a segment and, at the same time, to know that segments are designed to provide a special parameterization for every order arc. (On comparing the definition below with what we have done, we see that we do not require segments to be homeomorphisms: We want constant maps into hyperspaces to be segments when we consider spaces of segments in section 17. We will prove that nonconstant segments are homeomorphisms in 16.3.) 16.1 Definition. Let X be a compactum, let 31 c 2x, and let w be a Whitney map for 31. Let Ao, Al E Z. A function (7 : [0, 11 + 2 is said to be a segment in ‘l-f with respect to w from A0 to Al provided that D has the following four properties: (1) g is continuous; (2) u(0) = Ao and g(l) = AI; (3) w[u(t)] = (1 - t) . w[g(O)] + t. w[(~(l)] for each t E [0, l] (linearity); cr (t) 2 w h enever 0 5 ti 5 t2 5 1 (order preserving). (4) 4t11c When there will be no confusion, we will sometimes not refer to w: We will simply say that (T is a segment in ‘H from A0 to A1 ; however, we must be careful (see Exercise 16.12).
Results
about
Segments
Using the terminology in 16.1, the following we did at the beginning of the section.
theorem
sumarizes
what
16.2 Theorem. Let X be a compactum, let fl C 2”, and let w be a Whitney map for 5% Then every order arc in ?i is the range of a segment with respect to w.
RESULTS ABOUT SEGMENTS
129
Proof. The development that precedes16.1 is not affected by replacing 2x with ?l and assuming that w is a Whitney map for 3t. n
Next, we show how close we came to obtaining all segments at the beginning of the section. In fact, the next theorem showsthat we obtained all segmentsexcept for the constant maps of [0, l] into 2” (each such constant map is obviously a segment by 16.1). First, we prove the following lemma. 16.3
Lemma.
Any nonconstant segment is a homeomorphism.
Proof. Let X be a compactum, let ?l c 2x, and let w be a Whitney map for 3t; let 0 : [0, l] + 8 be a nonconstant segment with respect to w. By (1) of 16.1, g is continuous. Hence, it suffices to prove that g is one-to-one. Suppose, to the contrary, that u is not one-to-one. Then there are s, t E [0, l] such that s # t and U(S) = u(t). Thus, w[g(s)] = wig(t)]. Hence, by applying (3) of 16.1 to w[g(s)] and w[a(t)], it follows easily that
(t - s) . w[u(O)] = (t - s) . w[u(l)]. Thus, w[g(O)] = w[g(l)]. Also, by (4) of 16.1, a(O) c a(l). Hence, a(O) = a(l) by (1) of 13.1. Thus, it follows from (4) of 16.1 that 0 is a constant map. This is a contradiction. Therefore, 0 is one-to-one. n 16.4 Theorem. Let X be a compactum, let ?t c 2x, and let w be a Whitney map for 7-1. Let u : [0, l] -+ ?i be a nonconstant segment with respect to w from A0 to AI, and let a = U( [0, 11). Then, (Y is an order arc from A0 to Al and 0 = (w]o)-’ o X, where X is the unique increasing linear map of [0, l] onto w(o) = [w(Ao), w(Al)]. Proof. We first show that Q is an order arc from A0 to Al. By 16.3, D is a homeomorphism of [0, l] onto Q. Hence, a: is an arc with end points a(O) and a(1). Furthermore, by (4) of 16.1, a is a nest. Thus, by the definition in 14.1, Q:is an order arc. Since a(0) and o(l) are the end points of a:, we seefrom (2) of 16.1 that these end points are A0 and A,; also, by (4) of 16.1, A0 C Al. Therefore, Q is an order arc from A0 to A1 (cf. 15.1). Finally, we show that cr = (~]a)-’ o X, where X is as in the statement of the theorem. Note that X has the following formula:
X(t) = (1 - t) . w(Ao) + t. w(Al)
for each t E [0, 11.
Hence, by (2) and (3) of 16.1, we see that w o c = X. Now, note that since Q is an order arc from A0 to Al, W]CYis a one-to-one map of Q: onto
130
IV. ARCS IN HYPERSPACES
[ur(Ao),w(-41)] by 14.2. Thus, (w]&)-’ is well defined [w(Ao), UJ(A~)]. Th ere , f ore, since ?u 0 0 = X, clearly 0 = (w]o)-’
on all of 0 X. n
We prove two corollaries. The first corollary characterizes the ranges of segments; the second corollary is a uniquenessresult. Let X be a compactum, let 7-t c 2.Y, and let w be 16.5 Corollary. a Whitney map for 3c. A subset, S, of ‘U is the range of a segment with respect to w if and only if S is an order arc or S E FI (‘l-l). Proof. The “if” part is by 16.2 and the fact that every constant map of [0, I] into R is a segment with respect to w. The “only if” part is due to
16.4. W 16.6 Corollary. Let X be a compactum, let 7-i c 2”, and let 1~ be a Whitney map for ?i. Then a segment in 7-l with respect to w is uniquely determined by its range; in other words, if crl and LT~are segments in ‘H with respect to w such that c~i([O, 11)= ~2([0, l]), then ~1 = ~72. Proof. Let gi and 02 be as above, and let cy = oi([O, 11) = 02(\0, I]). Then, for each i = 1 and 2, o2 = (w]o)-’ o X as in 16.4 (with the proviso that X is constant if 0 is constant). Therefore, 01 = CTZ.n
In section 15, we obtained two important results about order arcs ~15.3 and 15.4. It is appropriate at this time to formulate these results in terms of segments. 16.7 Theorem. Let X be a compactum, and let w be a Whitney map for 2”. Let Ao, A1 E 2x. Then, (1) and (2) below are equivalent: (1) there is a segment with respect to w from A0 to A1 ; (2) A0 C A1 and each component of Al intersects Ao.
Obviously, (1) and (2) are both true if ‘40 = Al. So, assume from now on that A0 # Al. Then, by 15.3, (2) of our theorem is equivalent to the following statement: (*) there is an order arc, o, in 2x from A0 to Al. Thus, it suffices to prove that (1) of our theorem is equivalent to (*). Assume that (1) of our theorem holds. Then, since AC, # AI, we see from 16.4 that (*) holds. Conversely, assumethat (*) holds. Then, by 16.2, there is a segment, U, with respect to w such that a([O, 11)= cy, where ai is as in (*). Once we show that a(O) = AC, and g(l) = A 1, we will know that (I) of our theorem holds (cf. (2) of 16.1). Note that 0 is a homeomorphism of [O,l] onto cr by Proof.
ADDENDUM:
EXTENDING
WHITNEY
MAPS
131
16.3. Hence, a(0) and ~(1) are the end points of a; also, u(0) c a(1) by (4) of 16.1. On the other hand, since a: is an order arc from A0 to Al, A0 and Al are the end points of (Y and A0 c Al (recall 15.1). Therefore, we seethat a(O) = A0 and u(1) = Al. n 16.8 Theorem. Let X be a compactum, let ‘Ii C 2x, and let w be a Whitney map for 7-L If CTis a segment in ‘?f with respect to w such that a(O) E C(X), then o([O, 11) C C(X). Proof. Since the theorem is obvious if (T is constant, we assumethat (T is not constant. Then, by 16.5, a([O,l]) is an order arc; furthermore, the end points of a([O, 11) are ~(0) and (~(1) by 16.3, and a(0) c ~(1) by (4) of 16.1. Hence, (T([O,11) is an order arc from a(O) to a(l) (recall 15.1). Therefore, by our assumption that o(O) E C(X), we have by 15.4 that u([O, 11) c C(X). m
Our final result is the analogue of 16.7 for C(X): 16.9 Theorem. Let X be a compactum, and let w be a Whitney map for C(X). Let A,J, Al E C(X). Then there is a segment with respect to w from A0 to Al if and only if A0 C Al. Proof. Assume that there is a segment, 0, with respect to w from A0 to Al. Then, by (2) and (4) of 16.1, A0 C AI. Conversely, assume that A0 c A1 and that A0 # Al (the result is trivial if A0 = Al). Then, by 14.6, there is an order arc, o, in C(X) from A0 to Al. Hence, by 16.2, there is a segment, 0, with respect to w such that o([O, 11)= cy. We seethat D is a segment from A0 to Al by using 16.3 and (4) of 16.1 (as in the last part of the proof of 16.7). n
At first glance, one might suspect that 16.9 is a simple consequenceof 16.7 and 16.8. However, this is not the case (see Exercise 16.20).
Addendum:
Extending
Whitney
Maps
We stated most of the results in the section in terms of a Whitney map for ‘Fl, where 31 c 2x. On the other hand, the results are stated in the literature in terms of a Whitney map for 2x. This leads us to wonder if every segment in ‘H c 2x with respect to a Whitney map for ?f is a segment with respect to a Whitney map for 2x. We show that this is true in the corollary to the following theorem. The theorem is of independent interest.
132
IV. ARCS IN HYPERSPACES
16.10 Theorem. If X is a compactum, then any Whitney map for any closed subset of 2x can be extended to a Whitney map for 2x. Proof. for ‘fl. Let
Let Y-l be a closed subset of 2x, and let w be a Whitney K = 31 u FI(X)
map
u {X}.
Extend w to a Whitney map, w’, for Ic by letting w’({z}) = 0 for all (2) E Fi(X) and by letting w’(X) = 1 + sup w(N) if X 4 ‘+i. Then, w’ can be extended to a Whitney map for 2x by Theorem 3.1 of Ward [36]. n 16.11 Corollary. Let X be a compactum, and let ?f c Zx. If 0 : [0, l] + 3c is a segment with respect to a Whitney map for 3t, then o is a segment with respect to a Whitney map for 2x (and conversely). Proof. Let w denote a Whitney map for ‘l-i such that IT is a segment in ?f with respect to w. Let 6 = o([O, l]), and let g = WIG. Note that G is closed in 2x and that g is a Whitney map for G. Hence, by 16.10, g can be extended to a Whitney map, g’, for 2x. Clearly, 0 is a segment with respect to g’. The converse half of the corollary is evident from the fact that the restriction of a Whitney map for 2x to 3t is a Whitney map for 2. n
Regarding the extension theorem in 16.10, seeExercise 16.19.
Original
Sources
The result in 16.3 is 1.18 of [29]; 16.5, which includes 16.2 and the first part of 16.4, is 1.4 of [27]; the second part of 16.4 is 1.31.2 of [29]; 16.6 is part of the proof of 1.30 of [29] (which we,prove in 17.5); 16.7 is 2.3 of [16]; and 16.8 is 2.6 of [16]. Regarding the way results are stated in the papers just cited, seethe addendum above.
Exercises 16.12 Exercise. Let X be a compacturn, and let w be a Whitney map for 2x. Then, w2 = w . w is a Whitney map for 2x such that the only segments in 2x that are segments with respect to both w and w2 are the constant segments. 16.13 Exercise. Let X be a nondegenerate compacturn. Then every segment in 2x (with respect to some Whitney map for 2x) is a segment with respect to uncountably many Whitney maps for 2x.
EXERCISES 16.14 Exercise.
133
Define 0 : [0, l] + C([O, 11) as follows: u(t) = [O,t’] for all t E [0, 11.
Find a formula for a Whitney with respect to w.
map, w, for C(]O, 11) such that LTis a segment
16.15 Exercise. Let X be a compactum, phism of [0, l] into 2x such that a(tr)
c
g(t2) whenever
and let (T be a homeomor-
0 5 tr 5 tz 5 1.
Then there is a Whitney map, w, for 2” such that 0 is a segment with respect to 1~. [Hint: Find a way to use 16.10.1 Remark.
The result in 16.15 is a converse to 16.3.
16.16 Exercise. Let X be a compactum, let 3-1c 2.Y, and let w be a Whitney map for Ifl. Let u be a segment in 3-1 with respect to w, and let to, tr E [0, l] such that to _< tl. Then there is a unique map, cp, of [0, l] onto [to, tr] such that 0 o ‘p is a segment with respect to w. 16.17 Exercise. Let X be a compactum, let ‘?i c 2x, and let w be a Whitney map for ?l. If u is a segment in ?t with respect to w such that a(te) E C(X) for some to E [O,l], then a(t) E C(X) for all t E [to, 11. 16.18 Exercise. Let X be a compactum, let w be a Whitney map for If A, B E w-‘(to) such that A n B # 0 and C(X), and fix t, < w(X). A # B, then there is an arc, a, in w-r (t,) from A to B; furthermore, given a component, K, of A n B, (Y can be chosen to lie in C(A U B) so that K c L for all L E cr. [Hint: The proof is a balancing act with two segments.] Remark. The result in 16.18 essentially proves that if X is an arcwise connected continuum, then w -l(t) is arcwise connected for all t. See Chapter VIII. 16.19 Exercise. Give an example of a continuum X, a subset U of C(X), and a Whitney map w for 3c such that w can not be extended to a Whitney map for C(X). (Cf. 16.10.) 16.20 Exercise. Show that 16.9 is a direct consequence of 16.7, 16.8, and 16.10. (Recall comment following the proof of 16.9.)
134
IV. ARCS IN HYPERSPACES
17.
Spaces of Segments,
S,(R)
We have studied segments - “one at a time” - in the preceding section; now, we form spaces of segments and investigate their properties. We present an application that is particularly noteworthy: It is the lovely theorem which says that for any continuum X, 2~~ and C(X) are continuous images of the cone over the Cantor set. We give the definition for spaces of segments in 17.1. First, let us recall some standard notation and terminology. Let 1’ be a compacturn, and let Z be a metric space with metric dz. Then, 2’ denotes the space of all continuous functions from 1” into 2; the topology for Zy is the topology that is induced by the uniform metric, p, which is defined as follows: for any f, g E Zy , P(f,g)
= SUP {dz(f (y),dy))
: Y E Y).
We use the term uniform topology to refer to the topology on Zy or any subspace of Zy. The uniform topology for Zy is the compact-open topology [6, pp. 2692711. Thus, the uniform topology does not depend on the choice of the metric that induces the given topology on Z; also, convergence of sequences with respect to the uniform topology is uniform convergence. 17.1 Definition. Let X be a compactum, let ‘H C Zx, and let w be a Whitney map for 3t. The space of segmentsin 3t with respect to w is the space
S,(Z)
= {CJE ?t[OJl : o is a segment with respect to w}
with the uniform topology (defined above). Note: We often write &,(?-I) without mentioning w beforehand; when we do, it is to be understood that w is an arbitrary, fixed Whitney map for X.
Compactness Our first theorem about spacesof segmentsconcerns their compactness. The theorem is in 17.4; its proof is based on the special caseof the Arzela-4scoli Theorem that we state in 17.2. First, we recall the definition of equicontinuity for maps between compacta. Let (Y,dy) and (Z,dz) be compact metric spaces, and let F C Zy; then, F is said to be equicontinuous provided that for each E > 0, there exists 6 > 0 such that if dy(ylryz) < 6, then &(fh),
f (yz)) < E for all f E .K
COMPACTNESS
135
It is easy to see that since Y and Z are compacta, the notion of equicontinuity depends only on the topologies on I’ and 2 (and not on the given metrics dy and dz). 17.2 Arzela-Ascoli Theorem (special case). Let E’ and 2 be compacta, and let F C 2’. Then, 3 is compact if and only if F is equicontinuous and F is closed in 2’.
Proofs of the Arzela-Ascoli Theorem are in many texts; for example, see [6, p. 2671or [15, p. 2331. We use the following lemma in the proof of our theorem about, compactness of spacesof segments. At first, the lemma may appear to have no direct bearing on the compactness of spaces of segments ~ however, the lemma essentially proves that spacesof segmentsare equicontinuous. 17.3 Lemma. Let X be a compactum, let ‘R be a closed subset,of 25, and let w be a Whitney map for 31. Then, for any given E > 0, there is an Q(E) > 0 with the following property: If A, B E 3c such that, A c B and Iw(B) - w(A)1 < T(E), then H(A,B) < E. Proof. Suppose that, the lemma is false for some particular E > 0. Then, for each i = 1,2,. . ., there exist Ai, Bi E fl such that. iii c B;. lw(Bi) - w(A,)I < t, and H(Ai, B,) 2 E. Since 7-t is a compacturn (by 3.1 and 3.5), we can assumethat the sequences{A,}z”=, and {Bi};M=, converge in 31 to, say, A and B, respectively. Then we seethat A C B and w(A) = w(B); hence, by (1) of 13.1, A = B. However, since H(Ai, Bi) 2 E for each i, we also have that H(A, B) > E. Therefore, we have a contradiction. n 17.4
Let X be a compactum, and let 3c be a closed subset is compact.
Theorem.
of 2X. Then, S,(Z)
Proof. We show that S,(R) is equicontinuous and closed in ~[‘,‘l; then our theorem follows from 17.2 (since Fl, being closed in 2x, is a compactum). To prove that S,(X) is equicontinuous, let E > 0. Let q(~) be as guaranteed by 17.3. Let s = sup w(R). Choose 6 > 0 such t,hat 6.9 < V(E). Now, let tl, ta E [0, l] such that It1 - tzl < 6, and let D E S,,(R). We see at once from (3) of 16.1 that w[a(h)]
-
w[c(h)]
=
(t2
-
tl)
. w[a(O)]
-
(t2
-
t1)
. w[a(l)].
Hence, I+(tl)l
-
wb(tz)]l
=
It;?
5
1t.L -
-
t1 I t1I
Iw[40)] . s <
6 . s <
~~[41)11 7](E);
136
IV. ARCS IN HYPERSPACES
also, by (4) of 16.1, o(tl) C g(t2) or a(t2) C cr(tl). Therefore, since Q(E) is as in 17.3, we have that H(~(tl), u(tz)) < E. This proves that S,(R) is equicontinuous. Next, we prove that S,(%) is closed in 31[‘~~]. Let j E ~c[OJI such that some sequence, {a,}&, in S,(R) converges in 7-1[OJ] to j. We prove that j E S,(Y). Since j E ~[Oy’l, we know that j is continuous. Hence, it remains to prove that j satisfies (3) and (4) of 16.1. To prove that j satisfies (3) of 16.1, fix t E [0, 11. To verify the equalities below, recall 4.8 and use the continuity of w, the pointwise convergence of {Ui}zl to f, and the fact that each oi satisfies (3) of 16.1: w[j(t)]
=
w[Lim
ai(
= lim W[Ui(t)]
=
lim (1 - t) . W[Ui(O)] + t. UJ[ni(l)]
=
(1 - t) . w[Lim
=
(1 - t) . w[j(O)]
ai(
+ t . w[Lim
ui(l)]
+ t. w[j(l)].
This proves that j satisfies (3) of 16.1. To prove that j satisfies (4) of 16.1, let tl, t2 E [0, l] such that t 1 _< t 2. Then, since each (T, satisfies (4) of 16.1, we have that ui(tl) c ai for each i. Thus, since f(tl) = Lim ai and j(t2) = Lim cri(tz) (by 4.8), it follows easily that j(tl) c j(t2). This proves that j satisfies (4) of 16.1. We have now proved that j E S,(R). Therefore, we have proved that S,(x) is closed in M”l’l. n 17.5
Corollary.
If X is a compactum, then S,(2”)
and S,(C(X))
are compact. Proof.
Apply 17.4 (which can be applied to S,(C(X))
by Exercise
1.20). w
Our next theorem gives us an alternative way to think about spacesof segments. As a consequence,we will seethat spacesof segmentsare, from a topological standpoint, independent of the choice of Whitney map (17.8). The key idea is in the following definition: 17.6 Definition. Let X be a compactum, and let ?-l c 2x. We define the space of order arcs in 3-1,denoted by O(‘?i), and the closed space of
137 order arcs in Z, denoted by a(Z), O(~)={aE2?
as follows: a it is an order arc in ‘?I)
and D(N)
= O(Z)
u Fl(?l),
where 0(X) and ??(a) each has the topology obtained from the Hausdorff metric for 2R. Note: There can not be any confusion between our abstract use of the bar in ?!?(?f) and our usual way of using a bar to denote closure; besides, as is easy to check, 8(R) is, indeed, the closure of 0(R) in 2R. 17.7
S,(X)
Theorem.
Let X be a compactum, and let Z C 2”.
Then,
= D(?i).
Proof.
Define the function fw on S,(R)
as follows:
fw(u) = u([O, 11) for each u E S,(X). By 16.5, fw maps S,(R) onto B(Z). By 16.6, f,,, is one-to-one. Next, we prove that f,,, is continuous. We use the following notation: H denotes the Hausdorff metric for Z; HH denotes the Hausdorff metric for 23t induced by H as in 2.1; and p denotes the uniform metric for S, (3t) that is defined preceding 17.1. We prove that fw is nonexpansive with respect to HH and p; explicitly, we prove that (*) HH(~uJ(~I)> fd”2)) 5 P( (~r,gs) for all 01, u2 E S,(Z). To prove (*), fix 01, ~2 E S,,(R). Let r > p(ar,as). Then, by the way p is defined, H(al(t), an(t)) < r for each t E [O,l].
Thus,n([O, 11)~NH(f,~2([0,11))
and
02([0,11)
c
Nd~,m([o,
11)).
Hence,
by the first part of Exercise 2.9,
Thus, we have proved that HH(~~((T~), fw(cn)) < T for any r > p(ar,az). Therefore, we have proved (*). Clearly, (*) implies that fw is continuous. Finally, we prove that f;’ : B(X) + S,(R) is continuous. Let a E D(X), and let cy( E B(X) for each i = 1,2,. . . such that the sequence {oi}gr converges in ??(‘?I) to a. Let (T= f;‘(cu), and let oi = f;‘(ai) for each i. We prove the following fact: (#) There is a compact subset, C, of &,(‘?I) such that u E C and gi E C for each i.
138
IV. ARCS IN HYPERSPACES
To prove (#), let %’ = (Y U (U&ai), Whitney map for 3t’), and let
let w’ = ~1%’ (note that w’ is a
c = s,!(w); we show that C satisfies the conditions in (#). An easy sequenceargument shows that U’ is compact; hence, by 17.4, C is compact. Since 20’ = wllfl’ and 31’ C ?t, we seeeasily that c = {a’ E S,(U)
: o’([O, 11) c U’};
thus, C c SW(Y), cr E C, and pi E C for each i. This completes the proof of (#I. NOW, we use (#) to show that {f;‘(~i)}~r converges in S,,,(‘?Q to f;‘(a). Recall th at we have proved that fw is one-to-one and continuous on S,,,(U);hence, by (#), we have that (1) fw]C is a homeomorphism. Next, recall that o = f;‘(a) and pi = f;l(czi) for each i; hence, by (#), we have that (2) cr E fw(E) and cyi E fw(C) for each i. Finally, recall that {ai)E1 converges to a; hence, by (1) and (a), {f,~‘(cy~)}~~ converges in C to f;‘(a). Thus, since C C S,(x) by (#), we have shown that {f;l(cyi)}p”,I converges in S, (3t) to f;’ (a). Therefore, we have proved that f;’ is continuous. m Usually, SW,(Ufl) # SW,CW w hen w1 # w2 (e.g., see Exercise 16.12). Nevertheless, we have the following result: 17.8 Corollary. Let X be a compactum, and let ‘fl c 2x. Then, for (‘IY) zz SW,(31). any two Whitney maps wi and w2 for ?i, SW, Proof.
By 17.7, S,, (31) = 8(R)
S,(Zx),
S,(C(X))
When
and S,,(x)
M B(x).
X Is a Peano
n
Continuum
We prove a theorem about spacesof segments that is analogous to the Curtis-Schori Theorem in 11.3. Let X be a nondegenerate Peano continuum. Then (1) S,(2x) is the Hilbert cube, (2) &,(C(X)) is the Hilbert cube when there is no free arc in X, and (3) S,(C(X)) is an AR (i.e., a Hilbert cube factor by 11.4). 17.9
Theorem.
S,(Zx),S,(C(X))
WHEN X Is A PEANO CONTINUUM
139
Proof. By 17.7, we may as well prove the theorem for ??(2x) and n(C(X)); we base the proof on Toruriczyk’s Theorem in 9.3; we use ideas in section 11 to verify the assumptions in 9.3. The proof that ?!?(2x) and B(C(X)) are absolute retracts uses Exercises 17.13-17.15 and proceeds as follows. By Exercise 17.13, 8(2x) is a Peano continuum; hence, by 10.8, 2°(“x) is an AR. Thus, by Exercise 17.14, n(2*‘) is an AR. Therefore, since C(X) is a retract of 2x by 10.8, we see from Exercise 17.15 that ??(C(X)) is an AR. Next, we prove that the identity map on 8(2x) and the identity map on D(C(X)) is, in each case, a uniform limit of Z-maps. For use in the proof, we devote the next two paragraphs to obtaining results that are analogous to 11.2. For any closed subset, K, of X such that K” # 8, let (cf. Exercise 14.16) a,(2X)
= {a E a(29
: I-ICI > K}
(recall 14.16). We show that ??~(2~) is a Z-set in 8(2x). Let +z > 0. Then, by 11.2, there is a continuous function fc : 2x + 2x - 2$ such that fc is within e of the identity map on 2x. Now, let f: : 22x + 22x be the maps natural induced map in 13.3. Then, it follows easily that f:18(2x) ‘i7(2x) into ??(2x) -am and that fc,‘??(zx) is within E of the identity map on n(2x). This proves that am is a Z-set in ??(2x). For any closed subset, K, of X such that K” # 0 and such that I< contains no free arc in X, let D&C(X))
= {a E B(C(X))
: na > K}.
is a Z-set in fl(C(X)); the proof is similar to the proof Then, ~K(C(X)) for ??~(2~) in the preceding paragraph. Finally, we prove that the identity maps on 8(2x) and ??(C(X)) are uniform limits of Z-maps. We use the results that we just obtained to adapt the proof of 11.3 to the present situation. Let E > 0. Replace @‘e : 2x -+ 2x in the proof of 11.3 with @: : a(ax) --+ a(2x) given by @):(a) = {C~(E, A) : A E a} for each a E n(2”); replace 2;. in the proof of 11.3 with DK, (2x). Then, by making the obvious changes in the proof of 11.3, we see that G: is a Z-map and that @z is within e of the identity map on a(2x). Also, replacing CK,(X) in the proof of 11.3 with CK, (C(X)), we see as in the proof of 11.3 that +:(??(C(X)) is a Z-map of L?(C(X)) into a(C(X)) and that G:la(C(X)) is within E of the identity map on D(C(X)). n An example of a Peano continuum, X, such that &,(C(X)) is not a Hilbert cube is in Exercise 17.16 (the example is from [8, pp. 119-1201).
140
IV. ARCS IN HYPERSPACES
For a complete see [4].
Application:
characterization
of when
Mapping
S,(C(X))
the Cantor C(X)
is the Hilbert
Fan Onto
cube,
Zx and
The term Cantor fan is the usual name given to the cone over the Cantor set. We prove the theorem in 17.10. The Cantor fan is an arcwise connected continuum, and arcwise connectedness is preserved by continuous functions (8.28 of [28, p. 1331). Thus, 17.10 implies the result in 14.9 that 2x and C(X) are arcwise connected for any continuum X. Moreover, 17.10 is significantly stronger than 14.9: There are arcwise connected continua that are not continuous images of the Cantor fan (see Exercise 17.18 for an example). 17.10 Theorem. If X is a continuum, uous images of the Cantor fan.
then 2x and C(X)
are contin-
Proof. Let K denote the Cantor set, and let F = Cone(K) (the Cantor fan). We will use the following well-known result (an especially simple proof of the result is in [28, p. 1061): (*) Every compacturn is a continuous image of K. We first prove our theorem for 2x. We obtain a map, g, of K x [0, l] onto 2x, and then we use the quotient map of K x [0, l] onto F to complete the proof. Let w be a Whitney map for 2x (13.4). Let c = {CT E &(2X) Clearly, C is closed in S,(2x). there is a continuous function, as follows: g(z, t) = [f(z)](t)
: a(1) = X}.
Hence, by 17.5, C is compact. Thus, by (*), from K onto C. Define g : K x [0, l] -+ 2”
f,
for each (z,
t)
E K x (O,l].
We show that g is continuous and that g maps K x [0, 11 onto 2x. The proof of the continuity of 9 can be done with a straightforward sequence argument that uses the continuity.of f, the continuity of each f(z), and the fact that convergence in C is uniform convergence (as noted preceding 17.1). We omit the details and point out instead that the continuity of g is the direct consequence of the following general fact: The compactopen topology for continuous functions between Hausdorff spaces is jointly continuous on compact sets [15, p. 2231. Next, we prove that g maps K x [0, l] onto 2x. Let A E Zx. Then, since X is a continuum, we see from 16.7 that there is a segment, 0, with
ORIGINAL
SOURCES
141
respect to w from A to X. Note that G E C. Thus, since f(ic) exists z, E K such that f(z,) = 0. Hence,
= C, there
dzo,O) = [f(zo)l(o) = 40) = A. Therefore, we have proved that g maps K x [0, l] onto 2x. Now, we complete the proof for 2” as follows. Recall that F = Cone(K), let w denote the vertex of F, and let x : K x [0, l] -+ F denote the quotient map: 7r(z,t)
=
(z,t), { V,
if t # 1 ift=l.
Note that g is constant on T-‘(U) since g(z, 1) = [f(z)](l) = X for all is single-valued at each point of F. Thus, (z, 1) E K x [0, 11. Hence, goV’ since 7r is a quotient map and g is continuous, g o r-l is continuous (by 3.2 of [6, p. 1231 or 3.22 of [28, p. 451). Furthermore, since g maps K x [0,l] onto 2”, clearly g 0 7r-l maps F onto 2x. This completes the proof of our theorem for 2x. To prove our theorem for C(X), let
C’ = {u ES,(C(X)): a(l) = X}; then, repeat what we did for 2x using C’ instead of C (use 16.9 where we used 16.7). W Other results about continuous images of the Cantor fan, as well as results about continuous preimages of the Cantor fan, are in [2], [23], [25], and [31, p. 1951; also, see the comment immediately following 33.11. Many of the results are also discussed in [29, pp. 81-991 (the paper that is referred to in the footnote in [29, p. 2511 will not appear; see Comment after first question in section 82 here).
Original
Sources
The result in 17.3 is 1.5 of Kelley [16]; 17.4 in the form of 17.5 is from the first paragraph of the proof of 2.7 of [16]; 17.7 and 17.8 give full generality to 1.30 and 1.32 of [29]; 17.9 is from Eberhart-Nadler-Nowell [8]; 17.10 is due to Mazurkiewicz [22] although the proof we gave is from [16, p. 251.
Exercises 17.11 Exercise. For any continuum X, S,(2x) arcwise connected continua.
and S,(C(X))
are
142
IV. ARCS IN HYPERSPACES 17.12 Exercise.
Let X be a compactum, Sk(Z)
= (0 E SW(Z)
and let 31 c 2x. Let
: (T is constant}.
Then, S&(B) is a strong deformation retract of S,,,(R); hence, S,(Z) is of the same homotopy type as 7-l. (If Z is a space and Y c 2, then Y is said to be a strong deformation retract of Z provided that there is a continuous function h : 2 x [0, l] + 2 such that h(z, 0) = z for all .z E 2, h(z, 1) E Y for all z E 2, and h(y, t) = y for all (y, t) E Y x [0, 11.) [Hint: make use of fw in the proof of 17.7.1 17.13 Exercise. If X is a Peano continuum, then 8(2x) is a Peano continuum. (We used the result in the proof of 17.9, so don’t use 17.9. A hint follows.) [Hint: Let d be a convex metric for X (10.3), and let Hj denote the Hausdorff metric for 22x induced by Hd as in 2.1. Fix Q: E D(Zx) and E > 0. Prove that {p E 8(2x) : Hi(cr,P) < E} is arcwise connected.] 17.14 Exercise. 273(2x)
For any compacturn
X, F1(8(2~))
is a retract
of
[Him: Let fw : S,(2x) + a(ax) b e as in the proof of 17.7. The key idea for the proof comes from the following obvious fact: for each o E @2”), [y: = {[f,-+)](t) : t E [O,l]}. M a k e use of Exercise 11.5. Recall 14.7.1 17.15 Exercise. Let X be a compacturn. Then, ??(C(X)) of 8(2x) if and only if C(X) is a retract of 2x.
is a retract
Remark. There are continua, X, for which C(X) is not a retract is not always a retract of 8(2”). 2” [9]. Hence, by 17.15, ??(C(X))
of
17.16 Exercise. The space S,(C(S’)) is not the Hilbert cube. (See the comments following the proof of 17.9.) [Hint: Prove that the point {S’} of ??(C(S’)) is not a Z-set in Tj(C(S’)); recall 5.2 and our discussion of Z-sets in section 9.1 17.17 Exercise. A continuum, Y, is said to be arcwise decomposable provided that Y is the union of two arcwise connected, proper subcontinua. Any nondegenerate continuum that is a continuous image of the Cantor fan is arcwise decomposable. Hence, 2x and C(X) are arcwise decomposable whenever X is a nondegenerate continuum.
18. WHEN
C(X) Is UNIQUELY
ARCWISE
CONNECTED
143
17.18 Exercise. Let Y be the Warsaw circle ((4) of Figure 20, p. 63); then, Y is an arcwise connected continuum that is not a continuous image of the Cantor fan. (See the discussion preceding 17.10.) [Hint: Use 17.17.1 Remark. If X is a continuum, then any continuum that is a continuous image of 2” or C(X) must be arcwise connected by 14.9 (and 8.28 of [28, p. 1331). By 17.10 and 17.18, the Warsaw circle is an arcwise connected continuum that is not a continuous image of 2x or C(X) for any continuum X. Results about when there are continuous surjections between continua, 2x, and/or C(X) are in Chapter XII and in [29, pp. 241-2511. The idea of relating arcwise decomposability to continuous images of the Cantor fan is due to Bellamy [2]; 17.18 is Example II of [2, p. 171.
18.
When
C(X)
Is Uniquely
Arcwise
Connected
A space, Y, is said to be uniquely arcwise connected provided that for any p, q E Y such that p # q, there is one and only one arc in Y with end points p and q. In the part of section 7 entitled Knaster’s Question, we discussedsimilarities between C(X) and Cone(X) when X is a continuum. It is easy to determine when Cone(X) is uniquely arcwise connected: The necessary and sufficient condition is that X contains no arc. This condition is also necessary for C(X) to be uniquely arcwise connected (by 5.1); however, the condition is not sufficient for C(X) to be uniquely arcwise connected (by Exercise 14.19 and the fact that there are decomposable continua that contain no arc [28, pp. 28-301). In fact, we see from 14.19 t,hat for C(X) to be uniquely arcwise connected it is necessary that X be hereditarily indecomposable. In this section we show that this condition is also sufficient: C(X) is uniquely arcwise connected if and only if X is hereditarily indecomposable. The theorem is due to Kelley (16, p. 341. We remark that hereditarily indecomposable continua were at first thought to be anomalies. In fact, when Kelley proved the theorem just mentioned, he noted that only one example of such a continuum was known (footnote 14 in [16, p. 341). N ow, however, there are many examples of hereditarily indecomposable continua (see fourth paragraph following 7.2). Furthermore, hereditarily indecomposable continua are important in, for example, continuum theory and dynamical systems. Results in this section - including lemmas - will provide us with a general understanding of how the subcontinua of an hereditarily indecomposable continuum fit together.
144
IV. ARCS IN HYPERSPACES
The main theorem is 18.8. The results of 18.8; several of them are of independent
in 18.1-18.7 facilitate interest.
the proof
18.1 Proposition. A continuum, X, is hereditarily indecomposable if and only if whenever A and B are subcontinua of X such that A cl B # 0, then A c B or B c A. Proof. Assume that X is hereditarily indecomposable. Let A and B be subcontinua of X such that AII B # 0. Then, AU B is a subcontinuum of X. Hence, A U B is an indecomposable continuum. Therefore, A c B or
B c A. Conversely, assume that X is not hereditarily indecomposable. Then there is a decomposable subcontinuum, K, of X. Hence, there are proper subcontinua, A and B, of K such that K = A U B. Clearly, A II B # 0, A $ B and B q! A. W 18.2 Proposition. Let Y be an indecomposable continuum. If A is an arc in C(Y) such that ud = Y, then Y E A. Proof.
Let h be a homeomorphism of [O,l] onto A. Let t, = inf{ t E [O, l] : Uh([O, t]) = Y)
(to exists since Uh([O, 11) = Ud = Y). Clearly, for any t > t,, Uh([O, t]) = Y; hence, using 13.3 and (2) of Exercise 11.5, we see by a simple sequence argument that (a) Uh([O,t,]) = Y. Note that if t, = 0, then h(0) = Y by (a); hence, Y E A. Thus, we may assumefor the rest of the proof that t, > 0. For each t E [0, t,], let At = Uh([O, t]) and let Bt = uh([t, to]). Note the following three properties of At and Bt: by (a), we have that (b) At U Bt = Y for each t E [0, t,]; by (3) of 11.5, we have that (c) At and Bt are subcontinua of Y for each t E [O,t,]; and, by the definition of t,, we have that (d) At#Yforanyt 0, there is a sequence, {ti}gl, in [O,to) converging to t,. By using 13.3 and (2) of converges to Bt,. Thus, since Bt, = Y for each 11.5, we seethat {Bt,}&
STRUCTURE OF ARCS IN C(X) i = 1,2,..., YEd.D
WHEN X Is HEREDITARILY...
BtO = Y. Thus, since Bt, = h(t,),
h(t,)
145
= Y. Therefore,
In relation to 18.2, see Exercise 18.9.
Structure
of Arcs in C(X) When Indecomposable
X Is Hereditarily
Assume that X is an hereditarily indecomposable continuum. Let Ao, then, by 18.1, one of A0 or Al is contained in the other or -40 n Ai = 0. Hence, the next two propositions determine the structure of all arcs in C(X).
AI E C(X);
18.3 Proposition. Let X be an hereditarily indecomposable continuum, and let Ao, Al E C(X) such that A0 c Al and AC, # Al. If A is an arc in C(X) from A0 to Ai, then d is an order arc. Proof. For each A f A - (Ao}, we let A(AoA) A from AD to A; we let d(AoAo) = {Ao}. Now, let
denote the subarc of
: A E A}.
cr = {ud(A,A)
We prove (l)-(4) below (combining (3) and (4) proves our proposition): (1) (Yis a continuum; (2) Ao,Al E cu; (3) a = A; (4) cy is an order arc. Proof of (1): Let f : A -+ C(d) be given by
f(A) = d(AoA)
for each A E A.
Obviously, f is continuous. The union map u for C(d) is continuous by (2) of Exercise 11.5. Hence, 2t o f is continuous. Clearly, u o f maps d onto a. Therefore, (1) holds. Proof of (2): Since A0 = ud(AoAa),
clearly A0 E Q. We prove that
Al E Q as follows. Let M = (A E A : Al c ud(AoA)},
ni = {A E A : Al 3 ud(AoA)}
Clearly, A1 E M; also, since A0 c Al, A0 E N. Hence, M # 0 and N # 0. Next, note that M and N are closed in d (by an easy sequence argument using the continuity of u o f in the proof of (I)). We show that d = M UN.
146
IV. ARCS IN HYPERSPACES
Let A E A. Let Z = Ud(AsA). By (3) of 11.5, Z is a subcontinuum of X; also, since A0 C Ai, Zn Ai # 0. Thus, by 18.1, -41 c Z or Z c Al. Hence, AEM (if.41 CZ)orAEN(ifZcAi). ThisprovesthatdCMUN. Therefore, A = M UN. Now, since A is connected, we seefrom the properties of M and N just verified that M fl N # 0. Hence, there exists C E Jbi fl ni. Clearly (from the definitions of M and N), Al = Ud(AaC). Therefore, AI E cy. This completes the proof of (2). Proof of (3): Since A is an arc from A0 to Al, it suffices by (1) and (2) to prove that a c A. Let Y E LY,say I’ = Ud(AeB) for some B E A. By (3) of 11.5, Y is a subcontinuum of S. Hence, Y is an indecomposable continuum. Thus, by 18.2, Y E d(AaB) (note: if B = AO then, even though 18.2 does not apply, Y E d(AoB) since Y = As). Hence, I’ E A. Therefore, we have proved that a c A. This completes the proof of (3). Proof of (4): Let, E, F E A. Then, clearly, d(AoE) C d(AoF) or d(AcF) c d(AoE). Hence, Ud(A,E) c Ud(AcF) or Ud(AsF) c Ud(AsE). This proves that cr is a nest. By (l), cy is a subcontinuum of 2.’ and, by (2), (Y is nondegenerate. Therefore, by 14.7, a is an order arc. This proves (4). By (3) and (4), d is an order arc. n Let X be an hereditarily indecomposable contin18.4 Proposition. uum, and let Ao, Al E C(S) such that AenAi = 0. If A is an arc in C(X) from 40 to Al, then A = do u di, where do is an order arc from A0 to UA and dr is an order arc from ill to UA. Proof. Let Y = ud. By (3) of Exercise 11.5, Y is a subcontinuum of X. Thus, since X is hereditarily indecomposable, Y is an indecomposable cont,inuum. Hence, by 18.2, ‘I’ E A. Also, since Aond4i = 0, clearly Y # -40 and I’ # Al. Thus, there are subarcs, de and dr , of A from A0 to Y and from ill to Jr, respectively, Since A0 c 1-, we seefrom 18.3 that do is an order arc; similarly, di is an order arc. Clearly, A = do u di . W
Uniqueness
of Arcs in C(X) When Indecomposable
X Is Hereditarily
The propositions in 18.6 and 18.7 show that when X is an hereditarily indecomposable continuum, arcs in C(X) are uniquely determined by their end points.
UNIQUENESS
OF ARCS IN C(X) WHEN
X Is HEREDITARILY..
.
147
18.5 Lemma. Let X be an hereditarily indecomposable continuum! and let ‘w be a Whitney map for C(X) (2~exists by 13.4). If A, B E C(X) such that A n B # 0 and w(A) = w(B), then A = B. n Proof. By 18.1, A C B or B C A. Therefore, since w(A) = w(B), we have by (1) of 13.1 that A = B. n 18.6 Proposition. Let X be an hereditarily indecomposable continuum, and let K, L E C(X) such that K c L and K # L. Then there is one and only one arc in C(X) from K to L. Proof. By 14.9, there is an arc, A, in C(X) from K to L. Let B be any arc in C(X) from K to L. We show that A = D. We use a Whitney map, w, for C(X) (w exists by 13.4). By 18.3, A and t3 are order arcs. Hence, it follows from the definitions in 13.1 and 14.1 that w(d) = [w(K),w(L)]
= w(B).
Now, let A E A. Then, since w(d) = w(D), there exists B E B such that w(A) = w(B). Also, A n B # 0 since A and t? are order arcs from K to L (hence, An B > K). Thus, by 18.5, A = B and, hence, A E f?. This proves that A c B. By a similar argument, 0 c A. Therefore, A = f?. n 18.7 Proposition. Let X be an hereditarily indecomposable continuum, and let Ao, Al E C(X) such that A. n Al = 0. Then there is one and only one arc in C(X) from A0 to Al. Proof. By 14.9, there is an arc, d, in C(X) from A0 to Al. Let t? be any arc in C(X) from A0 to Al. We show that A = t3. Recall our assumption that A0 n A1 = 0. Hence, A = As U dr as in 18.4 and t? = a0 U 23r as in 18.4 (with ud replaced by UB). Note that W) n W) # 0 (since Ud > A0 and UZ? > Ao). Thus, by 18.1, Ud C Ul? or UB C ud, say ud C UB. Assume first that ud = UZ?. Then, by 18.6, do = &J and di = &. Hence, A = f?. Assume next that ud # Ua. Then, since ud c Ua, we know from 14.6 that there is an order arc, C, in C(X) from ud to UB. Hence, do U C is an arc in C(X) from A0 to UB, and di U C is an arc in C(X) from Al to UB. Thus, by 18.6, do U C = f?e and di U C = BI. Hence, A C D. Therefore, since A and 23 are arcs in C(X) with the same end points, it follows immediately that A = 13.n
148
IV. ARCS IN HYPERSPACES
The Characterization We prove Kelley’s
characterization
Theorem
theorem:
18.8 Theorem. Let X be a continuum. Then, X is hereditarily composable if and only if C(X) is uniquely arcwise connected.
inde-
Proof. Assume that X is hereditarily indecomposable. Let Ao, Al E C(X) such that Ac # AI. If ACJ C A1 or Al c Ao, then there is a unique arc in C(X) from Ao to Ai by 18.6. If A0 < Al and Al @ Ao, then Ao n Al = 8 by 18.1; hence, there is a unique arc in C(X) from As to Ai by 18.7. Therefore, C(X) is uniquely arcwise connected. The converse is due to Exercise 14.19; however, in the interest of selfcontainment, we present an independent proof of the converse. Assume that X is not hereditarily indecomposable. Then there is a decomposable subcontinuum, Y, of X. Let A and B be proper subcontinua of Y such that Y = A U B. Let p E A fl B (note that p exists since Y is connected). By using 14.6 twice, we seethat there is an order arc, (Y, in C(Y) from {p} to Y such that A E cr. Similarly, there is an order arc, p, in C(Y) from {p} to Y such that B E p. Clearly, A E (Y- p; thus, Q # /I. Therefore, C(X) is not uniquely arcwise connected. n More results about C(X) when X is an hereditarily indecomposable continuum are in [16] and [29] (see [29, pp. 686-6871for a compendium).
Original
Sources
All results except 18.1 are due to Kelley [16] or can be inferred easily from ideas in proofs in [16, p. 341. In particular, 18.2 is 8.1 of [16] (a stronger result is in 1.50 of [29, p. 1021); 18.5 is 8.3 of [16]; and 18.8 is 8.4 of [16].
Exercises 18.9 Exercise. A nondegenerate continuum, X, is indecomposable if and only if X arcwise disconnects C(X). [Hint for the “only if” part: Use that a nondegenerate, indecomposable continuum is irreducible about two points [28, p. 2031.1 Remark. The result in 18.9 is 8.2 of Kelley [16]; 18.9 is also true with C(X) replaced by 2” (4.3 of [27, p. 161). For related results, see [27, pp. 15241 or [29, pp. 357-3721; in particular, a characterization Of hereditarily indecomposable continua is in 4.11 of [27] (which is 11.15 of [29] - cf. Note 1 a.t the bottom of p. 371 of [29]).
REFERENCES
149
18.10 Exercise. A continuum, X, is hereditarily indecomposable if and only if every monotone increasing sequence of arcs in C(X) is contained in an arc. (A monotone increasing sequence of sets is a sequence, {Ai},“, of sets Ai such that Ai C AZ+1 for each i = 1,2,. . ..) [Hint for the “only if” part: Make use of 18.4. (The “only if” part does not follow from 18.8: The Warsaw circle in (4) of Figure 20, p. 63 is uniquely arcwise connected and, yet, is the union of a monotone increasing sequence of arcs.)] Remark. By 18.10 and 14.9, we can apply Theorem to see that C(X) has the fixed point property whenever X indecomposable continuum. The result is due to Rogers the proof just given is from Krasinkiewicz [18, p. 1801. see 22.17 and the comment following the proof of 22.17.
16 of Young [40] is an hereditarily 133, pp. 284-2851; For more details,
18.11 Exercise. Let X and Y be nondegenerate continua such that X is hereditarily indecomposable. Then, C(X) $ Cone(Y); also, C(Y) is not embeddable in Cone(X). Remark. Regarding the first part of 18.11, it is actually the case that Cone(Y) is not even embeddable in C(X) [30, p. 2371; moreover, Y x [0, l] is not embeddable in C(X) [18, p. 1821. For related results, see [29, pp. 1491521. 18.12 Exercise. The uniquely arcwise connected continuum in Figure 25, p. 158, can not be embedded in C(Y) for any hereditarily indecomposable continuum Y. 18.13 Exercise. Let X be an hereditarily indecomposable continuum, is an hereditarily and let w be a Whitney map for C(X). Then, w-‘(t) indecomposable continuum for each t E [0, w(X)]. [Hint: 18.5 yields a natural map of X onto w-l(t).] Remark. an hereditarily
In the terminology of Chapter VIII, 18.13 says that being indecomposable continuum is a Whitney property (27.1(a)).
References 1. 2.
David P. Bellamy, Indecomposable continua with one and two composants, Fund. Math. 101 (1978), 129-134. David P. Bellamy, The cone over the Cantor set - continuous maps from both directions, Proc. Topology Conference (Emory University, Atlanta, Ga., 1970) (J. W. Rogers, Jr., ed.), 8-25.
150
IV. ARCS IN HYPERSPACES
3.
K. Borsuk and S. Mazurkiewicz, Sur Z’hyperespace d’un continu, C. B. SOC. SC. Varsovie 24 (1931), 149-152.
4.
Doug Curtis and Mark Lynch, Spaces of order arcs in hyperspaces Peano continua, Houston J. Math. 15 (1989), 517-526.
5.
D. W. Curtis and R. M. Schori, Hyperspaces which characterize homotopy type, Gen. Top. and its Appls. 6 (1976), 153-165.
6.
James Dugundji, printing,
7. 8. 9.
of
simple
Topology, Allyn and Bacon, Inc., Boston, Mass., third
1967.
Carl Eberhart and Sam B. Nadler, Jr., The dimension of certain hyperspaces, Bull. Pol. Acad. Sci. 19 (1971), 1027-1034. Carl Eberhart, Sam B. Nadler, Jr., and William 0. Nowell, Spaces of order arcs in hyperspaces, Fund. Math. 112 (1981), 111-120. Jack T. Goodykoontz, Jr., Aposyndetic J. Math. 47 (1973), 91-98.
properties
of hyperspaces, Pac.
10. Jack T. Goodykoontz, Jr., C(X) is not necessarily a retract of 2”, Proc. Amer. Math. Sot. 67 (1977), 177-178. 11. Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton University Press, Princeton, New Jersey, 1948. 12. Alejandro Illanes, Cells and cubes in hyperspaces, Fund. Math. 130 (1988), 57-65. 13. Alejandro Illanes, Monotone and open Whitney maps, Proc. Amer. Math. Sot. 98 (1986), 516-518. 14. Ott-Heinrich Keller, Die Homoiomorphie der kompakten konuexen Mengen im Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. 15. John L. Kelley, General Topology, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1960. 16. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot. 52 (1942), 22-36. No O-dimensional set disconnects the hyperspace of 17. J. Krasinkiewicz, a continuum, Bull. Pol. Acad. Sci. 19 (1971), 755-758. 18. J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84 (1974), 1755186. 19. J. Krasinkiewicz, Shape properties of hyperspaces, Fund. Math. 101 (1978), 79-91. 20. K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. 21. A.Y. W. Lau and C. H. Voas, Connectedness of the hyperspace of closed connected subsets, Ann. Sot. Math. Pol. Series I: Comm. Math. 20 (1978)) 393-396.
REFERENCES 22. Stefan (1932), 23. Stefan Math.
151
Mazurkiewicz, SW l’hyperespace d’un continu, Fund. Math. 18 171-177. Mazurkiewicz, Sur le type c de l’hyperespace d’un continu, Fund. 20 (1933), 52-53.
24. Stefan Mazurkiewicz, SW le type de dimension de l’hyperespace d’un continu, C. R. Sot. SC. Varsovie 24 (1931), 191-192. Sur les images continues des continus, Proc. 25. Stefan Mazurkiewicz, Congress of Mathematicians of Slavic Countries (Warsaw, 1929), F. Leja, ed., 1930, 66-71. 26. M. h4. McWaters, Arcs, semigroups, and hyperspaces, Can. J. Math. 20 (1968), 1207-1210. 27. Sam B. Nadler, Jr., Arcwise accessibility in hyperspaces, Dissertationes Math. 138 (1976). 28. Sam B. Nadler, Jr., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 29. Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 30. Sam B. Nadler, Jr., Locating cones and Hilbert cubes in hyperspaces, Fund. Math. 79 (1973), 233-250. 31. Sam B. Nadler, Jr., Some problems concerning hyperspaces, Topology Conference (V.P.I. and S.U.), Lecture Notes in Math. (Ed. by Raymond F. Dickman, Jr., and Peter Fletcher), Springer-Verlag, New York, Vol. 375, 1974, 190-197. 32. J. T. Rogers, Jr., Dimension of hyperspaces, Bull. Pol. Acad. Sci. 20 (1972), 177-179. 33. J. T. Rogers, Jr., The cone = hyperspace property, Can. J. Math. 24 (1972), 279-285. 34 L. Vietoris, Kontinua zweiter Ordnung, Monatshefte fiir Math. und Physik 33 (1923), 49-62. 35. A.D. Wallace, Indecomposable semigroups, Math. J. of Okayama Univ. 3 (1953), l-3. 36. L. E. Ward, Jr., intending Whitney maps, Pac. J. Math. 93 (1981), 465-469. 37. T. Wazewski, Sur un continu singulier, Fund. Math. 4 (1923), 214235. 38. Hassler
Whitney,
18 (1932).275-278.
Regular families
of curves, I, Proc. Nat. Acad. Sci.
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IV. ARCS IN HYPERSPACES
39. Hassler Whitney, Regular families of curues, Annals Math. 34 (1933), 244-270. 40. Gail S. Young, The introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479-494.
V. Shape and Contractibility of Hyperspaces The chapter consists of two sections. In the first section we prove that, for any continuum X, 2x and C(X) have trivial shape in the sense of Borsuk [2]. We prove this theorem in a different form at the beginning of the section; namely, we prove that, for any continuum X, 2x and C(X) are nested intersections of absolute retracts. We use this form of the theorem to obtain some basic properties of 2x and C(X). In the second section of the chapter we are concerned with contractibility. We prove two general theorems about the contractibility of 2” and C(X). We use these theorems to determine some classes of continua, X, for which 2x and C(X) are contractible. More about contractibility of hyperspaces is in Chapter XIII. The two sections of the chapter are related by the following fact: A continuum (in this case, 2x or C(X)) has trivial shape if and only if the continuum is contractible with respect to every absolute neighborhood retract.
19.
2x and C(X)
A nested intersection each i.
as Nested is an intersection,
Intersections n&Yi,
where
of ARs Yi > Yi+i for
We prove that for any continuum X, 2x and C(X) are nested intersections of ARs (absolute retracts). We then derive several important properties of 2x and C(X). 153
154
V. SHAPE AND CONTRACTIBILITYOF
HYPERSPACES
The usual approach to the results in this section is to use inverse limits. However, we have chosen the present approach so that we can obtain the results in the simplest and most accessibleway possible. We begin with the following general lemma about continua. 19.1 Lemma. tinua.
Every continuum is a nested intersection of Peano con-
Proof. Let X be a continuum. We may assumethat X is contained in the Hilbert cube Ice [17, p. 2411. Let C denote the Cantor Middle-third set in [OJ], and let VI, U2, . . . be a one-to-one indexing of the components of [0, l] - C. There is a continuous function, f, of C onto X (7.7 of [22, p. 1061). By 9.1 and 9.2, Ic*3 is an AE; hence, f can be extended to a continuous function, g, of [O,l] into Ia. Now, for each i = 1,2,. . ., let Ki = [0, I] - lJj,,uj and let Yi = g(Ki). Since Ki 3 Ki+l for each i, clearly Yi > Y,+r for each i. We prove that X = f$ZrY% and that Yi is a Peano continuum for each i. Since K, > Ki+l for each i and since fIgr K, = C, Lim Ki = C (by the converges second part of Exercise 4.16); thus, by 4.7, the sequence{Ki}fZl in 2[“l’1 to C. Hence, by 13.3, the sequence {I$}& converges in 21m to g(C). Thus, since g(C) = f(C) = X, {I$}& converges in 2’O” to X. Therefore, since Yi > Yi+r for each i, we see that X = f$?rYi (by 4.7 and the second part of Exercise 4.16). Finally, we prove that Yi is a Peano continuum for each i. Fix i. We seeeasily that K; is a finite union of closed subintervals of [O,l]. Hence, Yi is a finite union of Peano continua (by 8.17 of [22, p. 1281). We show that I< is connected. To this end, note that
thus, since yi = g(Ki)
and g(C) = f(C) = X, we see that (1) yi = 9(C u [q&+1 Vj]) = x u [u,oo,~+1g(17j)]. Since aj f~ C # 0 for each j and since g(C) = X, we seethat
(2) g(Uj) n X # 0 for each j. Since X and each g(uj) are connected, we see from (1) and (2) that 1; is connected. Therefore, since Yi is a finite union of Peano continua (as noted above), 1; is a Peano continuum (since 10.7 implies that a connected metric space that is a finite union of Peano continua is a Peano continuum). a
19.2 Theorem. sections of ARs.
For any continuum X, 2” and C(X) are nested inter-
2x, C(X)
ARE ACYCLIC
155
Proof. If X = n&Xi then, clearly, C(X) = n&C(Xi) and 2x = fl&2X*. Therefore, the theorem follows from 19.1 and 10.8. n Throughout the rest of the section we use 19.2 to derive properties of 2” and C(X). We include somegeneral background for most of the properties.
2x, C(X)
Are Acyclic
The term acyclic refers to any homology theory or cohomology theory over a coefficient group for which the theory is continuous [7]. When the coefficient group is the integers, examples of such theories are Vietoris homology, tech homology, Tech cohomology, and Alexander-KolmogoroffSpanier cohomology. 19.3 Theorem. dimensions.
For any continuum X, 2” and C(X) are acyclic in all
Proof. Every AR is contractible (Exercise 19.11). Hence, every AR is acyclic ([2, p. 861 or [8, p. 301). Therefore, the corollary follows from 19.2 (since we are assuming that our homology and cohomology theories are continuous). n
2x, C(X)
Are crANR
Our result (which is in 19.6) involves homotopy and absolute neighborhood retracts. We first give the definitions and notation that we use and prove two lemmas. We refer the reader to [2] for more information. Let Y and 2 be spaces. A continuous function from Y x [0, l] into Z is called a homotopy (a continuous function from Y x [a, b] into 2 is also called a homotopy). For a homotopy h : Y x [0, l] -+ 2 and any t E [0, 11, we let ht denote the map of Y into 2 given by ht(y) = h(y, t) for all y E Y. We say that two maps f,g : Y + Z are homotopic provided that there is a homotopy h : Y x [0, l] + 2 such that ho = f and hl = g, in which case we say that h is a homotopy joining f to g. If a map f : Y + Z is homotopic to a constant map of Y into 2, then f is called an inessential map; otherwise, f is called an essential map. A space, Y, is said to be contractible provided that the identity map of Y is inessential. We describe what it means for a space to be contractible in the following way: The space can be continuously deformed, in itself, to a point. This intuitive description is a reasonable way to envision the notion of contractibility for compact spaces;however, the description leaves a lot to be desired when a space is not compact: R’ is contracted to the point zero
156
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
by the homotopy h given by h(z, t) = (1 - t) . 2 for all (2, t) E R’ x [0, 11; nevertheless, ht (R1) = R’ for each t < 1. with respect to Let Y and Z be spaces. We say that Y is contractible 2 (written Y is crZ) provided that every continuous function from Y into Z is inessential (i.e., homotopic to a constant map; note that we do not require that every continuous function from Y into 2 be homotopic to the Sameconstant map). We will use the following simple lemma in the proof of 19.6. 19.4 Lemma. every space Z.
A space, Y, is contractible if and only if Y is crZ for
Proof. Assume that Y is contractible. Then there is a homotopy h : Y x [0, l] + Y joining the identity map of Y to a constant map of Y. Hence, if f is a continuous function from Y into a space Z, we see that f o h : Y x [0, l] + 2 is a homotopy joining f to a constant map of Y into 2. Therefore, Y is crZ for every space 2. The other half of the lemma is obvious. n We discussedretracts, absolute retracts, and absolute extensors in section 9. We now define the following notions. retract (written A compactum, K, is called an absolute neighborhood ANR) provided that whenever K is embedded in a metric space, Y, the embedded copy, K’, of K is a retract of some neighborhood of K’ in Y. A extensor (written ANE) compactum, K, is called an absolute neighborhood provided that whenever B is a closed subset of a metric space, M, and f : B + K is continuous, then there is a neighborhood, U, of B in M such that f can be extended to a continuous function F : U + K. The following lemma is analogous to 9.1 and is due to Borsuk. 19.5 Lemma. ANE.
A compactum, K, is an ANR if and only if K is an
Proof. The lemma follows by adjusting the proof of 9.1. Assume that K c IO3 [17, p. 2411;T is a retraction from a neighborhood, W, of K in I” onto K; B, M, f, and g are as in the proof of 9.1; let U = g-l(W); then, r o (g]U) : U + K is an extension of f to the neighborhood, U, of B in M. For the converse, let K’ c Y be as in the last part of the proof of 9.1, and simply note that, since K is an ANE, there is a neighborhood, V, of K’ in Y such that the identity map for K’ can be extended to a continuous function r : V + K’. W
2’, C(X)
ARE UNICOHERENT
157
We write Y is crANR to mean that the space Y is contractible with respect to every ANR. We are ready to prove the following theorem: 19.6 Theorem.
For any continuum X, 2x and C(X) are crANR.
Proof. We prove the corollary for 2x; the proof for C(X) is similar. Let K be an ANR, and let f : 2x -+ K be continuous. By 19.2, 2x = fP2-l Y.2 where Y,E > 1’%+r and Yi is an AR for each i. By 19.5, I< is an ANE; hence, there is a neighborhood, U, of 2x in Yr such that f can be extended to a continuous function g : U + K. Since 2x is the nested intersection of the compacta Yi and since 2x C U, there exists n such that Y,, c U (by second part of Exercise 4.16 and by 4.7). Note that since I”, is an AR, Y, is crK (by Exercise 19.11 and 19.4); also, note that g is defined on all of Y, (since I$ c V). Hence, g1Y, : Y, + I( is inessential. Thus, since 2x c Y, and 912” = f, we see that f : 2” -+ K is inessential (for if h : Y, x [0, l] + K joins g]Y, to a constant map Ic : Y, + K, then h(2X x [0, l] joins f to the constant map lc(2x). Therefore, we have proved that 2x is crK. n We note that 19.6 can not be strengthened to say that 2x and C(X) are contractible. For example, let X be the continuum in Figure 25 (top of the next page), (X consists of two harmonic fans joined at a point): Neither 2x nor C(X) is contractible (Exercise 19.12). We remark that a lot of work has been done on contractibility in connection with hyperspaces. Wojdyslawski wrote the first paper about this [30]; Kelley obtained the first general results [12, pp. 25-271. We discusscontractibility of hyperspaces in section 20 (Wojdyslawski’s result is 20.14, and Kelley’s results are 20.1 and 20.12).
2x, C(X)
Are Unicoherent
We see from 19.6 that for any continuum X, 2x and C(X) are crS1 (where S’ is the unit circle). A weaker result that is useful and more geometrically appealing is in 19.8. First, we give a definition and make some comments about the definition. A continuum, X, is said to be unicoherent provided that whenever A and B are subcontinua of X such that AU B = X, then An B is connected. The notion of unicoherence has a clear geometric interpretation: A unicoherent continuum has no hole. For example, S’ and an annulus are not unicoherent. However, the interpretation is not without flaws: The 2-sphere is unicoherent (by Theorem 2 of [18, p. 506]), and the circle-with-a-spiral
158
V. SHAPE AND CONTRACTIBILITY
X with 2x, C(X)
OF HYPERSPACES
not contractible
Figure 25
in Figure 14, p. 51 is unicoherent. Thus, unicoherent continua need not be acyclic. The proof of the following lemma uses the notion of a lift; the definition of a lift and the relevant theorem about lifts are in Exercise 19.20. 19.7
Lemma.
If a continuum is crS’, then the continuum is unicoher-
ent. Proof. Let S: = ((2, y) E S’ : y 2 0}, and let S’ = { (2, y) E 5” : y 5 0). Note th at, since Si and S! are arcs, S: and Sl are AEs (by 9.1 and 9.2). Now, assumethat Y is a continuum that is not unicoherent. Then there are subcontinua, A and B, of Y such that A u B = Y and A fl B is not connected, say A n B = E/F (section 12). Let f : A n B + S: C-ISf.
WHITNEY
LEVELS
IN C(X) ARE
CONTINUA
159
be the continuous function that is defined by letting f(E) = (1,0) and f(F) = (-1,O). s ince S: and Si are AEs, we can extend f to continuous functions gi : A + S$ and g2 : B + Sl. Let g : Y + S’ be the continuous function that is defined by letting g/A = gi and glB = g2. Note the following fact: (1) g(A II B) = {(LO), (-1,O)) = g(A) n g(B). We show that g is an essential map. Suppose that g is an inessential map. Then, by the theorem in Exercise 19.20, g has a lift cp: Y + R’. Let (with exp as in Exercise 19.20)
M = exp[cp(A)n v(B)]. Since p(A) and p(B) are intervals in R’, q(A) n p(B) is connected; hence, M is connected. However, as we now show, M = {(l,O),(-1,O)). It is obvious that
n wMB)l. (2) exp[p(An B)] c M c exp[cp(A)l Since cpis a lift for g, we seefrom (1) that (3) exp[p(A n B)] = dA n B) = {Cl, 01, NO)) and that (4) e&p(A)] n exp[dB)l = g(A) n g(B) = {(LO), (-LO)). (2)-(4), M = {(LO), (-LO)). Th us, since we have previously proved that M is connected, we have a contradiction. Therefore, g is an essential map. We have shown that if Y is a continuum that is not unicoherent, then Y is not crS’. n BY
We are now ready to prove our result about unicoherent hyperspaces. 19.8 Theorem.
For any continuum X, 2x and C(X) are unicoherent.
Proof. By 19.6, 2x and C(X) are crS’; by 14.10, 2” and C(X) are continua. Therefore, by 19.7, 2x and C(X) are unicoherent. n
Whitney
Levels
in C(X)
Are Continua
Let X be a compactum, and let 3-1c 2x. A Whitney level for ?i is any where w is some Whitney map for subset of R that is of the form w-‘(t), 3c and t E [O,w(X)]. The next theorem provides the primary motivation for the study of Whitney levels. We will discussWhitney levels extensively in Chapters VIIIX; nevertheless, we include the theorem here as an immediate illustration of the applicability of 19.8. We note that the theorem differs from previous
160
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
theorems in two principal ways: It is only valid for C(X) (Exercise 19.16); it is concerned with proper subsetsof C(X). 19.9 Theorem. Let X be a continuum, and let w be a Whitney map for C(X). Then, w-‘(t) is a continuum for each t E [O,w(X)J. Proof. Fix t E [O,w(X)]. Let At = w-‘([O, t]) and let f?, = w-‘([t,w(X)]). Then, by using 14.6 (or 15.3), we see that dt and & are continua (cf. Exercise 15.14). Also, as is obvious,
Hence, by 19.8, At n Bt is a continuum. Therefore, since dt we have that w-‘(t) is a continuum. w
n Bt = we1 (t),
The theorem in 19.9 says that Whitney maps for C(X) belong to a wellstudied class of maps - the classof monotone maps. A monotone map is a continuous function f : Y + 2 such that f-‘(z) is connected for each z E 2 (sometimes it is required that f-‘(z) be a continuum for each z E 2 [29]). The terminology probably comes from the fact that a continuous function from R’ into R’ is monotone in the sensejust defined if and only if the function is monotone in the usual senseof real analysis. Regarding the existence and the nonexistence of monotone Whitney maps for 2x, see [4], [ll], and section 24. A natural generalization of Whitney maps and of 19.9 is in Exercise 19.18.
2x, C(X)
Have Trivial
Shape
When only considering continua, a number of seemingly different properties are actually equivalent to being a nested intersection of ARs. One such property is that of being crANR; thus, 19.2 and 19.6 are, in reality, equivalent results. We list a few more such properties (proofs of various equivalences are in [3], [lo], and [14, pp. 237-2391): having trivial shape, being a fundamental absolute retract, being absolutely neighborhood contractible, and being a weak proximate absolute retract. The prominent role of shape theory per se leads us to reformulate 19.2 as follows: 19.10 Theorem. shape. Proof. to 19.2. a
For any continuum X, 2x and C(X)
have trivial
According to comments just made, the theorem is equivalent
ORIGINAL
Original
SOURCES
161
Sources
The result in 19.2 for C(X) is 1.5 of Krasinkiewicz [13] (cf. 1.171 of [23, p. 1751); 19.3 is from various sources depending on the type of homology or cohomology that is used and on whether 2x or C(X) is considered (notably, Kelley [12, p. 271 for 2x, who asked about C(X), and Segal [27, p. 7081 for C(X); also, Lau [19] and McWaters [20, p. 12091; for more details, see [23, pp. 176-1791); 19.6 for C(X) is 1.6 of [13], and 19.6 for 2x is 1.183 of [23, p. 1801; 19.8 is from [24, p. 4121; 19.9 is from Eberhart-Nadler [6, p. 10321; 19.10 for C(X) is 1.9 of [13], and 19.10 for 2x is 1.184 of [23, p. 1801.
Exercises 19.11 Exercise. Every AR is contractible. (We used the result proofs above; show that the result follows easily from 9.1.) 19.12 Exercise. Let X be the continuum that 2” and C(X) are not contractible.
in
in Figure 25, p. 158. Prove
19.13 Exercise. Let X be a continuum, and let K be a subcompacturn of X. Then the containment hyperspace 2,x is a nested intersection of ARs. [Hint: If Y is a Peano continuum, then 24; is an AR (see the first part of the hint for Exercise 11.6).] Remark. The result in 19.13 is also true for CK(X) is an AR by Exercise 14.22 (cf. 14.23).
since CK(X)
itself
19.14 Exercise. Let X be a continuum, and let w be a Whitney map for 2x or C(X). Then, for each t E [O,w(X)], w-‘([&w(X)]) is a nested intersection of ARs. (Compare with the next exercise.) [Hint: Use 19.1, 16.10, and Exercise 11.8.1 Remark. We see that 19.2 is a special case of 19.14 by taking in 19.14 (and recalling 13.4).
t = 0
19.15 Exercise. In relation to Exercise 19.14, consider the unit circle S’. Show that for any Whitney map, w, for 2” or C(S’), there exists t > 0 such that w-l ([O, t]) is not a nested intersection of AI&. Remark. pp. 80-831.
Results
that are related to Exercises
19.13-19.15 are in [16,
162
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
19.16 Exercise. Show that Whitney maps for 2” need not be monotone (i.e., the analogue of 19.9 for 2x is false) by using x’ and w in (2) of Exercise 14.25. (See section 24.) 19.17 Exercise. Let E’ be the Hawaiian Earring (Figure 26): 1’ = where Ci is the circle in R2 with center at (0,l - 2-j) and radius 2-“. Is there a continuum, x’, such that 2.’ and/or C(X) is not cry? (Compare with 19.6 - the Hawaiian Earring is one of the simplest continua that is not an ANR.) UE_,ci,
19.18 Exercise. The following notion is a natural generalization of the notion of a Whitney map. Let x’ be a compactum, and let ?l C 2.‘; a size map for ?l is a continuous function u : ?l + [0, co) that satisfies the following two conditions:
Hawaiian Earring (19.17) Figure 26
EXERCISES (1) for any A, B E ‘H such that A
c
163
B, o(A) 2 a(B);
(2) u({x}) = 0 for each {z} E 7-1. Prove the result stated below (which generalizes 19.9). If X is a continuum and D is a size map for C(X), continuum for each t E [0, a(X)].
then a-‘(t)
is a
Remark. Size maps include Whitney maps and diameter maps. The point inverses of size maps are called size levels. All the size levels for C([O, 11) are completely characterized in [25]. So far, (251 is the only paper on size maps (except for those papers about Whitney maps). Open questions about size maps are in 83.15 and 83.16. 19.19 Exercise. Let X be a continuum, let u be a size map for 2x or C(X), and let t E [O,a(X)]. Then, (T-l(t) is a continuum if and only if 0-l ([0, t]) is a continuum. 19.20 Exercise. This exercise contains the basic theorem about lifts that we used when we proved the lemma in 19.7. We state the theorem in 19.20.1 and sketch its proof; we leave the details of the proof for the reader. First, we define the notion of a lift. Let exp denote the exponential map of R1 to S’ given by exp(t) = (cos(t),sin(t)) for each t E R1. Let Y be a space, and let f : Y + S’ be continuous. A lift for f is a continuous function cp: Y + R’ such that f = exp o cp. 19.20.1. Theorem. Let Y be a compactum, and let f continuous. Then, f is inessential if and only if f has a lift.
: Y -+ S’
be
Sketch of proof. Let p denote the uniform metric for (S’)* (section 17). Verify (l)-(3) below (from which the “only if” half of the theorem follows easily). (1) Any constant map of Y into S’ has a lift. (‘4
Ifa,
c (S’JY
such that p(gi, gs) < 2, then gi has a lift if and only
if gs has a lift.
(3) If h : Y x [0, l] + S’ is a homotopy, then there exist to = 0 < t1 < t2 < . . * < t, = l(n < 00) such that p(ht,, hti+,) < 2 for each i=O,l,..., n-l. To prove the other half of the theorem, assumethat the fact that [O,l] is contractible. n
f
has a lift and use
164
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
Remark. The theorem in 19.20.1 remains true for any topological space Y; this was proved by Eilenberg [7, p. 681 (the proof is also in [18, pp. 426-4271).
20.
Contractible
Hyperspaces
In the preceding section we gave an example of a continuum, X, such that 2d’ and C(X) are not contractible (Figure 25, p. 158). In this section we present some basic results about when 2” and C(X) are contractible. We prove two general theorems - 20.1 and 20.12 - and we determine some classesof continua for which 2” and C(X) are contractible. We recall from section 19 that a space, Y, is said to be contractible provided that the identity map of Y is inessential (i.e., homotopic to a constant map).
The Fundamental
Theorem
We begin with what we consider to be the most fundamental theorem concerning the contractibility of hyperspaces. The theorem shows that if one of 2x and C(X) .is contractible, then so is the other. The theorem also provides a useful way to determine whether 2x or C(X) is contractible namely, if (and only if) the space of singletons, Fl(X), is contractible in 2” or C(X). (If 2 is a space and Y C 2, then Y is said to be contractible in 2 provided that there is a homotopy h : Y x [0, l] + 2 such that hc is the inclusion map of Y into 2 and hi is a constant map of Y into 2.) Theorem. For any continuum X, (l)-(4) below are equivalent: 2” is contractible; C(X) is contractible; Fl (X) is contractible in C(X); Fl(X) is contractible in 2x.
20.1
(1) (2) (3) (4)
Proof. We first prove that (1) implies (2). Assume that (1) holds. Then there is a homotopy h : 2x x [O, l] -+ 2x such that ho is the identity map of 2” and hl is a constant map. Since 2x is arcwise connected (by (1) or by 14.9), it follows easily that we can assumethat hl (A) = X for all A E 2x. We define two functions, 3 and k, on 2x x [0, 11 as follows: for each (A, t) E 2x x [0, 11,
3(A,t)={h(A,s):OQ4t} and
lc(A, t) = u3(A, t).
THE FUNDAMENTAL
THEOREM
165
Since h is continuous, it follows easily that F maps 2.’ x [0, l] into 2”-’ and that 3 is continuous. Hence, by (1) and (2) of Exercise 11.5, we see that k maps 2” x [0, l] into 2x and that k is continuous. Also, (a) k(A, 0) = U3(A, 0) = h(A, 0) = A for each A E 2x and, since h(A, 1) = X for each A E 2x, (b) k(A, 1) = W(A, 1) = X for each A E 2x. Now, temporarily, fix A E 2x. Let (YA={k(A,t):O
= {h({a},t)
: a E A}
and f(4
t) = @(A,
t).
Since h is continuous, it follows easily that 6 maps 2” x [0, l] into 22x and that G is continuous. Thus, by (1) and (2) of Exercise 11.5, we see that f maps 2x x [0, l] into 2x and that f is continuous. Also, since h({z},O) = {x} for each {z} E PI(X), f(A, 0) = UG(A, 0) = U{ {a} : a E A} = A for each A E 2-y;
166
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
and, since h({z}, 1) = K for each {z} E Fi(X), f(A, 1) = UG(A, 1) = K for each A E 2”. Hence, the homotopy f shows that the identity map of 2x is inessential. Therefore, 2x is contractible. This proves that (4) of our theorem implies (1). .
X Contractible,
X Hereditarily
Indecomposable
We use the preceding theorem to show that 2x and C(X) are contractible for two diverse classesof continua X. Our results are in the following two corollaries. 20.2
Corollary.
If X is a contractible continuum, then 2aYand C(X)
are contractible. Proof. Let X be a contractible continuum. Then, since Fl(X) z X (Exercise 1.15), Fl(X) .is contractible (in itself). Hence, clearly, Fl(X) is contractible in C(X). Therefore, the corollary follows from 20.1. m 20.3 Corollary. If X is an hereditarily indecomposable continuum, then 2x and C(X) are contractible. Proof. Let X be an hereditarily indecomposable continuum. We show that Fi(X) is contractible in C(X) (and then appIy 20.1). By 13.4, there is a Whitney map, w, for C(X). We define h : Fl(X) x [0, w(X)] + C(X) asfollows. Let ({z}, t) E Fl(X) x [0, w(X)]; by considering an order arc in C(X) from {z} to X (14.6), we see from the continuity of w that there exists A,,t E w-‘(t) such that z E A,,,; furthermore, by 18.5, there is only one such A,,,; let
h({x), t) = ht. This defines the function h : Fl (X) x [0, w(X)] -+ C(X). It follows at once from the definition of a Whitney map in 13.1 that, for each {z} E Fl(X), h({x},O)
= {z} and h({x},w(X))
= X.
We prove that h is continuous. Let ({z}, t) E FI (X) x [0, w(X)], and let {({xcil, tz)lf% be a sequencethat converges in Fl(X) x [0, w(X)] to ({z}, t). We assume without loss of generality that the sequence {h({xi}, ti)}& converges to, say, B in C(X) (see3.1 and 3.7). Then, since xi E h({xi}, ti) for each i, we see that x E B; also, since w(h({xi},ti)) = ti for each i
PROPERTV (K) (KELLEY'S
PROPERTY)
167
and since 7u is continuous, we see that w(B) = t. Thus, B = /~({z},t). Therefore, we have proved that 11is cont,inuous. From the properties of h verified above, we have proved that Fl(S) is contractible in C(X). Therefore, by 20.1, 2dYand C(X) are contractible. n
Property
(6) (Kelley’s
Property)
We turn our attention to a useful sufficient condition for 2*Y and C(X) to be contractible. The condition is due to Kelley [12, p. 261and was called property (K) in [23, p. 5381. In 20.12 we show that if X is a continuum that has property (IC), then 2aYand C(X) are contractible. We give applications in 20.14 and 20.18. Let X be a continuum, and let d denote a metric for X. We say that X has property (K), or Kelley’s property, provided that for each E > 0, there is a 6 > 0 satisfying the following condition: (K) if p, q E ,Y such that d(p, q) < 6 and if il, E C(X) such that, p E A, then there exists B E C(X) such that q E B and H(il, B) < 6 (where H denotes the Hausdorff metric for C(X)). It is easy to seethat property (K) is a topological invariant. Even more is true: Property (6) is invariant under confluent, mappings (4.3 of [as]). We remark that open maps and monotone maps are confluent. We illustrate property (K) with the following examples. 20.4 Example. Any Peano continuum has property (K). To set this, let X be a Peano continuum with metric d, and let E > 0. Then there is a finite, open cover, U = {U,, . . . , UTL}, of X such that each cl; is connected and of diameter < E. Let S denote a Lebesgue number of U (Corollary 4d of [18, p. 241). Now, let p,q E X such that d(p,q) < S, and let, .4 CC(X) such that p E A. Then there exists j such that p, q E Uj. Let B = Uj U A. It follows easily that B E C(X), q E B, and H(A, B) < c. Therefore, we have proved that X has property (K). 20.5 Example. The sin(l/z)-continuum, pictured in (3) of Figure 20, p. 63, has property (K). However, slight modifications of the sin(l/s)continuum may produce continua that do not have property (K,): For example, see Figure 27 (top of the next page); in fact, of the uncountably many topologically different compactifications of a half-line with an arc as the remainder, the sin(l/z)-continuum is the only such compactification that has property (K) (2.5 of [21]).
168
V. SHAPE AND CONTRACTIBILITY
Modification
of Sin (i)-continuum
OF HYPERSPACES
without
Prop (K) (20.5)
Figure 27
20.6 Example. Any hereditarily indecomposable continuum has property (6). This follows at once from the uniform continuity of the homotopy h in the proof of 20.3. We conclude our examples by presenting a simple procedure for constructing continua that do not have property (K). We will use the procedure later (in 20.13). 20.7 Example. Let Y be a continuum that is not cik at some point yO; let A be an arc that is disjoint from Y, and let a, E A. Attach A to Y by identifying a, with y,,, and let X denote the resulting quotient space. Then, X is a continuum that does not have property (K). The fact that X is a continuum is due to 3.20 of [22, p. 431; the proof that X does not have property (K) is left as Exercise 20.19.
Theorem
about
Property
(K)
We prove that 2x and C(X) are contractible when X has property (K). We base the proof of the theorem on the function F, that we define as follows.
THEOREM ABOUTPROPERTY
(K)
Let X be a continuum, and let w be a Whitney by 13.4). Define F, on X x [0, w(X)] by letting F,,,(z, t) = {A E w-‘(t)
map for C(X)
= C,(X)
for each (z, t) E X x [0, w(X)]. We note the following preliminary 20.8 Proposition. X x [0, w(X)], F,(z,
(w exists
: 5 E A}
for each (z, t) E X x [0, w(X)]. It is useful to formulate containment hyperspaces C,(X): F,(z,t)
169
F,,, in terms of the
n w-‘(t)
fact about Fw:
Let X be a continuum. t) E 22x.
Then, for each (z,t)
E
Proof. Fix (2, t) E X x [0, w(X)]. Recall that since C(X) is compact (3.7), Cz(X) is compact (by Exercise 1.19). Thus, since F,(z,t) = Cz(X)n w-‘(t), we see that F,,,(z,t) is compact. Also, F,(s,t) # 0 (since there is an order arc, CY,in C(X) from {z} to X by 14.6, and since w(A) = t for some A E cr). Therefore, F,,,(z, t) E 22x. n The two lemmas that we prove next show that F, is continuous when X has property (K). We use the following terminology in the statements of the lemmas. and (M, do) be metric spaces, and let f be a Let (Y,~Y), (Z,dz), function from Y x Z into M. We say that f is equicontinuous in the first variable provided that for each c > 0, there exists 6 > 0 such that, for all 2 E 2, ddf(~l
l z),
f(312,z))
<
E whenever
dy
(YI,Y~)
<
6;
in other words, letting f,(y) = f(y, z ) for each y E Y and .z E Z, the family LfZ : z E Z} is equicontinuous (cf. definition preceding 17.2). Similarly, we say that f is equicontinuous in the second variable provided that for each E > 0, there exists 6 > 0 such that, for all y E Y, dll.i(f(y,a),f(y,zz))
<
c whenever dz(zl , ~2) < 6.
We do not assumethat X has property (K) in the following lemma. 20.9 Lemma, If X is a continuum, then the function F,:XxfO, 22x is equicontinuous in the second variable.
w(X)]+
170
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
As usual, H denotes the Hausdorff metric for 2”, and HH Proof. denotes the Hausdorff metric for 22x that is induced by H as in 2.1. Let J = [O,w(X)]. Let E > 0, and let 71(e) be as in 17.3. Fix ti, t2 E J such that Iti - t21 < V(E). We prove the lemma by proving that (*) H~(F,(z,tl),F,(z,tn)) < E for all 2 E X. Proof of (*). Fix p E X. Let Ai E F,,(p, ti). Since p E Al, there is an order arc, Q, in C(X) from (p} to X such that Al E o (use 14.6 twice unless Al = X). Since W(Q) = J, there exists A2 E CEsuch that w(A2) = tz. Since A2 E cr, p E AZ; thus, since ~(~42) = t2, we have that A2 E F,(p, tz). We show that H(A1, AZ) < c as follows: Since Al, A2 E CY,A, c A2 or AZ c AI; also, since ItI - tzl < q(e), 14-41)
-
74A2)I
=
It1
hence, by 17.3, we have that H(AI,Aa) F,(p, tz), we have shown that
-
t21
<
Q(E);
< E. Therefore, since A2 E
(1) Fw(p,tl) C NH(E,Fw(P,tz)). Similarly (by interchanging tl and t2 in the argument used to prove (l)), we have that F~(P, tl)). (2) F~(P, t2) C NH(f, By (1) and (2), HH(F,(P, tl), F,(p, tz)) < E (cf. Exercise 2.9). Therefore, we have proved (*). n 20.10 Lemma. If X is a continuum such that X has property (K), then F, : X x [0, w(X)] + 22x is equicontinuous in the first variable. Proof. Let d denote a metric for X; H denotes the Hausdorff metric for 2x , and HH denotes the Hausdorff metric for 22x that is induced by H as in 2.1. Let J = [O,w(X)]. Let E > 0. Let r](z) be as in 17.3. Since C(X) is compact (3.7) and w is continuous (13.I), w is uniformly continuous with respect to H (recall 3.1); hence, there exists y > 0 such that y < 5 and (1) Iw(K) - w(L)1 < q(H) whenever H(K,L) < y. Finally, since X has property (n), there exists 6 > 0 such that (2) if d(p,q) < 6 and p E A E C(X), then there exists B E C(X) such that q E B and H(A,B) < y. We prove the lemma by proving the following: (*) For all t E J, HH(F,(~, t), F,(q, t)) < E whenever d(p,q) < 6. Proof of (*): Fix t, E J, and fix p,q E X such that d(p,q) < 6. Let A E F,(p, to) (for the purpose of proving (6) below). Since p E A and
THEOREM ABOUT PROPERTY(K)
171
d(p, q) < 6, there exists B satisfying (2). By (2), H(A, B) < y; hence, by (11, IdA) - w(B)1 < v(;h Thus, since w(A)
= t, (because A E F,(p, to)), we have that
(3) Ito - w(B)1 < ~($1. Since B satisfies (2), q E B. Thus, it follows from 14.6 that there is an order arc, p, in C(X) from (q) to X such that B E ,!3 (use 14.6 twice unless B = X). Since w(p) = J, there exists C E 0 such that w(C) = t,. Since C E p, q E C; therefore, since w(C) = t,, we have that (4) c E Fw(q,tfJ). We show that H(A,C) < E. We start by showing Since B, C E p, B c C or C c B; also, by (3),
that H(B,C)
< 5:
b(C) - w(B)I = Ito - w(B)1 < 4;); hence, by 17.3, H(B,C) since y < $ (by choice),
< f. Therefore,
since H(A, B) < y (by (2)) and
(5) H(A, C) I H(A, B) + H(B, C) < y + f < E. By (4) and (5), we have proved that (6) Fwb, to) c NH(E, Fdq, to)). Similarly (by interchanging p and q in (2) and in the subsequent we have that (7) FuJ(q, to) c NH(% FdP, to)). By (6) and (7), HH(F,(~, t,), F,(q, we have proved (*). n
to)) < E (cf. Exercise
arguments),
2.9). Therefore,
As a consequenceof the lemmas that we just proved, we have the following relationship between property (K) and the function F,: 20.11 Proposition. Let X be a continuum, and let w be a Whitney map for C(X). Then, X has property (K) if and only if F, is continuous. Proof. The fact that property (K) implies F, is continuous follows from 20.9, 20.10, and the following general result: (#) Let (Y, dy), (2, d.z), and (M, do) be metric spaces,and let D denote the usual metric for Y x 2 (i.e., D = Jw); if f : Y x 2 + M is equicontinuous in each variable (separately), then f is uniformly continuous. The proof of (#) is simply done by taking the minimum of two deltas and using the triangle inequality (noting that dy 5 D and dz 5 D). Conversely, assumethat F, : X x [0, w(X)] + 22x is continuous (as usual, we assumethat the metric for 22x is the Hausdorff metric HH that
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V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
is induced, as in 2.1, by the Hausdorff metric H for 2x). Then, since X x [0, w(X)] is compact, F, is uniformly continuous. Therefore, using the definitions of F, and HH, it follows easily that X has property (K). n We are ready to prove our theorem about property (K) and contractibility. We note that the proof is, in reality, a generalization of what we did in proving 20.3. 20.12 Theorem. If X is a continuum that has property (K,), then 2x and C(X) are contractible. Proof. We show that FI (X) is contractible in 2x and apply 20.1. Let w be a Whitney map for C(X) (w exists by 13.4). By 20.11, F, : x x [O, w(X)] -+ 29 is continuous (recall 20.8). Let 2) = Fl (X) x [0, w(X)]. Define a function, h, on V as follows: h({z},t)
= UF,(z,t)
for each ({z},t)
E Z?.
Note that the natural map of Fl(X) onto X (i.e., {z} -+ Z) is continuous; thus, since F, is continuous, we see that h is a continuous function from 2, into 2x by using (1) and (2) of Exercise 11.5. Furthermore, using the definition of a Whitney map (13.1), we see that for each {z} E Fl(X), h({z), 0) = uF&,O)
= u{ {4}
= (~1
and W(z), w(X)) = uFw( 2, w(X)) = U{X}
= x.
Thus, h is a homotopy that showsthat Fl (X) is contractible in 2x. Therefore, by 20.1, 2x and C(X) are contractible. m The converse of 20.12 is false; the example below gives a classof continua for which it is easy to seethis. We remark that the continuum in Figure 27, p. 168 also shows that the converse of 20.12 is false (Exercise 20.20). 20.13 Example. Any contractible continuum that does not have property (K) showsthat the converse of 20.12 is false (by 20.2). We can obtain many contractible continua, X, that do not have property (K) as follows: Start with any contractible continuum, Y, that is not cik at some point y0 (e.g., the cone over any nonlocally connected compactum); then, let X = Y U A be as in 20.7. It is easy to see that the continuum X just constructed is contractible (deform A to y0 = a, with a homotopy that leaves the points of Y fixed; then deform Y to a point); and, as noted in 20.7, X does not have property (6).
X PEANO, X HOMOGENEOUS
X Peano,
173
X Homogeneous
We showed earlier that 2x and C(X) are contractible when X is a contractible continuum and when X is an hereditarily indecomposable continuum (20.2 and 20.3). We now use 20.12 to show that 2x and C(X) are contractible for two more classes of continua X. Our results are in 20.14 and 20.18. Since we have already proved the Curtis-Schori Theorem in 11.3, our first result below is of no substantive interest to us now; however, it is worth noting the easy proof based only on results in this section. 20.14
Corollary.
If X is a Peano continuum, then 2x and C(X) are
contractible. Proof. By 20.4, X has property (K). Therefore, the corollary follows from 20.12. n
We will show that 2x and C(X) are contractible when X is any homogeneous continuum. First, we prove the theorem in 20.17. The proof of 20.17 uses a completely new idea - upper semi-continuous set-valued functions. The theorem in 20.17 concerns a pointwise version of property (tc). We discussthese ideas in turn. Let (Y,Ty) and (Z,Tz) be topological spaces, and let F : Y + CL(Z) (USC) at p be a function. Then we say that F is upper semi-continuous (where p E Y) provided that whenever U E Tz such that F(p) C U, then there exists V E Ty such that p E V and F(y) c U for all y E V (Figure 28, top of the next page). If F is USCat every point of Y, then we say that F is USC.(For a discussion of possible origins for the terminology, see3.25 of [22, p. 461 and [22, p. 1031.) Let us note a general type of example of a USCfunction: If X and Y are compacta and f is a continuous function of X onto Y, then it is easy to see that f-’ : Y + 2x is USC. Upper semi-continuous functions are a unifying tool that can be used to prove a number of basic theorems; for applications to results about the Cantor set and space-filling curves, seeChapter VII of [22] and [22, pp. 1251271. We note the following easy-to-prove fact about USCfunctions. 20.15 Proposition. Let Y and 2 be compacta, let F : Y -+ 2z be a function, and let p E Y. Then, F is USCat p if and only if lim sup Flyi) c J’(P) wfi enever {yi)gi is a sequencein Y converging to p. We also note the following result; for a proof see ]18, pp. 70-711and [17, p. 3941(also, [17, pp. 207-2081).
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V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
F is USCat p Figure 28
20.16 Proposition. Let Y and Z be compacta. If F : Y -+ 2z is USC, then the points of continuity of F form a dense, Ga set in Y. We now turn our attention to the pointwise version of property (n), obtained by simply stating the definition of property (K) as before but holding p fixed (thus, 6 depends on E and p). Explicitly, let X be a continuum with metric d, and let p E X; we say that X has property (K) at p provided that for any E > 0, there exists S = 6(p,~) > 0 such that if q E X such that d(p,q) < b and if A E C(X) such that p E A, then there exists B E C(X) such that q E B and H(A, B) < E. For example, consider the continuum X in Figure 27, p. 168; X has property (K) at all points except the points on the upper half of the vertical arc on the left that lie below the top point. We see easily that a continuum has property (K) if and only if it has property (K) at each of its points (use Lebesgue numbers of covers [18, pp. 23-241). Also, clearly, if a continuum is cik at p, then the continuum has property (K) at p. We prove the following interesting result:
20.17 Theorem. If X is a continuum, then X has property (K) at each point of a dense, GJ set in X.
ORIGINAL
SOURCES
175
Proof. For each 2 E X, let F(z) = CZ(X) (where C,(X) is the containment hyperspace for x in C(X)). We see that F(z) is compact (by 3.7 and Exercise 1.19); and we see that F(z) is connected by using 14.6. Hence, F is a function from X into C(C(X)). We show that, F is USCby using 20.15. Let p E X, and let {xi}zo,l converge in X to p. Let il E lim sup F(xi). Then, A = Lim Aicj) where Ai E F(xicj,) for each J = 1,2,. . .. Thus, since xi(j) E Ai for each j and since {zicj,}~=i converges to p, we see that p E A; hence, A E F(p). Thus, we have shown that lim sup F(x,) c F(p). Therefore, by 20.15, F : X + C(C(X)) is USCat p. This proves that F is USC. It is easy to seethat F is continuous at x if and only if X has property (K) at 5. Therefore, since F is USC,the theorem now follows from 20.16. n
Now, we apply 20.12 and 20.17 to the class of homogeneouscontinua:
20.18
Corollary.
If X is a homogeneous continuum, then 2x and
C(X) are contractible.
Proof. By 20.17, X has property (K) at some point p. Thus, since X is homogeneous, it follows easily that X has property (K) at every point. Hence, X has property (K). Therefore, by 20.12, 2x and C(X) are contractible. n
An excellent survey (with proofs) of property (K) has been done by Acosta [l].
Original
Sources
Property (K) is 3.2 of Kelley [12], and property (K) at a point is from Wardle [28]; 20.1 (without (3)) is 3.1 of [12]; 20.2 is 16.8 of [23] but should be credited to Kelley (see16.9 of [23]); 20.3 for C(X) is in 2.3 of Krasinkiewicz [15] (a special case is in 3.5 of Rhee [26]), and 20.3 for 2.Y was noted in 1.76 of [23]; 20.4 was noted in the proof of 4.1 of [12]; 20.5 for the sin(l/x)continuum is from [12, pp. 26-271; 20.6 is from the proof of 3.5 of [26] and from 3.1 of [28]; 20.9-20.11 are from the proof of 3.3 of [12] (the “if” part of 20.11 is from 2.4 of [28]); 20.12 is 3.3 of [12]; 20.14 is due to Wojdyslawski [30] although the proof given here is from [12]; 20.17 is 2.3 of [28]; and 20.18 is 2.7 of [28]. The technique of using USCfunctions as in the proof of 20.17 has been used in other contexts; for example, see Fort [9].
V. SHAPE AND CONTRACTIBILITY
176
OF HYPERSPACES
Exercises 20.19 Exercise. do not have property
Prove that continua X that are constructed (K).
as in 20.7
20.20 Exercise. Prove that 2x and C(X) are contractible when X is the continuum in Figure 27, p. 168. (See the comment preceding 20.13.) 20.21 Exercise. Give an example of a compactification, line with an arc as the remainder such that 2x and C(X) tractible. (Compare with 20.5 and Exercise 20.20.)
X, of a halfare not con-
20.22 Exercise. Let X be a continuum. (K), then 2x and C(X) are contractible.
has property
If 2x or C(X)
20.23 Exercise. Let X be a continuum, let ‘U = 2x or C(X), and let w be a Whitney map for %. If ?i is contractible, then w-‘([t, w(X)]) is contractible for any t 2 w(X). [Hint: Make use of Exercise 11.5.1 20.24
Exercise.
Let X be a continuum, and let Y be a retract of X. (1) If 2x or (equivalently) C(X) is. contractible, then 2’ and C(Y) are contractible. (2) If X has property
(K), then Y has property
(K).
Remark. We see from (2) of 20.24 that if a Cartesian product, IIgiYi, of continua has property (K), then each Y, has property (K). However, surprisingly enough, there is a continuum, X, such that X has property (K) and yet X x X fails to have property (K): X consists of two oppositelydirected, disjoint spirals in R2 approaching the unit circle S’ (as in (8) of Figure 20, p. 63, without the “handle”). The proof that X x X does not have property (K) is in 4.7 of [28]. 20.25 Exercise. Let X be a continuum such that 2” or (equivalently) C(X) is contractible. If Y is a continuum that is a continuous image of X under an open map, then 2y and C(Y) are contractible. [Hint: Use 4.3 to prove the following preliminary result (whose converse is also true): Let f be a continuous function of a compactum, X, onto a compactum, Y; if f is an open map, then Lim f-r(&) = f-‘(B) whenever Lim Bi = B and Bi, B E 2y (i.e., f-’ : 2y -+ 2” is continuous (by 4.8 and 3.1)).]
177
REFERENCES
20.26 Exercise. Prove that monotone maps preserve property (n); that is, if X is a continuum that has property (K) and if f is a monotone map of X onto a continuum Y, then Y has property (K). However, show that monotone maps need not preserve property (K) at a point; that is, there are continua, X and Y, for which there is a monotone map, f, of X onto Y such that X has property (6) at a particular point, p, but Y does not have property (K) at f(p). 20.27 Exercise. Show that the analogue of 20.25 for monotone maps is false. [Hint: Find a monotone map of a contractible continuum onto the continuum in Figure 25, p. 158.1 20.28 Exercise. If X is a continuum erty (K), then X has property (K). [Hint: Make use of Exercise 11.5.1
such that 2x or C(X)
has prop-
Remark. Regarding converses of 20.28, the converse for 2x is false: The continuum with two spirals in the Remark following 20.24 shows this [5]. It is not known if the converse of 20.28 for C(X) is true. 20.29 Exercise. Let X be an hereditarily indecomposable continuum. If A is an arcwise connected subcontinuum of C(X), then A is contractible (in itself). [Hint: Let w be a Whitney map for C(X) (13.4). There exists M E A such that w(M) = lub(w[A]). Use 18.1, 18.4, and 18.8 to show that M > L for each L E A. Hence, for each L E A, there is a segment, (TL, in C(X) from L to n/f (16.9); moreover, cr~([O, 11) C A (18.8) and CJL is uniquely determined by L (16.5, 16.6, and 18.6). Define a homotopy; use 17.4 in proving that the homotopy is continuous.] Remark. The result in 20.29 is due to Krasinkiewicz [15, p. 1791. For results related to 20.29, see 123, pp. 115-1211. We use 20.29 in the proof of 22.17.
References 1.
2.
Gerard0 Acosta, Hiperespacios y la propiedad de Kelley, Tesis, Universidad Autonoma de Coahuila, Escuela de Matematicas, Saltillo, Coahuila, Mexico, 1994. K. Borsuk, Theory of Retracts, Monografie Matematyczne, Vol. 44, Polish Scientific Publishers, Warszawa, Poland, 1967.
178 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
V. SHAPE AND CONTRACTIBILITY
OF HYPERSPACES
K. Borsuk, Theory of Shape, Monografie Matematyczne, Vol. 59, Polish Scientific Publishers, Warszawa, Poland, 1975. X which has no confluent Wlodzimierz J. Charatonik, A continuum Whitney map for 2.‘, Proc. Amer. Math. Sot. 92 (1984), 313-314. Wlodzimierz J. Charatonik, Hyperspaces and the property of Kellq, Bull. Pol. Acad. Sci. 30 (1982), 457.-459. C. Eberhart, and S. 13. Nadler, Jr., The dimension of certain hyperspaces, Bull. Pol. Acad. Sci. 19 (1971), 1027--1034. Samuel Eilenberg, Transformations continues en circonfkewe et la topologie du plan, Fund. Math. 26 (1936), 61.-112. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, N.J., 1952. RI. Ii. Fort, Jr., E.qsential and non esscntzal fixed points, Amer. 3. Math. 72 (1950), 315-322. D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91-97. Alejandro Illanes, Monotone and open Whitney maps, Proc. Amer. Math. Sot. 98 (1986), 516-518. Amer. Math. Sot. .J. L. Kelley, Hyperspaces of a continuum, Trans. 52 (1942), 22-36. .J. Krasinkiewicz, Certain properties of hyperspaces, Bull. Pol. Acad. Sci. 21 (1973), 705.-710. .J. Krasinkicwicz, Curves whzch are continuous images of tree-like continua are movable, Fund. Math. 89 (1975), 233-260. J. Krasinkicwicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84 (1974), 1755186. J. Krasinkiewicz, Shape properties of hyperspaces, Fund. Math. 101 (1978), 79-91. K. Kuratowski, Topology, Vol. I, Acad. Press, New York, N.Y., 1966. K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. A. Y. W. Lau, Acyclicity and dimension of hyperspace of szlbcontintia, Bull. Pol. Acad. Sci. 22 (1974), 1139-1141. and hyperspaces, Can. J. Math. M. M. McWaters, Arcs, semigroups, 20 (1968), 1207-1210. Sam B. Nadler, Jr., Confluent images of the sinusoidal curve, Houston J. Math. 3 (1977), 515-519. Sam B. Nadler, Jr., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992.
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23. Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 24. Sam B. Nadler, Jr., Inverse limits and m&coherence, Bull. Amer. Math. Sot. 76 (1970), 411-414. 25. Sam B. Nadler, Jr., and Thelma West, Size levelsfor arcs, Fund. Math. 141 (1992), 243-255. 26. C. J. Rhee, On dimension of hyperspace of a metric co7ltinuum, Bull. Sot. Royale des Sci. de Liege 38 (1969), 602-604. 27. Jack Segal, Hyperspaces of the inverse limit space, Proc. Amer. Math. Sot. 10 (1959), 7066709. 28. Roger W. Wardle, On a property of J. L. Kelley, Houston J. Math. 3 (1977), 291-299. 29. Gordon Thomas Whyburn, Analytic Topology, Amer. Math. Sot. Colloq. Publ., Vol. 28, Amer. Math. Sot., Providence, RI., 1942. 30. M. Wojdyslawski, Sur la contractilite’ des hyperespaces des continus localement connexes, Fund. Math. 30 (1938), 247-252.
VI. Hyperspaces and the Fixed Point Property We discuss the fixed point property for the hyperspaces 2x and C(X). In the first section, we give preliminary results about the fixed point property in general. In the second section, we determine classes of continua, X, for which 2x and C(X) have the fixed point property (a few results are only for C(X)). We mention that there is a continuum, Y, such that Y has the fixed point property but C(Y) does not have the fixed point property. We discussedthis in the part of section 7 entitled Knaster’s Question. For details, see [35] or [32, pp. 292-2941.
21.
Preliminaries: Maps,
Brouwer’s Theorem, Universal Lokuciewski’s Theorem
A fixed point of a function, f, is a point, p, such that f(p) = p. A space, Y, is said to have the jixed point property provided that every continuous function from Y into Y has a fixed point. It is often difficult to determine that a spacehas the fixed point property; to paraphrase Bing, the fixed point property is elusive [l]. We present general results that will help us show that certain hyperspaces have the fixed point property. The most celebrated result about the fixed point property is the Brouwer theorem: 21.1 Brouwer point property.
Fixed
Point
Theorem
[5]. Any n-cell has the fixed
Different types of proofs of the Brouwer theorem are in various texts: For a proof using degree of a map, see [9, pp. 340-3411; for a proof using 181
182
VI. HYPERSPACES
AND THE FIXED
POINT
PROPERTY
multiple integrals, see [lo, pp. 468-4701;for a homological proof, see [13, p. 3011or [21, p. 41; and for a combinatorial proof, see [25, pp. 313-3141. For the next three results, recall the definitions of retract and absolute retract (AR) from section 9. 21.2 Proposition. If a space, Y, has the fixed point property, then every retract of Y has the fixed point property. Proof. Let Z be a retract of Y, and let T be a retraction of Y onto 2. Let f : Z + Z be continuous. Then, since f o r is a continuous function = p and for(Y) c 2, from Y into Y, for has a fixed point, p. Since for(p) clearly p E 2; thus, r(p) = p. Therefore, since f[r(p)] = p, we have that f(P)
= P. n
Let (Y, d) be a metric space, let 2 be a topological space, and let E > 0. A function f : Y + Z is said to be an e-map (with respect to d) provided that f is continuous and for each t E f(Y), diameterd[f-l(z)]
< E.
We will often consider the following condition: (#) There is an c-map from Y to Z for each E > 0. It is important to keep two facts in mind when we consider (#). First, the e-maps in (#) are assumedto be e-maps with respect to a fixed metric on Y; otherwise, (#) would be true for any Y for the following reason: For each E > 0, we may choose a metric, d,, for Y such that d, 5 Eand d, gives the topology on Y. Second, when Y is compact, (#) is a topological invariant; therefore, when Y is compact and we consider (#), we do not need to (and will not) mention the metric on to refer to a retraction that is also an We use the term c-retraction e-map. We note that if T, : Y + Y is an c-retraction with respect to a metric d for Y, then rC is within e of the identity map on Y. (Proof: If r,(y) = z, hence, d(y,z) < E and, thus, d(y,r,(y)) < E). then y,z E r;l(z); 21.3 Proposition. Let (Y, d) be a compact metric space. Assume that for each E> 0, there is a continuous function, fC, of Y into a subset, Ye, of Y such that Y, has the fixed point property and fc is within Eof the identity map on Y (the second condition holds, e.g., when fc is an E-retraction). Then, Y has the fixed point property. Proof. Let f : Y + Y be continuous. Let E > 0. Note that fc of/Y, is a continuous function from Y, into Y,. Hence, fc o f ]Y, has a fixed point,
21. PRELIMINARIES:
BROUWER’S
THEOREM,
UNIVERSAL
MAPS,.
. . 183
p,. Thus, since fL is within Eof the identity map on Y, d(f(Pc),PJ
= d(f(P~)7f~[f(Pr)l)
< E.
Now, from what we have shown, there is a sequence, {pi)Er, such that d(f(p;),pi) < i for each i = 1,2,. _.. Since Y is compact, somesubsequence of {pi)El converges to, say, p. It follows easily that f(p) = p. n 21.4
Corollary.
Every AR has the fixed point property.
Proof. By Urysohn’s metrization t,heorem [25, p. 2411,every AR can be embedded in the Hilbert cube I m = Hz_,[O, 11%. Hence, by 21.2, it suffices to prove that P has the fixed point property. We prove that I” has the fixed point property by using 21.2 and 21.3. For each n = 1,2,..., let 1,” be the n-cell in P given by I,”
= {(t$E1
E P
: t, = 0 for each i 2 n + l},
and let rn be the natural retraction of IO0 onto I,” given by rll((ti)zl)
= (tl,. . . ,t,,O,O,.
. .) for each (ti)zr
E I”.
We see that for each n, T,~ is a 2-“-retraction with respect to the metric d, for IO0 (defined in the introduction to Chapter II). Also, for each 11,I,” has the fixed point property by 21.1. Therefore, by 21.3, IO” has the fixed point property. n We begin a development that leads to Lokuciewski’s theorem in 21.7. We introduce several important notions along the way. Let Y and Z be topological spaces. A function u : Y -+ Z is said to be a universal map provided that u is continuous and u has a coincidence with every continuous function from 1. into Z (meaning that if g : Y -+ Z is continuous, then there exists y f Y such that g(y) = u(y)). We note that if u : Y + 2 is a universal map, then 2 has the fixed point property! The proof is simple: Let f : Z -+ Z be continuous; then there exists y E Y such that f o u(y) = u(y). A companion of this result is in the next proposition. Some other elementary yet enlightening facts about universal maps are in the exercises at the end of the section. 21.5 Proposition. Let Y be a compacturn. Assume that for each E > 0, there is a universal e-map, u,, of Y to a space, Z,. Then, Y has the fixed point property.
184
VI. HYPERSPACES AND THE FIXED POINT PROPERTY
Proof. Let f : Y + Y be continuous. Let E > 0. Since u, : Y -+ Z, is a universal map and since u, 0 f : Y + Z, is continuous, there exists p, E Y such that u, o f(p,) = ue(pe). Thus, letting d denote the metric on Y, we seethat since u, is an c-map,
Therefore, since Y is compact, it follows as in the last part of the proof of 21.3 that f has a fixed point. n We want a way to detect universal maps - a way that is applicable to hyperspaces. We will see in the next section that the condition in the following definition is particularly suitable. Let Y be a topological space, and let 2 be an n-cell (recall that 82 denotes the manifold boundary of 2). A function ‘p : Y -+ Z is said to be an AH-essential map provided that cpis continuous and (p(‘p-’ (82) can not be extended to a continuous function of Y into dZ. (Note: AH stands for Alexandroff-Hopf.) 21.6 Proposition. Let Y be a topological space, and let Z be an ncell. A map of Y into Z is universal if and only if it is AH-essential. Proof. Let Bn = {(zi)~!i E R” : & $ < l}. It is helpful to use the geometry of P; thus, we prove the proposition for Z = B”, and then we show how the result for Z = B” implies the proposition. Let f : 1’ + Bn be continuous. We prove each half of the equivalence in the proposition by arguing contrapositively. Assume first that f is not universal. Then there is a continuous function g : 1’ -+ B” such that g(y) # f(y) for each ME Y. Hence, for each y E Y, there is a unique convex, unbounded half-line, Hy, in R” that begins at g(y) and goesthrough f(y); clearly, H,ndB” consistsof one and only one point, which we denote by F(y). This defines a function F : Y + aBn. It is easy to see that F is continuous and that F(y) = f(y) for each y E fvl(aBn). Therefore, f is not AH-essential. Conversely, assumethat f is not AH-essential. Then, f) f -l(i3Bn) can be extended to a continuous function k : Y + dB”. It follows easily that f(y) # -k(y) for each y E Y. Therefore, since -k is a continuous function from Y into Bn, f is not universal. We have proved 21.6 for Z = Bn. Therefore, 21.6 now follows from two similar, easy-to-prove facts: Let Xi and X2 be topological spaces, let f’ : Xi -+ X2, and let h be a homeomorphism of Xs onto a spaceXs . Then (1) f’ is universal if and only if h o f’ is universal;
21. PRELIMINARIES:
BROUWER’S
THEOREM,
UNIVERSAL
MAPS,.
.
185
(2) f’ is AH-essential (here X2 is an n-cell) if and only if h o f’ is AHessential. (Note: The proof of (2) usesthe fact that h(aX2) = dXs [22, pp. 95-961). n
21.7 Lokuciewski’s Theorem [27]. Let Y be a compacturn. Assume that for each E > 0, there is an AH-essential e-map of Y to an n,-cell. Then, Y has the fixed point property. Proof.
In view of 21.6, the theorem is a special case of 21.5. n
We end the section with a proposition that enables us to know that certain maps that we will construct are AH-essential. Recall from section 19 the definitions of contractible with respect to a space and essential map (not to be confused with AH-essential map). 21.8 Proposition. Let Y be a topological space that is contractible with respect to an (n-1)-sphere for some n. Let 2 be an n-cell, and let g : Y -+ 2 be continuous. If there exists A c g-‘(a2) such that
glA : A + BZ is an essential map, then g is an AH-essential map (and conversely). Proof. Assume that g is not AH-essential. Then, by definition, g]g-‘(82) can be extended to a continuous function G : Y + 82. Since Y is contractible with respect to 82, G is inessential. Hence, by restricting a homotopy that joins G to a constant map, we see that Gig-‘(32)
: 9-l (82) + a2 is inessential.
Thus, since G]g-‘(aZ> = gig-‘(aZ>, + az is inessential. 919-l (az) : g-‘(az) Therefore, for any A c g-l (aZ), glA : A + dZ is inessential. This proves half of 21.8. To prove the converse half of 21.8, we note that any inessential map of any closed subset of Y into aZ can be extended to a continuous function of Y into dZ ([29]; for the easier casewhen Y is a metric space, see [4, p. 941 or [22, p. 871). Thus, if g : Y + Z is an AH-essential map, it is clear that g]g-‘(aZ) : g-‘(dZ) --+ dZ must be an essential map. n Open questions about universal maps and hyperspaces are in 83.23-
83.27.
186
VI. HYPERSPACES
AND THE FIXED
Original
POINT
PROPERTY
Sources
The first systematic study of universal maps was done in a seriesof papers by Holsztynski begining with [18] (seereferences in [17]); AH-essential maps have their origin in dimension theory, where they are called essential maps (e.g., see Theorem III.5 of Nagata [36, p, 591and compare with [18] and 21.6 here); Dyer [ll] may have been the first to use AH-essential maps in fixed pont theory; 21.2 is from 7 of Borsuk [3, p. 1551;21.4 is the corollary in [3, p. 1611(the main part - I” has the fixed point property - is due to Schauder [39, p. 521; for more historical information, see [lo, p. 4701); 21.5 is a consequenceof Proposition 3 and Lemma 1 of Holsztytiski [19]; 21.6 is 1.1 of Holsztynski [20, p. 1481although the “if” part was first proved by Lokuciewski [27], a special case of which is Lemma 1 of [ll]; 21.8 is essentially Proposition 10 of [19, p. 4341,where it is stated for the casewhere Y is a binormal, contractible space.
Exercises 21.9 Exercise. A topological space, 2, has the fixed point property if and only if the identity map on 2 is universal. A topological space, 2, has the fixed point property if and only if there is a universal map of some topological space to Z. 21.10 Exercise.
If 21: I’ -+ Z is a universal map, then ,u(Y) = Z.
21.11 Exercise. Any continuous function from a connected topological space onto an arc is a universal map. 21.12 Exercise. Give an example of a continuous function from [O,l] onto a simple triod that is not a universal map. 21.13 Exercise. Let Y and Z be compacta., and let f : k’ --+ Z be continuous. Assume that for each E > 0, there is an e-map, fC, of Z to a space, Z,, such that fc o f : Y + Z, is a universal map. Then, f is a universal map. 21.14 Exercise. Use 21.3 to show that the sin(l/z)-continuum in (3) of Figure 20, p. 63 has the fixed point property. Use 21.13 to prove that every continuous function from any continuum onto the sin(l/z)-continuum is a universal map. Remark. The second part of 21.14 is st,ronger than the first part and is true for any arc-like continuum. It is not known if arc-like continua are
22. HYPERSPACES
WITH THE FIXED
POINT
PROPERTY
187
the only continua for which the second part of 21.14 is true (Problem 1 of [26]; see the discussion about span in [30, pp. 254-2551).
22.
Hyperspaces
with
the Fixed
Point
Property
Our main results are about hyperspaces of the following types of continua: Peano continua, arc-like continua, circle-like continua, dendroids, and hereditarily indecomposable continua. We restrict our attention to the hyperspaces 2” and C(X).
Peano Continua We begin with Peano continua even though our theorem is eclipsed by the Curtis-Schori Theorem in 11.3. 22.1 Theorem. If X is a Peano continuum, then 2x and C(X) have the fixed point property. Proof. By 10.8, 2x and C(X) are ARs. Therefore, by 21.4, 2x and C(X) have the fixed point property. n
Arc-like
Continua
Interesting classesof continua are generated by the following idea: Start with a simple continuum P (usually a polyhedron), and consider all those continua, X, such that for each E > 0, there is an c-map, fc, of X onto P. Note that the decomposition spaces, D, = {f,-‘(p) : p E P}, are homeomorphic to P ([25, p. 1841or [30, p. 441). Hence, the statement that X admits an e-map onto P for each E > 0 means, in somewhat descriptive terms, that X becomes homeomorphic to P by identifying points of arbitrarily small sets in X; in this sense,X is “almost the same as” P. This leads to the following terminology. Let P be a given continuum; a continuum, X, is said to be P-like provided that for each e > 0, there is an e-map of X onto P. Thus, we have the notions of arc-like ([0, l]-like), circle-like (S’-like), n-cell-like (In-like), etc. Let us illustrate the notion of P-like with the continua in Figure 20, p. 63: The continua in (1) and (3) of Figure 20 are arc-like; the continua in (2), (4), and (5) are circle-like; the continuum in (6) is noose-like; and the continua in (7) and (8) are figure eight-like as well as theta curve-like. The reader will find it a pleasant exercise to verify what we have just said about the continua in Figure 20.
188
VI. HYPERSPACES
AND THE FIXED
POINT
PROPERTY
Arc-like continua are the simplest caseof P-like continua; nevertheless, arc-like continua have a long and distinguished history. Intriguing events that led to the interest in arc-like continua are discussed in [30, pp. 22& 2291. The events centered around continua all of which were eventually shown to be the same continuum; this continuum became known as the pseudo-arc. An in-depth treatment of the pseudo-arc is in the forthcoming book The Pseudo-arc by Wayne Lewis, to be published in 1998 by Marcel Dekker, Inc. The general theory of arc-like continua is covered in Chapter XII of [30]. For general interest we mention some properties of arc-like continua: Arc-like continua are one-dimensional, acyclic, a-triadic, embeddable in the plane, and have the fixed point property; arc-like continua are inverse limits of arcs; there are uncountably many, topologically different arc-like continua, and there is an arc-like continuum that contains every arc-like continuum. In 22.5 we show that C(X) has the fixed point property for any arclike continuum X. The propositions in 22.3 and 22.4 are two of the main ingredients in the proof of 22.5. We will also use 22.3 and 22.4 to prove other results. 22.2 Lemma. Let Y = A U 3 be a continuum, where A and B are subcontinua of Y such that An B is not connected, say A II B = EIF. Let S = Si U SZ be a simple closed curve, where Si and SZ are arcs such that Si 0 S2 = {p, q}. Let k : Y + S be a continuous function such that k(A) c Sl, k(B) c S2, k(E) = p, k(F) = q. Then, k is an essential map. Proof. We essentially proved this lemma when we proved 19.7: Let S: and Sl be as in the proof of 19.7. Let h : S + S’ be a homeomorphism such that h(S) = S:,h(Sz)
= S’,h(p)
= (l,O),h(q)
= (-170).
Let g = h o li : Y + S’. Then, by the proof of 19.7, g is an essential map. Therefore, since k = h-i o g, we see that k is an essential map (if there were a homotopy ‘p : Y x [0, l] + S joining k to a constant map, then h o cpwould be a homotopy joining g to a constant map). n We will make use of induced maps between hyperspaces. Let X and Y be compacta and let f : X + Y be continuous. The induced map f * : 2.’ -+ 2y is given by f*(A)
= f(A)
for each A E 2x;
ARC-LIKE
CONTINUA
189
the induced map f : C(X) + C(Y) is f*]C(X). Note from 13.3 that f’ and (hence) f are continuous. (Sometimes f * is denoted by 2f and f is denoted by C(f) .) 22.3 Proposition. Let X be a continuum. If f is any continuous function of X onto [O,I], then the induced map f : C(X) + C([O, 11) is an AH-essential map. Proof.
Let I = [0, 11. Recall from 5.1.1 that C(I) is a 2-cell and that
XT(I) = Fl(1) u Co(I) u Cl(I). We first find a subcontinuum, Y, of C(X) such that f^(Y : Y -+ &Z’(l) is an essential map. Since f(X) = I, there exist zo,zi G X such that f(zs) = 0 and f(si) = 1. By 14.6, there is an order arc, cq, in C(X) from {Q} to X for each i = 0 and 1. Let A = crc U ~1, let D = Fi(X), and let
jJ=duB. We show that the assumptions in 22.2 are satisfied: A is a continuum (since X E cron oi), B is a continuum (by Exercise 1.15), A n B = {{ze}, {zi}} (since og and or are order arcs), and (hence) y is a continuum; also, from the formula for j, j(d)
= Co(r)
u
Cl (I),
f(B)
= K(I),
f^({xo))
=
{o),
j(h))
= (1).
Therefore, by 22.2, we have that (#)f^]Y : Y + aC(l) is an essential map. Now, recall that C(X) is contractible with respect to S’ by 19.6. Therefore, by (#), we may apply 21.8 to conclude that f^ : C(X) -+ C(I) is an AH-essential map. n An application of 22.3 that has nothing to do with the fixed point property is in the addendum (22.18). 22.4 Proposition. Let X and Y be compacta, let d denote a metric for X, and let c > 0. If f : X + Y is an E-map with respect to d, then the induced map f” : 2x + 2y is an E-map with respect to Hd (as defined in 2.1). Hence, for any 7-1c 2x, f*]‘Ft is an &map with respect to Hd\‘ft x 7-L.
190
VI. HYPERSPACES
AND THE FIXED
POINT PROPERTY
Proof. Let K E f*(2X), and let A, B E (f*)-‘(K). We show that &(A> B) < e. Let a E A. Then, since f(A) = f(B), there exists b E B such that f(b) = f(u). Thus, since f is an e-map with respect to d, d(a, b) < E. Hence, we have proved that A c N~(E, B). A similar argument shows that B c Ncj(c, A). Therefore, Hd(A, B) < e (by Exercise 2.9). n We now prove our theorem about arc-like continua. 22.5 Theorem. point property.
If X is an arc-like continuum, then C(X) has the fixed
Proof. By the definition of arc-like, there is an e-map, fc, [0, l] for each z > 0. Recall that C([O, 11) is a 2-cell (5.1). By induced map fE : C(X) + C([O,11) is an AH-essential map; by fc is an e-map. Therefore, by Lokuciewski’s Theorem in 21.7, the fixed point property. W
of X onto 22.3, each 22.4, each C(X) has
It is not known if 2x has the fixed point property when X is an arc-like continuum. (See 22.9, 22.11, and [33, pp. 753-7541; also, see 83.23 for an approach that may solve the problem.) A stronger result than 22.5 is in Exercise 22.23 (seethe Remark following the exercise).
Circle-like
Continua
In 22.7 we prove that C(X) has the fixed point property for any circlelike continuum X. We proved a similar theorem for arc-like continua in 22.5. We note that there are circle-like continua that are also arc-like ([6] for the equivalence of arc-like and chainable, see [30, pp. 234-2371). In view of 22.5, we need only prove our theorem for circle-like continua that are not arc-like. The following lemma yields a convenient way to distinguish circle-like continua that are not arc-like from those that are arc-like. 22.6 Lemma. Let X be a nondegenerate continuum. If there is an inessential c-map fi : X -+ S1 for each c > 0, then X is arc-like (and conversely).
CIRCLE-LIKE
CONTINUA
191
Proof. By the theorem in Exercise 19.20, each f< has a lift cpc : X -+ R’. Since fc = expocp, and since fe is an E-map, it follows easily that cpf is an c-map. Now, let 6 = diameter (X), and note that 6 > 0. Since cpc is an c-map, clearly qe(X) is nondegenerate for each e 5 6. Hence, for each 6 5 6, qr is an e-map of the continuum S onto a nondegenerate interval. Therefore, X is arc-like (obtain an e-map of X onto [0, l] for each E 5 6 by composing (pc with a homeomorphism of P,(X) onto [O, 11). Conversely, assume that X is arc-like. Then, for each E > 0, there is an E-map, gc, of X onto [0, 11. Let fc = expog, for each E > 0. We see that each ft : X + S’ is an inessential e-map: fc is inessential since gr is a lift of fc (hence, 19.20.1 applies); f6 is an c-map since gr is an e-map and expl[O, l] is one-to-one. n Regarding the converse part of the lemma we just proved, a stronger result is true: Every arc-like continuum is crANR (12.47 of [30, p. 2631). 22.7 Theorem. If X is a circle-like continuum, then C(X) fixed point property.
has the
Proof. By the definition of circle-like, there is an e-map, fi, of X onto S’ for each E > 0. By 22.4, each induced map fc : C(X) + C(S’) is an E-map. By 22.5, we can assumefor the proof that X is not arc-like. Thus, since each fc : X + S’ is an e-map, 22.6 gives us a 6 > 0 such that fc is an essential map for each E < 6. Now, recall from 5.2 that C(S’) is a 2-cell and that aC(Sr) = Fl(Sl). S’mce fc : X + S’ is an essential map for each E< 6, clearly (by Exercise 1.15)
fc IF1 (X) : FI (X) -+ dC(S’)
.IS an essential map for each E < 6.
Thus, since C(X) is crS’ by 19.6, we have by 21.8 that fc : C(X) + C(S’) is an AH-essential map for each e < 6. Therefore, since fc is also an c-map, we can apply Lokuciewski’s Theorem in 21.7 to conclude that C(X) has the fixed point property. n As in the case of arc-like continua, we do not know if 2” has the fixed point property when X is a circle-like continuum (see83.23 for an approach that may solve the problem). The general approach used to prove 22.5 and 22.7 is useful in contexts other than hyperspaces; we illustrate this with Exercises 22.26-22.28.
192
VI. HYPERSPACES AND THE FIXED
A General
POINT
PROPERTY
Theorem
We prove a simple, general theorem that can be applied to a variety of continua. We illustrate the theorem with three examples (some more examples are in Exercise 22.19). Our main applications of the theorem are in the next part of the section, where we obtain results about hyperspaces of dendroids. 22.8 Theorem. Let X be a continuum. Assume that for each c > 0, there is a continuous function, jc, of X into a Peano continuum, X,, in X such that j< is within E of the identity map on X (e.g., when jc is an e-retraction). Then, 2x and C(X) have the fixed point property. Proof. Let d denote the metric for X. By assumption, fc is within & of the identity map on X with respect to d. Hence, we seeeasily that the induced map j: : 2-Y + 2xe is within c of the identity map on 2x with respect to Hd (as defined in 2.1). Also, 2xc has the fixed point property by 22.1. Therefore, we can apply 21.3 to conclude that 2dYhas the fixed point property. The proof for C(X) .is similar (using the induced maps jl : C(X)
-+ C(X,)).
n
22.9 Example. Let X be the sin(l/z)-continuum in (3) of Figure 20, p. 63. It is easy to see that there is an c-retraction of X onto an arc for each E > 0 (a horizontal projection to the right). Therefore, by 22.8, 2” and C(X) have the fixed point property (the caseof C(X) also follows from 22.5). 22.10 Example. The Warsaw circle, W, is in (4) of Figure 20, p. 63. Let X = W u U, where U is the bounded component of R2 - W; X is called the Warsaw disk. For each E > 0, there is an e-retraction of X onto a 2-cell (a horizontal projection to the right). Therefore, by 22.8, 2x and C(X) have the fixed point property. 22.11 Example. Let X be the indecomposable continuum that is often called the Buckethandle continuum: X is the closure in R2 of the one-toone, continuous image of [0, co) that weaves through the ‘<endpoints” of the Cantor Middle-third set as in Figure 29 (top of the next page). It can be seen from Figure 29 that, for each E > 0, there is a map, je, of X onto an arc in X such that jc is within e of the identity map on X (a formula for j< is in 2.7 of [31]). Therefore, by 22.8, 2x and C(X) have the fixed point property (the case of C(X) also follows from 22.5).
193
DENDROIDS
Buckethandle
continuum
Figure 29
Dendroids A dendroid is an arcwise connected, hereditarily unicoherent continuum (hereditarily &coherent means that each subcontinuum is unicoherent; we discussedunicoherence in section 19 preceding 19.7). Examples of dendroids are in Figures 22, p. 84,23, p. 92, and 25, p. 158. For general information we state some properties of dendroids: Dendroids are uniquely arcwise connected; every subcontinuum of a dendroid is a dendroid; dendroids are hereditarily decomposable (see [2, p. 171or 11.54 of [30, p. 2261); dendroids are one-dimensional (see 13.57 of [30, p. 3051); dendroids are acyclic, i.e., crS’ (see (f) of 12.69 of [30, p. 2711); and dendroids have the fixed point property [2] (a1so see the follow-up paper [16]). We do not know if 2x or C(X) has the fixed point property for all dendroids X. However, we will show that 2x and C(X) have the fixed point property for two kinds of dendroids - smooth dendroids and fans. We use the following notation in defining the notion of a smooth dendroid: Let X be a dendroid, and let x, y E X. If x # y, then we let xy denote the unique arc in X with end points x and y; if x = y, then xy = {x}.
194
VI. HYPERSPACES
AND THE FIXED
POINT
PROPERTY
Let X be a dendroid, and let p E X; then, X is said to be smooth at p provided that whenever {xcL}zl is a sequencein X converging to a point, x, in X, then Lim px+ = px. A dendroid that is smooth at some point is called a smooth dendroid [8]. The dendroid in Figure 22, p. 84 is a smooth dendroid. It is smooth at the point furthest to the right (but not at the point furthest to the left). The dendroid in Figure 23, p. 92 is also a smooth dendroid. On the other hand, the dendroid in Figure 25, p. 158 is not a smooth dendroid. For an interesting hyperspace characterization of smooth dendroids, see Theorem 7 of [28, p. 1161. A tree is a finite graph that contains no simple closed curve. Obviously, a tree is a locally connected dendroid. Fugate [15, p. 2611has proved the following theorem: If X is a smooth dendroid and 6 > 0, then there is a retraction, r,, of X onto a tree such that T, is within e of the identity map on X. Thus, we have the following result: 22.12 Theorem. If X is a smooth dendroid, then 2dYand C(X) have the fixed point property. Proof. By the theorem of F’ugate stated above, we can apply 22.8 to seethat 2’ and C(X) have the fixed point property. n It is not known if there is an e-retraction of every dendroid onto a tree for each E > 0 (the problem is mentioned in [15, p. 2611). If there were always such a retraction, then 22.12 would be true for all dendroids. A fan is a clendroid, X, for which only one point is a common end point of three or more arcs in X that are otherwise disjoint [7]. The harmonic fan in Figure 23, p. 92 is a smooth fan. The fan in Figure 30 (top of the next page) is not a smooth fan. Fugate [14, p. 1201has proved the following theorem: If X is a fan and E > 0, then there is a retraction, r,, of X onto a tree (which in this case is a simple n-od or an arc) such that T, is within Eof the identity map on X. Thus, we have the following result: 22.13 Theorem. property.
If X is a fan, then 2x and C(X) have the fixed point
Proof. By the theorem of F’ugate just stated, we can apply 22.8 to see that 2x and C(X) have the fixed point property. m The proofs of the next three theorems use the variations of 22.8 that are in Exercises 22.20, 22.21, and 22.22.
DENDROIDS
195
Non-smooth fan Figure 30
22.14 Theorem. Let X be a finite or countably infinite Cartesianproduct, where each coordinate space is a fan or a smooth dendroid. Then, 2x and C(X) have the fixed point property. Proof. Use Fugate’s theorems stated above 22.12 and 22.13 to apply the result in Exercise 22.20. n 22.15 Theorem. Let X = Cone(Y), where Y is a fan or a smooth dendroid. Then, 2x and C(X) have the fixed point property. Proof. Use Fugate’s theorems stated above 22.12 and 22.13 to apply the result in Exercise 22.21. n
Let Y be a topological space. The suspensionover Y, which we denote by C(Y), is the quotient space obtained from Y x [-1, l] by shrinking Y x { -1) and Y x { 1) to (different) points.
196
VI. HYPERSPACES AND THE FIXED POINT PROPERTY
We remark that if Y is a compactum, then C(Y) can be constructed geometrically in a manner similar to that of the geometric cone over Y (near the beginning of section 7). 22.16 Theorem. Let X = C(Y), where Y is a fan or a smooth droid. Then, 2x and C(X) have the fixed point property. Proof. Use F&ate’s theorems the result in Exercise 22.22. n
den-
stated above 22.12 and 22.13 to apply
We remark that the theorem about cones in 22.15 can be considered a corollary of the theorem about suspensions in 22.16 (see Exercise 22.24).
Hereditarily
Indecomposable
Continua
22.17 Theorem. Let X be an hereditarily indecomposable continuum. Then, C(X) has the fixed point property. Moreover, if 31 is an arcwise connected subcontinuum of C(X), then ‘?i has the fixed point property. Proof. Young [43, p. 8831 has proved the following theorem: If Y is a uniquely arcwise connected, contractible Hausdorff continuum, then Y has the fixed point property. Therefore, our result for C(X) follows from 18.8 and 20.3. Similarly, our result for 3-1 follows from 18.8 and Exercise 20.29. n
We note that 22.17 also follows from 14.9 and Exercise 18.10 by using the following theorem of Young [44, p. 4931: Any arcwise connected Hausdorff space in which every monotone increasing sequence of arcs is contained in an arc has the fixed point property. It is not known whether or not 2x has the fixed point property for any particular nondegenerate, hereditarily indecomposable continuum.
Addendum:
Dim[C(X)]
As promised earlier, we give an application fixed point property. 22.18 Theorem.
If Xis a nondegenerate
2 2 of 22.3 not related to the
continuum,
then dim[C(X)]
>
L.
Proof. Since X is a nondegenerate continuum, there is a continuous function, f, of X onto [O,l]. By 22.3, f : C(X) + C([O,l]) is an AHessential map. Thus, since aC([O,l]) x S’ by 5.1, S’ is not an extensor
ORIGINAL for the closed subsets of C(X). dim[C(X)] > 2. H
SOURCES
Therefore,
by Theorem VI 4 of [22, p. 831,
More about dim[C(X)] is in Chapter XI; of particular tion 73, which is devoted to a proof of 73.9.
Original
197
interest
is sec-
Sources
The result in 22.1 has been known since Wojdyslawski’s paper [42]; 22.2 is 1.3 of Krasinkiewicz [24]; 22.3 is extrapolated from the proof of 4.1 in [24]; 22.4 is 2.5 of [24], w h ere it is stated only for C(X); 22.5 is Theorem 3 of Segal [40], although the proof here is from [24]; 22.6 is 3.1 of [24], where it is stated for X being circle-like; 22.7 is 4.2 of [24]; 22.11 is 3.5 of [33]; 22.12 and 22.13 are 7.8.2 of [32]; 22.17 for C(X) is from Rogers [38, pp. 284-2851 (the second proof we mentioned is from Krasinkiewicz [23, p. 1801); 22.18 is Theorem 1 of Eberhart-Nadler [12].
Exercises 22.19 Exercise. Prove that 2” and C(X) have the fixed point property when X is the continuum in Figure 25, p. 158, the continuum in Figure 27, p. 168, and the continuum called the Infinite Ladder in Figure 31 (top of the next page). 22.20 Exercise. Let X = IIy=iXi, where n 5 00 and each Xi is a continuum. Assume that for each i and each E > 0, there is a continuous function, fi,c, of Xi onto a Peano continuum in Xi such that fi,c is within E of the identity map on X. Then, 2x and C(X) have the fixed point property. (Used in the proof of 22.14.) 22.21 Exercise. Let X = Cone(Y), where Y is a continuum. Assume that for each E > 0, there is a continuous function, fL, of Y onto a Peano continuum in Y such that fL is within E of the identity map on Y. Then, 2x and C(X) have the fixed point property. (Used in the proof of 22.15.) 22.22 Exercise. Let X = C(Y), where Y is a continuum. Assume that for each E > 0, there is a continuous function, fc, of Y onto a Peano continuum in Y such that fc is within z of the identity map on Y. Then, 2” and C(X) have the fixed point property. (Used in the proof of 22.16.) 22.23 Exercise. Let X be an arc-like continuum. If f is any continuous function of a continuum, Y, onto X, then the induced map f : C(Y) + C(X) is a universal map.
198
VI. HYPERSPACES
AND THE FIXED
POINT
PROPERTY
I
Infinite Ladder (22.19) Figure 31
[Hint: Use Exercise 21.13.1 Remark. Recall from the comment preceding 21.5 that the range of a universal map has the fixed point property; hence, 22.23 implies 22.5. Also, 22.23 implies that arc-like continua are in Class(W) (a continuum, X, is in Class(W) provided that if f is any continuous function of a continuum, Y, onto X and if A is any subcontinuum of X, then there is a subcontinuum, B, of Y such that f(B) = A). The fact that arc-like continua are in Class(W) follows immediately from 22.23 and Exercise 21.10. This result about Class(W) is due to Read [37]; 22.23 is 2.11 of [34]. See section 35. 22.24 Exercise. Let Y be a continuum. If 2’(‘) has the fixed point property, then 2Co”e(Y) has the fixed point property; if C&(Y)) has the fixed point property, then C(Cone(Y)) has the fixed point property. (Recall comment following the proof of 22.16.)
REFERENCES
199
22.25 Exercise. If X is an arc-like continuum, then FZ(X) has the fixed point property. [Hint: Use that Fz(X) is crS’ when X is an arc-like continuum; we sketch a proof of this fact: (1) X is crS’ since X is an inverse limit of arcs (12.19 and 12.47 of [30]); (2) X x X IS ’ crS’ by (l), 7.5 of [41, p. 2281,and 6.2 of [41, p. 2261; (3) t,he natural map of X x X onto F2(S) is an open map; (4) Fz(X) 1scrS’ by (3), 7.5 of [41, p. 2281, and 6.2 of [41, p. 2261. If X and Y are arc-like continua, then X x 1’ has t,he 22.26 Exercise. fixed point property. (See second comment following the proof of 22.7.) [Hint: Use that X x Y is crS’ (see (2) of Hint for 22.25).] If X is an arc-like continuum, the Cone(X) and C(X) 22.27 Exercise. have the fixed point property. (See second comment following the proof of 22.7.) If X is a circle-like continuum, then Cone(X) has the 22.28 Exercise. fixed point property. (See second comment following the proof of 22.7.)
References 1. 2.
R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76, (1969), 119-132. K. Borsuk, A theorem on fixed points, Bull. Pol. Acad. Sci. 2, (1954), 17-20.
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K. Borsuk, Sur les re’tractes, Fund. Math. 17, (1931), 152-170. K. Borsuk, Theory of Retracts, Monografie Matematyczne, Vol. 44, Polish Scientific Publishers, Warszawa, Poland, 1967.
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L. E. J. Brouwer, Beweis des Jordanschen Kuruensatzes, Math. Ann. 69, (1910), 169-175. C. E. Burgess, Chainable continua and indecomposability, Pac. J. Math. 9, (1959), 653-659.
6. 7. 8. 9.
-7. J. Charatonik, On fans, Dissertationes Math. 54, (1967). J. J. Charatonik and Carl Eberhart, On smooth dendroids, Fund. Math. 67, (1970), 297-322.
JamesDugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1967 (third printing). 10. Nelson Dunford and Jacob T. Schwartz, Linear Operators: Part I: General Theory, Interscience Publishers, New York, N.Y., 1964 (second printing).
200
VI. HYPERSPACES
AND THE FIXED
POINT
PROPERTY
11. Eldon Dyer, A fi~edpoint theorem, Proc. Amer. Math. Sot. 7, (1956), 662-672. 12. C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces,Bull. Pol. Acad. Sci. 19, (1971), 1027-1034. 13. S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, N.J., 1952. 14. J. B. Fugate, Retracting fans onto finite fans, Fund. Math. 71, (1971), 113-125. 15. J. B. Fugate, Small retractions of smooth dendroids onto trees, Fund. Math. 71, (1971), 255-262. 16. W. Holsztyliski, Fixed points of arcwise connected spaces,Fund. Math. 64, (1969), 289-312. 17. W. Holsztynski, On the product and composition of universal mappings of manifolds into cubes,Proc. Amer. Math. Sot. 58, (1976), 311-314. 18. W. Holsztynski, Une generalisation du the’ordme de Brouwer sur Ees points invariants, Bull. Pol. Acad. Sci. 12, (1964), 603-606. 19. W. Holsztynski, Universal mappings and fixed point theorems, Bull. Pol. Acad. Sci. 15, (1967), 433-438. 20. W. Holsztynski, Universality of the product mappings onto products of I” and snake-like spaces,Fund. Math. 64, (1969), 147-155. 21. Sze-Tsen Hu, Homotopy Theory, Acad. Press, New York, N.Y., 1959. 22. Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton University Press, Princeton, N.J., 1948. 23. J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84, (1974), 175-186. 24. J. Krasinkiewicz, On the hyperspacesof snake-like and circle-like continua, Fund. Math. 83, (1974), 155-164. 25. K. Kuratowski, Topology, Vol. I, Acad. Press, New York, N.Y., 1966. 26. A. Lelek, Some problems concerning curves, Colloq. Math. 23, (1971), 93-98. 27. 0. W. Lokuciewski, On a theorem on fixed points, Yen. Mat. Hayk 12 3(75), (1957), 171-172 (Russian). 28. Lewis Lum, Weakly smooth dendroids, Fund. Math. 83, (1974), lll120. 29. Kiiti Morita, On generalizations of Borsuk’s homotopy extension theorem, Fund. Math. 88, (1975), l-6. 30. Sam B. Nadler, Jr., Comtinuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992.
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31. Sam B. Nadler, Jr., c-SeEections, Proc. Amer. Math. Sot. 114, (1992), 287-293. 32. Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 33. Sam B. Nadler, Jr., Induced universal maps and some hyperspaces with the fixed point property, Proc. Amer. Math. Sot. 100, (1987), 749754. 34. Sam B. Nadler, Jr., Universal mappings and weakly confluent mappings, Fund. Math. 110, (1980), 221-235. 35. Sam B. Nadler, Jr. and J. T. Rogers, Jr., A note on hyperspaces and the fixed point property, Colloq. Math. 25, (1972), 255-257. Pub. Co., Am36. J. Nagata, Modern Dimension Theory, North-Holland sterdam, Holland, 1965. 37. David R. Read, Confluent and related mappings, Colloq. Math. 29, (1974), 233-239. 38. James T. Rogers, Jr., The cone = hyperspace property, Can. J. Math. 24, (1972), 279-285. 39. J. Schauder, Zur Theorie stetiger Abbildungen in Funktionalr&umen, Math. Z. 26, (1927), 47-65. 40. J. Segal, A fixed point theorem for the hyperspace of a snake-like continuum, Fund. Math. 50, (1962), 237-248. Analytic Topology, Amer. Math. Sot. 41. Gordon Thomas Whyburn, Colloq. Publ., Vol. 28, Amer. Math. Sot., Providence, R.I., 1942. Re’tractes absolus et hyperespaces des continus, 42. M. Wojdyslawski, Fund. Math. 32, (1939), 184-192. 43. G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Sot. 11, (1960), 880-884. 44. G.S. Young, The introduction of local connectivity by change of topology, Amer. J. Math. 68, (1946), 479-494.
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Part
Two
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VII.
Whitney
Maps
Part two of this book is mainly devoted to hyperspaces of continua. From now on the letter X will denote a nondegenerate continuum.
23.
Existence
and Extensions
Whitney maps were defined in section 13 where it was shown the existence of Whitney maps for the hyperspaces of compacta. Although in this chapter we are only considering hyperspaces of continua, it can be noted in Definition 13.1 that Whitney maps can be defined for the hyperspace, CL(Y), of an arbitrary topological space Y. However, the following theorem by Watanabe shows limitations for the existence of Whitney maps. 23.1 Theorem [37, Theorem 11. Let Y be a metric space. Then the following conditions are equivalent: (a) Y admits a Whitney map ,u : (CL(Y),H,) -+ R' for some bounded metric p for Y, (b) Y admits a Whitney map ~1: (CL(Y),Tv) (c) Y has the Lindeliif property, (d) Y is separable. Related to Theorem 23.1, Watanabe
+ R',
asked the following
question.
23.2 Question [37, Question 61. Let Y be a connected, locally nected metric space. Then does Theorem 23.1 hold for CLC(Y)?
con-
In [35], Ward recalled that certain hyperspace problems are more tractable if hyperspaces are regarded as a special type of partially ordered spaces. He certainly succeeded, he extended the notion of Whitney map to partially ordered spaces and, using some Nachbin’s results, in [36] he proved a useful extension theorem for Whitney maps. In Theorem 16.10 we showed how to prove the following theorem using Theorem 3.1 of Ward [36].
205
206
VII.
WHITNEY
MAPS
23.3 Theorem [36, Theorem 3.11 (see Theorem 16.10). Let X be a continuum. Let ?i be a nonempty closed subset of 2x. Then every Whitney map for 3c can be extended to a Whitney map for 2x. In particular, this theorem answered in the positive, the respective questions by Hughes and Nadler: “For a continuum X, can every Whitney map for C(X) be extended to a Whitney map for 2x?” ([3I, Question 14.71.51) and if “Y is a subcontinuum of X, can every Whitney map for C(Y) be extended to a Whitney map for C(X)“? For a nonempty closed subset 3c of 2x, in [18] it was considered the space of Whitney maps, W(R), defined on 7-1, with the “sup. metric”. Following Ward’s ideas a self-contained proof of Theorem 23.3 was given in [18, Theorem 4.21. This was done in order to prove the following result. 23.4 Theorem [18, Theorem 4.21. Let X be a continuum. Let ?k be a nonempty closed subset of 2x. Then there exists an embedding ‘p : such that cp(p) is an extension of /.J for each p E W(Z). W(U) -+ W(27
Exercises 23.5 Exercise. Let X be a continuum and let d denote a metric for X. Let p : F(X) + R’ be a uniformly continuous Whitney map, where F(X) is the space of finite subsets of X defined in 1.8. Then p can be extended to a Whitney map for 2x if and only if for every E > 0 there exists 6 > 0 such that if A, B E F(X), A c B and p(B) -p(A) < 6, then &(A, B) < 6 (compare with Lemma 17.3). 23.6 Exercise. Give an example of a continuum X and a Whitney map p for F(X) such that p can not be extended to a Whitney map for
2X. 23.7 Exercise.
Let X be a continuum.
Let p be a Whitney
map for
2x. For each A E 2x, define PA : 2x + R1 by PA(B) = p(A U B)p(B). Then PA is a Whitney map for 2x, and if A # B, then PA # pg. 23.8 Exercise [20, Theorem 1.21. Let X be a continuum. Let A be a compact subset of C(X) - ({X} U Fl(X)); A is said to be an anti-chain if A, B E A and A c B implies that A = B. Then A is a Whitney level for C(X) (see Definition 24.17) if and only if A is an anti-chain and A intersects every order arc Q in C(X) such that Q joins a one-point set and X. 23.9 Exercise [20, Proposition 1.31. Let X be a continuum. Let WL(X) = {d E C(C(X)) : A is a Whitney level for C(X)} (see Definition
24. OPEN AND MONOTONE WHITNEY
MAPS FOR 2’
24.17). By Theorem 19.9, each Whitney level for C(C(X)). Then WL(X) h as a metric as subspace WL(X) is called the space of Whitney levels and it and in [23]. Given A$ E WL(X), define A 5 B existsBEUsuchthatAcBanddefined
207
C(X) is an element of of C(C(X)). The space has been studied in [20] if for each A E A there
(a) if d c f?, then A = B, (b) A + a if and only if for every A E A, there exists B E B such that AcB#B, (c) A + B if and only if there exists a Whitney map p for C(X) and there are numbers 0 5 s < t < p(X) such that A = p-r(s) and 0 = p-‘(t), (d) A < t? or f? + A if and only if A n f? = 8, (e) A 5 U if and only if for every B E 8, there exists A E A such that A c B, (f) if {dn}r=r and {B,}~zI are sequences in WL(X) such that A, 5 23, for every n and A, + A and B, + B, then A < B.
24.
Open and Monotone
Whitney
Maps
for 2x
Monotone and open maps were defined after Theorem 19.9 and in Exercise 14.25, respectively. We repeat here these definitions for the convenience of the reader. 24.1
Definition.
A continuous function between continua f : X + Y
is said to be (a) monotone, provided that f-‘(y) is connected for every y E Y or, equivalently (see [40, (2.2) Chapter VIII] or Exercise 24.20), provided that f-‘(B) is connected for every subcontinuum B of f(X); (b) open, provided that f(U) is open in f(Y) X;
for each open subset U of
(c) confttient, provided that f is surjective and for each subcontinuum B of Y and each component C of f-‘(B), we have f(C) = B. Confluent maps were introduced by J. 3. Charatonik in [2]. In that paper he observed that monotone surjective maps and open surjective maps are confluent (Exercise 24.21). Eberhart and Nadler proved in [9] that, for a continuum X, Whitney maps for C(X) are monotone and open (see Theorem 19.9 or Exercise 27.6 and Exercise 14.25). However, Whitney maps for 2x need not have either of these properties, even when X = [0, l] (see Exercise 19.16). This fact is generalized in the following theorem.
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MAPS
24.2 Theorem. Let X be a continuum. Then there exists a Whitney map for 2x which is neither open nor monotone. Proof. Let d denote a metric for X. Fix two points p # q in X. Let E= v. Define A = cZx(B(c,p)) and B = cZx(B(c,q)). Then AnB = 0. Let 31 = {{a, b} E 2x : a E A and b E B}. For each point {a, b} E 3c, where a E A and b E B, define d{a, b}) = f + d(a,p) -t d(b,q). Since the function A x B + ‘2-fgiven by (a, b) + (a, b} is a homeomormap for phism, ?i is compact and Y is continuous. Hence u is a Whitney %. Notice that, if {a, b} E 7-l, then Y({a, b}) = 5 if and only if {a, b} = {P,9>. By Theorem 23.3, there exists a Whitney map p : 2x + R’ which extends u. Let U be the e-ball in 2” around {p, q}. If C E U - {p, q}, then C c A U B, C rl A # 0 and C n B # 0. Then C contains an element {a, b} of Ifl - {p,q). Hence CL(C)2 d{a,b)) > iIt follows that $ E p(U) c [&,p(X)] and U n p-‘(i) = {p, q}. Then p(U) is not open in [O,p(X)] and {p, q} is an isolated point of p-‘(i). Therefore, ~1is neither open nor monotone. n It is easy to construct confluent maps which are not monotone. However, for Whitney maps for 2x both concepts coincide as it is shown in the following theorem by W. J. Charatonik. 24.3 Theorem [6, Theorem 21. Let X be a continuum. For a Whitney map p : 2x + [0, p(X)], the following conditions are equivalent: (a) p is monotone, (b) p is confluent, (c) p-l ([0, t]) is connected for each t E [O,p(X)]. Proof. (J. R. Prajs) The equivalence (a) H (c) is included in Exercise 19.19. The implication (a) + (b) follows from Exercise 24.21. (b) + (c). Let t E [O,p(X)]. We will show that p-l([O, t]) is connected. Let C be a component of p-‘([O, t]). Since fi is confluent, p(C) = [0, t]. Then C n ,u-‘(0) # 0. Hence every component of cl-‘(IO, t]) intersects the connected set p-‘(O) = Fr(X). Therefore, p-‘([O, t]) is connected. n
24. OPEN
AND MONOTONE
WHITNEY
24.4 Corollary. Let X be a continuum. map for 2x is monotone.
MAPS
FOR 2’
209
Then every open Whitney
The following two theorems show that monotoneity of Whitney maps is intimately related to finite sets. 24.5 Theorem [22, Theorem 1.21. Let X be a continuum. Let p : 2.Y --+ R’ be a Whitney map. Then the following statements are equivalent: (a) p is monotone, (b) F,(X) n P-’ (P, 4) 1sconnected for every n = 1,2,. . and every t E P? P(X)l. Proof. Let d denote a metric for X. (a) =+ (b). Suppose, to the contrary, that there exists a positive integer n and there exists t E [O,p(X)] such that Fn(X) np-l ([O, t]) = 31 U K, where Z and K are nonempty disjoint compact subsetsof 2”. Since Fr (X) is a connected subset of F,(X) n p-l ([0, t]), we may assumethat Fr (X) C K. Then ‘14does not have singletons, so the number s = minp(‘fl) is positive and s 5 t. ForeachA={2l,...,z,)E31(mOsuchthatBH,(dA,A)n x = p)and the sets B(6A, zr), . . , B(6A, z,) are pairwise disjoint. Let U = U{ BH~ (by, A) : A E R}. Then U is an open subset of 2x. We will prove that (1) u n
p-l(~)
= 7f n p-l(s)
Clearly, Rnp-‘(s) c Unp-r(s). Now take an element B E Unp-‘(s). Then there exists A = {XI,. . . , zm} E Ifl such that Hd(B, A) < bA. Then, for each i 5 m, it is possible to find a point yi E B fl B(~,J, xi). Let Bo = {yr,. . ,ym}. Then BO c B and Hd(Bo,A) < 6.4. Thus Bo E F,(X) r-?p-‘([O, t]) = 7-tUK and B. 4 K. Hence Bo E ‘fl. By the definition of s, s 5 p(Bo) < p(B) = s. This implies that Bo = B. Then B E 3c rl pL-’ (s). This completes the proof of the equality (1). By (l), ?f n p-l(s) is an open and closed subset of p-‘(s). Since we are assuming that p-l(s) is connected, we conclude that p-‘(s) C 7-f. In particular, every element in p-‘(s) is finite. But s > 0 implies that /.~-l (s) has nondegenerate subcontinua of X. This contradiction completes the proof that (a) 3 (b). (b) 3 (a). By Theorem 24.3, we only have to prove that pL-l([O,t]) is connected for each t E [O,p(X)]. Since, for each n = 1,2,. . ., F,(X) n p-l([O, t]) is connected and contains the connected set PI(X), we obtain that the set C = U { F,(X) n p-‘([O, t]) :-n = 1,2,. . .} is connected. Since
210
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WHITNEY
MAPS
every element A in 2” can be approximated by finite sets contained in A, we conclude that C is dense in p-l ([0, t]). Therefore, p-‘([O, t]) is connected. n
24.6 Theorem [22, Theorem 1.41. If X is a Peano continuum, then a Whitney map /J for 2.’ is monotone if and only if (plF(X))-“(t) is connected for every t E [0, p(X)]. Proof.
NECESSITY. Let t E [O,/.J(X)]. Here, we will use that F(X) when X is locally connected ([22, Lemma I.31 or [7, Lemma 3.61). Then WConly need to prove that the sets 8’(X) n p-l ([O, f]) and are connected. By Theorem 24.5, F,(X) np-I ([0, t]) F(S)n~-‘([t,jL(-~)l) is connected for every n = I, 2,. . . Since F(X) n CL-1([0,t]) = U{F,(X) n j1-l ([O, t]) : 11= 1,2,. . .} , we conclude that F(X) n 11-l ([0, t]) is connected. is connected, let A = In order to prove that F(X) n p-‘([t,p(X)]) Define f : X + {z)l>~..,~n}, B = (41,. . . ,qm} E F(X) n p-‘([t,p(X)]). F(-71) flp-‘([t,p(X)]) by f(z) = AU {z}. Then f is continuous and A, AU (41) E f(X). Thus A and AU { q1} are in the same component of F(X) n 11-l ([t, p(X)]). Repeating this procedure we can conclude that A and AU B arc in the same component of F(X) n p-‘([t,p(X)]). Similarly, B and Therefore, .4 U B are in the same component of F(X) n p’([t,p(S)]). F(S) n p”-‘([t, p(X)]) is connected.
is unicoherent
SUFFICIENCY. Let d denote a metric for X. Let t E [O,p(X)]. We is connected it may assumethat t < p(X). In order to prove that p-l(t) = p-l(t). is enough to prove that C&X(F(X) n p-l(t)) Let il E p-‘(t) and 6 > 0. Take a component C of A. Then C # X. By Theorem 12.12, there exists a subcontinuum D of X such that c C D C Nd(c, C) and C # D. Notice that D is not contained in A. Choose a point p E D - A. Since F(A) is dense in 2”, there exists B E F(A) such that p(B U {p}) > t and Hd(A, B) < E. Define g : D + F(X) by g(z) = p(B U {z}). If z E C, then g(a) 5 p(A) = t. 011 the other hand, g(p) > t. From the connectedness of D, there exists x E D such that g(z) = t. Notice that Hd(A, BU {z}) < E and B U {x} E F(X) flp-l(t)). This completes the proof that cEz.y(F(X) n p-’ (t)) = p-l (t) and the proof of the theorem. n The proof of Theorem 24.6 is based in the unicoherence of F(X) when X is a Peano continuum ([22, Lemma 1.31or [7, Lemma 3.61). We do not know the answers to the following questions. 24.7 Question [22, Question 1.51. Is Theorem 24.6 true without the hypothesis that X is a locally connected continuum?
24. OPEN
24.8 Question tinuum X?
AND MONOTONE
WHITNEY
[22, Question 1.61. Is F(X)
MAPS
FOR 2’
211
unicoherent for every con-
With respect to open Whitney maps for 2”, we have the following equivalences. 24.9 Theorem [22, Theorem 1.11, compare with [29, p. 68, Theorem 11. Let X be a continuum. Let p : 2aY-+ J = [O,p(X)] be a Whitney map. Then the following statements are equivalent: (a) p is open, (b) the function t + p-l(t)
from J into 22x is continuous,
(c) if t E J and {tn}rzl is a sequencein [0, t) such that t, + t, then p-l(&) + p--((t) (in 22x), (d) the local minima of I-Loccur only on Fi (X), and (e) plF(X)
: F(X)
-+ J is open.
Proof. Let d denote a metric for X. We will only prove that (d) * (e). The rest of the proof is left as Exercise 24.22. Let U be an open subset of F(X). Let A = {al,. . . , a,} E U and t = p(A). Let E > 0 be such that A E F(X) n (B(2e,ai), . . . , B(2c, a,)) c U, B(c,ar) # X and the sets ClX(B(GW)), . . .1 clx (B(E, a,)) are pairwise disjoint. Let E = X - (B(e,ui) u ... u B(c,u,)), E = {A U {e} : e E E} and s = min{p(D) : D E E}. Clearly, & is a compact subset of F(X) and s > t. Let 6 > 0 be such that S < E and if G, K E 2x and Hd(G, K) < S, then b(G) - AWI < s - t. We will consider the case t > 0; the case t = 0 is left as Exercise 24.22. In the caset > 0, A is not a local minimum of ,u. Then there exists B E 2-Y such that H(A, B) < 6 and p(B) < t. For each i E (1,. . . ,n}, choose a point bi E B n B(6,ui). Define Bc = {bi,. . . ,bn}. Then CL(&) < t. Let C be the component of cl-x (B(E, al)) that contains bi. By Theorem 12.10, < 6 there exists a point c E C n Bdx (B( e,ui)). Since H~(Au{c},&U{C}) t. s, we have ,u(&, u {c}) > and AA u 1~)) L Consider the set D = {Be U {z} : x E C}. Notice that 2) is a connected subset of U. Then [b(Bs),p(&, U {c})] c p(D) c p(U). Therefore t is an interior point of p(U). Thus, p]F(X) is open. n In [31, Questions 14.63 and 14.641,it was asked whether for every continuum X there exists a monotone (or an open) Whitney map for 2x. W. J. Charatonik answered both questions in the negative with the following example.
212
VII.
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MAPS
24.10 Example [3]. Let S denote the unit circle in R”. Define functions f, g : [l, co) + R2 by f(t) = (1 + f) exp(it) and g(t) = (1 - f, exp(-it), where exp is the exponential map. Then define x = s u f(P, m)) u g([l, 0)). For this space X, W. J. Charatonik showed that there are no confluent Whitney maps for 2” and then there are neither open nor monotone Whitney maps for 2”. Generalizing Charatonik’s ideas, Awartani introduced the notion of skew compactification of the real line. He proved ([l]) that if X is such a compactification, then 2” admits no confluent Whitney map. On the other hand, the following theorem shows that for every Peano continuum X, 2x admits open Whitney maps (see Exercise 24.24). 24.11 Theorem [17]. If the metric for the continuum X is convex (see Theorem 10.3), then the Whitney map defined in Exercise 13.8 is open. In [5, Question 251,W. J. Charatonik asked the following question: Does there exist,s a continuum X which has a monotone (an open) Whitney map for 2aYand such that 2aYis not contractible? This question was solved in [22] where three plane dendroids Xi, X2 and Xa were constructed such that: 1. 2aYl is not contractible and 2”’ admits open Whitney maps, 2. X2 is contractible and then 2”2 is contractible (see Corollary 20.2), 2aY2admits monotone Whitney maps but there are no open Whitney maps defined in 2”~) 3. X3 is contractible and 2”~ does not admit monotone Whitney maps. These examples show that, even for plane dendroids, the contractibility of 2~~is not related to the existence of open (or monotone) Whitney maps for 2*Y. 24.12 Definition. Let X be a continuum. A nonempty proper closed subset A of X is said to be an R3-set of X provided that there exists an open subset U of X and a sequenceof components {C,}~Yr of U such that A c U and A = lim inf C,, If an R3-set of X is connected, then it is called an R3-continuum of X. For a simple example of an R3-set, consider the continuum illustrated in Figure 25, p. 158. The set consisting of the point which joins the two
24. OPEN AND MONOTONE WHITNEY
MAPS FOR 2’
213
harmonic fans is an R3-continuum. The concept of R3-continuum was introduced by Czuba in [8], the existence of an R3-set in a continuum implies that it is not contractible (see [5, Theorem 2] or Theorem 78.15). More about R3-sets is in sections 58 and 78. In the following theorem by W. J. Charatonik, it is shown that the existence of an R3-set in a continuum X, implies that 2x does not admit a monotone (or open) Whitney map. 24.13 Theorem [5, Theorem 241. If a continuum set, then it has no monotone Whitney maps for 2.v.
X contains an R3-
Proof. Let A # X be an R3-set of X. Let U be an open subset of X thath contains A and let {Cn}F=r be a sequence of components of U such that A = lim inf C,. Fix a point p E A. Let V be an open subset of X such that A C V C clx (V) c U. Notice that U # X. For each point z E Bd,y (V), 2 4 lim inf C,. Then there exists cz > 0 = 0 for infinitely many positive integers n. and such that cZ.x (B(2c,, z))nC, p $! cl,~(B(2~,,z)). By the compactness of Bdx(V), there is m = 1,2,. . and there are xi, . . . , x, E Bd,y(V) such that Bdx(V) c B(c,,,x1)lJ~~~U B(%m,
xm).
Let E> 0 be such that E < min{c,, , , . . ,E~,,,} and
B(c,p) c v - [cZx(B(t,, ,x1)) u ‘~~uclx(~(~,*,xm))l. Let p be any fixed Whitney map for 2”. By Lemma 17.3, there exists s > 0 such that, for each B E p-l ([0, s]), diameter (B) < E. Since p E lim inf C, there exists a sequence{pn}p=r of points in X such that pn -+ p and p, E C, for each n = 1,2,. . . (see Exercise 78.36). Since the closed sets {p,p,,p,+l, . . .} tend to {p}, there exists N = 1,2,. . . such that LJ({P,PN,PN+I,. . .I) < s. Foreachi E {l,... ,m}, let ni > N be such that C,, nclx(B(2E,xi)) =
0. Let cti be the component of &x(V) containing pni. Then Di n cl.~(B(2c,x,)) = 0. By Theorem 12.9, there exist disjoint compact subsets H, and Ki of clx(V) such that Di C Hi, cl~(V)ncl~y(B(2~,x,)) C I(, and Hi U Ki = clx(V). Define Hi = H, - Bd.y(V) = (X - Ki) n V. Then Hi is an open subset of X. Letd={B~~L-l([O,s]):B~HIU~~.UHmandBnH~#Oforeach i 5 m}. Clearly, A is a closed subset of p-l ([0, s]). Notice that the set {Pm,.. . ,pn,} is an element of A. Thus A is nonempty.
214
VII.
WHITNEY
MAPS
We claim that A is open in pL-’ ([0, s]). In order to prove this, it is enough toshowthatd=B,whereB={BECL-‘([O,s]):BCH:U...UH~and B II H,! # 8 for each i < m}. It is clear that f? c A. Now, take B E A. Suppose that B n Bdx(V) # 0. Then there exists i 5 m such that B n clx(B(~,zi)) # 0. Since diameter (B) < E, B C cl~(V) f~ clx(B(2~,zi)) c K,. This is impossible since B n Hi # 0. The contradiction proves that B n Bdx(V) = 0. Hence B E A. We have proved that A = B. Therefore, A is open in p”-‘([O, s]). If p is monotone, then pL-’ ([0, s]) is connected. Thus A = p-’ ([0, s]). This is impossible since if 2 is a point in X - U, then {z} E pL-’ ([0, s]) - A. This contradiction proves that p is not monotone and completes the proof of the theorem. H W. J. Charatonik has posed the following question. 24.14 Question [5, Question 261. Give a necesssaryand/or sufficient conditions for a continuum X to have a confluent (a monotone, an open) Whitney map for 2x. The known classesof continua X for which 2x does not have monotone Whitney maps are: (a) Awartani’s skew compactifications of the real line ([l]), and (b) continua having R3-sets (Theorem 24.13). And the known classesof continua X for which 2” has open Whitney maps are: (a) Peano continua (Theorem 24.11) and (b) arc-smooth continua (see Theorems 25.9 and 25.11). 24.15 Question [22, 3.61. If X is an hereditarily indecomposable continuum, does 2x have monotone Whitney maps? 24.16 Question. If X is an hereditarily indecomposable continuum, can 2x have open Whitney maps? 24.17 Definition. Let X be a continuum and let ?l = C(X) or 2x. A Whitney level for 7-f (respectively, positive Whitney level) is a set of the form p-‘(t), where p is a Whitney map for 3c and t E [O,p(X)) (respectively, t E (O>PW))). Whitney levels for C(X) have played an important role in the study of C(X). They are continua (Theorem 19.9), for each fixed Whitney map p the Whitney levels corresponding to p constitute a continuous decomposition
EXERCISES
215
of C(X) ([31, Theorem 14.441) and, as we will see in Chapter VIII, Whitney levels for C(X) share many properties with X. On the other hand, Whitney levels for 2-’ are not necessarily continua. Furthermore, Goodykoontz and Nadler have shown that if X is a simple closed curve, then there is a Whitney map p for 2eY and there is t > 0 such that p-‘(t) contains an arc with nonempty interior ([15, Example 4.151). In this case, the Whitney map p is not open. We have seen that when a Whitney map 1-1for 2~~ is open, then the respective levels p-‘(t) are continua and constitute a continuous decomposition on 2*Y. We will see in the next section that, under additional assumptions, the Whitney levels of an open Whitney map for 2” have very nice properties. 24.18 Question. Let X be a continuum. What topological do the Whitney levels of open Whitney maps for 2-Y have? Question 24.18 is very general and offers many possibilities; some of them in the following questions. 24.19 Questions. Whitney map for 2”,
Let X be a continuum.
(a) are positive Whitney their points? (b) if X is contractible, (c) if the Whitney (e) if the Whitney connected?
at each one of
levels of ,U also contractible?
levels of p are contractible,
(d) if X is locally connected, are Whitney
we mention
Suppose that p is an open
levels of p infinite-dimensional are the Whitney
properties
is X contractible?
levels of p locally connected?
levels of ~1 are locally connected,
then is X locally
Exercises 24.20 Exercise. and only if f-‘(B) f (W. 24.21 Exercise. are confluent.
A map between continua f : X --t Y is monotone if is a subcontinuum of X for every subcontinuum B of
Monotone
surjective
maps and open surjective
24.22 Exercise. Complete the proof of Theorem proof of the case t = 0 in the implication (d) + (e).
24.9, including
maps
the
216
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WHITNEY
MAPS
24.23 Exercise. Let X be a continuum and let /-I : 2x -+ R’ be a Whitney map. Suppose that A E 2x is such that p has a local minimum at A. Then A E F(X). 24.24 Exercise. Prove Theorem 24.11. [Hint: Let d denote a metric for X. Suppose that w has a local minimum at .4 = {al,. ..,a,} E F(X), where n > 1. Then there exists ze E X such that A C {z E X : d(z,zo) 5 WI(A)}. For each i E (1,. . . , n}, let t, = d(zo,az). Since the metric d is convex, there is an isometry CJ~: = ai. Define F : [O,l] + F(X) [0, ti] -+ X such that ai = 20 and ui(ti) = {a(~~),. . . ,a(&)} by F(t) and prove that w(F(t)) < w(A) for each t < 1.1 24.25 Exercise. Let X be a continuum. If C(X) c 3t c 2x and 3c II (F(X) - F’i (X)) # 0. Then there exists a non-open Whitney map for 7-l.
25.
Admissible
Whitney
Maps
25.1 Definition [15, Definitions 2.11. Let X be a continuum and let U = C(X) or 2-Y. A Whitney map p for ?i is called an admissible Whitney map for 7-lprovided that there is a (continuous) homotopy h : RX [0, 1] + ?t satisfying the following contitions: (a) for all A E ‘?f, h(A, 1) = A and h(A,O) E F*(X), (b) if p(h(A, t)) > 0 for some A E 3t and t E [0, 11, then p(h(A, s)) < ,u(h(A, t)) whenever 0 5 s < t. A homotopy h : 31.x [0, l] + 31 satisfying (a) and (b) is called an ,uadmissible deformation for 7-L. Notice that it is not required in (b) that h(A,O) be the same singleton for all A E 7-t. 25.2 Definition [25, Definition 2.11. Let X be a continuum and let U = C(X) or 2x. A Whitney map p for ‘l-i is strongly admissible if p is admissible and there is an p-admissible deformation h for ?l which satisfies the additional condition: (c) h({s},t) = {z} for each z E X and t E [0, l]. In [14, Section 21Goodykoontz considered an extension of the notion of admissible Whitney map. An extension to other spacesis presented in [33]. In [15], Goodykoontz and Nadler introduced admissible Whitney maps to obtain conditions under which positive Whitney levels are Hilbert cubes. Their paper was the first in determining positive Whitney levels for the hyperspace 2x. They proved the following theorems.
25. ADMISSIBLE
25.3 Theorem
WHITNEY
MAPS
217
[15, Theorem 4.11. Let X be a Peano continuum,
(a) if there is an admissible Whitney map p for 2x, then p-‘(t) Hilbert cube whenever 0 < t < p(X),
is a
(b) if there is an admissible map p for C(X) and if X contains no free arc, then ,u-‘(t) is a Hilbert cube whenever 0 < t < p(X). 25.4 Theorem
[15, Theorem 4.101. Let X be a Peano continuum,
(a) if there is an admissible Whitney map ~1for 2x, then I-L-‘([O, t]) and p-‘([t,,n(X)]) are Hilbert cubes whenever 0 < t < ,n(X), (b) if there is an admissible Whitney map p for C(X) and if X contains no free arc, then p-l ([0, t]) and pL-l ([t, p(X)] are Hilbert cubes whenever 0 < t < p(X). Using Theorem 25.3, Kato gave the following more precise version. 25.5 Theorem [25, Theorem 3.1 (i)]. Let X be a Peano continuum and let 31 = 2x or C(X). If ‘?f = C(X), assumethat X contains no free arc. If h is an admissible Whitney map for 3c, then
is a trivial bundle map with Hilbert cube fibers. 25.6 Theorem [25, Theorem 3.1 (ii)]. Let X be the Hilbert cube Q and let 7-l = 2x or C(X), then there is a strongly admissible map p for 3c such that
is a trivial bundle map with Hilbert cube fibers. Spheres in Euclidean spacesdo not have admissible Whitney maps (see Corollary 25.18). However Kato has extended Goodykoontz and Nadler’s ideas to obtain the following theorem. 25.7 Theorem [25, Theorem 3.51. Let X be the n-sphere S” (n = 1,2,. . .). Then there is a Whitney map ~1for ‘fl = 2x (n = 1,2,. . .) or C(X) (n = 2,3,. . .) such that, for some t E (0, p(X)),
AK1 ((03tl) : CL-l((0,tl) + v4tl is a trivial bundle map with X x Q fibers.
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MAPS
25.8 Definition. Let X be a continuum and let p E X. Then X is arcsmooth at p ([ll] and [12]) provided that there exists a continuous function CY: X -+ C(X) satisfying the following conditions: (4 4P) = {P)t (b) for each 3: E X - {p}, Q(Z) is an arc from p to z, and (c) if 2 E o(y), then o(x) c o(y). A continuum X is said to be arc-smooth provided X is arc-smooth at some point. For dendroids the arc joining two points is unique and the condition (c) is automatically satisfied. Then, in this case, the only requierement is the continuity of cr. Dendroids which are arc-smooth are called simply smooth dendroids. Generalizing the concept of arc-smoothnesson dendroids, Fugate,Gordh and Lum defined the concept of arc-smoothness for general continua in [ll] and [12]. In those papers they showed that arc-smooth continua coincide with the freely contractible continua of Isbell ([24]). Then arc-smooth continua can be contracted in a very nice way (see Exercise 25.35). Examples of arc-smooth continua include: compact convex subsets in Banach spaces, dendrites (locally connected dendroids, see Exercise 25.34) and cones over compact metric spaces. In [19], the following theorem was proved. 25.9 Theorem [19, Theorem 1.31. If a continuum X is arc-smooth, then there exists an admissible Whitney map /I for 2x such that p]C(X) is also admissible (Exercise 25.37). 25.10 Examples. The following continua are known to have admissible Whitney maps for 2aYand C(X) (Theorem 25.9): (a) cones over compact metric spaces([15, Theorem 2.15]), (b) arc-smooth continua. The following continua are known to ‘have strong admissible maps for 2x and C(X). (a) convex compact subsets of Banach spaces([25, Theorem 2.3]), (b) dendrites ([25, Theorem 2.4]), (c) finite collapsible polyhedra ([26, Theorem 3.11). Now, we will develop some of the basic theory of Whitney levels of admissible Whitney maps. Mainly, we will present results by Goodykoontz and Nadler ([15]). This results show that Whitney levels of admissible Whitney maps have very nice acyclicity properties.
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219
25.11 Theorem [15, Theorem 2.121. Let X be a continuum. If p is an admissible Whitney map for 2x, then ~1is open, and then p-i(t) is a continuum for each t E [O,p(X)]. Proof. Property (b) in Definition 25.1 implies that there is no local minima of p at any element A E 2x - pi(X). Appliying Theorem 24.9 we conclude that ~1is open. The second part of the theorem follows from Corollary 24.4. n 25.12 Theorem (compare with [25, Proposition 2.21). Let X be a continuum. Supposethat p is a strongly admissibleWhitney map p for ‘U = 2x or C(X). Then (a) X is a strong deformation retract of Z, (b) if ‘$f = C(X) and X7 is locally connected, then X is an AR, (c) if ?/ = 2x, then X is an AR. Proof. (a) is an immediate consequenceof condition (c) in Definition 25.2. (b) If X is locally connected, C(X) is AR ([41, pp. 190-1911or Theorem 10.8). Then X is a retract of an AR. Hence X is an AR. (c) From Lemma 2.1 in [13] it follows that: since X is a strong deformation retract of 2”, we have that X is locally connected. Thus 2” is an AR ([41, pp. 190-1911or Theorem 10.8). Since X is a retract of 2x, X is an AR. 4 Kato has asked the following question: 25.13 Question [25, Question 3.41. If X is a compact AR, is there a strongly admissible Whitney map for 2x or C(X)? We do not even know the answer to the following question. 25.14 Question. If X is a compact AR, is there an admissible Whitney map for 2x or C(X)? It would be interesting to answer Question 25.14 for the particular case when X is the Bing’s house. This space is illustrated in Figure 32 (top of the next page). Notice that X is a retract of the solid cylinder and then X is an AR. See comment after Question (14.43.12) in Chapter XV. It was shown in [19] an example of a contractible continuum X such that, for each Whitney map p for C(X) there is a number t E [O,p(X)] such that p-l(t) is not contractible. Since X is contractible, then C(X) also is contractible (Corollary 20.2). Hence, by Theorem 25.20, C(X) does
220
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WHITNEY
, Entrance
Entrance
MAPS
to Room 2
to Room 1
/
Bing’s house Figure 32
not have admissible Whitney maps. Therefore, contractibility of X is not a sufficient condition for the existence of admissible Whitney maps for C(X) (compare to Question 25.30). 25.15 Question [26, p. 3081. Let X be a compact connected ANR (or a polyhedron). Is there a Whitney map p for 8 = 2x or C(X) such that for some to E (0,/~(x)) there is a homotopy
h : P-l([o> to)) x [O, 11 + P-‘([Wo)) satisfying the following conditions: (a) for all A E p-l([O, to)), h(A, 1) = A and h(A,O) E Fl(X), (b) if p(h(A, t)) > 0 for some A E /.J-‘([O, to)) and t E [0, 11, then p(h(A, s)) < p(h(A, t)) whenever 0 5 s < t 5 1, (c) h({z}, t) = {z} for each x E X and t E [0, l]?
25. ADMISSIBLE
WHITNEY
MAPS
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25.16 Theorem [15, Theorem 2.31. Let X be a continuum. If there is an admissible Whitney map p for ?-f = 2x or C(X), then X is arcwise connected. Proof. Let h be an admissible deformation for R. By property (a) in Definition 25.1, h(3t x (0)) C Fl (X). Since 31 is arcwise connected (Theorem 14.9), then h(‘H x (0)) is an arcwise connected subset of Fl (X). = {z}. Fix a point {z} E Fl(X). By ( a) in Definition 25.1, h({z},l) Then ,u(h({z}, 1)) = 0. Thus it follows from (b) in Definition 25.1 that h({z}, t) E Fl(X) for every t E [0, l] (Exercise 25.31). Hence h({z} x [0, I]) is a path in Fl(X) joining {z} to a point in h(R x (0)). Therefore, Fl(X) is arcwise connected. n 25.17 Theorem [15, Theorem 2.41. Let X be a continuum. Assume that there is an admissible Whitney map p for ‘?i = 2x or C(X). Then, X is contractible if and only if 31 is contractible. Proof. Assume that 7-1is contractible. It was shown in the proof of Theorem 20.1 that there is a contraction G : Y-fx [0, l] + 3c such that for all A E ‘T-l, G(A, 0) = A, G(A, 1) = X and,ifO<s
f({zl,t) =
V(z), 1 h(G({x},2t
W,
- l),O),
ifOIt<
i,
if i 5 t 5 1.
It follows from Definition 25.1 (Exercise 25.31) that f maps FI (X) x [0, l] into Fl(X). Also, for each {z} E Fl(X), f({z},O) = {z} and f({z},l) = h(X, 0). Hence, Fl (X), therefore, X is contractible. This proves one half of Theorem 25.17. By Corollary 20.2, if X is contractible, then ?f is contractible (no assumption that there is an admissible Whitney map is necessary). W 25.18 Corollary [15, Corollary 2.51. If there is an admissible Whitney map for 2x or C(X) and X is a Peano continuum, then X is contractible. Proof. If X is a Peano continuum, then 2x and C(X) are contractible (Corollary 20.14). Then Corollary 25.18 follows from Theorem 25.17. n 25.19 Theorem [15, Theorem 2.71. Let X be a continuum. If p is an admissible Whitney map for 3t = 2x or C(X), then, for any to such that 0 < to < A-9 p-l (to) and CL-~([O, b]) are strong deformation retracts of p-‘([to, p(X)]) and of ‘l-t, respectively.
222
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MAPS
Proof. Let h : R x [0, I] + ?i be an p-admissible deformation for Fix to such that 0 < to < p(X), and let K = cl-‘([tc,p(X)]). Let A E K. Since h(A, 1) = A, p(h(A, 1)) > to and since h(A,O) E Fl(X), p(h(A, 0)) = 0. Thus, by the continuity of h and p, there exists tA E [0, l] such that p(h(A, tA)) = to; furthermore, since p(h(A, tA)) > 0, we have, by (b) in Definition 25.1, that there is only one such tA. Hence we can define a function 0 : K + [0, l] by letting, for each A E K, B(A) be the unique number in [0, l] such that p(h(A,B(A))) = to. We will prove that 8 is continuous. Suppose to the contrary, that 0 is not continuous at an element A E K. Since [0, l] is compact, there exists a sequence {An}~EP=l in K such that A, + A and 6(A,) + t for some t E + p(h(A, t)) implies that p(h(A, t)) = [0, l] -{B(A)}. But p(h(An,B(A,))) to. Thus t = B(A). This contradiction proves that 0 is continuous. Since 8 and h are Define T : K + ,u-‘(to) by r(A) = h(A,B(A)). continuous, r is continuous. If A E p-i(ta), then 8(A) = 1. Thus, r(A) = A. Therefore, r is a retraction from K onto ,u-'(to). Now, define G : K x [0, l] + K by
Ifl.
G(A, t) = h(A, to(A) + (1 - t)). By (b) in Definition 25.1, G(A, t) E K for every t E [0, 11. Clearly G is continuous, G(A, 0) = A and G(A, 1) = r(A) for every A E K. Moreover, G(A, t) = A for every A E p”-‘(to) and t E [O,l]. Hence, r is a strong deformation retraction. Therefore p-’ (to) is a strong deformation retract
of P-wo~ /4X)1) In order to prove that p-‘([O, to]) is a strong deformation retract of ?l, define R : 3t + p-‘([O, to]) by R(A) =
T(A), A >
ifAEK if A E p-l ([0, to])
And define F : 3c x [0, l] 4 ‘R by
G(A,9, WJ,t>= A1
ifAEK if A E p-‘([O, to])
Using R and F it is easy to show that h-l ([0, to]) is a strong deformation retract of ?I!. n Using Theorem 25.19, Goodykoontz and Nadler obtained the following more definitive version of half of Theorem 25.17.
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MAPS
223
25.20 Theorem [15, Theorem 2.81. Let X be a continuum. Assume that there is an admissible Whitney map ,LLfor R = 2” or C(X). If 31 is contractible, then 1-1-l(to) and ,&‘([(I, to]) are contractible for each
to E P,PWl. Proof. Fix to E [0,/~(x)]. If to = 0, then p-‘(to) is homeomorphic to X and hence, by Theorem 25.17, ,u-I (to) = pL-l ([0, to]) is contractible. Thus, we assume that to > 0. We also assume that to < p(X) since if to = p(-Y), p-‘(to) = (X}. By Theorem 25.19, ~-~([0,t0]) is a retract of Ifl, thus P-’ ([O,to]) is contractible. So we only have to prove that p-‘(to) is contractible. Let G : ?f x [O,l] -+ ?t be as in the proof of Theorem 25.17. Let K = p-‘([tO, p(X)]). It follows from the properties of G that GlK x [0, 11 maps K x [0, l] into K. Hence K is contractible. By Theorem 25.19, cl-‘(to) is a retract of K. Thus p-‘(to) is contractible. n 25.21 Theorem [15, Theorem 2.91. If there is an admissible Whitney map p for 2.Y or C(X) and if X is a Peano continuum, then p-‘(to) is an absolute retract for each to such that 0 < to < p(X). Proof. 091 that 25.19, /I-’
Since X is a Peano continuum, we have by the remark in [27, P(X)]) is an absolute retract. Therefore, by Theorem is an absolute retract. W
~l([t~,
The following general result determines many properties of positive Whitney levels when the Whitney map is admissible (see Theorem 30.3, Example 30.4, Example 46.2 and Example 53.8). 25.22 Theorem. Let X be a continuum. If p is an admissible Whitney map for R = 2x or C(X), then, for each to such that 0 < to < p(X), pL-’ (to) has all those properties which are common to r-images (see [6.1 in Chapter XV] for the definition of r-image) of all hyperspaces. In particular, p-‘(to) is an arcwise connected continuum which has trivial shape (and thus it is acyclic). Proof. By Theorem 25.19, p-‘(to) is an r-image of the continuum K = /J-~([~oJ~WI>. So, in order to prove Theorem 25.22 it is enough to prove that K is an r-image of the hyperspace C(K). Define f : C(K) t K by f(d) = ud. By Exercise 11.5, f is a well defined continuous function. Define g : K + C(K) by g(A) = {A}. Then g is continuous and g is a right inverse of f. Therefore, f is an r-map from C(K) onto K. Thus p-l (to) is an r-image of C(K).
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This completes the proof of Theorem 25.22 since the properties listed in the second part of Theorem 25.22 are known to be r-invariants and are known to be properties of all hyperspaces. n By Theorem 25.22, all positive Whitney levels have trivial shape when the Whitney map is admissible. We now show that X has trivial shape. 25.23 Theorem [15, Theorem 2.111. Let X be a continuum. If there is an admissible Whitney map p for 3t = 2” or C(X), then X has trivial shape. Proof. For each t E (0,/l-i(X)) let KCt = pP1([O,t]). By Theorem 25.19, Kt is a retract of ti. Thus, since ‘?i has trivial shape (Theorem 19.10), it follows easily that Kt has trivial shape (use [28, 2.11). Observe that Fl(X) = ll{K, : 0 < t < p(X)} Hence J’i (X) is a nested intersection of compacta having trivial shape. Thus, it follows easily that Fr(X) has trivial shape ([28, 2.11). Therefore, since Fl(X) is homeomorphic to X, _Y has trivial shape. n Recall that a cell-like map (or CE map) is a map whose point inverses all have trivial shape. So the following theorem is an immediate consequence of Theorems 25.11, 25.22 and 25.23. 25.24 Theorem [15, Theorem 2.121. Let X be a continuum. If /J is an admissible Whitney map for 2x or C(X), then p is an open CE map. In [15, Theorem 2.171, Goodykoontz and Nadler proved that, if X is a smooth dendroid, then every Whitney map for C(X) is admissible. As a partial inverse of this result we have the following. 25.25 Theorem. If X is a continuum with the property that every Whitney map for C(X) is admissible, then X is a dendroid. Proof. Let A = p-‘(to) be an arbitrary positive Whitney level for C(X). By Theorem 53.3 it is enough to prove that A is 2-connected. That is, we have to show that each map from 5”’ into A is null homotopic, where Sn is the n-sphere and 0 5 n < 2. By Theorem 25.22, p-‘(to) is arcwise connected. Then we may assumethat 1 5 n. Let 1 _
EXERCISES
225
Since S” is locally connected, Fi(Sn) is contractible in C(P) (Theorem 20.1 and Corollary 20.14). Then there exists a map G : Fr (P) x [0, I] + C(P) such that G(({p},O)) = {p} and G(({p}, 1)) = S" for every P E S”. Define F : 5’” x [0, l] 4 A by
Fb, t) = WW)(G(({P),
t))))-
The continuity of F follows from Exercise 11.5. Since, for every p E We conclude that f is Sn, F(p, 0) = f(p) and F(p, 1) = r(UC(f)(Sn)). homotopic to a constant map. Therefore, X is a dendroid. n 25.26 Remark. Answering questions in [15], Goodykoontz has shown in [13, Example 3.11an example of a non-contractible dendroid D such that every Whitney map for C(D) is admissible and such that no Whitney map for 2O is admissible (Exercise 25.33). The following questions remain open. 25.27 Questions [15, Questions 4.131. Let X be a continuum. Is the restriction to C(X) of every admissible Whitney map for 2x an admissible Whitney map for C(X)? If there is an admissible Whitney map for 2x, is there an admissible Whitney map for C(X)? 25.28 Question [15, Question 4.141. Let X be a continuum. If there is an admissible Whitney map for 2x, then must X be contractible? 25.29 Question. If X is a dendroid and there is an admissible Whitney map for 2x then must X be smooth? (smooth dendroids have been completely characterized in [16]) 25.30 Question. If X is a contractible dendroid, is there an admissible Whitney map for C(X)? By Theorems 20.1 and 25.20, an affirmative answer to this question would answer Question 41.3.
Exercises 25.31 Exercise. Let X be a continuum. Suppose that p is a Whitney map for Z = 2x or C(X) and h is an p -admissible deformation for 3-1.If h(A, t) E Fl(X) for some (A,t) E 7-l x [O,l], then h(A,s) E Fl(X) for each s < t. 25.32 Exercise. Show directly that if X = [0, l] x [O,l], then there is a strongly admissible Whitney map for 2x.
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MAPS
25.33 Exercise. If X is the continuum represented in Figure 25, p. 158, then there is a strongly admissible Whitney map for C(X) and there is no admissible Whitney map for 2*Y. 25.34 Exercise. A dendroid X is a dendrite if and only if X is smooth at each of its points. 25.35 Exercise. Let X be a an arc-smooth continuum at some point p E X. Let (Y : X -+ C(X) be a map satisfying the conditions of Definition 25.8. Consider a Whitney map p : C(X) + [O,l], with p(X) = 1. Define L : X + [0, 11by L(z) = ~(cY(x)). Define f : X x [0, l] -+ S by
f(s,t)= 2q the unique point y E o(z)
if L(2) < t such that L(y) = t,
if L(z) 2 t
Then f has the following properties: (a) f is continuous, (b) f(z, 0) = (p} and f(z, 1) = z for every z E X, and (cl f(f(z, s), t) = f(x,min{s, t)). In particular, X is contractible. 25.36 Exercise. Let X, p, Q, p, L and f be asin Exercise 25.35. Define G : 2” x [0, I] + 2x by G(A, t) = f(A x {t}) and define M : 2~~+ [0, l] by M(A) = sup{L(a) : u E A}. Let A E 2”. Then G and M have the following properties: (a) G and M are continuous, (b) if A E C(X) then G(A, t) E C(X) for every t E [0, 11, (c) G((z},t) E Fr(X) for every z E X and t E [O,l], (d) G(A, 1) = A and G(A,O) = {p},
(4 G(G(A, s>, t) = G(A, min{s, t)), (f) M(G(A, s)) = max{s, h!(A)}, (g) G(A, M(A)) = A, and (h) If s 5 t, then M(G(il,s)) 5 M(G(A,t)). 25.37 Exercise. Let X, G, and M be as in Exercise 25.36. Define w : 2” + [0, l] by M(A)
w(A) =
s0
P,(G:(A, t))dt.
And define h : 2aYx [O,1) -+ 2x by h(,4, t) = G(A, UK’(A)). h have the following properties:
Then w and
26. A METRIC
ON HYPERSPACES
DEFINED
BY WHITNEY
MAPS
227
(a) w is a Whitney map for 2x, (b) h is an w-admissible deformation of 2x, and (c) h(C(X) x [0, l] : C(X) x [O,l] + C(X) is an w]C(X)-admissible deformation of C(X) . Therefore, w (respectively, wjC(X)) is an admissible Whitney map for 2” (respectively, C(X)).
26.
A Metric
on Hyperspaces Defined Maps
by Whitney
Let X be a continuum. In [4], W. J. Charatonik observed that the Whitney map w defined in Theorem 13.4 has the following additional property: (1)
For every A, B E 2-x with A w(B
c
B and for every C E 2x,
u C) - w(A u C) < w(B) - w(A).
W. J. Charatonik has shown that Whitney maps with property (1) induce a nice metric on the hyperspace 2x (see Exercises 26.2-26.6). He posed the following question: 26.1 Question [4, Question 91. Given any continuum X and any Whitney map p : 2x + [0, p(X)], d oesthere exist an embedding h from [0, p(X)] into [0, oo) such that h o p is a Whitney map satisfying property (l)?
Exercises 26.2 Exercise. erty (1).
The Whitney map w defined in Theorem 13.4 has prop-
26.3 Exercise. not satisfy (1).
There is a Whitney map p for 21°~‘]such that p does
Let X be a continuum. Let p be a Whitney map for 26.4 Exercise. 2x that satisfies (1). For A, B E 2x, define D,(A, B) = maxMA
U B) - p(A),AA
U B) - ,@)I.
Then D, is a metric on 2* and D, is equivalent to the Hausdorff metric on 2x. Let D, be as in Exercise 26.4. Consider 2x with the 26.5 Exercise. metric D,, and let A c 2x be an order arc. Then ,u(d : A + [0, co) is an isometry.
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26.6 Exercise. Let D, be as in Exercise 26.4. Let x E A E 2x. Then DP( A, {x}) = p(A). In other words, the distance between a set and any point in the set does not depend on the choice of the point.
References 1. 2. 3. 4. 5. 6.
7.
8. 9.
10. 11. 12 13. 14.
M. M. Awartani, Whitney maps on hyperspaces of skew compactifications, Questions Answers Gen. Topology, 11 (1993), 61-67. J. J. Charatonik, Conjkent mappings and zlnicoherence of continua, Fund. Math., 56 (1964), 213-220. W. J. Charatonik, A continuum X which has no confluent Whitney map for 2x, Proc. Amer. Math. Sot., 92 (1984), 313-314. W. J. Charatonik, A metric on hyperspacesdefined by Whitney maps, Proc. Amer. Math. Sot., 94 (1985), 535-538. W. J. Charatonik, R’-continua and hyperspaces, Topology Appl., 23 (1986), 207-216. W. J. Charatonik, An open Whitney map for the hyperspace of the circle, General Topology and its Relations to Modern Analysis and Algebra VI. Proc. Sixth Prague Topological Symposium 1986, Z. Frolik, Editor, Heldermann Verlag Berlin (1988), 91-94. D. W. Curtis and N. T. Nhu, Hyperspaces of finite subsetswhich are homeomorphic to No-dimensional linear metric spaces,Topology Appl., 19 (1985), 251-260. S.T. Czuba, Ri-continua and contractibility of dendroids, Bull. Acad. Polon. Sci., S&. Sci. Math., 27 (1979), 299-302. C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces,Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 19 (1971), 102771034. V. V. Fedorchuk, On hypermaps, which are trivial bundles, Lecture Notes in Math., Springer, Berlin-New York, 1060 (1984), 26-36. J. B. Fugate, G. R, Gordh, Jr. and L. Lum, On arc-smooth continua, Topology Proc. 2 (1977), 645-656. J. B. Fugate, G. R. Gordh, Jr. and L. Lum, Arc-smooth continua, Trans. Amer. Math. Sot., 265 (1981), 545-561. 3. T. Goodykoontz, Jr., Some retractions and deformation retractions on 2x and C(X), Topology Appl., 21 (19851, 121-133. J. T. Goodykoontz, Jr., Geometric models of Whitney levels, Houston J. Math., 11 (1985), 75-89.
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and S. B. Nadler, Jr., Whitney levels in hyperspaces Trans. Amer. Math. Sot., 274 (1982),
continua,
671-694. 16. E. E. Grace and E. J. Vought, Weakly monotone images of smooth dendroids are smooth, Houston J. Math., 14 (1988), 191-200. and open Whitney maps, Proc. Amer. Math. 17. A. Illanes, Monotone Sot., 98 (1986), 516-518. 18. A. Illanes, Spaces of Whitney maps, Pacific J. Math., 139 (1989), 6777. 19. A. Illanes, Arc-smoothness and contractibility in Whitney levels, Proc. Amer. Math. Sot., 110 (1990), 1069-1074. 20. A. Illanes, The space of Whitney levels, Topology Appl., 40 (1991), 157-169. 21. A. Illanes, A characterization of dendroids by the n-connectedness of the Whitney levels, Fund. Math., 140 (1992), 157-174. 22. A. Illanes, Monotone and open Whitney maps defined in 2.Y, Topology Appl., 53 (1993), 271-288. 23. A. Illanes, The space of Whitney levels is homeomorphic to 12, Colloq. Math., 65 (1993), l-11. 24. J.R. Isbell, Six theorems about injective metric spaces, Comm. Math. Helv., 39 (1964), 65-76. 25. H. Kato, Concerning hyperspaces of certain Peano continua and strong regularity of Whitney maps, Pacific J. Math., 119 (1985), 159.-167. 26. H. Kato, On admissible Whitney maps, Colloq. Math., 56 (1988), 299309. 27. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot., 52 (1942), 22-36. images of tree-like co7128. J. Krasinkiewicz, Curves which are continuous tinua are movable, Fund. Math., 89 (1975), 233-260.
29. K. Kuratowski, Topology, Vol. II, Polish Scientific Publishers and Academic Press, 1968. 30. S. B. Nadler, Jr., Some basic connectivity properties of Whitney map inverses in C(X), Proc. Charlotte Topology Conference (University of North Carolina at Charlotte, 1974), Studies in Topology, Academic Press, New York, 1975, N. M. Stavrakas and K. R. Allen, Editors, 393410. 31. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N. Y., 1978.
230
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MAPS
32. S. B. Nadler, Jr. and T. West, Size levels for arcs, Fund. Math., 141 (1992), 243-255. 33. T. N. Radul, Whitney mappings for a space of inclusion hyperspaces, (Russian) Mat. Zametki, 52 (19921, 117-122. Traslation: Math. Notes, 52 (1992), 960-964. 34. R/I. van de Vel, On generalized Whitney mappings, Compositio Math., 52 (1984), 47-56. 35. L. E. Ward, Jr., A note on Whitney maps, Canad. Math. Bull., 23 (1980) 373-374. 36. L.E. Ward, Jr., Extending Whitney maps, Pacific J. Math., 93 (1981), 465-469. 37. T. Watanabe, A Whitney map and the Lindeliif property, Houston J. Math., 15 (1989), 147-151. 38. H. Whitney, Regular families of curves, I, Proc. Nat. Acad. Sci. U.S.A., 18 (1932) 275-278. 39. H. Whitney, Regular families of curves, Ann. of Math., 34 (1933), 244-270. 40. G. T. Whyburn, Analytic Topology, Amer. Math. Sot. Colloq. Publ., Vol. 28, Amer. Math. Sot., Providence, R. I., 1942. 41. M. Wojdyslawski, Re’tractes absoluset hyperespacesdescontinus, Fund. Math., 32 (1939), 184-192.
VIII. Whitney Properties and Whitney-Reversible Properties 27.
Definitions
In this chapter the letter IL will denote any Whitney map for C(S). As it was shown in Theorem 19.9 (see also Exercise 27.6) and Exercise 14.25, 11 is a monotone and open map. This implies that the map t -+ is continuous, or equivalently, the p-‘(t), from [O,p(X)] into C(C(X)) is a continuous monotone decomposition of family {/l-‘(t) : t E [O,p(X)]} C(X). Then it is natural to study the topological properties of the Whitney levels for C(X), and in particular, those properties induced by the space S to the Whitney levels. These properties are called Whitney properties (see Definition 27.1). On the other hand, if we consider a sequence {t,L}rZ1 in [0, /L(X)] such that t,, + 0 as n + 00, then /~-l(&) + p-‘(O). For each 71, p-‘(tll) is a continuum which we visualize as being a horizontal level in C(X) (see Figure 33, top of the next page). It follows easily form the definition of /J that p-‘(0)
= ({z}
: z E X} = Fl(X).
are homeomorphic in a natural way by asNote that X and Fi(X) sociating z with (~1. Thus it is also natural to study those topological properties which are preserved under the approximation of X by the levels p-l@,). This idea was formalized in [80], with the three notions of
231
232
VIII.
WHITNEY
PROPERTIES
AND WHITNEY-REVERSIBLE..
.
Whitney levels Figure 33
Whitney-reversible tion).
property (see (b), (c) and (d) in the following defini-
27.1 Definition P is said to be:
[66] and [80]. Let P be a topological property. Then
(a) a Whitney property provided that if a continuum X has property P, so does p-‘(t) for each Whitney map ,U for C(X) and for each t, 0 I t < P(X), (b) a Whitney-reversible property provided that whenever X is a continuum such that p-‘(t) has property P for all Whitney maps p for C(X) and all 0 < t < p(X), then X has property P, (c) a strong Whitney-reversible property provided that whenever X is a continuum such that p-‘(t) has property P for some Whitney map p for C(X) and all 0 < t < p(X), then X has property P,
EXERCISES
233
(d) a sequential strong Whitney-reversible property provided that whenever X is a continuum such that there is a Whitney map b for C(X) and a sequence{tn}~Tl in (0, p(X)) such that lim t, = 0 and p-l (tn) has property P for each n, then X has property P. Many authors have studied Whitney properties and Whitney-reversible properties. In Chapter XIV of [i’9] there is a complete discussion of what was known in 1978. The purpose of this chapter is to give a complete discussionof the known topological properties which are Whitney properties, Whitney-reversible properties, strong Whitney-reversible properties or sequential Whitneyreversible properties. Notice that: (a) a sequential strong Whitney-reversible property is a strong Whitneyreversible property, (b) a strong Whitney-reversible property is a Whitney-reversible property, and (c) the negation of a Whitney property is a sequential strong Whitneyreversible property. In [79, Question 14.561 it was asked if there is a Whitney-reversible property which is not a strong Whitney-reversible property. We will see in section 30 that the property of being an AH. has these characteristics. The following question remains open. 27.2 Question [79, 14.55.11. Is there a strong Whitney-reversible property which is not a sequential strong Whitney-reversible property? As we will seein this chapter, there are many Whitney properties which are not Whitney-reversible properties and vice versa. A table summarizing the results in the chapter is at the end of the chapter preceding the references.
Exercises 27.3 Exercise. Let A be a Whitney level for C([O, 11). Consider the function f : A + [0, l] given by f(A) = the middle point of A. Then f is a homeomorphism onto its image. Hence A is homeomorphic to an arc. 27.4 Exercise.
The property of being a circle is a Whitney property.
27.5 Exercise. The hyperspace C(X) some levels homeomorphic to [0, 11.
of the sin($)-continuum
has
234
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PROPERTIES
AND
WHITNEY-REVERSIBLE..
27.6 Exercise. Let X be a continuum. This exercise gives an alternative proof of the connectednessof Whitney levels. Here we do not use the unicoherence of C(X) (cornpare with the proof of Theorem 19.9). Let A be a Whitney level for C(X). Suppose that A = ?i u K, where ‘I-l and K are nonempty disjoint closed subsets of A. Define H = U?i and K = UK. Then H and K are nonempty disjoint closed subsets of X whose union is X. [Hint: Use Exercise 16.18 or Exercise 28.4.1 27.7 Exercise. Let X be a continuum. Let ,LLbe a Whitney map for C(X) and let A E C(X). Then (I3 E p-‘(t) : B c A} and (I3 E p-r(t) : B f~ A # 0) are connected.
28.
ANR
The interested reader can find the theory of Absolute Neighborhood Retracts (ANR) and Absolute Retracts (AR) in the classical monograph by Borsuk [5]. Some information about ARs and ANRs is in section 9. 28.1 Example [84, Example 501 or [86, Example 51. The property of being an absolute neighborhood retract is not a Whitney property. Let A0 = {(z,y,O)
E R3 : x2 + y2 = 1)
and, for each n = 1,2,. . ., let An = {(x, y, A)
E R3 : x2 + y2 = I}.
For each n = 1,2,..., let X, be the closed “bulging annulus” bounded by the circles A, and An+1 and contained in the 2-sphere in R3 with center (0, 0 , $$r) and radius 4%.
Let
x = Cl~3(U~~lXn) = (U~lW
UAo.
X is represented in Figure 34 (top of the next page). Assume the metric d for X is the usual Euclidean metric for R3 restricted to X, and let p denote the restriction to C(X) of the Whitney map constructed as in Exercises 13.5 and 13.6. Let
to= p(AO)(= p(A,) for each n = has shown that p-‘(to) is not locally
1,2,. . .).
contractible. The basic Petrus ideas used to do this are similar to those in Example 30.1, and the details
28. ANR
Being an ANR is not a Whitney
235
property
(28.1)
Figure 34
will not be done here. Since X Z S1 x [O,l], the example shows that being an absolute neighborhood retract is not a Whitney property (absolute neighborhood retracts are locally contractible, see Exercise 28.13). In [79, Question 14.571, it was asked whether the property of being an absolute neighborhood retract (ANR) is a strong Whitney-reversible property (or a Whitney-reversible property). It was suggested that the continuum X which is the union of a sequence of circles joined only by a point (the continuum called Hawaiian earring illustrated in Figure 26, p. 162), could be used to see that being an ANR is not a Whitney-reversible property. In her dissertation ([15, Theorem 161) Dilks used this continuum X to show that the properties of being an ANR, being locally contractible and having finitely generated homology groups are not strong Whitneyreversible properties. Next, we will use the powerful theorem by Lynch presented in Theorem 66.4 of the next chapter ([74]) and the same continuum X to show that the above mentioned properties are not Whitney-reversible properties.
236
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28.2 Example.
PROPERTIES
AND WHITNEY-REVERSIBLE..
.
The following properties:
(a) being an ANR, (b) being locally contractible, and (c) having finitely generated homology groups are not Whitney-reversible properties. Consider X the continuum illustrated in Figure 26, p. 162. Suppose that X = u{X, : n > I}, where each X, is a circle in R2 and p is the common point of all the circles. Let P be any Whitney map for C(X) and let t E (O,p(X)) be an arbitrary number. Define A = p-‘(t) and B = {A E A : p f A}. By Lynch’s theorem in [74] (Theorem 66.4), a is an AR. Since the sequence{X,}r=i tends to {p}, the set F = {n : p(X,) > t} is finite. For each n E F, let 3;, = {A E A : A c X,} = (&Z’(X,))-l(t) and 6, = ck&{A E 3-n : p $ A}). By Exercise 27.4, Fn is a circle. It is easy to show that G,, is an arc, the end points A,, and B, of 6, have the and the sets &, are pairwise disjoint. property that 6, n f3 = {A,,B,} Let A E A - B. Then p 4 A. Since p is the common point of all the So circles X, , there exists n such that A 9 X,. Then t = p(A) < p(X,). n E F and A E &,. This proves that A = B U (U{& : n E F}). Therefore, A is the union of the AR, t3, with a finite union of pairwise disjoint arcs joined to t3 only by their end points. Then it follows easily that A is an ANR ([5, Chapter V, 2.9]), A is 1ocally contractible (Exercise 28.13) and A has finitely generated homology groups. Since X does not have any of these properties, we conclude that these properties are not Whitney-reversible properties. 28.3 Theorem [75, Theorem 1.1 (i)]. The property of being an ANR is a Whitney property for l-dimensional continua.
Exercises 28.4 Exercise. In this exercise a more precise version of Exercise 16.18 is obtained. Let X be a continuum. Let A = ~1-l (to) be a positive Whitney level for C(X). Let A, B E A be such that A # B and AITB # 0. Let C be a component of AnB. By Theorem 14.6 there exist maps cr, ,0 : [0, I] + C(X) such that (Y(O) = C = p(O), a(1) = A, p(1) = B and, if s < t, then 4s) 5 49 and P(s) $ B(t). For each t E [0, 11, there exists st E [0, l] such that o(st) u /3(t) E A. Define q(t) = a(st) U@(t). Then q is a well defined map, ~(0) = A, r](l) = B and C c q(t) C A U B for every t E [O,l].
237
EXERCISES 28.5 Exercise. A nondegenerate monotone image of tinuum is an arc. [Hint: Let X be a continuum. Let (Y : [0, l] + X map. Let p : C(X) + R’ be a Whitney map. Define f f(4t)) = ~(4[0, tl)). Then f is a well defined one-to-one 28.6 Exercise. in A.
Let n be as in Exercise
an arc in a conbe a monotone : Imcr -+ R’ by map.]
28.4. Then n([O, 11) is an arc
28.7 Exercise. Let A, A and B be as in Exercise 28.4. If there is only one arc joining A and B in A, then A n B is connected. 28.8 Exercise. If X is a continuum and X is not hereditarily unicoherent, then some positive Whitney level for C(X) contains a circle. 28.9 Exercise. Let A, A, B, cr, p and q be as in Exercise 28.4. If there is only one arc joining A and B in A, then: (a) v(t) n B = P(t) for each t E [O,l], (b) {D E C(B) : An B c D} = p([O, l]), and (c) if s < t, then P(t) - p(s) is connected. 28.10 Exercise. Let X be a continuum. Let A be a Whitney level for C(X) and let p E X. By Lynch’s theorem in [74] (Theorem 66.4) the space a = {A E A : p E A} is an AR. Prove, without using Lynch’s theorem that Z? is locally connected. [Hint: Use Exercises 28.4 and 66.8.1 28.11 nonempty there are [Hint:
X.
Exercise. Let X be a continuum. Suppose that there exists a finite subset F of X such that X - F is not connected. Then locally connected Whitney levels for C(X). Use Proposition 10.7.1
28.12 Exercise. Let X be a continuum and let d denote a metric for Let {ur,~,. . .} be a countable dense subset of X. Then the map
f : X + [O,l]” embedding. 28.13 ible.
given by f(z)
Exercise.
Absolute
= [a,
neighborhood
e,.
retracts
. .] is an
are locally contract-
238
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PROPERTIES
29.
AND WHITNEY-REVERSIBLE..
.
Aposyndesis
29.1 Definition. A continuum X is said to be: (a) semi-uposyndetic, if for every p # q in X, there exists a subcontinuum M of X such that intx (M) contains one of the points p, q and X - M contains the other one, (b) aposyndetic, if for every p # q in X, there exists a subcontinuum M of X such that p E i&x(M) and q 4 M, (c) mutually uposyndetic, if for every p # q in X, there exist subcontinua M and iV of X such p E i&x(M), q E intx(N) and M f~ N = 0, (d) finitely aposyndetic, if for any p E X and any finite subset F of X such that p $ F, there exists a subcontinuum M of X such that p E intx (M) and M n F = 0, (e) countable closed set aposyndetic, if for any p E X and any countable closed subset F of X such that p $! F there exists a subcontinuum M of X such that p E intx(M) and M f~ F = 8, (f) zero-dimensional closed set aposyndetic, if for every p E X and any zero-dimensional closed subset F of X such that p $! F there exists a subcontinuum M of X such that p E intx (M) and M fl F = 0. The concept of aposyndesis was introduced by Jones in [46]. An excellent reference for aposyndesis properties is the book [93], where several authors surveyed the relationships among aposyndesisand other topological concepts. 29.2 Theorem [85, Propositions 9, 10, 11 and 121. The following are Whitney properties: (a) semi-aposyndesis, (b) aposyndesis, (c) mutual aposyndesis, and (d) finite aposyndesis. Theorem 29.3 and Example 29.4 answered questions by van Douwen and Goodykoontz (see [17, Questions 3 and 41). 29.3 Theorem [44, Theorem A]. Countable closed set aposyndesis is a Whitney property. 29.4 Example [44, p. 601. There exists a non-aposyndetic dendroid X such that every positive Whitney level for C(X) is zero-dimensional closed set aposyndetic. Therefore, even for dendroids, the following properties are not Whitney-reversible properties:
EXERCISES
239
(a) aposyndesis, (b) finite aposyndesis, (c) countable closed set aposyndesis, and (d) zero-dimensional closed set aposyndesis. Example 29.4 is represented in Figure 35 (next page). The continuum 2 is a dendrite without free arcs (every arc has empty interior) and X is the union of a sequence of copies of Z converging to another copy Za of Z, all of them joined by a point p. 29.5 Example (see [17, p. 471). Semi-aposyndesis desis are not Whitney-reversible properties. 29.6 Question [17, Question 31. Is zero-dimensional desis a Whitney property?
and mutual aposyn-
closed set aposyn-
Exercises 29.7 Exercise. erties.
Aposyndesis
and finite aposyndesis
are Whitney
prop-
29.8 Exercise. If X is a continuum and X is locally connected at a point p and p E A E C(X), then every Whitney level for C(X) which contains A is connected im kleinen at A. 29.9 Exercise. The continuum X in Figure 36, p. 241 is the union of a sequence {Cn}r=i of circles in the plane joined by a point and converging to a limit circle Cc. Prove that the positive Whitney levels of X are aposyndetic while X is not aposyndetic. 29.10 Exercise. Let X be a continuum. Let 13 be a closed subset of Then C(X) and let A E C(X) - B. Let p be a Whitney map for C(X). there exists a neighborhood U of A in C(X) and there exists 6 > 0 such then every order arc joining an that if 1s - p(A)! < 6 and s E [O,p(X)], element in U to an element in p-‘(s) is disjoint to f?. Deduce that C(X) is countable closed set aposyndetic. [Hint: Use Lemma 17.3.1
30.
AR
See comments preceding Example 28.1. In her dissertation, Petrus ([84, Example 461, [86, Example 2] or Example 30.1) discovered the important fact that there is a Whitney level A,
240
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
++ X
//
Some aposyndesis properties are not Whitney-reversible properties (29.4) Figure 35
30. AR
Aposyndesis
241
is not a Whitney-reversible
property
(29.9)
Figure 36
for the hyperspace C(D) of the plane unit disk D, such that A contains a 2-sphere S which is a retract of A. Thus her example showed that the properties listed in the next example are not Whitney properties: 30.1 Example [84, Example 461, [86, Example 21. The following erties are not Whitney properties: (a) being an AR, (b) being acyclic, (c) being arc-smooth, (d) being contractible, (e) being n-connected, (f) having the fixed point property,
and
(g) having trivial shape. The proof that (a)-(g) are not Whitney
properties
goes as follows:
prop-
242
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
Let S” be the unit 2-sphere in R” and let
Thus >y is a 2-cell. Let p denote the restriction to C(X) of the Whitney map constructed in Exercises 13.5 and 13.6 - the metric d for .Y is the usual Euclidean metric for R3 restricted to X x ,Y. Let to = p({(x,y,z)
E x : z = +i}).
We will define a homeomorphism g from S2 into p,-’ (to) such that g( S2) is a retract of p-‘(to). For each z0 E [-1, +l], let S’(z0) = {(x, y, z) E s2 : z = zo) Note that, from the formula for p, (1) and (2) below hold: (1) &9-;)) (2)
/G”(z))
= /@(+;)) >
t0
= to,
whenever -$ < z < +i.
For each (s,y,z) E S2 - {(O,O, -l), (O,O,+l)}, let (#) G(s, y, 2) denote the longitudinal great circle in S2 containing {(GYY,Z), (0,0,-l), (0,0,+1)) and let, (##) $(z, 9, .z) denote the unique point in G(z, y, z) n S’(a) which is nearest,to (2, y, 2). Now we define y. First, let (3) 9(0,0, -1) = S2(- $1 and g(O,O, +l) = S2(+$). Next, let (z,yz) E S--{(O,O,-l),(O,O,+l)). Then by (2),p(S2($)) > to. Thus, since S’( 5) is a circle, it, follows that, there is one and only one subarc .4(x, g, z) of S”(f) such that i~(A(.z, g, z)) = to and such that $(.c, ;y, z) is the midpoint of A(z, y, z). We define: (4) if (z,v,z) E S2 - {(O,O,-l),(O,O,+l)}, then g(z,y,z) = A(z,y,z) (as above). The formulas in (3) and (4) define a function g from S” into ,u-‘(to) (in particular, (1) implies that the values of & in (3) are members of ,K’(to)). It is easy to see that g is a homeomorphism. i.e., g is an embedding of S’ into p-‘(to). To show that g(S2) is a retract of 11-l (to), Petrus defines a continuous function f from ~~-l(to) onto S2 such that f o g is the identity mapping on S2 (i.e., S2 is an r-image of p-‘(to) - see [6.1 in Chapter XV]). The
30. AR
243
mapping f is defined with the help of the following fact, which is more general than (2): (5) if A is a subcontinuum of U{,!?“(z) : - $, 5 z 5 +i} such that il # S2 (- $) , A # S2 (+ i) and “projects along longitudinal great circles” onto S2(0), then p(A) > to. Instead of giving an analytical definition off, we will give a descriptive one (an analytical definition is in [84, p. 531 or [86, p. 2781). Let A E p-l(to). If (6) An (U{S”(a) : -1 I z 5 -+}) # 0, then let f(A) = (O,O, -1). If (7) A = S(+$), then let f(A) = (O,O, +l). since p(A) = to, we have by circles” onto a subarc A0 of Letting p(A) =
N ow, assume A is not as in (6) or (7). Then, (5) that A “projects along longitudinal great S2(0); let m(A) denote the midpoint of Ao. inf{z
: An S2(z) # 0},
we see that, since A does not satisfy (6), p(A) > -4. Also, note that p(A) 5 +i, as it would be true for any subcontinuum of X (actually, since ,4 does not satisfy (7), A is not contained in S(+t). Then p(A) < +f). Hence -1 5 2p(A) < +1 and so S2(2p(A)) “makes sense”. Let f (,4) denote that unique point in This completes the deG(m(A)) n S”(2p(A)) which 1s . nearest to m(A). scriptive definition of f. It can be seen that f is a continuous function from ,c’ (to) onto S2 and that f o g : S2 -+ S2 is the identity mapping. Therefore, g(S2) is a retract Since g is a of p-l(tO) (the mapping g o f being the desired retraction). homeomorphism, we have shown: (8) there is a retraction of ,~‘(to) onto a 2-sphere. Therefore, pcL-’ (to) does not have any of the properties (a) through (g). Since the disk has each one of these properties we conclude that properties (a) through (g) are not Whitney properties. The following theorem shows that it is impossible to obtain an example such as 30.1 (a) for l-dimensional continua. 30.2 Theorem a Whitney property
[75, Theorem l.l.(ii)]. The property for l-dimensional continua.
of being an AR is
The following general theorem shows that many properties, including the property of being an AR are not strong Whitney-reversible properties.
244
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PROPERTIES
AND WKITNEY-REVERSIBLE..
.
30.3 Theorem [25, Theorem 4.31. If P is a topological property which the Hilbert cube has but which the cone over some Peano continuum does not have, then P is not a strong Whitney-reversible property. Proof. Let K be the cone over some Peano continuum X such that K does not have property P. Then there is an admissible Whitney map p for C(K) (Theorem 25.9). By Theorem 25.3 (b), the positive Whitney levels for p are homeomorphic to the Hilbert cube. Then they have property P. Therefore, P is not a strong Whitney-reversible property. m 30.4 Example [25, Corollary 4.41. The property of being an AR is not a strong Whitney-reversible property. Let X be the continuum in Example 28.2. Let K be the cone over X and let q = (p,O) E K, that is, q is the point which corresponds to p in the base of the cone. Notice that K is not locally contractible at q. Then K is not an AR. Since the Hilbert cube is an AR (see [5, Chapter V, 2.13]), Theorem 30.3 implies that the property of being an AR is not a strong Whitney-reversible property. Notice that the example given in 30.4 is a 2-dimensional continuum. Goodykoontz and Nadler have observed that such an example is impossible in dimension one. 30.5 Theorem [25, Remarks 4.51. The property of being an AR is a sequential strong Whitney-reversible property for the classof l-dimensional continua. 30.6 Theorem. property.
The property of being an AR is a Whitney-reversible
Proof. Let X be a continuum such that every positive Whitney level is an AR. Then every positive Whitney level is a 2-connected locally connected continuum. By Theorems 53.3 and 52.2, X is a IocaIly connected dendroid. That is, X is a dendrite. Therefore, X is an AR ([5, Chapter V, Corollary 13.51). n In [79, Question 14.561,it was asked whether there is a Whitney-reversible property which is not a strong Whitney-reversible property. Combining results 30.4 and 30.6, we conclude that the property of being an AR is such a property. The property of being an AR is the first and the only known property having these characteristics.
EXERCISES
245
Exercises 30.7 Exercise. The property of having a basis of compact connected unicoherent neighborhoods is not a strong Whitney-reversible property. 30.8 Exercise. Let X be a disk. Then there exists a sequence of Whitney levels for C(X) such that A, converges to Fr(X) and an AR for any n.
31. 31.1 Theorem property (Exercise
Being
{dn}pz=1 A, is not
an Arc
[62,6.4 (a)]. The property 27.3).
of being an arc is a Whitney
31.2 Theorem [79, Corollary 14.501. The property of being an arc is a sequential strong Whitney-reversible property (Exercise 31.13). 31.3 Theorem [79, Corollary 14.521. The property of not containing an arc is a sequential strong Whitney-reversible property (Exercise 31.13). 31.4 Definition. A continuum X is said to be an arc-continuum each of its nondegenerate proper subcontinua is an arc. 31.5 Theorem uum is a Whitney property.
if
[16, Theorem 71. The property of being an arc-continproperty for the class of continua having the covering
Proof. For the definition of covering property, see Definition 35.3. Let X be a continuum. Let ~1be a Whitney map for C(X) and let t E (O,p(X)). Take a proper nondegenerate subcontinuum B of A = p-‘(t). Since X has : C E D} is a nondegenerate the covering property, the set B = U{C proper subcontinuum of X, and hence B is an arc. So 23 is a nondegenerate Hence, t3 is an arc, because subcontinuum of A n C(B) = (p]C(B))-l(t). Whitney levels of arcs are arcs (Theorem 31.1). n 31.6 Theorem. The property of being an arc-continuum is a sequential strong Whitney-reversible property. Proof. Let X be a continuum. Suppose that there is a Whitney map p for C(X) and there is a sequence{tn}rzr in (0, p(X)) such that lim t, = 0 and pel(tn) is an arc-continuum for each n. Let A be a nondegenerate proper subcontinuum of X. Then there is a nondegenerate subcontinuum B of X such that B n A = 0. Let N be a positive integer such that t, < min{~(A),~(B)} for every n > N.
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For each n 2 N, (&Z(A))-‘(t,) .is a subcontinuum of CL-’ (tn). Since p(B) > t,, there is a subcontinuum C of B such that p(C) = t,. Then C E p-‘(tn) - (,uJC(A))-l (tn). Since p(A) > t,, (,ujC(A))-’ (tn) is nondegenerate. By hypothesis, pL-l (tn) is an arc-continuum, then (pJC(A))-’ (tn) is an arc. Therefore, we can apply Theorem 31.2 and we conclude that A is an arc. n Dilks and Rogers ([16, Example 11) also showed that the covering property is an essential hypothesis in Theorem 31.5. 31.7 Example [16, Example 11. The property uum is not a Whitney property.
of being an arc-contin-
Exercises 31.8 Exercise. Let X be a continuum. morphic to [0, 11, then X is decomposable.
If C(X)
has a level homeo-
31.9 Exercise. Suppose that X is an arcwise connected continuum and it does not contain circles. Let p be a Whitney map for C(X). Fix a point p E X. Define f : X -+ [O,p(X)] by f(q) = p(the unique arc in X joining p and q). Then f is not necessarily continuous. 31.10 Exercise. Let f be as in Exercise 31.9. Let Y be a locally connected subcontinuum of X. Then f ]Y is continuous. [Hint: Y is locally arcwise connected ([81, Theorem 8.25]).] 31.11 Exercise. If X is a nondegenerate locally connected continuum without circles or simple triods (the definition of a simple triod is in the paragraph preceding Example 5.4), then X is an arc. [Hint: Fix a point p E X and define f as in Exercise 31.9. Let q E X be : z E X}. Now, define f’ as in Exercise 31.9 for such that f(q) = max{f(z) : 2 E X}. the choosen point q. Let y E X be such that f’(y) = max{f’(s) Prove that X is equal to the unique arc joining q and y.] 31.12 Exercise. an arc.
An atriodic
locally connected continuum
31.13 Exercise. Prove Theorems 31.2 and 31.3. [Hint: Use Theorem 52.2 and Exercise 31.12.1
is a circle or
32. ARC-SMOOTHNESS
32. Definition of arc-smoothness 25.37 there are some properties
247
Arc-Smoothness is in Definition 25.8. In Exercises of arc-smooth continua.
32.1 Example [84, Example 461, [86, Example 21 (Example property of being arc-smooth is not a Whitney property.
25.35 to
30.1). The
Since l-dimensional contractible continua are dendroids [lo, Proposition 11, the following theorem by Goodykoontz shows that being arc-smooth is a Whitney property for l-dimensional continua (see Exercise 25.35). This theorem generalizes the respective previous result by Petrus for dendrites ([86, Proposition 121). 32.2 Theorem smooth continuum
[23, Theorem 5.41. The property of being an arcis a Whitney property for the class of dendroids.
In [24] Goodykoontz constructed geometric models for Whitney levels of some non-locally connected continua. In this way he showed that the continuum in Figure 37 (a) (top of the next page) admits a Whitney map ~1 such that, p-‘(t) is arc-smooth for every t > 0. Using Petrus’ technique (Example 30.1) it is possible to prove that this continuum has non-contractible (and then non-arc-smooth) positive Whitney levels. In [39] Goodykoontz’s example was modified and it was shown that every positive Whitney level for the continuum in Figure 37 (b) is arc-smooth. In that continuum X, for each rational number z, with (m,n) = 1, a vertical segment of length i is constructed. Notice that, X is not contractible and then it is not arc-smooth (Exercise 25.35). 32.3 Example erty.
[39]. Arc-smoothness
33.
Arcwise
is not a Whitney-reversible
prop-
Connectedness
33.1 Theorem (see [79, Theorem 14.81). The property of being an arcwise connected continuum is a Whitney property (Exercise 33.15). 33.2 Example (Quinn, arcwise connected continuum 33.16).
[78, Example 31). The property of being an is not a Whitney-reversible property (Exercise
Quinn’s example in 33.2 ([78, Example 31) contains arcs. As we will see in this section a more radical example can be constructed.
248
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PROPERTIES
AND WHITNEY-REVERSIBLE..
(4 Arc-smoothness
.
lb) is not a Whitney-reversible
property (32.3)
Figure 37
33.3 Definition. A continuum X is continuum chainable if for each positive number E and each pair of points p # q in X, there is a finite sequence of subcontinua {AI, . . . , A,} of X such that diameter (Ai) < e, pEAl,qEA,andAinAi+i#Iforeveryi
Notice that each arcwise connected continuum is continuum chainable. However, even in the plane, continuum chainability is more general than arcwise connectedness. Recently, Hagopian and Oversteegen used inverse limits and a technique of Janiszewski to construct a continuum chainable plane continuum that does not contain any arc ([34]). 33.4 Theorem. For a continuum X the following statements are equivalent: (a) X is continuum chainable, (b) there is a Whitney map and there is a sequence {tm}~zl in (0, p(X)) such that tm + 0 and p-‘(tm) is continuum chainable for every m, (c) there is a Whitney map p for C(X) such that p-‘(t) is arcwise connected for every t > 0, and (d) every positive Whitney level for C(X) is arcwise connected.
33. ARCWISE
CONNECTEDNESS
249
Proof. Clearly, (d) + (c) and (c) =+ (b). (a) 3 (d). Let ~1be an arbitrary Whitney map for C(X) and let t > 0. Let A = p-l(t) and take A, B E A. Fix points p E A and q E B. By the continuity of ~1there exists c > 0 such that if diameter (C) < 6, then p.(C) < t. Since X is continuum chainable, there exists a sequenceof subcontinua {Al,. . . , A,} of X such that diameter (Ai) < E, p E Al, q E A, and Ai n A,+1 # 0 for every i < n. For each i 5 n, p(A2) < t. Let oi be an order arc in C(X), from A, to X. Then there exists an element Di E oi II A. Notice that A, C Di, An D1 # 0, Dl n D2 # 0,. . . , Dnel n D, # 0 and D, n B # 0. By Exercises 16.18 or 28.4 each one of the pairs {A, Dl}, (01, Dz}, . . . , {Dn-1, Dn} and {Dn, B) can be connected by a path in A. Hence, A and B can be connected by a path in A. Therefore, A is arcwise connected. (b) + (a). Let d denote a metric for X. Let 1-1be a Whitney map for C(X) and let {tm}~=i be a sequencein (O,p(X)) such that t, + 0 and p-l (tm) is continuum chainable for every m. Take two points p, q E X and E > 0. From Lemma 17.3, it follows that there exists t > 0 such that if C E C(X) is such that p(C) 5 t, then diameter (C) < 2. Let A, B E A be Take M > 1 such that TV < t and let A = II-‘( such that p E A and q E B. Since A is continuum chainable, there exists a finite sequence { di, . . . , dn} of subcontinua of A such that A E di, B E A,, diameter (di) < f (with the Hausdorff metric) and A, nd,+1 # 8 for every i < n. For each i 2 n, define Ai = U{D : D E A,}. By Exercise 11.5, A, E C(X). Notice that p E A c Al, q E B c A,. If D is an element in dindi+i then D c Ai n A,+l, then Ai n A,+1 # 0. Fix an element Di E A,, then diameter (Di) < $. Since diameter (AL) < a, A, C Nd( 6, Di). Since 5 F, d iameter (Ai) < E. This completes the proof diameter [Nd(i,Di)] that X is continuum chainable and the proof of the theorem. W As a consequence of Theorem 33.4, we have that the example of Hagopian and Oversteegen in [34] shows that even if every positive Whitney level is arcwise connected, the continuum X can be very far from being arcwise connected. 33.5 Example [34]. There is a plane continuum X such that every positive Whitney level for C(X) is arcwise connected and X does not contain arcs. 33.6 Corollary. The property of being continuum chainable is both a Whitney property and a sequential strong Whitney-reversible property.
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
33.7 Theorem [40, Lemma 1.31. If X is a continuum chainable and hereditarily unicoherent continuum, then X is arcwise connected (Exercise 33.17). By Theorems 33.4 and 33.7, the property of being arcwise connected is a sequential strong Whitney-reversible property for the classof hereditarily unicoherent continua. 33.0 Question. Is the property of being an arcwise connected continuum a sequential strong Whitney-reversible property for the class of continua which do not contain co-ods? Or equivalently (Theorem 33.4), if a continuum X is continuum chainable and it does not contain co-ods, is X arcwise connected? 33.9 Theorem [l, Theorem 2.61. Assume that p-‘(t) is hereditarily arcwise connected for each t E (O,p(X)), then X is an arc or a circle. Hence, hereditarily arcwise connectedness is a strong Whitney-reversible property. 33.10 Question. Is the property of being an hereditarily arcwise connected continuum a sequential strong Whitney-reversible property? 33.11 Definition [67]. A continuum X is said to be uniformly provided that there is a family ,F = {p : [0, l] + X} satisfying (a) for every two points x and y in X there is a path I-, in F p(O) = x and p(l) = y, (b) for each c > 0 there is k such that for each path p E 3 numbers 0 = to < tl < . . . < tk = 1 such that for each i E we have diameter [p([t*-1, ti])] 5 E. connected
pathwise
of paths such that there are { 1,. . . , k}
The notion of “uniformly pathwise connected” was introduced by Kuperberg in [67] where he showedthat a continuum X is uniformly pathwise connected if and only if X is a continuous image of the Cantor fan (the cone over the Cantor set). 33.12 Theorem [55, Corollary 3.21. The property of being uniformly pathwise connected is a Whitney property for the class of continua that have property (K). 33.13 Questions. Is the property of being uniformly pathwise connected a Whitney property? Is this property a Whitney-reversible property (strong Whitney-reversible property, sequential strong Whitney-reversible property)?
EXERCISES
251
33.14 Questions. Let Z be a fixed nondegenerate continuum. Is the property of being a continuous image of Z a Whitney (Whitney-reversible, strong Whitney-reversible, sequential strong Whitney-reversible) property? When Z is locally connected, the continuous images of Z are precisely the locally connected continua. Then for this case the Question 33.14 has been solved affirmatively (Theorems 52.1 and 52.2). In the case that Z is the Cantor fan (the cone over the Cantor set), Question 33.14 becomes Question 33.13 and in the case that Z is the pseudo-arc, Question 33.14 becomes Question 37.8.
Exercises 33.15 Exercise. Prove Theorem 33.1. [Hint: Use Exercises 16.18 or 28.4.1 33.16 Exercise. Construct continuum chainable continuum
an example of a non-arcwise (Example 33.2).
connected,
33.17 Exercise. Prove Theorem 33.7. [Hint: If p,q E X, let A = n{B E C(X) : p,q E B}. Then A is a subcontinuum of X, p,q E A and A is connected im kleinen at each of its points. Then, by Theorem 8.23 in [81], A is arcwise connected. This implies that A is an arc joining p and q.] 33.18 Exercise. which are continuous (a) m(Z) = m(W)
For a continuum 2 let m(Z) be the class of continua images of Z. Then: if and only if Z E m(W) and W E m(Z),
(b) in the particular case that Z is the harmonic fan (see Figure 23, p. 92) and W is the sin($)-continuum, show that %2(Z) is properly contained in m(W).
34.
Being
Atriodic
Definition of a triod and an n-od is in Exercise continuum is a continuum without triods.
12.20.
An atriodic
34.1 Example [88, Example 5.61. The property of being an atriodic continuum is not a Whitney property (see Theorem 34.5). This example is due to Rogers [88] and it is represented in Figure 38 (a) (top of the next page). Notice that for small t > 0, Whitney levels for C(X) are as the space represented in Figure 38 (b).
252
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PROPERTIES
AND WHITNEY-REVERSIBLE..
(4
.
lb)
Atriodicity
is not a Whitney property (34.1) Figure 38
34.2 Question. Is the property of being atriodic a Whitney property for continua having the property of Kelley? 34.3 Theorem [79, Theorem 14.491. The property of being an atriodic continuum is a sequential strong Whitney-reversible property (Exercise 34.8). 34.4 Theorem [79, Theorem 14.49.11. The property of not being a triod is a sequential strong Whitney-reversible property. In fact, if X is a triod, then there exists 6 > 0 such that p-‘(t) is a triod for each 0 5 t < 6. 34.5 Theorem [31, Theorem 3.21. The property of being atriodic and having k’(.) z 0 is a Whitney property. 34.6 Question.
Is the property of being an n-od a Whitney property?
34.7 Theorem [83, Theorem 2.41. The property of being atriodic and tree-like is a Whitney property.
EXERCISES
253
Exercises 34.8 Exercise. Let X be a continuum. Let Y E C(X) Then there exists 6 > 0 such that, for each 0 < t < 6, p-‘(t) n-od. 34.9 Exercise. Let X be a continuum. n-od, then there is t > 0 such that p-‘(t) Corollary 3.31).
be an n-od. contains an
Prove that if X contains an contains an (n - 1)-cell ([66,
34.10 Exercise. For each n 2 1 there is an atriodic continuum that there are Whitney levels for C(X) which contain n-ods.
35.
C*-Smoothness,
Class(W)
and Covering
X such
Property
35.1 Definition [79, Definition 15.51. For a continuum X, define C* : C(X) + C(C(X)) by C*(A) = C(A). A continuum X is said to be C*smooth at A E C(X) provided that the function C* is continuous at A. A continuum X is said to be V-smooth provided that C* is continuous on C(X), i.e., at each A E C(X). A continuum X is said to be absolutely C*-smooth ([28]) provided that if X is a subcontinuum of a continuum 2, then the function C* : C(Z) + C(C(2)) is continuous at X. 35.2 Definition. An onto continuous function between continua f : X -+ Y is said to be weakly confluent provided that for each subcontinuum K of Y, there exists a component C of f-‘(K) such that f(C) = K. The continuum X is said to be in Class(W), written X E Class(W), provided that every map from any continuum onto X is weakly confluent. It is easy to see (Exercise 77.3) that f is weakly confluent if and only if the induced map C(f) : C(Z) + C(Y) given by C(f)(A) = f(A) is surjective. The concept of Class(W) was introduced in 1972 by Lelek in his seminar at the University of Houston. 35.3 Definition. A continuum X is said to have the covering property, written X E CP, provided that no proper subcontinuum of p-‘(t) covers X for any Whitney map 1-1for C(X) and any t E [O,p(X)]. A continuum X is said to have the covering property hereditarily written X E CPH if every nondegenerate subcontinuum of X has the covering property. In [79, Theorem 14.73.211appeared a theorem by Hughes which says that if X is a continuum and X E CP, then X E Class(W). Later in [28,
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
Theorem 3.21 (see Theorem 67.1), Grispolakis and Tymchatyn proved that the following statements are equivalent: (a) X is absolutely C*-smooth, (b) X E Class(W), and (c) x E CP. Furthermore, Proctor in [87] showed that (a) (and then (b) and (c)) are also equivalent to the following property: (d) every compactification Y of [O,1) with remainder X has the property that C(Y) is a compactification of C([O, 1)). 35.4 Theorem [79, Theorem 14.55 (ii) and (iii)]. The property of being in Class(W) (and then having the covering property) is a strong Whitney-reversible property. 35.5 Example [29, Example 4.51 and Ill, Example 51. The property of being in Class(W) is not a Whitney property. The first example showing that being in Class(W) is not a Whitney property was given by Grispolakis and Tymchatyn in [29, Example 4.51. Since we know that being in Class(W) is equivalent to having covering property, we can use the following simpler example by W. J. Charatonik ([ll, Example 81) to show that being in Class(W) is not a Whitney property. Consider the spacesrepresented in Figure 39 (next page). Let the continuum X obtained by identifying the circle Si to the circle S’sby the identity map. In Exercise 35.18 it is asked to prove that X has CP and X has some non-unicoherent Whitney levels. Thus, by Exercise 35.13 there are someWhitney levels for C(X) which are not in Class(W). 35.6 Question. Is the property of being in Class(W) (or equivalently having covering property) a sequential strong Whitney-reversible property? 35.7 Theorem ney property.
[79, 14.76.81. The property of having CPH is a Whit-
35.8 Theorem. versible property.
The property of having CPH is a strong Whitney-re-
Proof. Let X be a continuum. Suppose that there exists a Whitney map p for C(X) such that p-‘(t) E CPH for each t > 0. Take a nondegenerate subcontinuum A of X. Notice that PIG’(A) is a Whitney map for C(A), and for each t E (O,,u(A)), p-‘(t) f~ C(A) is a positive Whitney level for C(A) and it is a subcontinuum of p-‘(t). Then
35. C*-SMOOTHNESS,
CLASS(W)
AND COVERING PROPERTY
(4
(b) Having covering property is not a Whitney Figure 39
property (35.5)
255
256
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
p-‘(t) n C(A) E CP. Since the covering property is a strong Whitneyreversible property (Theorem 35.4), we conclude that A E CP. Therefore, X E CPH. m With a similar argument as in Theorem 35.8, a positive answer to Question 35.6 would give a positive answer to the following question. 35.9 Question. Is the property of having CPH a sequential strong Whitney-reversible property? 35.10 Theorem [l, Theorem 2.91. The property of being a C*-smooth continuum is a sequential strong Whitney-reversible property. 35.11 Example [l, Remark 2.101. C*-smoothness is not a Whitney property (Exercise 35.15).
Exercises 35.12 Exercise. Prove directly, that the simple triod and the circle: (a) are not absolutely C*-smooth, (b) are not in Class(W), and (c) do not have the covering property. 35.13 Exercise. If X is a non-unicoherent continuum, then X is not in Class(W). [Hint,: Suppose that X = A U B, where A, B E C(X) and A II B is not connected. Fix a point p E A n B. Consider the space Y = Ao U Bo, where A0 is a copy of A, Bo is a copy of B and A0 rl Bo = 0. Let, 2 be the continuum obtained by identifying the copy of the point p in A0 with the copy of the point p in Bo. Define the natural map from 2 to X.1 35.14 Exercise. in Class(W).
Let X be a continuum. If X is a triod, then X is not
35.15 Exercise. The simple triod is useful to show that C*-smoothnessis not a Whitney property (Example 35.11). 35.16 Exercise. then X E CPH.
If X is an hereditarily indecomposable continuum,
35.17 Exercise. have CP.
An indecomposable continuum does not necessarily
36. TECH COHOMOLOGY
GROUPS,
ACYCLICITY
257
The continuum X of Example 35.5 has CP but it has 35.18 Exercise. some Whitney levels for C(X) which does not have CP. [Hint: Let A = p-l(t) be a Whitney level. Let Ai be the unique element in A which contains xi (i = 1,2). Then A is irreducible between Al and ilz. If p(each subarc of Si which joins two opposite points of 5’1) < t < k~(Si) then A is not unicoherent.]
36.
tech
Cohomology
Groups,
Acyclicity
36.1 Theorem [88, Corollary 4.121. If X is a circle-like continuum, then the Tech cohomology groups (over the integers) of X are the same as those of p-l(t) whenever 0 I t < p(X). 36.2 Theorem [89, Corollary 61. The property of being acyclic in dimension one (i.e., H’(.) z 0) is a Whitney property. 36.3 Theorem [89, Corollary 71. Being acyclic is a Whitney property for the class of one-dimensional continua. 36.4 Example [86, Example 21. The property of being acyclic is not a Whitney property (Example 30.1). 36.5 Theorem [SO, Corollary 4.31. If n > 1, the property of being acyclic in dimension n (i.e. HI”(.) M 0) is a sequential strong Whitneyreversible property.
37.
Chainability,
(Arc-Likeness)
37.1 Definition. By a chain in a topological space, we mean a finite collection of open sets Z4 = {VI,. . . , Un) such that Vi n U, # 0 if and only if ]i - j] 5 1. A member of l4 is called a link of U. 37.2 Definition. A continuum X is said to be chainable (some authors say snake-like or arc-like) provided that for each E > 0, there is a chain in X, covering X such that each link has diameter lessthan E. 37.3 Theorem [81, Theorems 12.11 and 12.191. For a continuum X, the following statements are equivalent:
(a) X is chainable, (b) X is arc-like (in the senseof Definiton 55.1 (c)) (c) X it is an inverse limit of arcs with onto bonding maps.
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AND WHITNEY-REVERSIBLE..
.
37.4 Theorem [62, 6.2 (a)]. The property of being a chainable continuum is a Whitney property. In [79, Question 14.571it was asked if the property of being chainable or circle-like is a strong Whitney-reversible property. Some partial answers to this question appeared in Theorems 2.4 and 2.5 of [l] and Theorems 5.5 (4) of [80]. Finally Kato in [47], using techniques of ANR and shape theory, solved this question by showing that both properties, chainability and circle-likeness, are sequential strong Whitney-reversible properties. 37.5 Theorem [47, Corollary 3.31. The property of being a chainable continuum is a sequential strong Whitney-reversible property. 37.6 Definition [70, p. 2721. A finite sequenceof nonempty subsets Xl,. . . ,X,, of a continuum X is said to be a weak chain provided that X,-inX,#IJforeveryi>l. Aweakchaind={Xi,...,X,}issaidtobe a refinement of a weak chain B = { ‘li , . . . , Yp} provided that each element X, of A is contained in some element Yk, of L3such that jki - kjjl < 1 if Ji - jl 5 1 (i,j = 1,. . . , m). A continuum X is said to be weakly chainable provided that there exists an infinite sequencedi, . . , A,,, , . . of finite open covers of X such that each A, is a weak chain, each element of A, has diameter less that i and A,,+) is a refinement of A, for each n 2 1. In [70] and [20], Lelek and Fearnley respectively defined the notion “weakly chainable” and they proved that a continuum X is weakly chainable if and only if X is a continuous image of the pseudo-arc. Using this characterization, Kato proved the following theorem, which is a partial answer to [79, Question 14.361. 37.7 Theorem [55, Corollary 3.11. The property of being weakly chainable is a Whitney property for the classof continua having property of Kelley. The general question about weak chanability remains open. 37.8 Questions [79, Question 14.36 and 14.571, (see Question 33.14). Is weak chainability a Whitney (Whitney-reversible, strong Whitney-reversible, sequential strong Whitney-reversible) property? 37.9 Theorem [16, Theorem lo]. Suppose that X is the inverse limit of arcs with open bonding maps. Then X is homeomorphic to each of its positive Whitney levels. Therefore, the property of being a particular continuum X constructed as an inverse limit of arcs with open bonding maps is a Whitney property. In particular, the property of being the Buckethandle continuum (see Example 22.11) is a Whitney property.
EXERCISES
259
Exercises 37.10 Exercise.
If X is a chainable continuum,
then X is atriodic.
37.11 Exercise. unicoherent.
If X is a chainable continuum,
then X is hereditarily
37.12 Exercise. Let X be a continuum. If X is chainable and decomposable, then there exists to > 0 such that p-‘(t) is an arc for each t > to ([66, Theorem 4.4 (a)]). [Hint: Suppose that X = AUB, where A, B E C(X){X). Then AnB is connected and if E E C(X) and p(E) > p(A), p(B), then An B C E.] 37.13 Exercise. This exercise shows that the “natural” definiton for weakly chainable continua is not a good one. Every continuum X has the following property: For each E > 0 there exists a weak chain A which covers X and diameter (U) < E for each U E A.
38.
Being
a Circle
38.1 Theorem [62, 6.4 (b)]. The property ney property (Exercise 27.4).
of being a circle is a Whit-
38.2 Theorem [79, Corollary 14.511. The property of being a circle is a sequential strong Whitney-reversible property (Exercise 38.4). 38.3 Theorem [79, Theorem 14.531. The property of not containing a circle is a sequential strong Whitney-reversible property (Exercise 38.5).
Exercises 38.4 Exercise. Prove Theorem 38.2. [Hint: Use Theorem 52.2 and Exercise 31.12.1 38.5 Exercise.
Prove Theorem 38.3.
39.
Circle-Likeness
39.1 Definition. A continuum X is said to be circle-like provided that for each E > 0, there exists a finite collection V = {VI,. . . , Vn} of open subsets of X, covering X such that: (a) & I-I vj # 8 if and only if ]i - j] 5 1 or i,j E (1, n}, (b) Vi has diameter less than e for each i = 1,. . . ,n.
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A continuum which is circle-like and not chainable will be called proper circle-like. It is well known and not difficult to prove that for circle-like continua, Definition 39.1 and Definition 55.1 (d) coincide (Exercise 39.9). For general facts about circle-like continua we refer the reader to [4] and [8]. 39.2 Theorem [62, 6.2 (a)]. The property of being a proper circle-like continuum is a Whitney property. 39.3 Theorem [88, Theorem 5.11. If X is a decomposable continuum which is both chainable and circle-like, then there exists to < p(X) such that p-‘(t) is not circle-like for any t > to (see Exercise 37.12). 39.4 Corollary. The property of being a circle-like continuum is not a Whitney property. For a particular example which ilustrates the Corollary 39.4 see Figure 40 (top of the next page). The space consists of two copies of the Buckethandle continuum (seeExample 22.11) joined by their end points. 39.5 Theorem [47, Corollary 3.51. The property of being a circle-like continuum is a sequential strong Whitney-reversible property (see comments preceding Theorem 37.5). 39.6 Theorem [47, Corollary 3.51. The property of being a proper circle-like continuum is a sequential strong Whitney-reversible property.
Exercises 39.7 Exercise. 39.8 Exercise. circle-like. 39.9 Exercise.
Circle-like continua are atriodic. The continuum in Figure 40 is both chainable and
Definition 39.1 and Definition 55.1 (d) coincide.
39.10 Exercise. If X is a circle-like continuum and A E C(X) is nonunicoherent, then A = X.
261
40. CONE = HYPERSPACE PROPERTY
Circle-likeness
is not a Whitney
property
(39.4)
Figure 40
40.
Cone = Hyperspace
Property
40.1 Definition. A continuum X is said to have the cone = hyperspace if there is a homeomorphism h : cone(X) + C(X) such that h maps the vertex of the cone to the point X in C(X) and maps the base of the cone onto the set of singletons in C(X).
property
A very complete discussion about the cone = hyperspace property is included in [79, Chapter VIII]. Additional information can be found in section 80. Answering a Question by Dilks and Rogers ([16, p. 636]), recently it has been proved in [45] the following theorem. 40.2 Theorem [45]. If X is a finite-dimensional continuum with the cone = hyperspace property, then X is homeomorphic to each of its Whitney levels. In particular, the property of being a finite-dimensional continuum with the cone = hyperspace property is a Whitney property.
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Exercise 40.3 Exercise. If a finite graph X has the cone = hyperspace property, then X must be an arc or a simple closed curve. [Hint: Prove that if a finite graph with the cone = hyperspace property is atriodic, then it is an arc or a circle.]
41.
Contractibility
41.1 Example [86, Example 21. The property of being contractible is not a Whitney property (Example 30.1).
The following theorem of Petrus is a particular case of Theorem 32.2. 41.2 Theorem [86, Proposition 121. The property of being contractible is a Whitney property for the class of dendrites. 41.3 Question [86, p. 2881,compare with Question 25.30. Is the property of being contractible a Whitney property for the classof dendroids? 41.4 Example [86, Example 51. The property of being locally contractible is not a Whitney property (Example 28.1). 41.5 Theorem [80, Theorem 3.11. The property of being contractible with respect to a fixed ANR Y (definition of “contractible with respect to” is in the paragraph preceding Lemma 19.4) is a sequential strong Whitneyreversible property. 41.6 Theorem 180,Corollary 3.21. Each one of the following properties is a sequential strong Whitney-reversible property: (a) having trivial shape, (b) being a fundamental absolute retract (=FAR), (c) being a weak approximate absolute retract, and (d) being absolutely neighborhood contractible. Proof. For continua it is known that each of the properties listed above is equivalent to contractibility with respect to every ANR (see for example [7], [37] and [64, 2.11). Therefore Theorem 41.6 follows from Theorem 41.5. n
Some properties which are equivalent to contractibility with respect to every ANR, but which are not listed in Theorem 41.6, may be found in the references in the proof of Theorem 41.6. Therefore these properties are sequential strong Whitney-reversible properties.
41. CONTRACTIBILITY
263
In [80, Example 3.71, it was shown that for the dendroid X illustrated in Figure 25, p. 158, every positive Whitney level for C(X) is contractible (Exercise 41.16), and then Whitney levels for C(X) are contractible with respect to any space (Lemma 19.4). In particular they are contractible with respect to X. However X is not contractible and, then it is not contractible with respect to itself. Then we have the following two results. The second one says that it is not possible to extend Theorem 41.5 to spaceswhich are not ANR. 41.7 Example [80, Example 3.71. The property of being contractiblr is not a Whitney-reversible property, even for dendroids. 41.8 Example [80, Example 3.71. The property of being contractible with respect to a fixed continuum is not a Whitney-reversible property. 41.9 Question [25, Remarks 4.51. Is the property of being contractible a strong Whitney-reversible (or sequential strong Whitney-reversible) property for the class of Peano continua? 41.10 Definition. Let A4 be an ANR. A continuum X is said to be extendable with respect to M provided that for any closed subset,A of X and any continuous function f : A + M, there exists a continuous extension f: x -9 A4 off. The following theorem is useful to prove that the property of having dimension 5 n is a sequential strong Whitney-reversible property (see Theorem 45.1) 41.11 Theorem [47, Theorem 2.11. The property of being extendable with respect to a fixed ANR is a sequential strong Whitney-reversible property. The following example answered Question 18 of [14]. 41.12 Example [41, Example 2.21. There exist a contractible continuum X and a positive Whitney level A for C(X) such that A has non-contractible hyperspace C(A). Therefore, the property of having contractible hyperspace is not a Whitney property. The example is the cone over the compact space Y represented in Figure 41 (a) (top of the next page). The space Y is a union of a circle S and a sequenceof arcs converging to S.
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264
WHITNEY PROPERTIESAND WHITNEY-REVERSIBLE...
Y
. . Q Q
s
’
l
.
0
(4
(b)
Having contractible hyperspace is not a Whitney property (41.12) Figure 41
Exercises 41.13 Exercise. A proper nonempty closed subset A of a continuum fixed provided that if H : X x [0, l] -+ X is X is said to be homotopically a map such that H(x,O) = x for each z E X, then H(A x [0, 11) = A. If X contains a homotopically fixed subset then X is not contractible. 41.14 Exercise. Every R3-subset (seeDefinition 24.12) of a continuum X is homotopically fixed. 41.15 Exercise. finite graphs.
Question 41.9 has a positive answer for the class of
41.16 Exercise. Positive Whitney levels for the space of Figure 25, p. 158 are contractible and X is not contractible.
42. CONVEX METRIC
Convex
42.
265
Metric
Convex metrics are defined in section 10. 42.1 Theorem [l, Theorem 3.11. Let X be a continuum. Assume that there is a sequence {tn}Fzr such that t, -+ 0 as n + 00, and the restriction of the Hausdorff metric to p-‘(tn) is convex, then the original metric d for X is convex (Exercise 42.5). 42.2 Example [l, Example 3.21. There is an arc X, a convex metric d for X, a Whitney map p for C(X) and a number to E (O,p(X)) such that the Hausdorff metric restricted to p-‘(t) is not convex for any t E (0, to). 42.3 Question. Characterize those Peano continua X for which it is possible to define a Whitney map p for C(X) such that for each t E [0, p(X)), the Hausdorff metric restricted to p-‘(t) is convex (see Exercise 42.6). 42.4 Question. C(X).
The same question
as 42.3 but with
2x instead
of
Exercises 42.5 Exercise.
Prove Theorem 42.1.
42.6 Exercise. Prove the following particular answer to Question 42.3. Let X be a circle. Prove that for each metric d for X, and for each Whitney level .4 for C(X), the Hausdorff metric Hd restricted to A is not convex. [Hint: Let p E X and A E A be such that d(p, A) = max{d(z, B) : 2 E X and B E d}. Let J E A be such that p is in the interior of J. Prove that if J’ E A and Hd(J, J’) is small, then Hd(A, J) = Hd(A, J’).] 43.
Cut Points
43.1 Definition. A point p in a continuum that X - {p} is disconnected.
X is a cut point provided
43.2 Example. The property of having a cut point is not a sequential strong Whitney-reversible property. Define C = [-l,O] x [-l,l], ZO = ((0) x [-l,l])U{(x,y) E R2 : 0 <x 5 l’and /sin($) -yJ < f}. For each 72, let Z, = ((0 7&L-r) + (g-r, $b) : (X,Y) E Zo}. Finally, define X = CU(U{Z, : n 2 I}), the continuum X is represented in Figure 42 (top of the next page). Clearly, X is a continuum without cut points.
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X C A,
A2
A3
Having cut-points is not a sequential strong Whitney-reversible property (43.2) Figure 42
Let p be any Whitney {Olx[ t~--$;&;&l 3 Then
map for C(X). For each n 1 1, let A, = Z, and let t, = p(A,,) and A, = pL-’ (k). =
Wenwill show t;at each A,, is a cut point of A,. First, note that every subcontinuum B of X that intersects Z, - A, and X - Z, contains A, and then B = A, or B $ A,. This implies that A, = {B E A, : B C : m # n})}. These two sets are nonempty, Z,}u{B E A, : B c Cu(u{Z, closed in A, and their only common element is A,. Hence, A, is a cut point of A,. Therefore, each A, has cut points and X does not have cut points. Thus, the property of having cut points is not a sequential strong Whitney-reversible property. 43.3 Question. ible property?
Is the property of having cut points a Whitney-revers-
EXERCISES
267
Exercises 43.4 Exercise. property.
The property
of having cut points is not a Whitney
43.5 Exercise ([35, Theorem 2.11). Let X be a continuum. Let A be a Whitney level for C(X). Let A E A. Then A is a cut point of A if and only if there exist nonempty disjoint open subsets U and V of X such that X - A = U u V and such that, for any B E A, B c A u U or B C A U V. 43.6 Exercise. Let X be a continuum. If A E C(X) and A and Z?are Whitney levels for C(X), with A E An 23,then A is a cut point A if and only if A is a cut point of B. Thus, being a cut point of a Whitney level is independent of the Whitney map.
44.
Decomposability
Notions related to decomposable and indecomposable continua were defined after Theorem 7.2. 44.1 Theorem [57]. The property of being an hereditarily indecomposable continuum is a Whitney property (Exercise 18.13). 44.2 Theorem [66, Theorem 3.41. The property of being a decomposable continuum is a Whitney property (Exercise 44.10) 44.3 Corollary [79, Theorem 14.46 (l)]. The property of being an indecomposable continuum is a sequential strong Whitney-reversible property. 44.4 Example [66, Example 5.41. The property of being an indecomposable continuum is not a Whitney property, even for circle-like continua (Exercise 44.14). 44.5 Theorem [66, Theorem 4.31. The property of being an indecomposable chainable continuum is a Whitney property. Since chainable continua have the covering property ([66, Theorem 4.21 or [79, Lemma 14.13.1]), Theorem 44.6 generalizes Theorem 44.5. 44.6 Theorem [85, Proposition 281. The property of being an indecomposable continuum is a Whitney property for continua with the covering property (Exercise 44.11).
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44.7 Theorem [35, Theorem 3.11. Let X be a continuum. If there is a Whitney map p for C(X) such that, for some to E [O,p(X)) the level h-i (to) is decomposable and irreducible (about two members of 11-l (to)), then X is decomposable. 44.8 Theorem [79, Theorem 14.54 (l)]. The property of being an hereditarily indecomposable continuum is a sequential strong Whitneyreversible property (Exercise 44.12). 44.9 Theorem [1, Theorem 2.21. The property of being an hereditarily decomposablecontinuum is a sequential strong Whitney-reversible property (Exercise 44.13).
Exercises 44.10 Exercise. Prove Theorem 44.2. [Hint: Let X be a continuum. Suppose that X = A u B, where A and B are proper subcontinua of X and let C = p-i(t) be a Whitney level for C(X). Letd={C~C:C~A#0}andB={C~C:C~B#0}. Then A and B are subcontinua of C with nonempty interior (see Exercise 27.7). If A or B are properly contained in C, then C is decomposable. Then, we may assumethat A = C = B. If p(A) < t, enlarging A with an order arc, we may assumethat p(A) 2 t. Let di = C(A)nd. Then di is a nonempty subcontinuum of C and every element in C can be joined by an arc to an element in di. Then C is decomposable (see [81, Theorem 11.15]).] 44.11 Exercise.
Prove Theorem 44.6.
44.12 Exercise.
Prove Theorem 44.8
44.13 Exercise. Prove Theorem 44.9. [Hint: Suppose that Y is an indecomposable subcontinuum of X. Let t > 0 be such that p-‘(t) rl C(Y) is nondegenerate. Apply Exercise 80.21 to obtain a minimal subcontinuum A of p-‘(t) n C(Y) with the property that U{A : A E A} = Y.] 44.14 Exercise. Let X be the continuum obtained by identifying two points a and b in different composants of the pseudo-arc. Then X is an indecomposable continuum but Whitney levels for C(X) are decomposable.
45.
Dimension
In [60, Theorem 31, the problem of whether the property of being of dimension < n is a Whitney-reversible property was partially solved by
EXERCISES
269
Koyama. He proved that, assuming additional hypothesis on the continuum X, the answer to this problem is affirmative. In [80, Theorem 2.71 it was shown that no additional conditions are necessary and it was asked (180, Question 2.10)) if the property of being of dimension 5 n is a sequential Whitney-reversible property. As a consequence of his theorem on extendability with respect to a fixed ANR (Theorem 41.11) and the theorem relating dimension 5 n with exandability with respect to the n-sphere (see Theorem VI, 4 in [36]), Kato obtained the following affirmative answer. 45.1 Theorem [47, Corollary 2.21. For any given n < co, the property of having dimension 5 n is a sequential strong Whitney-reversible property. 45.2 Corollary (compare with [80, Theorem 2.91). Let X be a continuum. If there exists a Whitney map ~1for C(X) and there exists a sequence I n {tm}~=r of elements of (0, p(X)) such that t, -+ 0 and dim[p-‘(t,)] for each m, and for some fixed n < co, then dim[X] = 1. Proof. By Theorem 45.1, dim[X] 5 n. If dim[X] 2 2, then X contains a subcontinuum Y such that dim[Y] = 2 (Theorem 72.6). By Theorem 73.10, there exists to > 0 such that dim[(p]C(Y))-‘(t)] = 00 for every 0 < t < to. Then dim[p-l(t,)] = 00 for infinitely many positive integers m. This contradicts the hypothesis and completes the proof of the corollary. n
45.3 Question [80, Question 2.111. Is the property of being finite-dimensional a strong (or, sequential strong) Whitney-reversible property? A positive answer of Question 73.12 implies that the answer to the questions contained in 45.3 is positive.
Exercises 45.4 Exercise. If X a 2-cell or the harmonic fan (Figure 23, p. 92), then every positive Whitney level for C(X) is infinite dimensional. 45.5 Exercise. Find an example of a continuum X with the property that for each Whitney map /I for C(X), there exist 0 < s < t < p(X) such that dim[p-l(s)] < dim[h-l(t)]. 45.6 Exercise. s < t.
The same as in Exercise 45.5 but with t < s instead of
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Fixed
Point
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46.1 Example [66, Example 5.61. The property of having the fixed point property is not a Whitney property (Exercises 46.8 and 46.9). 46.2 Example [25, Corollary 4.41. The property of having the fixed point property is not a strong Whitney-reversible property. By Theorem 30.3, it is enough to find a locally connected continuum X such that its topological cone K does not have the fixed point property. In [59], Knill has contructed a family of continua with the property that their topological cones do not have the fixed point property. Some of them are locally connected. Therefore, the property of having the fixed point property is not a strong Whitney-reversible property. 46.3 Question. Whitney-reversible
Is the property property?
of having the fixed point property
a
46.4 Definition [91]. For a continuum X and a continuous function f : X + X, define the fixed point set off as {x E X : f(x) = x}. A continuum X is said to have the complete invariance property (CIP) provided that for each nonempty closed subset A of X there exists a continuous function f : X -+ X such that A is the fixed point set of f. For a continuum X and a continuous function F : X + C(X) the fixed point set of F is the set {z E X : x E F(x)}. A continuum X is said to have the complete invariance property for continuum-value maps (MCIP) ([26]) provided that for each nonempty closed subset A of X there exists a continuous function F : X + C(X) such that A is the fixed point set of F. 46.5 Question. property?
Is the property
of having
CIP
(MCIP)
a Whitney
46.6 Questions. Is the property of having CIP (MCIP) a Whitney-reversible (strong Whitney-reversible, sequential strong Whitney-reversible) property?
Exercises 46.7 Exercise.
Fixed point property
is preserved
under retractions.
46.8 Exercise. Show that having the fixed point property tial strong Whitney-reversible property.
is a sequen-
47. FUNDAMENTAL
271
GROUP
46.9 Exercise. Let X be the Warsaw circle illustrated in (4) of Figure 20, p. 63. Then X has the fixed point property and there are Whitney levels for C(X) which are circles. Thus, having the fixed point property is not a Whitney property. 46.10
Exercise.
46.11 Exercise.
If a continuum
X has CIP, then X has MCIP.
The arc and the circle have CIP and MCIP.
47.
Fundamental
Group
47.1 Example [86, Example 81. The property of having trivial fundamental group is not a Whitney property, even for contractible continua. In [43], it was studied the fundamental group of spaces of the form p-l(.s, t), where p is a Whitney map for C(X) and X is a Peano continuum. These results are mentioned in section 68. We do not know if results in [43] are valid for Whitney levels. In particular, we do not know the answer to the following questions. 47.2 Question. Is the property of having trivial Whitney property for the class of Peano continua?
fundamental
group a
47.3 Questions. Is the property of having trivial fundamental group a Whitney-reversible property for the class of arcwise connected continua? and for Peano continua? This question is related to the results in sections 53 and 68.
Exercises 47.4 Exercise. The Warsaw circle in (4) of Figure 20, p. 63 has trivial fundamental group and it admits Whitney levels which have non-trivial fundamental group. 47.5 Exercise. The answer to Question 47.2 is positive for the particular case of finite graphs. 47.6 Exercise. The property of not having trivial fundamental is not a Whitney property even for the class of finite graphs.
48.
group
Homogeneity
48.1 Definition. A continuum X is homogeneous, if for every two points p, q E X there exists a homeomorphism f : X -+ X such that
f(P)
= 4.
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In [90, Theorem 111, Rogers showed that for the circle of pseudo-arcs, there is a positive Whitney level which is not homogeneous. Hence homogeneity is not a Whitney property. In the same direction, later in [12], W. J. Charatonik showed that for the 2-sphere there are non-homogeneous positive Whitney levels. 48.2 Example [90, Theorem 111(another example was given in [12]). Homogeneity is not a Whitney property. As it is shown in [25, Corollary 4.41, from Theorem 30.3 it follows that homogeneity is not a strong Whitney-reversible property. However, as we will seenext, with the results developed in the same paper [25], it is possible to prove that homogeneity is not a Whitney-reversible property. 48.3 Example. Homogeneity is not a Whitney-reversible property. Let D be the a dendrite without free arcs (see, for example 10.37 in [81] or the continuum 2 in Figure 35, p. 240). By Theorem 2.17 in [25] every Whitney map for C(D) is admissible. Then by Theorem 25.3 (b), every positive Whitney level for C(D) is homeomorphic to the Hilbert cube which is homogeneous (see [56]). S’mce every dendrite contains points that are end points and points that are not end points, D is not homogeneous. Therefore, the property of being homogeneousis not a Whitney-reversible property. 48.4 Question [12, p. 3121. Does there exist a homogeneous continuum X such that for every Whitney map p for C(X), there exists a t with p-i (t) non-homogeneous? 48.5 Question [12, p. 3121. If X is a homogeneouscontinuum and p is a Whitney map for C(X), are most (e.g., a dense Gh-set) Whitney levels of p homogeneous? 48.6 Question [12, p. 3121. Is there a characterization in terms of other topological properties of homogeneouscontinua where every Whitney level of every Whitney map is homogeneous? By Theorems 38.1, 56.1 and 61.2, the circle, the pseudo-arc and the solenoids have this property.
Exercise 48.7 Exercise. If a continuum X has a free arc (seeparagraph preceding Theorem 11.2 for definition of free arc) and X is different from the circle, then there are positive Whitney levels for C(X) that are not homogeneous. Therefore:
49. IRREDUCIBILITY
273
(a) all the positive Whitney levels of a dendrite X are homogeneous if and only if the X does not have free arcs. (b) the unique homogeneouscontinuum with free arcs for which homogeneity is a Whitney property is the circle. [Hint: For the first part, see Exercise 31.11. For (a) see explanation of Example 483.1
49.
Irreducibility
49.1 Definition. A continuum X is said to be irreducible about a snbset Z of X provided that no proper subcontinuum of X contains 2. A continuum X is irreducible provided that X is irreducible about {p, (I} for some p and q in X, in which case sometimes X is said to be irreducible betweenp and q. Answering a question by Hughes it was proved in [19] that if a positive Whitney level for C(X) is irreducible, then X is irreducible. This result is generalized in Theorem 49.3. 49.2 Lemma (compare with [19, Lemma 2.71). Let X be a continuum. Let A be a Whitney level for C(X) such that A = f? U a, where B is a subcontinuum of A, cy is an arc with end points .4s and Bo, and t3 n a = {Ao}. Then Bo g u(d - {Bo}). Proof. We may assumethat Bo nAl # 0 for someAI E A - {Bo}. By Exercise 28.4, there is an arc X in A, joining A1 and Bo such that AnB, # 8 for every A E X. Let A* E (Y be such that A1 is not in the subarc y of X that joins AZ and Bo. Then A2 separates A1 and Bo in A. Thus A’L E X and A2 n Bo # 0. Notice that y is the only arc in A that joins A2 and Bo. By Exercise 28.7, A2 n Bo is connected. Let P,Q be as in Exercise 28.4. Then: (4 v(PJl) = 7, (b) r](t) n B. = P(t) for each t E [O,l], (c) {D E C(Bo) : A2 n B. c D} = {P(t) : t E [0, l]}, and (d) if s < t, then p(t) - /3(s) is connected. For each t E [O,l), let Ct = Bo - q(t) = Bo - /3(t) and Q = cl.~(C~). Let t E [0, 1) be fixed and let A E A be such that A n Ct # 0. We claim that A E ~((t, I]). Suppose to the contrary that A Q!~((t, 11). Let p E A n Ct. If A +! v([O, l]), then AZ separates A and B. in A. Since p E An Bo. By Exercise 28.4 there is an arc 0, joining A and Bo in A such that p E B for each B E IT. Notice that A2 E D then p E A2 n B. c q(t). This is a contradiction since p E Ct. Now, if A E q([O, t]), let A = v(r),
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where T 5 t. Then p E V(T) n Bs = /3(r) c /3(t) c q(t). This is again a contradiction which completes the proof that A E q((t, I]). We analize two cases: (1) There exists t E [0, 1) such that Dt is indecomposable. Fix a number r E (t, 1). Let C = cZ~(p(r) - P(t)). Then C is a subcontinuum of /3(r) n Dt. Since 0 # Be - p(r) c Dt - C. Hence C is a proper subcontinuum of Dt. Let E be the composant of Dt that contains 0 Given s E [r, l), reasoning as above, cZx(p(s) - P(t)) is a proper subcontinuum of Dt that contains C. Then clx(p(s) - p(t)) c E. Since C, is a nonempty open subset of B,J and Dt c Bo, Ct is a nonempty open subset of Dt. Since intD, (E) = 0, we can choose a point q E ~3, - E. If there exists an element A E A - {Bo} such that q E A, then there exists se > t such that A = q(se). Then q E ~(so)flBo = p(so) and q 4 ,8(t). Thus q E E. This contradiction proves that q 6 BO - u(A - {Bo}) and completes the proof for this case. (2) For each t E [0, l), Dt is decomposable. For each t E [0, l), let Dt = Ft U Gt, where Ft and Gt are proper subcontinua of Dt. Since Ct is not contained in Ft, 0 # Ct - Ft C Bo (/3(t) U Ft). Then P(t) U F t is a proper subset of Bo. Similarly, p(t) U Gt is a proper subset of Bo. We will prove that q(t) n Ff = 0 or v(t) n Gt = 0. Suppose to the contrary that q(t) n Ft # 0 and q(t) rl Gt # 0. Then P(t) n F, # 0 and p(t)nG, # 0. Then P(t)uF t and p(t) uG, are subcontinua of BO containing AZ n Bo. By (c) there exist tl, tz E [0, l] such that p(ti) = P(t) U Ft and /3(t2) = p(t) U Gt. We may assume that tl 5 t2. From the preceding paragraph t2 < 1. But Bo = ,0(t) u Ft U Gt c ,8(t2) $ P(1) = Bo which is a contradiction. Therefore, we may assumethat v(t) n Gt = 0. Since /3(t)UFtlJGt = Be, BO is connected and /3(t)flGt = 0, we have p(t)nFt # 0. From (c), there exists st E [0, 1) such that P(t) U Ft = P(st). Now, we will show that the family F = {Gt : t E [0, 1)) has the finite intersection property. Take tl, t2,. . . , t, E [0, 1). We may assume that HenceF Sf, I: St2 I ‘.. <St,,. ThenO#Bo-P(st,)cGt,n...nG,,. has the finite intersection property. Choose a point y E nF. Then y E Bo - (U{q(t) : t E [O,l)}). Let A E A be such that y E A. Given t E [O, l), since 3 f Ct, A = q(r) for somer E (t, I]. This implies that A = Bo. Therefore, y E Be-U(d-{Bo}). This completes the proof for case (2) and the proof of the lemma. n 49.3 Theorem [38]. Let X be a continuum. If there is a Whitney map p for C(X) and there is to E (O,p(X)) such that p-’ (to) is irreducible
49. IRREDUCIBILITY
275
about a set with n elements, then X is irreducible about a set with m elements for some m 2 n. Proof. Let ‘D = p-‘(to). Suppose that 2) is irreducible about a finite subset C = {Ei,..., E,} but ‘D is not irreducible about any proper subset of C, where m 5 n. The following three facts are easy to show: (a) if A E C(X) has the property that El (Exercise 49.10),
U
...
U
E,
c A then A = X
(b) let E E 27, A E C(X) and p E E n A be such that, p $! U(D then A c E or E c A (Exercise 49.11),
-
{E}),
(c) if a: is an arc in V containing Ei and Ej, with i # j, then Ei and EJ are the end points of cy (Exercise 49.12). To prove the theorem, it is enough to seethat there exist points qi E Ei such that X is irreducible about {ql,. . . , qm}. Suppose that this is not true. First, we will prove that, for each i, there exists a point pi E Ei U(V - {Ei}). Suppose also that such pi does not exist for all i. Then we may assumethat for El,. . . , Ekvl there exist points with the mentioned property and that for Ea, . . . , E, there are no such points (1 5 k 5 m). Choose points xk E &, . . . , x,,, E E,,,. Since X is not irreducible about the set {pi,. . . ,&&-I, xk,. . . , zm}, then there exists a proper subcontinuum A of X containing it. Enlarging A if necessary (using order arcs), we may assume that to < p(A) < p(X). Then A is not contained in Ei for any i. C A. Since A # X, by (a), we may assumethat Ek By (b) El u... U&-i is not contained in A. Let J = {i E {k, . _. , m} : Ei g A). Given i E J, let Ai E V be such that zi E Ai C A. Then Ei # Ai. By Exercise 28.4, there exists an arc pi in V, such that the end points of pi are Ei and Ai and xi E D for every DE pi. If i # k, then Ai # Ek and Ei # En. Then Ek is not an end point of the arc : i E J - {k}}) pi. By (c), Ek $ pi. Then the set A = (2, n C(A)) U (U{pi is a subcontinuum of V, C - {Ek} c A and & $! A. Let A0 be the first point in A belonging to the arc ok, walking form Ek to Ak and let o be the subarc of Pk that joins A0 and Ek. Then A U Q is a subcontinuum of V that contains C. Thus 2) = A u o. By Lemma 49.2, Ek g U(2> - {Ek}). Th is contradicts the choice of k and completes the proof of the existence of the points pl , . . . , p,. Since we are assuming that X is not irreducible about the set there exists a proper subcontinuum B of X such that . ,p,,}, .,p,,,} C B. We may assume that t < p(B) < p(X). By (b), IF&l;‘.‘- , U E,, C B. By (a), B = X. This contradiction compietes the proof of the theorem. n
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49.4 Corollary. The property of not being irreducible about a set with at most n elements is a Whitney property. 49.5 Corollary. The property of not being irreducible about a finit,e set is a Whitney property. 49.6 Example [38, Example 2.11. There exists a continuum X, there exists a Whitney map ~1for C(X) and there exists a to E (0,/1(X)) such that p-‘(te) is irreducible about a countable closed subset, but X is not, irreducible about a countable closed subset. 49.7 Questions [38, Question 2.21. Is the property of being irreducible about a countable closed subset a Whitney-reversible (strong Whitneyreversible, sequential strong Whitney-reversible) property? 49.8 Example [19, Example 3.21. The property of being irreducible is not a Whitney property. The example presented by Eberhart and Nadler for showing that irreducibility is not a Whitney property is the example in Figure 38, p. 252. Notice that X is irreducible, small positive Whitney levels for C(X) are not irreducible and large Whitney levels for C(X) are arcs, and hence, irreducible continua. 49.9 Question. Is there an irreducible continuum X such that every positive Whitney level is not irreducible.7 In other words, is the property of not being irreducible a Whitney-reversible property?
Exercises 49.10 Exercise. Prove (a) in the proof of Theorem 49.3. IHint: Consider (p)C(A))-1 (to).] 49.11 Exercise. Prove (b) in the proof of Theorem 49.3. [Hint: There is an element B E 2) such that A c I3 or B C A.] 49.12 Exercise. Prove (c) in the proof of Theorem 49.3. [Hint: Suppose that a(s) = Ei, a(t) = Ej and 0 5 s < t < 1. Let A be a proper subcontinuum of D that contains C - {Ej). Then A U a([O, t]) is a proper subcontinuum of 2) and it contains C.]
50.
Kelley’s
Property
Property of Kelley (= property (K)) was defined in section 20. Kelley introduced the now called Property of Kelley in [57, 3.21. He used this property for studying the contractibility of hyperspaces (Theorem
50. KELLEY'S PROPERTY
277
20.12). In [92, p. 2951, Wardle asked the question whether the property of Kelley is a Whitney property. This problem has turned out to be very difficult. It has been solved only for some particular cases (see Theorems 50.3 and 50.7). 50.1
Question
[92, p. 2951. Is the property of Kelley a Whitney prop-
erty? Trying to solve Question 50.1, Kato has found a surprising result. He has proved ([50, Corollary 3.31) that in order to obtain an affirmative answer to Question 50.1 it is enough to obtain a positive answer to the following question. 50.2 Question [55, p. 11471. If a continuum X has property (K), is it true that X x [0, l] has property (K)? 50.3 Theorem [13, p. 91. The property of Kelley is a Whitney property in the classof continua having the covering property hereditarily (Exercise 50.10). 50.4
Theorem.
The property of Kelley is a sequential strong Whitney-
reversible property. Proof. Let X be a continuum and let d denote a metric for X. Suppose that there is a Whitney map p for C(X) and there is a sequence {tn}r?.i in (0, p(X)) such that t, + 0 and each of the continua A, = pL-‘(tn) has the property of Kelley. We need to prove that X has the property of Kelley. Suppose, to the contrary that this is not true. By Exercise 50.8 there are a point p E X and an E > 0 such that for every m > 1, there are a point qm E X and a subcontinuum A,,, of X with the property that d(p,q,) < A, p E A, and if B E C(X) and q,,, E B, then Hd(A,,B) 2 6. From Lemma 17.3, there exists n > 0 such that if C E C(X) and p(C) < 7, then diameter (C) < 5. Choose N such that tN < n. For each m 2 1, let f),, be an element of dN such that qm E D,. Taking subsequencesif necessary, we may assumethat A,, + A and D,, -+ D for some .4 E C(X) and D E dN. Since q,,, + p, p E D. Then A U D is a subcontinuum of X. Since diameter (D) < 5, Hd(A, A U D) < 5. Since dN has the property of Kelley at D, there exists 6 > 0 such that if E E dN, Hd(D, E) < b and D E A E C(dN), then there exists B E C(dN) such that E E B and HH~ (A, f?) < 5. Define A = C(A U D) n AN = (p]C(il U D))-‘(TV). Then A is a subcontinuum of AN and D E A. Let M be such that Hd(An4, ‘4) < 5 and
278
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WHITNEY
PROPERTIES
AND WHITNEY-REVERSIBLE...
Hd(D, DM) < 6. Thus there exists B E C(dN) such that DM E B and HHd(d,B) < 2. Define BO = U{B : B E B}. Then Bs is a subcontinuum of X (Exercise 11.5 (3)) and qM E Bo. Since AU D = U{B : B E A}, Hd(A u D, Bo) < $ (Exercise 11.5 (2)). Thus Hd(AM, Bo) < E. This contradicts the choice of qM and completes the proof of the theorem. n 50.5 Definition [52]. Let X be a continuum and let d denote a metric for X. Let F = {pl,p2,... ,pn} be a finite sequence in X, F is said to be an c-chain (6 > 0), if d(pi,pi+l) < c for each i < n. Let a E A E C(X). Let U be a finite open covering of A. For each zcE A, we consider the set Chain,,(U) of all finite weak chains (see Definition 3’7.6) {Vi, Us, . . . , Urn} ofUwithaEUiandxEi7,. Let E> 0. Consider the following conditions (A, a, c)*: there exists J(e) > 0 such that if b E B(~(E), a), then for each x f A and C > 0 there is a finite open covering U of A with mesh(U) < E (diameter (U) < E for each U E U) such that if { U1, U2, . . . , Urn} E Chain,,(U), then thereisac-chainb=bi,bq . . . . bf(‘),bi,bIj ,..., b22),..., b&,bk . . . . bi,“‘of points of X such that Hd({b:},clx(Uj)) < e for each j = 1,2,. . . ,m and k=1,2 ,..., i(j). (K) * with respect to (A, a) if A continuum X is said to have property X satisfies the condition (A,a,c)* for each t > 0. Also X is said to have property (K)* if X has property (K)* with respect to each (A,a) (a E A E C(W). Property (K)* was introduced by Kato in [52]. This property has some similarities with property (K). Kato has shown that if a continuum X has property (K.)*, then X has property (6) ([52, Proposition 1.11). He also showed that if X is the union of a circle S with two spirals spiraling around S in opposite senses,then X is a continuum having property (K) and not having property (K)* ([52, Example 3.61). 50.6 Theorem [52, Theorem 2.61. The property of having property (K)* is a Whitney property. 50.7 Theorem [52, Corollary 2.91. The property of KeIley is a Whitney property for continua having property (K)*.
Exercises 50.8 Exercise. A continuum X with metric d has property (K) if and only if for every point p E X and for every E > 0, there exists 6 > 0 with the property that if A E C(X), p E A and q E B(6,p), then there exists B E C(X) such that q f B and Hd(A, B) < E.
51. ~CONNECTEDNESS
279
50.9 Exercise. A continuum X has property (K) if and only if for every point p E X, for every subcontinuum A of X and for every sequence {P~},“,~ such that p E A and pn + p, there exists a sequence {An}?=* with pn E A,, for each n, and A, -+ A. 50.10 Exercise. Prove Theorem 50.3. [Hint: Let A = p-‘(t) be a Whitney level for C(X). Let B E A, let B E C(d) and let {B,}$!?O=l be a sequence in A such that B, + B and B E f?. Let D = uB. Then t? = (plC(D))-l(t). Let {D,l}~zl be a sequence in C(X) such that D, + D and D, n B, # 0. Define Suppose that B, + C for some C E C(d). B, = (plC(D, U B,))-l(t). Then UC = D.]
51.
A-Connectedness
51.1 Definition. A continuum X is said to be A-connected ([58, p. 851) provided that for any two points a and b of X there exists an irreducible continuum of type A from a to b, i.e., an irreducible continuum from a to b whose indecomposable subcontinua have empty interiors. A continuum X is said to be b-connected ([32, p. 1171) if for each two points a, b E X there is an hereditarily decomposable continuum containing a and b. Originally, Hagopian called X-connected continua to 6 -connected continua. An explanation about the origin of this terminology can be found in [32]. Note that each continuum which is h-connected is X-connected. These two concepts coincide for plane continua ([33, Theorem 21). 51.2 Example [ll, 5.11. The property of being a d-connected continuum is not a Whitney property. The example 51.2 was constructed by W. J. Charatonik and answered a question by Krasinkiewicz and Nadler ([66, p. 1791). H is example is a continuum X which is a compactification of the half-ray S = [0, CQ), the remainder of X is a simple triod T. The continuum X has the additional property that every subcontinuum of X is the limit of subcontinua of S. Notice that X is hereditarily decomposable and irreducible. Then X is b-connected. However, the small Whitney levels for C(X) (t near to 0) are compactifications of S whose remainders contain 2-cells. Then they are irreducible between the end point of S and any point in the 2-cell. Since the square is not hereditarily decomposable, we conclude that the small Whitney levels for C(X) are not &connected. Charatonik also used this example to show that the covering property is not a Whitney property. He was not the first author to present such an example. The sameexample was previously constructed by Grispolakis and Tymchatyn ([29, Example 4.51) to show that being in Class(W) is not a Whitney property (see the preceding paragraph to Theorem 35.4).
280
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51.3 Question erty?
PROPERTIES
AND WHITNEY-REVERSIBLE..
.
[ll, Question 71. Is X-connectedness a Whitney prop-
51.4 Question [79, Question 14.571. Is X-connectedness (konnectedness) a Whitney-reversible property?
Exercises 51.5 Exercise. nected continua.
There are X-connected continua which are not &con-
51.6 Exercise. A metric compactification X of [0, 1) is &connected if and only if the remainder of X is hereditarily decomposable. 51.7 Exercise. The positive Whitney levels of the continuum represented in Figure 43 are S-connected continua.
A continuum with B-connected Whitney levels (51.7) Figure 43
52. LOCAL CONNECTEDNESS
52.
Local
281
Connectedness
52.1 Theorem [78, Theorem 31. The property of being a locally connected continuum is a Whitney property (Exercise 52.5). 52.2 Theorem [79, Theorem 14.471. The property of being a locally connected continuum is a sequential strong Whitney-reversible property (Exercise 52.7).
Exercises 52.3 Exercise. If X is a locally connected continuum, U is a connected open subset of X and A is a Whitney level for C(X), then A1 = {A E A : are open and arcwise connected AcU}anddZ={A~d:AnU#O} subsets of A. [Hint: Locally connected continua are locally arcwise connected ([81, Theorem 8.251).] 52.4 Exercise. Let X be a continuum. If U is an open subset of a Whitney level A, then UU is not necessarily an open subset of X. 52.5 Exercise.
Prove Theorem 52.1.
52.6 Exercise. A continuum X is connected im kleinen at each of its points if and only if for each E> 0, there exists a finite closed cover U of X such that for each A E U, A is connected and diameter (A) < E. 52.7 Exercise. Prove Theorem 52.2. [Hint: Use Exercise 52.6 and Exercise 11.5.1 52.8 Exercise. The property of being hereditarily locally connected continuum is not a Whitney property. 52.9 Exercise. Let X be a continuum. If there is a sequence {tn}rzl such that t, + 0 and p-‘(tn) is hereditarily locally connected for each n, then X is an arc or a circle. Then being hereditarily locally connected is a sequential strong Whitney-reversible property. [Hint: See Exercise 31.11.1
53.
n-Connectedness
53.1 Definition. -4 metric space Y is n-connected if, for every 0 5 i 5 n, each continuous function f : S” + Y is homotopic to a constant map, where Si denotes the i-sphere. A space k’ is m-connected if it is n-connected for every 71.
282
VIII.
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
A metric space is locally n-connected, if for each y E Y and each neighborhood U of y in Y, there exists a neighborhood V of x in U such that each continuous function f : Si + V is homotopic in U to a constant map, for every 0 < i 5 n. 53.2 Example. The property of being n-connected is not a Whitney property (Example 30.1). The following unexpected result was obtained in [40]. 53.3 Theorem [40]. Let X be a continuum. Then the following statements are equivalent: (a) X is a dendroid, (b) every positive Whitney level for C(X) is 2-connected, and (c) every positive Whitney level for C(X) is oo-connected. As a consequenceof Theorem 53.3, we have the following result. 53.4 Theorem. For each n 2 2, the property of being an n-connected continuum is a Whitney-reversible property. Proof. Let n 2 2 and let X be a continuum such that each positive Whitney level for C(X) is n-connected. Then every positive Whitney level for C(X) is 2-connected. By Theorem 53.3, X is a dendroid. In order to prove that X is n-connected, let 1 5 i 5 n and let f : Si -+ X be any map. Since Y = f(S) is a locally connected subset of X, then Y is a dendrite. Since dendrites are contractible (Exercises 25.34 and 25.35), it follows that f is homotopic to a constant map. Therefore, X is n-connected. n 53.5 Question. Let n >_ 2, is the property of being an n-connected continuum a strong Whitney-reversible property? Since 0-connectedness is the same as arcwise connectedness and this property is not a Whitney-reversible property (Example 33.5), Theorem 53.4 can not be extended to 0-connectedness. Since 1-connectednessis the same as having trivial fundamental group, the question whether Theorem 53.4 can be extended to n = 1 is included in Question 47.3. 53.6 Theorem [51, Corollary 2.21. The property of being a locally lconnected continuum is a Whitney property for the classof ANR continua contained in a 2-dimensional manifold.
283
EXERCISES 53.7 Example [51, Example 3.11. The property connected continuum is not a Whitney property.
of being a locally
l-
53.8 Example [25, Corollary 4.41. For each n 2 1, the property of being a locally n-connected continuum is not a strong Whitney-reversible property. Since the Hilbert cube is locally n -connected, by Theorem 30.3, it is enough to find a Peano continuum X such that its topological cone K is not locally n-connected. Consider the continuum X in Example 28.2. Then X is a Peano continuum. Let q = (p, 0) E K, that is, q is the point which corresponds to p in the base of the cone. Since K is not locally l-connected at q, we conclude that local 1-connectedness is not a strong Whitney-reversible property. 53.9 Question. ible property?
Let n 2 2, is local n-connectedness
a Whitney-revers-
Exercises 53.10
Exercise.
A contractible
space is oo-connected.
53.11 Exercise. Prove that the arc is the unique 2-connected continuum which is homeomorphic to each of its positive Whitney levels. [Hint: Use Exercise 80.21.1 53.12 Exercise. This exercise is a particular case of Proposition 2.6 of [49]. Let X = ~1 Ua2 ~a3 be a finite graph, where each CY~ is an arc joining the points pi and pf~ and cri n crj = {pi,ps} if i # j. If ~1is a Whitney map for C(X) and t is near to p(X), then A = p-‘(t) can be retracted to a 2-sphere and then it is not 2-connected. [Hint: A = A1 U A2 U A3 U 131U Us, where di = {A E A : oi C A} and f?j = clc(x)({A E A : pj 4 A}). Each di is homeomorphic to a tetrahedron and each L?, is homeomorphic to a 2-cell.]
54.
Planarity
54.1 Theorem [88, Corollay 4.101, [65, Corollary 3.3.(ii)]. For the class of circle-like continua the property of planarity is a Whitney property. 54.2 Theorem [88, Corollary 4.81, 165, Corollary 3.31. For the class of circle-like continua the property of non-planarity is a Whitney property. 54.3 Question.
Is planarity a Whitney-reversible property?
284
VIII.
WHITNEY
PROPERTIES AND WHITNEY-REVERSIBLE..
.
Exercises 54.4 Exercise. ties.
Planarity and non-planarity are not Whitney proper-
54.5 Exercise. Characterize the finite graphs such that their Whitney levels are planable. 54.6 Exercise. Characterize the finite graphs such that their small Whitney levels (t near to 0) are planable. [Hint: See Example 65.4.1 54.7 Exercise. Find an infinite family of continua F such that for every X E F’, every Whitney level for C(X) is planable.
55.
P-Likeness
In the following definition, we extend the definition of P-like given in section 22 to a family of continua.
55.1 Definition. (a) Let Y be a continuum and let P be a given collection of continua. Then Y is said to be P-like provided that for each c > 0, there is an e-map fe from Y onto a member 2, of P. (b) Let P be a given family of continua. A continuum Y is said to be weak P-like provided that for each E > 0 there is a (not necessarily surjective) e-map fe from Y into a member 2, of P. (c) If P consists only of the unit interval [0, l] clearly, weak P-like and P-like continua are the same and they are called arc-like (Exercise 55.4). (d) In the case that P consists only of the unit circle in R2, then P-like continua are called circle-like. (e) If P is the collection of trees (finite acyclic connected graphs), then weak P-like and P-like continua are the same and they are called tree-like. 55.2 Theorem [47, Corollary 3.21. Let P be a family of ANR continua. Then the property of being a weak P-like continuum is a sequential strong Whitney-reversible property. Generalizing previous results ([80, Theorem 5.31 and [60, Theorem 2]), Kato proved the following theorem.
EXERCISES
285
55.3 Theorem [53, Theorem 2.11. Let P be a family of compact connected ANRs. Then the property of being a P-like continuum a is sequential strong Whitney-reversible property.
Exercises 55.4 Exercise. If P consists only of the unit interval [0, 11,then weak P-like and P-like continua are the same. 55.5 Exercise. Theorem 55.3 is not true if the elements in P are not assumedto be AN%. [Hint: Consider the continuum X in Figure 44 (a). Small Whitney levels (t near to 0) for C(X) are homeomorphic to the continuum Y represented in Figure 44 (b). Then there are no maps from X onto Y. Therefore, X is not {Y}-like.]
X
(b)
(4
P-likeness is not a sequential strong Whitney-reversible property (55.5) Figure 44
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WHITNEY
PROPERTIES AND WHITNEY-REVERSIBLE...
56.
Pseudo-Arc
By [3, p. 441, we may say that the pseudo-arc is the only hereditarily indecomposable chainable continuum. To know more about this interesting continuum we refer the reader to the paper [73]. Its construction can be found in [81, Exercise 1.231. 56.1 Theorem Whitney property.
[18, p. 10321. The property
of being a pseudo-arc
56.2 Theorem [79, Theorem 14.54 (2)]. The property pseudo-arc is a sequential strong Whitney-reversible property.
is a
of being a
Exercise 56.3 Exercise.
57.
Prove Theorems
Pseudo-Solenoids
56.1 and 56.2.
and the Pseudo-Circle
57.1 Definition. A pseudo-solenoid is any hereditarily indecomposable proper circle-like continuum. A pseudo-circle is a planar pseudosolenoid. More information about pseudo-solenoids can be found in [2, p. 7911 and [88, p. 5801. It is known that there is only one pseudo-circle ([21]). 57.2 Theorem (see [79, Remark 14.241). The property particular pseudo-solenoid is a Whitney property.
of being any
57.3 Theorem [79, Theorem 14.54 (3)]. The property being any particular pseudo-solenoid is a sequential strong Whitney-reversible property. 57.4 Theorem (see [79, Remark pseudo-circle is a Whitney property.
Definition
14.241). The property
58.
R3-Continua
of R3-continuum
is in Definition
of being the
24.12.
58.1 Example [41, Example 2.11. The property continuum is not a Whitney property.
of containing
no R3-
58.2 Question. Is the property of containing Whitney property for the class of dendroids?
no R3-continuum
a
287
EXERCISE 58.3 Question. property?
Is the property
of containing
58.4 Question. Whitney-reversible
Is the property property?
no R3-set
of containing
a Whitney
an R3-continuum
a
Exercise 58.5 Exercise. Whitney property.
The property
59.
of having
Rational
an R3-continuum
is not a
Continua
59.1 Definition. A continuum X is said to be ratzonal provided that each point p of X has a local base L$, such that the boundary of each member of & is at most countable. 59.2 Question. Is the property ney-reversible property?
of being a rational
continuum
a Whit-
continuum
is not a
Exercises 59.3 Exercise. Whitney property.
The property
59.4 Exercise. The property a Whitney property.
60.
Shape
of being a rational
of not being a rational
continuum
is not
of Continua
For notions related to this section we refer the reader to the books by Borsuk [7] and by Mardesic and Segal [77]. If F= {G,, f;+‘} is an inverse sequence, we will denote by lip{ G,, fz+‘} to its inverse limit. 60.1 Theorem [65, Corollary 3.3(i)]. If X is a circle-like continuum, then the shape of X is equal to the shape of p-l(t) whenever 0 I t < p(X). Theorem 60.1 has been extended by Kato to strongly winding curves. A curve X (i.e. a l-dimensional continuum) is said to be a strongly winding curwe if there is an inverse sequence F = {Gn, f,“+‘} of finite graphs such that X = l?{G,,, f, n+1} and 3 satisfies the following condition: if S is a simple closed curve in Gn+i,
then f,“+‘(S)
= G,.
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VIII.
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PROPERTIES
AND WHITNEY-REVERSIBLE..
.
Clearly, every tree-like continuum and every circle-like continuum are strongly winding curves. 60.2 Theorem [49, Theorem 3.51. If X is a strongly winding curve, then the shape of X is equal to the shape of p-‘(t) whenever 0 < t < p(X). Besides Theorem 60.1, Kato also obtain the following theorem as a corollary of Theorem 60.2. 60.3 Theorem [49, Corollary 3.71. If X is a tree-like continuum, then Sh@‘(t)) is trivial for every 0 5 t 5 p(X). Theorem 60.3 can also be obtained as a consequenceof the following theorem by Lynch. 60.4 Theorem [75, Theorem 1.1 (iii)]. For I-dimensioual continua the property of having trivial shape is a Whitney property. 60.5 Example [86, Example 21. The property of having trivial shape is not a Whitney property (Example 30.1). 60.6 Definition. A compact subspace Y of the compact metric space Z is said to be an approximate strong deformation retract of Z provided that for any ANR M containing 2, there exist mappings T : 2 x [0, ok) -+ M and D : Z x [0, co) x [0, l] -+ M satisfying the conditions: (a) ~(9, t) = y for each y E Y and each t E [O,oo), (b) for any neighborhood U of Y in M there is a real to 2 0 such that T(Z x [to, oo)) c U, and (c) D(y, t, s) = y for each y E Y, each t E [O,oo) and each s E [0, l], D(z, t,O) = z, D(z, t, 1) = T(Z, t) for each z E Z and each t E [O,oo), (d) for any neighborhood V of 2 in M there is a real tl > 0 such that D(Z x [t1,00) x [O,11) c v. When Y is an approximate strong deformation retract of Z, the inclusion from Y to 2 is a (strong) shape equivalence (e.g., see[7] or [77]). Hence Sh(Y) = Sh(Z). A compact metric space Y lying in the Hilbert cube Q is said to be movable provided that for every neighborhood V of Y in Q there is a neighborhood U of Y in Q such that for any neighborhood W of Y in Q there is a homotopy cpw : U x [0, l] -+ V satisfiying the following conditions: ‘pw(y, 0) = y and cpw(y, 1) E W for every point y E U. A compact metric space Y lying in the Hilbert cube Q is said to be n-movable provided that for every neighborhood V of Y in Q there is a
60. SHAPE OF CONTINUA
289
neighborhood U of Y in Q such that for any neighborhood W of Y in Q, any compact metric space A with dim[A] 5 n and any map f : A -+ U, there is a homotopy cpw : A x [0, l] + V satisfiying the following condition:
c~w(a,O)= f( a1 and cpw(u, 1) E W
for every point a E A.
Similarly, “pointed movable” and “pointed n-movable” are defined (see [7] or [77]). It is well known that those properties are topological properties. Clearly, “(pointed) movable” implies “(pointed) n-movable” for each n > 1. 60.7 Theorem [47, Theorem 1.31. Let X be a continuum. Let 3c = C(X) or 2”. If p is any Whitney map for 3t, then p-‘(t) is an approximate strong deformation retract of p-’ ([s, t]) for 0 5 s 5 t 5 p(X). In particular, Sh(b-‘(t)) = Sh (/~-l([s, t])). 60.8 Theorem [47, Corollary 1.51. The property of being l-movable is a Whitney property. 60.9 Theorem [54, Theorem 2.11. The property of being pointed lmovable is a sequential strong Whitney-reversible property. 60.10 Theorem [54, Corollary 2.51. The property of being pointed movable is a sequential strong Whitney-reversible property for curves (= ldimensional continua). 60.11 Theorem [54, Theorem 3.11. The property of being nearly lmovable is a Whitney property. 60.12 Theorem [54, Theorem 3.21. The property of being nearly lmovable is a sequential strong Whitney-reversible property. 60.13 Question [54, Problem 3.51 ([60, p. 3161). Is the property of being pointed movable a (sequential) strong Whitney-reversible property? Let X be Borsuk’s non-movable Peano continuum [6]. Is there a Whitney map p for C(X) such that p-‘(t) is movable (or 2-movable) for each t > O? 60.14 Theorem [48, Theorem 1.11. The property of being movable is not a Whitney property. More precisely, there exists a movable curve X and a Whitney map w for C(X) such that for some 0 < t < w(X), w-‘(t) is not 2-movable. Hence, the property of being (pointed) 2-movable is not a Whitney property. 60.15 Theorem [47, Corollary 1.10 (3)]. The property of being an approximately m-connected continuum (A(F) is a sequential strong Whitneyreversible property.
290
VIII.
WHITNEY
GO.16 Theorem ial groups pro-H,(.) properties.
PROPERTIES
AND WHITNEY-REVERSIBLE..
.
[47, Corollary 1.10 (4)]. The properties of having trivand fin(.) are sequential strong Whit,ney-reversible
60.17 Theorem [48, Corollary 2.21. The property of being l-shape connected is a Whitney property. 60.18 Definition (see [77]). The fundamental dimension, Fd(X)? of a continuum X is defined as Fd(X)
= min {dim[Z] : 2 is a continuum such that Z has the same shape as X)
60.19 Theorem [47, Corollary 1.10 (S)]. The property of having fundamental dimension 5 n is a sequential strong Whitney-reversible property. 60.20 Example [86, Example 21 (Example 30.1). The property of having fundamental dimension 5 n is not a Whitney property. For definitions of FAR and FANR, we refer the reader to [77]. 60.21 Example [86, Example 21. The property of being an FAR is not a Whitney property. 60.22 Theorem [47, Theorem 4.11. The property of being an FAR is a Whitney property for l-dimensional continua. 60.23 Theorem [60, Theorem 11, [80, Corollary 3.21 (Theorem 41.6). The property of being an FAR is a sequential strong Whitney-reversible property. 60.24 Question [47, (5.4)]. Is the property of being an FANR a Whitney property for l-dimensional continua? 60.25 Question [60, p. 3161. Is the property of being an FANR a strong Whitney-reversible property?
61.
Solenoids
61.1 Definition. For each p = 2,3,. . ., let fP : S’ + S’ (S’ is the unit circle in R2) be given by f”(z) = 9’ for each t E S1, where zp is the pth power of z using complex multiplication.
62.
291
SPAN
For a given p, let fi>E”=,, where each X, = S’ and each fi = fP.
cp = yxz,
The continuum C, is called the p-a&c solenoid. A solenoid is any continuum of the form l@{Xi, fP(i)}&, {1,2,3 ,... }foreachi=1,2 For some properties
where p(i) E
,.... of solenoids see [81, 2.81.
61.2 Theorem [66, 4.51. The property is a Whitney property.
of being any particular
solenoid
61.3 Question [79, Question 14.571. Is the property of being a particular solenoid a Whitney-reversible (strong Whitney-reversible, sequential strong Whitney-reversible) property?
Span
62.
62.1 Definition. Let X be a continuum and let d denote a metric for projection for i = 1,2. X. Let pi : X x X -+ X be the it”-coordinate The snrjective span g*(X) (respectively, surjective semi-span o;(X)) of X is the least upper bound of the set of all real numbers Q which satisfy the following condition: there exists a continuum (respectively, pi(C=) The span a(X)
C, c X x X such that pr(C,) = pz(C,) = X = X) and d(z, y) 2 cy for each (z:, y) E C,.
and the semi-span oc(X)
of X are defined by
g(X)
=
sup{a*(A)
: A E C(X)},
go(X)
=
sup{a;(A)
: A E C(X)}.
and
Koyama in [61] proved that zero and zero semi-span are sequential strong Whitney-reversible properties. Kato extended Koyama’s results with the following theorem. 62.2 Theorem [53, Theorem 3.11. Let X be a continuum and let d denote a metric for X. Let r E {u* , ai, o, (TO}. If there exists a decreasing sequence {tn}rzl in (0, h(X)] such that lim t, = t and Iim r(fi-l
(tn)) = 0,
then ‘(F1
(Q) = 0,
292
VIII.
WHITNEY
PROPERTIES
AND WHITNEY-REVERSIBLE..
where (Y is a real number with o > 0 and r@‘(s)) Hausdorff metric Hd.
.
is defined using the
The concept of span was introduced by Lelek in [71]. It is known that if X is a chainable continuum, then c(X) = 0 ([71, p. 2101). The most important problem about span is if the converse of this implication is true
([721). 63.
Tree-Likeness
The definition of tree-like continua is in Definition 55.1 (e). 63.1 Theorem [63, 4.21, [79, Theorem 14.151. The property of being an hereditarily indecomposable tree-like continuum is a Whitney property. 63.2 Theorem [47, Corollary 3.31. The property of being a tree-like continuum is a sequential strong Whitney-reversible property.
64.
Unicoherence
Definitions of unicoherence and hereditary unicoherence precede Lemma 19.7 and follow Example 22.11, respectively. More results related to unicoherence of Whitney levels are in section 68. 64.1 Example [66, Example 5.41, [88, Example 5.61, [79, Example 14.121. The property of being unicoherent is not a Whitney property, even for circle-like continua (the example in Exercise 44.14 is a unicoherent continuum with some non-unicoherent Whitney levels). 64.2 Example [66, Example 5.51. Non-unicoherence is not a Whitney property (Exercise 64.11). 64.3 Theorem [79, Theorem 14.46 (2)]. Unicoherence is a sequential strong Whitney-reversible property (Exercise 64.9). 64.4 Definition. If 2 is a topological space, let be(Z) denote the number of components of 2 minus one, if this number is finite, and bc(2) = 00, otherwise. The multicoherence degree, r(Y), of a connected space Y is defined by r(Y) = sup {be(.4 fl B) : A, B are closed connected subsets of Y and Y=A
U B}.
Notice that Y is unicoherent if and only if r(Y) = 0.
EXERCISES
293
64.5 Theorem [42, Theorem A]. The property of having multicoherence degree 5 n is a Whitney property for the class of Peano continua. 64.6 Corollary. Peano continua.
Unicoherence
is a Whitney
property
for the class of
Exercises 64.7 Exercise.
Verify Example 64.1.
64.8 Exercise. The continuum X in Example 35.5 is a unicoherent continuum such that C(X) has non-unicoherent positive Whitney levels. 64.9 Exercise.
Prove Theorem 64.3.
64.10 Exercise. There are unicoherent A for C(X) such that r(d) = CO.
continua ,Y and Whitney
levels
64.11 Exercise. The noose presented in Example 5.3 is a non-unicoherent continuum with some unicoherent Whitney levels. The following
table summarize
the results of this chapter.
294
VIII.
WHITNEY
Table
Topological
Properties
PROPERTIES AND WHITNEY-REVERSIBLE...
Summarizing
Chapter
VIII
TABLE
SUMMARIZING
Topological Properties
Whitney Property yes
no
CHAPTER
295
Whitney Strong Sequential Reversible Whitney- Strong W. Reversible Reversible yes
Contractible, Dendroids Contractible for Peano Contractible Hyperspace 1
VIII
(41.12 1
no
yes
no
yes
no
296
VIII.
WHITNEY
Topological Properties
PROPERTIES
Property
AND WHITNEY-REVERSIBLE..
.
Reversible
1
TABLE SUMMARIZING
Topological
Properties
Property
CHAPTER VIII
Whitney Reversible
Strong Whitney-
297
Strong W.-
298
VIII.
WHITNEY
Topological Properties
PROPERTIES
Pro
Un. Path. Con. Prop. (r;)133.121
AND WHITNEY-REVERSIBLE..
REFERENCES
299
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E. Abo-Zeid, Some properties (1978) 301-312.
2.
B. J. Ball and R. B. Sher, ~rn~e~~~ng circle-l&e J. Math., 25 (19731, 791-805.
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R. H. Bing, Concerning hereditarily J. Math., 1 (1951), 43-51.
4.
R. H. Bing, Embedding circle-like Math., 14 (1962), 113-128.
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K. Borsuk, On a locally connected non-movable continuum, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 17 (1969), 425-430.
7.
K. Borsuk, 1975.
8.
C. E. Burgess, Chainable Math., 9 (1959), 653-659.
9.
J. H. Case and R. E. Chamberlin, Characterizations tinua, Pacific J. Math., 10 (1960), 73-84.
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Proc., 3
continua,
Pacific
in the plane, Canad. Mat., 44, PWN
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Warszawa,
Mat., 59, PWN
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of tree-like
J. con-
10. J. J. Charatonik, Problems and remarks on contractibility of curves, General Topology and its Relations to Modern Analysis and Algebra IV, Proc. Fourth Prague Topological Symposium, 1976; Part B Contributed Papers, Society of Czechoslovak Mathematicians and Physicists, Prague 1977 (1977), 72-76. 11. W. J. Charatonik, Some counterexamples concerning Bull. Polish Acad. Sci. Math., 31 (1983), 385-391.
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12. W. J. Charatonik, Homogeneity is not a Whitney property, Math. Sot., 92 (1984), 311-312. 13. W. J. Charatonik, On the property Notes in Math., Springer, Berlin-New 14. W. J. Charatonik, (1986)) 207-216.
Ri-continua
of
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Kelley in hyperspaces, Lecture York, 1060 (19841, 7-10.
and hyperspaces,
Topology
Appl.,
23
15. A. M. Dilks, Structure of hyperspaces, Dissertation, Tulane University, New Orleans, La., (1980); J. T. Rogers, Jr., Director of Dissertation. 16. A.M. Dilks and J.T. Rogers, Whitney stability and contractible spaces, Proc. Amer. Math. Sot., 83 (1981), 633-640.
hyper-
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AND WHITNEY-REVERSIBLE..
.
17. E. IS. van Douwen and J. T. Goodykoontz, Jr., Aposyndesis in hyPerwces and 6’ech-Stone remainders, General Topology and Modern Analysis, L. F. McAuley and M. M. Rao, Editors, Academic Press, Inc., New York, 1981, 43-52. 18. C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces,Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 19 (1971), 1027-1034. 19. C. Eberhart and S. B. Nadler, Jr., Ir+rcducible Whitney ,!cvels,Houston J. Math., 6 (1980), 355-363. 20. L. Fearnley, Characterizations of the continuous images of the pseudoarc, Trans. Amer. Math. Sot., 111 (1964), 380-399. 21. L. Fearnley, The pseudo-circle is unique, Trans. Amer. Math. 149 (1972), 45-64.
Sot.,
22. J. T. Goodykoontz, Jr., Aposyndesis and hyperspaces, Collection: Topology Conference, 1979 (Greensboro, N.C., 1979), Guilford College, Greensboro, N.C., 1980, 129-135. 23. J. T. Goodykoontz, Jr., Arc-smoothness in hyperspaces, Topology Appl., 15 (1983), 131-150. 24. J. T. Goodykoontz, Jr., Geometric models of Whitney levels, Houston J. Math., 11 (1985), 75589. 25. J. T. Goodykoontz and S. B. Nadler, Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Sot., 274 (1982), 671-694. 26. J. T. Goodykoontz and S.B. Nadler, Jr., Fixed point sets of continuum valued mappings, Fund. Math., 122 (1984), 85-103. 27. J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only, (I), Houston J. Math., 5 (1979), 483502. 28. J. Grispolakis and E. D. Tymchatyn, Weakly confluent mappings and the covering property of hyperspaces, Proc. Amer. Math. SOC., 74 (1979), 177-182. 29. J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only, (II), Houston J. Math., 6 (1980), 375387. 30. J. Grispolakis and E. D. Tymchatyn, Spaceswhich accept only weakly confluent mappings, Proceedings of the International Conference of Geometric Topology, PWN-Polish Scientific Publishers Warzawa 1980, 175-176.
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31. J. Grispolakis and E. D. Tymchatyn, On a characterization of W-sets and the dimension of hyperspaces, Proc. Amer. Math. Sot., 100 (1987), 557-563. 32. C. L. Hagopian, Mappings theorems for- plane continua, Topology Proc., 3 (1978), 117-122. 33. C. L. Hagopian, Mapping theorems for plane continua, Topology Proc., 3 (1978), 117-122. 34. C. L. Hagopian and L. G. Oversteegen, Continuum chainability without arcs, Houston J. Math., 21 (1995), 407-411. 35. C. B. Hughes, Some properties of Whitney continua in the hyperspace C(X), Topology Proc., 1 (1976), 209-219. 36. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Ninth printing, 1974. 37. D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math., 64 (1969), 91-97. 38. A. Illanes, Irreducible Whitney levels with respect to finite and countable subsets, An. Inst. Mat. Univ. Nat. Autonoma Mexico, 26 (1986), 59-64. 39. A. Illanes, Arc-smoothness is not a Whitney-reversible property, Aportaciones Mat. Comun., 8 (1990), 65580. 40. A. Illanes, A characterization of dendroids by the n-connectedness of the Whitney levels, Fund. Math., 140 (1992), 157-174. 41. A. Illanes, R3-continua in hyperspaces, Houston J. Math., 20 (1994), 529-538. 42. A. Illanes, M&coherence of Whitney levels, Topology Appl., 68 (1996), 251-265. 43. A. Illanes, The fundamental group of Whitney blocks, Rocky Mountain J. Math., 26 (1996), 1425-1441. 44. A. Illanes, Countable closed set aposyndesis and hyperspaces, Houston J. Math., 23 (1997), 57-64. 45. A. Illanes, Cone = hyperspace property, a characterization, preprint. 46. F. B. Jones, Aposyndetic continua .and certain boundary problems, Amer. J. Math., 63 (1941), 545-553. 47. H. Kato, Shape properties of Whitney maps for hyperspaces, Trans. Amer. Math. Sot., 297 (1986), 529-546. 48. H. Kato, Movability and homotopy, homology pro-groups continua, J. Math. Sot. Japan, 39 (1987), 435-446. 49. H. Kato, Whitney (1987), 367-381.
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50. H. Kato, Shape equivalences of Whitney continua of curves, Canad. J. Math., 40 (1988), 217-227. 51. H. Kate, On local 1-connectednessof Whitney continua, Fund. Math., 131 (1988)) 245-253. 52.
53. 54. 55. 56. 57. 58. 59.
Kate, On the property of Kelley in the hyperspace and Whitney continua, Topology Appl., 30 (1988), 165-174. H. Kate, On some sequential strong Whitney-reversible properties, Bull. Polish Acad. Sci. Math., 37 (1989), 517-524. H. Kato, Movability and strong Whitney-reversible properties, Topology Appl., 31 (1989), 125-132. H. Kato, A note on continuous mappings and the property of J. L. Kelley, Proc. Amer. Math. Sot., 112 (1991), 1143-1148. 0. H. Keller, Die homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum, Math. Ann., 105 (1931), 748-758. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot., 52 (1942), 22-36. B. Knaster and S. Mazurkiewicz, Sur un prob&me concernant les transformations continues, Fund. Math., 21 (1933), 85-90. R. J. Knill, Cones, products and fixed points, Fund. Math., 60 (1967), H.
35-46. 60. A. Koyama, A note on some strong Whitney-reversible properties, Tsukuba J. Math., 4 (1980), 313-316. 61. A. Koyama, Zero span is a sequential strong Whitney-reversible property, Proc. Amer. Math. Sot., 101 (1987), 716-720. 62. J. Krasinkiewicz, On the hyperspace of snake-like and circle like continua, Fund. Math., 83 (1974), 155-164. 63. J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math., 84 (1974), 175-186. 64. J. Krasinkiewicz, Curves which are continuous images of tree-like continua are movable, Fund. Math., 89 (1975), 233-260. 65. J. Krasinkiewicz, Shape properties of hyperspaces, Fund. Math., 101 (1978), 79-91. 66. J. Krasinkiewicz and S. B. Nadler, Jr., Whitney properties, Fund. Math., 98 (1978), 165-180. 67, W. Kuperberg, Uniformly pathwise connected continua, Studies in Topology, N. Stavrakas and K. Allen, Editors, Academic Press, Inc., New York, 1975,315-324. 68. K. Kuratowski, Topology, Vol 11,Polish Scientific Publishers and Academic Press, 1968.
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69. A. Y. W. Lau, Whitney continuum in hyperspace, Topology Proc., 1 (1976), 187-189. 70. A. Lelek, On weakly chainable continua, Fund. Math., 51 (1962), 271282. 71. A. Lelek, Disjoint mappings and the span of spaces, Fund. Math., 55 (1964), 199214. 72. A. Lelek, Some problems concerning curves, Colloq. Math., 23 (1971), 93-98. 73. W. Lewis, The pseudo-arc, Contemp. Math., 117 (1991), 103-123. 74. M. Lynch, Whitney levels in C,(X) are ARs, Proc. Amer. Math. SOC., 97 (1986), 748-750. 75. M. Lynch, Whitney properties for l-dimensional continua, Bull. Polish Acad. Sci. Math., 35 (1987), 473-478. 76. M. Lynch, Whitney levels and certain order arc spaces, Topology Appl., 38 (19$-H), 189-200. 77. S. MardesiC and J. Segal, Shape Theory, North-Holland Mathematical Library, North-Holland, Amsterdam, 1982. 78. S. B. Nadler, Jr., Some basic connectivity properties of Whitney map inverses in C(X), Studies in Topology, N. Stavrakas and K. Allen, Editors, Academic Press, New York, 1975, 393-410. 79. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 80. S. B. Nadler, Jr., Whitney-reversible properties, Fund. Math., 109 (1980) 235-248. 81. S. B. Nadler, Jr., Continuum Theory, An introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 82. S. B. Nadler, Jr. and T. West, Size levels for arcs, Fund. Math., 141 (1992)) 243-255. 83. L.G. Oversteegen and E.D. Tymchatyn, On atriodic tree-like continua, Proc. Amer. Math. Sot., 83 (1981), 201-204. 84. A. Petrus, Whitney maps and Whitney properties of C(X), Dissertation, Tulane University, New Orleans, La., (1976); J. T. Rogers, Jr., Director of Dissertation. of C(X), Topology 85. A. Petrus, Whitney maps and Whitney properties Proc., 1 (1976), 147-172. 86. A. Petrus, Contractibility of Whitney continua in C(X), General Topology Appl., 9 (1978), 275-288.
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.
87. C. W. Proctor, A characterization of absolutely C*-smooth continua, Proc. Amer. Math. Sot., 92 (1984), 293-296. 88. J. T. Rogers, Jr., Whitney continua in the hyperspace G(X), Pacific J. Math., 58 (1975), 569-584. 89. J. T. Rogers, Jr., Applications of a Vietoris-Begle theorem for multivalued maps to the cohomology of hyperspaces,Michigan J. Math., 22 (1976), 315-319. 90. J.T. Rogers, Jr., Solenoids of pseudo-arcs, Houston J. Math., 3 (1977), 531-537. 91. L. E. Ward, Jr., Fixed point sets, Pacific J. Math., 47 (1973), 553-565. 92. R. W. Wardle, On a property of J. L. Kelley, Houston J. Math., 3 (1977), 291-299. 93. General Topology and Modern Analysis, L.F. McAuley and M.M. Rao, Editors, Academic Press, Inc., New York, 1981.
IX.
Whitney 65.
Finite
Levels Graphs
In section 5 we constructed models for the hyperspace C(X) of some finite graphs X. In this section we will see tools that can be used to construct models for Whitney levels of finite graphs. We need some conventions. We defined the notion of a finite graph in section 5; it is convenient here to reformulate the notion as follows: 65.1 Conventions. A finite graph G is a continuum that can be written as the union of a finite family 3, the elements of 3 are arcs or circles, each circle C in 3 has a choosen point which is called the end point of C. Any two elements of 3 can intersect only at their end points. The elements of 3 are the segments of G. The end points of the segmentsare the vertices of G. The order, o(v), of a vertex v is defined by:
o(v)
=
(the number of segmentsof the graph which are arcs and contain v)
+
2 (the number of segmentsof the graph which arc circles and contain v).
An end point of G is a vertex of order 1 and a ramification point of G is a vertex of order > 3. For finite graphs different from the circle, we ask that all the vertices of the graph are of order different from 2. This condition eliminates unnecessary segmentsand vertices. In this section the letter G denotes a finite graph different, from the circle and p denotes a Whitney map for C(G). We assumethat the metric d for G is the metric of arc length and each segment of G has length equal to one. For each segment J in G we identify J with a closed interval [(O)J, (l)~]. 305
306
IX. WHITNEY
LEVELS
Notice that it is possible that (0)~ = (1)~. We write 0 (respectively, 1) instead of (0)~ (respectively, (1)~) if it causesno confussion. Let G be a finite graph. A subgraphof G is a subcontinuum H of G such that H is the union of some segmentsof G or H is of the form H = {v}, where u is a vertex of G. Then, by definition, the subgraphs of G are nonempty and connected. A small Whitney level for C(G) is a set of the form IL-’ (t), where 0 < t < min{p(H) : H is a segment of G}, and a large Whitney level for C(G) is a set of the form p-‘(t), where max{p(H) : H is a proper subgraph of G} < t < p(G). If A E C(G) and E > 0, define Q(c,A) = (2 E G : there exists a E A such that d(s,a) 5 6). For a proper subgraph S of G, define !7Jk = (B E C(G) : S c B c Q(l, S) and B
n Iv(1, S) is connected)
if p(S) < t < p(Q(l,S)). Notice that Q(l,S) and m,(t) = ms np-l(t), is t,he union of S and all the segmentsin G intersecting S. Let Ii,. . . ,I,. be the segments in G such that, for each i, I, is an arc and Ii intersects S in exactly one of its end points (we may assumethat Ii n S = { (0)li }) and let J1, . . , J, be the segmentsin G such that, for each j, SnJj = { (0)~~, (l).r, } (here, it is possible that (0)~~ = (1)~) ). Notice that (Exercise 65.30) each element A E 9&(t) can be written in . the form (*) 24= SU (U{[O,Cti] 1 15 i 5 T}) U (U{[O,Cj] U [dj, l] 1 1 5 j 5 S}), where 0 5 a, 5 1 for each i and 0 5 cj 5 dj 5 1 for each j. Let .4 ED?s(t). We say that A is in the relative interior (A ~RI(ms(t))) of c331s (t) if 0 < a, < 1 for each i, and 0 < cj < dj < 1 for each j. If A does not belong to the relative interior, we say that A is in the relative boundary RB(Ms(t)) of mm,(t). Then RB(!.%‘s(t)) = !JJ?s(t)- RI(!Yk(t)). Sets of the form 332swere introduced by Duda in [8]. The following theorem is useful for studying Whitney levels of finite graphs. 65.2 Theorem [26, Proposition 2.41 and [24, Theorem 1.11. Let n = is homeomorphic to [0, lln-’ and homeomorphisms 2s + r. Then (m,(t) onto [0, l]+l send RB(337s(t)) to the boundary of [O,lln-‘. from 9&(t)
It is possible to extend the definition of 9&(t) for S = 8. Let J be a segment of G. Define MeJ - { A E C(G) : A c J and A - {(O)J,(~)J} is connected} and Q(l,S,J) = J. If 0 < t < p(J), define %$(t) = 9J$ n p”-’ (t). We write 3;ne (respectively, 9&(t) and Q( 1,0)) instead of ‘?J$ (respectively, mm,J(t)and Q(l, 0, J)) if it causesno confusion. Notice that (Exercise 65.31) !lJlo(t) is an arc.
65. FINITE
GRAPHS
307
If S is the empty set or S is an acyclic graph without end points (of G) then S is called a fine subgraph of G. A maximal fine subgraph is a fine subgraph of G which is not contained propertly in another fine subgraph of G. Thus (Exercise 65.32) S is a maximal fine subgraph if and only if S is a fine subgraph of G which contains all the ramification points of G. The following theorem is easy to prove (Exercise 65.33). 65.3 Theorem. Let G be a finite graph. If A = p-‘(t) is a positive Whitney level for C(G) and 3 = (S : S is a fine subgraph of G for which : S E 3}. 9X,(t) is defined}, then A = U{m.s(t) 65.4 Example. In this example we describe the small Whitney levels of a finite graph G. Let A = p-‘(t) be a small Whitney level for C(G) and let 3 be as : S E 3’). By the in Theorem 65.3. By Theorem 65.3, A = U{ms(t) definition of !.DIs(t), for each S E 3, either S = 0 or p(S) < t. Since A is a small Whitney level, then either S = 0 or S is a one-point set which contains a ramification vertex of G. For each segment J of G, 0 < t < p(J). Then m:(t) is an arc contained in A (Exercise 65.31). Notice that, for two different segments J and L of G, !&(t)n m;(t) = 0. If v is a ramification point of G. Then ~({v}) = 0 < t < p(Q(1, {v})). Then !lX{,}(t) is a subset of A. If A E C(G) and A contains two different vertices, then A contains a segment of G. Thus A $! A. This proves that if v and w are two different vertices of G, then !JX{,,l(t)n %2{,)(t) = 8. Let u be a ramification point of G. Let 11,. . . , I, be the segmentsof G which are arcs and contain v. Let J, , . . , J,. be the segmentsof G which are circles and contain ‘u. Then n = s + 2r is the order of w and, by Theorem 65.2, M{,)(t) is homeomorphic to the cube [0, l]“-‘. Let J be a segment of G and let 21be a ramification point of G. If u E J, since P(V) < t < p(J), taking an order arc from {v} to J, it can be shown that there exists A E !DIt,l(t) n 9X;(t). On the other hand if !TLT{,,(t) n !%$(t) # 0, then there exists a subcontinuum A of G such that v E A c J. Thus v E J. In the case that J is an arc, there is a unique element A E C(G) such that v E A c J. In this case A is an end point of the segment ?%$(t) and A E RB(!XXf,l(t)) (the rest of the “coordinates” of A in %l{,l(t) are equal to zero). In the case that J is a circle, there are exactly two elements pi, A2 E C(G) such that v E A, c J and Ai - (v) is connected. In this caseAl and AZ are the end points of !JXi (t) and Al, AZ E
RBW{tg
(t)).
Therefore, !Bl~,l(t) J is an arc, then !JJ$,)(t)
n DZi(t) # 8 if and only if r~ E J. Furthermore, if n 9X;(t) contains exactly one element which is
308
IX.
WHITNEY
LEVELS
an end point of the segment m;(t) and a point in RB(mf,l(t)). If J is a circle, then Im{,,l(t) n !JYli(t) contains exactly two elements which are the end points of the segment !JJ$(t) and they are points in RB(lmf,l(t)). Therefore, we may say that for obtaining a model for the small Whitney levels of a finite graph, we need to change each ramification point ZI for a cube of dimension o(w) - 1. All these cubes must be joined by arcs in the sa.me way that vertices are joined in G. In Figure 45 we illustrate a small Whitney level for a particular graph G. From Example 65.4, the small Whitney levels for a finite graph G are all homeomorphic. Moreover, small Whitney levels are homotopically equivalent to G. This fact was generalized by Kato in the following theorem. 65.5 Theorem [26, Proposition 2.31. Let G be a finite graph. Suppose that G contains circles. Let p be a Whitney map for C(G) and let
G
Small Whitney level
4-dimensional cube
Small Whitney levels for finite graphs (65.4) Figure 45
65. FINITE
GRAPHS
309
to = min(p(S) : S is a circle in G). Then for each t E [O,to), /.~-i (t) is homotopically equivalent to G. Example 65.4 shows that models for small Whitney levels for C(G) are easy to construct. This is not true for other Whitney levels for C(G). For some particular graphs, it is possible to obtain information from the large Whitney levels for C(G). The most general result is the following easy to prove theorem (Exercise 65.34). 65.6 Theorem. Let G be a finite graph. If G has a cut point, then large Whitney levels for C(G) are contractible.
It is not known if the converse of Theorem 65.6 is true. 65.7 Question [24, Question 3.51. Let G be a finite graph. Is it true that if large Whitney levels for C(G) are contractible, then G has cut points?
In [26] are analyzed the large Whitney levels of the family of graphs described in the following theorem. 65.8 Theorem [26, Proposition 2.61. Let G = G(m) = U{.& : 1 5 i 5 m} (2 5 m), where each Li is an arc from ~11to wz and Li fl Lj = (~1, ~2) if i # j. Then each large Whitney level for G(m) is homotopically equivalent to the (m - l)-sphere Sm-’ (see Exercise 53.12).
Another family for which large Whitney levels for C(G) have been analyzed is described in the following definition. 65.9 Definition [24, Definition 3.11. The finite graph G is said to be a fruit tree if the following condition holds: if y is a circle in G, then y is a segment of G. 65.10 Theorem [24, Theorem 3.21and [25, Theorem 2.61 (seeExercise 65.36). For a finite graph G the following statements are equivalent: (a) G is a fruit tree, (b) there exists a positive Whitney level A for C(G) which is homeomorphic to some cube [0, I]“, and (c) each large Whitney level for C(G) is homeomorphic to the cube [0, l]“, where n = (number of end points of G) + 2(number of circles in G) -1.
By Example 65.4, a finite graph is completely determined by its small Whitney levels. That is, if we know the small Whitney levels for C(G) then we know the finite graph G. This is not true for large Whitney levels.
310
IX. WHITNEY
LEVELS
Using Theorem 65.10, it is possible to construct two different fruit trees with homeomorphic large Whitney levels (Exercise 65.37). However, large Whitney levels also contain a lot of information of the graph as it is shown in the following two theorems. 65.11 Theorem [24, Theorem 2.71. Suppose that G and H are finite graphs without cut points, then G and H are homeomorphic if and only if large Whitney levels for C(G) are homeomorphic to large Whitney levels for C(H). For a generalization of the next theorem, see Theorem 68.10. 65.12 Theorem [24, Theorem 3.31. G is a circle if and only if large Whitney levels for G are not unicoherent. 65.13 Question. Characterize those finite graphs G for which large Whitney levels are homotopically equivalent to the n-dimensional sphere S”. It would be interesting to solve this question for the particular case
n = 2. In [26] and [29], it was studied the fundamental dimension of Whitney levels for C(G), where G is a finite graph. In [29], Kato defined an index i(G) of G such that if p is any Whitney map for C(G), then (Fundamental Dimension of p-‘(t))
5 i(G) - 1, for every t E [0, p(G)]
Since finite graphs are l-dimensional compact connected polyhedra, it is natural to ask if for polyhedra of greater dimension it is possible to define an index with the properties as those of i. This problem was solved in the negative by Kato in [28]. He showedthat if P is any n -dimensional compact connected polyhedron (n > 2) and m > 2, then there exists a Whitney level A for C(P) such that the m-sphere is homotopically dominated by A. In particular, (Fundamental Dimension of A) 2 m. In [25] some properties of the dimension of Whitney levels of finite graphs are studied. The results in [25] are based in the combinatorial formula described in the following theorem. 65.14 Theorem 125, Lemma 1.11. Let G be a finite graph. Let A = p-‘(t) be a positive Whitney level for C(G). If A E A, then dimA[d]
=
max{dim[ms]
- 1 : 5’ is a fine subgraph of G,
A E ms and S $2 A 4
Q(LS)),
where dimA[d] is the dimension of A at A (Exercise 65.38).
65. FINITE GRAPHS 65.15 Corollary (Exercise 65.39).
[25, Corollary
1.21. dimA[d]
311 does not depend on A
65.16 Theorem [25, Theorem 1.51. Let G be a finite graph and let p be a Whitney map for C(G). If 0 < t < T < 1, then dim[p-‘(t)] 5 dim[p-‘(r)] (Compare with Exercise 45.6). In [39, 1.51, Rogers proved that, for any continuum X and any Whitney level B for C(X), dim[B] + 1 5 dim[C(X)]. In the case of finite graphs, the following theorem shows exactly for which elements A E A (A is a positive Whitney level for C(G)), the formula dimA[d] + 1 = dimA[C(G)] holds. 65.17 Theorem [25, Theorem 1.71. Let G be a finite graph. Let A be a positive Whitney level for C(G) and let A E A. Then dimA[d] + 1 < dimA[C(G)], and the following statements are equivalent: (a) A is a fine subgraph of G, and (b) dimA[d] + 1 < dimA[C(G)]. 65.18 Theorem [25, Theorem 1.81. Let G be a finite graph. If A is a positive Whitney level for C(G) then dim[d] = dim[C(G)] - 1 if and only if there exists A E A such that A contains all the ramification vertices of G and A is not a fine subgraph of G. In [26, Proposition 2.41, Kato showed that Whitney levels for finite graphs are polyhedra. Then the following question arises naturally. 65.19 Questions. Given a polyhedron P, consider the finite graphs G which admit a Whitney level A such that A is homeomorphic to P. Is it possible to determine those finite graphs G? Is it possible to say how many of those finite graphs G there are ? Is it possible to give an upper bound for the number of those finite graphs G? Two particular orems.
answers
to Questions
65.19 are the following
three the-
65.20 Theorem [25, Theorem 2.71. Let G be a finite graph. There is a Whitney level for C(G) which is homeomorphic to [0, l]* if and only if G is a simple triod or a noose. 65.21 Theorem [25, Theorem 2.91. Given a polyhedron only finitely many graphs G such that P is homeomorphic Whitney level for C(G).
P there exist to a positive
65.22 Theorem [21, Theorem 21. For each finite graph G there is only a finite number of topologically different Whitney levels for C(G).
312
IX. WHITNEY
LEVELS
65.23 Question. Given a finite graph G, how many topologically different Whitney levels are there for C(G)? (Compare with Question 65.28)
In [9], Duda characterized those polyhedra which are homemorphic to the hyperspace of an acyclic finite graph. This is the motivation of the following question. 65.24 Question. Characterize those polyhedra which are homeomorphic to some Whitney level of an acyclic finite graph.
The following theorem shows how a Whitney map induces a nice decomposition in the hyperspace C(G). 65.25 Theorem [21, Theorem 11. Let G be a finite graph. Let p be a Whitney map for C(G). Suppose that {Te,Ti,. . . ,T,} = {p(H) : H is a subgraphofG}U(0),whereTo
such that (a) (p(A,T) = A and cp(A, s) c cp(A, t) if s < t and A E p-‘(T), and (b) for each t E (Ti-i,Z’i), I&-‘(T) x {t} is a homeomorphism from p-‘(T) x {t} onto p-‘(t). Two natural questions are: What is the class of continua that can be obtained as positive Whitney levels? What is the class of continua that can be obtained as positive Whitney levels of finite graphs? The first question as been solved by W. J. Charatonik in the following theorem. 65.26
Theorem
[4]. Every continuum is a positive Whitney
level of
some continuum. With respect to the second question, as Kato showed in [26, Proposition 2.41, positive Whitney levels of finite graphs are polyhedra. By Exercise 65.41 not every polyhedron can be obtained as positive level of a finite graph. However, the following theorem by Kato shows that, homotopically speaking, every polyhedron can be obtained as a positive Whitney level of a finite graph. 65.27 Theorem [27, Theorem 3.31. For every compact connected ANR S, there exist a finite graph G and a Whitney level A for C(G) such that A is homotopically equivalent to X.
EXERCISES
313
65.28 Question [27, Question 11. Let G be a finite graph. Let A be a Whitney level for C(G). What is the homotopy type of A? How many homotopy types do Whitney levels for C(G) admit? 65.29 Question [27, Question 21. Let X be a continuum. Does there exist a curve (l-dimensional continuum) Y such that Y admits a Whitney level f3 for C(Y) with the property that ShB = ShX?
Exercises 65.30 Exercise.
Prove the claim (*) in 65.1.
65.31 Exercise.
Sets of the form 9$(t)
are arcs.
65.32 Exercise. Let G be a finite graph. Then S is a maximal fine subgraph of G if and only if S is a fine subgraph of G which contains all the ramification points of G. All the maximal fine subgraphs have the same number of segments. 65.33 Exercise. Let G be a finite graph. If A = p-l(t) is a positive Whitney level for C(G) and F = {S : S is a fine subgraphs of G for which : S E F}. ms(t) is defined}, then A = ~{9Jls(t) 65.34 Exercise. Let G be a finite graph. If G has a cut point, then large Whitney maps for C(G) are contractible [Hint: Use Theorem 66.41 65.35 Exercise. If G is a fruit tree, then every ramification point is a cut point of G and there exists a unique maximal fine subgraph T. Furthermore, if A is a large Whitney level for C(G), then every element in A contains T. 65.36 Exercise. If G is a fruit tree, then each large Whitney level for C(G) is homeomorphic to the cube [0, lln, where n = (number of end points of G) + 2(number of circles in G) - 1. 65.37 Exercise. Find all the fruit trees G such that their large Whitney levels are homeomorphic to the cube [0, 117. 65.38 Exercise.
Prove Theorem 65.14.
65.39 Exercise. Let G be a finite graph. If A is a Whitney level for C(G) and A E A, then dimA[d] does not depend on the Whitney map which is used to define A.
314
IX. WHITNEY
LEVELS
65.40 Exercise. Prove the following particular caseof Theorem 65.26. Every finite graph is homeomorphic to a positive Whitney level of a continuum. 65.41 Exercise. There exists a polyhedron which is not homeomorphic to a Whitney level of a finite graph. [Hint: Let P = D U L, where D is the unit disk in the plane centered at the origin and L is the segment [-1, l] x {-l}.]
66.
Spaces of the Form
Q(X,
t) Are ARs
As usual X denotes a continuum, let d denote a metric for X and let p denote a Whitney map for C(X). Since the product of a positive constant by a Whitney map is again a Whitney map, we may assumethat p(X) = 1. = {A E p-‘(t) : Let E E C(X) be such that p(E) _< t. Define c~(X,t) E c A}. In [31], Lynch proved that the sets of the form CE(X, t) are ARs. His proof uses a Dugundji-type construction. This section is devoted to present Lynch’s Theorem. For proving the main theorem of this section, Lynch used a convex structure defined on certain subspacesof the space of order arcs. Here, we present a simpler convex structure given by W. J. Charatonik in [3]. For each s E [O,l], let A,(X) = {d E C(C(X)) : A is an order arc joining an element in p-l(s) with X}. For each A E As(X) and for each t E [s, 11,let d(t) be the unique element in A f? p-‘(t). 66.1 Lemma.
Consider the metric defined on As(X) by
D(d,B)
= sup{Hd(d(t),Ij(t))
: t E [s, 11).
Then D is equivalent to the Hausdorff metric HH~ restricted to As(X). Proof. Clearly, D is a metric for A,(X) (Exercise 66.5). Let T(D) and be the respective topologies on As(X) defined by D and HH~. If A, f? E As(X) are such that D(d, a) < E, then Hd(d(t), B(t)) < E for every t E [s, 11. Thus A c N(c,Ij) and 13c N(c,d). Then HHd(d,a) < E. Hence, r(H) c T(D). Now, let E > 0. By Lemma 17.3, there exists n > 0 such that if A, B E C(X), A C B and jp(A) --p(B)1 < r,~,then H(A, B) < f. Let S > 0 be such that b < f and if A, B E C(X) and Hd(A, B) < 6, then Ip(A) - p(B)1 < 7. Take d,f? E A,(X) such that HH~ (d, f?) < 6. Given t E [s, 11, there = exists a(r) E .13,with r E [s, 11such that Hd(d(t),a(r)) < 6. Then It-r] MA(t)) - ~(a(~))1 < 7). Then Hd(a(r),D(t)) < 5. Thus Hd(d(t),B(t)) < c. This proves that D(d, f3) < c. Hence T(D) c r(H). W
r(H)
66. SPACES OF THE FORM
66.2 Construction
315
t) ARE ARs
CE(X,
[3]. For each n = 1,2,. . ., define
A, = {(tl,. . . ) tn) E [O,l]n : t1 + . . . + t, = 1). For T = (tr,. . . ,tn) E In, define M(T) For each s E [0, 11. Let
M,
= {(A,.. . ,A,) E (As(X))”
M
=
= max{tr,.
. . , tn}.
: Al(s) = ... = An(s)),
and
u{M,xA,:n=1,2,...}. E M> associate to & the element
Given E = ((Al, . ..,dn),(tl,.-.,tn))
((Al,. . . , A,, {Al(s)}, {Al(s)}, . . .), (tr, . . . , k, O,O, . . .)) in (C(C(X)))”
X P.
Consider this product with the metric o*(((&,B2,.
. .), (~,Tz,.
. .)),
((C,,Cz,.
. .I,
(wru2,.
. .)))
=
In this way we have a metric defined on M. Define c : M + A,(X) by g(((dl,...,d,),(tl,...,t,)))
=
{A1 ((1 - 731)s+ rtl) U . . . ud,((l-rt,)s+rt,):rE[O,66.3 Theorem.
c has the following properties:
(a) E is well defined, (b) c(((d,.
. . 74, (tl, . . . , tn>)> = 4
(c) c(((d1,. . . , AA (tl, . . . , tn))) = Wd,(l)>. . . ,&(n,), (t,(l), . . . ,t,(n,))) for every permutation u : { 1,. . . , n} + { 1, . . . , n}, (4 e( ((Al, . . . ,A,), (tl , . . . , b-1,0, &+I, . . . , tn))) = e(((dl,...,di-l,di+l,...,d,),(tl,...,ti-l,ti+l,...,t,))), (e) c is continuous.
1
M(T)]} .
316
IX. WHITNEY
LEVELS
Proof. (a) If T E [0 , h], then 0 5 rt, <_ 1. Thus each set of the form di(( 1 - rt,)s + rti) contains the common set &(s). If r = &, then rtj = 1 for somej and dj( (1 - rtj)s f Y-Q)= X. Therefore, for each to X (4 T) E M, c((d, T)) is an order arc from the common value Ai (see Exercise 66.6). (b) Since c(((d,. . . ,A), (ti,. . . ,tn))) is an order arc from d(s) to X and it is contained in -4, it is equal to A. (c) and (d) are easy to check (Exercise 66.7). (e)Let(d,T)EM,whered=(di ,..., d,)andT=(ti ,..., &),and let 6 > 0. By Lemma 17.3, there exists n > 0 such that 77< f and if rl,B E C(X), A c B and p(B) - p(A) < 77,then Hd(A, B) < f. In the proof of Lemma 66.1 it was shown t,hat there exists p > 0 such that if ID, < p, then D(D,E) < f & E As(X) and HHd(DD,&) Since M(T) > 0 and ti +. . . + t, = 1, then there exists 6 > 0 such that S<~andifV=(211,...,~n)~Inand~+l~+...+~<6, then: (1) &]ti-~li]
< M(V)
(3) ]I - (211+...+v,)]
< 2M(T)
n, and ]&
- -----I Mf”,
< z, and
< M(T)v 4 .
Take (a, U) E M, where D = (&, . . . ,&) and U = (~1,. . . ,um). Suppose that D*((d,T), (B, U)) < 6. We will prove that HHd(&((d,T)), e(@, U))) < 6. Define V = (VI,. . . , w,), where vi = Ui if i 6 m and vi = 0 otherwise. < ,j. Thus Since D*((d, T), (0, U)) < 6, then b..++h$!d+...+b+ inequalities (1)) (2) and (3) hold for V. . .+wrr) < v < M(V) = UiO Ifm > n, then u,+i+.. .+um = I-(vi+. for someic 5 71. This implies that M(V) = M(U). In the case that m 5 n, it is clear that M(V) = M(U). Thus, in any case, M(V) = M(U). Take a typical element E = d1((1-rtl)s+rtl)U...Ud,((l--rt,)s+rt,) < f, there exists Since I’& - -!in c(d, T), where T E [0, &I. M(u) 1 ] such that (T R( < 5. REP&j Define F = B1((1-Ru~)s+Ru~)U~~4J&((1--Ru,)~+R~,) E e(B,v). We will show that Hd(E, F) < E. Let p E di((1 - rt,)s + rti), where i 5 n, be an element in E. < 6 < &, HHd(di,f?i) < $. Then If i 5 m, since o*((d,T),(Z3,U)) D(d,,&) < 5. Thus Hd(di((l
- rti)s + rti),Bi((l
- rti)~ + rti)) < f.
66. SPACES OF THE FORM CE(X, t) ARE ARs
317
Since I(1 - rt,)s + rtl) - ((1 - RQ)S + &)I 5 17% - Ruil 5 T+ - uil + uilr - RI < z (by (l)), the choice of 17implies that H&?i((l
- r&)s + ?q,&((l
- RUi)S + RU*)) < 5.
Then p E N(e, F). If m < i, then wi = 0. By (l), & < :. Then [(I - rti)s + rt, s] < z. From the choice of q, Hd(di((l - rt,)s + rti),di(s)) < 5. Thus mce HHd(dr,Z?i) < p, D(di, ai) < 1. Then p E P E N+W)). s N(E, B,(s)) c N(E, F). Th is completes the proof that E C N(c, F). Now, let p E &((l - &)s + RUG) be an element in F, where i 5 m. If i 5 n, proceeding as before, we conclude that p E N(E, E). If i > n, I(1 I &(l-(vl+~..+v,,)) < 5 Rui)s+Rui-s( 5 Rui 5 R(u n+l+f.‘.+u,) (by (2) and (3)). By the choice of n, Hd(&((l - Ru,)s + Rui),&(s)) < f. Since HHd(di,8r) < p, D(di, &) < 5. So p E N(c, dr (s)) C N(c, E). This completes the proof that F C N(E, E). Therefore, Hd(E, F) < 6. This proves that E(d,T) c iV(e,E(B, U)). Similarly, E(a, U) C N(E, c(d,T)). Hence HH~(E(A,T), (.?(a, U)) < E. This completes the proof that e is continuous. n 66.4 Theorem [31]. Let X be a continuum. Then each set of the form A = c~(X,t) (where p(E) 2 t) is an AR. Proof. As usual, let d denote a metric for X. We will prove that A is an absolute extensor for metric spaces(see Theorem 9.1). Take a metric space (2, p), a closed subset A c 2 and a map g : A -+ A. We need to prove that g can be extended to a map G : Z + A. Let AE = {a : 23is an order arc from E to X} and let s = p(E) Given p E Z - A, define BP = (2 E Z : P(P, 2) < $P(P,A)I. Let U = {V, : cr E J} be a locally finite open refinement of {B, : p E Z - A}. Let P = {& : a E J} be a partition of unity subordinated to U. For each CYE 3, choosea point p, E U,, alsochoosea point a, E A such that p(pcl, a,) < 2p(pL1,A), finally chooseZ?, E A.5 such that f?,(t) = g(au) (That is, g(a=) E 13,). Define G : Z -+ d, as follows: If p E A, define G(p) = g(p). IfpE Z-A,letcq,..., on be those elements in 3 such that #ni (p) > 0. Then define 4l=
WI,
I . . . , Ban, da, (PI,. . . , ban (P)) and G(P)
= Wt).
318
IX. WHITNEY
LEVELS
First, we will show that G is continuous on 2 - A. Take a point p E z A. Let U be a neighborhood ofp in Z such that U~IA = 0 and there is only a finite number of elements ol,. . , , (Y, of J’ such that UnU,, # 0. Given q E U, the set {o E J : &(q) > 0) is contained in ((~1,. . . , (Y,}. By Theorem 66.3 Cc) and (4, G(q) = (WL, , . . . , &, , kl (q), . . . , &, (q)))(t). Since the function q + (&, (q), . . . ,&,(q), O,O, . . .) from U in Im is continuous, by Theorem 66.3 (e), we conclude that G is continuous on U. Hence G is continuous on Z - A. Finally, we will show that G is continuous on BdZ(A). Take a point a E Bdz(A) and let E > 0. By Exercise 66.8 there exists q > 0 such that, for each pair A, B E A, if B c N(q, A), then Hd(A, B) < E. Let 6 > 0 be such that if 6 E A and p(b, a) < 96, then Hd(g(b), g(u)) < c. Take p E 2 - A such that p(a,p) < 6. Suppose that Q E J is such that &(p) > 0. Then p E U, and U, c B, for some q E 2 - A. Notice that p(q,p)
< $p(q, A) 5 $p(q,p)
+ ~P(P, a). This
implies
that
dq,p)
< 6 and
p(q,A) < 26. Th en, for every y, 25E B,, ph 2) L P1ylq) + dq, z) < 26 and P(Y, A) I P(Y, q) + p(q, 4 < 36. Thus Aa, aa) 5 P(~,P) + P(P~P~) + d~db) < 36+2 p (p,, A) < 96. Hence Hd(g(a), a,(t)) < n. Thus a,(t) C NC777 da)). Let cq,...,c+ G(P) = (WL,
be the elements in 3 for which q&;(p) > 0.
> . . . , B,, , &, (P), . . . , &,
(p)))(t)
is ofthe
form &,
(rl)U.
Since * LJ
&, (rn) and it is an element in the level A = p-‘(t), it follows that ri 5 t for each i. Then D,; (ri) C D,;(t) c N(q, g(a)). Hence G(p) C N(q, g(a)). Thus %(G(P), G(a)) < E. This completes the proof that G is continuous on Bdz(A) and the proof of the theorem. n
Exercises 66.5 Exercise.
Verify that D in Lemma 66.1 is a metric.
66.6 Exercise. Let A be an order arc from A to B, and let T be as in 66.2. Then the map t + d(t) from [p(A),p(B)] onto A is continuous. Therefore, the set e(d, T) in Theorem 66.3 is compact and connected. 66.7 Exercise.
Verify (c) and (d) in Theorem 66.3.
66.8 Exercise. Let X be a continuum and let d denote a metric for X. Let A be a Whitney level for C(X). Then for every c > 0 there exists 6 > 0 such that, for each pair A, B E A, if B c N(6, A), then Hd(A, B) < E (compare with Exercise 13.10). 66.9 Exercise. Let X be a continuum. Let E E C(X). Let cE(X) be the containment hyperspace for E in C(X) as it was defined in Chapter
67. ABSOLUTELY C*-SMOOTH,
CLASS(W)
319
AND COVERING...
II. By Exercise 14.22 CE(X) is an AR. Prove that CE(X) n p-‘([O, retract of CE(X). Therefore, CE(X) n p-‘([O, t]) is an AR.
t]) is a
66.10 Exercise. Let X be a continuum. If X has a cut point, then, for all t in some neighborhood of 1, p-l(t) is an AR.
67.
Absolutely
C*-Smooth, Class(W) Property
and Covering
Definitions of Absolutely C*-Smooth, Class(W) and Covering Property are given in section 35. The notion of Class(W) was introduced by Lelek in 1972. Since then several authors studied which continua are in Class(W) and have looked for different characterizations of those continua in Class(W). Some of the abundant results related to Class(W) are the following: (a) all hereditarily indecomposable continua are in Class(W) (Cook, [5]) (Exercise 67.14), (b) chainable continua are in Class(W) (Read, [38]), (c) non-planar circle-like continua are in Class(W) (Feuerbacher, [13]), (d) metric compactifications of the half line [O,oo) which have remainder in Class(W) are in Class(W) (Grispolakis and Tymchatyn, [15, Theorem 3.4]), (e) atriodic continua with trivial first Tech cohomology are in Class(W)
(Davis,PI), (f)
if X is in Class(W), then X is irreducible (Hughes, [34, Theorem 14.73.11 and Theorem 67.1). Two generalizations to the notion of Class (W) appeared in [7] and [33]. The concept of Covering Property was introduced in [30, section 61. In that paper the problem was raised of characterizing continua with this property. In [34, Chapter 141 it was included a discussion about covering property. In the next theorem it is shown that being in Class(W) and having the covering property are equivalent and they are also equivalent to two more properties. The beauty of this theorem resides in the way that it comprises four notions defined in very different contexts. A characterization of continua with the covering property in terms of irreducibility of Whitney levels appeared in 14.73.3 of [34]. Regarding next theorem: Implication (a) + (d) was proved by Hughes ([34, Theorem 14.73.211). Hughes asked if the converse implication was true ([34, Question 14.73.251). Implication (a) + (b) was proved in [14, Theorem 2.21. The equivalence
320
IX.
WHITNEY
LEVELS
between (b) and (c) was showed by Proctor in [37]. Finally (b) + (d) and (d) + (a) were proved by Grispolakis and Tymchatyn in [16, Theorem 3.21. 67.1 Theorem. alent:
For a continuum X the following statements are equiv-
(a) X E CP, (b) X is absolutely C*-smooth, (c) every compactification Y of [0,03) with remainder X has the property that C(Y) is a compactification of C([O, oo)), and (d) X E Class(W). Proof. (a) + (b). Supposethat X E CP. Assume that X is a subcontinuum of a continuum 2. Consider the function C* : C(Z) -+ C(C(2)) given by C*(A) = C(A) (the set of subcontinua of A). We need to prove that C* is continuous at x. Take a sequence{X,)r=, in C(Z) such that X;, --t X. Since C(C(2)) is compact, in order to show that C*(X,,) + C*(X) we may assumethat C*(X,) + A and we will prove that A = C*(X). Given A E A, by Exercise 78.36, there is a sequence {A,}~E?=l in C(Z) such that A, -+ A and A, E C*(X,) for each n = 1,2,. . .. Since A, c X,, for each n = 1,2,. . ., it follows that A c X. Therefore, A c C*(X). Now, take A E C*(X). Let p : C(Z) + R’ be a Whitney map. Let Then A E C. to = p(A) and C = (PIG’(X))-‘(to). If to = p(X), then A = X. Then A = lim X, E lim C*(X,) = A. Thus A E A. If to < p(X), then there exists N 2 1 such that to < n(X,) for each n 2 = (/.&‘(X,))-’ (to). Then N. For each n 1 N, define B, = ,u-l (to)fK’(X,) 0, is a subcontinuum of C(Z), Since C(C(Z)) is compact, there exists a subsequence{f3,,,}~& of {&,}~=i and there exists f3 E C(C(Z)) such that t?,, + D. Since &, c C*(Xnr,), 0 C A, Thus t? c p-‘(to) fl C(X) = C. Hence, B is a subcontinuum of C and C is a Whitney level for C(X). Since ,13,, + Z?, by Exercise 11.5, U&, + Uf?. But U&, = X,, -+ X. Thus X = uB. Then B covers X. Since we are assuming that X E CP, we conclude that t? = C. Hence, A E a c A. This ends the proof that C*(X) = A. Therefore, X is absolutely C*-smooth. (b) 3 (c). Suppose that X is absolutely C*-smooth. Take a compactification Y of [O,co) with remainder X. We will prove that C(Y) is a compactification of C([O, 00)). First of all, by Exercise 67.19, Y is a
67. ABSOLUTELY C*-SMOOTH,
CLASS(W)
AND COVERING...
321
metrizable continuum, let p be a compatible metric for Y. Formally speaking, there is an embedding h : [0, co) + Y and X is homeomorphic to Y - h([O, 00)). Then h([O,co)) is an open subset of Y and the function C(h) : C([O, 00)) + C(Y) g’rven by C(h)(A) = h(A) (the image of A under h) is an embedding also. Then, as usual, we identify each of the points p E [O,oo) with h(p). In this way we may think that [O,oo) and X are subspaces of Y and C([O, 00)) is a subspace of C(Y). Let A = &(y)(C([O, co))). Then A is. a compactification of C([O, 00)). We will prove that A = C(Y). Let A E C(Y). It is easy to show that A must satisfy one of the following three conditions: (i)
,4 c [0, co). In this case >4 E d,
(ii) A = X U [t, co) f or some real number t. In this case A is the limit of the sequence A, = [t, t + n] and then A E A, (iii) A c X. For each n = 1,2,. . ,, let U,, be a finite family of open subsets of Y which covers X and with the property that diameter (U) < f and U n X # 8 for every U E U,,. Since X c cly ([n, CQ)), there exists m, E [n, 00) such that [n, m,] intersects every element in U,,. Let X, = [n, m,]. Then X c N(A, XtL). In order to show that X, + X, let E> 0. Since [n, co) U X tends to X, there exists N > 1 such that H,(X, [n, co) u X) < E and & < E. Then, for each n 2 N, X c N(c,Xn) and X, c N(E,X). Then H,(X, X,) < E for every n > N. Hence X, + X. Since we are assumingthat X is absolutely C*-smooth, C(X,) -+ C(X), then A = lim A, for some A,, E C(X,) c C([O,co)). Hence A E A. This completes the proof that C(Y) is a compactification of C([O, 00)). (cl * (4. SuPpose that every compactification Y of [0, co) with remainder X has the property that C(Y) is a compactification of C( [0, 00)). In order to prove that X E Class(W), take a continuum 2 and an onto mapf:Z-+X. By Exercise 67.18, there is a metric compactification W of [0, 1) with remainder 2. Define an equivalence relation N on W by setting p N q if and only if p = q or p,q E 2 and f(p) = f(q). It is easy to show that the equivalent classesof N form an upper semicontinuous decomposition. Then ([35, Theorem 3.101) the space Y = W/N is a continuum. Let II : W -+ Y be the natural map. Since 2 = II-‘(II(Z)), 2 is closed in W and II]2 has the same fibers that f we have that (see Exercise 67.20) II(Z) is homeomorphic to X then we will identify X with II(Z). Since [0, oo) is a II-saturated dense open subset of W and II][O, 03) is the identity map, we have that II( [0, 00)) is a densesubset of Y which is homeomorphic to [0, oo).
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Then Y is a compactification of [0, 00) with remainder X. By hypothesis, ckT,Y,(c(~([o, 00)))) = C(Y). We are ready to prove that f is weakly confluent. Take a subcontinuum B of X. Then there exists a sequence {A,}rfl of subcontinua of [O,co) such that B = limlI(A,). Taking a subsequence if necessary we may assumethat A, + A for some subcontinuum A of W. Since II(A) = B C X, A c II-‘(X) = 2. Then A is a subcontinuum of Z such that f(A) = II(A) = B. This proves that f is weakly confluent. (d) + (a). Let d denote a metric for X. Suppose that X $ Cl’. Then there exists a Whitney map ~1 for C(X), there exists a number t E [O,p(X)] and there exists a proper subcontinuum d of the Whitney level B = /~-l(t) such that X = u{A : A E A}. Since F,(X) = p-i(O) and we have that t # 0 and t # p(X). We will prove that ix> = P-‘Mm x 4 Class(W). Fix an element B E D - A. Let e > 0 be such that &(A, B) > E. By Exercise 66.8 there exists 6 > 0 such that if C, D E t3 and C c N(6, D), then Hd(C, D) < 6. Then, for each A E d, A g N(6, B) and B g N(6, A). Since B is compact, there exists a finite family of open subsets VI,. . . , U,,, such that B c U1 U . . . U U,,, , diameter (Vi) < 6 and Vi f~ B # 8 for each i= l,...,m. For each i = l,... ,m, define di = {A E A : A n U, = 0). Then A, is compact. If there exists A E A - (A1 U. . . U A,,), then An Ui # 0 for each i = l,..., m. This implies that B c N(6,A). This contradiction proves that A = -Al UB.-ud,. For each i = l,... ,m, define Xi = u{A : A E A,}. By Exercise 11.5. Xi is compact. Since X = u{A : A E A}, X = Xi U . . . U X,. Consider the space Y = (u{X,
x {i} : i E (1,. . ,m}}) U ((X - N(b, B)) x [l,m])
The space Y is considered as a subspaceof the continuum X x [l, m]. Clearly Y is a metric compact space. Let f : Y + X be the restriction of the natural projection from X x [l,m] into X. Notice that f is onto. In order to prove that f is not weakly confluent, first we will show that Y is a continuum. Suppose to the contrary that Y is not connected. Then Y can be written in the form Y = H U K, where H and K are nonempty disjoint closed subsets of Y. Given an element A E A, there exists i E {l,..., m} such that A E Ai. Then A x {i} C X, x {i} C 1’. Since A x {i} is a connected subset of Y. Then A x {i} C H or A x {i} C K. Define 3t
=
K: =
{>4 E A : there exists i E (1,. . . ,m} such that A x {i}
c
H} and
{A E A : there exists i E (1,. . . ,m} such that A
c
K}.
x
{i}
67. ABSOLUTELY C*-SMOOTH,
CLASS(W)
AND COVERING..
.
323
Thend=%UK. Let A E A. Since A g N(6, B), there is a point a E A - N(S, B). Then (Ax {l,..., ml) u ({al x [km]) is a connected subset of Y. This implies that 7-l II K = 0. Choose a point (p, t) E H and let i E (1, , . , n] be such that p E Xi. Then there exists A E Ai such that p E A. Thus the set A x {i} is a connected subset of Y that is contained in H. Hence A E E. This shows that ?f is nonempty. Similarly, K is nonempty. Now, we will prove that 3f is closed. Take a sequence {A,,)~=i in E which converges to an element A E A. For each n = 1,2,. . ., there exists i, E {l,..., m} such that A, x {in} c H. Taking a subsequence if necessary, we may assume that there is a j E { 1, . . . , m} such that i, = j foreveryn=1,2,.... Then A, x {j} -+ A x {j}. Since A, x {j} C H for each n > 1, A x {j} c H. Thus A E 31. Therefore ‘R is closed. Similarly, K is closed. This contradicts the connectednessof A and proves that Y is a continuum. Now, supposethat g is weakly confluent. Then there exists a subcontinuum D of Y such that f(D) = B. Then D c U{Xi x {i} : i E (1,. . . ,m}}. Since D is connected, there exists i E { 1, . . . , m} such that D C Xi fl {i}. Then B = f(D) c Xi. This is impossible since B fl Vi # 8. Therefore, f is not weakly confluent. Hence X $! Class(W). This completes the proof that (d) + (a) and the proof of the theorem. w In [20], Grispolakis and Tymchatyn introduced the notion of W-set. A subcontinuum A of X is said to be a W-set in the continuum X provided that for every onto map f : Y + X, (where Y is a continuum) there exists a subcontinuum B of Y such that f(B) = A. Notice that X E Class(W) . a W-set in X. They showed the if and only if for every A E C(X), A 1s following characterization of W-sets. 67.2 Theorem [20, Theorem 2.11. Let X be a continuum and let d denote a metric for X. Let A be a proper subcontinuum of X. Then A is not a W-set in X if and only if there exists some E > 0 and a neighborhood IJ of A such that (1) for each z E U there exists a continuum B in X such that 2 f B, B II Bdx(U) # 0 and A g N~(E, B), and (2) for each decomposition of Bdx (U) = R U S into disjoint closed sets R and S, there exists a continuum B of X such that B n R # 8, BnS#OandAgiVNd(e,B).
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Grispolakis and Tymchatyn also showed the following local version of part of Theorem 67.1. 6’7.3 Theorem [20, Theorem 2.31. Let A be a subcontinuum of the continuum X. Then the following are equivalent: (i) A is a W-set in X, (ii) for every Whitney level A for C(X) such that A E A and for every subcontinuum B of A such that X = U{B : B E B} we have A E B, and (iii) if X is contained in the Hilbert cube and {Y;Z}~Z”=lis a sequence of arcs in the Hilbert cube such that Y, -+ X, then A E lim inf C(kl,). In this way Grispolakis and Tymchatyn added the following intrinsic characterization of Class(W) to Theorem 67.1. 67.4 Theorem [20, Corollary 2.21. Let X be a continuum and let d be metric for X. Then X is in Class(W) if and only if for every subcontinuum A of X, for each E> 0 and each neighborhood U of il we have that either (1) there exists z E U such that for every continuum B C c1.y(U) which contains z and intersects Bd.y(U) we have that A C N~(E, B) or (2) there exists a decomposition of Bdx(U) = R U 5’ into disjoint nonempty closed sets R and S such that for every continuum K C cl.y(U) which intersects R and S we have that A C Nd(c, K). Recently, Ma&s in [32] has defined the following notion which is related to the covering property. 67.5 Definition [32]. Let X be a continuum. A closed subset A of C(X) is a minimal closed couer provided that A consists of nondegenerate proper subcontinua of X, X = Ud and A is minimal with respect to these properties in the sensethat no proper closed subset of A covers X. Ma&s obtained some results on the structure of X in terms of its minimal closed covers. 67.6 Theorem [32, Theorem 51. A continuum X is a graph if and only if every minimal cover is finite. 67.7 Theorem [32, Theorem 71. If X is a continuum such that all the minimal closed covers are countable, then X is hereditarily locally connected. 67.8 Question [32, p. 2041. Let X be a continuum. If every Whitney level for C(X) is a minimal closed cover, is X hereditarily indecomposable?
EXERCISES
325
67.9 Question [32, p. 2041. Let X be a continuum. If every minimal closed cover for C(X) is connected, is X hereditarily indecomposable? 67.10 Question [32, p. 2041. Let X be a continuum. Is it true that X is hereditarily locally connected if and only if all its minimal closed covers are totally disconnected?
Exercises 67.11 Exercise. Given a class of maps M, a continuum X is said to be in Class(M) provided that all maps from continua onto S are in class M. Let M1 be the class of monotone maps. Then X is in Cla.ss(Ml) if and only if X is a one-point set. [Hint: Project an appropiate subcontinuum of X x [0, l] onto X.1 67.12 Exercise. Let Mz be the classof open maps. Then a continuum X is in Class M2 if and only if X is a one-point, set. 67.13 Exercise. Let M3 be the class of confluent maps. Then a continuum X is in Class(Ms) if and only if X is hereditarily indecomposable (151). 67.14 Exercise. Class(W).
Every hereditarily indecomposable continuum is in
67.15 Exercise. Let X be a continuum. Let ‘u.: C(C(X)) --f C(X) be the union map defined in Exercise 11.5. Then X has the covering property hereditarily if and only if for each Whitney map p and each t E [0, p(X)], PIw-l@)) : W-l(t)) -j P-‘h4w1) is a homeomorphism ([36, Corolary IS]). 67.16 Exercise. Let X be a metric compactification of the half-open interval [O,oo). Let A be a Whitney level for C(X). Then uclc:(Au)(d n C([O, co))) = X. If A E C([O, 00)) n A and 0 $ A, then A separates A. If B is a subcontinuum of A and Uf? = X, then A n C([O, co)) c 8. 67.17 Exercise. Let X be a metric compactification of the half-open interval [0, m). Then X E CP if and only if C(X) = cZccx,C([O, oo)) ([2, Proposition 41). 67.18 Exercise. Let 2 be a continuum, then there exists a metric compactification of [0, 1) with remainder 2. [Hint: By Exercise 28.12, we may assume that 2 is contained in the Hilbert cube [0, l]O”. Fix a point z. E 2. For each n 2 1, there is a finite set
326
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LEVELS
F C 2 such that 2 c Nd, (A, F) c [0, l]O”, where di is the metric in (0, l]03. Since Nd, ($, F) is an arcwise connected subset of [0, 1]03, there exists a map an : [A, $1 + NdI(irF) , such that an(&) = ~0 = (w,( $) and E (O,l]” x [O,l] : F c Im(cr,). Then define W = (2x {O})~(~{{(a,(t),t)
t E [A, Al> : n 2 11.1 67.19 Exercise. If Y is a compactification of [O,oo) such that the remainder of Y is a (metrizable) continuum, then Y is metrizable. [Hint: By Urysohn’s metrization theorem ([ll, Corollary 9.2, Ch. IX]) it is enough to prove that 2’ has a countable basis.]
67.20 Exercise. In the proof of (c) 3 (d) of Theorem 67.1 prove that II(Z) is homeomorphic to X. [Hint: Use the Transgression Theorem ([ll, Theorem 3.2, Ch. VI]).] A continuum X is decomposable if and only if there 67.21 Exercise. exists a finite minimal closed cover. Every minimal closedcover of [0, l] is finite ([32, The67.22 Exercise. orem 41). [Hint: Let A be a minimal closed cover of [0, 11,then there exists E > 0 such that 6 < i and diameter (A) > 6 for every A E A. Let [as, be] E A be such that a0 = 0. Let bi = max{b E [0, l] : there exists a E [0, l] such that bo E [a, b] E d}. Let bz = max{b E [0, l] : there exists a E [0, l] such that bl E [a, b] E A}. Then b2 2 26.1 67.23 Exercise. Every minimal closed cover of [0, l] is contained in a Whitney level for C([O, 11). Find a continuum X and a minimal closed cover of X which is not contained in a Whitney level for C(X).
68.
Holes in Whitney
Levels
As usual, the letter p denotes a Whitney map for C(X). In this section we suppose that p(X) = 1. A Whitney block is a set of the form pL-l(s, t), where ~1is a Whitney map for C(X) and 0 L: s < t 5 1. As we have seen in Theorem 19.8, for each continuum X, C(X) and 2x are unicoherent. For Whitney levels the situation is different. The following observation was done by Rogers in [40]: “As we go higher into the hyperspace C(X), no new one-dimensional holes are created, and perhaps some one-dimensional holes are swallowed”. This intuitive statement has found several formulations. In [40, Theorem 51, Rogers proved the following theorem.
68. HOLES IN WHITNEY
LEVELS
327
68.1 Theorem [40, Theorem 51. Let X be a continuum. If p is a Whitney map for C(X) and 0 2 s 5 t 2 1, then there exists a monomorphism y* : H’(p-l(t)) where H’(Y) of Y.
-+ Hl(p-l(s)),
denote the reduced nth Alexander-tech
cohomology group
With respect to fundamental group, the following results were shown in
WI. 68.2 Theorem [23, Theorem A]. If X is a Peano continuum and if 0 5 q < T < s < t 5 1, then there exists an epimorphism
where, for an arcwise connected space 2, xl(Z) group of 2.
denotes the fundamental
68.3 Theorem [23, Theorem B]. If X is a Peano continuum and if 0 5 s < t 5 1, then ~1(CL-’(s, t)) is finitely generated. 68.4 Theorem [23, Theorem C]. If X is a Peano continuum. Then the following statements are equivalent: (a) X is a circle, (b) ~1(P-‘(~7 t)) is a non-trivial group for every 0 5 s < t 5 1, and (c) For each T < 1, there exist r < s < t < 1 such that 7r1(p-l (s, t)) is a non-trivial group. 68.5 Question 123, Question 6.21. Is Theorem 68.2 true for Whitney levels? That is: if X is a Peano continuum and 0 5 s < t 5 1, then does there exist an epimorphism
A positive answer to Question 68.5 would give a positive answer to Question 47.2 68.6 Question [23, Question 6.21. Is Theorem 68.3 true for Whitney levels? That is: are the fundamental groups of the positive Whitney levels of a Peano continuum finitely generated? Rogers statement about the one-dimensional holes also can be expressed in terms of multicoherence degree defined in section 64. The following results were proved in [22].
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68.7 Theorem [22, Theorem A]. If S is a Peano continuum and if 0 5 s < t 5 1, then +-l(s)) 2 r(p-l(t)). 68.8 Theorem [22, Theorem B]. If A is a positive Whitney level for C(X) and X is a Peano continuum, then r(d) is finite. 68.9 Theorem [22, Theorem C]. Let X be a Peano continuum. If A is a Whitney level for C(X) and m is an integer such that 0 < m 5 r(X), then there exists a Whitney level A for C(X) such that r(d) = m. 68.10 Theorem [22, Theorem D]. For a Peano continuum the following statements are equivalent:
(a) X is a circle, (b) there exists a Whitney map w : C(X) -+ R’ such that w-‘(t) unicoherent for each 0 _
is not
(c) every positive Whitney level for C(X) is not unicoherent. Theorem 68.7 is not true for non-locally connected continua as it is noted in Exercises 64.7 and 64.8. An example as it is asked in Exercise 64.10 also shows that Theorem 68.7 is not true for non-locally connected continua. 68.11 Question [22, Question lo]. Is Theorem 68.9 true without the hypothesis of local connectedncss.
All the results in this section are for l-dimensional holes. For n-dimensional holes, where n 2 2, the situation is different. As we seein Example 30.1, there are Whitney levels for the hyperspace of a disk which have 2dimensional holes. Furthermore, by Theorem 65.8, there are finite graphs which have Whitney levels with n-dimensional holes, for all n 2 2. 68.12 Question [23, Question 6.11. What could be an appropriate version of Theorem 68.9 for the fundamental group of Whitney blocks? 68.13 Question [23, Question 6.21. Is Theorem 68.4 true for Whitney levels instead of Whitney biocks? 68.14 Question [23, Question 6.31. Is the implication (b) 3 (a) in Theorem 68.4 true for every arcwise connected continuum (instead of Peano continuum)?
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68.15 Question [23, Question 6.41. Is the following greater dimensional version of implication (c) + (a) in Theorem 68.4 true: Let X be a Peano continuum and let p be a Whitney map for C(X). If for each T < 1, there exist T < s < t 5 1 and there exists an integer n 2 1 such that n,(p-‘(s, t)) is a non-trivial group, then is X a connected finite graph? 68.16 Question [23, Question 6.51. Characterize those finite graphs which satisfy the assertion in Question 68.15 (compare with Question 65.7).
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E. Abo-Zeid, Some properties of Whitney continua, Topology Proc., 3 (1978), 301-312.
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W. J. Charatonik, Some counterexamples concerning Whitney leuels, Bull. Polish Acad. Sci. Math., 31 (1983), 385-391.
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W. J. Charatonik, Convex structure on the space of order arcs, Bull. Polish Acad. Sci. Math., 39 (1991), 71-73.
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W. J. Charatonik, Continua as positive Whitney levels, Proc. Amer. Math. Sot., 118 (1993), 1351-1352.
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H. Cook, Continua which admit only the identity mapping onto nondegenerate subcontinua, Fund. Math., 60 (1967), 241-249.
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J.C. Davis, Atriodic acyclic continua and class W, Proc. Amer. Math. Sot., 90 (1984), 477-482.
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E. Duda, Continuum covering mappings, General Topology and its Relations to Modern Analysis and Algebra VI. Proc. Sixth Prague Topological Symposium 1986, Z. Frolik, Editor, Heldermann Verlag Berlin (1988)) 173-175.
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R. Duda, On the hyperspace of subcontinua of a finite graph, I, Fund. Math., 62 (1968), 265-286.
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R. Duda, On the hyperspace of subcontinua of a finite graph, II, Fund. Math., 63 (1968), 225-255. 10. R. Duda, Correction to the paper “On the hyperspace of subcontinua of a finite graph, I”, Fund. Math., 69 (1970), 207-211.
11. J. Dugundji, Topology, Allyn and Bacon, Inc., 1966. 12. C. Eberhart, Intervals of continua which are Hilbert cubes,Proc. Amer. Math. Sot., 68 (1978), 220-224. 13. G. A. Feuerbacher, Weakly chainable circle-like continua, Dissertation, Univ. of Houston, Houston, 1974.
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14. J. Grispolakis, S.B. Nadler, Jr. and E. D. Tymchatyn, Some properties of hyperspaceswith applications to continua theory, Canad. J. Math.. 31 (1979), 197-210. 15. J. Grispolakis and E. D. Tymchatyn, Continua which admit only certain classesof onto mappings, Topology Proc., 3 (1978), 347-362. 16. J. Grispolakis and E. D. Tymchatyn, Weakly confluent mappings and the covering property of hyperspaces, Proc. Amer. Math. Sot., 74 (1979), 177-182. 17. J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only, (I), Houston J. Math. 5 (1979), 483502. 18. J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only, (II), Houston J. Math. 6 (1980), 375387. 19. J. Grispolakis and E. D. Tymchatyn, Spaceswhich accept only weakly confluent mappings, Proceedings of the International Conference of Geometric Topology, PWN-Polish Scientific Publishers Warzawa 1980, 175-176. 20. J. Grispolakis and E. D. Tymchatyn, On a characterization of W-sets and the dimension of hyperspaces, Proc. Amer. Math. Sot., 100 (1987), 557-563. 21. A. Illanes, Whitney blocksin the hyperspace of a finite graph, Comment. Math. Univ. Carolinae, 36 (1995), 137-147. 22. A. Illanes, Multicoherence of Whitney levels, Topology Appl., 68(1996), 251-265. 23. A. Illanes, The fundamental group of the Whitney blocks, Rocky Mountain J. Math., 26 (1996), 1425-1441. 24. A. Illanes and I. Puga, Determining finite graphs by their large Whitney levels, Topology Appl., 60 (1994), 173-184. 25. A. Illanes and R. Torres, The dimension of Whitney levels of a finite graph, to appear in Glasnik Mat. 26. H. Kato, Whitney continua of curves, Trans. Amer. Math. (1987), 367-381.
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X.
General Properties of Hyperspaces 69.
Semi-Boundaries
69.1 Definition [14]. Let X be a continuum. Let A, B E C(X) such that A 5 B. The semi-boundary of C(A) in C(B) is defined by Sb(A, B)
=
be
{C E C(A) : there exists an order arc g in C(B) such that II Q = C and D g A for every D E c~- {C}}.
In the case that B = X, we write Sb(A) instead of Sb(A,X). If C E Sb(A, B) and u is an order arc with the properties mentioned in the de& nition of Sb(A, B), th en there exists an one-to-one map (Y : [0, l] + C(B) such that cr(0) = C, o(s) 5 a(t) ifs < t and a(s) is not contained in A for each s > 0. Such a map 0 will be named a removing map for C in B. Notice that, in the terminology of Exercise 15.18, we may write Sb(A, B) = {C E C(A) : C is order arcwise accesible from C(B) - C(A)}. In this section we prove several results for Sb(A). With some obvious small changes, we have that these results are also true for Sb(A, B) (Exercise 69.6). Let us consider some examples (Exercise 69.7): (a) Sb({z}) = {z} for each 2 E X, (b) A E Sb(A) for every A E C(X) - {X}, (c) if A E C([O, 11) - {[O,l]} and A is nondegenerate, then Sb(A) is an arc and A is an end point of this arc if and only if 0 f A or 1 E A, (d) if X is a circle and A E C(X) - {X}, then Sb(A) is an arc and A is not an end point of this arc, (e) let X be the harmonic fan. Then there exists A E C(X) - {X} such that Sb(A) is not closed in C(A). 333
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X. GENERAL PROPERTIES OF HYPERSPAGES
The following theorem describes some elementary properties boundaries. Its proof is easy and it is left as Exercise 69.8.
of semi-
69.2 Theorem [14, Theorem 1.21. Let X be a continuum. Let A E C(X) - m (a) B E Sb(A) if and only if there exists a map cy : [0, l] -+ C(X) such that Q(O) = B and o(s) is not contained in A for each s > 0, (b) if B E Sb(A) and B c D c A, then D E Sb(A),
(c) Sb(A) is arcwise connected, (4 WA) c B~c(x,(W)L (e) if X is the sin(i)-continuum and A is the limit segment of X, then WA) # B&(x)(CtA)), and (f) ifB,DEC(X),BnD#O,B-D#0#D-BandEisacomponent of B n D, then E E Sb(B, B u D) n Sb(D, B U D). The following theorem is useful for determining when a subcontinuum is in a semi-boundary. 69.3 Theorem [14, Theorem 1.31. Let X be a continuum. Let A E C(X) - {X} and B E C(A). Let {B,}~c=l c C(X) be a sequence such that B,, + B. Then each one of the following conditions implies that B E Sb(A): (a) if B, is not contained in A and B,+l c B, for each n, (b) if B, is not contained in A and B, n B # 0 for each n, and (c) if B, E Sb(A) and B, n B # 0 for each n. Proof. (a) For each n 2 1, applying Theorem 14.6, there exists a map 7 Bn71, ~~(a) = B, and s 5 t an : [$, ;I + Wn) such that o.(q) implies that (Y,(S) c an(t). Define Q!: 0,l + C X by
o(t)
=
@l(t),
if t E [&,
B,
ift=O
i] for some n 2 1, and
It is easy to show that the following properties hold: (1) cy is continuous, (2) ifs 5 t, then (Y(S) c a(t), and (3) if s > 0, then o(s) is not contained in A. Thus cr([O, 11) is an order arc, B = m([o, 11) and, for every D E 4[0,11) - {B), D is . not contained in A. Therefore, B E Sb(A). (b) For each n 2 1, let C,, = B u B, u B,+l U Bn+2 U . . .. Then C,, is a subcontinuum of X, C,, is not contained in A, C,+l c C, and C., -+ B. By (a) we obtain that B E Sb(A).
69. SEMI-BOUNDARIES
335
(c) Let d be denote a metric for X. For each R. > 1, there exists a removing map cr, for B, in X. Choose t, > 0 such that Hd(crY,(t,), B,) < i. Then a,(tn) is a subcontinuum of X, a,,&) is not contained in A, an(tn) n B # 0 for each n, and crn(tn) + B. Applying (b), we obtain that B E Sb(A). n 69.4 Theorem [14, Theorem 1.4). Let X be a continuum. If A E C(X) - {X} and B E Sb(A), then there exists a minimal element (with respect to the inclusion) C E Sb(A) such that C c B. Proof. By the Brouwer Reduction Theorem (see[18, p. 611or Exercise 80.19) it is enough to show that the intersection of a countable nest of elements in Sb(A) n C(B) is an element in Sb(A) II C(B). Since this follows from Theorem 69.3 (c), Theorem 69.4 is true. n
Given an element A E C(X) - {X}, minimal elements in Sb(A).
we denote by m(A) the set of
69.5 Theorem [14, Theorem 4.21. Let X be a continuum. Let n > 1. Then X contains n-ods if and only if there exists E E C(X) such that m(E) has at least n elements. Proof. NECESSITY. Suppose that n 2 2. Let A, B E C(X) be such that B - A has at least n components. Let D1, . . . , D, be components of B - A. By Exercise 12.15 applied to the continuum B, clx (Di) n A # 8 and AU Di E C(X) for each i. Fix an open subset U of X such that A C U and Di - clx (U) is nonempty for every i. Let E be the component of clx(U) such that A c E. Given i E (1,. . , n}, E n clx(Di) # 0. We will show that there is an element C E Sb(E) such that C c clx (Di). In the case that E C clx (Di), since E E Sb(E), we can set C = E. In the case that E - clx(Di) # 8, let C be a component of E nclx(Di). Then we can apply Theorem 69.2 (f) to obtain that C E Sb(E, E U clx(Di)) c Sb(E). By Theorem 69.4, there exists a minimal element Mi of Sb(E) such that Afi C C C cZx(Di). We will show that Mi is not contained in A. Suppose to the contrary that Mi c A. Let cr : [0, l] + C(X) be a removing map for Mi in X. Since o(0) c A c U, there exists t > 0 such that a(t) c U. Then A U a(t) is a continuum contained in U. Thus A U a(t) c E. This contradicts the choice of (Y. Hence Mi - A # 0. Therefore Mi n Di # 8. Thus A&i,. . . , M, are pairwise different. This completes the proof of the necessity.
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X. GENERAL
PROPERTIES
OF HYPERSPACES
SUFFICIENCY. Let E E C(X) - {X} be such that Sb(E) has at least n minimal elements El,. . . ,E,. By Exercise 14.20, we may assume that An B has a finite number of components for every A, B E C(X). For each i E (1,. . .) n}, choose a removing map q for Ei in X. Given i E { 1,. . . , n}, for every j # i, Ej is not contained in Ei. Since ~(0) = Ei, there exists t, > 0 such that Ej is not contained in cri(tz) for every j # i. Given i # j, we will show that there exists s > 0 such that oi (3) n q(Q) c E. If EifIaj(tj) = 8, then in fact there is as > 0 such that oi(s)floj(tj) = 0. Thus we may assume that Ei II czj(tj) # 0. Let Ci, . . . , C, be the components of Ei n aj (tJ). In order to prove the existence of s, suppose to the contrary, that there is no such s. Let k be a positive integer. Let C be the union of the components of oi( i) n cq (tj) which do not intersect Ei. Then C is a compact set disjoint from Ei. Thus there exists zk E (0, i), such that (Yi(zk) n C = 0. By the assumption, we can choosea point z in oi(i&) ncrj (tj) - E. By construction of C, the component Dk of c~i(i) flczj(tj) which contains 2 meets Ei. Since 0# DknE, C DknEincrj(tj), thereexistslk E {l,...,r}suchthat Dk fl Cl, # 0. Since Cl, c Ei n ai C ai rl ajftj), we conclude that cl, C Da. Let 10 E { 1,. . . , r} be such that lo = lk for infinitely many k. Suppose that kl < kg < . . . are such that 10 = lg,,, for each m. Then the following properties are true for each m 2 2: Dk, E C(X), Dk,,, is not contained in E, Cl, C Dk, and Dk, is &) rl oj (tj). a component of oi (&) n crj (tj) which is contained in CY~( Thus Dk_ C .&,,-,. Let D = Dkl nDk, fI.. . = lim &. Since Dk, C oi( &) and CY~(&) + Eiy D C Ei C E. By Theorem 69.3 (a), D E Sb(E). Since E.i is miGma1, D = Ei. This is a contradiction since D c oj(tj). This proves the existence of s. Therefore, given i E { 1, . . . , n}, we may choose si E (0, ti) such that oi(si)noj(t,)CEforeveryj#i. DefineB=EUcq(sl)U...lJcr,(s,). Then B is an n-od. n
Exercises 69.6 Exercise. Reformulate Theorems 69.2 (a), (b), (c), (d), 69.3,69.4 and 69.5 for spacesof the form Sb(A, B) instead of Sb(A). 69.7 Exercise.
Check the examples before Theorem 69.2.
70. CELLS IN HYPERSPACES
337
Prove Theorem 69.2.
69.8 Exercise.
69.9 Exercise ([14, p. 671). A continuum is hereditarily indecomposable if and only if Sb(A) = {A} for every A E C(X) - {X}.
Let X be a continuum. Let A E C(X) - {X} (a) if A is indecomposable, then C(X) - {A} is arcwise connected if and only if every composant Ii’ of A contains an element of Sb(A), and 69.10
Exercise.
(b) if A is decomposable, then C’(X) - {A) is arcwise connected if and only if Sb(A) # {A}. 69.11 Exercise [14, Theorem 5.11. For a continuum X, the following statements are equivalent: (a) X is locally connected, (b) if A E C(X) - {X} and p E Bd,~(~4), then {p} E Sb(A), and Cc) W-4 = Wqx,(W)) for every A E C(X) - {X}.
70.
Cells in Hyperspaces
In studying the structure of hyperspaces, it is useful to know if they contain cells or Hilbert cubes. In this direction Mazurkiewicz in [21] proved for every continuum X, 2dYalways contains Hilbert cubes (see Theorem 14.12). In [24] it was given a sufficient condition in order that C(X) contains Hilbert cubes. This result was extended in [13] where it was proved that a necessary and sufficient condition in order that C(X) contains Hilbert cubes is that X contains oo-ods (see Exercise 70.3). From Exercise 14.19 and Theorem 18.8 it follows that C(X) contains a 2-cell if and only if X contains a decomposable subcontinuum. Rogers in [26] proved that if X contains nods, then C(X) contains n-cells (seeExercise 14.18). The latter suggested the question asked in [25, Question 1.1471, namely if the fact that C(X) contains n-cells implies that X contains n-ods (the question for n = 3 was proposed by Ball). A positive answer to this question was given in [13]. The purpose of this section is to present the proof that the existence of n-cells in C(X) implies the existence of n-ods in X. 70.1
Theorem
contains n-cells
[13, Theorem 1.91. Let X be a continuum. Then C(X) if and only if X contains n-ods (n 2 2).
Proof. Let d denote a metric for X. The sufficiency was asked to be proved in Exercise 14.18. In order to show the necessity we assumethat X does not contain n-ods and then we will see that C(X) does not contain n -cells. This proof is based in the classical theorem of Dimension Theory
338
X. GENERAL
PROPERTIES
OF HYPERSPACES
which says that: An n-cell can not be separated by any of its (n - 2)IV.41). Then we need t,o dimensional subsets ([12, Corollary to Theorem estimate the dimension of some subspacesof G(X). This will be done by proving a seriesof auxiliary results. We need the following conventions. The letters A and B denote subcontinua of X such that A $ B and C denotes a subcontinuum of Sb(A, B) which is upperly closed with respect to A. That is, C is a subcontinuum of Sb(A, B) with the property that if D E C, E E C(X) and D C E c A, then E E C. Given m 2 1, we say that C has property S(m) if it is not possible to construct m-ods with the elements of C. This means that if D, E E C and D C E then E - D has at most m - 1 components (for m = 1, this means that D = E).
(A) For each D E Sb(A, B), there exists E> 0 such that if E E Sb(A, B) and Hd(D, E) < E, then D n E # 0. Since we are assumingthat X doesnot contain n-ods, by Theorem 69.5, the set M = {M E C(X) : A4 is a minimal element of Sb(A, B)} is finite. Let Ml = {M E M : M is not contained in D}. Since MI is finite, the set C = {E E C(X) : E contains some element in Mi } is closed in C(X). Since D $ L, tl iere exists E > 0 such that BH~(E, D) does not intersect C. If E E Sb(A, B) and Hd(D, E) < 6, then E 4 C. By Theorem 69.4, there is a minimal element Al of Sb(A, B) such that M C E. Then A4 4 Ml and A4 c D. Therefore, En D # 0. (B) Sb(A, B) is closed. Let D E C(X) and let {Dk};i”=l be a sequence in Sb(A,B) such that Dk + D. Since each Dk is contained in A, D C A. From (A), we may assume that Dk n D # 0 for every k. By Theorem 69.3 (c), we conclude that D E Sb(A, B). (C)Let D E C(A) - {A}. Then B&(C(D) nC) = Sb(D, A) nC. Let E E Sb(D, A) n C. Then E E C,. E c D and there exists a map cr : [0, l] -+ C(A) such that a(O) = E and o(s) is not contained in D for each s > 0 (Theorem 69.2 (a)). Since C is upperly closed with respect to A, each a(i) is in C-C(D). Since o(i) + E, we conclude that E E Bdc(C(D)nC). Now take E E B&(C(D) nC). Then E is a subcontinuum of D, E E C and there exists a sequenceof elements { Ek}& of C such that Ek + E and Ek # C(D) for each k. By (A), since C c Sb(A, B), we may assumethat EI, n E # 8 for each k. By Theorem 69.3 (b), E E Sb(D,A). Therefore, E E Sb(D, A) n C. This completes the proof of (C). (D)Let G be a closed subset of X such that A rJJG. Suppose that the set C(G n A) = {D E C(X) : D c G n A} is nonempty. Then there is a finite number of components Ci, . . . , Ck of G n A such that
70. CELLS IN HYPERSPACES
B&(C(GnA)
nC) = (Sb(Ci,A) u...uSb(Ck,A))
339
nC.
By Theorem 69.5, there is only a finite number of minimal elements in Sb(A, B). Suppose that Ml,. . . , Mk are the minimal elements in Sb(A, B) contained in G n A. Let Gi be the component of G n A which contains ngi. If D E C(G n A) n C, then D c G n A and D E C. By Theorem 69.4, there exists a minimal element in Sb(A, B) which is contained in D. Then this minimal element is some Mi. Hence D is contained in some Ci. Thus D E (C(C,) u . . . u C(C,)) n C. This proves that C(G n A) fl C C (C(Ci) U . . . U C(Ck)) n C. The other inclusion is clear. If i # j, then either C(Ci)nC = C(Cj)-nC or (C(Ci)rlC)rl(C(Cj)nc) = 0. Hence B&(C(G n A) n C) = B&(C(C1) n C) u. .. U B&(C(Ck) n C). Applying (C) we obtain that Bdc(C(G n A) n C) = (Sb(Cr , A) U .. U Sb(Cn, A)) n C. (E)If C has property S(m) (m > 2) and D E C(A) - {A}, then C n Sb(D, A) is a subcontinuum of Sb(D, A) which is upperly closed with respect to D and C n Sb(D, A) has property S(m - 1). Since C is upperly closed with respect to A, Theorem 69.2 (b) implies that C n Sb(D,A) is upperly closed with respect to D. Then each order arc from an element in C n Sb(D,A) to D is contained in C n Sb(D,A). This implies that C n Sb(D, A) is arcwise connected and, by (B), Sb(D, A) is closed. Thus C n Sb(D, A) is a subcontinuum of Sb(A, B). Suppose that C n Sb(D, A) doesnot have property S(m- 1). Then there exist E, F E C n Sb(D, A) such that E c F and F - E has at least m - 1 components. Proceeding as in Exercise 12.20, there are m - 1 subcontinua E =G1 n...nG,-l, of F such that F =G1 u...uG,-~, Gl,..., G,-i Gi n Gj = E if i # j, and Gi - E is nonempty for each i. Fix points
pi E Gi - E. Since E E Sb(D,A), there exists a map cy : [0, l] + C(A) such that a(O) = E, (Y(S) C$a(t) if s < t and (Y(S) is not contained in D for each s>O. Lettc>Obesuchthat{pi,... ,pk} n a(to) = 0. Since X does not contains n-ods, @(to) n D has a finite number of components. Let C be the n D) - C is closed component of a(&,) n D such that E c C. Then (I and it does not intersect E. Thus there exists ti > 0 such that tr < to and a(tl) n (cr(to) n D - C) = 0. Then a(tl) n D c C. Let Fl = FUa(tl). Since C is upperly closed with respect to A, we have that the continua FI and C belong to C. And F; - C = (G1 - C) U. *. U (G,-1 - C) u (a(h) - C). Since pi E Gi - C and 0 # a(tl) - C, the sets G1-C,... ,G,+r-C, a(ti)-Carenonempty. Sinceclx(Gi-C)n(a(ti)C) c Dn(a(t,)-C) = 0 and clx(cr(tl)-C)n(Gi-C) c a(tl)n(D-C) = 0,
340
X.
GENERAL
PROPERTIES
OF HYPERSPACES
the sets Gi - C,. . , G,-1 - C, o(ti) - C are mutually separated. Hence Ei is a.n m-od in C. This contradicts the fact that C has property S(m) and completes the proof that C n Sb(D,A) has property S(m - 1). (F) If C has property S(m), then dim[C] 5 m - 1 (m 2 1). The proof of (F) will be done by induction. For m = 1. If D is an element of C, then D c A. Since C is upperly closed with respect to A, A E C. Since C has property S(l), A - D does not have components. Then A = D. This proves that C = (A}. Hence dim[C] = 0. NOW suppose that the claim in (F) is true for m - 1 and m 2 2. Then we are assuming that C has property S(m) and we need to show that dim[C] 5 m - 1. Take an element D E C and an open subset U of C(X) such that D E U. We will find an open subset, V of C such that D E V c U n C and dim[B&(V)] 5 m - 2. By (A), we may assumethat far each E E U n C, D n E # 0. Let Ui,.. . , Uk be open subsets of X such that D E (r/l, . . . , Uk) C U. Take an open subset U of X such that D C U C c1.x(U) C UI U . . . U uk. Given i E (1,. . . ,lc}, let Ui = {E E C(X) : E n Vi # 0} and let Gi = A - U,. We will show that dim[Bdc(Ui n C)] < m - 2. Since points in D n Vi belong to A - Gi, A g Gi. Since the set Si = C (L&nC) = {E t C : E c Gi)={E~C:EcGinA),inordertoshowthat dim[Bdc (Ui nc)] < m - 2, we only need to prove that dim[Bdc (Gi)] 5 m - 2. We may assumethat & is nonempty. Then we may apply (D) and we obtain a finite number of components Cl,. . . , Ck of G, n A such that Bdc(!&) = (Sb(Ci,A)U...U Sb(Ck,A))nC. By (E), C fl Sb(Ci,A) is subcontinuum of Sb(C,, A) with is upperly closed with respect to C,, and C n Sb(Ci, A) has property S(m- 1). Then by the induction hypothesis, dim[C fl Sb(Ci, A)] 5 771 - 2. Therefore ([la, Theorem III 2]), dim[Bdc(Gi)] 5 7n - 2. Let Cc be the component of U n A which contains D. Let C = clx (Co). Define
Then V is an open subset of C. ThenEcUnAand Given E E (U n U1,. . . , UnUk)flC,EEUnC. En D # 0. Thus E c Co. This proves that D E (U n U1,. . . , U n uk) rY C C(C) n C. Hence D E intc(C(C) n C). Therefore D E V. If C # A, by (C), Bdc(C(C) n C) = Sb(C, A) n C. By (E), we can apply the induction hypothesis to Sb(C,A) n C and we conclude that dim[Bdc(C(C) nC)] 5 m - 2. In the case that C = A, the set Bdc(C(C) n C) = Bdc(C) is empty. In any case dim[Bdc(C(C) n C)] 5 m - 2.
EXERCISES
341
Since B&(U) C mc(cnUl)u... u B&(C n&) u B&(C(C) n C), by [12, Theorem III 21,we have that dim[Bdc(V)] 5 m - 2. Finally, it is clear thatI/‘CUnC. This completes the proof of the properties of Y. Therefore dim[C] 5 m- 1. This completes the induction and the proof of (F). (G)C(X) does not contain n-cells. Supposeto the contrary that there exists an n-cell D contained in C(X). Take two elements D # E in ID. We may assumethat E is not contained in D. Fix a point p E E - D. Then there exists a neighborhood U of D in C(X) such that p 4 F for every F E U. Let f be an n-cell such that D E & and & c D n U. Let A = u{F E C(X) : F E E}. Then A E C(X) and p 4 A. Since X does not contain n-ods, by Exercise 70.5, 5 n - 2. Then Sb(A,X) has property S(n - 1). By (F), dim[Sb(A,X)] f is not contained in Sb(A,X). Fix an element G E E- Sb(A,X). Since dim[Sb(A, X)] I n - 2, Sb(A, X) nD does not separate ‘D ([12, Corollary to Theorem IV 41). Thus there exists a map a : [0, l] + (D- Sb(A, X)) such that o(O) = G and o(1) = E. Let to = max{t E [0, l] : a(t) c A}. Then to < 1, and it is easy to show that a(ts) E Sb(A, X). This is a contradiction which proves (G) and completes the proof of the theorem. H It is easy to prove (seeExercise 34.9) that if X contains an n-od (n > 2), then there is a Whitney level A for C(X) which contains an (n - 1)-cell. It is not known if the converse is true. 70.2 Question. Let X be a continuum. If there exists a Whitney level A for C(X) which contains an n-cell (n 2 2), then does X contain an (n + 1)-od?
Exercises 70.3 Exercise. Hilbert cubes.
If a continuum X contains co-ods, then C(X) contains
70.4 Exercise. Let X be a continuum. If dim[C(X)] is finite and A is an indecomposable subcontinuum of X, then C(X) - {A} is not arcwise connected. [Hint: Use Exercise 69.10.1 70.5 Exercise. If a continuum X does not contain n-ods (n 2 2), then Sb(A) has property S(n - 1) for every A E C(X) - {X}.
342
X. GENERAL 70.6 Exercise
PROPERTIES
OF HYPERSPACES
([14, Theorem 4.31). If a continuum X is atriodic, then
Sb(A) is a one-point set or an arc for every A E C(X) - {X). 70.7 Exercise. answer.
71.
Prove that for n = 1, Question 70.2 has a positive
Neighborhoods
of X in the Hyperspaces
Properties of local connectednessin hyperspaces of continua have been studied by several authors ([2], [3], [4], [5], [6], [7], [8], [9], [lo], [ll], [17] and [22]). From the point of view of local connectedness the element X has a privileged position. The hyperspaces C(X) and 2x are always locally arcwise connected at X (Corollary 15.5). Then it is natural to ask: when does X has a neighborhood L4in C(X) which is locally connected at each of its points? A complete answer to this question was given by Eberhart in [6, Theorem 1.71. 71.1 Theorem [6, Theorem 1.71. The continuum X has a neighborhood U in C(X) which is locally connected at each of its points if and only if there is a locally connected Whitney level for C(X) (Exercise 71.13). 71.2 Theorem [6, Theorem 2.11 (Exercise 71.14). Let X be a continuum. Any of the following conditions implies that X has a neighborhood U in C(X) which is locally connected at each one of its points: (a) there is a finite subset F of X such that all sufficiently large continua in X meet F, (b) any two sufficiently large continua in X have a point in common, and (c) each sufficiently large continuum in X contains a point at which X is connected im kleinen. For the hyperspace 2x the answer is very simple and it is given in the following theorem. 71.3 Theorem (Exercise 71.15). Let X be a continuum. Then X has a neighborhood U in 2x which is locally connected at each of its points if and only if X is locally connected. Eberhart ([6, Example 3.51) gave an example of a smooth dendroid for which conditions (a), (b) and (c) of Theorem 71.2 fail. An example with similar properties was also given in [23, Example 1.81. Recall that a continuum X is said to be locally contractible at a point p, provided that for every neighborhood U of p in X, there exist a neighborhood V of p, a point u E U and a map H : V x [0, l] -+ U such that H(w,O) = w and H(v, 1) = u for every u E V.
71. NEIGHBORHOODS
OF X IN THE HYPERSPACES
343
Answering a question by Dilks (119, Question ill]), Kato in ([16, Example 3.11) gave an example of a chainable continuum X such that C(X) and 2x are not locally contractible at X. Dilks’ question was also answered in [15], where an example of a continuum X was given such that C(X) and 2x are not locally contractible at any of their points. Kato also showed the following results. 71.4 Theorem [16, Proposition 2.11. Let X be a continuum. If 2AY is locally contractible at X, then C(X) is also locally contractible at X (Exercise 71.16).
Kato in [16, Example 3.21 gave an example of a dendroid X such that C(X) is locally contractible at X but 2x is not locally contractible at X. Thus the converse of Theorem 71.4 is not true. 71.5 Theorem 116,Proposition 2.31 (Exercise 71.16). Let X be a continuum. Let p be a Whitney map for Z = C(X) or 2x, then ‘l-l is locally contractible at X if and only if there exists 0 5 to < p(X) such that pL-‘([t,p(X)]) is contractible for any to < t < p(X).
Recently, Lopez has studied conditions for which X has a closed neighborhood U in the hyperspace 3c = C(X) or 2” such that U is homeomorphic to an n-cell (n 2 2) or to the Hilbert cube. He has given some combinatoric conditions on X which imply that X has a closed neighborhood in C(X) homeomorphic to an n-cell. For the particular case n = 2, he proved the following result. 71.6 Theorem [20]. The continuum X has a closed neigborhood homeomorphic to a 2-cell in C(X) if and only if C(X) has a Whitney level which is homeomorphic to either an arc or a circle.
Lopez also showed the following results. 71.7 Theorem [20]. Let X be a continuum. Then X has a closed neighborhood homeomorphic to the Hilbert cube in 2” if and only if X is locally connected (Exercise 71.17). 71.8 Theorem [20]. If X is a locally connected continuum, then S has a closed neighborhood homeomorphic to the Hilbert cube in C(X) if and only if X is not a finite graph.
In [23] Montejano-Peimbert and Puga-Espinosa studied someconditions on a smooth dendroid X in order that sets of the form {A E C(X) : p E A and p(A) 2 t} are homeomorphic to the cone over the set {A E C(X) :
344
X. GENERAL
PROPERTIES
OF HYPERSPACES
p E A and p(A) = t}, where p is a point of smoothness of X and 1-1is a Whitney map for C(X). Let X be a continuum. An element A E C(X) is said to be stable if, for every continuous function f : C(X) 4 C(X) sufficiently close to the identity map, A E f(C(X)). Thus A is unstable if and only if {A} is a Z-set in C(X). In [l], Curtis gave a characterization of the elements in C(X) which are stable. Applying this characterization to the element X in C(X), Curtis obtained the following theorem. 71.9 Theorem [l, p. 2591. The continuum X E C(X) only if X is a finite graph with no cut points.
is stable if and
Exercises 71.10 Exercise [6, Theorem 1.21. Let X be a continuum. If C(X) is connected im kleinen at an element A E C(X) and ~1is a Whitney map for is connected im kleinen at A. C(X), then p-‘&(A)) Let E > 0. By [Hint: Let d denote a metric for X. Let A = p-‘(p(A)). Exercise 66.8, there exists b > 0 such that if B E A and B C Nd(S, A), then Hd(A, B) < e. Let U be a closed connected neighborhood of A in C(X) such that U c BH~(~, A). Let V = An C(U{B : B E U}).] 71.11 Exercise. There are continua X with the property that X is the only element of local connectedness of C(X). 71.12 Exercise [6, Theorem 1.21. Let X be a continuum. If A = p-‘(t) is a locally connected Whitney level for C(X) with p(X) > t, then p-‘(t, p(X)] is locally connected. [Hint: Let d be a metric for X. Let B = p-l(t, p(X)], B E 13and c > 0. Given an element A E C(B) II A, there is a closed connected neighborhood CA of A in A such that if DA = ~(0 : D E CA}, then I < p(B) and Hd(A, DA) < 5. It is easy to prove that there exists 6 > 0 such that 6 < E, and if E E BH~(~, B), then there exist F E C(E) n A and A E C(B) n A such that F E CA. Thus B U DA U E E C(X) and Hd(B U DA U E, B) < E. Then use order arcs to connect E and B in BH~(E, B).] 71.13 Exercise. Prove Theorem 71.1. [Hint: Use Exercises 71.10 and 71.12.1 71.14 Exercise.
Prove Theorem 71.2 (Compare with Exercise 28.11).
71.15 Exercise.
Prove Theorem 71.3.
REFERENCES
345
[Hint: Let d be a metric for X. For the sufficiency use Exercise 15.10. For the necessity, let p E X, let e > 0. By hypothesis there exists 6 > 0 such that 6 < 6 and 2x is locally connected at A = {p}U(X -Bd(b,p)). Let U be an open connected subset of 2x such that A E Z.4and 24 C BH,, ( $, A). Let U = U{B : B E U} and let C be the component of U which contains p. Then C = Bd(i,p) n U.] 71.16 Exercise.
Prove Theorems
71.4 and 71.5
71.17 Exercise. Prove Theorem 71.7. [Hint: Use Theorem 71.3 and Theorem 11.3).]
References 1.
D. W. Curtis, Stable points in hyperspaces of Peano continua, Topology Proc., 10 (1985), 259-276.
2.
C. Dorsett, Local connectedness, connectedness im kleinen, and other properties of hyperspaces of h& spaces, Mat. Vesnik, 16 (1979), 113123.
3.
C. Dorsett, Local connectedness Palermo, 31 (1982), 137-144.
4.
C. Dorsett, Connectedness 11 (1982), 31-36.
5.
C. Dorsett, Connectivity properties in hyperspaces Fund. Math., 121 (1984), 189-197.
6.
C. Eberhart, Continua with locally connected Whitney ton J. Math., 4 (1978), 165-173.
7.
J.T. Goodykoontz, Jr., Connectedness im kleinen and local connectedness in 2x and C(X), Pacific J. Math., 53 (1974), 387-397.
8.
J. T. Goodykoontz, Jr., More on connectedness im kleinen and local connectednes in C(X), Proc. Amer. Math. Sot., 65 (1977), 357-364.
9.
J. T. Goodykoontz, Jr., Local arcwise Houston J. Math., 4 (1978), 41-47.
10
K. Hur, J.R. Moon and C. J. Rhee, Connectedness im kleinen and local connectedness in C(X), Honam Math. J., 18 (1996), 113-124.
11 II.
K. Hur, J. R. Moon and C. J. Rhee, Connectedness im kleinen and components in C(X), Bull. Korean Math. Sot., 34 (1997), 225-231.
in hyperspaces,
Rend.
Circ.
Mat.
im kleinen in hyperspaces, Math. Chronicle,
connectedness
12. W. Hurewicz and H. Wallman, Dimension sity Press, ninth printing, 1974.
and product spaces, continua, Hous-
in 2x and C(X),
Theory, Princeton
Univer-
346
X. GENERAL
PROPERTIES
OF HYPERSPACES
13. A. Illanes, Cells and cubes in hyperspaces, Fund. Math., 130 (1988), 57-65. 14. A. Illanes, Semi-boundaries in hyperspaces,Topology Proc., 16 (lggl), 63-87. 15. A. Illanes, A continuum having its hyperspacesnot locally contractible at the top, Proc. Amer. Math. Sot., 111 (1991), 1177-1182. 16. H. Kato, On local contractibility at X in hyperspacesC(X) and zx, Houston J. Math., 15 (1989), 363-369. 17. H. Katsuura, Concerning generalized connectedness im kleinen of a hyperspace, Houston J. Math., 14 (1988), 227-233. 18. J. L. Kelley, General Topology, Springer Verlag, Graduate Text in Mathematics, Vol. 27, New York, Heidelberg, Berlin. 19. W. Lewis, Continuum theory problems, Topology Proc., 8 (1983). 361394. 20. S. L6pez, Hyperspaces which are locally euclidean at the top, preprint. 21. S. Mazurkiewicz, Sur le type de dimension de l’hyperespace d’un continu, (French) C. R. Sot. SC. Varsovie, 24 (1931), 191-192. 22. A. K. Misra, C-supersets, piecewise order-arcs and local arcwise connectednessin hyperspaces,Questions Answers Gen. Topology, 8 (1990), 467-485. 23. L. Montejano-Peimbert and I. Puga-Espinosa, Shore points in dendroids and conical pointed hyperspaces,Topology Appl., 46 (1992), 4154. 24. S.B. Nadler, Jr., Locating cones and Hilbert cubesin hyperspaces,Fund. Math., 79 (1973), 233-250. 25. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 26. J. T. Rogers, Jr., Dimension of hyperspaces,Bull. Acad. Polon. Sci., S&. Sci. Math. Astronom. Phys., 20 (1972), 177-179.
XI.
Dimension 72.
Previous
of C(X)
Results about Dimension Hyperspaces
of
One of the most interesting problems in hyperspace theory is to determine the dimension of C(X). A complete discusion of what was known up to 1978 appeared in Chapter II of [21]. Next, we list the main results related to this topic. 72.1 Theorem [18] (Corollary 14.13). For every continuum X, 2x contains a homeomorphic copy of the Hilbert cube and then dim[ 2x] = 00. 72.2 Theorem [9, Theorems 5.4 and 5.51, [21, Theorem 1.109 and Remark 1.1101. If X is a locally connected continuum, then dim[C(X)] < 00 if and only if X is a finite graph (see Exercise 72.9). In the case of finite graphs a very simple combinatorial formula, involving the order of the vertices of X can be obtained for calculating dim[C(X)], see [9,5.4 and 5.51 and Exercise 72.9. The first part of the following theorem was proved in Theorem 22.18. 72.3 Theorem [5, Theorem 11. If X is a continuum, then dim[C(X)] > 2. Hence, [5, Corollary l], if X is a chainable or circle-like continuum, dim[C(X)] = 2. 72.4 Theorem [5, Theorem 21, [21, Theorem 2.21. If X is an hereditarily indecomposable continuum, then dim[C(X)] = 2 or dim[C(X)] = oo. The following 72.6.
theorem is a consequence of Theorem 72.4 and Theorem
347
348
XI. DIMENSION OF C(X)
72.5 Theorem such that dim[X]
[24, Theorem 51, [21, Theorem 2.31. If X is a continuum 2 3, then dim[C(X)] = 00.
Then it is natural to ask if, for every continuum X, dim[X] = 2, implies that dim[C(X)J = 00 This problem has been one of the most important questions in hyperspace theory for many years. It appeared in [20, 2.21, [21, Question 2.41, [17, Question 1061, [26, Question 361 and [8]. Some authors have offered partial solutions of this problem. They have proved that the implication is true when X is a continuum satisfying additional conditions such as: (a) X is arcwise
connected ([24, Corollary
31);
(b) X contains a subcontinuum which is homeomorphic to the product of two nondegenerate continua ([24, Corollary 21, Exercise 14.21); (c) dim[X]
2 3 ([24, Theorem 51);
(d) X is hereditarily
indecomposable
(IS, Theorem 21);
(e) the rank of the first tech cohomology group of X is finite ([8, Theorem 2.21). The problem has been finally solved by Levin and Sternfeld in [15]. Their solution depends on a clever way of modifying previously developed methods. One of the keys of their proof (Lemma 73.8 below) has its roots in a result which appeared on Kelley’s paper 19, Theorem 7.81 in 1942. As in many other results in hyperspaces, a fundamental tool of the proof is the following result by Bing. 72.6 Theorem [l, Theorem 51. Any (n + 1)-dimensional continuum contains an n-dimensional hereditarily indecomposable continuum. The main object of this chapter is to reproduce Levin’s and Sternfeld’s proof. In order to make the proof as self-contained as possible, we will prove some elementary results of dimension theory. At the end of the chapter we include a discussion of dim[C(X)] when dim[X] = 1.
Exercises From Exercise 72.7 to 72.10 we assume that X is a finite graph. We adopt the conventions and the notation of section 65. In particular, we assume that X is different from a circle and all the vertices of X are end points or ramification points. 72.7 Exercise [2, 5.11. For X as above, C(X) subgraph of X} ( compare with Exercise 65.33).
= U{Dls
: S is a fine
73. DIMENSION
OF C(X)
FOR ~-DIMENSIONAL
CONTINUA
X
72.8 Exercise [2, 7.11. For X as above, if S,T are fine subgraphs X such that S 5 T, then dim[%JZs] < dim[MT].
349 of
72.9 Exercise [2, 7.21. Let X as above. Let s be the number of segments of X, let ‘u be the number of vertices of X and let e be the number of end points of X. Then dim[C(X)] = 2(1 + s - V) + e. [Hint: Use Exercises 65.32, 72.7 and 72.8.1 72.10 Exercise. Let X as above. Then the dimension of C(X) determined by the maximum integer n such that X contains an n-od.
73.
Dimension
of C(X)
for 2-Dimensional
Continua
is
X
We start with a definition. 73.1 Definition. Define
Let Y be a compact metric
di (A) = sup{diameter The following
space and let A E 2’.
(C) : C .is a component
lemma shows some properties
of A}.
of di .
73.2 Lemma. Let Y be a compact metric space and let A E 2=. Then: (a) in the definition of di (A) the word “sup” can be changed by “max” , (b) if dl(A) < c, then there exist pairwise disjoint closed subsets Al,...,A,,ofYsuchthatA=A1U...UA,anddiameter(A,)<~ foreveryi=l,...,n, (c) if di (A) < c and A c U, where U is an open subset of Y, then there exists an open subset V of Y such that A c V c cly(V) c U and dl(ClY(V))
<
E.
Proof. (a) It is a consequence of the compactness of C(A) (Exercise 73.15). (b) We may assume that A is not connected. For each p E A, let C, be the component of A such that p E C,. Since diameter (C,) < 6, there exists an open subset UP of Y such that CP c VP, diameter (cly (UP)) < E and A g U,,. Apply the Cut Fence Wire Theorem (Theorem 12.9) to the space A, to the component C, of A and to the closed subset A - Up of A. Then there exists an open and closed subset W, of A such that C, c W, c A f~ Up. Notice that diameter (W,) < E.
350
XI. DIMENSION
OF C(X)
By the compactness of A, A can be covered by a finite union of sets WI,... , W,, where each Wi is of the form W,, , for some i. Defining Al = W,, Ag = W, -WI,.. W+l), we obtain the .,A, = W, - (WI U-..U desired family of subsetsof A. (c) Take a finite collection of subsets Al, . . . , A, of A as in (b). Using the normality of the space Y, one can obtain open subsets Ui ,..., U,, of Y such that cZy(Ur),. , . , cly (U,) are pairwise disjoint and Ai C Vi C Cly(Ui) C U, for each i. Since each diameter (Ai) < E,we may assumethat diameter (dy (Ui)) < E for every i. Define V = U1 U . . . U U,. Then, for each component C of cZy(V), C c cly(Ui) for some i. Thus diameter (C) < E. This proves that dr (cly (V)) < 6 and completes the proof of the lemma. n
73.3 Definition. A continuous function between compact metric spaces f : Y + X is said to be light provided that f-‘(f(g)) is totally disconnected for each y E Y. 73.4 Theorem [15, Theorem 1.11. Let X be an n-dimensional continuum, with n < 00. Then there exists an n-dimensional hereditarily indecomposablecontinuum Y and a (not necessarily onto) light map f : Y -+ X. Proof. Since dim[X x [0, I]] = n + 1 ([7, lines 13-14, p. 341). By Theorem 72.6 there exists an n-dimensional hereditarily indecomposable subcontinuum Y of X x [0, 11. Let II : X x [0, l] + X be the projection and let f be the restriction of II to Y. In order to prove that f is light, let A be a subcontinuum of a set of the form f-‘(f(y)) = Y n [{f(y)} x [0, l]]. Then A is a subcontinuum of Y (so A is indecomposable) and A is contained in an arc, thus A is a one-point set. Hence f-‘(f(y)) is totally disconected. Therefore f is light. n The proof of the following lemma is left as Exercise 73.16. 73.5 Lemma. Let f : Y + X be a light map, where X, Y are compact metric spaces. Then, for each e > 0, there exists 6 > 0 such that if A c X and diameterx(A) < 6, then di [f-‘(A)] < 6. 73.6 Lemma. Let Y be a compact metric space. Then dim[Y] 2 2 if and only if there exist two disjoint closed subsetsFl and F2 of Y and there exists t > 0 such that every closed subset L of Y which separates FI from Fz in Y must satisfy c&(L) 2 c
73. DIMENSION
OF C(X)
FOR ~-DIMENSIONAL
CONTINUA
X
351
Proof. SUFFICIENCY. Suppose that dim[Y] 5 1. Then Y has a basis of neighbohoods f? such that, for each element U of B, dim[B&(U)] 5 0. Take two disjoint closed subsets Fl and FJ of Y. For proving the sufficiency, it is enough to show that there exists a closed subset L of I’ such that L separates Fl from F2 in Y and dl(L) = 0. Since Fl is compact, there exists a finite number of elements U1, . . . , U, of t3 such that Fl c U1 u ... U U,, and cly(U1 U .. . U Un) Cl FZ = 8. Define L = Bdy (VI U . . u U,,). Then L separates Fl from F2 in Y and L c Bd~(U~)u...uBdy(U,). By [7, Theorem111 21dim[Bdy(U1)U...U Bdy (Un)] 5 0. We conclude that dl (L) = 0. NECESSITY. Suppose, to the contrary, that there are no such Fl, F2 and E. We will prove then that dim[Y] 5 1 by showing that 2’ has a basis of neighborhoods I3 such that, for each element U of B, Bdy(U) = 0 or Bdy (U) is O-dimensional. We will need the following: (a) let A be a closed subset of Y, let U be an open subset of Y such that .4 c U and let c > 0. Then there exist open subsets V and W of 1’ such that A c V c cly (V) c W c U and if P is an open subset,of 1r and cly(V) c P c W, then dl(Bdy(P)) < E. We will prove (a). Let A, U and E be as in the hypothesis of (a). By the assumption at the beginning of the proof of the necessity, there exists a closed subset L which separates A from Y - U in Y and dl (L) < E. Let V and VI be disjoint, open subsetsof Y such that A C V, Y-U C VI and Y - L = V U V,. Since L c U - A, Lemma 73.2 (c) implies that there exists an open subset R of Y such that L C R C cZy(R) C U - A and dl(cl~(R)) < E. Define W = VU R. Observe that Bdy(V) c L C R, so cly(V) C W. Now, if P is an open subset of Y such that cl=(V) c P C W, then Bdy(P) c cly(W) - cly(V) c cZy(R). Thus dl(Bdy(P)) < 6. This completes the proof of (a). Now, we are ready to prove that dim[Y] 5 1. Let p be a point in Y and let 6 > 0. Let VI and WI be open subsets of Y obtained by applying (a) to the closed set {p}, the open set B(6,p) and the positive number 1. NOW, let V2 and W2 be open subsets of Y obtained by applying (a) to the closed set cZy(Vl), the open set WI and the positive number f. In general, for every n 2 2, let V, and U;, be open subsetsof Y obtained by applying (a) to the closed set cl~(V,,-~), the open set Wtlel and the positive number i. Notice that, for every n 2 2, cly(Vnwl) c V, c cly(Vn) c W, c Wtlvl.
352
XI. DIMENSION
OF C(X)
Define V = U{Vn : n = 1,2,. . .}. Clearly, p E V c B(d,p). Given n > 1, notice that V,, c IV,, for every m 2 1. Thus cly (l/,,) c < +, for each n = 1,2.. .. Hence dl(Bdy(V)) = V C W, and d~(Bdy(V))
0. Thus Bdy (V) is a compact totally disconnected subset of Y. Then is the empty set or it is O-dimensional (see Theorem 12.11). Therefore, dim[Y] 2 1. This completes the proof of the lemma. n
Bdy(V)
73.7 Remark. Let S > 0. Consider the cover U of the n-dimensional Euclidean space Rn formed by all the hypercubes of the form:
A = [&I, 6(al + l)] x . . . x [6an, s(a, + l)], where (~1,. . . , a,, run on the integers. Observe that each hypercube A only can intersect (and intersects) the hypercubes of U contained in the hypercube [s(ai - l), 6(ni + 2)] x . . . x [s(a, - l),S(a, + 2)], w h’rch is . formed of exactly 3” hypercubes of U. This proves that A intersects exactly 3n - 1 of the other elements of U. Thus, we have proved the following fact. 73.7.1. For each n = 1,2,. . ., there exists m 2 1 (m = 3n - 1) such that for each c > 0, there exists a closed cover U of R” such that each element of U intersects at most m of the other elements of 2.4,diameter (A) < c for every A E U and each compact subset of R” is contained in a finite number of elements of U. The number 3” - 1 is not the smallest value for m. For instance, if we put rectangles in R2 as bricks are usually put in a wall, instead of 32 - 1 we obtain that 6 suffices in the fact above. 73.8 Lemma [15, Lemma 1.31. Let X and Y be continua and let f : Y -+ X be a light map. Let K be a subset of C(Y) such that K is a decomposition of Y and no element of K: is a one-point set. Define g : Y + C(X) by g(y) = f(K), where y E Y and K is the element of K which contains y. Suppose that g is continuous and g(Y) is a finite-dimensional subset of C(X). Then, for each 6 > 0, there exists a closed subset 2 of Y such that dr(Z) < E and 2 intersects every element of Ic. Proof. Let d denote a metric for X. Let ‘l-l = {f(K) E C(X) : K E K}. Since R is finite-dimensional, there exists Ic 2 1 and there exists an embedding h : ?t + R”. Let m E { 1,2,. . .} be as in 73.7.1 applied to Rk. By the lightness of f we have that no element in ?l is a one-point set. Let X = min{diameter (H) : N E R}. Then X > 0.
73. DIMENSION OF C(X)
FOR ~-DIMENSIONAL
CONTINUA
X
353
Let 6 > 0 be as in Lemma 73.5; we may assume also that 6 < A. We will prove the following claim: (a) if H E ‘?-l, then it is impossible to cover H with m sets of diameter less than or equal to 26. In order to prove (a), suppose to the contrary that H c B1 U . . U B,, where diameter (Bi) 5 26, for each i. We may assume that each Bi is closed, nonempty and it is contained in H. Since H is connected, applying a typical argument of connectedness, we may assume also that, for each i < m, &+I n (B1 u . . u Bi) # 0. Then, an inductive argument shows that, for each i, diameter (B1 U . . . U &) < 2id. In particular, we obtain that diameter (H) 5 2m6. This contradicts the choice of X and completes the proof of (a). By the choice of m there exists a finite closed cover of h(z) such that each of the elements of the cover intersect at most m of the other elements in the cover and each one of them has small diameter. Using the uniform continuity of h-’ : h(X) -+ ‘H, we can obtain a similar cover for 31. Thus there exists a finite closed cover {VI,. . ,V,} of 7-1,such that, for each i, Vi # 0, diameter (Vi) < t and Vi intersects at most m of the other Vj'S.
Notice that if A,B E Vi, then Hd(A,B) < i. Thus B($,p) n B # 0, for every p E A. Finally, we will inductively define Z = Z1 U . . U Z,, where each Zi will be a closed subset of Y, dl (Zi) < E, Zi will intersect every element I< E K such that f(K) E ‘Di and Z1, . . . , Z, will be pairwise disjoint. If we are able to construct such a set Z, we will have: Z will be closed in Y, dl (Z) < E and, for each I< E Ic, f(K) E Vi, for some i, so Z n K # 0. In order to construct Z1, fix an element Al E VI and choose a point x1 E AI. Define Z1 = f-‘(clx(B(g,sl)))ng-‘(V1). Then Z1 is closed in Y. By the choice of 6, dl(Z1) < E. n f(K) # 0. Take a Let K E K: be such that f(K) E VI, then B(i,zl) point p E K such that f(p) E B( i, x1). Thus p E Z1 n K. This proves that Z1 satisfies the required properties. NOW, suppose that Z1, . . . , Zj-1 have been defined, where each Zi is of the form Zi = f-‘(cl*y((B( :, XL))) n 9-l (Vi), where z, is a point in Ai and Ai is an element of Vi. Let J = {i < j : Vi n Vj # 0}. Then J has at most m elements. Fix an element A, E Vj. By (a), it is possible to choose a point xj E ndx(B($,xc,)) = 0, A, - (‘J{clx(B(h, xi)) : i E J}). Then cl.y(B($,zj)) for every i E J.
354
XI.
DIMENSION OF C(X)
Define Zj = f-‘(clx(B($, x3))) n g-‘(ID,). Clearly, Zj is closed in 1’ and dl(Zj) < E. Let K E K be such that f(K) E Vj. Then B(i,zj) I-Jf(K) # 0. This implies that Zj n K # 8. Finally, if i < j and there exists a point y E Zi nzj, then g(y) E z)i nDj . Thus Vi nDj # 8, SOi E J. Hence, cEx(B($,z:j)) ndx(B(g,zi)) = 0. Thisisacontradiction,sinceyEf-l(~l,~(B(~,z~)))nf-‘(clx(B(~,si))). HenceZinZj=(bifi<j. This completes the proof of the existence of Z. n 73.9 Theorem [15, Theorem 2.11. If X is a continuum such that dim[X] = 2, then dim[C(X)] = 03. Proof. Suppose that dim[C(X)] < 03. By Theorem 73.4, there exist a 2-dimensional hereditarily indecomposable continuum Y and a light map f : Y -+ X. By Lemma 73.6, there exist two disjoint closed subsets F,, Fz of Y and there exists E > 0 such that every closed subset L of Y which separates FI from FJ in ‘I’ must satisfy dl (L) 2 E. Let 1-1: C(1-) -+ R’ be a Whitney map. From Lemma 17.3, it follows that there exists t > 0 such that, for each A E p-‘(t), diameter (A) < E. Since Fl and F2 are disjoint, the number T = min{dy(p, q) : p E Fl and q E Fz} is strictly positive. Let K: = p-‘(t). Since Y is hereditarily indecomposable, K is a continuous decomposition of Y ([9, proof of 8.51, Exercise 73.20). Then f(K) = {f(K) E C(X) : K E K} is a finite-dimensional closed subset of C(X). By Lemma 73.8 there exists a closed subset 2 of Y such that d*(Z) < T and 2 intersects every element of K. By Lemma 73.2 (b), there exist pairwise disjoint closed subsets Z1, . . , Z,, of Y such that Z = Z1 U . . . U Z, and diameter (Zi) < T for every i = l,...,n. Define G1 = Fl U (u{ Zi : Zi n Fl # 0)) and G:! = FZ U (U{ & : Zi n Fl = 01). Since diameter (Zi) < T, for every i, we see that G1 and G2 are disjoint closed subsets of Y. By normality of Y, there exists an open subset V of Y such that G1 c V c dy (V) c Y - G2. Then L = Bdy (V) is a closed subset of Y which separates G1 and Gz. By the choice of F, and F2, dl (L) 2 E. Thus L has a component 11/1 such that diameter (M) 2 E.
73. DIMENSION
OF C(X)
FOR ~-DIMENSIONAL
CONTINUA
X
355
Since K is a decomposition of Y, there exists an element K E Ic such indecomposable, K C M or that K fl A4 # 0. Since Y is hereditarily M C K. But diameter (K) < E 5 diameter (M), so K c M. Since 2 n L = 0, 2 n K = 0. This contradicts the choice of 2. This contradiction proves that dim[C(X)] = oo and completes the proof of the theorem. W Levin and Sternfeld have also done several improvements of the result above. Next, we give the result in which they show how the infinitedimensionality can be detected in Whitney levels. Notice that only a bit of extra work is necessary. 73.10 Theorem [15, Theorem. 2.21. Let X be a 2-dimensional continuum and let p : C(X) -+ R’ be a Whitney map. Then for all sufficiently small t, dim[p-l(t)] = 03. Proof. Let dy denote a metric for Y. By Theorem 73.4, there exists a 2-dimensional hereditarily indecomposable continuum Y and a light map f : Y + x. By Lemma 73.6, there exist two disjoint closed subsetsPi, F2 of Y and there exists E > 0 such that every closed subset L of Y which separates Fl from F2 in Y must satisfy dl(L) > E. Let T = min{dy(p, q) : p E FI and q E Fz}; then r > 0. By Lemma 73.5, it is possible to find 6 > 0 such that if A c X and diameter-y (A) < 6; then dl [f-’ (A)] < E. Since f is light, p(f(Y)) > 0. From Lemma 17.3, there exists to > 0 such that to < p(f(Y)) and, for each 0 < t < to and for each A E p-’ (t), diameter (A) < 6. Let 0 < t < to. We will show that dim[p-l(t)] = 00. Suppose, to the contrary, that dim[p-l(t)] < 00. Notice that Kc is the inverse Let Ke = {A E C(Y) : f(A) E p-‘(t)}. under the map A + f(A) from C(Y) into C(X). Thus Its image of p-‘(t) is closed in C(Y). For each point u E Y, let cr = {A E C(Y) : y E A}. It is easy to show that a is closed and connected. Since Y is hereditarily indecomposable, if A,B E Q, then A c B or B c A. Then Q is an order arc in C(Y). Since ~(f({~l)) = 0 and P(W)) > 4 Q rl Ice # 0. Then there exists the largest element KO E o n Ke. If L E Ku and KO c L then L E cy, so KO = L. This proves that Ko is a maximal (with respect to inclusion) element in Ke. Define Ic = {K E X0 : K is a maximal element of Ku}. Since Y is hereditarily indecomposable, if, K, L E K: and K n L # 8, then K c L or
356
XI. DIMENSION
OF C(X)
L C K, by the maximality of the elements of K, we have K = L. Thus, by the paragraph above, K: is a decomposition of Y. Given K E K, p(f(K)) = t > 0, so K is not a one-point set. Define 7-1 = {f(K) E p-‘(t) : K E K}. Then 3c is finite-dimensional. Define g : Y + p-‘(t) by g(y) = f(K), where K is the element of K: which contains y. In order to apply Lemma 73.8, we need to prove that g is continuous. To this end, let {yn}rzl be a sequence in Y and let y E Y be such that Y,~ + y. For each n = 1,2, _. ., let K, E K be such that yta E K, and let K E K be such that y E K. We may assume that K, + Ku for some KO E C(Y). Then y E Kc and f (K,) -+ f (l(o). Thus f(Ko) E p-‘(t). By the choice of K, KO c K. Then f(Ko) c f(K) and f(Ko), f(K) E p-‘(t). This implies that f(Ko) = f(K). Hence dyn) = fWn) + f(Ko) = f(K) = s(y). Applying Lemma 73.8 to the positive number r, we obtain that there exists a closed subset 2 of Y such that dr(Z) < r and 2 intersects every element of K. By Lemma 73.2 (b), there exist pairwise disjoint closed subsets 2, of Y such that 2 = Zr U . . . U 2, and diameter (Zi) < r for Zl,..., every i = 1,. . . , 72. From now on, the proof of the theorem is analogous to the proof of Theorem 73.9. Define Gi = FiU(U{& : ZinFr # 0)) and Gz = FzU(U{Zi : ZinFl = 01). Since diameter (Zi) < T, for every i, we have Gr and Gs are disjoint closed subsets of Y. By normality of Y, there exists an open subset 1’ of Y such that Gr c V c cly (V) c Y - Gs. Then L = Bdy (V) is a closed subset of Y which separates Gr and Gz. By the choice of Fl and F2, dl(L) >_ c. Thus L has a component M such that diameter (M) 2 c. Since ic is a decomposition of Y, there exists an element K E K such K C M or that K n A4 # 0. Since Y is hereditarily indecomposable, M c K. But diameter (K) < E 5 diameter (M), so K C M. Since 2 n L = 0, then 2 n K = 0. This contradicts the choice of 2. This contradiction proves that dirn[p-l (t)] = 00 and completes the proof of the theorem. n 73.11 Corollary (Exercise 73.17, compare with [21, Corollary 1.861). Let X be a continuum such that 2 5 dim[X] < 00. Let ~1be a Whitney map for C(X). Then there exists to > 0 such that dim[p-l(t)] = 00 whenever 0 < t < to. The following question is still open.
EXERCISES
357
73.12 Question [21, Question 1.871 (compare with Question Does Corollary 73.11 remain valid if the assumption that dim[X] is deleted?
45.3). < 00
Looking for another improvement of Theorem 73.9, Levin and Sternfeld have found that actually the I-dimensional subcontinua of X are responsible for the infinite dimensionality of C(X). They proved the following result. 73.13 Theorem [14, Theorem 2.11. Let X be a 2-dimensional cont,inuum and let n be a positive integer. Then X contains a l-dimensional continuum T,, with dim[C(T,)] > n. Although the general idea of the proof of Theorem 73.13 has some similitudes with the idea of the proof of Theorem 73.10, it is more complicated and we will not include it here. Answering Question 1.1 of [14], very recently Levin has extended Theorem 73.13 as follows. 73.14 Theorem [13, Theorem 1.11. Let X be a 2-dimensional continuum, then X contains a l-dimensional subcontinuum T with dim[C(T)] = co.
Exercises 73.15 Exercise.
Prove Lemma 73.2 (a).
73.16
Prove Lemma 73.5.
Exercise.
73.17 Exercise. Prove Corollary [Hint: Use Theorem 72.6.1
73.11.
73.18 Exercise. If X is an arcwise connected continuum and X contains a nondegenerate indecomposable subcontinuum, then X contains an co-ad. [Hint: Indecomposable continua have infinitely many composants and each of them is dense ([22, Theorem 11.15 and Exercise 5.20 (a)]) 73.19 Exercise. Prove Theorem 73.14 for the particular wise connected continua. [Hint: Use Theorem 72.6 and Exercise 73.18.1
case of arc-
73.20 Exercise. If X is an hereditarily indecomposable continuum, then for each Whitney level A for C(X), A is a continuous decomposition of X. Let f : X + A be the function f(p) = (the unique element A E A such that p E A). Then f is continuous and open.
358
74.
XI. DIMENSION
Dimension
of C(X)
OF C(X)
for l-Dimensional
Continua
X
After Levin’s and Sternfeld’s Theorem (Theorem 73.9), for a continuum X, dim[C(X)] is undetermined only for the case dim[X] = 1. Next, we will discuss some results and questions for this case. In some l-dimensional continua, such as finite graphs, the dimension of C(X) is determined by the maximum integer n such that X contains an n-od (Exercise 72.10). Continua without, n-ods (n 1 2) are exactly the hereditarily indecomposable ones (Exercise 14.19). Theorem 73.14 improved the following previous result by Lewis. 74.1 Theorem [16, p. 2943. There is a l-dimensional decomposable continuum X such that dim[C(X)] = 03.
hereditarily
in-
Lewis’ example has infinitely-generated cohomology groups, and he asked if a l-dimensional continuum with finitely-generated cohomology groups can have infinite-dimensional hyperspace. This question has been answered by Rogers who showed the following result. 74.2 Theorem [25, Theorem 11. If X is a l-dimensional, hereditarily indecomposable continuum and H’(X) (the first Tech cohomology group with integer coefficients of the space X) has finite rank, then dim[C(X)] = 2. Rogers’ result has been generalized by Grispolakis
and Tymchatyn.
74.3 Theorem [6, Theorem 3.41. If X is an atriodic that a’(X) has finite rank, then dim[C(X)] = 2.
continuum
such
74.4 Questions. Let X be a continuum. If n > 2 is an integer such that X has n-ods but X has no (n + 1)-ods and k’(X) has finite rank, then dim[C(X)] < co? Is dim[C(X)] = n? On the other hand, Oversteegen lowing result. 74.5 Theorem [23, Corollary uum, then C(X) is 2-dimensional.
and Tymchatyn
have obtained
2.51. If X is an atriodic
We finish this chapter with two questions
tree-like
the fol-
contin-
by Rogers.
74.6 Questions [17, Question 1071 (this question also appeared in [6, p. 5621). If dim[X] = 1 and X is a planar and atriodic continuum, is dim[C(X)] = 2? Is C(X) embeddable in R3?
REFERENCES
359
Krasinkiewicz in [lo] (see [21, Theorem 2.81) has shown that if X is an hereditarily indecomposable planar continuum, then C(X) is embeddable in R4. Hence, by Theorem 72.4, dim[C(X)] = 2. Tymchatyn ([27]) has improved this theorem by proving that, under the same hypothesis, C(X) is embeddable in R3 and dim[C(X)] = 2. Therefore, the answer to Question 74.6 is yes if X is hereditarily indecomposable. In the case that X is an atriodic locally connected continuum, we have that, X is an arc or it is a simple closed curve (Exercises 31.11 and 31.12). Then C(X) is embeddable in R2 and dim[C(X)] = 2. Thus the answer to Question 74.6 is also yes for locally connected continua. 74.7 Question [17, Question 1081. If dim[X] = 1 and X is an hereditarily decomposable and atriodic continuum, is dim[C(X)] = 27
References 1.
R. H. Bing, Higher-dimensional hereditarily indecomposablecontinua, Trans. Amer. Math. Sot., 71 (1951), 267-273.
2.
R. Duda, On the hyperspace of subcontinua of a finite graph, I, Fund. Math., 62 (1968), 265-286.
3.
R. Duda, On the hyperspace of subcontinua of a finite graph, II, Fund. Math., 63 (1968), 225-255.
4.
R. Duda, Correction to the paper: “On the hyperspace of subcontinua of a finite graph, I”, Fund. Math., 69 (1970), 207-211.
5.
C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces,Bull. Acad. Polon. Sci., SQ. Sci. Math. Astronom. Phys., 19 (1971), 1027-1034. J. Grispolakis and E. D. Tymchatyn, On a characterization of W-sets and the dimension of hyperspaces, Proc. Amer. Math. Sot., 100 (1987)) 557-563.
6.
7. 8. 9.
W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, ninth printing, 1974. H. Kato, The dimension of hyperspacesof certain 2-dimensional continua, Topology Appl., 28 (1988), 83-87. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot., 52 (1942), 22-36.
10. J. Krasinkiewicz, On the hyperspaces of certain plane continua, Bull. Acad. Polon. Sci., %r. Sci. Math. Astronom. Phys., 23 (1975), 981-983.
360
XI. DIMENSION
OF C(X)
11. P. Krupski, The hyperspaces of subcontinua of the pseudo-arc and of the solenoids of pseudo-arcs are Cantor manifolds, Proc. Amer. Math. SOL, 112 (1991), 209-210. 12. M. Levin, Hyperspaces and open monotone maps of hereditarily indecomposable continua, Proc. Amer. illath. Sot., 125 (1997), 603-609. 13. M. Levin, Certain finite hyperspaces, preprint.
dimensional
14. M. Levin and Y. Sternfeld, Hyperspaces Fund. Math., 150 (1996), 17-24.
maps and their application of two dimensional
15. M. Levin and Y. Sternfeld, The space of subcontinua corltinu?Lm is infinite dimensional, Proc. Amer. (1997), 2771-2775. 16. W. Lewis, Dimensions of hyperspaces continua, Proc. Geometric Topology publishers, Warzawa, 1980, 293-297. 17. W. Lewis, 394.
Continuum
to
continua,
a 2-dimensional Math. Sot., 125
of
of hereditarily indecomposable Conf., PWN - Polish Scientific
theory problems, Topology Proc., 8 (1983), 361-
18. S. Mazurkiewicz, Sur l’ensemble des continus pe’aniens, (French) Math., 17 (1931), 273-274.
Fund.
19. S. Mazurkiewicz, Sur l’hyperespace Math., 18 (1932), 171-177.
Fund.
d’un continu,
(French)
20. S. B. Nadler, Jr., Some problems concerning hyperspaces, Topology Conference (V. P. I. and S. U., 1973), Lecture Notes in Math., vol. 375, Springer Verlag, New York, 1974, R. F. Dickman, Jr. and P. Fletcher, Editors, 190-197. 21. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 22. S. B. Nadler, Jr., Continuum Theory, An introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 23. L. G. Oversteegen and E. D. Tymchatyn, On atriodic Proc. Amer. Math. Sot., 83 (1981), 201-204.
tree-like continua,
24. J. T. Rogers, Jr., Dimension of hyperspaces, Bull. Acad. Polon. SCr. Sci. Math. Astronom. Phys., 20 (1972), 177-179. 25. J. T. Rogers, Jr., Weakly confluent maps and finitely-generated mology, Proc. Amer. Math. Sot. 78 (1980), 436--438.
Sci., coho-
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26. J.T. Rogers, Jr., T+ee-like curves and three classical problems, in: Open problems in topology, North Holland, Amsterdam, New York, Oxford, Tokio, 1990, J. van Mill and G. M. Reed, Editors, 303-310. 27. E. D. Tymchatyn, Hyperspaces of hereditarily indecomposable plane continua, Proc. Amer. Math. Sot., 56 (1976), 300-302. 28. G. T. Whyburn, Analytic Topology, Amer. Math. Sot. Colloq. Pub]., vol. 28, Amer. Math. Sot., Providence, R. I., 1942.
This Page Intentionally Left Blank
XII.
Special Types Maps between Hyperspaces 75.
of
Select ions
75.1 Definition. Let X be a continuum and let I? c 2”. A continuous function f : I? + X is called a selection for r provided that for each A E JT, f(A) E A. We introduced selections in Exercises 5.12 and 5.13. Observe that a selection for 2” or C(X) can be seen as a special kind of retraction of the hyperspace onto Fl (X). The dual to the notion of a selection -a coselectionis defined in [5.22] of Chapter XV. Definition 75.1 was given by Michael in [56, p. 1541. Michael considers more general spaces than metric ones. A discussion about selections can be found in Chapter V of [60]. For the case of l? = 2x, the problem of the existence of selections is completely solved by the following theorem. 75.2 Theorem [56, 1.9 and 2.71. For a continuum X, the following statements are equivalent: (a) there is a selection for 2x, (b) there is a selection for Fz(X),
and
(c) X is homemorphic to the interval [0, 11. The implications (a) + (b) and (c) =P (a) in Theorem 75.2 are immediate. The implication (b) =P (c) is left as Exercise 75.19. Notice that if Y is any topological space, the notion of selection can be defined in the same way as in Definition 75.1 for subspacesof CL(Y). The implication (b) =S (a) of Theorem 75.2 was generalized for locally compact 363
364
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
separable metric spaces in [50, Theorem 21. In [59], van Mill and Wattel proved the following theorem. 75.3 Theorem [59, Theorem 1.11. Let Y be a compact Hausdorff space. Then the following statements are equivalent: (a) Y is orderable, (b) there is a selection for F!(Y), and (c) there is a selection for CL(Y). A topological space Y is called weakly orderable (abbreviated WO) provided that there is a linear order 5 on Y’ such that for each p E Y, the sets and {q E Y : p 3 q} are closed. It is easy to see that if Y {4EY:q5Pl is WO, then Y is a Hausdorff space and Fz(Y) admits a selection (Exercise 75.20). It is not known if the converse is true. 75.4 Question [59, p. 6051. Let Y be a Hausdorff space. Is Y a WO if and only if Fs(Y) admits a selection? 75.5 Definition. A continuum X is said to be selectible provided that there exists a selection for C(X). Selectible continua have not been characterized. In this direction, the most general result is the following theorem. 75.6 Theorem is a dendroid.
162, Lemma 31. If a continuum X is selectible, then X
Proof. Assume that X is a continuum such that there is a selection f : C(X) +x. Given a subcontinuum Y of X and given a subcontinuum K of Y, C Y. For each y E Y, f(K) E K c Y. This implies that f(C(Y)) f{(g)) = y. Therefore, f(C(Y)) = Y and Y is also selectible. Since C(Y) is arcwise connected (Theorem 14.9) and f is continuous, we conclude that Y is arcwise connected. Hence f(C(Y)) = Y and Y is arcwise connected for each subcontinuum Y of X. In particular, X is arcwise connected. Now, suppose that X is not hereditarily unicoherent. Then there exist subcontinua A and B of X such that A II B is not connected. Since A and B are arcwise connected, it is easy to show that A U B contains a circle S. By Example 5.2, C(S) is a 2-cell whose manifold boundary is the circle Fl (S). Let g denote the restriction of f to C(S). Then g(C(S)) = S. Let j : S -+ Fl(S) be given by j(z) = {z}. Then jog: C(S) + Fl(S) is amap such that j o g({z}) = {z} for each z E S. Therefore, j o g is a retraction
365
75. SELECTIONS from the 2-cell C(S) onto its manifold boundary Corollary la of [48, p. 3141. Thus X is hereditarily X is a dendroid. n
Fi(S). This contradicts unicoherent. Therefore,
75.7 Definition. A selection s : C(X) + X is said to be rigid provided that if A, B E C(X) and s(A) E A c B, then s(A) = s(B). Continua which admit rigid selections have been completely characterized by Ward who proved the following result. 75.8 Theorem [65]. A continuum X is a smooth dendroid if and only if there exists a rigid selection for C(X). An important class of selectible dendroids is the class of smooth dendroids (Exercise 75.21). In Exercise 75.25 it is asked to prove that there are nonselectible fans. In [60, Question 5.111it was asked if each contractible dendroid X is selectible. This question was answered in the negative by MaCkowiak with the following example. 75.9 Example [53]. There exists a contractible and nonselectible dendroid X. The dendroid X is represented in Figure 46 (next page). X is the union of a simple triod T and two sequencesof subcontinua {A,}~fl and {B,}p!l such that A,, + T and B, + T. The dendroid X was rediscovered in [44]. In this paper, X was presented as a nonselectible dendroid for which there exists a retraction from C(X) onto Fi(X). Recently, J. J. Charatonik, W. J. Charatonik, Omiljanowski and Prajs have shown that, in fact, there exists a retraction from 2x onto Fr(X) ([23, Theorem 5.591). The proof that X is contractible is left as Exercise 75.27. We will prove that X is nonselectible. Suppose, to the contrary, that there exists a selection f : C(X) + X. For this example we adopt the following convention. Given two points p # q in X, let (p,q) be the unique arc in X which joins p and q. By Exercise 75.24, f(T) = ps and f((a,b)) = b. Let d be the metric induced by R3 on X. Suppose that d(a, b) = 2 and d(po, e) = 2. Let b > 0 be such that Hd(A, B) < 6 implies that d(f(A), f(B)) < i. Let N 2 1 be such that d(bN,b), d(hN,a) < g and Hd((eN,c,v),(a,b)), &((po,m),T), d(eN,e) and WN,e)< fM(fN,sw), (a,b)), K~Po,cN),T), 6. Let 00, (~1,CJZ: [0, l] -+ C(X) be the maps defined by: go(t)
=
(a, b) U @PO + (1 -
m(t)
=
(eN,cN)
m(t)
=
(fN,
gN)
t)e,
4 ,
u
@PO
+
(1
-
t)eN,eN),
u
@PO +
(1
-
t)fN,
fN)
366
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
Al
A2
PO B2
Bl
PO
PO
T e
PO -I
a
A contractible and nonselectible dendroid (75.9) Figure 46
75. SELECTIONS
367
Notice
that ac(0) = (a,b), a0(1) = T, m(0) = (eN,cN), n(1) = 02(o) = (fjV,gN), “a(1) = @O,gN) and Hd(go(t)roi(t)) < 6 for each i = 1,2 and each t E [0, 11. Then f(u0(0)) = Let R = {p E X : d(p,a) 5 $ or d(p,b) I f}. and f(ae(1)) = f(T) = po $! R. Let to = max{t E f((a, bj) = b E R [0, l] : f(ao(t)) E R}. Then to < 1. Since f(ae(to)) E UO(~O)n R c T n R, f(ao(to)) E b,b). We will analize the case d(f(~~(te)), a) 5 i. The case d(f(ao(to)), b) i i is similar. Since d(f(oe(te)), f(ar(te))) < $, d(a, f(crr(to))) < f. Since (J&,&v),
f(m
(to))
E 01 (to)
c
( PO,
CN)
and
d(a,
f(ul
(to)))
<
i,
f(gl
(to))
E
(CN,
brv)
-
ibN). Since
d(f(al
(l)),pO)
=
d(f(al
(I)),
f(aO(l)))
<
iT
f(al
(1))
4
teN7
w).
We have obtained that f(ar(to)) E (cN, b,v) - {bN} and f(ar(l)) $ (eN, cN). Thus there exists ti E (to, l] such that f(gr(ti)) = biv. + d(bN,b) < f. Thus Then d(f(uo(tl)),b) I d(f(uo(tl)),f(al(tl))) f(oo(tl)) E R. Th is contradicts the choice of to and completes the proof that C(X) admits no selections. 75.10 Definition. Let X be a dendroid, given two points p and q in X (p # q) the unique arc joining p and q in X is denoted by pg. The dendroid X is said to be of type N (between points p and q, p # q) provided there exist in X: two sequencesof arcs p,pL and qnqk and points px E q,,qk - {qn, qk} and 4:: E P,PL - {P,, ~3 such that the following conditions are satisfied: P9
=
lim P,P~ = lim 91X4h,
P
=
limp, = lim p; = lim pz,
9 =
limq, = limqk = limq:.
75.11 Question [60, Question 5.111and [ll, Problem 8.61. What is an internal characterization of selectible dendroids (of selectible fans). 75.12 Questions [ll, Question 8.71, [8, Question 111. Does there exist a contractible and nonselectible dendroid which is (a) planable, (b) hereditarily contractible, (c) a fan, (d) has at most two ramification points?
In [52] it was was shown that neither monotone nor open maps preserve selectibility of dendroids (see Exercise 75.26). The following questions remain open. 75.13 Questions [22, Question 14.141. Is selectibility invariant under maps of fans that are (a) light and open, (b) open, (c) light and confluent?
368
XII.
SPECIAL TYPES OF MAPS BETWEEN HYPERSPACES
75.14 Question [22, Question 14.16). Do there exist a nonselectible fan and a light open map defined on it such that the image is a selectible fan? The above question is a particular 75.15 Question preserve selectibility
case of a more general one.
[22, Question 14.171. What (nonselectibility) of fans?
kind of confluent
maps
75.16 Question [9, Question 3.161 (some related questions can be found in [8, Question 261). Does there exist a nonselectible fan which is not of type N? 75.17 Question [B, Question 291. Assume a dendroid X is the inverse limit of an inverse sequence {X,, fn}rzi of selectible dendroids (of dendrites, of selectible fans). Under what conditions with respect to the bonding maps f,a is the dendroid X selectible? In particular, is selectibility of X, transferred to X if the bonding maps are monotone?
Exercises 75.18 Exercise.
the compact metric space Y = Ii U . . U where each I, is an arc and the sets Ii,. . . , In, {pi}, . . . , {pm} are pairwise disjoint. Then ([64]) the number of different selections for 2y is equal to In
u
{Pl)
u
...
Consider
u
{PTn),
[Hint: Use Exercise 5.12 (1) to prove that for each L C (1,. . . ,n} and M~{1,...,m}withL#0orM#0,theset{A~2~:Afll~#0ifand only if i E L and pj E A if and only if j E M} admits exactly 2]L] + ]M] selections.] 75.19 Exercise. Let X be a continuum. If there is a selection s for Fz(X), then X is homeomorphic to the interval [0, 11. The proof of this exercise can be done by following the steps (a) through (c) below: (a) define, for each p, q E X, p 4 q if and only if s({p, q}) = p. For each p E X, let [t,p] = {q E X : q 4 p} and [p,+] = {q E X : p 4 q}. Then [+,P], b, +I are closed subsetsof X such that X = [t, p]U[p, +] and [C,p] n [p, --+I = {p}. Therefore, [t,p] and [p, +] are connected, (b) ifp+qandq*r,thenp+r, [Hint: Assume that q # T. Then r $ [t, q]. Consider the connected set
{S(-I?c7))E x : 2 E I+,411.1
EXERCISES
369
(c) let p : C(X) + R’ be a Whitney map. Define f : X + f(p) = p([t,~]). Then f is a one-to-one map. 75.20 Exercise. If Y is a WO, then Y is a Hausdorff is a selection for Fz (Y). 75.21 Exercise
[62, Theorem
11. Smooth dendroids
R’ by
space and there are selectible.
75.22 Exercise [62, p. 3711. Let X be a locally connected continuum. Then: X is selectible if and only if X is a dendrite. [Hint: Use Exercise 25.34.1 75.23 ible.
Exercise.
There are non-contractible
dendroids
that are select-
75.24 Exercise. Let X be a dendroid. Suppose that there exist two in C(X) such that A,, + A, B,, -+ A, A,, n sequences {A,}El, {&)El B,, = {a,} for each n 2 1, and a, + (1 for some a E A. Suppose that f : C(X) -+ X is a selection. Then f(A) = a. [Hint: Suppose that f(A) # u. Fix a large enough number N. Let a,/3 : [0, l] + C(X) b e maps such that o(O) = {a~) = p(O), o(1) = AN, p(1) = BN and ifs 5 t, then a(s) C a(t) and p(s) C b(t). Consider the connected set A = {AN U ,0(t) E C(X) : t E [0, l]} U {BN U a(t) E C(X) : t E [0, l]}. Then each element in A is close to A.] 75.25 Exercise. Each dendroid of type N is nonselectiblc. a nonselectible fan. [Hint: Use Exercise 75.24.1
Construct
75.26 Exercise [52]. Selectibility is not preserved under either open or monotone maps. Nonselectibility is not preserved under either open or open maps. [Hint: Find the appropriate maps in Figure 47 (next page), the first open map is obtained by identifying symmetric points.] 75.27 Exercise.
The dendroid
in Example 75.9 is contractible.
75.28 Exercise. Let X be a continuum and let d denote a metric for X. An e-selection for Fz(X) ([Sl]) is a map (T : Fz(X) -+ X such that a(A) E Nd(c,A) for each A E F2(X). If, for each E > 0, there is an c-selection for Fz(X) then X does not contain a simple triod. [Hint: Suppose that X contains a simple triod T. Consider the subcontinuum A of Fz(X) consisting of all the elements of F2(X) of the form
370
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
Selectibility and nonselectibility are not preserved under either open or monotone maps (75.26) Figure 47
76. RETRACTIONS
BETWEEN
HYPERSPACES
371
{a, b}, where a is an end point of T and b is in the union of the two legs of T which do not contain a.] 75.29 Exercise. If X is a locally connected continuum and, for each c > 0, there is an c-selection for Fz(X), then X is an arc (in fact, as it is proved in [61, Corollary 2.51, this result is true for an arcwise connected continuum instead of a locally connected continuum). [Hint: Use Exercise 31.11.1
76.
Retractions
between
Hyperspaces
76.1 Definition. A 2 a subcontinuum of a continuum X is said to be: (a) a retract of X if there exists a continuous function T : X -+ 2 (called retraction) such that T(Z) = z for every z E 2, (b) a deformation retract of X if there exist a retraction r : X + Z and a map H : X x [0, l] + X such that H(z, 0) = 2 and H(z, 1) = T(X) for every z E X, (c) a strong deformation retract of X if there exist r and H as in (b) with the additional property that H(z, t) = z for every z E Z. Let X be a continuum. In [36], Goodykoontz studied the relationships among the following statements: (1) C(X) is a retract of 2x, (2) C(X) is a deformation retract of 2”, (3) C(X) is a strong deformation retract of 2.Y, (4) Fl (X) is a retract of 2sY, (5) Fl (X) is a deformation retract of 2dY, (6) PI (X) is a strong deformation retract of 2”, (7) Fl(X) is a retract of C(X), (8) Fl(X) is a deformation retract of C(X), and (9) Fl(X) is a strong deformation retract of C(X). If X is locally connected, then (Exercise 76.23) each of the statements (l), (2) and (3) hold. Furthermore, in the casethat X is locally connected, then each of the statements (4) through (9) is equivalent to the statement that X is an absolute retract ([36, Corollary 2.61). In the case that X is not locally connected, Goodykoontz studied the problem of determinig whether any of the statements (l), (2), (4), (5), (7), (8) and (9) imply any of the others (by Exercises 76.24 and 76.25, (3) and (6) never hold in the casethat X is not locally connected). In the following table we summarize what is known and what is not known (each entry in the table refers to the question of whether the statement of the column of
XII.
372
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
the left implies the statement in the row at the top). This table is similar to Table 1 of Goodykoontz ([36, p. 1301). We only have added, with bold faced letters, the information that was discovered after [36].
* I (1) I (2) I (11 I ves I no I t (2j ) >es ] yes i (4) 110 no
(4)
no no yes
EH I10
;;;
no
(8)
no
(9)
no
76.12
I (5) I (7)
I
(8)
(9)
II no II no I4 no 1 no 1 no 1 no no --yes d ~~ d--z IlO no, 76.3 yes no yes yes no yes yes
no no ?, 76.4 ?
no, 76.3 ? yes
76.2 Theorem. Let X be a continuum. If Fi(X) is a deformation retract of 2”, then Pi(X) is a deformation retract of C(X). Proof. Let I‘ : 2x -+ F,(X) and H : 2” x [0, l] + 2* be maps such that T({P)) = {P> for each p E X and H(A,O) = A, H(A, 1) = r(A) for each A E 2.‘. Define F : C(X) x [O,l] -+ C(X) by:
F(A,t)
=
u{H(A,s)
: 0 5 s < 2t},
i ~{H(A,s):2t-l<s
if t E [0, f] iftE[$,l]
We will verify some properties of F: If t = i, both definitions of F give the value U{H(A,s) : 0 < s 5 1). Since H is continuous, the set {H(A, s) : 0 5 s < 2t) is a connected subset of 2” that contains the element H(A,O) = A E C(X). Then, by Exercise 15.9 (2), F(A,t) E C(X) f or each A E C(X) and each t E [O, f). By a similar argument (H(A,l) E Fl(X)), F(A,t) E C(X) for each A E C(X) and each t E [$, l]. Therefore, F is well defined. It is easy to check that F is continuous and F(A,O) = A and F(,4,1) = r(A) for each A E C(X). Therefore, F, (X) is a deformation retract of C(X). n 76.3 Example [44]. There is a selectible dendroid X such that FI (X) is not a deformation retract of C(X). The dendroid X is represented in Figure 48 (next page). X is the union of two sequences {An}F!l and converges to the (B,,}F=, and the set Y = c&U cd, the sequence {A,}F!l set af U cct and the sequence {B,,,}~!i converges to the set eb U cd. In [44, pp. 68-691, it was described a selection for C(X). In Exercise 76.26 it is asked to give a geometric argument to show that Fl (X) is not a deformation retract of C(X).
76. RETRACTIONS
a
BETWEEN HYPERSPACES
373
e
a’ A selectible dendroid for which Fl(X) is not a deformation C(X) (76.3) Figure 48
retract
of
374
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
76.4 Question. If X is the dendroid described in 76.3, is FI(X) a retract of 2x? A positive answer to this question would solve two of the four open questions included in the table preceding Theorem 76.2. 76.5 Definition. Let X be a continuum. A mean is a retraction from F2(X) onto X. By Exercise 76.27 we may also think of means as maps m : X x X + X such that m(z, y) = m(y, z) for every (2, y) E X x X and m(x,x) = x for each 5 E X. Continua which admit a mean are far from being characterized. There are many open questions about means. Two excellent references on this topic are [lo] and [23]. Here, we wiIl mention somerecent important results about means and we will offer a list of open questions. The following example is the first known l-dimensional and indecomposable continuum admiting a mean. This example answersquestion 5.48 of [23]. 76.6 Example [John Franks, unpublished]. The dyadic solenoid C2 admits a mean. Let C2 be the dyadic solenoid as it is defined in Definition 61.1. Define m : CZ x C2 -+ CZ by: ~((z~,Q,Q,. . .), (yl, 5/2,y3,. . .)) = (~y2,z3y3,. . .). By the definition of Cz, s$,+~ = IC, and yi+, = yn for every n 2 1. Then (~,+~y~+l)~ = z,y, for every n 2 1. This implies that m(z, y) E C2 for each (z, y) E C2 x C2. Clearly, m is continuous and m(z, y) = m(y, z) for each (2, y) E C2 x C2. Since m((21,22,23,.-.),(21,~2,~3,...))
= (&&...)
= (~l,zZ,...),
m(z,z)
=
x for each x E C2. By Exercise 76.27, we conclude that CZ admits a mean. Bacon proved that a continuum admitting a mean must be unicoherent [3]. Bacon in [2] also showed that the sin($)-continuum does not admit a mean. This gave the first example of an acyclic continuum without means. Recently, Bell and Watson ([4]) provided a criterion for the non-existence of a mean on a certain classof continua (which include the sin( $) -continua). For related results see[l]. Kawamura and Tymchatyn ([46]) extended Bell’s and Watson’s criterion to a more general class of continua. Next we will present Bell’s and Watson’s criterion. We denote the product [0, l] x [0, l] by l?. The diagonal of r is denoted by A. 76.7 Lemma [4, Corollary 3.21. If m : r + [0, l] is a mean, then there is a continuum I< c I? - m-l ((0, 1)) which intersects A and which also intersects the manifold boundary C = ({O,l} x [O,l]) U ([O,l] x {O,l})
0f r.
76. RETRACTIONS
BETWEEN
HYPERSPACES
375
Let A = m-l (0) and B = m-‘(l). Then A and B are compact, Proof. nonempty and disjoint. Let U and Ir be open subsets in F such that clr(U) n clr(V) = 0, A c U and B c V. Let W be the component of U which contains the point (O,O), let D be the component of I? - W which contains the point (1,l). By Exercise 76.31, I? - D is connected. From the unicoherence of I? (Theorem 3, p. 438, section 57, Chapter VIII of [49]), Bdr(D) = clr(I?-D)nD is connected. Define K = B&(D). From Exercise 76.30, Bdr(D) c Bdr(W) c Bdr(U). Then K c I’ - m-‘({O,l}). Since (0,O) $! D and (1,1) E D, every connected subset of F containing (0,O) and (1,1) intersects K. Therefore, A and C intersect K. n 76.8 Definition [4, Definition 3.31. Let X be a continuum and let cl denote a metric for X. A continuous function f between two subspacesof X is said to be an e-idy map if for all z, d(z, f(z)) < E. A sequenceof arcs {anbn}~!i is said to be strongly convergent to an arc ab if for every E > 0 there exists N 2 1 such that for each n > N, there exists an e-idy map h : ab + anbn such that h(a) = a, and h(b) = b,. Notice that order is relevant in this definition, i.e., (anbn}F& strongly converging to ab is not, the same as {anbn}Fz, strongly converging to bu. 76.9 Lemma [4, Lemma 3.41. Let X be a continuum. Let m : XXX + X be a mean. Assume that X contains an arc A = ab and two sequences of arcs {a,c,}r& and {anen}Fz?=1both strongly converging to ab, that X is such that every subcontinuum containing c,, and e, contains a,. Then, for every subcontinuum K of A x A such that K n {(z, z) : z E A} # 0 and K n ({al x 4 # 0, we have that K rl m-l (a) # 0. Proof.
Let d denote a metric for X. We consider the space X x X with
the metric P((W ~1, (z, Y)) = m={d(w
x), d(v, ~1).
Suppose, to the contrary, that there exist a subcontinuum K1 of A x A and points p, q E A such that {(p, p), (a, 4)) c Ki and KI is disjoint from m-‘(a). Let Ka = {(~,a) E A x A : (5,~) E K,}. Since m is symmetric, Kzflm-‘(a) = 0. Let K = KiUKz. Since (p,p) E K,nK,, K is connected, symmetric and a 4 m(K). Let E > 0 be such that B(26,a) U m(K) = 0. By uniform continuity of m, let 6 > 0 be such that 6 < E and p((u,v), (s, y)) < 26 implies that d (m(u,v),m(z, y)) < E. Choose n 2 1 and &idy maps h : ab -+ a,c, and k : ab + anen such that h(a) = a, = k(a), h(b) = cn and k(b) = e,,. Put
B = {(h(z),h(y)) : (xcl~) E K)
u i(G),
h(y)) : (2,~) E Wu
376
XII.
SPECIAL TYPES OF MAPS
BETWEEN
HYPERSPACES
Notice that each of the sets of this union is a continuum. The first one and the second one have the point (h(a), h(q)) = (a,,, h(q)) = (k(a), h(q)), and the second one and the third one have the point (k(q), h(a)) = (k(q),a,) = (k(q), k(a)). Thus B is a subcontinuum of X x X. Notice that h(p) = m(h(p), h(p)) E m(B) and k(p) = m(Q), k(p)) E where h(p)c, (respectively, k(p)e,) m(B). Let C = m(B)Uh(p)c,,Uk(p)e,, is the subarc of a,c, (respectively, a,e,) joining h(p) and c, (respectively, k(p) and e,). Then C is a subcontinuum of X containing c, and e,. By hypothesis, a, E C. This implies that a, E m(B). Let z E B be such that a, = m(z). We analize the casein which z is of the form z = (k(z), h(y)), with (5,~) E I<, the other casesare similar. Since d(z, k(z)) and d(y, h(g)) < 6, by the choice of S, d(m(z, y), a,) = d(m(z, Y)7m(z)) < E. Since d(u, a,) = d(u, k(u)) < E, d(a,m(a, y)) < 2-5 Thus m(z, y) E m(K) n B(26, u). This contradicts the choice of 6 and completes the proof of the lemma. n 76.10 Theorem (The No Mean Theorem) [4, Theorem 3.51. Let X be a continuum. Suppose that X contains an arc A = ub and four sequences {a,,~,}~!~, {a,e,}F==,, {fnbn}rzl and {gnbn}Fzl of arcs with each of these sequencesstrongly converging to A, and X is such that for every n, every subcontinuum containing cn and e,, contains a,, and every subcontinuum containing f,, and gn contains b,. Then X does not admit a mean. Proof. Suppose, to the contrary, that there is a mean m : X x X + X. Since A is an absolute retract, let r : X -+ A be a retraction. Put M = (r o m)l(A x A). Then M is a mean on A. By Lemma 76.7, there exists a subcontinuum I( of A x A such that K II M-’ ({a, b}) = 8 and, since a and b have symmetric status, we may assumethat K II {(z, 2) : 2 E A} # 8 and K n ({a} x A) # 0. By Lemma 76.9, K n M-‘(u) # 0. This contradiction proves the theorem. 1 76.11 Example [46, Theorem 2.51. Let X be an hereditarily unicoherent continuum which contains a pseudo-arc. Then X admit,s no mean. 76.12 Example 123, Example 5.531. There exists a smooth dendroid admitting no mean. The dendroid X is defined as X = (A0 U.41 UAz U.. .) U (BlUBzU.. .), where A0 = ub is an arc and A, -+ A0 and B, -+ Ao. In Figure 49 (next page) a typical A, is represented. The point a belongs to each A,, (m = 0, 1,2,. . .), bcrn)-+ b and ccrn) + b. On the other hand, each A, is a smooth dendroid, A, = T, U (U{R?) : n > 1)) U (U{S~m’ : n > l)),
76. RETRACTIONS
BETWEEN
377
HYPERSPACES
a
\ f
f ntrn) I f
(m)
e
5
\
(ml
I
I I
0-N
l
n (m)
I
I dtm)
wn
J
Cm) Rntm)
ON en
“n I
:
d,(m) ’
/ ~
/
I
/ / L (m)
(m) /
/
l (m)
Sn
A smooth dendroid admitting no mean (76.12) Figure 49
Am
378
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
R(‘“) -+ T, and S?’ -+ T,. Each of the sets Tm, RF) and Sim’ are tryods. The end points of T, are a, btm) and ccm); the end points of Rim) are a, b?’ and him,“)and the end points of Sim) are a, 9irn’ and cim). The continuum B, is a dendroid which is symmetric to A, with respect to the line in R3 containing the points a and dcm). The dendroid B, is constructed in such a way that A, n B,, = TnE and Ak II B, = {u} if lc # 712. In Exercise 76.33 it. is asked to give a geometric argument to show that X does not admit a mean, and in Exercise 76.32 it is asked to show that for every smooth dendroid Y, pi(Y) is a strong deformation retract of C(Y). In particular, for the dendroid of Example 76.12, pi(X) is a strong deformation retract of C(X). 76.13 Question [57, p. 1961. Let X be a locally connected subcontinuum of R3. If X admits a mean, then is X contractible? 76.14 Question [2, p. 131. Is there an acyclic locally connected continuum that admits no mean? 76.15 Question [lo, Question 201. Does the Buckethandle continuum (Example 22.11) admit a mean? 76.16
Question
[2, p. 131. Is the arc the only arc-like continuum that
admits a mean? 76.17 Question [2, p. 131. Is the arc the only arc-like continuum containing an open dense half line that admits a mean? 76.18 Question [23, Question 5.441. Does there exist a dendroid X such that it admits a mean and for which there is no retraction T : 2” + S(X).
It is known ([5, Proposition 21) that selectible dendroids are uniformly pathwise connected (see Definition 33.11). Then the following question naturally arises. [23, Question 5.491. If a dendroid X admits a mean, then is X uniformly pathwise connected? 76.19
Question
76.20 Question [23, Question 5.501. If X is a dendroid with the property of Kelley, then does X admit a mean? By the main result of [28] (see Theorem 78.30) dendroids with property of Kelley are smooth.
EXERCISES
379
Exercises 76.21 Exercise. Give an example of a continuum X such that X is not a dendroid and Fi (X) is a strong deformation retract of 2x. 76.22 Exercise. Let X be the cone over the Cantor set. Then Fi(X) is a retract of 2x. Furthermore, for every subcontinuum Y of X, F,(Y) is a retract of 2’. Since every smooth fan can be embedded in X ([22, Proposition 11.2]), then, for every smooth fan Y, Fl(Y) is a retract of 2’. [Hint: As usual suppose that the Cantor set is contained in [0, l] x { 0) C R2 and the vertex of the cone is the point (i, 1). Given A E 2.’ such that the vertex of X is not in A, consider the minimal quadrilateral C, such that A c C and two sides of C are horizontal segments and the other two sides of C are segments contained in X.1 76.23 Exercise [36, Corollary 2.31. If X is a locally connected continuum, then each of the statements (l), (2) and (3) at the beginning of this section holds. [Hint: Let d be a convex metric for X (see section 10). For each r > 0 and A E 2x, define C(r,A) = {x E X : d(a,z) < r for some a E A}. Let (Y : 2.’ + R’ defined as a(A) = inf{r 2 0 : C(r,A) is connected}. Then define H : 2*’ x [0, l] -+ 2x by H(A, t) = C(ta(A), A).] 76.24 Exercise [36, Theorem 2.2). The continuum X is locally connected if and only if C(X) is a strong deformation retract of 2x. [Hint: Suppose that H : 2x x [0, l] + 2x has the properties asked in Definition 76.1 (c). Let p E X. If q E X and q is close to p, then U{ff({P, 413t) : t E [O,111is a connected subset of X (see Exercise 15.9 (2)) and this set is close to the set {p}.] 76.25 Exercise [36, Corollary 2.71. Let X be a continuum. If Fl(X) is a strong deformation retract of 2x, then X is locally connected. 76.26 Exercise [44]. Give a geometric argument to show that, for the example X described in 76.3, Fl(X) IS . not a deformation retract of C(X). IHint: The subcontinuum cd U ef can not be moved under a map H as in Definition 76.1 (b).] 76.27 Exercise. A continuum X admits a mean if and only if there exists a map m : X x X -+ X such that m(x:, y) = m(y,z) for each (z, y) E X x X and m(z, z) = z for each z E X.
380
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
76.28 Exercise. Let X be a continuum. If there is a retraction from 2.’ onto Fl(X) then X admits a mean. The converse is not true. [Hint: The dyadic solenoid is not arcwise connected.] 76.29 Exercise.
If X is an AR, then X admits a mean.
76.30 Exercise. If Y is a locally connected space and D is a component of a subset U of Y, then Bdy (D) c Bdy (U). 76.31 Exercise. If Y is a connected spaceand D is a component of the complement in Y of a connected subset E of Y, then Y - D is connected. 76.32 Exercise. If X is a smooth dendroid, then Fl(X) deformation retract of C(X). [Hint: Use the map defined in Exercise 25.35.]
is a strong
76.33 Exercise. Give a geometric argument to show that Example 76.12 does not admit a mean. [Hint: Let d be the metric in R 3. Suppose that there exists a mean m : Fs(X) + X. Let A N B means A is close to B (Hd(A, B) is small). Fix very large integers M and n. Take a pair of points {p, q} in AM = R~“‘US~M’. We will move {p, q} along different positions. This movements will always be done in such a way that p and q will have always the same height, with respect to Figure 49, p. 377. Since A4 and n are very large, for each position of {p, q} there is a point z E AC,= ab such that z has the same height than p and {p, q} - (2). Then m( {p, q}) - {z} and (height of NP, ql)) N Sleight ofp). First movement: p moves from ji”’ to bL”’ and q from fi”’ to c$?‘. At the beginning, m({p,q}) = m({fi”‘}) = &MI E AM. Since all the time (height of m({p,q})) N (height of p) then m({p,q}) # {a}. Then m({p,q}) E AM. At the end, m({&“‘,c~M’}) E uL~)&~) Udi”)c&M’. In the case that m({&“‘,~~M)}) E zli”)bkM), it follows that m({lci”), dlM’)) E z~i’~)bk~) and rn({/~!~~‘,v~~‘}) - {IcAM’}. In particular m({g (M), h(M)}) N {g(M)}. Second movement: p moves from bk”) to hk”) and q from ci”’ to ei”’ and then to v$,~). Since p does not reach the height of ft”), m( (p, q}) remains in RLhf). Then m( { hL”‘, viA4’}) belongs to the segment zui”)hiM). Third movement: p moves from hL”’ to .wk”) and then returns to h$,“) and q from vk”’ to SAM’. Since p does not reach the height of ec”), Then m({g~“‘,h~M’}) is close to hi”). m({PT 41) remains in zi”)hAM). This implies that m({g, h}) - {h}. This is a contradiction.
77. INDUCED
Analyze the case m({b~“),c~?}) tion using BM instead of AM.]
77.
381
MAPS
E dh’*‘cL”)
Induced
and obtain a contradic-
Maps
77.1 Definition. Given a (not necessarily surjective) continuous function between continua f : X -+ Y, the induced maps associated to f are the functions: 2f : 2x + 2’ and C(f) : C(X) + C(I’)
given by
2f(A) C(f)(A)
=
f(A) for each A E 2x (the image of A under f) and
=
f(A) for each A E C(X).
Previously, we denoted 2f by f* and C(f) by f^. Note that, by Lemma 13.3, each of the functions 2f and C(f) is continuous. Let f be a continuous function from X onto Y and let M be a class of maps. Consider the following three statements:
(*) f E M, (C*) C(f) E M, and (2*) 2f E M. A lot of work has recently been done to find interrelations between statements (*), (C*) and (2*) for several classesof maps ([12], [14], [15], [16], [17], [18], [19], [20], [21], [24], [25], [26], [33], [38], [39], [40], [41], 1421 and [45]). An excellent survey on induced mappings is [14]. Here we only discussdetails of the interrelations between (*), (C*) and (2*) for the most common classesof maps. Namely, confluent, light, monotone, open and weakly confluent maps (see Definitions 24.1, 35.2 and 73.3, respectively). The following table gathers what is known about possible implications among (*), (C*) and (2*) for confluent, light, monotone, open and weakly confluent maps.
382
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
77.2 Theorem [41, Theorem 6.31 (f or a generalization see[24, Theorem 4.201). For the class of confluent maps, (C*) +- (*) and (2*) + (*) hold. Proof. We only prove that (C*) + (*). The proof that (2*) + (*) is similar. Suppose that C(f) is confluent. Let B be a subcontinuum of Y. Let D be a component of f-‘(B). Then Fl(D) is a subcontinuum of C(f)-‘(Fr (B)). Let 2) be the component of C(f)-‘(Fr (B)) which contains Fl(D). Since C(f) is confluent, C(f)(D) = Fl(B). Let DO = U{E : E E D}. Then D c DO c f-‘(B) and f(Do) = B. By Exercise 15.9 (2), DO is connected. Then D = DO. Thus f(D) = B. Therefore, f is confluent. n 77.3 Question [41, Problem 31, [24, Question 4.25]. Does there exist a map f such that 2f is confluent while C(f) is not’? 77.4 Theorem (see[41, Theorem 4.31; compare also 140,Theorem 3.21). For the class of open maps, (C*) + (*) * (2*). Proof. Let d (respectively, p) denote a metric for X (respectively, Y). (C*) 3 (*) (the proof that (2*) 3 (*) is similar). Suppose that C(f) is open. Let U be an open subset of X and let p E U. Let U = (A E C(X) : A C U}. Then U is an open subset of C(X). Thus C(f)(U) is an open subset of C(f)(C(X)) and C(f)({p}) E C(f)(U). Let c > 0 be such that B~p(~C(f)(bl)) n C(f)(C(X)) c C(f)(U). We claim that Bp(c,f(p)) 17f(X) c f(U). Let f(q) E B,(c,f(p)). It follows that C(f)({q}) E C(f)(U). Then there exists A E U such that C(f)({q}) = C(f)(A). This means that {f(q)} = f(A). Fix a point a E A.
Then f(a) = f(q) and f(a) E f(A) c f(u).
Thus f(q) E f&J).
This proves that f is open. (*) + (2*). Suppose that f is open. Let U be an open subset of 2x and let A E U. Let e > 0 be such that B,qd (2e, A) c U. For each point p E A, since f(Bd(~,p)) is an open subset of f(X), there exists 6, > 0 such that B,(26,, f(p))nf(X) c f(Bd(~,p)). Since f(A) is compact, there exist n 2 1 andpi,... ,P, E A such that f(A) c B,(&, , f(m)) U . . . U BJb,, f(pn)) and A C Bd(c,pl) U ..- U Bd(qp,). Let 6 = min{b,, , . . . ,&+,}. We claim that BH,(6, f(A)) n 2f(ZX) c 2f(U). Let D E 2x be such that H,(f(D),f(A)) < 6. Given y E f(D), there exists a E A such that p(y, f(a)) < 6. Then there exists i E (1,. . . , n} such that p(y, f(pi)) < 26,, . Choose uy E Bd(e,pi) such that f(uy) = y. Then y E f(Bd(c,pi)). n}, there exists vi E f(D) such that p(~i, f(pi)) < ForeachiE {l,..., 6 < 2Spi. By the choice of Spi, there exists zi E Bd(e,pi) such that f(%) = Vi.
77. INDUCED
383
MAPS
Define B
=
CZX({Uy
E x
: y E f(D)})
u
(21,.
. . ,~n~
E TY.
{Rr..
.3 %I>
Since f is continuous, f(B)
= f (cZx({u,
c dY({f(U~)
E X : !/ E f(W))) E k’ : v E f(W))
u Uf(D)
= f(D).
On the other hand,
Thus f (II) = f(B). By construction, B C Nd(2c, (~1,. . . ,pn}) C Nd(2E, A). ,z}), it follows that A C Nd(2e, B). {Pl,... ,%I} c Nd(f,{xl,.-. Hd(A,B) < 2~. Therefore, B E U. We have proved that 2f(D) E This proves that 2f(U) is open in 2f(X) and completes that proof implication (*) * (2*). n
Since Thus 2f(U). of the
In Exercise 77.38 it is asked to show an example of an open map f such that C(f) is not open. In [45] it was proved the following result,. 77.5 Theorem [45]. If f : X + Y is a surjective map between con: C(C(X)) + C(C(Y)) is open and Y is nondegenerate, tinua, C(C(f)) then f is a homeomorphism. 77.6 Theorem [24, Proposition 6.101(for a generalization see[24, Theorem 7.21). If 2f . 2x -+ 2y is weakly confluent, then f is weakly confluent too. Proof. Let B be a subcontinuum of Y. Since 2f is weakly confluent, Let A = there exists a subcontinuum A of 2x such that 2f (A) = Fl(B). U{D E 2x : D E A}. Then A is a closed subset of X and f(A) = B. Let C be a component of A. We claim that f(C) = B. Suppose, to the = contrary, that f(C) # B. Fix a point q E B - f(C). Then C O f -‘({q}) 0. We can apply the Cut Wire Fence Theorem (12.9) to the compact Then there exist two metric space A and the closed sets C and f -‘({q}). disjoint closed nonempty sets H and K such that A = H U I<, C c H and f-‘({q))
c K.
Define3C={DEd:DOH#0}andX:={DEd:DCI<}. Clearly, 3t and Ic are disjoint and closed in 2~~and A = 3c U Ic. Since there exists an element E E A such that 2f (E) = {q}, we have that E c
384
XII.
SPECIAL TYPES OF MAPS BETWEEN HYPERSPACES
f-‘(q) c K. Thus K # 0. F’1x a point x E C. Then there exists D E A such that x E D. Then D fl H # 8. Thus 3t # 0. The properties of ‘?i and K imply that A is not connected. This is a contradiction and proves that f(C) = B. Therefore, f is weakly confluent. n 77.7 Questions [24, Question 6.121. For the class of weakly confluent maps, does (C*) imply (2*)? Does (2*) imply (C*)?. 77.8 Definition [24, section 31. Let X be a continuum. Let K be a subcontinuum of X and let p E K. The continuum K is arcwise approximated at the point p provided that there is a sequenceof arcwise connected subcontinua I(,, of X such that p E K, for each n 2 1 and K, -+ K. A subcontinuum K of X is said to be arcwise approximated provided that it is arcwise approximated at each point of K. In particular, this condition is satisfied in the case when K c K, for each n 2 1. Then we say that K is strongly arcwise approximated. In other words, a continuum K C X is strongly arcwise approximated provided that there is a sequenceof arcwise connected subcontinua K, of X such that K = limK,, C n{K, : n 2 1). The continuum X is said to have the (strong) arc approximation property provided every subcontinuum of X is (strongly) arcwise approximated. The arc approximation property was introduced by W. J. Charatonik for the study of confluence of induced maps. He obtained the following result. 77.9 Theorem [24, Theorem 4.41. Let f : X t Y be a confluent surjective map between continua. If C(Y) (respectively, 2’) has the arc approximation property, then C(f) (respectively, 2f) is confluent. Theorem 77.9 is a generalization of the following result by Hosokawa. 77.10 Theorem [38, Theorem 2.51and [40, Theorem 4.41. Let f : X + Y be a confluent surjective map between continua. If Y is locally connected, then C(f) and 2f are confluent. In [24] W. J. Charatonik studied properties of continua which have the arc approximation property. He offered the following sequenceof interesting questions.
77.11 Questions [24, Questions 3.591. (a) for what dendroids X does the hyperspace C(X) arc approximation property?
have the (strong)
77. INDUCED
MAPS
385
(b) does C(X) have the strong arc approximation property for every hereditarily arcwise connected continuum X? (c) does 2.’ have the (strong) arc approximation property for every smooth fan X or for every smooth dendroid X? 77.12 Questions [24, Questions 3.601. Suppose that X is an arcwise connected continuum and it has the strong arc approximation property. Do the hyperspaces C(X) or 2x also have the strong arc approximation property? 77.13 Questions [24, Question 3.611. For what continua X do the hyperspaces 2” and C(X) have the arc approximation property? 77.14 Questions [24, Question 3.621. Are the three statements: (a) the continuum X has the arc approximation property, (b) the hyperspace 2x has the arc approximation property, (c) the hyperspace C(X) has the arc approximation property, equivalent ? If not, what implications between them are true? 77.15 Question [24, Question 3.631. Let X be a continuum and let the hyperspace C(X) have the arc approximation property. Does it follow that every arc component of X is a dense subset of X? 77.16 Questions [24, Question 3.641. Assume the continua X and Y have the (strong) arc approximation property. (a) Does it follow that the product X x Y has the (strong) arc approximation property? (b) What if Y = [O, l]? 77.17 Definition. A surjective continuous function between continua f : X -+ Y is said to be: (a) semi-confluent provided that for each subcontinuum L of Y and for every two components Ki and K2 of f-‘(L), either f(K1) C f(K2) or f(Kd c f(Kl), (b) OM-map (respectively, MO-map) if there are maps g and h, where g is open and h is monotone, such that. f = g o h (respectively, f = 11o g), (c) near-homeomorphism (respectively, near-monotone, near-OM) if it is the uniform limit of homeomorphisms (respectively, monotone maps, OM-maps) from X onto Y. 77.18 Questions 124, Question 4.191(Compare with [24, Question 4.18 and Example 4.171). Let f : X -+ Y be any (confluent) map between hereditarily indecomposable continua. (a) Is C(f) confluent? (b) Is 2f weakly confluent?
386
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
77.19 Questions [24, Questions 5.21. For the class of semi-confluent maps, does (C*) imply (2*)? Does (2*) imply (C*)? 77.20 Question 121,Question lo]. Let f : X + Y be a surjective map where X is a dendroid and Y is a continuum. Suppose that C(f) is open, does it follow that f is a homeomorphism? 77.21 Questions [15, Question 5.11. Are lightness of C(f) and 2f equivalent conditions for a map between arcwise connected (in particular, locally connected) continua? 77.22 Questions [41, Problem 21. For the classof MO-maps, does (*) imply (C*)? Does (C*) imply (*)? Does (2*) imply (*)? If f is open and surjective, then is C(f) an MO-map? 77.23 Questions [18, Question 5.91. Do there exist locally connected (a) continua, (b) curves X and Y and an open surjective map f : X + 1’ such that, C(f) is not near-open? More generally, 77.24 Questions [18, Question 5.101. Under what conditions (concerning the domain and/or the range space) (a) openness,(b) near-openness of a surjective map between continua f : X + Y implies near-openness of the induced map C(f) : C(X) + C(Y)? 77.25 Questions [18, Question 5.161. (a) Does near-openness of 2f imply near-opennessof C(f)? (b) If not, under what conditions concerning the structure of either the domain or the range space (or both) the implication holds? 77.26 Question [18, Question 5.171. Does there exist a surjective map between continua f : X + Y such that C(f) is near-open while 2f is not? 77.27 Questions [18, Questions 5.32-5.341. For the class of nearmonotone (or near-OM) maps, which of the following implications hold (C*) j (*), (2*) 3 (*), (C*) + (2*) and (2*) + (C*)? 77.28 Question [14, Question 5.51. Suppose that C(f) is a nearhomeomorphism (in particular, C(X) and C(Y) are homeomorphic). Does it imply that 2f is a near-homeomorphism? The same question if X = Y.
EXERCISES
387
Exercises 77.29 Exercise. For the class of homeomorphisms, on p. 381, are equivalent.
(*), (C*) and (2*),
77.30 Exercise. ([60, 0.67.51). If f,g : X -+ Y are homotopic maps between continua, then C(f) and C(g) (respectively, 2f and 29) are homotopic. The converse is not true. 77.31 Exercise. Let f : X + Y be a surjective map between continua. Then f is weakly confluent if and only if C(f) is surjective; 2f is a surjective map. 77.32 Exercise [51] and [40, Theorem 3.51. For the class of monotone maps, (*), (C*) and (2*), on p. 381, are equivalent. 77.33 Exercise (Hosokawa and Kawamura) [40, Example 5.11. For the class of confluent maps, (*) + (C*) and (*) + (2*). [Hint: Consider the continua represented in the Figure 50 (next page). The continuum Y is obtained by identifying antipodal points in the circle contained in X. Let f : X + Y be the quotient map. Fix a Whitney map /A : C(Y) -+ R’. Fix a positive small t. The elements of p-‘(t) are arcs. Let B = cZc(u)({B E p-l(t) : q 4 B}).] 77.34 Exercise [24, Example 4.121. Let X be the continuum represented in Figure 14, p. 51. Consider the space Y obtained by identifying antipodal points in the circle S1 c X. Let f : X + Y be the natural quotient map. Prove that Y is homeomorphic to X, f is confluent and 2f is not confluent. In fact in [24, Example 4.121 it is proved that C(f) is confluent. Thus this is an example where f and C(f) are confluent but 2f is not confluent. [Hint: Consider (2f)-‘(Fl (Y)).] 77.35 Exercise [15, Theorem 3.101. Let f : X + Y a map between continua. If 2f is light, then C(f) is light. If C(f) is light, then f is light. 77.36 Exercise [15, Example 3.81. Let f : S’ -+ S’ be given by f(z) = z2, where S1 is the unit circle in R 2. Then f is light and C(f), 2f are not light.
388
XII.
SPECIAL
TYPES
OF MAPS
BETWEEN
HYPERSPACES
X
n\\
Confluence is not preserved under induced maps (77.33) Figure 50
EXERCISES
389
77.37 Exercise [15, Example 3.91. Let Xr be the sin(t)-continuum and let Lr = (0) x [-1, l] be the limit segment of Xr. Let X2 be the image under the reflexion of the plane R2 with respect to the line x = 1. Let L2 = (2) x [-1, l] be the limit segment of X2. Put X = X1 U X2. Let Y be the space obtained by identifying each point (0, t) in X with the point (2, t). Let f : X -+ Y be the natural quotient map. Then C(f) is light while 2f is not. 77.38 Exercise [38, Example 3.21. Let Y = [0, I] x [0, l] and let X = Y U (-2 E R2 : z E Y}. Let f : X -+ Y be given by
Then f is open and C(f) is not open. [Hint: Let A = ((0) x [-I, 11)U ([-I, l] x (0)). Find subcontinua B of Y such that B is close to f(A) but I3 is not the image of subcontinua of X which are close to A. Then C(f) is not open at the point A.]
X
H Weakly confluence is not preserved under induced maps (77.39) Figure 51
390
XII.
SPECIAL
TYPES
0~ MAPS
BETWEEN
HYPERSPACES
77.39 Exercise [54, p. 2361 (see [24, Example 6.81). Let f : ,y -+ T’ be the map illustrated in Figure 51, p. 389. Then f is weakly confluent while C(f) and 2f are not. [Hint: Consider 8 c C(Y) defined by 8 = FI (ab) u ({bc : c E bd}).] 77.40 Exercise.
If C(f) is weakly confluent, then f is confluent.
77.41 Exercise.
If f is an MC-map, then 2f is an MC-map.
References 1.
2. 3.
M. M. Awartani and D. W. Henderson, Compactijkations of the ray with the arc as remainder admit no n-mean, Proc. Amer. Math. Sot., 123 (1995), 3213-3217. P. Bacon, An aye& continuum that admits no mean, Fund, Math, 67 (1970), 11-13. P. Bacon, Unicoherence in means, Colloq. Math., 21 (1970), 211-215.
4.
M. Bell and S. Watson, Not all dendroids have means, Houston J. Math., 22 (1996), 39-50.
5.
J. J. Charatonik, Contractibility and continuous selections, Fund. Math., 108 (1980), 109-118. J. J. Charatonik, On continuous selections for the hyperspace of subcontinua, Elefteria, 4B (1986), 335-350. J. J. Charatonik, Some problems on selections and contractibility, Third Topology Conference (Trieste 1986), Rend. Circ. Mat. Palermo (2) Suppl., 18 (1988), 27-30. J. J. Charatonik, On continuous selectionsfor the hyperspace of subcontinua, Colloquia Mathematics Societatis Janos Bolyai. Topology P&s (Hungary). North Holland, Amsterdam-New York, 1993, 55 (1989), 91-100.
6. 7.
8.
9.
J. J. Charatonik, Conditions related to selectibility, Math. Balkanica, (N.S.) 5 (1991), 359-372.
10. J. J. Charatonik, Some problems concerning means on topological spaces,in: Topology, Measures and Fractals, C. Bandt, J. Flachsmeyer and H. Haase, Editors, Mathematical Research vol. 66 Akademie Verlag Berlin 1992; 166-177. 11. J. J. Charatonik, On acyclic curves, a survey of results and problems, Bol. Sot. Mat. Mexicana (3) vol. 1, (1995) l-39. 12. J. J. Charatonik, Properties of elementary and of some related classes of mappings, preprint.
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391
13. J. J. Charatonik, Homogeneous means and some functional equations, to appear in Math. Slovaca. 14. J. J. Charatonik, Recent results of induced mappings between hyperspaces of continua, to appear in Topology Proc. 15. J. J. Charatonik and W. J. Charatonik, Lightness of induced mappings, to appear in Tsukuha J. Math. 16. J. J. Charatonik and W. J. Charatonik, HereditariLy weakly confluent induced muppings are homeomorphis?ns, to appear in Colloq. Math. 17. J. J. Charatonik and W. J. Charatonik, Inducible mappings between hyperspaces, to appear in Bull. Polish Acad. Sci. Math. 18. J. J. Charatonik and W. J. Charatonik, Limit properties of induced mappings, preprint. 19. J. J. Charatonik and W. J. Charatonik, Atomicity of mappings, to appear in Internat. J. Math. Math. Sci. 20. J. J. Charatonik and W. J. Charatonik, Confluence lifting property, preprint. 21. J. J. Charatonik, W.J. Charatonik and A. Illanes, Openness of induced mappings, preprint. 22. J. J. Charatonik, W. J. Charatonik and S. Miklos, Confluent mappings of fans, Dissertationes Math. (Rozprawy Mat.), 301 (1990), l-86. 23. J. J. Charatonik, W. J. Charatonik, K. Omiljanowski and J. R. Prajs, Hyperspace retractions for curves, Dissertationes Math. (Rozprawy Mat.), 370 (1997), l-34. 24. W. J. Charatonik, Arc approximation property and conf?uence of induced mappings, to appear in Rocky Mountain J. Math. 25. W. 3. Charatonik, Openness and homogeneity on induced mappings, to appear in Proc. Amer. Math. Sot. 26. W. J. Charatonik, Induced near-homeomorphisms, preprint. 27. D. W. Curtis, A hyperspace retraction theorem for a class of half-line compactifications, Topology Proc., 11 (1986), 29-64. 28. S. T. Czuba, On dendroids with Kelley’s property, Proc. Amer. Math. Sot., 102 (1988), 728-730. 29. E. K. van Douwen, Mappings from hyperspaces and convergent sequences, Topology Appl., 34 (19901, 35-45. 30. C. Eberhart, A note on smooth fans, Coiloq. Math., 20 (1969), 89-90. 31. V.V. Fedorchuk, Exponentials of Peano continua-fibre version variant, (Russian) Dokl. AN SSSR, 262 (1982), 41-44. 32. V. V. Fedorchuk, On open mappings, (Russian) Uspekhi Mat. Nauk, 37 (1982), 187-188.
XII.
392
SPECIAL TYPES OF MAPS BETWEEN HYPERSPACES
33. V. V. Fedorchuk, On hypermaps, which are trivial bundles, Lecture Notes in Math., Springer, Berlin-New York, 1060 (1984), 26-36. 34. S. Fujii and T. Nogura, Characterizations of compact ordinal spaces via continuous
selections,
preprint.
35. J. T. Goodykoontz, Jr., A non-locally connected continuum X such that C(X) is a retract of 2x, Proc. Amer. Math. Sot., 91 (1984), 319-322. 36. J. T. Goodykoontz, Jr., Some retractions and deformation retractions on 2-Y and C(X), Topology Appl., 21 (1985), 121-133. 37. Y. Hattori and T. Nogura, Continuous selections on certain spaces, Houston J. Math., 21 (1995), 585-594. 38. H. Hosokawa, Induced mappings between hyperspaces, Bull. Tokyo Gakugei Univ., 41 (1989), l-6. 39. H. Hosokawa, Mappings of hyperspaces induced by refinable mappings, Bull. Tokyo Gakugei Univ., 42 (1990), l-8. 40. H. Hosokawa, Induced mappings between hyperspaces, II, Bull. Tokyo Gakugei Univ., 44 (1992), 1-7. 41. H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math., 21 (1997)) 42.
43. 44. 45. 46.
239-250.
H. Hosokawa, Induced mappings on hyperspaces, II, preprint. A. Illanes, A continuum X which is a retract of C(X) but not of 2”, Proc. Amer. Math. Sot., 100 (1987), 199-200. A. Illanes, Two examples concerning hyperspace retraction, Topology Appl., 29 (1988), 67-72. to appear in A Illanes, The opennessof induced maps on hyperspaces, Colloq. Math. K. Kawamura and ED. Tymchatyn, Continua which admit no mean, Colloq. Math., 71 (1996), 97-105.
47. W. Kuperberg, Uniformly pathwise connected continua, Studies in TOpology (Proc. Conf. Univ. Norh Carolina, Charlotte); Academic Press, New York (1975), 315-324. 48. K. Kuratowski, Topology, Vol. I, Polish Scientific Publishers and Academic Press, 1966. 49. K. Kuratowski, Topology, Vol. II, Polish Scientific Publishers and Academic Press, 1968. 50. K. Kurat.owski, S. B. Nadler, Jr. and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Acad. Polon. Sci., S&. Sci. Math. Astronom. Phys., 18 (1970), 5-11. 51. A.Y.W. Lau, A note on monotone maps and hyperspaces, Bull. Acad. Polon. Sci., S6r. Sci. Math. Astronom. Phys., 24 (1976), 121-123.
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52. T. MaCkowiak, Continuous selections for C(X), Bull. Acad. Polon. Sci., S&. Sci. Math. Astronom. Phys., 26 (1978), 547-551. 53. T. MaCkowiak, Contractible and nonselectible dendroids, Bull. Polish Acad. Sci. Math., 33 (1985), 321-324. 54. M. M. Marsh and E. D. Tymchatyn, Inductively weakly confluent mappings, Houston J. Math., 18 (1992), 235-250. 55. J.R. Martin and S.B. Nadler, Jr., Examples and questions in the theory of fixed point sets, Canad. J. Math., 31 (1979), 1017-1032. 56. E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Sot., 71 (1951), 152-182. 57. G. J. Michaelides, A note on topological m-spaces, Colloq. Math., 32 (1975), 193-197. 58. J. van Mill, J. Pelant and R. Pol, Selections that characterize topological completeness, Fund. Math., 149 (1996), 127-141. 59. J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Sot., 83 (1981), 601-605. 60. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 61. S. B. Nadler, Jr., c-selections, Proc. Amer. Math. Sot., 114 (1992), 287-293. 62. S.B. Nadler, Jr. and L.E. Ward, Jr., Concerning continuous selections, Proc. Amer. Math. Sot., 25 (1970), 369-374. 63. T. Nogura and D. Shakhmatov, Characterizations of intervals via continuous selections, to appear in Rend. Circ. Mat. Palermo. 64. T. Nogura and D. Shakhmatov, Spaces which have finitely many continuous selections, preprint. 65. L. E. Ward, Jr., Rigid selections and smooth dendroids, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 19 (1971), 1041-1044. 66. L. E. Ward, Jr., Local selections and local dendrites, Topology Appl., 20 (1985), 47-58.
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XIII. More Contractibility Hyperspaces 78.
More
Contractibility
on Contractible
on of
Hyperspaces
vs. Smoothness
in Hyperspaces
As it was noted in Exercise 25.35, arc-smooth continua are contractible. We start this chapter by introducing a weaker version of arc-smoothness which is equivalent to the contractibility in hyperspaces. 78.1 Definition. A continuum X is said to be weak arc-smooth at p E X provided that there exists a continuous function (Y : X + C(X) satisfying the following conditions: (9 4~) = (~1, and (ii) for each 2 E X - {p}, o(z) is an arc from p to X is said to be weak arc-smooth provided that such that X is weak arc-smooth at p. Clearly, if X is arc-smooth at a point p, then X p. The converse is not true as it is noted in Exercise 78.2 Theorem. equivalent: (a) C(X)
For a continuum
X,
2. there exists a point p is weak arc-smooth 78.37.
the following
statements
at
are
is contractible,
(b) there is an element B E C(X) and there is a map cy : X + C(C(X)) such that for each 2 E X, o(z) is an arc from {z} to B (in the case that B = {b} E Fl(X), we ask that o(b) = {B}), 395
396
XIII.
(c) C(X) (d) C(X)
MORE ON CONTRACTIBILITY
is weak arc-smooth, is weak arc-smooth
OF HYPERSPACES
and at X.
Proof. (a) + (d) is easy to prove (Exercise 78.38), (d) * (c) and (c) (b) are immediate. We only prove (b) + (a). Let B E C(X) and let cy : X + C(C(X)) be a map such that ~(2) is an arc from {z} to B for each {z} E Fl(X) - {B}, and, in the case that *
B = {b} E Fl(X),
a(b) = {B}.
Fix a Whitney map p for C(X) that p(O) = B, p(l) = X and p(s) Lemma 14.2). For each 2 E X, let D, = U{A
and fix a map p : [0, l] + C(X) such P(t) if s 2 t (see Theorem 14.6 and
c
: A E Q(Z)}.
By Exercise
11.5, D, E
C(X). For each eIement A E Q(Z) let [A,s] = u(C that joins {z} and A}. Define G : Fl (X) x [O,l] -+ C(X) by
( [A747
: C is in the subarc of (Y(Z)
where A E a(x) is choosen in such a way that
W(x), t) =
A[A, xl) = WDz), Dz u PW - l),
ifO
If A,E E a(x) are such that p([A, z]) = 2tp(D,) = p([E,x]). Then since O(Z) is an arc, we have that one of the continua [A,x] and [E,x] is contained in the other. Since their size is the same, we conclude that [A,x] = [&xl. Thus G({x},t) is well defined for t E [0, $1. On the other and D, Up(Z(i) - 1) = D, Up(O) = hand, p([B,x]) = ~(0~) = 2(i)p(D,) D,. This proves that G({x}, 2) = D, with any of the two definitions of G. Therefore, G is well defined. Notice that G({z},O) = {x} and G({z}, 1)) = X for every x E X. We will show that G is continuous on the set Fl(X) x [0, f]. Suppose that G is not continuous on this set. Then there exist a sequence{ {z,}}~=~ in Fl (X), converging to an element {x} E Fl (X) and a sequence{ tn}rZ1 in [0, $1 converging to a number t E [0, i], such that G({xn}, tn) fi G({x}, t). Put G({G&J = [An, x,]. Let (An,z,) be the subarc of a(~,) that joins A,, and {z~}. Taking subsequencesif necessary, we may assume that G({zn}, tn) + D, for some D E C(X) - {G({x},t)}, and (A,,x,) -+ A for some A E C(C(X)). Since (An,x,) c cy(x,,) and cu(x,) + a(x), then A c a(x). Hence A is a subarc (or a one-point subset) of Q(Z) that contains the point {z}.
CONTRACTIBILITY
vs. SMOOTHNESS IN HYPERSPACES
397
Let A0 and {z}
be the end points of A. By Exercise 11.5 G({z,},&) = (An, zn)} tends to U{E : E E A} = [Ao,~]. Then 2t&Dz,) = p([A,,x,]) + p([Ao,z]). Since a(~,,) + a(z), + 2tp(D,). applying again Exercise 11.5, D,” + D,. Then 2t,p(D,,,) Thus p([Ao,x]) = 2tp(D,). This proves that G({z},t) = [Ao,z]. Thus G( {z,}, tn) + G( {z}, t). This contradiction proves the continuity of G on the set Fi(X) x [0, i]. It is easy to check that G is continuous on the set Fi (X) x [f , 11. Therefore, G is continuous. We have proved that Fi (X) is contractible in C(X). By Theorem 20.1, we conclude that C(X) is contractible. n
[A,,z,]
=
u{E
: E E
A natural question is whether arc-smoothness is equivalent to contractibility of hyperspaces. An attempt to solve this question in the positive was done by Dilks in [22]. Unfortunately, her proof contains a mistake. Goodykoontz has offered the following variatons to this question, all of them are still open. 78.3 Questions [30, Question 6.41. Does there exist a continuum X such that C(X) is contractible but not arc-smooth? If X is contractible, is 2x (respectively, C(X)) arc-smooth? If X has property (n), is 2x (respectively, C(X)) arc-smooth? 78.4 Question [30, Question 6.51. Let X be a continuum. arc-smooth, is C(X) arc-smooth?
If 2x is
78.5 Questions 130, Question 6.61. Let X be a continuum. If C(X) is arc-smooth, is 2” arc-smooth? In particular, if X is a continuum such that C(X) and the cone over X are homeomorphic, is 2x arc-smooth?
78.6 Questions [30, Question 6.71. Let X be a continuum. If 2x (respectively, C(X)) is arc-smooth at some point, is 2x (respectively, C(X)) arc-smooth at X? In Theorem 8 of [24], Eberhart proved that if X is a dendroid smooth at a point p, then C,(X) = {A E C(X) : p E A} is homeomorphic to the Hilbert cube if and only if p is not in the interior of a finite tree in X. Goodykoontz asked if this theorem can be generalized to arc-smooth continua. 78.7 Question [30, Question 6.101. If X is an arc-smooth continuum at a point p and p is not in the interior of a finite tree in X, is C,(X) homeomorphic to the Hilbert cube?
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The following theorem it is easy to prove (Exercise 78.39) 78.8 Theorem. equivalent.
For a continuum X, the following statements are
(a) C(X) is contractible, (b) there is a map a: : X + he(X) c C(C(X)) such that, for each 2 E X, o(z) is an order arc from z to X, where A,,(X) is the space of order arcs in C(X), that join an element of Fi (X) and X. Theorem 78.8 follows from Theorem 78.2. It also follows from the next theorem by Curtis. 78.9 Theorem [16, Theorem 5.41. Let X be a continuum. The hyperspaceC(X) is contractible if and only if there exists a lower semi-continuous set valued function F : X -+ C(C(X)), such that for every x E X we have: {x),X E F(X), and (b) if M E F(z), then there is an arc between {z} and M contained in F(Z) n C(M). (a)
Suppose that X is a dendroid, given points p, q E X, let pq denote the unique arc in X joining them if p # q, and pq = {p} if p = q. The dendroid X is said to be pointwise smooth ([20, (2.2), p. 1981)provided that for each point z E X, there exists a point p(z) E X such that for each sequence {~~}p=i in X, converging to 5, we have p(z)xn --+ p(z)z. Czuba has asked in [20, (3.10), p. 2021whether pointwise smooth dendroids are hereditarily contractible (the converse was proved in [20, (3.10) p. 2021). Since each subcontinuum of a pointwise dendroid is again a pointwise smooth dendroid (Exercise 78.40), the problem is equivalent to the problem of contractibility of pointwise smooth dendroids, which is an open problem. 78.10 Question [20, (2.2), p. 1981. If X is a pointwise smooth dendroid, is X contractible? Using Theorem 78.9, W. J. Charatonik proved the following result. 78.11 Theorem Ill]. Pointwise smooth dendroids have contractible hyperspaces C(X) and 2x. As a consequence of Theorem 78.11, W. J. Charatonik obtained the following corollary.
R3-S~~s
399
78.12 Corollary [II, p. 4111. A dendroid X is pointwise smooth if and only if each subcontinuum Y of X has contractible hyperspaces 2’ and C(Y). Related to these results W. J. Charatonik questions.
has raised the following
two
78.13 Question [ll, Question 11. Characterize those continua such that their subcontinua have contractible hyperspaces. Note that there are such continua which are not dendroids, for example the sin( $)-continuum. In particular: is it true that such continua are exactly those which do not have subcontinua containing R3-continua? (see Definition 24.12, for dendroids it follows from [20, (3.9), p. 2021 and from Corollary 78.12). 78.14 Question [ll, Question 21. Given a pointwise X, does there exist a retraction r : C(X) -+ Fl(X)?
smooth dendroid
If the answer to Question 78.14 is positive, since contractibility is preserved under retractions, Theorem 78.11 implies that Question 78.10 has a positive answer.
R3-Sets The following theorem gives one of the most useful conditions contractibility of C(X).
for non-
78.15 Theorem [12, Corollary 41 and [3, Theorem 3.21. Let X be a continuum. If X contains an R3-set (see Definition 24.12), then C(X) is not contractible. Proof. Let K be an R3-set in X. Then K # X and there exist an open subset U of X and a sequence of components {Cn}F=i of U such that K = liminf C,, c U. Suppose that C(X) is contractible. By Exercise 78.35, there exists a map G : X x [0, l] + C(X) such that (a) G(z,O) = {z} and G(z, 1) = X for each z E X, and (b) if s < t and 2 E X, then G(z,s) c G(z,t). Fix a point p E K. By Exercise 78.36, there exists a sequence of points {pn}~Tl of X such that p, -+ p and pn E C, for each n 1 1. Define to = max{t E [0, l] : G(p, t) c K}. Since G(p, 1) = X, to < 1. Since G(p, to) c U, there exists ti > to such that G(p, tl) c U. Then there exists N 2 1 such that G(p,,tl) c U for each n 2 N. c U for each n > N. By property (b), {pn} = G(p,,O) c G(p,,tl) Thus G(p,, ti) c C,. Therefore, G(p, ti) = lim G(p,, ti) c lim inf C, = K.
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This contradicts the choice of to and completes the proof of the theorem. n 78.16 Example [12, Example 51. There exists a dendroid X such that C(X) is not contractible and X does not contain an R3-set. The dendroid X is represented in Figure 52 (next page). Each A, is a dendroid consisting of a triad and two sequencesof segments. The sequenceof dendroids {A,}r!l converges to the arc ab. Each B, is a dendroid similar to A,. In Exercise 78.44 it is asked to prove the properties of X. Consider the following statements: (a) X contains an R3-continuum, (b) C(X) contains an R3-continuum, (c) 2.’ contains an R3-continuum, (d) X contains an R3-set, (e) C(X) contains an R3-set, and (f) 2” contains an R3-set. It is known that (b) + (a) ([37, Example 2.1]), (a) 3 (c) ([12, Theorem 31, Exercise 78.48), (d) + (e) ([3, Theorem 3.71) and (d) + (f) ([3, Theorem 3.61). 78.17 Questions. Which of the following implications are true (c) 9 (b), (4 * (4 (W Quest ion 20]), (e) 3 (d), (f) j (e) and (f) 3 (d)? In Theorem 6 of [12], W. J. Charatonik, claimed that implication (a) 3 (b) is true. His proof has a mistake which is discussedin Exercise 78.46. So the question whether this implication is true or not is still open. In Theorem 1.5 of [37], using Charatonik’s Theorem, it was claimed that for dendroids (a), (b) and (c) are equivalent. The situation is that, for dendroids, (a) and (c) are equivalent and (b) implies (a). So the following question has special interest. 78.18 Question.
Does (a) imply (b) for the casethat X is a dendroid?
Spaces of Finite We defined symmetric products F,,(X) and 1.8, respectively.
Subsets and the hyperspace F(X)
in 1.7
78.19 Problem. Characterize the continua X for which F(X) (F,(X)) is contractible. With respect to Problem 78.19, the following facts are known: (a) if X is locally connected, then F(X) is an AR ([17, Lemma 3.6]),
SPACES OF FINITE
SUBSETS
A dendroid X such that C(X) is not contractible an R3-set (78.16) Figure 52
401
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(1~) if X is contractible, then F(X) is contractible (Exercise 78.49), and (c) F(X) is arcwise connected if and only if X is arcwise connected ([17, Lemma 2.31). 78.20 Question. Let X be a continuum. If F,(X) is contractible for some 11 2 2, then is X contractible? If Fa(X) is contractible, then is X contractible’? Related to Question 78.20, by [SO], F,(S1) is not contractible for any n 2 1. Generalizing results of [36, Theorem 2.61 and [28], Macias in [47] has proved that F,,(X) is unicoherent for each 71 2 3 and for each continuum S. He also has shown that r(Fx(X)) 5 1 for each continuum X. (r( ) represent,s the multicoherence degree defined in 64.4.) For locally connected continua X, if X is unicoherent, then Fz(X) is also unicoherent [28] and if X is not unicoherent, then T(F~ (X)) = 1 ([36, Theorem 1.61). Recently, Castaiieda in [4] has shown an example of a unicoherent continuum X such that Fz(X) is not unicoherent.
Admissibility 78.21 Definition [53]. Let X be a continuum, let d denote a metric for X and let 2 E X. The total fiber at the point z, is the set T(z) = {A E C(X) : z E A}. An element A E F(z) is said to be admissible at 2 if, for each E > 0, there is a 6 > 0 such that for each y E B(6, z), there exists B E F(y) such that Hd(A,B) < 6. The continuum X is said to be admissible if for each 2 E X and for each Whitney level A for C(X), there exists A E A n 7(z) such that A is admissible at z. Admissibility was introduced in [53]. The relations between this concept and the contractibility of hyperspaces have been studied in [2], [3], [32], [33], [34], [35], [43], [50], [51], [52], [53], [54], [55] and [56]. Here, we will only show that contractibility of C(X) implies admissibility of X and a counterexample showing that the converse of this implication is not true. 78.22 Theorem [51, Proposition is contractible, then X is admissible.
1.81. Let X be a continuum.
If C(X)
Proof. Let d denote a metric for X. Let Suppose that C(X) is contractible. By Exercise 78.35, there exists a map G : C(X) x [0, l] + C(X) such that: (a) G(A,O) = A and G(A, 1) = X for each A E C(X), and (b) G(A, s) c G(A, t) if A E C(X) and 0 < s 5 t 5 1.
MAPS
PRESERVING
HYPERSPACE
CONTRACTIBILITY
403
is admissible at 5 for each Let z E X. We will see that G({z},t) t E [0, 11. In order to do this, let E > 0. Let b > 0 be such that if t E [0, l] and Hd(A,B) < 6, then Hd(G(A,t),G(B,t)) < E. ThenforeachyEB(6,z),G({y},t)EF(y) andHd(G({z},t),G({~/),t))< c. This proves that G({z}, t) is admissible at 2. GivenaWhitneyleveldforC(X),sinceG({z},O)={~}andG({z},1)= X, there exists t E [0, 11such that G({z}, t) E d. Thus X is admissible. n The following example is a slight modification of Example 6.3 of [16]. 78.23 Example [16, Example 6.31. There is an admissible continuum X such that C(X) is not contractible. The continuum X is illustrated in Figure 53 (b) ( next page). The fundamental part of the contruction of X is the continuum 2 represented in Figure 53 (a). The continuum 2 is the union of a triod T and a sequence of arcs converging to T. In the continuum X, each 2, is a topological copy of 2 with its respective triod Tn. The sequences{Z,l}~zp=l and {Tn}rcl both converge to ab. Each Y,, is symmetric to 2, with respect to the point b. In Exercise 78.51 it is asked to show that X is admissible and C(X) is not contractible.
Maps
Preserving
Hyperspace
Contractibility
As it is observed in Exercises 20.25 and [50, Theorem 3.11, contractibility of hyperspaces is preserved by open surjections and maps with right homotopy inverses, respectively. Using Theorem 78.9 Curtis extended these results with the following theorem. 78.24 Theorem [16, Proposition 7.11. Let g : Y + X be a map between continua, and supposethere exists a lower semi-continuous set-valued function @ : X + Y such that 29 o @ : X -+ X is a single-valued function (therefore continuous) which is homotopic to the identity map defined on X. Then if C(Y) is contractible, so is C(X). 78.25 Definition [25]. A continuous function between continua g : X + Y is refinable if for every E > 0 there exists an onto map f : X -+ Y such that diameter (f-‘(y)) < E for each y E Y, and sup{&(f(z),g(z)) : x E X> < E, where dy denotes a metric for Y. In [39, Theorem 2.11, Kato proved that refinable maps preserve Kelley’s property and he asked the following question.
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z 7
. .
a*
1
. I!
.
T (4
a I
An admissible
continuum
X such that C(X) Figure 53
is not contractible
(78.23)
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405
78.26 Question [39, Question 2.61. Let X and Y be continua. Suppose that g : X + Y is a refinable map between continua and C(X) is contractible. Is C(Y) contractible?
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on Kelley’s
Property
The following questions are still open (seeRemark after Exercise 20.28). 78.27 Questions [48, Questions 16.371. Let X be a continuum. If X has property (K), then does C(X) have property (K)? If 2x has property (K), then does C(X) have property (K)? The first part of Question 78.27 has been affirmatively solved for the property (K)* defined in 50.5 (see [41, Theorem 2.61). Kato ([42, Corollary 3.31) has discovered that a positive answer to Question 50.2 (if X has property (K), then does X x [0, I] have property (K)?) implies a positive answer to the first question of 78.27. 78.28 Questions [41, Question 21. If X is an hereditarily indecomposable continuum, does X have property (K)*? (see Definition 50.5). Is it true that C(X) has property (K)? In [7, Problem 61, J. J. Charatonik and W. J. Charatonik have proposed the following problem. 78.29 Problem erty (K).
[7, Problem 61. Characterize dendroids having prop-
An important step in the solution of Problem 78.29, was given by Czuba with the following result. 78.30 Theorem smooth.
[21]. If a dendroid X has property (K), then X is
For dendroids with property (K) hereditarily (a continuum X has propif every subcontinuum of X has property (n)), Questions 78.29 have been completely answered. erty (K) hereditarily
78.31 Theorem [49, Theorem 2.11. A dendroid X is a dendrite if and only if X has property (K) hereditarily. Recently, in [l] it has been showed the following generalization of Theorem 78.31.
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78.32 Theorem (1, Theorem 4.11. Let X be an arcwise connected continuum. Then X has property (K) hereditarily if and only if X is hereditarily locally connected. 78.33 Question [7, Question 71. Let X be a dendroid having property (k), then is X the limit of an inverse sequence of finite trees with confluent bonding maps? The notion of property of Kelley can be extended to Hausdorff in the following natural way.
continua
78.34 Definition [14]. Let Y be a compact connected Hausdorff space. We say that Y has the property of Kelley if for any point p E X, for any continuum A containing p and for any open subset U of C(Y) with A E U, there exists an open subset U of X such that p E U and if q E U, then there exists L E C(Y) such that q E L and L E U. As it was shown in the proof of Corollary 20.18, homogeneous continua have property (6). Recently, W. J. Charatonik has shown in [14] that there is a homogeneous Hausdorff continuum Y such that Y does not have the property of Kelley.
Exercises 78.35 Exercise. Let X be a continuum. The hyperspace C(X) is contractible if and only if for each Whitney map p for C(X) with p(X) = 1, there exists a map G : C(X) x [0, l] + C(X) such that: (a) G(A, 0) = A and G(A, 1) = X for each A E C(X), (b) G(A,s) c G(A,t) if A E C(X) and 0 < s < t < 1, and (c) for each t E [0, 11, G(p-‘([O, t]) x {t}) = p-‘(t) and for each A E
/.-‘([t, 111,G(A, t) = A. 78.36 Exercise. Let X be a continuum and let {C,,}~=r of nonempty subsets of X. Then p E lim inf C,, if and only a sequence of points {p,)z!r of X such that p, -+ p and p, n 2 1. [Hint: Let d denote a metric for X. For each n 2 1, let p, E R’ : q E C,} + i.] that d(p,p,) < inf{d(p,q) 78.37 smooth at [Hint: the y-axis
be a sequence if there exists E C, for each E C, be such
Exercise. There is a. continuum X such that X is weak arca point p and X is not arc-smooth at p. Consider the continuum which is obtained by rotating around the continuum illustrated in Figure 54 (top of next page).]
EXERCISES
407
P ,. ..
Weak arc-smoothness
does not imply arc-smoothness
(78.37)
Figure 54
78.38
Exercise.
78.39 Exercise. 78.40 Exercise. is again a pointwise
C(X
Prove the implication
(a) =S (d) of Theorem 78.2.
Prove Theorem 78.8. Each subcontinuum smooth dendroid.
of a pointwise
smooth dendroid
78.41 Exercise. [50, Theorem 4.21. Let X and Y be continua, then x Y) is contractible if and only if C(X) and C(Y) are contractible. [Hint: See Exercise 20.24.1
78.42 Exercise. and only if C(C(X))
Let X be a continuum. is contractible.
Then C(X)
is contractible
if
78.43 Exercise. Let X be a continuum. If K is an R3-set of X, then X is not locally connected at any point of I<.
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78.44 Exercise. The dendroid in Example 78.16 doesnot have R3-sets and it does not have contractible hyperspaces C(X) and 2x. [Hint: It is impossible to move the point c under any homotopy G : X x [O,l] -+ C(X) with the properties stated in Exercise 78.35.1 78.45 Exercise. If the continuum X contains an R3-set, then F,(X) and F(X) are not contractible. 78.46 Exercise. Let X be a continuum. Let K be an R3-continuum of X. Let U be an open subset of X and let {C,}~Zp,, be a sequence of components of U such that K = lim inf C,, and K C U. In the proof of Theorem 6 of [12], it is claimed that lim inf C(C,) is a subcontinuum of X. Show that for the case that I< = ef U cd in Example 76.3 (Figure 48, p. 373), lim inf C(C,) is not connected. [Hint: K is an isolated point of lim inf C(C,).] 78.47 Exercise. Let X be a continuum. If U is a nonempty open subset of X and C is a component of U, then 2c is a component of 2”. 78.48 Exercise [12, Theorem 31. Let X be a continuum. If K is an R3-continuum of X, then 2K is an R3-continuum of 2aY. 78.49 Exercise [29, Proposici6n 3.31. Let X be a continuum. Fl(X) is contractible in F(X) if and only if F(X) is contractible.
Then
78.50 Exercise [50, Theorem 3.11. Let X be a continuum. Then contractibility of C(X) is preserved by maps with right homotopy inverses. [Hint: Use Exercise 77.30.1 78.51 Exercise. The continuum X in Example 78.23 is admissible and C(X) is not contractible. [Hint: For each b,, each vertical segment containing b, is admissible at b,. The horizontal segmentscontaining b are admissible at b. In order to show that C(X) is not contractible, observe that the set {b} can not be moved under any homotopy G : C(X) x [0, l] + C(X) with the properties stated in Exercise 78.35.1
References 1. 2.
G. Acosta and A. Illanes, Continua which have the property of Kelley hereditarily, preprint. B.S. Baik, K. Hur, P. K. Lim and C. J. Rhee, Hyperspace contractibility of type sin($)-continua, J. Korean Math. Sot., 29 (1992), 15-42.
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B.S. Baik, K. Hur and C.J. Rhee, I?-sets and contractibility, J. Korean Math. Sot., 34 (1997), 309-319. E. Castafieda, A unicoherent continuum for which its second symmetric product is not unicoherent, preprint. J. J. Charatonik, The property of Kelley and confluent mappings, Bull. Polish Acad. Sci. Math., 31 (1983), 9-12. J. J. Charatonik, Some problems on generalized homogeneity of continua, Lecture Notes in Math., Springer, Berlin-New York, 1060 (1984), l-6. J. J. Charatonik and W. J. Charatonik, ley, Topology Appl., 29 (1988), 73-78.
Fans with the property
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8.
J. J. Charatonik and W. 3. Charatonik, Smoothness and the property of Kelley, preprint. 9. W. J. Charatonik, Hyperspaces and the property of Kelley, Bull. Acad. Polon. Sci., SQ. Sci. Math., 30 (1982), 457-459. 10. W. J. Charatonik, On the property of Kelley in hyperspaces, Lecture Notes in Math., Springer, Berlin-New York, 1060 (1984), 7-10. 11. W. J. Charatonik, Pointwise smooth dendroids have contractible hyperspaces, Bull. Polish Acad. Sci. Math., 33 (1985), 409-412. 12. W. J. Charatonik, Ri-continua and hyperspaces, Topology Appl., 23 (1986), 207-216. 13. W. J. Charatonik, Some functions on hyperspaces of continua, Topology Appl., 22 (1986), 211-221. 14. W. J. Charatonik, A homogeneous continuum without the property of Kelley, to appear in Proc. Amer. Math. Sot. 15. W. J. Charatonik and W. Makuchowski, Smoothness of hyperspaces and of Cartesian products, preprint. 16. D. W. Curtis, Application of a selection theorem to hyperspace contractibility, Canad. J. Math., 37 (1985), 747-759. 17. D. W. Curtis and N. T. Nhu, Hyperspaces of finite subsets which are homeomorphic to No-dimensional linear metric spaces, Topology Appl., 19 (1985), 251-260. 18. S. T. Czuba, R2-continua and contractibility of dendroids, Bull. Acad. Polon. Sci., %r. Sci. Math., 27 (1979), 299-302. 19. S. T. Czuba, Ri-continua and contractibility, Proceedings of the International Conference on Geometric Topology (PWN, Warszawa, 1980), 75-79. 20. S.T. Czuba, On pointwise smooth dendroids, F’und. Math., 114 (1$X31), 197-207.
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21. S. T. Czuba, On dendroids with Kelley’s sm., 102 (1988), 728-730.
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22. A. M. Diiks, Structure of hyperspaces,Dissertation, Tulane University, New Orleans, La., (1980); J. T. Rogers, Jr., Director of Dissertation. 23. A. M. Dilks and 3. T: Rogers, Jr., Whitney stability and contractible @perspaces,Proc. Amer. Math. Sot., 83 (1981), 633-640. 24. C. Eberhart, Intervals of continua which are Hilbert cubes,Proc. Amer. Math. Sot., 68 (1978), 220-224. 25. J. Ford and J. W. Rogers, Refinable maps, Colloq. Math., 39 (1978), 263-269. 26. J. B. Fugate, G. R. Gordh Jr. and L. Lum, On arc-smooth continua, Topology Proc., 2 (1977), 645-656. 27. J.B. Fugate, G.R. Gordh Jr. and L. Lum, Arc-smooth continua, Trans. Amer. Math. Sot., 265 (1981), 545-561. 28. T. Ganea, Symmetrische Potenzen topologischer Rcume, (German) Math. Nachr., 11 (1954), 305-316. 29. R. M. Garcia, El hiperespacio de subconjuntos finitos de un continua, (Spanish) Tesis de Licenciatura, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, 1995. 30. J. T. Goodykoontz, Jr., Arc-smoothness in hyperspaces,Topology Appl., 15 (1983), 131-150. 31. H. Hosokawa, Some remarks on the atomic mappings, Bull. Tokyo Gakugei Univ., 40 (1988), 31-37. 32. K. Hur, S. W. Lee, P. K. Lim and C. J. Rhee, Set-valued contractibility of hyperspaces,J. Korean Math. Sot., 29 (1992), 341-350. 33. K. Hur, R. Moon, C. J. Rhee, Connectedness im kleinen and local connectednessin C(X), Honam Math. J., 18 (1996), 113-124. 34. K. Hur, R. Moon, C. J. Rhee, Connectednessim kleinen and components in C(X), Bull. Korean Math. Sot., 34 (1997), 225-231. 35. K. Hur and C. J. Rhee, On spaceswithout Ri-continua, Bull. Korean Math. Sot., 30 (1993), 2955299. 36. A. Illanes, Multicoherence of symmetric products, An. Inst. Mat. Univ. Nat. Autonoma Mexico, 25 (1985), 11-24. 37. A. Illanes, R3-continua in hyperspaces,Houston J. Math., 20 (1994), 529-538. 38. W. T. Ingram and D. D. Sherling, Two continua having a property of J. L. Kelley, Canad. Math. Bull., 34 (1991), 351-356. 39. H. Kato, Concerning a property of J. L. Kelley and refinable maps, Math. Japonica, 31 (1986), 711-719.
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40. H. Kato, Generalized homogeneity of continua and a question of J. J. Charatonik, Houston J. Math., 13 (1987), 51-63. 41. H. Kato, On the property of Kelley in the hyperspace continua, Topology Appl., 30 (1988), 165-174.
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42. H. Kato, A note on continuous mappings and the property Kelley, Proc. Amer. Math. Sot., 112 (1991), 114331148. 43. I. S. Kim, R. S. Kim and C. J. Rhee, W-regular convergence continua, Bull. Korean Math. Sot., 31 (1994), 105-113.
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44. P. Krupski, The property of Kelley in circularly chainable and in chainable continua, Bull. Acad. Polon. $ci., Ser. Sci. Math., 29 (1981), 377-381. 45. P. Krupski, The property of Kelley, local end points and homogeneous continua, Houston J. Math., 10 (1984), 215-225. 46. P. Krupski, Berlin-New
Open images of solenoids, Lecture Notes in Math., Springer, York, 1060 (1984), 76-83.
47. S. Ma&s, On symmetric products of continua, to appear in Topology APP~. 48. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. and I. Puga-Espinosa, Shore points and dendrites, 49. V. Neumann-Lara Proc. Amer. Math. Sot., 118 (1993), 939-942. 50. T. Nishiura and C. J. Rhee, Contractibility of the hyperspace continua, Houston J. Math., 8 (1982), 119-127. 51. T. Nishiura and C. J. hyperspaces, Topology 52. T. Nishiura and C. J. Kyungpook Math. J.,
Rhee, An admissible condition Proc., 8 (1983), 303-314. Rhee, Contractible hyperspaces 24 (1984), 1433154.
of
sub-
for
contractible
of
subcontinua,
53. C. J. Rhee, On a contractible hyperspace condition, Topology Proc., 7 (1982), 147-155. 54. C.J. Rhee, Obstructing sets for hyperspace contraction, Topology Proc., 10 (1985), 159-173. 55. C. J. Rhee, M-set and contraction of C(X), des Sciences de Liege, 56 (1987), 55-70. 56. C. J. Rhee, Admissible fibers and T-admissibility ogy Appl., 64 (1995), 75-83.
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57. M. van de Vel, Invariant arcs, Whitney levels, and Kelley Trans. Amer. Math. Sot., 326 (1991), 749-771.
continua,
412
XIII.
MORE
ON CONTRACTIBILITY
OF HYPERSPACES
58. R. W. Wardle, On a property of J. L. Kelley, Houston J. Math., 3 (1977), 291-299. 59. M. Wojdyslawski, Re’tractes absolus et hyperespaces des continus, (bench) Fund. Math., 32 (1939), 184-192. 60. W. Wu, Note SW les produits essentielssyme’triques des espacestopologiques, I, (French) Comptes Rendus des SBancesde 1’Academie des Sciences, 16 (1947), 1139-1141.
XIV.
Products, Cones and Hyperspaces
79.
Hyperspaces
Which
Are Products
In this section we will discuss, for a continuum X, when the hyperspace C(X) is a product. We say that C(X) is a product, or write C(X) M Y x 2 when C(X) is homeomorphic to the Cartesian product of nondegenerate continua Y and Z. If X is a locally connected continuum and X does not contain free arcs, then C(X) is homeomorphic to the Hilbert cube (Theorem 11.3). Thus there are many continua X for which C(X) is a product. Answering a Question in 3.19 of [12], in [5, Example 4.11 it was shown that there is a non-locally connected continuum X such that C(X) is homeomorphic to x x [O,11. Combining Theorems 5.4 and 5.5 of [8] and 9.7 of [3], the following theorem follows. 79.1 Theorem (see paragraph above). Let X be a locally connected continuum. If C(X) is a finite-dimensional product, then X is an arc or a circle (and conversely). Answering Question 2.0 of [12], recently it has been proved in [5] that the hypothesis of local connectednessis not necessaryin Theorem 79.1. The rest of this section is devoted to present the proof of the following theorem. 79.2 Theorem [5, Theorem 3.81. Let X be a continuum. Then C(X) is a finite-dimensional product if and only if X is an arc or a circle. The steps for the proof of Theorem 79.2 are the following. 1. We assumethat X is a continuum, C(X) is finite-dimensional and C(X) = Y x Z. Then Y and Z are arcwise connected (Exercise 79.11). 413
414
XIV.
PRODUCTS,
CONES AND HYPERSPACES
We also assumethat X is not locally connected. We define the notion of wrinkles in C(X) and we prove that, since X is not locally connected, then C(X) contains a wrinkle (Theorem 79.4). 2. The notion of a wrinkle is specifically defined for hyperspaces. We define, for a topological space W the notion of folds in W. We prove that, since C(X) contains a wrinkle, then C(X) contains a fold (Theorem 79.7). 3. We prove (Theorem 79.8) that if Y x Z contains a fold, then either Y or Z contains a fold. Then there is an arc (Y in Y x Z (E C(X)) such that Y x Z has a fold at every point of cy. 4. We show (proof of Theorem 79.10) that if C(X) has a fold at every point of an arc cy, then X contains mods for every m 1 1. This will be a contradiction since C(X) is finite-dimensional (see Theorem 70.1). The contradiction will prove that X is locally connected. Thus we can apply Theorem 79.1 to obtain Theorem 79.2. The proof presented here of Theorem 79.2 is shorter than the original one presented in [5]. The main difference is that we substitute the results about arcs of wrinkles (Lemma 3.2, Lemma 3.3, Lemma 3.4, Theorem 3.5 and Theorem 3.6 of [5]) for the simpler arguments contained in Lemma 79.9 and Theorem 79.10.
Wrinkles At this moment we are not assuming that C(X) is a finite-dimensional Cartesian product. This is done because we want specify the necessary hypothesis in each of the theorems. 79.3 Definition [5, Definition 1.21. Let X be a continuum. We say that C(X) has a wrinkle at an element A E C(X) - Fi(X) if there exist a point a E A, a sequence {a,}~& in X - A and a sequence {An}rzl in C(X) such that a, + a, A,, -+ A, Al > AZ > . . ., a, E A,, for each n > 1 and, if B E C(A1) and {n 2 1 : a,, E B} is infinite, then A C B. 79.4 Theorem [5, Theorem 1.31. Suppose that X is an hereditarily decomposable continuum and there exists M > 1 such that X contains no M-ods. Then C(X) has a wrinkle if and only if X is not locally connected. Proof. The necessity is left as Exercise 79.18 and it will not be used for the proof of Theorem 79.2. So we only prove the sufficiency. Suppose that X is not locally connected. Let d denote a metric for X. Then there is an open subset U of X and there is a component C of U such that C is not open. Fix a point a E C - intx(C). Let E > 0 be such that B(E,~) c U. For each n 2 1, choose a point a,, E B(;,a) - C. Then a, + a.
FOLDS
415
Let A = {B E C(X)
: {n 2 1 : a,, E B} is infinite}.
Fix a Whitney map p : C(X) + R1. Since X E A, A # 0. So there exists A E &(x)(d) such that p(A) = min{p(B) : B E clccx,(d)}. We will show that C(X) has a wrinkle at A. Since B = {B E C(X) : a E B} is a closed subset of C(X) and A C 13, &(,x)(A) c B. Then a E B for every B E cZc(.u)(d). In particular, a E A. If A = {a}, let B E A such that Hd(A,B) < 6. Then B C B(e,a) C U. This implies that B c C. Thus C contains infinitely many a,. This contradicts the choice of a, and proves that A # {u}. Since X is hereditarily decomposable, there exist two proper subcontinua D and E of A such that A = D U E. By the choice of A, the sets D and E do not belong to A. Then there exists N > 1 such that, a, $! D and a, 4 E for each n > N. Thus a, $ A for each n > N. Let {B,}p--i be a sequence in A such that B,, + A. For each n 2 1, let
A,=AuB,uB,+~u.... By Exercise 11.5, A, E 2x, and A, is connected since u E B, n A for each m >_ 1. Thus A, E C(X). Notice that Al > A2 > ... and A, + A. Let ni > N be such that a,, E B1 c AI. Let n2 > n1 be such that a,,, E B2 C AZ. Proceeding in this way it is possible to construct a sequence N < nl < 712 < ... such that a,,,, E A, for every m 2 1. Notice that a,,,, $Aforeverym>l. Finally we will show that the continuum A, the point a, the sequence {A,}$l in C(X) and the sequence of points (a,,}~=, in X satisfy the properties in the definition of wrinkle at A. Let B E C(A1) be such that {m > 1 : a,, E B} is infinite. Notice that B E A and a E B. Foreachm>1,{u,,,,u,,+,,...}~A,. Then{k>l:a,,E4,nB} is infinite. Since X contains no M-ods, by Exercise 14.20, A,, n B has at most M - 1 components. Thus there exists a component D, of A, n B such that {k 2 1 : utlb E Dn} is infinite. Then D, E A. such that D,,,? + Bo Let (Dm,.}~~l be a subsequence of {Dm}~=l for some Bo E C(X). Then Bo E clccx,(d) and p(A) 5 p(Bo). Since D m, c Anr n B, BO c A rl B. Then ,u(Bo) 5 p(A). Then B. c A and p(Bo) = p(A). Thus Bo = A and Bo c B. Hence A c B. This completes the proof that C(X) has a wrinkle at A. m
Folds 79.5 Conventions. The symbol P* denotes the subset of R2, defined by P* = J u (u{Jn : n > l}), where J, = [0, l] x {A} and J = [0, l] x (0).
416
XIV.
PRODUCTS,
CONES AND HYPERSPACES
We also define P = P* U JO, where .Js = (0) x [O,l]. For each p = (x,0) E J and n > 1, define p(n) = (5, k). Then p(n) -+ p. Let 0 = (0,O) E R2. If U is an open subset of a continuum X and p is a point of U, we denote by comp(U,p) the component of U containing p. 79.6 Definition [5, Definition 2.21. A continuum W has a fold at a point w E W if there exists a continuous function f : P + W such that f (0) = w and for each p E J - (01, there exists an open subset U of W such that f(p) E U and comp(U, f(p)) does not intersect {f(p(n)) : n > 1). The converse of the implication contained in the following theorem is also true ([5, Theorem 2.41). Here, we only prove the implication that we will use in this section. 79.7 Theorem [5, Theorem 2.41. Suppose that X is an hereditarily decomposable continuum and there exists M 2 1 such that X contains no M-ods. Let A E C(X) - Fi(X). If C(X) has a wrinkle at A, then C(X) has a fold at A. Proof. Let d denote a metric for X. Suppose that a is a point in A, {a,}~xI is a sequencein X - A and {An}FcI is a sequencein C(X) and they satisfy the conditions in the definition of a wrinkle at A. Fix a Whitney map /L : C(X) + [0, l] such that p(X) = 1. For each n 2 1, by Theorem 16.9 there exists a map cn : [0, l] + C(X) such that o,, is a segment in C(X) with respect to ~1from {a,} to A,. Let S,(C(X)) be the space of segments in C(X) with respect to p. By Theorem 17.4, S,(C(X)) is compact. Then there exists a subsequence and there exists a map B : [0, l] -+ C(X) such that ~~n,l~1 of {Gltp,l 0 is a segment in C(X) with respect to p and o,,~ -+ u, with the uniform metric p defined in section 17. Then
Since ~(o,*~,a) + 0, we conclude that ~(0) = {a}. Similarly, ~(1) = A. Therefore, cr is a segment from {a} to A. By Exercise 79.19, there exists a map 0: : [0, l] + C(X) such that a(O) = A and o(k) = A,, for each Ic > 1. Define F: ‘P --+ C(X) by
~(1 - t),
if s = E for some k 2 1, if s = 0,
4s),
if (t,s) E (0) x [0, 11.
gnk(l
F(t,s)
=
- t),
FOLDS
417
If (t, s) is of the form (t,s) = (0, i), then LT,,(~ - t) = A,, = a($). If (t,s) = 0, then a(1 - t) = A = a(O). This proves that F is well defined and F(O) = A. We will seethat FlP* is continuous. : lc > 1). In It is clear that FlP* is continuous at the points of U{Jk order to show that FIT’* is continuous at a point p = (T, 0) E J, let c > 0. Let 6s > 0 be such that if Ir - t( < 60, then Hd(a(l -r), o(1 -t)) < f. Let K 2 1 be such that p((~,cr~~) < 5 for every k 2 K. Let b = min{&, k}. Let (t, s) E P* be such that Ir - tl < 6 and s < 6. Then s = 0 or s = i for some k > K. - r),g(l - t)) < i. If s = 0, then Hd(F(p), F(t,s)) = Hd(a(l If s = i for some k > K, then Hci(F(p), F(t, .$)I = Kd4 &(a(1
- r),fr(l
-t))
- ~1,gnk (1 - t)) 5 E t + Hd(cr(1 - t),c&(1 -t)) < 2 + 5 = E.
In both casesHd(F(p), F(t, s)) < c. This proves that Fl’P * is continuous at p and completes the proof that FI’P* is continuous. Since a is continuous, we conclude that F is continuous. Now, let p = (t,O) E J - (0). We will show that there exists an open subset U of C(X) such that F(p) E U and comp(U, F(p)) rl {F@(k)) : k > 1) = 0. Let E = F(p). Then p(E) = p(g(l -t)) = tp(a(0)) + (1 - t)p(a(l)) = (1 - t)p(A) < p(A). Thus E is properly contained in A. Fix a point z E A - E. Let U be an open subset of X such that E c U and z 4 cl~(U). Let Ui = {B E C(X) : B c 17). Then Ui is an open subset of C(X) and cl~(x~(Ui) c {B E C(X) : B c cl,y(U)}. Let Cr = comp(Ui, E) and let S = u{B : B E &cx,(Cr)). By Exercise 15.9, S E C(X). Notice that E c S and z $ S. Notice that if k 2 1 and F(p(k)) E Ci, then {a,,} = crlle(0) c ~~~(1 t) = F@(k)) c S. Thus a,, E S. Let No = {k 2 1 : F(p(k)) E Ci}. We will show that Ns is finite. Suppose to the contrary t,hat No is infinite. By the previous paragraph {k 2 1 : a,, E S} is infinite. Since X contains no M-ods, A,, nS has a finite number of components (Exercise 14.20). Then there exists a component B of A,, n S such that the set {k 2 1 : a,, E B} is infinite. By the definition of {a,}r!i, A C B. This is a contradiction since B c S and z $ S. Hence No is finite. Since (ftnk : k E No} does not intersect A, there exists an open subset VofXsuchthatE~Vccl~(V)cU-{a,,,:kENc}. Define U = {B E C(X) : B c V}. Then U is an open subset of C(X) and E E U C Ur. If there exists k > 1 such that F(p(k)) E comp(Z4,E) c
418
XIV.
PRODUCTS,
CONES AND HYPERSPACES
Cl, then a,, E F(p(k)) c V and k E No. This contradicts the choice of V
and proves that, comp(U, E) n {F(p(k)) construction of U. Therefore, C(Jri) has a fold at A. n
: k > 1) = 0. This completes the
79.8 Theorem [5, Theorem 2.51. Let X be a continuum. Supposethat C(X) x Y x 2. Suppose that Y x 2 has a fold. Then Y has a fold or 2 has a fold. Proof. Let 7ry and ~/TZbe the respective projections from Y x Z onto Y and 2. Let f : P + Y x 2 be a map such that f(o) = (yo,zo) and, for each p E J - {O}, there exists an open subset UP of Y x 2 such that f(p)
E4
and cow&,
f(p))
n U(P(~))
: 12 2 I)=
0.
For each p E P, let f(p) = (y(p), z(p)). Let q = (1,O). Let Yq and Z, be open subsetsof I’ and Z, respectively, such that (y(q), z(q)) E ‘I< x Z, c U,. Since comp(Yq, y(q)) x comp(Z,, z(q)) c comp&, f(s)), (cow(Y,, y(q))x comp(Z,,z(q))) n {f(q(n)) : n > 1) = 0. This implies that the set of positive integers is equal to
in 2 1: y(q(n))4 compOl’,, y(4)))U{n 2 1: z(q(n))4 cow(Z,,z(q))). Then we may assumethat the set N*= {n > 1 : y(q(n)) $comp(Yq, y(q))} is infinite. Under this assumption, we will show that Y has a fold. Suppose that N* = {nl, n2, . .}, where n1 < n2 < . .. Let I
=
{t E [0, l] : there exists an open subset V of 1’ such that y(t,O) E V and the set {k L 1 : Y(C ;)
E comp(V, y(t, 0))) is finite}.
Since {k L 1 : ~(1, &) E comp(Yqb,y(q))) = {k 2 1 : y(q(m)) E comp(Y*,y(q))} = 0, 1 f I. Given an open subset V of Y such that y(O,O) E V, there exists S > 0 such that y(([O,S] x [O,S]) n P) c V. Since Y((P~~l x P>4) r-lv is connected, y(([O, 61x [0,6]) n P) C comp(V, y(O,O)). Thus {k > 1 : ~(0, $-) E comp(V, y(O,O))} is infinite. This proves that 0 q! I. We will show that I is open in [O,l]. Let t E I. Let V be as in the definition of 1 for the number t. Let 6 > 0 be such that ((t-6, t+6) x [O,Q)n P c y-‘(V). Define L = (t - 6, t + 6) n [0, l] and let K 2 1 be such that & < 6. Given s E L, the connectednessof y(L x (0)) implies that y(s, 0) E comp(V, y(t, 0)). If k 2 K, then y(L x {A}) is a connected subset of V, so
FOLDS y(L x {&I)” implies that
419
comP(V, y(t, 0)) = 8 or y(L x { *})
C comp(V, y(t, 0)). This
{k 2 1:
Y(S, $)
E comp(V, y(s,O)))
=
{k 2 1:
Y(S, $1
E comp(V, y(hO))}
C
{k>l:
Il(~,-$~
E comp(ll;y(t,O))}U{l,...,K-1).
Thus s E I. We have proved that L c I. Therefore, I is open in [0, 11. Define r = max([O, l] - I). Then r < 1. Define pe = (r, 0). We will show that Y has a fold at y(po). For each k 2 1, let rk = max{t E [O,11: ~(4 $)
E y([O, 11x (0))) U {r}
We claim that rk -+ r. Suppose, to the contrary, that the sequence{rk}r?i does not converge to r. Since T 5 rk for each k 1 1, there exists se E (r, l] and there exists a subsequence{rh,,,}E’i of {rk}r=r such that Tk, -+ SO. We may assume that r < fk, for every m 2 1. Then y(rk,, &) E y([O, l] x (0)). Thus there exists (sm,O) E J = [O,l] x (0) such that y(rk,, k) = y(sm,O). We may also assumethat s, -+ s for some s E (0, 11. Then y(s,O) = y(se,O). By the choice of T, we have that se E 1. Let V be as in the definition of I for the point SO.Let 6 > 0 be such that
(((s -6,s+6)
x [0,6))flP)U
(((so - 6,so +6) x [0,6)) flP) c y-‘(v)
Let A4 2 1 be such that, for each m 2 M, * < 6, Irk,,, - SO]< 6 and Is, - s] < S. Then th e sets y(((s - 6,s + S) n [c, 11) x (0)) and ~(((so are connected, both contain the point y(sm, 0) = ~,~0+4n[o,1l)x(~)) ,,,, $-) and the first one contains the point y(s,O) = y(so,O). Thus they are contained in comp(V,y(se,O)). In particular, y(se, k) E ?drk
comp(V, ds0,O)). Hence {k 2 1 : y(s 0, &) E comp(V, y(se, 0))) is infinite. This contradicts the choice of V and completes the proof that rk --+ T. For each m 2 1, let V, = B(i,y(r,O)) c Y and let C, = camp (Vm, y(r,O)). Since r 4 I, the set N,, = {k 2 1 : y(r, $) E C,} is infinite.
420
XIV.
PRODUCTS,
CONES AND HYPERSPACES
Then a sequence kl < kz < . . . can be chosen in such a way that, for each m 2 1, rk, < 1 and k, E N,. Then y(r;$--) E C,. For each m 2 1, we will define three coltinua A,, B,, and D, in X. In order to do this, we wil1 identify C(X) and Y x 2. Then we will think that the points in Y x Z are subcontinua of X. For each m > 1, define A, = f(r, T&h JL = b(r,O),z(r,
&)I,
and
D, = U{E E C(X) : E E cLy(C,,) x {z(r, $-)}}. For each m > 1, define cry, : [O,l] + Y by a,(s) = ~((1 - s)rk- + sr,&). Then o,(l) = ry(A,) and am([O, 11) tends to {y(r, 0)) (in C(Y)). By Exercise 15.9, D, is a subcontinuum of X. Notice that A,,, U B, c D,. Let A = f(r,O). Then A,,, + A, B, + A, D, + A and ry(B,) = dr, 0) = TY (%+I) for each m 1 1. By Exercise 79.20, there exists a map y : [0, l] + Y such that y(k) = a,(O) for each m 2 1 and y(O) = ry(A). Define F : P + Y by
F(t~ s, =
Y((l - rkm)t + rk,, , &),
if s = A,
y((1 - r)t + r,O),
if s = 0, ift=O.
i Y(S),
If t = 0 and s = A, then ~((1 - rk,,,)t + rp,, $)
= y(rg,,
$-)
=
@n(Q) = Y($). If t = 0 and s = 0, then ~((1 - r)t + r, 0) = g(r, 0) = ry(A) = y(O). This proves that F is well defined. Clearly, F is continuous. Let p = (t,O) E J - (0). In order to finish the proof that Y has a fold at the point y(r, 0), we need to prove that there exists an open subset U of Y such that F(p) E U and comp(U, F(p)) n {F(p(n)) : n _> 1) = 0. Suppose, to the contrary, that there is no such a U. First, we will show that for each m > 1, F(p) # F(p(m)). Since 0 < t , r$,,, < (1 - rk, )t + rk, . By the definition of rk, , y( (1 - rk,,,)t + rk, , $-) fj! y([O,l] X (0)). Then ~((1 -rk,)t+rk,,,, F(p(m)) # F(P). Let me 2 1 be such that F(p(mo)) ~1 > 0 be such that
k)
# ~((1 -r)t+r,O).
That is,
E comp(B(1, F(p)), F(p)).
Bh F(P)) n {F(P(~)), . . . , F(p(mo))) = 0. Let ml > 1 be such that F(p(ml)) E comp(B(ci, F(p)), F(p)). ml > me. Let 62 > 0 be such that 62 < cl, 4 and B(Q, F(P)) n {F(P(~)), . . . , F(p(ml))l = 0.
Let
Then
PROOF
OF THE MAIN
421
THEOREM
Proceeding in this way, sequencescl, ~2,. . . and mo < ml < . . can be constructed such that F(p(m,)) E comp(B(cj, F(P)), F(P)) and Q -+ 0. Define u = (1 - r)t + T and p* = (u, 0). Then u E 1. Then there exists an open subset V of Y such that y(u,O) E V and the set {k > 1 : y(u, &) E comp(V, y(u, 0))) is finite. Notice that F(p) = y(u,O) E V. Then there exists 6 > 0 such that B(E, F(p)) c V and y( ( (U - c, u + E) x [O,e)) n P) C V. < E and Let js > 1 be such that, for each j 2 jc, cj < E, $mJ rkm,
-I-<E.
For each j 2 jo, B(c,,F(p)) c If, so F(p(mj)) ; cow (B(~j,F(p)), F(P)) C comp(K F(p)). Then ~((1 -v+,,~ )t+w,,, , ~1 E comp(K F(p)). 1
Since
I(1
-
Y(((u-h~+f)nP?ll) F(p(mj)).
rk,,
)t
+
Tk,,
-
uI
=
It(r
-
rk,,
) -
(r
-
rk,
1
)I
<
c,
is a connected subset of V which contains
x {*I,
Hence this set is contained in comp(V, F(p)). In particular, Y(% - l
nkm,
)E comp(V,
Thus for each j > jc, y(u, $-
F(P)).
) E comp(V, y(u,O)).
This contradicts
the choice of V and completes thz’proof of the existence of U. Therefore, Y has a fold at y(r, 0). n
Proof
of the Main
Theorem
79.9 Lemma. Let X be a continuum. Suppose that C(X) has a fold at an element A E C(X). Let f : P + C(X) be as in the definition of a fold at A. If f(l,O) - A # 8, then, for each M 2. 1, X contains an M-od. Proof. Let d denote a metric for X. Choose a point po E f(l,O) - A. Let E > 0 be such that B(2c,po) II A = 8. For each n > 1, let P, = C BH,,(E,A) and ([0, “1 x [0, i]) rl P. Let N > 1 be such that f(P,) f(l,ni) E B~~(c,f(l,O)) for each n 2 N. Define B = U{c : C E I}. By Exercise 15.9, B E C(X). Notice that B c N(E, A). For each n > N, f(1, k) n B(e,po) # 0. Then f(1, i) B # 0. Suppose that X contains no M-ods. By Theorem 69.5 there are at most M - 1 minimal elements {El,. . . , Ek} in the semi-boundary of B. C B. Then we can define t, = max{t E For each n 2 N, f(&,A) $11 : f(t,;) c B}. Since f(1, $) - B # 0, t, < 1. Thus, by Theorem
422
XIV.
PRODUCTS,
CONES AND HYPERSPACES
69.2, the continuum C,, = f(tn, i) is in Sb(B). Hence, by Theorem 69.4, there exists i, E (1,. . . , Ic} such that Ein C C,. This implies that there is an element E in {El,. . . , Ek} which is contained in infinitely many C,. Then there exist positive integers ni < ns < . . . such that E c C,_ for every m 2 1. We may assume that t,,,, + t for some t E [ $, 11. Let p = (t, 0). Let C = f(p). Notice that 0 < t < 1 and E c C. By the choice of f, there exists an open subset U of C(X) such that f(p) E U and comp(U, f (p)) does not intersect {f (p(n)) : n 2 1). Let 60 > 0 be such that BH~ (co, C) c U. Let 6 > 0 be such that ((t - 6, t + - tl < d 6) x [0,6)) nP C f-‘(BHd(Eo,C)). Let m > N be such that It,, and & < S. Let L be the segment in R2 which joins the points (t, --$ if these points coincide). Then f(L) c and (in,,, , h m) (or L = {(t, 5)) BH,,(Q,C). In particular C,,,, = f(tn,, $-) E BH~(EO,C). Since E c C,,, I-I C, Exercise 79.21 implies that C,, “E cOmp(BH,(cs, C), C). Then f(L) C comp(BH,(co, C),C). This implies that f (p(n,)) = f (t, &) E comp(U, C). This contradicts the choice of U and completes the proof of the lemma. w 79.10 Theorem [5, Theorem 3.71. Let X be a continuum. If C(X) is a finite-dimensional product, then X is locally connected. Proof. Suppose that C(X) M Y x 2, where Y and Z are finitedimensional nondegenerate continua. Then ~ -
there exists M 2 1 such that X contains no M-ods (Theorem 70.1),
X is hereditarily decomposable ([12, Theorem 2.51, Exercise 79.15), E- and 2 are arcwise connected ([12, Lemma 2.11, Exercise 79.11), for each A E C(X), C(X) - {A} is arcwise connected ([12, Lemma 2.31, Exercise 79.11). Suppose that X is not locally connected. By Theorem 79.4, there exists A E C(X) - Fi(X) such that C(X) has a wrinkle at A. By Theorem 79.7, C(X) has a fold at A. We identify C(X) with Y x 2. Let wy and ~2 be the projection maps from 1’ x 2 onto Y and 2, respectively. By Theorem 79.8, we may assume that Y has a fold at a point y E 1’. Let f : P + Y be a map as in the definition of a fold at y. Then f (0) = Y. Fix a one-to-one map (Y : [0, l] + 2. Let A = (y,o(O)) and B = (y,o(l)). Then A and B are different subcontinua of X. Then we may assumethat B - A # 0. Then there exists r > 0 such that (f (r, 0), o(l)) -
A # 0.
EXERCISES
423
Define g : P -+ Y x 2 by g(t,s) = (f(tr,s),cr(t)). We will check that g satisfies the properties of the maps used in the definition of a fold at (Y, 40)) = A. First, notice that g is continuous and g(0) = (f(o), a(0)) = (g, o(0)). By the choice Let p = (t,O) E J - (0). Then (tr,O) E J - (0). of f, there exists an open subset U of 1’ such that f(tr,O) E U and comp(U, f(tr, 0)) does not intersect {f(tr, i) : n > 1). Let U = U x 2. Then U is an open subset of Y x Z and g(p) E U. Let C = comp(U,g(p)). Suppose that there exists n 2 1 such that g(t, i) E C. Since C c U is connected and g(p) E C, KY(C) is a connected subset of In particular, U and f(tr,O) E KY(C). Then Q(C) c comp(U, f(tr,O)). ry(g(t, i)) = f(tr, i) E comp(U, f(tr, 0)). This contradicts the choice of U. Hence we have t,hat comp(U,g(p)) does not intersect the set {g(p(a)) : n > 1). This completes the proof that g has the properties of the maps used in the definition of a fold at (g,a(O)). Therefore, C(X) has a fold at
A. is not contained in A = g(O), by Lemma Since g(l,O) = (f(r, O),dl)) 79.9, X contains m-ods for every m > 1. This contradiction ends the proof of the theorem. n 79.2 Theorem [5, Theorem 3.81. Let X be a continuum. Then C(>X) is a finite-dimensional product if and only if X is an arc or a circle. Proof. NECESSITY. If C(X) is a finite-dimensional cartesian product, by Theorem 79.10, X is locally connected. Then, Theorem 79.1 implies that C(X) is an arc or a circle. The sufficiency is immediate from 5.1.1 and 5.2. n
Exercises 79.11 Exercise [12, Lemmas 2.1 and 2.31. Let X be a continuum. If C(X) M Y x 2, then Y and 2 are arcwise connected and C(X) - {,4} is arcwise connected for every A E C(X).
{X}
79.12 Exercise. If X is an indecomposable continuum, then C(X) has an uncountable number of arc-components. [Hint: Use Proposition 18.2 and Theorem 11.15 of [13].]
79.13 Exercise. If B is a (nondegenerate) proper indecomposable subcontinuum of a continuum X and K is a composant of B, then either C(K) = {A E C(B) : A c K} is an arc component of C(X) - {B} or there exists an element D E Sb(B) (seeDefinition 69.1) such that D C K. [Hint: Use Proposition 18.2.1
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79.14 Exercise. Let X be a continuum. Suppose that there exists an integer M > 1 such that X does not contain M-ods. Let B be a (nondegenerate) indecomposable subcontinuum of X, then C(X) - {B} has infinitely many arc components. [Hint: Use Theorem 69.5.1 79.15 Exercise. Let -Y be a continuum. If C(X) z I’ x Z and dim[C(X)] is finite, then X is hereditarily decomposabie. 79.16 Exercise. Find all the wrinkles for C(X) when X is the sin(t)continuum and when X is the harmonic fan. 79.17 Exercise. If X is an hereditarily indecomposable continuum, then C(X) has a wrinkle at every element in C(X) - (F1 (X) U {X}). 79.18 Exercise. Prove the necessity in Theorem 79.4. [Hint: Use Exercises 14.18 and 14.20.1 79.19 Exercise. Let X be a continuum. Let {A7L}rTt0=1 be a sequence in C(X) such that A1 > AZ > ... and let A = n{A,, : n 2 1). Then there exists a map 0 : [0, l] + C(X) such that a(i) = A,, for each n 2 1. 79.20 Exercise. This exercise is used in the proof of Theorem 79.8. Let X be a continuum. Let {A,,}~=l, {B,,}zzl and {Dm}zzl be three sequencesin C(X) and let x : C(X) + Y be a map such that A, -+ A, for each B,, + A, D,, + A, A, U B, c D,, and n(B,) = r(B,+l) m > 1. Suppose that there is a sequence of maps {a,}zzl from [0, l] into 1’ such that (w,( 1) = r(Am), for each m 2 1, and (Y,([O, 11) tends to {7r(.4)} (in C(Y)). Tl len there exists a map y : [0, l] --+ 1’ such that, y(h) = a,,,(O) for each m 2 1 and y(O) = r(A). 79.21 Exercise. Let X be a continuum and let d denote a metric for X. If E > 0, Hd(A, B) < E and An B # 8, then B E comp(BHd(E, A), A).
80.
More
on Hyperspaces
and Cones
80.1 Conventions. For a continuum X, we denote the cone over X by Cone(X), the vertex of Cone(X) is denoted by u(X) and the base of Cone(X) is denoted by B(X). That is, B(X) = {(x,0) E Cone(X) : 5 E X}. The continuum X is said to have the cone = hyperspace property provided that there exists a homeomorphism h : C(X) + Cone(X) such that h(X) = v(X) and hlFl(X) is a homeomorphism from Fl(X) onto B(X). A homeomorphism h with the properties described above is called a
80. MORE
ON HYPERSPACES
AND CONES
425
Rogers homeomorphism. By Exercise 80.15, for a Rogers homeomorphism h, we can ask that h({z}) = (x,0) for each x E X. 80.2 Theorem. Suppose that the continuum X has the cone = hyperspace property and dim(C(X)) is finite. Then: (a) dim(X) = 1 (see comments to Theorem 7.2), (b) dim[C(X)] = dim[Cone(X)] = 2 ([ll, Lemma 8.0]), (c) X is atriodic (Theorem 70.1), (d) every composant of X is arcwise connected ([15, Theorem 11,Exercise 80.16), (e) every proper subcontinuum of X is unicoherent (Exercise 80.17), (f) every proper subcontinuum of X is arcwise connected(Exercise 80.18), (g) every proper nondegenerate subcontinuum of X is an arc ([15, Theorem 71, Exercise 80.21), (h) if X is decomposable, then X is a circle or an arc ([15, Corollary 11, Exercise 80.22), (i) if X is indecomposable, every composant of X is a one-to-one continuous image of [0, co) or R’ ([lo, Corollary 5.51, Exercise 80.23). Figure 55 consists of three continua. We will discuss these continua in relation to the properties in 80.2 later (80.10 and 80.33). In the proof of Theorem 1 of [2] it was claimed that if a continuum X has property (K) and each arc component of X is the one-to-one continuous image of the real line such that the images of [0, 00) and (-co, 0] are dense in X, then every proper nondegenerate subcontinuum of X is an arc. We do not know how justify this claim. Using this Theorem and adding the hypothesis that X has the covering property, in Corollary 3 of [2], it is obtained that X has the cone = hyperspace property. In the following theorem we show similar conditions that imply that X has the cone = hyperspace property. 80.3 Theorem. Let X be a continuum with property (K). Suppose that each composant of X is the one-to-one image of the real line such that the images of [O,oo) and (-oo, 0] are dense in X. Then X has the cone = hyperspace property. Proof. For the proof of this theorem we will use that continua are a-connected ([13, Theorem 5.161). That means that a continuum can not be written as the union of more than one and at most countably many nonempty, mutually disjoint, closed subsets.
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CONES AND HYPERSPACES
lb)
(c)
The cone = hyperspace property does not imply property of Kelley (80.10) Figure 55
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ON HYPERSPACES
AND CONES
427
Given a composant K of X, fix a map hK : R1 + K such that hK is one-to-one, surjective map and hK([O, co)) and hK( (-co, 01) are dense in are X. By Exercise 80.24, for each t E R1, hK([t,cm)) and hK((-m,t]) dense in X. First we show that every proper nondegenerate subcontinuum of X is an arc. In order to do this, let B be a proper nondegenerate subcontinuum of X. Then there is a composant K of X such that B C K. Let h = hK. Let T = h-‘(B). If T has an unbounded component C, by the density of h(C), it follows that B = X. This contradiction proves that the components of T are compact. If T is unbounded, by Exercise 80.25, T is the union of a countable family {Tn}~zto=l of nonempty, pairwise disjoint, compact subsets of R1. Then the sets h(T,) are compact, nonempty, pairwise disjoint and their union is B. This contradicts the a-connectednessof B and proves that T is bounded. Then T is compact. Thus hlT : T + B is a homeomorphism. Therefore, T is a nondegenerate, subcontinuum of R’. Then T and B are arcs. Fix a Whitney map p : C(X) + [0, l] such that p(X) = 1. Define G : Cone(X) -+ C(X) by G(z, s) = U{A E C(X) : p(A) = s and 2 E A}. By Exercises 15.9 and 28.4, G(z,s) E C(X) for each (5,s) E Cone(X). Notice that G(z, 1) = X for every z E X. Then G is a well defined function. Since X has property (KC),G is continuous (see Proposition 20.11). Notice that G(s,O) = {z} for every 2 E X. Given a point zr E X and a number s E [0, l), let K be the composant and of X such that 2 E K. Let h = hK and 3: = h(t). Since h([t,m)) h((-oo, t]) are both dense in X, there exist 2r > t and u < t such that
AW, 4)) = dh(b, tl)) = sWe claim that h([u, u]) = G(a, s). Clearly, h([u, v]) c G(z, s). In order to prove the opposite inclusion, let B E C(X) be such that p(B) = s and is an arc. Then there exists a < b x E B. As we saw before, h-l(B) such that h([a, b]) = B. Notice that t E [a, b]. Since p(h([a, t])) 5 p(B) = dh(b, tl)), u < - a. Similarly, b 5 w. Thus B c h([u,v]). We have proved that h( [u, v]) = G(z, s). In particular, this implies that G(x, s) # X for every x E X and s < I. Now, we prove that G is one-to-one. Suppose that G(x, s) = Gfy, T) and s, T < 1. Let K be the composant of X such that G(x, s) C K. Let h = hK. Let u,,wz,tz,uy,vy,ty E R’ be such that h(t,) = x, p(h([u+,&])) = s, dW27~11))= s, h(t,) = Y, dh(by,ty])) = T,Ah([$,,qJ) = T. Then G(z, s) = h([uz,v,]) and G(y,r) = h([uy, ~1). Since h is one-to-one, u, = uy and u, = vy.
428
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CONES AND HYPERSPACES
If t, # t,, assume, for example that t, < t,. Then s = p(h([u5,t2])) < P(N%&1)) = T = Pw!/~~yln < P(Wz, IJ~])) = s. This contradiction proves that t, = t,. This implies that 5 = y and s = r. Therefore, G is one-to-one. Finally, we show that G is surjective. Let B be a proper subcontinuum of X. Let K be the composant of X which contains B and let h = hK. Then there exists an interval [a, b] such that. h([a, b]) = B. Define g : [a, b] --+ R’ by g(t) = ,u(h([a,t])) - p(h([t,b])). Since g is continuous, there exists to E [a,b] such that g(ts) = 0. Then G(h(to),p(h([a, to]))) = B. Therefore, G is a Rogers homeomorphism. Hence, X has the cone = hyperspace property. n It is known ([14, Theorem 21and [15, p. 2801or Theorem 80.4, respectively) that solenoids and the Buckethandle continuum have the cone = hyperspace property. Notice that the Buckethandle continuum does not satisfy the hypothesis of Theorem 80.3. Thus, conditions in Theorem 80.3 are sufficient but not necessary for a continuum to have the cone = hyperspace property. 80.4 Theorem [2, Corollary 121. Let X be a continuum. If X is the inverse limit of arcs with open bonding maps, then X has the cone = hyperspace property. Recently, the following characterization has been obtained. 80.5 Theorem [6]. Let X be a finite-dimensional continuum. Then the following statements are equivalent: (a) X has the cone = hyperspace property, (b) there is a selection s : C(X) - {X} -+ X such that, for every Whitney level A for C(X), s]d : A + X is a homeomorphism, and (c) there is a selection s : C(X) - {X} + X and there exists a Whitney map ~1: C(X) --t [0, l] such that p(X) = 1 and SIP-~(~) : p-‘(t) --t X is a homeomorphism for every t E [0, 1). The implication (b) + (c) is immediate, (c) + (a) was proved by Sherling in [17, Theorem 3.11 and it is left as Exercise 80.26. The following Corollary answers questions by Dilks and Rogers ([2, p. 6361) and it is left as Exercise 80.27. 80.6 Corollary [6]. If X is a finite-dimensional continuum, X is not a circle and X has the cone = hyperspace property, then X has the covering property and it is Whitney stable (a continuum X is Whitney stable if it
80. MORE ON HYPERSPACES
429
AND CONES
is homeomorphic to each of the Whitney levels for C(X) 14.39.11).
[ll,
Definition
The notion of type N was defined for dendroids in Definition 75.10. It can be extended to any continuum in the following way. 80.7 Definition. Let A be a proper subcontinuum of the continuum X. Then X is said to be of type N at A if there exist four sequencesof {B,,}p=r, {C,}~=r and {O,l};P=l of X, there exist subcontinua {An}rEl, two points a # c in A and there exist sequencesof points {a,,}~zP=1 and {cn)~& in X such that a, + a, c, + c, A, -+ A, B, -+ A, C,, + A, D, -+ A and A, n B, = {a,) and C, n D, = {cn) for every n 2 1. Another consequence of Theorem 80.5 is the following result which is also left as Exercise 80.28. 80.8 Corollary [6]. Let X be a finite-dimensional continuum. If X has the cone = hyperspace property, then X is not of type N at any of its proper subcontinua. 80.9 Question [6]. Suppose that X is an indecomposable finite-dimensional continuum such that every nondegenerate proper subcontinuum of X is an arc and X is not of type N .at any of its proper continuum. Then does X has the cone = hyperspace property? If the answer to this question is affirmative, then we would have an intrinsic characterization of finite-dimensional continua which have the cone = hyperspace property. Answering a question by Dilks and Rogers ([2, p. 636]), it has recently been found the following example. 80.10 Example [6]. There is a finite-dimensional continuum X such that X have the cone = hyperspace and X does not have the property (K). The continuum X is pictured in Figure 55 (b), p. 426. It is a modification of the Buckethandle continuum illustrated in Figure 55 (a). For contructing X, we alternate complete “legs” with shortened “legs”. Notice that X does not have property (K). 80.11 Question. Give an intrinsic characterization of finite-dimensional continua with the cone = hyperspace property. See Question 80.9. Answering Question 8.14 of [ll], in [2, Theorem 61 and [17, Example 4.11 two non chainable and non circle-like continua were constructed with the cone = hyperspace property. Sherling’s example in [17, Example 4.11 has been shown to have property (K) ([7]).
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In Theorem 80.2, we showed the known properties of the finite-dimensional continua with the cone = hyperspace property. Now consider a finitedimensional continuum X for which C(X) is homeomorphic to Cone(X) (not necessarily with a Rogers homeomorphism). The following theorem shows that X satisfies several properties mentioned in Theorem 80.2. 80.12 Theorem. Suppose that X is a finite-dimensional continuum such that C(X) is homeomorphic to Cone(X), then: (a) dim(X) = 1 (see comments to Theorem 7.2), (b) dim[C(X)] = dim[Cone(X)] = 2 ([ll, Lemma 8.0]), (c) X is atriodic (Theorem 70.1), (d) if X is hereditarily decomposable, then X is one of the subcontinua described in Figure 20, p. 63. Hence, from now on, we may assume that X is not an hereditarily decomposable continuum. (e) X contains exactly one (nondegenerate) indecomposable subcontinuum Y ([16, Theorem 81, Exercise 80.30). Let Y be the unique (nondegenerate) indecomposable subcontinuum of x. (f) Y is a finite dimensional continuum such that C(Y) is homeomorphic to Cone(Y) and X -Y is arcwise connected (combine 111,8.20.41 and
PI).
Let h : C(Y) + Cone(Y) be a homeomorphism. (g) h(1’) = u(Y) ([ll, 8.20.31, Exercise 80.31), (h) h(Fi(Y)) = B(X) ([lo, Theorem 5.71). Therefore, Y has the cone = hyperspace property. Hence, Y has properties mentioned in Theorem 80.2. By Theorem 80.12 (f), (g) and (h), an intrinsic characterization of the finite-dimensional continua with the cone = hyperspace property (Question 80.11) will lead to an intrinsic characterization of those finite-dimensional continua X for which C(X) is homemorphic to Cone(X). Recently; Macias [9, Theorem 41 has proved the following result which corrects an error in [ll, p. 3331. 80.13 Theorem [9, Theorem 41. Let X be a locally connected continuum; then C(X) is homeomorphic to the cone over a finite-dimensional continuum 2 if and only if X is an arc, a circle or a simple n-od.
Let T, denote a simple n-od. A model for the hyperspace C(T’,) has been discussedin section 5. This model is represented in Figure 9, p. 42. Macias showsthat C(Tn) is homeomorphic to the cone over h/l,, where M,
EXERCISES
n
431
arcs \
(4 Hyperspaces
of nods
are cones (80.13)
Figure 56
is the continuum represented in Figure 56 (a). In Exercise 80.32 it is asked to prove that C(Ts) is homeomorphic to the cone over Ms. J. J. Charatonik asked the following question: If 2 is a continuum such that C(T,) is homeomorphic to Cone(Z), then must 2 be homemorphic to Mn? Charatonik’s question has been solved in [l, Theorem 6.21, with the following theorem. 80.14 Theorem [l, Theorem 6.21. If n = 3 or 4 and C(T,) x Cone(K,) for some compact metric space Kn, then K,, x M,. However, for each n 2 5 there is a continuum L, such that C(T,) PZ Cone(L,) but L, is not homemorphic to M,.
Exercises 80.15 Exercise. Let X be a continuum. If X has the cone = hyperspace property, then there exists a Rogers homeomorphism h : C(X) + Cone(X) such that h({s}) = (x,0) for each 2 E X.
XIV.
432
PRODUCTS,
CONES AND HYPERSPACES
80.16 Exercise. Prove Theorem 80.2 (d). [Hint: Identify C(X) and Cone(X), identifying {z} and (x,0) for each x E X. Suppose that 4 is a proper subcontinuum which contains two point,s 2 and y. Let r : Cone(X) - u(X) + B(X) be the projection. Since r(C(A)) is arcwise connected, there is an arc in X which joins z and y.] Exercise. Prove (e) of Theorem 80.2. [Hint: Suppose that X contains a proper non-unicoherent continuum. Use Theorem 80.2 (d) and the fact that composants are dense to contruct a triod in X. This contradicts Theorem 80.2 (c).] 80.17
Prove (f) of Theorem 80.2. 80.18 Exercise. [Hint: Suppose (f) is not true. Let d denote a metric for X. Then X contains a free arc. Then there exists E > 0 such that if A E BH~ (E, X), then A has a point of local connectednessof X. Then C(X) is locally connected at every A E B~,,(c,x) (Theorem 71.2). Since C(X) z Cone(X), this implies that X is a locally connected, atriodic continuum. By Exercise 31.12 X is an arc or a circle. Thus, X is hereditarily arcwise connected.] 80.19 Exercise. Prove the Brouwer’s Reduction Theorem: In a second countable space Y let K be a nonempty family of closed subsetsof Y with the property that for each sequence& c K1 c Kz c . . . of elements of ic, there exists K E K such that K, c K for each n 2 0. Then K contains a maximal element (i.e. an element K of K which is not contained in any other element of K). Formulate also the respective version of this theorem for minimal instead of maximal elements. [Hint: Let {U, : n > 1) be a countable basis for the topology of Y. Choose an element KO E K. Inductively define Kn+l E K with the properties that K, c Kn+l and Kn+l n V,+l # 0 if such Kn+l exists; otherwise define K,+l = K,.] 80.20 Exercise. Dendroids contain maximal arcs. [Hint: Let X be a dendroid. In order to apply Exercise 80.19, take a point a E X and a sequence{bn}F& of points of X such that ab, $ a&+~ for every n > 1. Let b be a limit point of the sequence {bn}pE1. For each : m > n}). Since B, is a subcontinuum of X, n 2 1, let B, = cly.(~{b,b, B, is arcwise connected. Then B, is decomposable. Let B, = A, U C,, where A, and C, are proper subcontinua of B,. We may assumethat C, contains infinitely many b,. Then b, E A, - C,. Let p E C,, be such that b,p n C,, = {p}. Then B, = b,p U C,, and b E C,. This implies that b, E ab. Thus ab, c ab for every n 2 1.1
EXERCISES 80.21 Exercise. The only atriodic quence, prove (g) of Theorem 80.2. 80.22
Exercise.
433 dendroid
is the arc.
As a conse-
Prove (h) of Theorem 80.2.
80.23 Exercise. Prove (i) of Theorem 80.2. [Hint: Let K be a composant of X. Fix two points p # q E K. Let p : C(X) -+ R’ be a Whitney map. For each T E K, let rp denote the unique arc joining p and r in X if T # p, and rp = {p} if T = p. For = P(TP) if rp fl pq # {p}, and f(~) = -p(TP) if each r E K, define f(r) rpflpq = {p}. Then f is one-to-one, f]a is continuous for every arc cyC I<. The inverse f-’ : f(K) -+ I< is continuous. Since cl,y(K) = .Y, f(K) is an unbounded interval in RI.] 80.24 Exercise. Suppose that there exists a one-to-one map h : R’ -+ X such that h([O,oo)) and h((-co,O]) are dense in X. Then, for each t E R’, h([t, oo)) and h((-co, t]) are dense in X. 80.25 Exercise. Let T be an unbounded closed subset of R’ which does not have unbounded components. Then T is the union of a countable family {Tn}~Jl of nonempty pairwise disjoint compact subset,sof R’ 80.26 Exercise [17, Theorem 3.11. Suppose that there is a selection s : C(X) - {X} + X and there exists a Whitney map ~1: C(X) -+ [0, 1] such that p(X) = 1 and s]p-‘(t) : p-‘(t) + X is a homeomorphism for every t E [0, 1). Then X has the cone = hyperspace property. [Hint: Define h : C(X) + Cone(X) as follows: h(A) =
(44,~(4), v(X),
if A # x ifA=X’
1
80.27 Exercise. Prove Corollary 80.6. [Hint: Use Theorems 80.2 and 80.5.1 80.28 Exercise. Prove Corollary 80.8. [Hint: see Exercise 75.24.1 80.29 Exercise. {v(W),
Coned)
Let X be a continuum. For each p - {P) h as a.t most two arc components.
80.30 Exercise. Prove (e) in Theorem 80.12. [Hint: Use Exercises 79.14 and 80.29.1
E
Cone(X) -
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80.31 Exercise. Prove (g) in Theorem 80.12. [Hint: Use Exercise 80.30.1 80.32 Exercise. If !fy is a simple triod, then C(Ts) is homeomorphic to t,he cone over the cont,inuum A,fs illustrated in Figure 56 (b), p. 431. 80.33 Exercise. Without using Corollary 80.8, prove that the continuum X illustrated in Figure 55 (c), p. 426 satisfies properties (a) through (g) and (i) in Theorem 80.2 and X does not have the cone = hyperspace property. The continuum X is a modification of the Buckethandle continuum and it is constructed by enlarging and folding the “legs” of the Buckethandle continuum represented in Figure 55 (a). [Hint: Suppose that there is a Rogers homeomorphism h : C(X) -+ Cone (X) such that h({z)) = (x,0) for each 5 E X. Then there is t E (0,l) such that a E h-‘(b, t) c ab and h-‘(b,s) c ab for each s 5 t. This implies that It-’ ({u} x [0, 1)) fl h-’ ({b} x [0, 1)) # 0 which is a contradiction.]
References 1.
F. D. Ancel
2.
A. M. Dilks and J. T. Rogers, Jr., Whitney stability and contractible hyperspuces,Proc. Amer. Math. Sot., 83 (1981), 633-640.
3.
R. Duda, On the hyperspuce of subcontinua of a finite graph, I, Fund. Math., 62 (1968), 265-286 (also: see correction, Fund. Math., 69 (1970), 207-211).
4.
A. Illanes, Hyperspuceshomeomorphic to cones, Glasnik Mat., 30(50) (1995)) 285-294.
5.
A. Illanes, Hyperspuceswhich are products, Topology Appl., 79 (1997), 229-247.
6.
A. Illanes, Cone = hyperspaceproperty, a characterization, preprint.
7.
W. T. Ingram and D. D. Sherling, Two continua having a property ,J. L. Kelley, Canad. Math. Bull., 34 (1991), 351-356.
8.
.J. L. Kelley, Hyperspuces of a continuum, Trans. Amer. Math. 52 (1942), 22-36.
9.
S. Macias, Hyperspuces and cones, Proc. (1997), 3069-3073.
and S.B. Nadler, Jr., Cones that are cells, and an upplicution to hyperspuces,preprint.
Amer.
Math.
of
SOC.,
SOL, 125
10 S.B. Nadler, Jr., Continua whose cone and hyperspuce are homeomorphic, Trans. Amer. Math. Sot., 230 (1977), 321-345.
REFERENCES
435
11. S. B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 12. S.B. Nadler, Jr., Continua whose hyperspace is a product, Fund. Math., 108 (1980), 49-66. Theory, An introduction, Monographs 13. S. B. Nadler, Jr., Continuum and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N. Y., 1992. 14. J. T. Rogers, Jr., Embedding the hyperspaces of circle-like plane continua, Proc. Amer. Math. Sot., 29 (1971), 165-168. 15. J.T. Rogers, Jr., The cone = hyperspace property, Canad J. Math., 24 (1972), 279-285. with cones homeomorphic to hyperspaces, 16. J. T. Rogers, Jr., Continua General Topology Appl., 3 (1973), 283-289. 17. D. D. Sherling, Concerning the cone = hyperspace property, Canad. J. Math., 35 (1983), 1030-1048.
This Page Intentionally Left Blank
XV.
Q uest ions
The chapter has three sections. The first two sections concern questions in the 1978 book Hyperspaces of Sets [56]; the last section contains other questions, some of which are original with the book. After the references for the three sections, we include a list of all the literature related to hyperspaces of continua that we found that appeared since 1978.
81.
Unsolved
and Partially
Solved
Questions
of [56]
(0.61 of [56]) Definition. “The members of a class A of continua are provided that if X, Y E A and C(X) z C(Yr), then said to be C-determined x x Y.” “What are some other classesof continua, (0.62 of [56]) Questions. besides those in (0.59) and (0.60), whose members are C-determined? In particular, what about the class of chainable continua? What about the class of circle-like continua? Recently [439], it has been shown that the members of the class of smooth fans are C-determined. What about the class of fans?” Comment. 1.
Classesof continua known to be C-determined: (a) finite graphs different from the arc, (Duda, [18, 9.11 or [56, Theorem 0.591)) (b) hereditarily indecomposable continua, (Nadler, [56, Theorem 0.601), (c) smooth fans (Eberhart and Nadler, [21, Corollary 3.3]), (d) indecomposable continua for which all the nondegenerate proper subcontinua are arcs (Ma&as, [53, Theorem 3]), (e) compactifications of [0, oo) (Acosta, [l]). 437
438 2.
XV.
QUESTIONS
Classes of continua known to be non-C-determined: (a) chainable continua @lanes, [45, Example 3]),
(b) fans (Illanes, [46]). In [45, ExampIe 31 it is considered a particular chainable continuum X and it is constructed another chainable continuum Y by adding a segment at one end point of X. It is proved that X is not homeomorphic to Y and C(X) x C(Y). This is the only known way to construct two nonhomeomorphic chainable continua X and 1’ such that C(X) z C(Y). So the following question naturally arises. Question [45, Remark and Question 51. Are there five chainable continua Xl,..., Xs such that Xi is not homeomorphic to Xj and C(X,) z C(Xj) if i # j? (1.27.3 of [56]) Question. (note: S(w) and l?(X) in [56] are respectively, Sw(2ay) and Tj(2x) here) “The spaces S(w) and l?(X) have never been investigated before [except for (1.29)]. It would be interesting to obtain more information about them. Some results in this direction are in (1.30), (1.32) and (1.203.3). The last of these implies that I’(X) and 2” have the same homotopy type. If X is a locally connected continuum, then is I(X) an absolute retract” [see (1.96)] and, more specifically, is I(X) M 103 [see (1.97)]? More generally [see (1.97)], is I(X) M 2x for any continuum X?” Comment. Now there are some results in the literature about the spaces S(w) and I(X). Besides the results in sections 17 and 66 the reader can find more information about S(w) and I’(X) in [7], [lo], [23] and [51]. In [lo] a characterization of Peano continua for which S,(C(X)) is homeomorphic to the Hilbert cube is given. The question: is l?(X) M 2” for any continuum X? is still open. (1.86 of [56]) Corollary. “Let Y be any continuum such that 3 I dim[Y] < co. L et p be any given Whitney map for C(Y). Then, there exists to > 0 such that dim[p-r(t)] = cc whenever 0 < t < to. Hence, dim[C(X)] = co.” (1.87 of [56]) Question. the assumption that dim[Y] Comment.
Unsolved,
“Does the first part of (1.86) remain valid if < cc is deleted?” see Corollary
73.11 and Question
73.12.
81. UNSOLVED AND PARTIALLY SOLVED QUESTIONS OF [56]
439
(1.117 of [56]) Question. “([85] and [87]). What is the structure of those polyhedra which are homeomorphic to C(X) for non-acyclic finite graphs?” “For what classes of continua Y is it true (1.119 of [56]) Question. that Cone(Y) is embeddable in C(Y) [see (1.121)]? We remark that by letting Y be the join at a point of two (otherwise disjoint) hereditarily indecomposable continua, we obtain a decomposable continuum Y such that Cone(Y) is not embeddable in C(Y); for a proof, see [244, Example l] .” (1.123 of [56]) Question. “[244, Problem 11. Can C(Z) ever (topologically) contain the Cartesian product of two continua when Z is an hereditarily indecomposable continuum [see (1.125)]?” “For what
(1.126 of [56]) Question.
that Y x Y is embeddable
in C(l’)
classes of continua
Y is it true
[see (1.128)]?”
(1.129 of [56]) Question. “Is an arc the only chainable continuum 1’ such that Y x Y is embeddable in C(Y) [see (1.128) and, for example, (lO.lO)]?” (1.130 of [56]) Question. “Is there a circle-like continuum Y x Y is embeddable in Co’)? [see (1.128); also by (lO.lO), homeomorphic to Y x 1’ for any circle-like continuum ET].”
Y such that C(Y) is not
(1.144 of [56]) Question. “[Jack T. Goodykoontz, Jr.]. What are necessary and sufficient conditions for C(X) to be locally connected at hl E C(X)?” Comment. Even when there is no a characterization for local connectedness at elements in C(X), a lot of work has been done about local arcwise connectedness and connectedness im kleinen on C(X), see [13], [14], [15], [16], [20, Theorem 1.21, [25], [33], [34], [49] and [54]. (1.186 of [56]) Definition. have a hyper-onto representation {P,, fn}p& such that (i)
X = l.{P,,
“[log, p. 1331. A continuum
X is said to provided that there is an inverse sequence
fn)El,
(ii) P, is a compact polyhedron for each n = 1,2, . . ., and -+ C(P,) maps C(P,+l) onto C(P,,)
for each n =
440
XV. An inverse sequence satisfying
representation
QUESTIONS (i), (ii), and (iii) is called a hyper-onto
for X.”
(1.188 of [56]) Definition. “[log, p. 1291. Let, X be a continuum. We say that C(X) has a minimal representation provided that there exist one-dimensional compact connected polyhedra PI, P2, . . . , P,, . . . such that where 7n maps C(P,+l) onto C(P,%) for each C(X) = ~~wmJ,Yn~~I, n = 1, 2, . . ..” (1.200 of [56]) Question. “[log, p. 1341. Does every continuum have a hyper-onto representation?” (1.201 of [56]) Question. “[log, p. 1341. For any continuum X, does C(X) have a minimal representation? (if the answer to (1.200) is “yes,” then by (1.189) the answer to (1.201) is “yes.“)” (1.193 of [56]) Theorem. “[106, Lemma 51. If a mapping of a continuum X onto a circle is essential, then it is weakly confluent [see (0.67.4) and (1.202)].” “What conditions characterize weakly con(1.202 of [56]) Question. fluent mappings of circles onto circles? The converse of (1.193), for X a circle is false [see (0.67.4)].” (2.17 of [56]) Question. “For what continua X is C(X) dimensionally homogeneous?The answer for 1ocaIly connected continua is given in (2.16). Also, by (2.15) C(X) is ’ dimensionally homogeneousif dim[C(X)] = 2 (as is the case, for example, when X is chainable or circle-like [by (2.1)] or when X satisfies the hypotheses of (2.8)). I conjecture that C(X) is dimensionally homogeneous if X is any Cartesian product of (nondegenerate) continua the second part of (1.103) may be helpful in proving this.” (3.5 of [56]) Question. “[176, Problem 41. If X is a continuum such that C(X) is embeddable in R3, then must X be embeddable in the plane [see (3.6)]?” (3.10 of [56]) Theorem. (note: X connected in [56] is J-connected here). “[134, Theorem 4, p. 2341. Let X be a X connected continuum. Then: C(X) can be c-mapped into the plane for each E > 0 if and only if X is chaninable or circle-like.”
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
441
“[134, p. 2341. Does (3.10) remain valid (3.11 of [56]) Question. without the assumption of X connectedness? The “if” part of (3.10) is valid without the assumption of X connectedness as can be seen using (1.169), (1.162), (0.54), and (0.55).” (3.12 of [56]) Question. “[250, 2.71. When, if ever, does a continuum (X,d) have the property that (C(X),Hd) or (2x,Hd) is isometric to a subset of &?” (4.4 of [56]) Question. “[250, 3.11. When is there a mapping from 2x onto Y? Of special interest is the case when Y = X.” (4.5 of [56]) Question. “[250, 3.11. When is there a mapping from C(X) onto Y? Of special interest is the case when Y = X.” (4.6 of [56]) Question. “When is there a mapping from X onto C(X)? By (4.1) and (4.2) if X is locally connected or if X contains an open set with uncountably many components, then there is a mapping from X onto C(X).” (note: a space is said to be g-contractible (4.11 of [56]) Question. provided there is a mapping of the space onto itself which is homotopic to a constant map) “[250, 3.101. Is C(X) g-contractible for any continuum X? A partial solution will be given in (4.12). Of course if X is a locally connected continuum, C(X) is contractible [by (16.18)]. However, let us note that C(X) and 2” are not always contractible [see (16.22)].” (4.15 of [56]) Question. (note: a mapping f : X + Y from X onto Y is said to be non-alternating provided that for no two points y, z E Y do there exist disjoint nonempty open subsets Vi and Uz of X such that X \ f-‘(y) = Ui U U;l and z E [f(Ul) n f(Uz)]) “When does there exists an onto monotone, or open, or confluent or non-alternating, etc. mapping between two of the spaces X, 2”, and C(X)? See (6.8) for a related question.” (5.6 of [56]) Question. C(X)?”
“For what continua X is there a selection for
(5.11 of [56]) Question. “For what dendroids D is there a selection for C(D)?" “I do not know if every contractible dendroid D admits a selection for C(D)."
442
XV.
Comment. Selections Question 5.6 of [56] is the This question is far to be [56] has been solved in the Some related questions are
QUESTIONS
are discussed in section 75. By Theorem 75.6, same as the first part of Question 5.11 of [56]. solved. The second part of Question 5.11 of negative by MaCkowiak with the Example 75.9. in section 75.
(5.13 of [56]) Question. (note: an 5’4 space is a Hausdorff space which admits a selection for each covering of it by mutually disjoint nonempty compact subsets). “[225, p. 1551. What spaces are Sq spaces? See (5.14) for a partial answer.” It is easy to see that an Sd space can not contain a simple closed curve. This suggests the following question. (5.15 of [56]) Question. “[347, p. 1091. If a (metric) S4 space, then must it be a dendrite? see below.”
continuum
is an
(5.16 of [56]) Question. “Can a continuum which is an Sq space contain an indecomposable continuum ? In (5.14) we showed that a continuum which is an Sq space can not contain an hereditarily indecomposable continuum.” (5.17 of [56]) Question.
“Is every dendrite
an Sq space?”
(5.18 of [56]) Definition. “Call a Hausdorff space a u-space (c-space) provided that it admits a selection for each upper semi-continuous (continuous) decomposition of it into nonempty compact subsets (as usual, the selection’s continuity is in terms of the Vietoris topology, not the decomposition topology).” “What
(5.19 of [56]) Question. are c-spaces?” (5.20 of [56]) Definition. selection continuum provided there is a selection from p-l(t)
continua are u-spaces?
What continua
“A continuum x’ is called a Whitney map that, for every Whitney map p for C(X), into x’ for each t E [0, p(X)].”
(5.21 of [56]) Question. “What continua are Whitney map selection continua? What chainable continua and what circle-like continua are Whitney map selection continua?” Comment. Theorem 80.5.
A partial
answer
to question
5.21 of [56] is contained
in
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
443
(5.22 of [56]) Definition. “Let X be any continuum. A function f : X + 2x is called a coselection for X provided that f is a mapping and z E f(x) for each z E X. If f is a coselection for X and f(z) E A for each 2 E X [some A c 2.‘], then f is called a A-coselection for X. If f is a A-coselection for X where A c [2x \ (Fl(X) U {X})], then f is called a non-trivial coselection for X. If f is a coselection for X, then the mesh of f , denoted mesh(f ), is defined by: mesh(f) = l.u.b.{diam[f (z)] : 2 E X) If X is a continuum such that given any E> 0 there exists a non-trivial coselection [resp., A-coselection] for X of mesh lessthat e, then X is called a coselection space [resp., A-coselection space].” (5.23 of [56]) Question. “When do coselections of the various types defined in (5.22) exist? What continua are coselection or C(X)-coselection spaces?” Comment. The following questions (6.1-6.29 of [56]) are related to retractions between hyperspaces. This topic is discussedin section 76. We refer the reader to the papers cited in Chapter XII for more information. (note: a mapping T : X + Y is called an r-map (6.1 of [56]) Question. provided there is a mapping g : Y + X such that I- o g = ly.) “Given any two of the spacesX, 2x and C(X), w hen is there an r-map between them? Also, see (6.28) and (6.29).” “[243, p. 4131. What are necessary and suffi(6.2 of [56]) Question. cient conditions in order that the space of singletons Fl (X) of a continuum X be a retract of 2x or of C(X)?” (6.3 of [56]) Question. “[250, 3.71. For what continua X is C,(X) a retract of 2”? Recall that Cl (X) means C(X) as it is naturally embedded in 2x by inclusion. Some important material related to this question is in (6.10). In particular, the example in (6.10.2) may be of interest to the reader at this point.” (6.7 of [56]) Question. “Is Cl (X) a monotone open retract of 2x when X is any locally connected continuum? By (6.6), C(X) is always such an r-image when X is locally connected. Also, by (6.12), C,(X) is always a monotone retract of 2x when X is locally connected. However, the retract defined in the proof of (6.12) is not necessarily open - see the paragraph following the proof of (6.12).”
444
XV. QUESTIONS
(6.8 of [56]) Question. “Same as (6.1) except require the r-map to be monotone, or open, or confluent, etc.” (6.10.3 of [56]) Questions. “Let X be the familiar sin($)-continuum defined in (1) of (8.22). Is Cl(X) a retract of 2”? In fact, except for the arc, it is not known whether or not Cl(Y) is a retract of 2’ for any given chainable continuum Y. Is Cl (I”) a retract of 2’ when Y is the pseudo-arc? Analogous questions are open when +etract” is replaced by “r-image”; in particular, [camp., (6.29)], it is not known whether or not C(X) is an Timage of 2x if X is the continuum in (6.10.2). No information is available for circle-like continua which are not the circle. Also, see (6.10.1).” Comment. In [9], Curtis then C(X) is a retract of 2”.
showed
that if X is the sin($)-continuum,
(6.14 of [56]) Question. “For what continua X is there a retraction from Fi(X) U r onto some r, PI(X) C r C Cl(X)?” (6.15 of [56]) Question.
“[see (6.17.1)].
(6.16 of [56]) Question.
“Take r = C1 (X) in (6.14).”
(6.17.1 of [56]) Question.
“What
Take r = F,(X)
continua
in (6.14).”
X admit a mean?”
“For what continua X is there a mapping (6.20 of [SS]) Questions. from Fp(X) into Cl (X) which is the identity on J’l (X)? Does such mapping exist for the familiar sin($)-continuum in (1) of (8.22)? We call such a mapping a pseudo-mean.” Comment. Let X be the sin(i)-continuum. Since C(X) is a retract of 2” (Curtis, [9]), then C(X) admits a pseudo-mean. (6.21 of [56]) Question. (note: So is the sin( i)-continuum) ‘LFor what continua X does C(X) admit a mean? I do not know of any continuum X for which C(X) does not admit a mean. I do not know if C(So) admits a mean, where So is as in (1) of (8.22) [note: C(So) M Cone($) - see Chapter VIII]. It would be interesting to know if C(X) admits a mean when X is the continuum in (6.10.2). It is easy to see that the proof in (6.10.2) shows there is no retraction from Cz(X) [see (0.48)] onto Cl(X). However, this does not seemto imply there is no retraction form Fz(C(X)) onto Fl (C(X)) [camp., (0.71.2)].”
81. UNSOLVED
AND PARTIALLY
SOLVED QUESTIONS OF [56]
445
Comment. Among other results, in [9], Curtis showed that: (a) there are continua Y such that C(Y) does not admit a mean ([9, Example 9.21)) (b) if X is the sin(i)-continuum, then there is a retraction r : 2x + C(X). Then the map {A, B} --t r(AuB), from Fs(C(X)) onto C(X) is a mean. (6.22 of [56]) Question. “For what continua X is Fl (C(X)) a retract of 2c(x)? Clearly, by using (1.48) and (1.49), Fl (C(X)) is a retract of C(C(X)) for any continuum X.” (6.23 of [56]) Question. “For what continua X is C(A) an r-image of C(X) for all subcontinua A of X?” (6.24 of [56]) Question. “For what continua X is 2A an r-image of 2x for all subcontinua A of X?” (6.25 of [56]) Question. “For what continua X is C(A) x C(X) for each nondegenerate subcontinuum A of X? Obviously [by (0.52)], any continuum X, all of whose nondegenerate subcontinua are homeomorphic to X, is such an X (for example, the arc and the pseudo-arc). Also [by (0.55)], a circle is such an X. Henderson [140] has proved that a decomposable continuum which is homeomorphic to each of its nondegenerate subcontinua must be an arc.” (6.26 of [56]) Question. “For what decomposablecontinua X is C(A) z C(X) for each nondegenerate subcontinuum A of X? If such an X contains an arc, then X must be an arc or a circle [as is easily seen using (1.92) and (1.109)]. Can there be such an X which contains no arc? It would be interesting if the answer to this question were “no.” The answer would then be a generalization of Henderson’s Theorem mentioned in (6.25). Also, if this generalization could be proved without using Hendreson’s Theorem, a new proof for his Theorem would be obtained. There is some hope for obtaining a hyperspace proof of Henderson’s Theorem since the arc structure for C(X) is a strong feature which is not available when working just in X as Henderson did. In particular, (14.8.1) might be helpful; also, see (11.18). Henderson’s proof is difficult and a new proof of it would be of interest. To obtain his result, it may of course be assumed that X is hereditarily decomposable (instead of only decomposableas assumedabove). Thus, there are many arcs in C(X) “parallel to and near Fl (X)” by (14.8.1).” (6.27 of [56]) Question. ‘Same as (6.25) and (6.26) except replace C(X) and C(A) by 2x and 2A respectively.”
446
XV.
QUESTIONS
(6.28 of [56]) Question. “Let X be a continuum. Which of the following four statements are equivalent to each other? (6.28.1) Fi(X) is a retract of 2x. (6.28.2) Fi (X) is a retract of C(X). (6.28.3) X is an r-image of 2x. (6.28.4) X is an r-image of C(X). By (6.4), they are all equivalent when X is a locally connected continuum.” Comment. The first example of a continuum X such that X is a retract of C(X) but not of 2x was shown in [35]. Thus 6.28.1 of [56] is not equivalent to 6.28.2 of [56]. Example 76.12 is a smooth dendroid X such that X is a strong deformation retract of C(X) and X does not admit a mean (then C(X) is not a retract of 2.Y). Then, for the continuum X in Example in 76.12, it would be interesting to determine if X is an r-image of 2x. (6.29 of [56]) Question. “Let X be a continuum. Is Ci (X) being a retract of 2x equivalent to C(X) being an r-image of 2x? A continuum worth investigating in this connection is the continuum (6.10.2) [see (6.10.3)].” (7.4 of [56]) Question. “[293, p. 2481. For what continua X is C(X) a quasi-complex? The only examples for which it has been proved that C(X) is not a quasi-complex are the two above.” (7.7 of [56]) Question. “[280, p. 2831. When does C(X), uum, have the fixed point property?” Comment. hyperspaces.
In Chapter
VI we discussed
X a contin-
the fixed point property
(7.8 of [56]) Question. “[285, p. 2341. If X is a tree-like does C(X) have the fixed point property?”
in
continuum,
(7.8.1 of [56]) Question. “Does C(D) have the fixed point property for every dendroid D [see (7.8.3)]? Partial answers are given in (7.8.2). It is known [39, Theorem 21 that dendroids have the fixed point property. Hence, a negative answer to the question would be a “one-dimensional answer” to Knaster’s question [camp., (7.3)]. In this connection, it would be of interest to know whether or not C(X) has the fixed point property for every onedimensional continuum X with the fixed point property [camp., (7.2) and the paragraph following it].”
81. UNSOLVED AND PARTIALLY SOLVED Q~JESTIONS OF [56]
447
“[442, p. 2611. If X is a dendroid with metric (7.8.3 of [56]) Question. d, then is there, for each E > 0, a finite graph X, c X and a retraction T , from X onto X, such that d(r,(z), x) < E for each 2 E X? By using the proof of (7.8.2), an affirmative answer to this question would give an affirmative answer to the first question in (7.8.1). It would also give an affirmative answer to the last question in (7.10).” Comment. Two questions found in Remark 3.9 of [59].
related
to Question
7.8.3 of [56] can be
“[250, 3.81. When does 2x, X a continuum, (7.9 of [56]) Question. have the fixed point property? This question was in part the motivation for (6.3) -.- see (6.10).” “Does 2x have the fixed point property for all (7.10 of [56]) Question. hereditarily indecomposable continua X [camp., (7.6) above]? What about chainable [camp., (7.1)] or circle-like [camp., (7.5)] continua? What about all dendroids [see (7.8.2) and (7.8.3)]?” Comment. Theorem
In [58], it was proved the following [58, Theorem 3.31. If X = 12{D,,
result. fn}, where D,, is a den-
drite and fn is a quasi-monotone map (an onto map between continua f : X -+ Y is quasi-monotone provided that for any continuum K in Y with nonempty interior, f-‘(K) has only a finite number of components and each of these maps onto K under f) of D,,+l onto D, for each n = 1,2,. . ., then 2” and C(X) have the fixed point property. As a consequence, it is shown that if X is the Buckethandle (Example 22.11), then 2 a’ has the fixed point property.
continuum
(7.11 of [56]) Question. “Does 2’ have the fixed point property, where Y is as in (7.3)? I conjecture that the answer is “no” and, in fact, that retractions can be defined from 2y onto 2z and from 2’ onto Ci (2) where Z C Y is as in (7.2); then of course, (7.2) can be applied.” Comment. The continuum 2 in Question 7.11 of [56] is the circle with a spiral represented in Figure 20 (6), p. 63. The continuum Y is the union of 2 with a disk which has as its boundary the circle contained in 2. In [9], Curtis showed that there is a retraction from 2’ onto C(Z). It is known that C(Z) does not have the fixed point property ([56, Theorem (7.2)]). Then 2z does not have the fixed point property and Z is the first known example with this characteristic. If it were possible to find a retraction
448
XV.
QUESTIONS
from 2y onto 2’, then 2’ would not have the fixed point property. The important difference between Y and Z is that Y has the fixed point property while Z does not have it. (7.12 of [56]) Question. “Are the following for any continuum X: (7.12.1) 2x has the fixed point propertry; (7.12.2) C(X) has the fixed point property?”
two statements
equivalent
(7.13 of [56]) Question. ‘LFor what cont,inua X does p-i(t) have the fixed point property for every Whitney map p for C(X) and for each t E [O,,U(X)]? For some examples showing that X can have the fixed point property and c1-l (t) may not, see (14.30) and (14.43.6). Let us also note that, when X is a chainable continuum, p-‘(t) has the fixed point property for each Whitney map ,LLfor C(X) an each t [as follows using (14.4)].” (8.11 of [56]) Question. cone = hyperspace property?”
“What
finite-dimensional
continua
have the
Comment. In section 80 we discuss some aspects of spaces having the cone = hyperspace property. In particular Theorem 80.5 gives a characterization of finite dimensional continua having the cone = hyperspace property in terms of the existence of a particular selection from C(X) - {X} onto X. A positive answer to Question 80.9 would give an intrinsic characterization of these continua. (note: the continuum X is said to be a C-H (8.12 of [56]) Question. to Cone(X)). “[camp., continuum provided that C(X) is homeomorphic C-H continua?” (8.33)]. What are the finite-dimensional Comment, By Theorem 80.12, an important part of Question 8.12 of [56] has been reduced to give answers to Question 8.11 of [56]. (8.15 of [56]) Question.
“What
are the indecomposable chainable Ccontinuum [defined in (1.209.3)] is an example of one, as was noted by Rogers 1280, p. 2801.”
H continua [also see (8.31)]? The “Buckethandle” (8.16 of [56]) Question.
“What
are the indecomposable circle-like Cand the solenoids [by
H continua [also see (8.32)]? The “Buckethandle” (S.l)] are examples of such continua.” (8.31 of [56]) Question.
“[244, Problem
C-H continua [camp., (8.14) and (8.15)]?”
21. What
are the chainable
81. UNSOLVED
AND PARTIALLY
(8.32 of [56]) Question. [camp., (8.14) and (S.lS)]?” (8.33 of [56]) Question. tinua?”
SOLVED
“What
QUESTIONS
OF [56]
449
are the circle-like C-H continua
“[244, Problem 21. What are the C-H con-
“If X is a C-H continuum, must there exist, (8.34 of [56]) Question. a homeomorphism h : Cone(X) + C(X) such that h[B(X)] = Fi(X)?” “For what continua X does there exist a (8.35 of [56]) Questions. continuum Y such that C(X) M Cone(Y)? If X is any given such continuum, what can be said about the class of continua Y such that C(X) = Cone(Y)? Answers to these questions would even be of interest under the additional assumption that dim[C(X)] < oo. Under this assumption I can show that if X is a locally connected continuum and Y is a continuum such that C(X) M Cone(Y), then X and Y must be arcs or circles. The proof useslocal cut points and ideas and results in [85]. For examples related to this and the two questions above, we refer the reader to the proof of (10.2) and to (10.16).” Comment. The two questions contained in 8.35 of [56] are still open. Recently, Ma&s [52, Theorem 41 has corrected an error contained in the previous paragraph by showing the following result: Theorem [52, Theorem 41. Let X be a locally connected continuum; then C(X) is homeomorphic to the cone over a finite-dimensional continuum 2 if and only if X is an arc, a circle or a simple n-od. (8.36 of [56]) Question. “For what continua X is 2x M Cone(X)? Two comments are appropriate. First: if X is not arcwise connected and h : Cone(X) + 2x is any homeomorphism, then h(v) E C(X); this follows from (11.3). Second: If X is a locally connected continuum and 2.’ M Cone(X), then Cone(X) M 1” by (1.97).” (8.37 of [56]) Question. ‘Same as (8.35) with C(X) Recall, dim[2”] = 00 by (1.95).”
replaced by 2x.
(9.4 of [56]) Question. (note: Sus(X) denotes suspensionof X) “[252]. For what infinite-dimensional continua X is C(X) M Sus(X)? For example note that, since C(I”) M 10° [by the second part of (1.98)], C(1”) M Sus(1”). In (10.16) we will give another example.”
450
XV.
QUESTIONS
(9.5 of [56]) Questions. ‘&For what continua X does there exist a continuum Y such that C(X) z Sus(Y) [see (lO.lS)]? If X is any given such continuum, what can be said about the class of continua Y such that C(X) x Sus(Y)? If X and Y are continua such that C(X) z Sus(Y) and dim[C(X)] < 03, then must X be an arc or a circle, equivalently must Y be an arc?” (9.6 of [56]) Question. “For what continua X is 2x M Sus (X)? It appears that (9.2) can be modified by replacing C(Y) with 2”. If so, then (9.3) with C(Y) replaced by 2’ would be valid, and we would know that the continua X such that 2x z Sus(X) must be arcwise connected.”
C(X)
(9.7 of [56]) Question. replaced by 2~~.”
“Same as the first two questions
Given a continuum X, Comment. ofX is the quotient space C(X)/Fi (X). where it was proved that HS(X) has chainable continuum X. The following
in (9.5) with
the hyperspace suspension, HS( X), This notion was introduced in [57], the fixed point property for every questions are open:
Questions [57, Question 4.11. For what circle-like continua X does HS(X) have the fixed point property? If a circle-like continuum X has the fixed point property, then does HS(X) have the fixed point property? Questions [57, Question 4.21. For what continua X is HS(X) homeomorphic to the usual suspension Sus(X) of X? If C(X) is homeomorphic to the cone over X, then is HS(X) homeomorphic to S(X)? This last question would have an affirmative answer if Question 8.34 of [56] has an affirmative answer. “[241, 3.191. If X is any locally connected (10.21 of [56]) Question. “The continuum satisfying (10.20.3), then must C(X) be a product?” question above is stated in the context of locally connected continua. It may in fact be true that if C(X) is a product, then X is locally connected. If this is true and if the answer to the above question is “yes,” then a complete characterization of those continua X such that C(X) is a product would be obtained.”
In [43, Example 4.11 it was shown that there exists a nonComment. locally connected continuum X such that C(X) is homeomorphic to X x
[O,11.
81. UNSOLVED
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SOLVED
QUESTIONS
OF [56]
451
“For what continua X is 2x a product? Note (10.22 of [56]) Question. that if X is any locally connected continuum, then 2.’ is a product because 2-Y z I” [by (1.97)]. If X is a continuum such that 2x is a product, then must X be locally connected [camp., comment near end of (10.21)]?” “If X is a continuum such that C(X) x I” (10.26 of [56]) Question. 2-Y, then must X be locally connected?”
z
(10.27 of [56]) Question. “If X is a continuum such that C(X) M 2~‘, then must X be locally connected? If the answer is “yes”, then by (1.97) and (1.98) we would know all such continua X.” (9.2 of [56]) Theorem. “[252, 2.41. Let Y be any continuum. If Al,Az E C(Y) such that C(Y) \ {Ai} is arcwise connected for each i = 1 and 2, then C(Y) \ {Al, As} is arcwise connected.” (11.17 of [56]) Question. “How can (9.2) be generalized? For example: Is it still true with the two sets Al and A2 replaced by n sets, n finite? What about countably many? What about a collection {Ax : X E A} which is a compact zero-dimensional subset of the hyperspace? The last of these questions is somewhat motivated by (2.15).” Comment. Theorem 9.2 of [56] has been generalized to finitely many An’s by (Ward, [67, Theorem 1.11, see also [32, Corollary lo]) and it has also been generalized for a countable family {A,,}:& which is closed in C(X) (Illanes, [38, Corollary 2.21; this result has been recently rediscovered by Hosokawa in [32, Corollary 121). The question about a compact zerodimensional subset of C(X) is still open. Definition. Let X be a continuum. Let C c 2x. A member A of Cr (X) is said to be arcwise (segmentwise) accessiblefrom C \ C,(X) [55] provided there is a homeomorphism (segment) cr : [0, l] + C such that u(t) E C \ Cl(X) for all t < 1,and ~(1) = A. We will say that A is accessible beginning with Ii’ [55] provided there is a CTas above such that u(0) = K. Comment. With respect to Questions 12.1, 12.5, 12.19, 12.22, 12.23, 12.24, 12.25 and 12.26 of [56], Grispolakis and Tymchatyn have proved the following results: Theorem [30, Corollary 4.21. Let X be an hereditarily decomposable continuum and let p be a point of X which belongs to a subcontinuum of X with finite rim-type at p. Then p is arcwise accessiblefrom C’s(X) \ Cr (X).
452
XV.
QZJESTIONS
Theorem [30, Corollary 4.31. Let X be a rational continuum of finite rim-type. Then every point of X is arcwise accessible from Cx(X) \ Cr (X). Theorem [30, Theorem 4.51. Let X be a Suslinian continuum. Then the set D of points of X which are arcwise accessible from C’s(X) \ Cr (X) is dense in X. Moreover, the set X \ D contains no (nondegenerate) continuum. Theorem [30, Theorem 4.71. Let X be a chainable hereditarily decomposable continuum. Then the set of points of X which are arcwise accessible from C,(X) \ Ci (X) is dense in X. (12.1 of [56]) Question. “[234, 1.31. Which arcwise accessible from 2” \ Ci (X)?”
members
of C,(X)
(12.5 of [56]) Question. “[234, 1.21. When is a member arcwise accessible from 2dY \ Ci (S)‘?”
are
of Fr(X)
“[234, 6.11. For any hereditarily decompos(12.19 of [56]) Question. able continuum X, must there be a point 2 E X such that {x} is arcwise accessible from 2x \ Cr (X)?i” (12.22 of [56]) Question. “[234, 6.41. What are conditions, which are at the same time both necessary and sufficient, in order that {xc} be arcwise accessible from 2” \ Ci (X) for a given point 20 of a continuum X? What about such conditions when X is rational or when X is hereditarily decomposable? For hereditarily decomposable continua, is the converse of (12.14) true?” Definition.
For a given continuum
X, let
C,(X)
=
{AE 2x : A has at most n components},
AA[X]
=
{x E X : { x } is arcwise accessible from 2x \ Cr (X)}.
(12.23 of [SS]) Question.
AA[X]
“[234, 6.51. For what rational
and
continua X is
= X?‘”
“What is the Bore1 type [185, p. 471 of (12.24 of [56]) Question. when X is a rational continuum or, more generally, when X is any continuum? In fact, is AA[X] always a Bore1 set?”
AA[X]
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
453
(12.25 of [56]) Question. “[234, 6.61. What hereditarily decomposable continua X have the property that each [or someor no] singleton is arcwise accessiblefrom 2aY\Cr (X)?l It may be that each hereditarily decomposable continuum has some arcwise-accessiblesingleton; see (12.19).” “[234, 6.71. Is there a continuum X such (12.26 of [56]) Question. that, for some 20 E X, (~0) is arcwise accessiblefrom 2x \ Cl(X) but not from C?(X) \ Ci (X)?” “Given a continuum X and an .4 E Ci (.Y), (12.32 of [56]) Question. what can be said about a K E [2x \C, (X)] such that -4 is arcwise accessible from 2.’ \ Ci (X) beginning with I
6.91. What subcontinua of 2dYcan be if A is nondegenerate, any nondegenA. This would certainly be true if one is a touching set at A.”
(14.35 of [56]) Question. “[ZSS, Question 31. How are other properties of continua reflected in the continua p-*(t)?” Comment. 14.35 of [56].
Chapters VIII
and IX are plenty of answers to Question
(14.36 of [56]) Question. “[176, section 61. For a given topological property, determine whether or not it is a Whitney property. In particular, is homogeneity (resp., X connectednes, weak chainability) a Whitney property?”
454
XV.
QUESTIONS
Comment. Homogeneity is not a Whitney property (Rogers, [62, Theorem 111 and W. J. Charatonik, [6]), see Example 48.2. X connectedness of [56] is &connectedness defined in 51.1, where a different notion of Xconnectedness is defined. It is known that &connectedness is not a Whitney property (W. J. Charatonik), see Example 51.2. It is not known (see Question 51.3) whether, X-connectedness is a Whitney property. The question whether weak chainability is a Whitney property (see Theorem 37.7 and Question 37.8) is still unsolved. (14.39 of [56]) Question. “For what continua X is it true that for some Whitney map (all Whitney maps) p for C(X), p-l(t) z X for all t E [O,p(X))?” (14.39.1 of [56]) Definition. “A continuum X is said to be weak Whitney stable (Whitney stable) provided that for some Whitney map (all Whitis ney maps) p for C(X), p-l(t) ’ 1romeomorphic to X for all t E [0, p(X)).” Comment. Dilks and Rogers have proved ([12, Theorem lo]) that if ,Y is a continuum and X is the inverse limit of arcs with open bonding maps, then X is Whitney stable. This result has recently been generalized in Theorem 80.5 where it has been proved that if X is a finite-dimensional continuum which have the cone = hyperspace property, then X is Whitney stable (see Theorem 80.4). Using Theorem 53.3 it can be proved that the unique Whitney stable 2-connected continuum is the arc. “Is there a continuum which is weak Whit(14.42.2 of [56]) Question. ney stable but not Whitney stable ? By (14.42.1), the Hilbert cube is not Whitney stable; it may be weak stable - see (14.38). Is there a finitedimensional continuum which is weak Whitney stable but not Whitney stable?” Comment. By Theorems 25.3 and 25.10, the Hilbert cube is weak Whitney stable. This answers the first part of Question 14.42.2 of [56], the last question in 14.42.2 of [56] is still unsolved. (14.42.3 of [56]) Question. “(camp., [249, P. 4, p. 4071) If a continuum X is weak Whitney stable [resp., Whitney stable], then is every (nondegenerate) subcontinuum of X weak Whitney stable [resp., Whitney stable]? In other words: Are these properties hereditary? It might be that the Hilbert cube is weak Whitney stable [see (14.42.2)]; if so, then the property weak Is the property weak Whitney stable Whitney stable is not hereditary. continua? Note [resp., Whitney stable] h ereditary for finite-dimensional
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
455
that the continua mentioned at the beginning of (14.42) are hereditarily Whitney stable.” Comment. Since there are subcontinua of the Hilbert cube which are not weakly stable (e.g., a 2-cell) and the Hilbert, cube is weak Whitne? stable (Theorems 25.3 and 25.10), we seethat the property of being weakly Whitney stable is not hereditary. (14.43 of [56]) Definition.
“[267, p. 311. We say that a continuum X maps provided that there exist Whitney maps p and p’ for C(X) such that, there is some member A of one of the collections admits
essentially
diflerent
Whitney
{p-‘(t) or {(IL’)-l(t)
: 0 < t < p(X)} : 0 5 t 5 p’(X)}
such that A is not homeomorphic to any member of the other collection.” “What continua admit essentially differ(14.43.11 of [56]) Question. ent Whitney maps? Are the arc and the circle the only locally connected continua which do not admit essentially different Whitney maps? We note that the Hilbert cube admits essentially different Whitney maps [camp., (14.38) and (14.42.1)]. Are the arc and the circle the only finite-dimensional locally connected continua which do not admit essentially different Whitney maps? What chainable or circle-like continua do not admit, essentially different Whitney maps?” “If X is an absolute retract such that, (14.43.12 of [56]) Question. dim[X] > 2, then does X admit essentially different Whitney maps? In particular: Are there Whitney maps /L and p’ for C(X) such that p-‘(t) is contractible for each t E [0, p(X)] but (p/)-l (to) is not contractible for some to E [0, p’(X)]? This is true for the Hilbert cube [camp., (14.38) and (14.42.1)].” Comment. Suppose that X is an absolute retract such that dim[X] > 2. By Theorem 72.6, X is not hereditarily decomposable. This implies that X is not a dendroid. By Theorem 53.3, there is a Whitney map ,LL’for C(X) and there is t E (0, p(X)) such that ($)-l(t) is not 2-connected (a.nd then (p’)-‘(t) .1snot contractible). Then an affirmative answer to Question 14.43.12 of [56] would be obtained if it is possible to construct a Whitney map p for C(X) such that p-l(t) is contractible for each t E [O,p(X)]. Notice that if the answer to Question 25.14 is affirmative, then such a p can be constructed.
456
XV.
QUESTIONS
Reasoning as in the previous paragraph and using Theorems 25.9 and 25.21, it can be shown that if X is an arc-smooth continuum and dim[?r] > 2, then X admits essentially different Whitney maps. (14.55.1 of [56]) Question. “Is there a strong Whitney-reversible property P, which is not a sequential strong Whitney-reversible property? In particular, what about the properties in (14.55) (being irreducible, being in Class(W), having CP)?” Comment. By Theorem 49.3, irreducibility is a sequential strong Whitney-reversible property. The rest of the questions contained in Question 14.55.1 are still open (see Questions 27.2 and 35.6).
“For a given topological property determine (14.57 of [56]) Question. whether or not it is a strong Whitney-reversible property [or, Whitneyreversible property]. Specifically, what about the following properties: Acyclic, absolute neighborhood retract, absolute retract, chainable, circlelike or proper circle-like, contractibility, hereditarily decomposable”, X connected, one-dimensiona12, being a particular solenoid, weakly chainable? Perhaps S = z X,,, where each *Yn is the circle in R” with center (9,
0)
11= 1
and radius k, can be used to seethat being an absolute neighborhood retract is not a Whitney-reversible property; for t > 0, all but finitely many of the “holes” in X seem to disappear in p-i (t) .” Comment. - Acyclicity
is a sequential strong Whitney-reversible property (Nadler, see Theorem 36.5), - Being an absolute neighborhood retract is not a Whitney-reversible property (see Example 28.2, where it is shown that, for the continuum X suggested in Question 14.57 of [56], every positive Whitney level is an ANW, - Being an absolute retract is not a strong Whitney-reversible property (Goodykoontz and Nadler, see Example 30.4), - Chainability is a sequential strong Whitney-reversible property (Kato, see Theorem 37.5), - The properties of being a circle-like and a proper circle-like continuum are sequential Whitney-reversible properties (Kato, seeTheorems 39.5 and 39.6), - Contractibility is not a strong Whitney-reversible property (Nadler, see Example 41.7), - Being hereditarily decomposable is a sequential strong Whitney-reversible property (Abo-Zeid, see Theorem 44.9),
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
457
- X connectedness of [56] is b-connectednesshere (Definition 51.1). A different definition of X-connectednessis given in 51.1. It is not known if 6connectednessis a (strong, sequential strong) Whitney-reversible property (seesection 51), -Beingofdimension
(14.71.1 of [SS]) Question. “Are the conversesof the results in (14.71) true? In other words: If Y and 2 are continua such that W[C(Y)] (respectively, W[2y]) is homeomorphic, or both homeomorphic and algebraically isomorphic, to W[C(Z)] (respectively, W[2z]), then must Y and Z be homeomorphic?”
458
XV.
QUESTIONS
Comment. In [39, Theorem 5.61, it was proved that W[C(Y)] is homemorphic to the Hilbert space & for every continuum Y, In [37, Theorem 3.11, it was proved that if there exists a homeomorphism ‘p : W[C(Y)] -+ W[C(Z)] (or ‘p : W[2y] + W[2’]) which p reserves products and a natural order defined on W[C(Y)] (or W[2y]), then the continuum Y is homeomorphic to the continuum 2. The “both homemorphic and algebraically isomorphic” part of Question 14.71.1 of [56] is still open. Questions. Is W[2y] homemorphic to & for every continuum this question true for locally connected continua?
Y? Is
(14.71.2 of [56]) Question. “For what continua X are IV[C(X)] lV[2”] homeomorphic and/or algebraically isomorphic?”
and
(14.71.4 of [56]) Question. “What other topological properties, sides those in (14.71.3), do the spaces W[C(X)] and W[2”] possess? example, are these spaces topologically complete?”
beFor
Comment. By [37, Proposition 1.11, for every continuum and LV[ax] are topoIogicalIy complete. “[Bruce (14.73.26 of [56]) Question. pings preserve the covering property? In property, then does every monotone image do not preserve CP, or even CPH, since [by (14.73.17)] an d can be openly mapped
X, W[C(X)]
Hughes]. What classes of mapparticular, if X has the covering of X?3 Note that open mappings non-planar solenoids have CPH onto a circle [by (1.207.5)].”
Comment. In Theorem 3.9 of [28], it was proved that monotone maps preserve the fact of being in Class (I/t’). Then by Theorem 67.1, monotone maps preserve the covering property. (14.73.27 of [56]) Question. ering property or have CPH?”
“What
classes of continua
have the cov-
“Let X be a continuum. Call C(X) a (14.73.28 of [56]) Definition. Whitney hyperspace [resp., we& Whitney hyperspace] provided that given any [resp., some] Whitney map /A for C(X) and any t E [O,p(X)), there exists a continuum Yt,fi such that
If C(X) is a Whitney hyperspace [resp., weak Whitney hyperspace], then we say that X has a Whitney hyperspace [resp., X has a weak Whitney hyperspace] .”
81. UNSOLVED
AND PARTIALLY
SOLVED
QUESTIONS
OF [56]
459
(14.73.29 of [56]) Question. “For what classesof continua X is C(X) a Whitney hyperspace or a weak Whitney hyperspace [see (14.73.30)]?” (14.73.30 of [56]) Question. “Is there a continuum X such that C(X) is a weak Whitney hyperspace but not a Whitney hyperspace?” (14.73.31 of (561) Definition. “Let X be a continuum. Call C(X) an invariant Whitney hyperspace provided that for each Whitney map p for wo? P-w
/-4X)1) = C(X)
for each t E [0,/~(x)). If C(X) is an invariant Whitney hyperspace, then we say that X has an invariant Whitney hyperspace.” (14.73.34 of [56]) Question. “For what classesof continua X is C(X) an invariant Whitney hyperspace? Note the following facts: If X E CPH and if X is Whitney stable [see (14.39.1)], then it follows easily from (14.73.11) that C(X) is an invariant Whitney hyperspace. Hence, an arc, a pseudo-arc [also see (14.73.15)], any particular pseudo-solenoid [also see (14.73.16)], and any non-planar solenoid [also see paragraph following (14.73.1?)] has an invariant Whitney hyperspace. So does the circle, by (1.213.2). The result in (14.73.33) leads to the following question which is more specific than the one above: Which indecomposable chainable or circle-like continua have an invariant Whitney hyperspace?” “What are classesof C*-smooth continua (15.14 of [56]) Question. besides those given in (1.207.8), (14.76.6), (15.11), and (15.13)?” Comment.
More properties of C*-smooth can be found in [28].
“If X is a homogeneousC*-smooth contin(15.21 of [56]) Questions. uum, then must X be indecomposable? If, in addition, X is planar, then must X be hereditarily indecomposable?” (16.34 of [56]) Question. “What are necessary and/or sufficient conditions in terms of X in order that 2x [equivalently, by (16.7) C(X)] be contractible?” Comment. See sections 20 and 78, where several conditions for contractibility of the hyperspaces are given.
460
XV.
QUESTIONS
(16.37 of [56]) Question. “If X E property (K), then does 2” E property (K)? If X E property (IC), then does C(X) E property (PG)? If 2x E property (K), then does C(X) E property (K)? If C(X) E property (/G)? then does 2x E property (K)? The converse of the second question is true and is stated and proved in [329, 2.81; the proof shows the converse of the first question is also true.” Comment. W. J. Charatonik has shown ([4]) that two opposite spirals in R2 approaching the unit circle S1 Exercise 20.24), then X and C(X) h ave property (K) and property (K). The second and third questions in 16.37 of (see paragraph after Questions 78.27).
if X consists of (see remark after 2x does not have [56] are still open
(16.38 of [56]) Question. “What kinds of mappings preserve property (K)? In (16.29) we saw that confluent mappings preserve property (IC); hence, so do monotone [329, 4.41 and open [329, 4.51 mappings. Also, Wardle [329, 2.91 has shown that retractions preserve property (K).” Comment. (see Definition
In [48, Theorem 2.11, Kato showed 78.25) preserve property (K).
that refinable
maps
(16.40 of [56]) Question. “If X is a continuum whose hyperspaces are contractible, then what kinds of images of X have contractible hyperspaces? For example, open images do by (16.39).” Comment. See Theorem 78.24 for a.class of maps which preserve hyperspaces contractibility. See also Question 78.26. “When is 2y or C(Y) homogeneous for a (17.5 of [56]) Question. (locally connected) Hausdorff continuum Y? I do not know an example of a non-metric Hausdorff continuum Y such that 2’ or C(Y) is homogeneous [see (17.6)].” “If Y is a locally connected homogeneous (17.6 of [56]) Question. Hausdorff continuum, then is it necessarily true that 2’ is homogeneous? Since the Cartesian product of homogenoeus spaces is homogeneous, it follows from [167] that any uncountable product 2 of closed real number intervals is a homogeneous Hausdorff continuum. I do not know if 2’ or C(Z) is homogeneous for these product spaces 2.”
81. UNSOLVED AND PARTIALLY SOLVED QUESTIONS OF [56]
461
“[255]. Let 2 be a subset of a metrizable real (18.1 of [56]) Definition. topological vector space L. We will assume 2’ has the “Hausdorff metric”. Now let cc(Z) = {A E 2’ : A is convex} with the restricted Hausdorff cc-hyperspace of Z.”
metric.
The space cc(Z)
will be called the
“[256, Problem 3.51. Is cc(e,) M f?,? Since this (18.9 of [56]) Question. question was asked in [256], a related result has appeared - see (19.40).” (18.22 of [56]) Question. “For what 2-cells X in R” is cc(X) x Ioo? By (18.21.1) we know that if X is a 2-cell in R2 with polygonal boundary such that (1) each member or cc(X) has arbitrarily small infinite-dimensional neighborhoods or (2) every maximal convex subset of X is a 2-cell [see (18.29)] holds, then cc(X) M I”. This is a partial answer to the questions in (18.23) through (l&25).” Comment. In [ll], Curtis, Quinn and Schori characterized polyhedral 2-cells X in R2 such that cc(X) M 10°. They proved that, for a polyhedral 2-cell X in R2, cc(X) M I” if and only if X does not contain an alternating sequence of three collinear points of local non-convexity. (18.23 of [56]) Question. “[256, Problem 2.41. If X is a 2-cell in R2 such that (1) of (18.22) holds, then is cc(X) M 10°? A pertinent example is in (18.20).” (18.24 of [56]) Question. “[256, Problem 2.51. If X is a 2-cell in R2 such that (2) of (18.22) holds, then is cc(X) M P?” (18.25 of [56]) Question. “[256, Problem 2.41. If X is a 2-cell in R2 such that every maximal convex subset of X is a point or a 2-cell, then is cc(X) w Ic”? A pertinent example is in (18.20). Also, the converse question for 2-cells in R2 has a negative answer by (18.19).” (18.26 of [56]) Question. cc(X) M p?”
“If X is a starshaped
2-cell in R2, then is
(18.37 of [56]) Question. “Do there exist three convex Hilbert cubes in & whose pairwise intersections are Hilbert cubes, whose total intersection is nonempty and finite-dimensional, and whose union is a Hilbert cube?”
462
XV. Comment.
QUESTIONS
The following comment and questions appeared in section
4 of [21] ([l] in [21] is [2] here and [15] in [21] is [64] here): “In [l], Anderson conjectured that if Qi, Qz and Qi n Qz are Hilbert cubes, then Qi U Qx is a Hilbert cube. He verified this for the case when Qr flQa is a Z-set in each of Qi and Q2 [l]. Handel has shown that Qi nQz need only be a Z-set in Qi , and Sher’s example ([I5]) indicates that Handel’s result may be the best possible in the general setting. In this section we will inquire into the exent to which Anderson’s conjecture holds when Qr and Q2 are embedded nicely in some hyperspace. Specifically we ask the following two questions. Question. Let p,q E X, C{,}(X) and C{,,(X) C{,l (X) a Hilbert cube?
C{,}(X),
a metric continuum, and assume that
n Cj,l (X) are Hilbert cubes. Is Cipl (X)
u
We call a closed subset A of C(X) convex in C(X) if A, B E A and -4 C C c B, C E C(X) implies that C E A. Sets of the form CA(X) are special types of convex sets, and the previous Question can be asked more generally: Question. Assume that A, B are convex sets in C(X) (X is a continuum) such that A, B and A n f? are Hilbert cubes. Then is A U f3 a Hilbert cube?” “[256, Problem 3.31. Let W be an open (18.39 of [56]) Question. convex subset of P. What are necessary and sufficient conditions on VI; M I”?” in order that P\W ‘
82. SOLVED
QUESTIONS
OF [56]
(19.8 of [56]) Question. 71,m > 2?”
Ys a(P)
(19.9 of [56]) Question. homogeneous?”
“Is a(R3), more generally a(P)
(19.10 of [56]) Question. (19.11 of [56]) Question. (or for any) n 2 2?”
homeomophic to a(P)
463 for each
for 122 3,
“Is a(&) homogeneous?” “Is a([,) homeomorphic to u(P)
for some
(19.12 of [56]) Q uestion. The samequestions as (19.8) through (19.11) with u( ) replaced by any of the spaces s( ), t( ), or e( ). Also, I do not know if all these spacesare mutually homeomorphic. In particular: Is a(@) homeomorphic to s(R2)? (19.20 of [56]) Question. “For what dendroids D is u(D) U Fl (D) compact [as a subspaceof C(D)]? More generally: For what arcwise connected continua X is u(X) U Fl (X) compact?” (19.32 of [56]) Question. “The same as (19.8) through (19.12) with the spacesconsidered there replaced by any of the spaces hi( ), ch( ), or P( ). Also [camp., (19.4) and (19.9)], I do not know if hi(R2), ch(R2), or P(R2) is homogeneous.” (19.33 of [56]) Question. “Is P(R2) [resp., hi(R2), ch(R2) ] homeomorphic to &? I do not know if these spaces are homogeneous. I also mention that that I do not know the answer to the question at the beginning of (19.33) when R2 is replaced by any of the spaces 0’. Note that P(fi*), hi(R*), and ch(R*) are topologically complete.”
82.
Solved
Questions
of [56]
(1.44 of [56]) Lemma. “[250, 3.111. If X contains an open subset with uncountably many components, then so does C(X).8” (1.46 of [56]) Question.
“[250, 3.131. Is the converse of (1.44) true?”
Comment. In [56, p. 2511, it was announced a paper that was never published which contains positive answers to Question 1.46 and 1.47 of [56]. Later, in [49, Corollary 3-81, Katsuura gave an affirmative answer to Question 1.46 of [56]. Then, by [56, Theorem 1.411,the answer to Question 1.47 of [56] is also affirmative.
464
XV.
QUESTIONS
(1.45 of [56]) Theorem. “If X contains an open set with uncountably many components, then there is a mapping of C(X) onto the cone over the Cantor middle-thirds set.8” (1.47 of [56]) Question.
“Is the converse of (1.45) true?*”
(1.70 of [56]) Theorem. “Let X be an hereditarily indecomposable continuum. If a dendroid D is embeddable in C(X), then D is a smooth dendroid. More specifically, if A c C(X) is a dendroid, then A is smooth at UA.” (1.74 of [56]) Question. true?3” Answer.
Yes (Grispolakis
“Is the converse of the first part of (1.70)
and Tymchatyn),
see Theorem 4.1 of [29].
(1.147 of [56]) Question. “If C(X) contains an n-cell for some 71 > 3, then does X contain an n-od? B. J. Ball asked me this question for the case when n = 3. If the answer is “yes”, then by (1.100) we could have a characterization of those continua X such that C(X) contains an n-cell.” Answer.
Yes, see Theorem 70.1.
(1.103 of [56]) Theorem. “[244, Theorem 61. Let X be a continuum which contains subcontinua Y, Yr, Ys, . . ,I$,. . . such that (1) Y; fl Y # 0 and yi g Y for any i = 1,2,. . .; (2) (1; \ Y) n (Yk \ Y) = 0 whenever j # k; (3) diam[x] + 0 as i -+ co. Then, C(X) contains a Hilbert cube.” (1.148 of 1561) Question. “If C(X) contains a Hilbert cube, then does X contain continua Y, Yr , Ys, . . . , Yi, . . . satisfying (1) through (3) of (1.103)?” Answer. No ([36, p. 641). H owever, in [36, Theorem 2.91 it was proved that, for a continuum X, the following are equivalent: (a) C(X) contains a Hilbert cube, (b) X contains an oo-od, and (c) there is a sequence of continua Y, Yr, Ys, . . . which satisfies (1) and (2) in Question (1.148). (2.4 of [56]) Question. “[250, 2.21. If X is any two-dimensional tinuum, then must dim[C(X)] = oo?”
con-
82. SOLVED QUESTIONS OF [5G] Answer.
465
Yes (Levin and Sternfeld), seeTheorem 73.9.
“[93, Question 31. Is there a one-dimensional (2.7 of [56]) Question. hereditarily indecomposable continuum X such that, dim[C(X)] = 00 [see 2.8]? By (2.2), this is equivalent to asking: Is there a one-dimensional hereditarily indecomposable continuum X such that X admits a monotone open image of dimension greater than one?” Answer.
Yes (Lewis), seeTheorem 74.1.
(3.7 of [56])
C(X)
Question. “[250, 2.11. If S is a continuum such that, is embeddable in R”, then must X be one-dimensional?”
Answer.
Yes (Levin and Sternfeld), see Theorem 73.9.
“[347, p. 1091. What compact totally-discon(5.24 of [56]) Question. nected Hausdorff spacesadmit a (continuous) selection from their space of nonempty closed subsets?” Answer. See the characterization given by van Mill and Wattel in Theorem 75.3. (6.10.1 of [56]) Question. “Is there a non-locally connected continuum X such that there is a retraction from 2x onto Cl(X)? In particular, what, about X = (SP)l as defined in (8.22)? Some other special cazes are asked about in (6.10.3).” Answer. Yes. The first example of a non-locally connected continuum Y such that C(Y) is a retract of 2’ was given by Goodykoontz in [27] (1’ is the harmonic fan). The continuum X = (SP)l is the circle with a spiral represented in the Figure 20 (6), p. 63. For this continuum S, Curtis in [9] has shown that C(X) is a retract of 2”. (8.14 of [56]) Question. “Must a finite-dimensional indecomposable C-H continuum be chainable or circle-like?” Answer. No. Two examples have been given, the first one by Dilks and Rogers in [12, Theorem 61 and the second one by Sherling in [65, Example 4.11.
466
XV.
QUESTIONS
(8.20 of [56]) Some Comments and Questions. “Let X be a finitedimensional C-H continuum. As we will see in (8.23), it is completely known what X must be if X is hereditarily decomposable. So, for our purposes here, assume X contains a nondegenerate indecomposable subcontinuum 1’. Note that, by (8.18), X contains only one such subcontinuum. In general, a nondegenerate subcontinuum of a C-H continuum need not be a C-H continuum. This is easily seen by examining various subcontinua of the continua listed in (8.23) ( one such example is given in 1281, Example 11). However, I do not know the answer to the following questions. (8.20.1) Question. Must Y be a C-H continuum? (8.20.2) Q ues t’ Ion. If h : Cone(X) -+ C(X) is a homeomorphism, then must. h[Cone(Y)] = C(Y)?” Answer.
Yes to both questions,
see [42].
(10.9 of [56]) Question. “[241, 2.01. If product, then must X be an arc or a circle? is “yes” when X is locally connected. We will under other conditions, the most general one (10.12)].” Answer.
C(X) is a finite-dimensional We have seen that the answer show that the answer is “yes” being when X is a-triadic [see
Yes, see section 79.
(11.18 of [56]) Question. “Assume X is any chainable continuum which is not an arc; then, must X contain a subcontinuum Y such that C(Y) \ {E} is not arcwise connected for some subcontinuum E of Y? Of course [see (ll.l)], the question is only open for the class of hereditarily decomposable chainable continua. 2 This may seem like an off-beat question, but, an affirmative answer may lead to a hyperspace proof of Henderson’s result [140]. Hopefully, a “simpler” proof than the one Henderson gave would evolve. Some material which may lead to generalizations of Henderson’s result is in (6.23) through (6.27).” Answer.
Yes (Grispolakis
and Tymchatyn),
see Theorem 4.9 of [30].
“[234, 6.21. Is there an hereditarily decom(12.20 of [56]) Question. of it has a cut point?‘” posable continuum such that no subcontinuum Answer. Yes (Thomas, and Grispolakis 53 of [66] and Example 5.1 of [30].
and Tymchatyn),
see pp. 50-
“[234, 6.31. Is there an hereditarily (12.21 of [56]) Question. posable continuum which contains no rational continuum?‘”
decom-
82. SOLVED
Answer.
QUESTIONS
OF [56]
Yes (Grispolakis and Tymchatyn),
467
see 5.2 of [30].
“Let X be the “Buckethandle” continuum (14.22 of [56]) Questions. [defined in (1.209.3)]. Let p be a Whitney map for C(X) such that p(X) = 1. Is there a homeomorphism h : Cone(X) + C(X) such that the diagram [where p is the standard projection mapping]
Cone (X)
commutes? It is known that Cone(X) and C(X) are homeomorphic [280, p. 2801, and that any cone-to-hyperspace homeomorphism must take the base and the vertex of the cone onto /l-‘(O) and X E C(X) respectively [by (8.7)].” Answer. Yes (Dilks and Rogers), see[12, Theorem 111. In fact, it has been recently proved in [47] that if X is a finite-dimensional continuum with the cone = hyperspace property, then there exists such a homeomorphism h : Cone(X) + C(X). (14.38 of [56]) Question. “Let X denote any compact subset of a Banach space such that dim[X] > 1. Let d be the metric for X obtained from the norm on the Banach space. Let pd denote the restriction to C(X) of the Whitney map constructed (using d) as in (0.50.1). Is pi’(t) a Hilbert cube whenever 0 < t < p(X)?” Answer.
Yes (Goodykoontz and Nadler), seeTheorem 25.3.
‘&Is there a Whitney-reversible (14.56 of [56]) Question. which is not strong Whitney-reversible property?”
property
468
XV.
QUESTIONS
Answer. Combining Example 30.4 and Theorem 30.6, we conclude that the property of being an AR is a Whitney-reversible property which is not a strong Whitney-reversible property. Being an AR is the only known topological property having these characteristics. (14.63 of [56]) Question. “Given any continuum map for 2.’ which is monotone?”
X, is there a Whitney
Answer. No to both questions 14.63 and 14.64 of [56] (W. J. Charatonik), see Example 24.10, see also Theorem 24.13. (14.64 of [56]) Question. map for 2” which is open?”
“Given any continuum
X, is there a Whitney
(14.69 of [56]) Question. Y is a circle [see (14.64)]?”
“IS there an open Whitney
map for 2y where
Answer. Yes (W. J. Charatonik and Illanes). In fact, for every Peano continuum Y, there is an open Whitney map for 2’ (see Theorem 24.11 and Exercise 24.24). “Is the mapping f : W[2”] + W[C(X)] (14.71.5 of [56]) Question. onto W[C(X)], where f(w) denotes the restriction of w to C(X) for each w E W[2x]? In other words: Can every Whitney map for C(X) be extended to a Whitney map for 2x?” Answer. Yes (Ward); see the more general version that we proved in 16.10 using Ward’s Theorem 3.1 of [68]. In fact, following Ward’s ideas, it was proved in [37, Theorem 4.21 that if X is a continuum and ‘7-t is a closed subset of 2sy, t.hen there is an embedding 4 : W[3t] + W[2x] such that 4(p) is an extension of p for each p E W[31]. “[179, section 61. What classes of continua (14.73 of [56]) Question. does the covering property characterize?” Answer. Continua with covering property several ways, see Theorem 67.1.
have been characterized
in
(14.73.22 of [56]) Question. “[Bruce Hughes]. If X E CP, then is /~-l(t) unicoherent for each Whitney map ,u for C(X) and each t E [O,p(S))? Note that, since X is unicoherent when X E CP [by the second part of (14.14.1)], the answer to the question above is “yes” when t = 0. Let us also note that if X E CP, then by (14.73.3) p-‘(t) is irreducible
82.
SOLVED
QUESTIONS
for each t E [0,1.1(X)) - h owever, unicoherent [see X in (14.29)l.l’ Answer.
OF
[56]
an irreducible
No (W. J. Charatonik),
469
continuum
need not be
see Example 35.5 ([5, Example
81).
(14.73.23 of [56]) Question. “[Bruce Hughes]. If there is a Whitney map /.L for C(X) such that cl-‘(to) is irreducible for some to E [O,p(X)), then must X be irreducible? Partial answers are in (14.55.1).” Answer. Yes (Eberhart and Nadler), generalization, see Theorem 49.3.
see [22, Theorem
1.11. For a
(14.7324 of [56]) Question. “[Bruce Hughes]. If X is any decomposable unicoherent continuum which is not a triod [in the sense of (0.21)], then does there exist a Whitney map ~1for C(X) such that p-l (to) is irreducible for some to > O? Affirmative answers to (14.73.23) and (14.73.24) would give a hyperspace proof of Sorgenfrey’s Theorem [306, Theorem 3.21.” Answer.
No (Eberhart
and Nadler),
see [22, Example 3.11.
(14.73.25 of [56]) Question. (([Bruce Hughes]. Is the converse of (14.73.21) true? Lelek [194, Problem l] has asked for a characterization of those continua which are in Class(W). An affirmative answer to (14.73.25) would give a hyperspace characterization. It is known that any continuum in Class(W) is unicoherent and not a triod (thus, by [306, Theorem 3.21, irreducible).” Answer. Yes (Grispolakis and Tymchatyn), see Theorem 67.1 in which it is proved that being in Class(W) is equivalent to having covering property. (14.76.9 of [56]) Question.
“Is CP a Whitney
property?”
Answer. No to both questions 14.76.9 and 14.76.10 of [56] (Grispolakis and Tymchatyn, and W. J. Charatonik), see Example 35.5. (14.76.10 of [56]) Question. “Is being in Class (W) a Whitney property? If the answer is “yes”, (14.73.5) can then be used to give an affirmative answer to (14.73.25).” (15.15 of [56]) Question. “What are the C*-smooth circle-like continua [see (lS.lS)]?’ A partial answer was given in (14.76.6), i.e., non-planar circle-like continua are C*-smooth.”
470
XV.
QUESTIONS
Answer. In Theorem 4.1 of [28], Grispolakis, Nadler and Tymchatyn have shown that for a non-chainable circle-like continuum X, the following are equivalent: (i) X E CPH; (ii) X E CP; (iii) X is in Class (W); (iv) X contains no local separating continuum; (v) X is C*-smooth; and (vi) X is not weakly chainable. (15.18 of [56]) Question. “Must a C*-smooth herent [hence, by (15.6), hereditarily unicoherent]?’ continua the answer is “yes”.” Answer. of [28].
Yes (Grispolakis,
continuum For arcwise
Nadler and Tymchatyn),
be unicoconnected
see Corollary
3.4
(16.36 of [56]) Question. “If X and Y are continua whose hyperspaces are contractible, then are the hyperspaces of X x Y contractible? The converse is true, as can be seen by using the mapping induced by the projection of X x Y onto a factor.” Answer.
Yes (Nishiura
83.
and Rhee), see Exercise 78.42.
More
Questions
General
Spaces
83.1 Question. Let (Y,T) be a topological space. If (CL(Y),Tv) metrizable, then must (Y,T) be compact? (Yes if (Y, T) is a Ti-space, see Theorem 3.4.) 83.2 Question. Let (Y,T) be a TO-space. If able, then must (Y, T) be metrizable? (Yes if (Y,T) is a Ti-space, see Theorem 3.4.)
(CL(Y),Tv)
is
is metriz-
83.3 Question. Let (Y,T) be a countably compact Hausdorff space. Is L-convergence for sequences in CL(Y) equivalent to TV-convergence in
CL(Y)? (Yes if (Y,T) is compact Hausdorff. In fact, yes if compact and regular, see Theorems 4.4 and 4.6.)
(Y,T) is countably
83.4 Question. If (Y,T) is a metrizable space that is not separable, then is there a metric, p, for Y such that TH, g TV and TV $JT~I,? (If (Y,T) is a metrizable, separable space, then is there a metric, p, for Y such that TH, C TV.)
GEOMETRIC
MODELS
471
83.5 Question. If (Y,T) is a totally disconnected, metrizable space, then is (CL(Y), TV) totally disconnected? Is (2x, TV) totally disconnected? (Yes, if (Y,T) .IS a1so compact, by 3.5, 8.6 and 12.11.)
Geometric
Models
83.6 Question. Find geometric models for 2y when Y is any zerodimensional, infinite, Hausdorff compacturn with a dense set of isolated points. When are these models homeomorphic? (In the metric case, there is one common model, seeTheorem 8.9.) 83.7 Question. For what Hausdorff continua, Y, is 2’ or C(Y) a Tychonoff cube? (In the metric case, 2’ z IM if and only if Y is a nondegenerate Peano continuum; also, C(Y) M IO0 if and only if Y is a nondegenerate Peano continuum and there is no free arc in Y, seeTheorem 11.3.)
Z-Sets 83.8 Question. For what continua X do 2x and/or C(X) have Torunczyk’s property (= identity is a uniform limit of Z-maps.) (Known if X is a Peano continuum, see.the Curtis-Schori Theorem 11.3.) 83.9 Question. What hyperspaces, 2” or C(X) when X is a continuum, have the property that every countable, closed subset,of the hyperspace is a Z-set in the hyperspace’? (Known if X is a Peano continuum.) 83.10 Question. For what continua, X, is Fi(X) In 2x? (Known if X is a Peano continuum.)
Symmetric
a Z-set in C(X)?
Products
83.11 Question. For what continua, X and Y, is it true that if Fz(X) M Fz(Y), then X M Y? 83.12 Question. For what continua, X, is F,(X) z P? (It is known that F,(Im) z Im ([24, Corollary 5 of section 31.) 83.13 Question. For what continua, X, is F,(X) (of two nondegenerate continua)? (See, e.g., Theorems 6, 8, and 9 of [63].)
a Cartesian product
472
XV.
QUESTIONS
83.14 Question. Is there a one-dimensional continuum, X, such that the a-sphere S’ can be embedded in Fl(X)? (S” can not be embedded in the Cartesian product of any two l-dimensional metric spaces, see [3].)
Size
Maps
Definition. Let X be a continuum, and let 3t = 2x or C(X). mnp for 3-1 is a continuous function (T : 3-1 + [0, 00) such that (1) ~(~4) 5 a(B) whenever A,B E ‘l-t such that A c B, (2) a({~}) = 0 for all z E X, and {z} E ‘7i. A size level in 7t is a point inverse of a size map for xc.
A size
Note: size levels in C(X) are continua (the proof is similar to the proof of Theorem 19.9.; see Exercise 19.18.)
Theorem [61]. Assume that 2 is a continuum, and consider the following three conditions: (1) 2 is a planar AR; (2) cut points of 2 have component number two; (3) any true cyclic element of 2 contains at most two cut points of 2. Any size level in C(X) for an arc X satisfies (l)-(3); conversely, if 2 satisfies (l)-(3), then 2 is a diameter level in C(X) for some arc X. 83.15 Question. Characterize those continua that are size levels in C(X) when X is a simple closed curve (a simple triod, any finite graph). For any continuum C(N(X))
X and for Z(X)
= 2x or C(X),
let
= (0 : ?-1(X) -+ [0, m) : 0 is a size map for N(X)}
and let C(C(U(X))) x(%(X)) topologized
= {A E N(‘,Q(X))
: A is a size level for B(X)};
is topologized by the uniform metric, by the Hausdorff metric for 31(%(X)).
and C(l(Z(X)))
is
83.16 Question. If X is any nondegenerate continuum, then is c(u(x)) = eg I~ c(c(7-q~))) M e2? (It was proved in Theorem 5.6 of [39] (respectively, [40]) that the space of Whitney maps (respectively, Whitney levels) of a continuum X is homeomorphic to e,.)
THE
SPACE OF WHITNEY
LEVELS
The Space of Whitney
FOR 2’
Levels
473
for Zx
For a continuum X, let WL(2-‘) = {A E 22x : there exists a Whitney map p : 2x -+ [0, 00) and there exists t E [0, 00) such that A = p-‘(t)}. 83.17 Question [41, Question 2.101. If WL(2x) is arcwise connected, then does there exists a monotone Whitney map for 2x? 83.18 Question [41, Question 2.111. Is WL(2x) connected space for every continuum X?
- {Fi (X)} an arcwise
83.19 Question [41, Question 2.121. Is WL(2x) every continuum X?
a connected spacefor
83.20 Questions [41, Question 2.131. What other topological properties does the space WL(2x) possess? For example, if X is a locally connected continuum, then is WL(2x) locally connected? If X is a locally connected continuum, then is WL(2x) homemorphic to &? (see comment to Question 83.16.)
Aposyndesis 83.21 Questions [26, Questions 1 and 21 or [17, Questions 1 and 21. Let X be a continuum. Is C(X) (respectively, 2x) zero-dimensional closed set aposyndetic? (see Definition 29.1). In [44, Theorem B], it was shown that 2x is countable closed set aposyndetic. An affirmative answer to the following question might help to answer Question 83.21. 83.22 Question [44, Question 51. Let X be a continuum. If B is a zero-dimensional closed subset of C(X) and A E C(X) is not a one-point set such that .4 4 L3, then does there exists a Whitney level A for C(X) suchthatdnf3=0andAEd? Related to Question 83.22, we remark that Krasinkiewicz ([50]) has shown that, for every continuum X, C(X) can not be separated for any of its zero-dimensional subsets.
Universal
Maps
83.23 Question [60, Problem 1.51. Let Y = [0, l] or S1. If f maps a continuum X onto Y, then must the induced map 2f : 2ay + 2’ be universal? An affirmative answer to this problem for the case that X is
474
XV.
arc-like or circle-like 21.5 and 22.4.
QUESTIONS
would solve part of Question
7.10 of [56] by applying
83.24 Question [GO, Problem 1.61. If Y is a simple n-d and f is a universal map of a continuum X onto Y, then must C(f) : C(X) + C(Y) be universal? A positive answer would yield a different proof of 73.9 (see comment after Problem 1.6 of [60]). 83.25 Question [60, Problem 1.91. Let X be a Peano continuum with the following property: Whenever f is a map of X onto a continuum Y such that C(f) : C(X) + C(Y) is universal, f must be universal. Then is X a dendrite? (Known if X is one-dimensional, seeTheorem 1.8 of [60]). 83.26 Question [60, Problem 1.111. If X is a Peano continuum such that every monotone image of X has the fixed point property, must dim[X] 5 l? If the answer is yes, then the answer to Question 83.25 is yes (by Theorems 1.8 and 1.10 in [60]). 83.2’7 Question [60, Problem 1.131. A continuum Y is said to be in Class(U) if for every map f on any continuum X onto Y, C(f) : C(*X) -+ C(Y) is universal. Determine an intrinsic characterization of Class(U).
References 1. 2.
3.
4. 5. 6. 7. 8. 9.
G. Acosta, Continua with unique hyperspace, preprint. R. D. Anderson, Topological properties of the Hilbert cube and the injinite product of open intervals, Trans. Amer. Math. Sot., 126 (1967), 200-216. K. Borsuk, Remark on the Cartesianproduct of two l-dimensional co71tinua, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 23 (1975), 971-973. W. J. Charatonik, Hyperspaces and the property of Kelley, Bull. Acad. Polon. Sci., Ser. Sci. Math., 30 (1982), 452-459. W. J. Charatonik, Some counterexamples concerning Whitney levels, Bull. Polish Acad. Sci. Math., 31 (1983), 385-391. W. J. Charatonik, Homogeneity is not a Whitney property, Proc. Amer. Math. Sot., 92 (1984), 311-312. W. 3. Charatonik, Convex structure on the space of order arcs, Bull. Polish Acad. Sci. Math., 39 (1991), 71-73. W. J. Charatonik and A. SW& On size mappings, preprint. D. W. Curtis, A hyperspace retraction theorem for a class of half-line compactifications, Topology Proc., 11 (1986), 29-64.
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10. D. W. Curtis and M. Lynch, Spaces of order arcs in hyperspaces Peano continua, Houston J. Math., 15 (1989), 517-526.
of
11. D. W. Curtis, J. Quinn and R. M. Schori, The hyperspace of compact convex subsets of polyhedral 2-ceil, Houston J. Math., 3 (1977), 7-15. 12. A. M. Dilks and J. T. Rogers, Jr., Whitney stability and contractible hyperspaces, Proc. Amer. Math. Sot., 83 (1981), 633-640. 13. C. Dorsett, Local connectedness, connectedness im kleinen, and other properties of hyperspaces of RQ spaces, Mat. Vesnik, 16 (1979), 113123. 14. C. Dorsett, Local connectedness Palermo, 31 (1982), 137-144. 15. C. Dorsett, Connectedness 11 (1982), 31-36.
in hyperspaces,
Rend.
im kleinen in hyperspaces, Math.
16. C. Dorsett, Connectivity properties Fund. Math., 121 (1984), 189197.
in hyperspaces
Circ.
Mat.
Chronicle,
and product spaces,
Jr., Aposyndesis in hy17. E. K. van Douwen and J. T. Goodykoontz, perspaces and Tech-Stone remainders, General Topology and Modern Analysis, L. F. McAuley and M. M. Rao, Editors, Academic Press, Inc., New York, 1981, 43-52. 18. R. Duda, On the hyperspace of subcontinua Math., 62 (1968), 265-286.
of a finite graph, I, Fund.
19. R. Duda, Correction to the paper “On the hyperspace of subcontinua of a finite graph, I”, Fund. Math., 69 (1970), 207-211. 20. C. Eberhart, Continua with locally connected Whitney continua, HOUSton J. Math., 4 (1978), 165-173. 21. C. Eberhart and S.B. Nadler, Jr., Hyperspaces Amer. Math. Sot., 77 (1979), 279-288.
of cones and fans, Proc.
22. C. Eberhart and S.B. Nadler, Jr., Irreducible J. Math., 6 (1980), 355-363.
Whitney levels, Houston
23. C. Eberhart, S. B. Nadler, Jr. and W. 0. Nowell, Jr., Spaces of order arcs in hyperspaces, Fund. Math., 112 (1981), 111-120. 24. V. V. Fedorchuk, Covariant functors in the category of compacta, absolute retracts and Q-manifolds, (Russian) Uspekhi Mat. Nauk 36:3 (1981), 177-195. Traslation: Russian Math. Surveys, 36:3, (1981), 211-233. 25. J. T. Goodykoontz, Jr., Local arcwise Houston J. Math., 4 (1978), 41-47.
connectedness
in 2x and C(X),
476
XV.
QUESTIONS
26. J. T. Goodykoontz, Jr., Aposyndesis and hyperspaces, Collection: ToPoloi?~ Conference, 1979 (Greensboro, N.C., 1979), Guilford College, Greensboro, N.C., 1980, 129-135. 27. J.T. Goodykoontz, Jr., A non-locally connected continuum X such that C(X) is a retract of 2 x, Proc. Amer. Math. Sot., 91 (1984), 319-32‘2. 28. J. Grispolakis, S. B. Nadler, Jr. and E. D. Tymchatyn, Some properties ?f hyperspaces with applications to continua theory, Canad. J. Math., 31 (1979), 197-210. 29. J. Grispolakis and E. D. Tymchatyn, Embedding smooth dendroids in hyperspaces, Canad. J. Math., 31 (1979), 130-138. 30. J. Grispolakis and E.D. Tymchatyn, Irreducible continua with degenerate end-tranches and arcwise accesibility in hyperspaces, Fund. Math., 110 (1980), 117-130. 31. M. Handel, On certain sums of Hilbert cubes, General Topology Appl., 9 (1978), 19-28. 32. H. Hosokawa, Arcwise connectedness of the complement in a hyperspace, Tsukuba J. Math., 20 (1996), 479-486. 33. K. Hur, J. R. Moon and C. J. Rhee, Connectedness im kleinen and local connectedness in C(X), Honam Math. J., 18 (1996), 113-124. 34. K. Hur, J. R. Moon and C. J. Rhee, Connectedness im kleinen and components in C(X), Bull. Korean Math. Sot., 34 (1997), 225-231. 35. A. Illanes, A continuum X which is a retract of C(X) but not of 2x, Proc. Amer. Math. Sot., 100 (1987), 199-200. 36. A. Illanes, Cells and cubes in hyperspaces, Fund. Math., 130 (1988), 57-65. 37. A. Illanes, Spaces of Whitney maps, Pacific J. Math., 139 (1989), 6777. 38. A. Illanes, Arcwise disconnecting subsets of hyperspaces, Houston J. Math., 16 (1990), l-6. 39. A. Illanes, The space of Whitney levels, Topology Appl., 40 (1991), 157-169. 40. A. Illanes, The space of Whitney levels is homeomorphic to 12, Colloq. Math., 65 (1993), l-11. 41. A. Illanes, Monotone and open Whitney maps defined in 2x, Topology Appl., 53 (1993), 271-288. 42. A. Illanes, Hyperspaces homeomorphic to cones, Glasnik Mat., 30 (50) (1995)) 285-294. 43. A. Illanes, Hyperspaces which are products, Topology Appl., 79 (1997), 229-247.
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63. R.M. Schori, Hyperspacesand symmetric products of topological spaces, Fund. Math., 63 (1968), 77-88. 64. R. Sher, The union of two Hilbert cubesmeeting in a Hilbert cube need not be a Hilbert cube, Proc. Amer. Math. Sot., 63 (1977), 150-152. 65. D. D. Sherling, Concerning the cone = hyperspace property, Canad. J. Math., 35 (1983), 1030-1048. 66. E. S. Thomas, Jr., Monotone decompositions of irreducible continua, Dissertationes Math., (Rozprawy Mat.) 50 (1966), l-74. 67. L. E. Ward, Jr., A cyclic connectivity theorem for hyperspaces,General Topology Appl., 10 (1979), 291-295. 68. L.E. Ward, Jr., Extending Whitney maps, Pacific J. Math., 93 (1981), 465-469.
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482
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QUESTIONS
64. D. W. Curtis, Growth hyperspaces Math. Sot., 238 (1978), 271-283.
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65. D. W. Curtis, Hyperspaces homeomorphic Math. Sot., 75 (1979), 126-130.
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QUESTIONS
128. J. Grispolakis and E. D. Tymchatyn, Spaces which accept only weakiy confluent mappings, Proceedings of the International Conference of Geometric Topology, PWN-Polish Scientific Publishers Warszawa 1980, 175-176. 129. J. Grispolakis and E. D. Tymchatyn, On a charac&i&ion of W-sets and the dimension of hyperspaces, Proc. Amer. Math. Sot., 100 (1987), 557-563. 130. P.M. Gruber, The space of compact subsets of Ed, Geom. Dedicata,8 (1979)) 87-90. 131. P. M. Gruber, The space of convex bodies, Handbook of Convex Geometry, Vol. A, B, Noth Holland, Amsterdam, 1993, 301-318. 132. P. M. Gruber and G. Lettl, Isometrics of the space of compact subsets of Ed, Studia Sci. Math. Hungarica, 14 (1979), 169-181. 133. P.M. Gruber and G. Lettl, Isometrics of the space of convex bodies of Ed, Bull. London Math. Sot., 12 (1980), 455-462. 134. P. M. Gruber and R. Tichy, Isometrics of spaces of compact or compact convex subsets of metric manifolds, Monatshefte Math., 93 (1982), 117-126. 135. B.-L. Guo and K. Sakai, Hyperspaces of CW-complexes, find. Math., 143 (1993), 23-40. 136. Y. Hattori and T. Nogura, Continuous selections on certains spaces, Houston J. Math., 21 (1995), 585-594. 137. R. E. Heisey and J. E. West, Orbit spaces of the hyperspace of a graph which are Hilbert cubes, Colloq. Math., 56 (1988), 59-69. 138. H. Hosokawa, Some remarks on the atomic mappings, Bull. Tokyo Gakugei Univ., 40 (1988), 31-37. 139. H. Hosokawa, Induced mappings between hyperspaces, Bull. Tokyo Gakugei Univ., 41 (1989), l-6. 140. H. Hosokawa, Mappings of hyperspaces induced by refinable mappings, Bull. Tokyo Gakugei Univ., 42 (1990), l-8. 141. H. Hosokawa, Induced mappings between hyperspaces, II, Bull. Tokyo Gakugei Univ., 44 (1992), l-7. 142. H. Hosokawa, Arcwise connectedness of the complement in a hyperspace, Tsukuba J. Math., 20 (1996), 479-486. 143. H. Hosokawa, Induced mappings on hyperspaces, Tsukuba J. Math., 21 (1997), 239-250. 144. H. Hosokawa, Induced mappings on hyperspaces, II, preprint. 145. K. Hur, S. W. Lee, P. K. Lim and C. J. Thee, Set-valued contractibility of hyperspaces, J. Korean Math. Sot., 29 (1992), 341-350.
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of
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of
symmetric
indecomposable
continua
products
of continua,
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256. A. K. Misra, Embedding partially ordered topological spaces in hyperspaces, Publ. Math. Debrecen, 37 (1990), 137-141. 257. E. V. Moiseev, Spaces of closed hyperspaces of growth and inclusion, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3 (1988), 54-57. Translation: Moscow Univ. Math. Bull., 43 (1988), 48-51. 258. L. Montejano, Convexity in the hyperspace of compact convex sets, An. Inst. Mat. Univ. Nat. Autonoma Mexico, 26 (1986), 145-159. 259. L. Montejano, The hyperspace of compact convex subsets of an open subset of R”, Bull. Polish Acad. Sci. Math., 35 (1987), 793-799. 260. L. Montejano-Peimbert and I. Puga-Espinosa, Shore points in dendroids and conical pointed hyperspaces, Topology Appl., 46 (1992), 4154. 261. S. B. Nadler, Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 262. S. B. Nadler, Jr., A fixed point theorem for hyperspace suspensions, Houston J. Math., 5 (1979), 125-132. 263. S.B. Nadler, Jr., Continua whose hyperspace is a product, Fund. Math., 108 (1980), 49-66. 264. S. B. Nadler, Jr., Whitney-reversible properties, Fund. Math., 109 (1980) 235-248. 265. S. B. Nadler, Jr., Universal mappings and weakly confluent mappings, Fund. Math., 110 (1980), 221-235. 266. S. B. Nadler, Jr., Induced universal maps and some hyperspaces with the fixed point property, Proc. Amer. Math. Sot., 100 (1987), 749-754. 267. S. B. Nadler, Jr., Continuum Theory, An introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 268. S. B. Nadler, Jr., e-selections, Proc. Amer. Math. Sot., 114 (1992), 287-293. 269. S. B. Nadler, Jr., Some noncompact hyperspaces with the almost fixed point property, Rocky Mountain J. Math., 22 (1992), 691-696. 270. S.B. Nadler, Jr., New problems in continuum theoy, Continuum theory and dynamical systems, Lecture Notes in Pure and Appl. Math., 149, T. West, Editor, Marcel Dekker, New York, 1993, 201-209. 271. S.B. Nadler and J. Quinn, Continua whose hyperspace and suspension are homeomorphic, General Topology Appl., 8 (1978), 119-126. 272. S.B. Nadler, Jr., J. Quinn and N.M. Stavrakas, Hyperspaces of compact convex sets, Pacific J. Math., 83 (1979), 441-462.
494
XV.
QUESTIONS
273. S. B. Nadler, Jr. and T. West, Size levels for arcs,Fund. Math., 141 (1992), 243-255. 274. V. Neumann-Lara and I. Puga-Espinosa, Shore points and den&tea, Proc. Amer. Math. Sot., 118 (1993), 939-942. 275. T. Nishiura and C. J. F&e, Cut points and the hyperspace of a&continua of C(X), Proc. Amer. Math. Sot., 82 (1981), 149-154. 276. T. Nishiura and C. J. Rhee, Contractibility of the hyperapace of au& continua, Houston J. Math., 8 (1982), 119-127. 277. T. Nishiura and C. J. Rhee, An admissible condition for contractible hyperapaces, Topology Proc., 8 (1983), 303-314. 278. T. Nishiura and C. J. Rhee, Contractible hyperapaces of a&continua, Kyungpook Math. J., 24 (1984), 1433154. 279. T. Nogura and D. Shakhmatov, Characterizations of intervals tinuous selections, to appear in Rend. Circ. Mat. Palermo 280. T. Nogura and D. Shakhmatov, tinuous selections, preprint. 281. J. Oledzki,
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496
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310. T. Watanabe, A Whitney map and the LindelCf property, Houston J. Math., 15 (1989), 147-151. manifolds which arise from the 311. M. M. Zarichnyi, Infinite-dimensional direct limits of ANRs, (Russian) Uspekhi Mat. Nauk, 39 (1984), 153154.
Special A, cl(A)
0 CL(X) TV (Sl,...,&) M
CLC(X) 2”
ii 3
dZ I-=, &a
32 32
3 3 3 5 6
dim(Y)
32 32 41 52
F(X) S’
Ga
15 16 16 19 19
lim inf Ai, lim sup Ai, lim A,
21
6 x I”
32 32
THY TV
H,
TH
“f Cone(Y),
4
v
52 66 67 76 77
G(Y) W) N(lr 1 AR AE cd(r,
81 83 86 97 98 99 104 106 125
-4)
cik int(B), B0 Y = EIF Bd
9 10 10 11 14
4x7 A) Nd(r, 4 Hd Hd
(r,
2$, CKW
6 6 6 6 7
C(X) I4 KC-Y)
oHd
Symbols
ClC 4P) diamd c(V 2X(K), P swwfl) WV, crZ ANR 497
C(X;
mo
A-)
125 134 134 137 156 156
498
SPECIAL SYMBOLS
ANE crANR.
156 157
exp FW USC
163 169
A.4[X]
173 188
WC(Wl, cc(Z)
195 206 231
a(W, s(W, t(M), e(M), P(M), NW, hi(M), L(M), a* W(X)), C(W(X)))
f’, f^ W)
WL(X) p (for Chapter
VIII)
C*
253
CP, CPH, X E CP, X E CPH H”(.)
x 0
253 257 270
CIP, MCIP Chain,,(U) (-4, a, f)*
278 278
Fd(X)
290
FAR, FANR
290 291 291 292
=, c’(X), r(Y)
go’(X),
IT(X), CT&c)
o(w) ‘JJb, ms(t), JWJ-%(~)), FWJ-%(~)>, “0” i(G) CE
(x’,
t)
L3(W Xl (2) WA, W, WA) 744 dl (A) 2f, C(f) C(X) z I’ x 2 comma v(X)
P)
305 306 310 314 314 327 333 335 349 381 413 416 424
mesh(f)
443 449
Sus(X) HS(W
WL(29
W2”l
450 452 457 461
462 472 473
Index A
ANR,
Absolute
extensor,
Absolute
neighborhood
Absolute
Absolute
crANR,
77
extensor,
with respect
to an, 263
156 Anti-chain,
retract,
156
Aposyndetic,
retract,
76
Appoximate
C*-smooth,
157
extendable
neighborhood
Absolutely
206 238, 473 strong deformation
retract,
253
Approximated
Accessible arcwise,
continuum,
Admissible
deformation
strongly,
free, 85
402
generalized,
Whitney
114
order, 110
216
pseudo-, 286
map, 216
(see Order arc)
216
Arc approximation
AE, 77 AH-essential
384
end point of, 32
Admissible
Admissible
(arcwise),
Arc, 32
451
155
b-h
288
AR, 76
127
segmentwise, Acyclic,
156
Arc-continuum,
map, 184
Arc-like,
ANE, 156 499
property, 245
187, 257, 284
384
500
INDEX
Arc-smooth,
218
Buckethandle
weak, 395 Arcwise
continuum,
192
Buried in X, 126
accessible, 127, 451
(see Segmentwise
accessible)
Arcwise
approximated,
Arcwise
connected, 80
uniquely,
384
C Cantor
143
fan, 140
(see Uniformly
pathwise
con-
Middle-third
nected)
set, 65
Arcwise
decomposable,
Arcwise
disconnects,
Atriodic,
set, 65
142 126
251
cc-hyperspace,
461
C-determined,
437
CE map, 224 Cell, n-, 32 manifold
B
boundary
of, 32
Cell-like map, 224
Ball
Chain, 257 generalized,
10, 81
c-, 257
open Hd-, 16
weak, 258
Base
(see Nest) of cone, 52
Chainabie,
257
of n-fin, 39
continuum
of co-fin, 48
weakly,
Bin&s
house, 219
Block (Whitney), Boundary,
C-H 326
semi-, 333
258
continuum,
448
(see Cone = hyperspace
98
manifold boundary
chainable, 248
warty) of I”, 32
Choice function, Cik, 83
46
INDEX Circle, 9
501 Component,
99
pseudo-, 286
Composant,
104
sin(l/z)-,
Cone, 51
62
Warsaw, Circle-like,
62
base of, 52
187, 259, 284
geometric,
proper, 260
52
vertex of, 52
Circle-with-a-spiral,
51
Cone = hyperspace
property,
261,
424
Class
(see C-H
M, 325
Confluent
0, 474
continuum)
map, 207
semi-, 385
IV, 198, 253
Connected
Clopen, 65 Closed space of order arcs, 137
s-, 279
Comb(null),
A-, 279, 453
50
im kleinen, 83
Compact countably, Compact
Containment
23
[Theorems:
convex sets
remainder Compactum,
Complete,
(see Intersection
of, 67
Continuum,
31
Continuum
Complete invariance
property
for continuum-valued (MCIP),
270
maps
101
chainable,
Contractible, crANR,
270
hyperspace)
31
Hausdorff,
70
15 (2.14)
(CIP),
1.19, 11.2, 11.6,
66.4, 66.91.
67
Pelczynski,
32
11.7, 14.22-14.24, 19.13,
space of, 461 Compactification,
hyperspace,
155 157
g-, 441 in 2, 164
248
502
INDEX with respect to Z (crZ),
156
Convergence
retract
(see Strong. . . )
L-, 21
Dendroid,
TV-, 22 Convergent,
375
space, 443 Countable closed set aposyndetic, 238
194
Diameter
map, 106
Different
Whitney
compact, 23
..)
32
fundamental,
253, 319
maps
(see Essentially. Dimension,
hereditarily,
398
type N, 367
443
Covering property,
smooth,
smooth,
Convex metric, 80 Coselection,
193
pointwise strongly,
Countably
Deformation
290
zero, 64
253, 319
Dimensionally
crZ, 156
homogeneous,
440
Disconnected
C*-smooth,
253
absolutely,
(see Totally
disconnected)
253
c-space, 442 (see u-space)
E
Cut point, 265
Earring,
Hawaiian,
162
e-chain, 257 e-idy map, 375
D d-connected,
c-map, 182
Decomposable arcwise,
continuum,
61,267
of arc, 32 of circle in finite graph, 305
142
hereditarily, Deformation
End point
279
of finite graph, 305
61, 268
(~-admissible),
216
of x, 103
503
INDEX
Equicontinuous,
134
Finite graph, 33, 305
in first (second) variable, 169 c-retraction,
182
c-selection,
fine subgraph
369
ramification
Essential map, 155 AH-,
subgraph
184
Essentially
Whitney
maps, 455
point of, 305 of, 306
ANR, 263 (of a function),
Extensor
Finite subsets, Finitely
with respect to an
Extension
of, 307
vertex of, 305
different
Extendable
end point of, 305
space of, 7, 400
aposyndetic,
238
Fixed homotopically,
264
Fixed point 77
of a function, property,
(see Absolute
Extensor)
181
181
set, 270 Fold, 416 Follicle, 48
F
Free arc, 85
Factor Hilbert
Fruit tree, 309 cube, 89
Fundamental
dimension,
Fan, 194 Cantor,
G
140
Harmonic,
92
Fiber
Ga, 19 g-contractible,
(see Total fiber)
441
Generalized
arc, 114
Figure eight, 44
Generalized
ball, 10, 81
Fin, 39
Graph (finite),
cc, 48 Fine subgraph,
33, 305
end point of, 305 307
fine subgraph
of, 307
290
504
INDEX
ramification point of, 305
Homotopically fixed, 264
subgraph of, 306
Homotopy, 155
vertex of, 305 Ga-set, 19
joining f to g, 155 Hyper-onto representation, 439 Hyperspace, 3
H
cc-, 461
H, f&i, 11 (2.5), 13
containment, 32
Hair, 48
intersection, 125
Hairy point, 46
invariant Whitney, 459
Harmonic fan, 92
Whitney, 458
Hausdorff continuum, 101
Hyperspace suspension,450
Hausdorf metric, 11 (2.5), 13
I
Hawaiian Earring, 162 Hereditarily decomposable, 61, 267 Hereditarily indecomposable, 61, 267 Hereditarily unicoherent, 193 Hilbert cube, 32 [Theorems: 9.3, 11.3, 11.9.1,
Indecomposable continuum, 61, 267 hereditarily, 61, 267 Induced maps, 188,381 Inessential map, 155 co-connected, 281 m-fin, 48
14.12, 14.13, 14.21, 15.6,
Infinite Ladder, 197
15.7, 15.15(3), 17.9,
Intersection hyperspace, 125
25.3-25.7, 70.3, 71.7, 71.81. Hilbert cube factor, 89 Homogeneous, 122, 271 dimensionally, 440 Homotopic, 155
(see Containment hyperspace) Invariance property (see Complete. . . ) Invariant Whitney hyperspace, 459
INDEX
505
Irreducible, 273
Limit (of sets), 21
Isolated point, 65
Limit inferior (of sets), 20
Isometry, 15
Limit superior (of sets), 21 Link (of chain), 257
K
Locally connected, 75
K.
(see Cik) (see Property
(K), Property
Locally n-connected, 282
(4’) Kato’s index, 310
M
Kelley’s property, 167, 406 at a point, 174 hereditarily, 406 (seeProperty (IC)*)
M Class(M), 325 p-admissible deformation, 216 p-admissible Whitney map, 216
L Ladder, infinite, 197 Large Whitney level, 306
Manifold boundary of In, 32 (see Boundary) Map (= continuous function)
&connected, 279, 453
AH-essential, 184
A-coselection, 443
cell-like, 224
space, 443
confluent, 207
L-convergent, 21
diameter (diamd), 106
Level
c-idy, 375
size, 163
E-, 182
Whitney, 214
c-retraction, 182
Lift, 163
essential, 155
Light map, 350
Fw, 169
Like
induced, 188, 381 (see Arc-, Circle-, P-, Tree-)
inessential, 155
506
INDEX
light, 350
Minimal member, 110
monotone, 160, 207
Minimal representation, 440
MO-, 385
Moebius band, 9
near-, 385
MO-map, 385
non-alternating, 441
Monotone increasing sequence of
OM-, 385 open, 118, 207
sets, 149 Monotone map, 160, 207
quasi-monotone, 447
near, 385
r-, 443
quasi-, 447
refinable, 403 semi-confluent, 385
Movable, 288 n-, etc., 288
size, 162
Multicoherence degree, 292
union, 9 (1.23), 91 (11.5)
Mutually aposyndetic, 238
universal, 183
Mutually separated, 97
weakly confluent, 253
(see Separated)
Whitney, 105 z-, 79 Maximal fine subgraph, 307
N N-cell, 32
Maximal member, 110
manifold boundary of, 32
Mean, 374
[Theorems: 14.18-14.20,
pseudo-, 444 Mesh of f, 443 Metric convex, 80
70.1, 71.61. (see Hilbert cube) N-connected, 281 locally, 282
Hausdorff, 11 (2.5), 13
Near-homeomorphism, 385
uniform, 134
Near-monotone, 385
Minimal closed cover, 324
Near-OM, 385
507
INDEX
Nest, 111 from
Order arc, 110 A0 to Al, 111
begins in 3t, 122
Nested intersection, 153
from A0 to Al, 119
N-fin, 39
stays in ?f, 122
N-fold symmetric product, 6
Order arcs, space of, 136
N-movable, 288
Order of vertex, 305
N-od, 105
Orderable, weakly, 364
simple, 39 (see Od) Non-alternating map, 441
P
Nondegenerate, 32
P-adic solenoid, 291
Non-trivial coselection, 443
Pathwise connected, 80
Nontrivial sequence,26
uniformly, 250
Noose, 36
Peano continuum, 75
(N - l)-sphere, 32
Pelczynski compactum, 70
Null comb, 50
Perfect, 65 P-like, 187 P-like, 284
0 Od
weak P-like, 284 Pointed movable, 289
T-b-,105
Pointwise smooth dendroid, 398
simple n-, 39
Positive Whitney level, 214
OM-map, 385
Proper circle-like, 260
near, 385
Property (n), 167, 405
Open ball (see Ball) Open map, 118, 207
at a point, 174 hereditarily, 405 Property (K)*, 278
508
INDEX
Property
2
Retraction,
(see Z-set) Pseudo-arc,
76
c-, 182
286
(see R-map)
Pseudo-circle,
286
R-image, 443
Pseudo-mean,
444
R3-continuum,
Pseudo-solenoid,
286
212
R-map, 443
Puns, 46 (5.14), 60, 188
Rogers homeomorphism,
425
R3-set, 212
Q Quasicomponent
(qc), 99
Quasi-monotone
map, 447
S & space, 442 Segment w.r.t.
R
map, 128
Ramification
point, 305
Rational
continuum,
Refinable
map, 403
Refinement
287
of compactification, 67
Removing
Retract,
accessible, 451
(see Accessible) Selectible, 364
E-, 369
Representation
minimal,
Segments, space of, 134
Selection, 46, 363
map, 333
hyper-onto,
Segment of finite graph, 305
Segmentwise
(of weak chain), 258
Remainder
Whitney
rigid, 365 439 440
Selection continuum,
76
(see R-image,
(see Coselection) 442
(see Selectible, Strong defor-
mation retract)
Whitney Semi-aposyndetic,
map) 238
509
INDEX
Semi-boundary,
333
Semi-confluent Semi-span,
Smooth dendroid,
map, 385
pointwise,
291
Snake-like,
Separated
mutually,
257
pseudo-, 286
98
Space
Sequence
c-, 442
convergence
(of sets), 20
Sq, 442
monotone increasing
To-, 7
sequence of sets, 149 nontrivial,
26
reversible
property,
two-point
space, 7
Simple closed curve, 32 Simple n-od, 39 (see N-od)
Space of compact convex sets, 461 233
Space of finite subsets,
7, 400
Space of order arcs, 136 Space of segments,
134
Space of singletons,
6
Space of size levels, 472
Simple triod, 39 (see Od, Triad) Sin( l/x)-circle,
62 62
space of, 6
Size level, 163, 472 (see Whitney
levels, 207, 473
Space of Whitney
maps, 457
Sphere, (n - l)-, 32 Spoke (of simple n-od), 39 Stable, 344
level)
Size map, 162, 472 (see Whitney
Space of Whitney
Span, 291
Sin(l/z)-continuum,
Small Whitney
TI-, 6 u-, 442
Sequential strong Whitney-
Singletons,
398
Solenoid, 291
in X by B, 98
Sierpinski
194
(see Whitney Standard
map)
level, 306
stable)
simple n-od, 39
Strong arc approximation property,
384
510
INDEX
Strong deformation
retract,
142,
371
Totally bounded,
approximate, Strong
288
Totally
Whitney-reversible
prop-
erty, 232 Strongly
arcwise
Strongly
convergent,
Strongly
winding
simple, 39 (see Od) To-space, 7 (1.11)
31
Ti-space,
31
6
Type N, 429
Surjective
semi-span,
Surjective
span, 291
dendroid,
291
367
U
195
hyperspace,
284
Triod, 105
curve, 287
306
Symmetric
Pee-like,
375
Subgraph,
Suspension,
set, 453
fruit, 309 map), 216
mated, 384
Subcontinuum,
u
450
product,
Class(fi),
6
Unicoherent,
474 157
hereditarily,
T Table summarizing
Chapter VIII,
294
Topological
Uniform
metric,
Uniform
topology,
Uniformly
Tame (simple n-od in aP), TV-convergence,
101
Tree, 194
approxi-
Subcompactum,
15 (2.12)
disconnected,
Touching
admissible (Whitney
Strongly
Total fiber, 402
22
boundary,
(see Boundary)
43
193 134 134
pathwise
con-
nected, 250 Union map, 9, 91
98
[Theorems: 15.9(2)].
1.23, 11.5, 11.8,
511
INDEX
Uniquely arcwise connected, 143 Universal map, 183, 473
Weakly confluent map, 253 (seeClass(W), W-set), 253
Unstable, 344
Weakly orderable, 364
Upper semi-continuous (USC),173
Whitney block, 326
u-space, 442
Whitney hyperspace, 458
(see c-space)
V Vertex
invariant, 459 weak, 458 Whitney level, 159, 214 small, large, 306
of cone, 52 of finite graph, 305 of simple n-od, 39 order of, 305 Vietoris topology, 3
w
(see Size level, Space of Whitney levels) Whitney map, 105 admissible, 216 essentially different, 455 p (for chapter VIII),
231
selection continuum, 442
W Class(W), 198, 253 -set, 323 Warsaw circle, 62 Warsaw disk, 192 Weak arc-smooth, 395 Weak chain, 258 Weak P-like, 284 Weak Whitney hyperspace, 458 Weak Whitney stable, 454 Weakly chainable, 258
space of, 457 strongly admissible, 216 (see Size map) Whitney property, 232 Whitney-reversible property, 232 sequential strong, 233 strong, 232 Whitney stable, 428 weak, 454 Wild (simple n-od), 43 Winding curve, strongly, 287
512
INDEX
WO, 364 Wrinkle, 414
W-set, 323
Z Zero-dimensional, 64 Zero-dimensional closed set aposyndetic, 238 Z-map, 79
Z-set, 78
