TOPICS IN GENERAL TOPOLOGY
North-Holland Mathematical Library Board of Advisory Editors:
M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, F.P. Peterson, I.M. Singer and A.C. Zaanen
VOLUME 41
NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO
Topics in General Topology
Kiiti MORITA and Jun-iti NAGATA Osaka Kyoiku University Japan
1989 NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, lo00 AE Amsterdam, The Netherlands Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010. U.S.A.
Library of Congress Catalogingin-Publication Data Morita, Kiiti, 1915Topics in general topology / Kiiti Morita and Jun-iti Nagata. p. cm. -- (North-Holland mathematical library: v. 41) Includes bibliographies and index. ISBN 0-444-70455-8 I. Topology. I. Nagata, Jun-iti, 1925- 11. Title. 111. Series. QA611.M675 1989 5 14--dc20 89-9374 CIP
ISBN: 0 444 70455 8
0 ELSEVIER SCIENCE PUBLISHERS B.V., 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic. mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. (North Holland), P.O. Box 103,1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.
Printed in The Netherlands
PREFACE
The primary purpose of this book is to provide an advanced account of some aspects of general topology for students as well as working mathematicians in the field relevant to the subject. Although some of the topics discussed here are quite new, all of them are not necessarily the newest, because what we want to present here is not a . collection of newest research papers that could become out of date rapidly, but rather a collection of results that represent recent developments of general topology and could be a foundation for coming developments. This book does not necessarily cover all aspects of modem general topology, because what we intend here is not to give an encyclopedic exposition of the subject but to give a wider scope of various trends in recent developments of general topology while avoiding overlap with other books of the same type such as the Handbook of Set Theoretic Topology. We assume the reader to have a rudimentary knowledge of set theory, algebra and analysis while a little more is presupposed from general topology which may be obtained in the undergraduate course (e.g. J. Nagata’s book, Modern General Topology). Thus graduate students (and some undergraduate students as well) with sufficient knowledge of basic general topology would feel little difficulty in reading the book. We would be especially happy if our book could help them in writing their theses. This book consists of fifteen chapters, and each chapter is written independently from the others (with a few exceptions). Thus the reader could begin with any chapter, though he is advised to begin with Part I for topics divided into two chapters. This book was planned and edited under support of the “Symposium of General Topology”. Thus we conclude this preface with our thanks to the staff of the “Symposium”; especially to Prof. Y. Yasui, Mr. K. Yamada, Mr. H.Chimura, Miss K. Iwata, Mr. T. Nagura and Mr. S. Okada, who assisted us in editing the manuscripts written by different authors. Kiiti MORITA Jun-iti NAGATA Editors
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CONTENTS PREFACE
V
CONTENTS
vii
OF MAPPINGS 1 CHAPTER 1 EXTENSIONS by Kiiti Morita Introduction 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem References
1 2 10 25 35 38
CHAPTER 2 EXTENSIONS OF MAPPINGS 11 by Takao Hoshina Introduction 1. Preliminaries 2. C*-, C-, P"- and P-embeddings 3. Unions of C*-embedded subsets 4. C*-embedding in product spaces 5. Homotopy extension property References
41 42 51 62 69 75
78
CHAPTER 3 NORMALITY OF PRODUCT SPACES 1 by Masahiko Atsuji Introduction 1. Fundamental results 2. The first of Morita's three conjectures 3. The second and third of Morita's three conjectures References
81 82 92 109 116
OF PRODUCT SPACES 11 CHAPTER 4 NORMALITY by Takao Hoshina 1. Products of normal spaces with metric spaces 2. Products of normal spaces with generalized metric spaces References
121 140 158
viii
Contents
CHAPTER 5 GENERALIZED PARACOMPACTNESS by Yoshikazu Yasui Introduction 1. Preliminaries 2. Characterizations of paracompactness 3. Characterizations of submetacompactness 4. Characterizations of metacompactness 5. Characterizations of subparacompactness 6. Shrinking properties 7. Examples References
161 162 164 175 178 184 186 191 198
CHAPTER 6 THE TYCHONOFF FUNCTOR AND RELATED TOPICS by Tadashi Ishii Introduction 1. The Tychonoff functor 2. Product spaces and the Tychonoff functor 3. w-Compact spaces 4. A space X such that r ( X x Y ) = z ( X ) x r ( Y ) for any space Y 5 . A space X such that r ( X x Y ) = r ( X ) x r ( Y ) for any k-space Y 6. Products of w-compact spaces 7. w-Paracompact spaces and the Tychonoff functor 8. A generalization of Tamano’s theorem 9. Rectangular products and w-paracompact spaces References
203 204 207 209 215 218 22 1 224 230 234 24 1
7 METRIZATION I CHAPTER by Jun-iti Nagata Introduction 1 . Metrizability in terms of g-functions 2. Metrizability in terms of base of point-finite rank 3. Metrizability in terms of hereditary property 4. Some other aspects References
245 245 251 259 265 27 1
8 METRIZATION I1 CHAPTER by Yoshio Tanaka Iritroduction 1. Spaces which contain a copy of S, or S, 2. Spaces dominated by metric subsets
27 5 276 284
Contents
3. 4. 5. 6.
Spaces with a-hereditarily closure-preserving k-networks Spaces with certain point-countable covers Quotient s-images of locally separable metric spaces Ic-Metrizable spaces and b-metrizable spaces References
CHAPTER 9 GENERALIZED METRIC SPACES I by Jun-iti Nagata Introduction 1. Lagnev space and K-space 2. Developable space 3. M-space and related topics 4. Universal spaces References CHAPTER 10 GENERALIZED METRIC SPACES I1 by Ken-ichi Tamano Introduction 1. Review of basic results 2. Closure-preserving collectionsand definitionsof various stratifiable spaces 3. Expandability, extension property, and sup-characterization of stratifiable spaces 4. Classes of M,-spaces 5. Embeddings 6. Closed maps and perfect maps 7. Related topics 8. Problems References CHAPTER 11 FUNCTION SPACES by Akihiro Okuyama and Toshiji Terada 1. Introduction and notation 2. Some properties of C,*(X) 3. Some properties of C , ( X ) 4. Some properties of C,(X) 5. Topological properties and linear topological properties 6. Topological properties and /-equivalence References
ix
287 29 1 30 1 304 311
315 315 326 337 351 365
367 368 371 382 388 396 400 405 406 407
41 1 416 417 427 436 447 457
X
Contents
CHAPTER 12 N-COMPACTNESS AND ITS APPLICATIONS by Katsuya Eda, Takemitsu Kiyosawa and Haruta Ohta Introduction 1. Conventions and notation 2. N-compactness and N-compactifications 3. N-compactness vs. realcompactness 4. N-compactness of k,X 5. Rings and lattices of continuous functions 6. Applications to abelian groups 7. Applications to non-Archimedean Banach spaces 8. Problems References
459 460 46 1 467 474 479 483 497 516 518
13 TOPOLOGICAL GAMES AND APPLICATIONS CHAPTER by Yukinobu Yajima Introduction 1. The games and winning strategies 2. K-scattered spaces 3. Closure-preserving covers by compact sets 4. Covering properties of product spaces 5. The games in product spaces 6. Applications to dimension theory References
523 524 529 536 543 549 553 560
14 CATEGORICAL TOPOLOGY CHAPTER by Ryosuke Nakagawa Introduction 1. Categories and functors 2. Monomorphisms and epimorphisms 3. Diagrams and limits 4. Complete categories 5. Factorizations of morphisms 6. Reflective subcategories 7. Characterization theorem 8. (E, M)-categories 9. Epireflective vs. bireflective in Top 10. Separation axioms and connectedness 11. Simplicity of epireflective subcategories 12. Topological functors 13. Topological categories References
563 565 569 575 579 582 588 592 598 602 606 61 1 614 618 622
Contents
xi
CHAPTER 15 TOPOLOGICAL DYNAMICS by Nobuo Aoki Introduction 1. Orbit structures 2. Expansive behaviours 3. Expansivity and dimension 4. Pseudo-orbit-tracing property 5. Coordinate systems and topological stability 6. Representations of maps with hyperbolic coordinates 7. Chain components and decompositions 8. Markov partitions and subshifts 9. Topological entropy References
625 627 635 658 667 680 687 695 707 727 736
SUBJECT INDEX
741
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 1
EXTENSIONS OF MAPPINGS I
Kiiti MORITA Professor emeritus, University of Tsukuba, Ibaraki, 305, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . 1. Generalized uniform spaces and semi-uniform spaces 2. Completeness, completions and extensions of spaces. 3. Extensions of continuous maps from dense subspaces Appendix. A generalization of the Ascoli Theorem . References.. . . . . . . . . . . . . . . . . . .
............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............
1 2 10 25 35 38
Introduction Let X and Y be topological spaces, A a subspace of X and f:A + Y a continuous map. In case there exists a continuous map g : X + Y such that g I A = f (that is, g ( x ) = f ( x ) for all x in A), g is called an extension (or a continuous extension) off over X ;in this casefis said to be extendable (or continuously extendable) over X,or more precisely, to be extendable to a continuous map g from X to Y. Here it is to be noted that the image of g should lie in the range space of the original mapf; if the range of g is admitted to be enlarged beyond Y, thenf: A + Y is always extended to a continuous map from X to the adjunction space X u, Y (e.g. see Morita [198l]). In genera1,fis not extendable over X. For example, if Xis the closed unit interval Z = [0, 11, A = Y = (0, 1) and f : A + Y is the identity map A, then f is not extendable over X. 1, :A To discuss conditions under which a given continuous map f:A + Y is extendable over X , is one of the most important problems in topology. The purpose of this chapter and the next is to investigate this problem. We distinguish two cases, according to the subspace A being dense or not. This chapter is devoted to the study of the case of A being dense.
K. Morita
2
A basic result in this case is a theorem asserting that if A is a dense subspace of a metric space X , then every uniformly continuous map from A to a complete metric space Y is extendable to a uniformly continuous map from X to Y. As is well known, this theorem was generalized to the case of uniform spaces. However, those spaces which are obtained as uniform spaces are restricted to the class of completely regular spaces. The most satisfactory results in the extension theory concerned here are obtained in the realm of regular spaces. Therefore, we need a generalization of uniform spaces so that it defines any regular space. Such a generalization is provided by semi-uniform spaces introduced by Morita [1951]. In this chapter we shall first discuss semi-uniform spaces in the framework of generalized uniform spaces with applications to strict T,-extensions of spaces (Sections 1 and 2), and then establish the most general extension theorem for continuous maps from dense subspaces (Section 3). In the Appendix, it will be shown that semi-uniform spaces are available for obtaining a generalization of the Ascoli Theorem.
1. Generalized uniform spaces and semi-uniform spaces 1.0. Notation and terminology. Let X be a set, let x E X, A c X,B c X , and let d and A? be collections (or families) of subsets of X.
ud
nd
= ~ { A I A E ~ } , = n{AIAEd}, d v A? = {A u BI A E d , B E A?}, d A W = {AnBIAEd,BEa}, d A B = {AnBIAEd}, d < W o for each A E d there exists B E A? such that A c B o d is a refinement of A? ( d refines A?), St(B, at) = U { A E d l A n B # S}, St(x, d ) = St({,}, d ) = U { A E d l x E A } , St"+'(B, d ) = St(St"(B, d),d ) , n = 1, 2 , . . . , d is a cover of X o Uat = X.
Let X be a topological space and Y a subspace of X. C1 A = ClxA = the closure of A in the space X, C1,B = the closure of B c Y in the subspace Y, Int A = Int,A = the interior of A in the space X, Int,B = the interior of B c Y in the subspace Y, c1 d = {CI A I A E d } , d is an open (resp. a closed) cover of X o U d = X and each A E d is open (resp. closed) in X.
Extensions of Mappings I
Let f:X + Y be a map and d (resp. (resp. Y).
fW)=
{f(A)IA
E
a) a
3
collection of subsets of X
4, f - W
=
{f-'(B)lBE a}
Let us begin with recalling the definition of uniform spaces. There are several ways to approach the theory of uniform spaces. Here we adopt the approach of J. W.Tukey [1940].
1.1. Definition. A uniformity (or uniform structure) on a set Xis a nonempty family 0 of covers of X satisfying the following conditions: (I)
If diE 0 for i = I , 2, then there exists W 3 < dl A d 2 .
(2)
If d
(U)
For each d d.
E
0 and 8 is a cover of X such that d < E
E
0 such that
W,then a E 0.
@ there exists W E 0 which is a star-refinement of
Here a cover W of X is called a star-refinement of a cover d if {St(B, a)I B E W } < d. A set X together with a uniformity 0 on X is called a uniform space and will be denoted by (X, 0). By weakening the axiom (U) gradually we have the concepts of semiuniform spaces and generalized uniform spaces.
1.2. Definition. A semi-uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) from Definition 1.1 and the following: (SU)
For each d E 0 there exists W ment of d in a.
E
0 which is a local star-refine-
Here a cover W of X is called a local star-refinement of another cover d of X in a family 0 of covers of X, if for each B E W there exist dBE 0 and A E d such that St(B, dB) c A. A set X together with a semi-uniformity 0 on X is called a semi-uniform space and will be denoted by ( X , 0).
1.3. Definition. A generalized uniformity on a set X is a non-empty family 0 of covers of X satisfying conditions (1) and (2) of Definition 1.1 and the following:
K. Morita
4
(GU)
For each d E @ the collection { I n t A I A which is refined by some W E @.
Here we define for B by
(3)
t
E
d } is a cover of X
X the set Int, B, the interior of B with respect to 0,
I n t B = {x E XI there exists d
E
@ such that St(x, d ) c B}
A set X together with a generalized uniformity '-D on X is called a generalized uniform space and will be denoted by (X, 0).
1.4. Proposition. Every uniform space is a semi-uniform space and every semi-uniform space is a generalized uniform space.
Proof. The first statement is obvious since a star-refinement is a local star-refinement. To prove the second let @ be a semi-uniformity. Let 9 E @ be a local star-refinement of d in 0.Then for each B E 9 there exist 9 E @, A ~d such that St(B, 9)c A. This shows that B c I n t A . Hence 9 refines {IngA I A E d } . 0 1.5. Definition. A generalized uniformity (a semi-uniformity or a uniformity) @ on a set Xis called T I ,resp. Hausdorf, when condition (TI), resp. (H), below is satisfied:
(TI)
For any distinct x, y E X there exists d y $ S t k d).
E
@ such that
(H)
For any distinct x, y E X there exists d St(x, d )n St( y , d ) = 8.
E
0 such that
Clearly (H) implies (TI). Conversely, if @ is a semi-uniformity, then (TI) implies (H). Such a naming will be justified by Theorem 1.10 (c). In the discussion about generalized uniformities it is sometimes useful to deal with generalized uniformity bases.
1.6. Definition. (a) A subfamily CDo of a generalized uniformity @ is called a base for CD if for each d E @ there exists W E 0,such that W < d ,and a subbase for @ if the family { d lA . . A d n l d CD, i ~i = 1, . . . , n; n = 1, 2, . . .} is a base for @. (b) A nonempty family CD of covers of a set X is called a generalized uniformity base (resp. a semi-uniformity base or a uniformity base) if @ satisfies (1) and (GU) (resp. (SU) or (U)).
-
5
Extensions of Mappings I
1.7. Propasition. (a) A basefor a generalized uniformity (resp.semi-uniformity or uniformity) is a generalized uniformity base (resp. semi-uniformity base or uniformity base). (b) Let 0 be a generalized uniformity base (resp. semi-uniformity base or uniformity base). Then the family 0'of those covers of X which are refined by some members of 0 is a generalized uniformity (resp. semi-uniformity or uniformity) and 0 is a base for W ; W is said to be generated by 0.
The proof is straightforward and is left to the reader. 1.8. Examples. (a) Let (X,e) be a metric space. For any positive number > 0, let us define an open cover @(E)of X by
E
@(E)
= {U(x; E ) ( XE X} where U(x;
E)
=
{ y E Xle(x, y ) c E}.
Then @(+E)is a star-refinement of @(E),because V (y ; $ 8 ) n U(x; + E ) # 8 implies U( y ; ) E ) c U(x; E ) ; that is, St(U(x; +E), @ ( ; E ) ) c U(x; E ) E a(&). Therefore, @,,(p) = {@(E)I E > 0} is a base for a uniformity on X , which is called the metric uniformity induced by e and will be denoted by 0(4). (b) Let R be the set of all real numbers and e the Euclidean metric on R (that is, p(x, y ) = I x - y I). With the notation from (a), let us put 4Vn = {U(n
+ k; +(1
- 2P))lk
= 1, 2,
. . .} u @(l/2") f o r n = 1,2,
... .
Then we have
St(U(n
+ k; +(I
c V((n - 1)
St(U(x; 2-"), @,)
- P)), @),
+ (k +
1); +(l - 2-@+'))) for m > k
c U(x; 2 4 " - ' ) ) for rn
> max(x
+
+2 1, n).
Hence @, is a local star-refinement of en-, in 0,where 0 = {@,,In = 1, 2, . . .}. Therefore 0 is a base for a semi-uniformity0' on R. However, 0' is not a uniformity. 1.9. Proposition. Let 0 be a generalized uniformity base on a set X . Then the
of ) all the subsets G of X such that I n t G = G is a topology on collection ~ ( 0 X which is called the topology underlying 0. If0 is a base for a generalized uniformity W , then 7 ( 0 ) = T(W).Moreover, I n k is the interior operation in ( X t(@)).
K. Morita
6
Proof. It suffices to verify the conditions: (i) (ii) (iii) (iv)
I n t A c A and IntJ = X, A i c A2 * I n t A , c IntA,, Int,(B, n B2) = In@, n Int&, Int,(Int,B) = Int,B, where A, A l , A * , B, B , , B2 are subsets of X.
(i) and (ii) are obvious. If x E X and St(x, di) c Bi, i = 1, 2 for d l , dzE 0, then St(x, d ) c BI n B2 for d E 0 with d < di A d2.This proves (iii) by (ii). Next, for each d E 0, let us put Intd
=
{IngA IA E d}.
(1.1)
Then I n b d E 0 by (2) and (GU). To prove St(x, Int,d) c Int,[St(x, d ) ] ,
(1.2)
let y E St(x, I n t d ) . Then there exists A E d such that x, y E I n t A . Since y E Int,A, there exists W E 0 such that St( y , %) c A. Since x E Int,A c A, we have St(y, W) c A c St(x, d).This shows that y E Int,[St(x, d ) ] . Thus, (1.2) holds. Let x E Int,B. Then there exists d E 0 such that St(x, d ) c B. Hence we have by (1.2), St(x, Int,d) c Int,[St(x, d ) ]c Int,B. This shows that I n t B c Int,(Int,B), which proves (iv) by (i).
0
1.10. Theorem. Let ( X , a) be a generalized uniform space. Then the following hold in the topological space ( X , ~(0)).
(a)
{St(x, d )I d each x E X .
(b)
If B c ,'A
(c)
The space (A', ~(0)) is Ti or Hausdorffaccording as 0 is Ti or Hausdorff.
E
0}is a base for n b h ( = neighborhoods) of x for
the closure C1 B of B equals n{St(B, d )I d E a}.
Proof. (a) Suppose that G c X is open and that x E G. Since I n t G = G, there exists d E 0 such that St(x, d ) c G. On the other hand, I n t d is an open cover of X by Proposition 1.9. Thus {St(x, &)Id E 0}is a base for nbds of x.
Extensions of Mappings I
7
(b) By (a) we have Cl B = {x E XI St(x, d)n B #
0 for all d E 0}.
Since St(x, d)n B # 0 iff x E St(B, d), the assertion (b) is proved. (c) follows readily from (a).
0
1.11. Proposition. If 0 is a semi-uniformity on a set X , then {St2(x,d)I d E 0}is a base for nbds of each point x E ( X , ~(0)). Proof. Let d E 0.Let d,and d2be local star-refinements of d and dlin 0 respectively. For each x E X there exists A, E d, with x E A 2 . Let us first find W,E 0,A , E d, such that &(A,, a,)c A, and then $3, E 0,A E d such that St(A,, W,) c A. Thus, for W E 0 with W < W,A W,we have St2(x,93) c St(St(A,, a,),9,) c St(A,, W,) c A c St(x, d). Thiscompletes the proof by Theorem 1.10 (a). 1.12. Proposition. Every generalized uniformity 0 on a set X has a base Q0 which consists of open covers of ( X , ~(0)). Proof. = { I n g dI d E 0}(see (1.1)) is a desired base for 0,since Int,d is an open cover of the space ( X , ~(0)) by Proposition 1.9. 0
1.13. Definition. Let (X,t)be a topological space. A generalized uniformity (resp. uniformity base) 0 on the set X is said to be compatible with the =)T. Sometimes, by a generalized uniform space ( X , 0) topology of Xif ~ ( 0 we refer to a topological space X together with a generalized uniformity compatible with the topology of X. In applications the following theorem is fundamental. Indeed, in view of Proposition 1.12, it provides a method of introducing a generalized uniformity on a given topological space. 1.14. Theorem. Let ( X , t )be a topological space and a non-empty family of open covers of X which satisfies condition (1) in Definition 1.1. (a) is a generalized uniformity base compatible with the topology ifl {St(x, 4V) 1% E O 0 }is a base for nbds of x for each x E X . (b) #o is a semi-uniformity (resp. uniformity) base compatible with the topology if {St(x, 4V) I 9 E Q0] is a base f o r nbrls of x for each x E X and satisfies (SU) (resp. (U)).
8
K. Morita
Proof. The “only if” part of (a) follows directly from Theorem 1.10. Conversely, suppose that {St(x, @) 1% E Q0} is a base for nbds of x for each x E X. Then for B c X,the interior Int B of B in the space ( X , T) coincides with I n h B defined by (3). Hence CPo satisfies (GU) and ~ ( 0= ~r. )This proves the “if” part of (a). The assertion (b) follows readily from (a). 0 To discuss the problem: “when does a topological space admit a generalized uniformity compatible with the topology?”, we give the following definition. 1.15. Definition. A topological space Xis called weakly regular if for x and for any open set G with x E G we have Cl{x} c G.
E
X
Clearly every T,-space as well as every regular space is weakly regular. One of the important properties of semi-uniform spaces is shown in the following theorem. 1.16. Theorem. Let X be a topological space. (a) If X admits a generalized uniformity (resp. semi-uniformity) compatible with the topology, then X is weakly regular (resp. regular). (b) If X is weakly regular (resp. regular), then the family of all covers of X which are rejined by open covers of X is a generalized uniformity (resp. semi-uniformity) on X compatible with the topology.
Proof. Let 0 be a compatible generalized uniformity on the space X . Let x E X and G be an open set of X with x E G. Then there exists an d E @ such that St(x, d ) c G. By Theorem 1.10 we have Cl{x} c St(x, d ) ,that is, CI { x} c G. Thus X is weakly regular. Suppose further that 0 is a semi-uniformity. Then by Proposition I . 11 {St2(x,d )I d E 0 }is a base for nbds of x for each x E X . Since by Theorem 1.10 we have C1 St(x, d ) c St2(x, d),it follows that Xis regular. Thus, (a) is proved. Next, let CP, be the family of all the open covers of X. Then CPo satisfies condition ( I ) . Let x E G where G c Xis open. Let X be weakly regular. Then Cl{x} c G and hence @o = {X - Cl{x}, G} is an open cover of X . Hence % ’ , E Q0 and St(x, @), c G. This shows that {St(., @) I @ E Q0} is a base for nbds of x for each x E X. Hence by Theorem 1.14, 0, is a generalized uniforrhity base compatible with the topology. Therefore, 0 which is generated by O0, is a generalized uniformity on X compatible with the topology.
Extensions of Mappings I
9
Finally, suppose that X is regular. Then the family defined in the preceding paragraph satisfies condition (SU).To see this, let 4 be any open cover of X. Then for each x E X there exists U(x) E such that x E U(x). Since Xis regular, there is an open nbd V ( x )of x such that CI V(x) c U(x). Let us put V = { V ( x )Ix E X} and Wx = {X - CI V(x), U(x)} for x E X. Then we have St( V ( x ) ,Wx)c U(x). This shows that Y is a local star-refinement of 4 in (Po. Hence, by Theorem 1.14, O0 is a semi-uniformity base compatible with the topology of X. 0 1.17. Remark. A topological space X admits a uniformity compatible with the topology iff X is completely regular (the proof is found in Tukey [ 19401 and in textbooks on general topology). The family @ described in (b), however, is not always a uniformity; indeed @ is a uniformity iff X is paracompact regular. This is an essential difference between uniformities and semi-uniformities. 1.18. Definition. Let ( X , 0)and (Y, Y) be generalized uniform spaces. A map f from X to Y is called a uniformly continuous map (or a uniformitypreserving map) from ( X , @) to (Y, Y) if f-’(W) E @ for each W E $. A uniformly continuous mapf: (X,@) --* (Y, Y) is called a uniform isomorphism or unimorphism iffis a bijection from X to Y and the inversef-’ off is also a uniformly continuous map from (Y, Y) to (X,a). If O0(resp. Yo)is a base for @ (resp. Y)J: X + Y is a uniformly continuous map from ( X , @) to ( Y , Y) if for each W E Yothere exists d E such that cc9 c f-’(W). The uniformly continuous maps are often used in this form. The naming “continuous” is justified by the following proposition. 1.19. Proposition. A uniformly continuous map f:( X , @) + ( Y , Y) induces a continuous map f:(X,~(0)) --* ( Y , ~(‘4’)). Proof. For each W E Y and each x E X we have f(St(x,f-’(W)) c St(f(x), 9 ) . This proves the continuity off by Theorem 1.10. 0 1.20. Definition. Let X c Y. Then a generalized uniform space ( X , @) is called ’a subspace of a generalized uniform space (Y, Y) if Y 1 X = {W A XI W E Y } is a base for 0.A subspace (X, @) of (Y, Y) will be denoted by (X,YI X ) .
10
K. Morita
Products of generalized uniform spaces can be defined analogously as in the case of uniform spaces but the definition is omitted here because we have no oDportunity of using them in this chapter. 1.21. Remarks. (a) The concepts of generalized uniformities and semiuniformities were introduced by Morita [ 195I] for topological spaces, under the name of T-uniformities and regular T-uniformities agreeing with the topology. Indeed, in that paper we started our study by taking Theorem 1.14 as the definition of these uniformities and established Theorem 1.16. (b) A. K. Steiner and E. F. Steiner [1973] gave a topology-free axiomatization to regular T-uniformities, which they renamed semi-uniformities;the axiom (SU) is due to Morita [1951]. A topology-free axiomatization was given by Herrlich [ 1974bl to T-uniformities agreeing with the topology, which we call, in the text, generalized uniformities. (c) Herrlich [I9741 introduced the concept of nearness spaces. As was pointed out by Herrlich [1974b], the category of nearness spaces and nearness-preserving maps is equivalent to the category of generalized uniform spaces and uniformly continuous maps. (d) T-uniformities in the sense of Morita with a certain condition are discussed by Rinow [1967], [I9751and Poppe [1974]. In Poppe [1974], spaces with such T-uniformities are called generalized uniform spaces, and semiuniform (resp. generalized uniform) spaces are called regular (resp. weakly regular) generalized uniform spaces. For various generalizations of uniform spaces, see Herrlich [ 1974bl.
2. Completeness, completions and extensions of spaces Among various generalizations of uniform spaces the most important is the class of semi-uniform spaces, which will play a fundamental role in the subsequent sections of this chapter. Although our main concern lies in semi-uniform spaces, we shall discuss completeness and completions for generalized uniform spaces. Throughout this section let us assume that ( X , 0)is a generalized uniform space and let CP,, = {%!a I a E a} be a base for CP which consists of open covers of x. Filters, filter bases and their convergence will be used in what follows (for the definitions see any textbook on general topology). The collection of all (not necessarily open) nbds of a point x is called the neighborhoodfilter of x and will be denoted by %!(x).
11
Extensions of Mappings I
2.1. Definition. A filter base d is called Cauchy, resp. strictly Cauchy, with respect to Oowhen (I), resp. (2), below holds: (1)
For each a E R there exist A E d and U E 9YUsuch that A c U .
(2)
For each a E R there exist A E d , U E St(A, %& c u.
aUand /3 E R such that
The words “with respect to (Do’’ are omitted unless there is a fear of ambiguity. 2.2. Examples. (a) Let (X,e) be a metric space and let U ( x ; E ) and (Do(@) be the same as in Examples 1.8 (a). Let {a,,}be a sequence of points of X and A, = {aiI i 2 n } , n = 1,2, . . . . {a,,}is called Cauchy if for any E > 0 there exists no such that e ( x i , xi) < E for i, j 2 no. Hence the filter base {A,} is Cauchy with respect to (Do(@) iff the sequence {a,} is Cauchy. (b) The nbd filter is a Cauchy filter. (c) Let d ( x ) be a collection which consists of a single-point set {x}, where x E X. Then d ( x ) is a strictly Cauchy filter base; because for each a E R there exists U E 4?la with x E U and for such U there exists j? E R with St(x, aP) c U since {St(x, as) I /3 E R} is a base for nbds of x. The filter % ( x ) is not always a strictly Cauchy filter unless X is a regular space.
2.3. Proposition. Let (Do be a semi-uniformity base. Then afilter base d is strictly Cauchy if it is Cauchy.
Proof. Since A c St(A, GVP), every strictly Cauchy filter base is Cauchy. Conversely, let d be a Cauchy filter base. For a E R there exist A E d and VE such that A c V , where ”pcs is a local star-refinement of au. For V there exist y E R and U E 9ZU such that St(V, aY) c U . Hence we have St(A, aY) c St(V, ‘By)c U . This shows that d is strictly Cauchy. 0
-
2.4. Definition. A strictly Cauchy filter base d is said to be equivalent to A? in notation, if for each a E R another strictly Cauchy filter base A?, d and A E d there exist /3 E R and B E W such that
--
2.5. Umma. (a) Ifd (b) Ifd
--
Let d,B and V be strictly Cauchyfilter bases. W,then A? d. W and A? V, then d 5%.
-
K. Morila
12
Proof. (a) Let a E R and B E A?. Since d is strictly Cauchy, there exist E R, A, E d , U, E %a such that St(A,, t?Za0) c U,. Since d A?, there c St(A,, %%).Since B n B, # 0, exist /3, E R, B, E A? such that St(B,, and Bo c St(A,, 42%) c V,, we have U, c St(B, %a). This shows that d. St(Ao, 42%) c U, c St(B, @a), that is, A? (b) This assertion follows directly from Definition 2.4 0
-
a,
-
-
Since d d for any strictly Cauchy filter base d , the equivalence defined above is an equivalence relation. 2.6. Proposition. I f 0,is a semi-uniformity base and d , 93 are strictly Cauchy Jilter bases, then the following are equivalent.
-
(i) d A?. (ii) For each a E R and A E d there exists B E A? such that B c St(A, ea). (iii) For each a E R there exist A E d , B E A? and U E %a such that A v B c U (that is, d v A? is a CauchyJilter base). Proof. (i) * (ii) is a direct consequence of Definition 2.4. (ii)-(iii). Assume (ii) and let a E R. Since d is strictly Cauchy, there exist A E d , U E %a, /3 E R such that St(A, %)) c U. Then by (ii) there exist B E A? such that B c St(A, %,). Hence we have A u B c St(A, '4!lp)c U . This shows that (iii) holds. (iii)*(i). Assume (iii) and let A E d , a E R. Since d v A? is Cauchy by (iii), d v A? is strictly Cauchy by Proposition 2.3. Hence for a there exist A, E d , B, E 93, U E @a and /3 E R such that St(A, u B,, 4Yp) c U. Since A, c U and A, n A # 0,we have A n U # 8 and hence U c St(A, %a), and consequently
St(&,
@p)
This shows that d
-
c St(A0 u Bo, %p)
c U c St(A, %a).
A?.
-
2.7. Lemma. (a) Ifd and A? are strictly Cauchy Jilter bases and d A?, then nCl d = nCl 93. (See Section 1.0.). (b) Ifd is afilter base and converges to x E X , then x E n C l d . (c) Ifd is a strictly Cauchy Jilter base and x E n C l d , then d converges to x.
(d) If X is a T,-space and d is a strictly Cauchy Jilter base, then n C l d is either empty or composed of a single point.
Extensions of Mappings I
- a,
13
= n{St(A, S ) I A E d,u E R}. d and u E Q, we have
Proof. (a)ByTheorem 1.lOwehavenCl d Since d
for each A
nci
w
E
= n{st(B, % p ) ~B E 8, p
E
n} c
st(A, %J
and therefore nC1 W c n{St(A, %u) I A E d , u E Q} = nCl d , that is, nCl c nCl d . Since W d by Lemma 2.5 we have also nCl d c nC1 W,and therefore nC1 d = nCl $3. (b) Is well known and easy to prove. (c) Let a E R. Since d is strictly Cauchy, there exist A E d ,p E R, U E eU such that St(A, %& c U.Since x E nCl d = n{St(A, a a ) l A E d ,a E Q}, we have x E St(A, %p) c U . Since x E U , we have U c St(x, %u) and hence A c St(A, %p) c St(x, %u). Since {St(x, 9 )I u E R} is a base for nbds of x, d converges to x. (d) Suppose that nCl d # 8 and that x E nCl d .Let y E X with y # X. Then there exists a E R such that y 4 St(x, aU). As was shown in the proof of (c), there exist A E d , /? E R such that St(A, %p) c St(x, qU). Hence we have y # St(A, %$). Since nCl d = n{St(A, % u ) l A E d,u E Q}, this shows that y 4 ncl d. 0
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2.8. Definition. For a strictly Cauchy filter base d the collection {St(A, 9 )I A E d , a E R} is a filter base and the filter generated by it is called the star-filter of d with respect to and will be denoted by S t ( d ; moo). Every star-filter is Cauchy. The nbd filter %(x) of x is St(d(x); O0) (see Examples 2.2 (c)).
2.9. Proposition. Let d and W be strictly CauchyJilter bases. Then d tfSt(d; 0 0 ) = St(W; 00). Proof. This follows directly from Definition 2.4 and Lemma 2.5.
-W 0
It follows from Proposition 2.9 that there is a bijection between the set of all the equivalence classes of strictly Cauchy filter bases and the set of all the star-filters. To express an important property of star-filters we need two more definitions.
2.10. Definition. A Cauchy filter % is called a weak star-Jilter if for each F E 9 there exists u E R such that U E and U E % imply U c F, that is, n %u) c F.
u(%
K. Morira
14
2.11. Definition. A nonempty collection W of subsets of X is called a stack if B E W and B c C imply C E W. A stack is called Cauchy if it contains at least an element of 42, for each u. A Cauchy filter (resp. Cauchy stack) d is called a minimal CauchyJilter (resp. minimal Cauchy stack) if it contains no proper subfamily which is a Cauchy filter (resp. stack). 2.12. Proposition. Every star-Jilter is a weak star-filter. Proof. Let d be a strictly Cauchy filter base and let 9 = S t ( d ; Q0). Let F E 9. Then there exist A E d , u E R, such that %(A, 42,) c F. Let U E 9 n 42,. Since d v 9 has the finite intersection property, we have U n A # 0 and hence U c St(A, 42,) c F. 0 2.13. Proposition. A Cauchy stack is a minimal Cauchy stack i f i t is a weak star-Jilter. Proof. Assume that W is a weak star-filter and that W' c W is a Cauchy stack. Then for each u E R there exists U, E @' n 42,. Let B be any element of W.Since g is a weak star-filter, there exists u E R such that u(g n 42,) c B. Since U, E a' n 42, and W' c W, we have U, c U(W n 42,) c B. Since 9.3' is a stack and U, E W', we have B E W'.Hence W c W'. Thus W is a minimal Cauchy stack. Conversely, assume that 43 is a minimal Cauchy stack. Let us put
G, = lJ(W n 42,)
for u E R.
Since W is Cauchy, we have W n 42, # 0 and hence G, # 0 for each a E R. Let B be any element of W.Suppose that B $ G, for all u E R, that is, (X - B) n G, # 0 for all u E R. Then there exists, for each u, an element n& such I that (X - B) n U, # 0. Let us put U, E #
=
{A c XI A contains U, for some u E R}.
Then A?' is a Cauchy stack and W' c W.Since W is a minimal Cauchy stack, we have = W. On the other hand, it follows from the definition of g'that &?I
(X - B) n A #
0
for all A E W = W.
However, this is a contradiction since B is itself an element of 9.Thus we have B, 3 G, for some u E R. If 42y < A 42, (a, 8, y E R), then G, c G, n G,. Therefore W is a filter. Thus W is a weak star-filter. 0
Extensions of Mappings I
15
2.14. Corollary. A weak star-filter is a minimal Cauchyfilter. In general, a minimal Cauchy filter is not necessarily a weak star-filter and a weak star-filter is not necessarily a star-filter. These filters, however, are coincident for some cases which we are interested in.
2.15. Theorem. Let Oobe a semi-uniformity base. Then every Cauchyjilter contains a weak star-filter, and the three types of Cauchy filters - star-filters, weak star-filters and minimal Cauchyfilters - are all coincident.
Proof. A Cauchy filter 9 is strictly Cauchy by Proposition 2.3 and hence it contains the star-filter S t ( 9 ; (Do). In particular, if 9is a minimal Cauchy filter, then 9 = S t ( 9 ; O,,). From Proposition 2.12 and Corollary 2.14 it follows that the three types of Cauchy filters mentioned in the theorem are all coincident. There are two more cases in which the minimal Cauchy filters are precisely the weak star-filters; one is given in the next theorem and the other in Theorem 2.36 at the end of this section.
2.16. Theorem. Suppose that every open cover of X is refined by some cover in (Do. Then every Cauchyjilter converges. Therefore the three types of Cauchy filters mentioned in Theorem 2.15 are all coincident.
Proof. Let 9be a Cauchy filter. Suppose, on the contrary, that % does not converge to any point of X.Then for each x E X there is an open nbd U(x) of x such that V(x) Ffor all F E %. Then %o = { U ( x )I x E X} is an open cover of X and hence a0> @a for some a E R. On the other hand, since 9 is a Cauchy filter, 9contains some element Uof %a, and U c U(x) for some x E X. But this is in contradiction with the property of V(x). Therefore, 9 converges to a point x of X,that is, 9 3 @(x). If % is a minimal Cauchy filter, then 9 = @(x). On the other hand, we have % ( x ) = St(d(x); Oo) (see Examples 2.2 (c)). 0 2.17. Definition. X i s said to be complete with respect to (Do (or simply, Oo is complete) if every weak star-filter with respect to Ooconverges to a point in X . 2.18. Lemma. Let 0, be another base for O which consists of open covers of X . Then X is complete with respect to OoijTX is complete with respect to (D,. Hence in case X i s complete with respect to Oo,we shall say that ( X , O ) or (D is complete.
K. Morita
16
Proof. Since every cover in CDI is refined by some cover in CD, and vice versa, a filter is a weak star-filter with respect to 0,iff it is a weak star-filter with respect to CDl. From this the lemma follows immediately. 0 Theorem 2.16 immediately yields the following theorem. 2.19. Theorem. Zf(X, CD) is a generalized uniform space such that every open cover of X is contained in CD, then ( X , 0)is complete. The following theorem, together with Theorem 2.19, is important in Section 3. 2.20. Theorem. Let ( X , 0)be a semi-uniform space. Then the following conditions are equivalent. (a)
( X , 0)is complete.
(b)
Every Cauchy filter base converges.
(c)
For every Cauchyfilter base d , we have nCl d #
0.
{a
Proof. We shall assume that CDo = I a E R} is a base for CD as before. It suffices to prove the theorem for CD,. (a)*(b). Let d be a Cauchy filter base. Then d is strictly Cauchy (see Proposition 2.3) and S t ( d ; CD,) is a weak star-filter (see Proposition 2.12). By (a) S t ( d ; CDo) converges to some point x in X , that is, S t ( d ; CDo) = %!(x). This shows that each nbd of x contains %(A, eU) for some A E d , a E R. Hence d converges to x. (b)o(c) is a direct consequence of Lemma 2.7. (b)*(a) is obvious. 0 2.21. Example. Let ( X , e) be a metric space and @(e)its metric uniformity (see Examples 1.8 (a)). Then a metric space ( X , e) is complete iff the uniform space ( X , CD(e)) is complete. Thus the concept of completeness above is a generalization of the completeness for metric spaces. The following construction of completions of generalized uniform spaces is a highly elaborated application of Hausdorff’s method for completing metric spaces. Let us denote by E the set of all the weak star-filters with respect to CDo which do not converge to any point in X . Let us put
x*
=
XUE
Extensions of Mappings I
17
and for any open set G of X let us define
G* = G u { ~ E E I G E ~ } . Then we have the following lemma.
2.22. Lemma. Let G and H he open sets of X . (a)
G c H implies G * c H * ; G* n X = G .
(b)
(G n H ) *
=-
G* n H*.
The proof is straightforward and is left to the reader. From Lemma 2.22 it follows that the collection { G * I G open sets of X} is a base for a topology of X * and we shall consider X * as a topological space with this topology.
2.23. Lemma.
=
{ U * I U E an} is an open cover of X * .
Proof. Let 9 E E. Since 9 is a Cauchy filter, there exists U E U E 9. This shows that 9 E U*.
with
0
2.24. Lemma. Let y E X * and y E G * for an open set G of X . Then there exists a E R such that St( y , a:) c G * .
Proof. Since the lemma is obvious for y E X , assume that y E X * - X and Since 9 is a weak star-filter, that y = 9 E E n G*. Then we have G E 9. there exists a E R such that U ( 9 n an)c G. Suppose that 9, X E U * with U E an. Then U E 9 n and hence U c G. On the other hand, U E X . Therefore G E X , that is, X E G*. If x E X n U* = U , then x E U c G . Thus, we have St( y , a:) c G * . 2.25. Lemma. St(G*, a:)
c
[St(G, an)]* for any open set G of X .
Proof. If G* n U* # 0 with U E ea,then G n U # 0 and hence U c St(G, an). Therefore, U* c [St(G, aa)]*. This proves the lemma. 0 Let us now put @$ =
{a: I a E R}.
2.26. Lemma. (a) @$ is a generalized uniformity base on X * which is compatible with the topology.
18
K. Morita
(b) Let @* be a generalized uniformity generated by @$. Then ( X * , @*) does not depend on the choice of a base (Do, and is called the completion of ( X , 0). (c) If@ is a semi-uniformity, then @* is a semi-uniformity. The same holds for uniformities.
Proof. (a) If %, < +!$ A C?Zy, then % : < %$ A 4.3‘: Hence (a) follows from Lemma 2.24, by virtue of Theorem 1.14. = {V,1 fl E W} be another base for 0 which consists of open (b) Let covers of X.As was shown in the proof of Lemma 2.18, a filter in Xis a weak star-filter with respect to iff it is a weak star-filter with respect to (Do. Hence, even if we use @I instead of O0, we obtain the same set X * and the same topology on X*. If 43, < Vpand Vp< %, then we have % : < V,* and V,* < % .: This shows that 0: = {4$‘I a E R} and @? = {V,*1 fl E W} generate the same generalized uniformity which is @*. is a semi-uniformity base. Hence (c) Let @ be a semi-uniformity.Then for each a E R there is a local star-refinement%, of%, in (Do.For each V E 42, there exist 6 E R, U E %, such that St(V, %a) c U . Hence we have, by ): c [St(V, %a)]* c U*. This shows that %$ is a Lemma 2.25, St(V*, % local star-refinement of % : in 0;. Therefore 0; is a semi-uniformity base. 0 2.27. Lemma. (a) I f A is a weak star-filter with respect to 0$, then A‘ = { A c XI there exists U* E A n 43: with a E R such that U c A } is a weak star-filter with respect to (Do. (b) I f 9 is a weak star-filter with respect to a,, then 9*= { B c X * I there exists U E 9 n %, with a E R such that U* c B} is a weak star-filter with respect to @$. (c) [A’]*= A, [9*]’ = 9 for A in (a) and 9 in (b).
e}.
Proof. (a) First we observe that A‘ n %, = { U c XI U* E A n To see this, let U E A‘ n 42,. Then there exist /IE R, V E %, such that V c U , V* E A n 42;. Since V* c U*, we have U* E A n .% : Thus A ’ n %a c { U I U* E A n q}. Since the converse inclusion holds by the definition of A’,we have A’ n %, = { U (U* E A n 4%:). Let A E A‘. Then there exists U E A’ n 92, such that U c A. Since U* E A and A is a weak star-filter, there exists fl E R such that U ( A n 92;) c U*. Hence we have U ( A ’ n 42,) c U c A. This shows that A‘ is weak star. Let a, B E $2. Then there exists y E R such that %, < A %., If we define G, = U ( A ’ n 4 ‘ 2,) and define G,, G, similarly, then G, c G, n G,. Since any element of A’ contains some GayA‘ is a filter.
Extensions of Mappings I
19
Since A' n %a # 0 for all a E R, A' is Cauchy, and hence a weak star-filter with respect to Qo. (b) Similarly as in the proof of (a) we can prove that 9*n 922 = { U* I U E 9 n %a} and also that 9* is a weak star-filter with respect to Q:. (c) follows readily from (a) and (b). 0 2.28. Lemma. Let A and A' be the same as in (a) of Lemma 2.27.
(a)
I f A' converges to x in X , then A converges to x in X*.
(b)
I f A' does not converge to any point in X , then A' E E and defines a point y in X * - X and A converges to y in X*.
Proof. (a) Let G be any open set of X with x E G*. Then x E G and, since A' converges to x in X , there exists M' E A' with M' c G. Since M' E A', there exists U E A' n %a for some a E R such that U c M'. Hence we have U c G, and consequently U* c G * with U* E A n .% : This shows that A converges to x in X*. (b) Let y E G* where G is an open set of X. Then there exists a E R such that St( y , 922)c G*. Let y E U,+ with U, E Then Uo E A'. Since A' n = { U I U * E A n a:}, we have U,*E A. Since U,* c G*, this shows that G * E A. Thus, A converges to y in X*. 0
As further properties of X* we have the following lemmas.
2.29. Lemma. For any open set G of X we have G* = X * - Cl(X - G )
where CI means the closure operation in the space X*.
Proof. Since X * - G * is closed in X * and (X* - G*) n X = X - G, it is obvious that CI(X - G ) c X * - G*. Conversely, let y E X * - G* and let y E H * for an open set H of X. Then H * n (X - G ) # 0, because, otherwise we would have H n (X - G) = 0 and hence H c G which implies H* c G*. Therefore y E Cl(X - G). 0 2.30. Lemma. For x
E
X * the following holds:
n{St(x, %*)la
E
R}
=
i
{x}
ij-XEX* -
Cl,{x}
if x E
In particular, each point of X * - X is closed.
x.
x,
20
K. Morira
Proof. Let x E X and y E X* - X. Suppose that y is defined by a weak star-filter 9 in X which converges to no point in X. Then there exists a E R such that St(x, an) 4 9.If x E U E an,then U 4 9, that is, y 4 U*. Hence y 4 St(x, 4%';). Therefore n{St(x, a:) I a E R} c X, and n{St(x, 43:) I a E R} = a E n} = ci,{x>. n{st(x, an)[ On the other hand, the relation y 4 St(x, 99:) implies that x 4 St( y, 43:). Therefore we have n{St( y , )I %: a E Q} c X * - X. If z E X* - X and z # y, then the weak star-filter Y which defines z is different from 9. Hence there exists a E R such that U ( 9 n an) 4 3, which shows that z 4 St( y , %.): This proves the lemma. 0 2.31. Definition. A topological space Y is called an extension of a topological space X if Y contains X as a dense subspace. An extension Y of X is called a T,-extension of X if { y } is a closed set of Y for each y E Y - X, and called strict if { E , ( G ) I G open sets of X} is a base for the open sets of Y , where
E y ( G ) = Y - CI,(X - G ) . The results which have been obtained hitherto show that the completion ( X * , a*) satisfies the conditions in Theorem 2.32 below. 2.32. Theorem. Let ( X , #) be a generalized uniform space. Then the completion ( X * , #*) of (X,#) is characterized as a generalized uniform space ( Y , Y) satisfying conditions (i) to (iii):
(i) (Y, t(Y)) is a strict T,-extension o f ( X , ~(0)). (ii) For each a E R the collection E Y ( a n )= { E y ( U ) I U E '%a} is an open cover of Y and 'Po = {EY(42n) I a E R} is a base for Y. (iii) ( Y , Y) is complete. Here (Do = {an I a E R} is a base for Q, as before. Moreover, i f ( X , #) is a semi-uniform space (resp. a uniform space), then ( X * , #*) is a semi-uniform space (resp. a uniform space). Proof. Let (i)-(iii) hold for (Y, Y). Let us first observe the following lemma. 2.22'. Lemma. Let G and H be open sets of X .
(a)
G c H implies E,(G) c E , ( H ) ; E y ( G ) n X = G ;
(b)
E,(G n H ) = E , ( G ) n E , ( H ) .
The proof of Lemma 2.27, as a careful examination will show, is based on Lemma 2.22 only. Hence, analogously we have the next lemma on the basis of Lemma 2.22'.
21
Extensions of Mappings I
2.27'. Lemma. (a) Zf A is a weak star-jilter with respect to Y o , then R,(A) = { A c XI there exists E y ( U )E A n E Y ( q a )with a E R such that U c A } is a weak star-jilter with respect to (Do. (b) Zf 9 is a weak star-jilter with respect to O0, then E y ( 9 ) = { B c Y I there exists U E 9 n %a with a E R such that E y ( U ) c B} is a weak star-jilter with respect to Yo. ( 4 EY[RX(&I = M&49-)l = 8. A
9
Let y E Y - X and %( y ) the nbd filter of y . Since { y } is closed in Y and hence { y } = Cl{ y } = (){St( y , Ey(%J) I a E R}, %( y ) does not converge to any point of X . Similarly as in the proof of Lemma 2.28, we can prove that R,[%(y)] does not converge to any point in X . Hence R X [ % ( y ) E] 9 and defines a point z in X* - X . Let us put cp(y) = z. For any x E X we put cp(x) = x . Thus cp defines a map from Y to X*. Let z E X * - Xbe defined by a weak star-filter 9 with respect to (Dowhich does not converge in X.Then E y ( 9 )is a weak star-filter with respect to Y o . Since (Y, Y ) is complete, E y ( 9 ) converges to a point y E Y - X , that is, E y ( 9 ) = % ( y ) . Then by Lemma 2.27' we have R,[%(y)] = 8. Hence cp( y ) = z. This shows that cp is surjective. If y , , y, E Y - X and y , # y,, then %( y , ) # %( y,) and hence It,[%( y , )] # Rx[%( y,)], and consequently we have cp( y , ) # cp( y2). Thus cp is a bijection. Let G be any open set of X , and let y E Y - X . Then we have Y E EY(G) 0 EY(G) E W Y ) * G E R,[Q(y)l*
cp(Y) E G * .
This shows that cp(E,(G)) = G*. Therefore cp:(Y,Y) + (X*, a*) and cp-': ( X * , @*) + (Y, Y ) are both uniformly continuous, and cp is a uniform isomorphism which leaves each point of X fixed. This completes the proof of the theorem. The following is an application of Theorem 2.32.
2.33. Theorem. Let X and Y be weakly regular spaces and let Y be a strict TI-extension of X . Then there exists a generalized uniformity @ on X compatible with the topology, such that Y is homeomorphic to the underlying space of the completion of ( X , 0). Proof. Let Yobe the collection of all the open covers of Y which consist of the sets of the form E y ( G )with G open in X . Since { E y ( G )I G open in X } is a base for the open sets of Y, every open cover of Y is refined by some member of Yoand hence Yois a base for Y , where Y is a generalized uniformity on Y such that every member of Y is refined by some member of Yo; of course,
K. Moriia
22
Y is compatible with the topology of Y since Y is weakly regular (see Theorem 1.16). By Theorem 2.19, (Y,Y) is complete. is a generalized uniformity base on Let Q = {*lr A XI *lr E Y o }Then . X compatible with the topology. Let CD be the family of covers of X which are refined by some covers in B0.Then Q, is a generalized uniformity on X compatible with the topology, and (Do is a base for Q,. Therefore by Theorem 2.32 there is a uniform isomorphism between (Y, Y) and ( X * , @*). 0
Since a regular space is a strict extension of each of its dense subspaces, we have the following theorem from Theorem 2.33. 2.34. Theorem. Let X be a regular TI-space. Then every regular TI-space which contains X as a dense subspace can be obtained as the completion of a semi-uniform space ( X , a) with CD as a semi-uniformity compatible with the topology of x. The following theorem will be used in the next chapter.
2.35. Theorem. Let ( X , (D) be a generalized uniform space and (X*, a*) its 1 a E R} be a base for Q, consisting of open covers of completion. Let (Do = {aa X , and {G, I ilE A} a collection of open sets of X . Let us put Wu = { G A I i l ~ A }
u ( U I U n ( X - U { G A l l ~ A }#) ~ , U E @ ~U} E, R . (a) If for each u E R the cover Wuis refined by some 4fpwith p E R, then = tU(G, 13, E AH*. U{G? I A E (b) In case each of covers aU, a E R, is afinite cover of X , then the converse of (a) holds.
Proof. (a) Since we have clearly U{G: 11 E A} c [U{G, I 1 E A}]*, it suffices to prove the converse inclusion. From the assumption in (a) we have U{G: 11 E A} u St(X - UG,, @!,*) = X*,a E R and consequently X * - U{Gf 11E A} c St(X - UG,, 4)2:'
for a E R.
(2.2)
Therefore it follows from Theorem 1.10 that X* - U(G:IAEA}
c Cl(X - U { G , I L E A } ) .
(2.3)
Consequently, by Lemma 2.29 we have the desired inclusion [U{Gn I 1 E All*
= U { G f I 1 E A}.
(2.4)
23
Extensions of Mappings I
(b) Suppose that U{Gt 11 E A} = [U{G, 11 E A}]*. Since the implications (2.4)*(2.3)*(2.2) hold in the proof of (a), and for U E 42,, ( X - UG,) n U # 0 iff (X - UG,) n U* # 0, we have U { W * l W E K } = X*. Now Theorem 2.36 below shows that W, is refined by some 42p, /3 E R. 0
2.36. Theorem. Let ( X , O ) and Oo = {42, I u E R} be the same as in Theorem 2.35. Assume, in addition, that each 42, with u E R is afinite cover of X . (a)
Every Cauchy stack with respect to m0(see Definition 2.1 1) contains a minimal Cauchy stack (= a weak star-flter by Proposition 2.13).
(b)
Every open cover of X * is refined by
(c)
X* is compact.
422
with some u E R.
Proof. (a) Let W be a Cauchy stack. Let r be the family of all the Cauchy stacks which are contained in W.Then there exists a minimal element in r, ordered by incluson. To prove this, let (W(1) I 1 E A} be a subfamily of such that A is a linearly ordered set of indices and that W(1) =I W ( p ) if 1 < p in A. Let us put cp,(l) = { U l U
E
W(1) n %,},
u
E
R.
Since 42, is a finite set and cp,(1) # 0 for all 1 E A, there exists 1, E A such that cp,(1) = cp,(1,) for 1 > 1,. Now, let us put W o = n(W(1) 11 E A}. Then W ois clearly a stack, and for u E R we have W on 42, = cp,(L,). Because, if U E cp,(A,), then U E 42, n W(1)for 1 > 1,and hence for all 1 E A and consequently cp, (1,) c gon 42,. Thus W ois a Cauchy stack and Wo t W ( 1 ) for all 1 E A. Therefore by Zorn's lemma there exists a minimal element W'in r. W'is a minimal Cauchy stack. This proves (a). (b) Let X be any open cover of X*. Then there exists u E R such that 42; < 2.To see this, assume, on the contrary, that 42: 4 2 for all a E R. Then for each u E R there exists U, E 42, such that U$ 4: H for all H E 2, that is, U,* n ( X * - H ) #
0
for all H E X .
Let us put W = {B c X * I B n ( X * - H) # 0 for all H E X } .Then W is a stack in X * and W n 422 # 0 for all u E R since U$ E W.By applying (a) to the generalized uniform space ( X * , O*), one sees that W contains a weak star-filter W'. Since ( X * , 0*)is complete, there exists y E X * such that 93' coincides with the nbd filter of y in X*. Hence each nbd of y meets every
K. Morita
24
element X * - H with H E &'. This shows that y E CI(X* - H). Since H i s open in X*, we have y E X* - Hfor all H E &',which, however, contradicts the assumption that M is an open cover of X*. Finally, (c) is a direct consequence of (b). 0 In concluding this section, we shall show that Shanin's compactification, which is a generalization of the Wallman compactification, is obtained as the completion of a certain generalized uniform space. Let X be a weakly regular space and Y a base for the open sets of X satisfying conditions below: (i) X E Y, (ii) if G, H E Y, then G n H E 9, (iii) if x E G for G E Y, then there exist Gi E Y, i = I , . . . , k such that x 4 Gi for 1 < i < k and G u (U{GiI i = 1, . . . , k } ) = X . Let be the collection of all the finite open covers of Xconsisting of open sets from 9.Then Qois a generalized uniformity base on X compatible with the topology by virtue of Theorem 1.14. The collection @ of all the covers of X which are refined by members of Qois a generalized uniformity and O0 is a base for @. Let ( X * , @*) be the completion of (X, 0). Then by Theorem 2.36 the space X * is compact. By Theorem 2.35 we have [U{Gili = I , .
. . ,n}]*
= U{Gi*li = 1 , .
..,n}
f o r G i E Y , i = 1, . . . , n. 2.37. Theorem (Shanin [1943]). Let X be a weakly regular space and 9 a base for the open sets of X satisfying conditions (i), (ii) and (iii) above. Then there exists a compact space Y with the properties below: (a)
Y is a TI-extensionof X ,
(b)
{ E,(G) I G E Y} is a base for the open sets of Y,
(c)
&(GI u . . . u G,) = E Y ( G , )u . . . u E,(G,) holds for any finite number of elements G I ,. . . , G, E Y.
Moreover, such a space Y is essentially unique.
Proof. Let 9 = {X - GI G E Y}. Then 9 is a base for the closed sets of X. Shahin's theorem is stated in terms of 9.The existence of Y with the properties mentioned in the theorem is provided by X * constructed above. Conversely, suppose that Y has the properties mentioned in the theorem.
Extensions of Mappings I
25
Let y E Y - X and y E E , ( G ) for some G E S. Since C1,B is a base for the closed sets of Y and { y } is a closed set in Y , there exists a collection { F , I I E A } such that F i e B for all A E A and { y } = n{CI,F,IAEA}. Since n{ClyFi1AE A} c E,(G) and Y is compact, there is a finite set {A,, . . . , A,} such that n { C l , F , , J i= 1, . . . , n} c E,(G). Then by taking intersections with X we have n{F,,li = 1, . . . , n} c G. Let us put q 0 ={ X - F J i = 1, . . . , n } ~ { G } . T h e n ~ ~ i s a n o p e n c o v e r o f X a n d q0E (Do. Therefore we have St(y, E y ( q 0 ) )c E y ( G ) . On the other hand, if x E X and St(x, 42) c G for 4 E (Do and G E 9,then St(x, Ey(42)) c E,(G). Therefore, Yo = { E Y ( 4 2 ) ) qE (Do} is a generalized uniformity base on Y compatible with the topology by Theorem 1.14. Since { E , ( G ) I G E S}is a base for the open sets of Y , every open cover of Y is refined by some cover Ey(42)with 42 E (Do, Yois complete by Theorem 2.19. Thus Theorem 2.32 applies to the present case and we obtain Theorem 2.37 0 2.38. Remarks. (a) The completeness and completions for spaces with T-uniformities were discussed in Morita [1951J by means of star-filters, and it was proved there that completions of uniform spaces and Shanin’s compactification are obtained as such completions. (b) The concept of weak star-filters was introduced by Rinow [1967]; it was defined also by Harris [1971] as round Cauchy filters, and an equivalent concept of minimal Cauchy stacks was given by Herrlich [1974b]. Rinow [ 19671 announced that replacement of star-filters by weak star-filters yields important results such as Theorems 2.32 and 2.33 (which he stated for T, -spaces), although the completions of semi-uniform spaces are the same as those in Morita [1951]. Our description in this section follows the line of thoughts in Morita [1951] in the main. (c) For corresponding results on nearness spaces, see Herrlich [1974]. (d) Some of the results in this section hold for spaces with T-uniformities. In Morita [1951], Theorem 2.37 (Shanin’s Theorem) was obtained by means of completions for the case that X is a topological space which is not necessarily weakly regular.
3. Extensions of continuous maps from dense subspaces Throughout this section let X be a topological space and A a dense subspace of X unless otherwise specified. As in Section 2 for an open subset G of the subspace A we define an open subset E,(G) of X by E,(G)
=
X - Cl,(A - G).
(3.1)
26
K. Morita
Then the following hold for open subsets G, H of A:
E,(G) n A = G; E,(G) = U { M c XI M open in X and M n A
=
G},
if G c H, then E,(G) c E,(H); E,(G n H ) = E,(G) n E,(H).
(3.2) (3.3)
For a collection Y of open subsets of A, we define
The following is a fundamental theorem in this chapter, which enables us to deduce all the extension theorems for continuous maps as well as for uniformly continuous maps.
3.1. Theorem (Morita [1951]). Let f : A + Y be a continuous map where Y is a regular Hausdorf space. Let Y be a complete semi-uniformity on Y compatible with the topology and Yo = {Y,,la E R} a base for Y which consists of open covers of Y. Let us put H ( Y ~ )=
n { u w f - w )i a
(3.5)
E QI.
Then there exists uniquely a continuous map g :H ( Y o ) Y which is an extension o f f ; moreover, if Ypis a local star-refinement of Y, in Y o ,then Ex(f-'(V8)) A H ( Y o ) is a refinement ofg-I(Y,).
Proof. Let x E H ( Y o ) .Since x E UE,( f - ' ( f for aeach ) ) a E R, there exists V , E % such that x E Ex(f - I ( V,)). Then the collection { V,I a E Q} has the finite intersection property. To prove this, let { a I , . . . , a,} be any finite subset of R. Since x E (){Ex(f - I ( & , ) ) I i = 1 , . . . , n} and A is dense, we have n{E,(f-'(V,,))Ii = 1 , . . . , n} n A # 8 and by (3.2) n { f - ' ( V J l i = 1, . . . , n } # 8. Thus n{V,, 1 i = 1, . . . ,n } # 8. Therefore the collection d
=
{ V , , n . . - n V , n l a i ~ R , i =, .l . . , n ; n = 1,2 , . . .}
of subsets of Y constitutes a Cauchy filter base with respect to Y o .Since (Y, Y) is complete, d converges to a point of Y (see Theorem 2.20 and Lemma 2.7). This point will be denoted by g(x). Then we have g ( x ) = nC1 d If { W, I a
E
=
n{Cl (&, n . . . n V.,)lai E R, i = 1 , . . . , n ; n = 1,2 , . . .}.
Q} is another collection of subsets W, E YEwith a
E
R such that
Extensions of Mappings I
xE
E , ( f - ' ( K ) )then , the collection of subsets of Y W = { W . , n * * - nW a n I a i ~ R , i I=, . . . , n ; n =
27
1,2, . . . }
is also a Cauchy filter base with respect to Y o . For any ai E R, i = 1, . . . , n and E R,we have x
E
n{Ex(f-'(K,)li = 1, . . . , n} n E,(f-'(q))
and hence, by an argument which has been used above, one sees that n { K , l i = 1 , . . . , n } n W, # 8, that is, W, c St(K, n . . . n Kn,V,). This shows by Proposition 2.6 that d is equivalent to W. Hence by Lemma 2.7 we have nCl d = nCl W.Therefore the value g(x) does not depend on the choice of the collection { K} and is determined uniquely by x. If x E A, thenf(x) E V, for all a E R and since {St(y, Va) I a E R} is a base for nbds of y for each y E Y,we have g(x) = f(x) for x
E
A
.
Now, let Vpbe a local star-refinement of Vain Y o .Let W E V,.Then there exist 6 E R and V E Vasuch that St( W, V6)c V. Then we obtain from the definition of g g(E,(f-'(W))
n H(Yo))c CI,W c st(w, "y) c V .
This shows that the open cover E,(f-'(Vp)) A H ( Y o )refines g-'(V#). This implies the continuity of g. Because for x E H ( Y o ) we have
g-wm c St(g(x), m.
g(St(x9 E*(f-'(%)) A WYO)))= g(St(x9
Since {St(g(x), Va) I a E R} is a base for nbds of g(x), g is continuous at x. Since Y is Hausdorff, such an extension map g is unique. Here we make a supplement to the above theorem.
3.2. Lemma. Let X,A , Y, Y , Yoand f:A + Y be the same as in Theorem 3.1. Let Yl = {W,ly E r} be a subbase for Y and Y o = {W,, A . . * WJyi E r, i = 1, . . . , n; n = 1, 2, . . .>.Then
H(\Y,) =
nwwf-l(-Wj))I Y E ri.
Proof. By a repeated application of (3.3) we have
A
28
K. Morita
Therefore we have the desired equality.
0
The following is the first application of Theorem 3.1.
3.3. Theorem. Let ( X , #) be a generalized uniform space, ( A , # I A ) a dense subspace of ( X , #), and ( Y , @) a complete semi-uniform Hausdorffspace. Then every uniformly continuous map f from ( A , # I A ) to,( Y , Y) can be extended to a uniformly continuous map from ( X , #) to ( Y , Y).
Proof. Let Yo = { VaI a E R} be a base for Y which consists of open covers of Y. Since f is uniformly continuous, for each a E R there exists an open cover %a E # such that %a A A < f -'(Va). Let U E %fa. Then there is V E Va such that U n A c f - ' ( V ) . Hence by (3.2) and (3.3) we have U c E,(U n A) c E , ( f - ' ( V ) ) . Therefore %a < Ex(f -'(Va))and hence X = Uaac U E x ( f - ' ( V m ) )that , is, U E x ( f - ' ( V a ) )= X for each a E 0. With the notation in Theorem 3.1 we have H ( Y o ) = X and hence Theorem 3.1 shows that there exists a map g : X -, Y such that if Vbis a local < g-'(Va).Since%@< Ex(f -'(V@)), star-refinementof Va,then Ex(f -'(V@)) we have < g--'(Va).This shows that g : (X, (0) + (Y, Y) is a uniformly continuous map, completing the proof. 0
3.4. Remark. As was mentioned in Section 1, the category of nearness spaces and nearness-preserving maps is equivalent to that of generalized uniform spaces and uniformly continuous maps. In the former category Herrlich [ 1974al proved a theorem corresponding to Theorem 3.3. The following are concerned with semi-uniformities (see Section 2).
3.5. Corollary. Let ( X , #) be a semi-uniform Hausdorffspace. Then any two complete semi-uniform Hausdorff spaces each of which contains ( X , 9)as its dense subspace are uniformly isomorphic by a uniform isomorphism which leaves invariant each point of X .
Extensions of Mappings I
29
Proof. Let ( q ,'Pi), i = 1, 2, be complete semi-uniform Hausdorff spaces each of which contains (X, #) as a dense subspace. By Theorem 3.3 the inclusion map from ( X , #) into (Y,, Y , ) is extended to a uniformly continuous map f : ( Y ,, Y I ) --* ( K ,Y 2 ) ,and similarly we have a uniformly continuous map g : ( Y , , Y,) + ( Y , , Y l ) such that g ( x ) = x for X E X . Then g f : Yl + Y , is a continuous map which coincides with the identity map on the dense subspace X and hence g f = 1 y, since Y, is Hausdorff. Similarly f o g = 1 ,. This proves Corollary 3.5. 0 0
0
3.6. Corollary (=Theorem 2.34). Let X be a dense subspace of a regular Hausdorfl space Y. Then Y is obtained as the completion ( Y , Y ) of a semiuniform space ( X , #), where Y is a complete semi-uniformity on Y which is compatible with the topology of Y and # = Y I X . (See Theorems 1.16 and 2.19.)
Proof. Let ( X * , #*) be the completion of the semi-uniform space (X, #). Then X * is a regular TI-spaceby Theorem 2.32. Applying Corollary 3.5 to the semi-uniform space (Y, Y ) and ( X * , #*), we obtain Corollary 3.6 immediately. 0 Returning to the general case, the necessity condition for extendability of continuous maps will be discussed.
3.7. Lemma. Let f : A + Y be a continuous map. I f f is extendable over B with A c B c X , then the following hold: (a)
If #' is an open cover of
(b)
I f 9 is a family of closed subsets of Y with n c i , f - l ( s ) = 0.
Y , then UE,( f
-I(#'))
3
B.
09 = 8,
then
Proof. Let g : B -, Y be a continuous extension off. Let H c Y be open and F c Y closed. Then f - ' ( H ) = g - ' ( H ) n A , f - ' ( F ) = g - ' ( F ) n A . Since g - ' ( H ) is open in B and g - ' ( F ) is closed in B, we have, by applying (3.2) to EB, g - ' ( H ) c E B ( f - I ( H ) ) = B n E , ( f - ' ( H ) ) c E x ( f - ' ( H ) ) , and C1, f - ' ( F ) c g-'(F). Thus (a) and (b) follow from these relations. 0 Now Theorem 3.1, together with Lemmas 3.2 and 3.7, leads us to the following, which is the main theorem in this section. 3.8. Theorem. Let f : A + Y be a continuous map, where Y is a regular Hausdorfspace. Let Y be a complete semi-uniformity on Y compatible with
K. Morifa
30
the topology, and Y o= { Va1 a E R} a subbase for Y which consists of open covers of Y , and let us put H ( Y ~ ) = n { u E X ( f - m )I a E q. Then the following hold: (a)
f is extended to a continuous map g : H ( Y o ) + Y
(b)
H(Yo) is the largest subspace of X which contains A and over which f is extendable.
(c)
H ( Y o ) = { x E XI f ( @ ( x ) A A ) converges}, where @(x) is the nbdfilter of x. From now on, H ( Y o ) will be denoted by H( f ).
Proof. (a) follows from Theorem 3.1 and Lemma 3.2. (b) Suppose that g : B + Y is an extension off, where A c B c X. By Lemma 3.7, for each a E R we have B c U E x ( f -I(%)) and hence B c H(Yo). (c) Let x E X - A. Suppose that f ( @ ( x ) A A) converges to a pointy of Y. For each a E R, there exists V E Vawith y E V. Sincef ( @ ( x ) A A) converges to y, there is a nbd U of x such that f ( U n A) c V. Hence we have x
E
u = E A U n A ) = E x ( f - I ( V ) E E x ( f -I(%)),
that is, x E UEx(f -'(Va)). Therefore x E H ( Y o ) . Conversely, if x E H ( Y o ) ,then f ( @ ( x ) A A) converges to g ( x ) by (a).
0 As corollaries to Theorem 3.8 we obtain the following theorems.
3.9. Theorem. Let ( Y , Y ) be a complete semi-uniform Hausdorff space, f : A + Y a continuous map, and {VaI a E R} a subbase for Y which consists of open covers of Y. Then f is extendable over X iy Ex(f -'(Va)) is a cover of X for each a E R. 3.10. Theorem (see Dugundji [ 1966, p. 2 161). Let f : A + Y be a continuous map, where Y is a regular Hausdorff space. Then f is extendable over X iff the filter base f ( @ ( x ) A A ) converges for each x E X . The following is a dual form of Theorem 3.9. 3.11. Theorem. Let X , A , Y , Y and {Va1 a E R} be the same as in Theorem 3.9. Then a continuous map f : A + Y is extendable over X i y f o r each a E R we have n { c i , f - l ( Y - V ) IV E
va}= 0.
(3.6)
Extensions of Mappings I
31
If we require the validity of condition (3.6) for all covers Y E Y . Theorem 3.1 1 becomes a theorem proved by Wooten [1973], which he derived from the following theorem.
3.12. Theorem (Wooten [1973]). Let (Y, Y) be a complete semi-uniform Hausdorff space. Then a continuous map f : A + Y is extendable over X iff U{Int,Cl,f-'(V)l V E Y } = X f o r each Y EY. Proof. For any open set V of Y we have E,(f-'(V))
=
x - Cl,(A
- f-'(V))
c
cl,f-'(v),
that is, Ex(f - I ( V ) ) c Int,Cl,f-'(V). Hence the condition in Theorem 3.9 implies the condition in Theorem 3.12 for each open cover Y E Y, and hence for all Y E Y , since any cover in Y is refined by an open cover in Y. To prove the converse of this implication, let Wand V be open sets of Y such that C1, W c V , and put H = Int,Cl, f -'(Cl, W). Then we have A n H c A n Cl,f -'(Cl, W) = C1, f -'(Cl, W) =
f -I(Cl,w) c f -I(v),
and hence H c E,(H n A ) c Ex(f - ' ( V ) ) . This shows that if Ypis a local star-refinement of Yain Y then {Int,Cl, f - I ( W) 1 W E V p }< Ex(f -'(YE)). Thus, the condition in Theorem 3.12 implies the condition in Theorem 3.9.
0 As was proved by Theorems 1.16 and 2.19 the collection of all the open covers of a regular TI-space Y generates a complete semi-uniformity Yl on
Y compatible with the topology. If is a base for the open sets of Y,then the collection of all the open covers of Y consisting of members of 4?is clearly a base for Yl. Hence we have the following theorem which is free from the terminology concerning semi-uniform spaces. 3.13. Theorem. Let be a basefor the open sets of a regular Hausdorflspace Y. Then a continuous map f : A + Y is extendable over X i#lJE,( f -'(Y))= X for any 3 ' c 9d with UY = Y .
The dual form of Theorem 3.13 reads as follows.
3.14. Theorem (Herrlich [1967]). Let f :A + Y be a continuous map where Y is a regular Hausdorfspace. Let d be a base for the closed sets of Y . Then f is extendable over X z y n{Clxf I B E W } = 0 for any 9d c d with
na
=
0.
32
K. Morita
3.15. Remark. (a) The assumption of regularity for the image space Y in Theorem 3.13 cannot be weakened to Hausdorffness. To see this, let us consider the set Roof all non-negative real numbers and define two topologies t and t' : t is the usual Euclidean topology and t' = {G u (H - D) I G, H E t}, where D = { l / n l n = 1, 2, . . .}. Let us put X = (!&, t), A = &,- D , Y = (R,,t')anddefinef:A+ Y b y f ( x ) = x f o r x E A . T h e n A is dense in X , Y is a Hausdorff space which is not regular, andf is continuous. But f is not extendable over X . To prove this, suppose that there exists a continuous map g : X + Y with g I A = f. Then we have g ( x ) = x for x E D also and hence g is the identity map. Therefore, we have g([O, r)) [0, s) - D for any r, s > 0. This shows that g is not continuous at x = 0, contrary to the continuity of g . On the other hand, UE,( f -I(&')) = X for any open cover &' of Y. To see this, let us consider the base A9 = { [0, a ) - D I a > 0 } u {(a, b)10 < a < 6 ) for t'. Since E,(f-'([0, a ) - D)) = [0, a) and Ex(f - I @ , b)) = (a, b), we have UE,( f -I(&')) = X for any open cover 8 ' of Y consisting of elements in A9, and hence for any open cover 8 'of Y. Thus, the condition in Theorem 3.13 is satisfied. (b) Theorem 3.3 is not true if we assume (Y, Y ) to be a complete generalized uniform space. Indeed, let 0 and Y be the collections of all covers refined by open covers of X and Y in (a) respectively. Then f :(A, 0 I A) + (Y, 4')' is a uniformly continuous map which is not extendable to a uniformly continuous map on (X, 0). (c) It is to be noted that f is extendable over A u F for any finite subset F of D, in particular over A u {x} for each x E X - A, although f is not extendable over X . This fact is interesting in view of Proposition 3.16 below.
+
3.16. Proposition (Bourbaki and DieudonnC [1939]). Let Y be a regular T,-space. Then a continuous map f : A + Y is extendable over X i f f f is extendable over A u {x} for each x E X - A.
Proof. This proposition follows readily from Theorem 3.8 (c).
0
In case the range space Y is compact Hausdorff, the family of all the finite open covers of Y generates a complete semi-uniformity(in fact, a uniformity) on Y compatible with the topology of Y. Hence the following is an immediate consequence of Theorem 3.9. 3.17. Proposition (Eilenberg and Steenrod [1952]). Let Y be a compact Hausdorffspace. Then a continuous map f :A + Y is extendable over X ifffor anyfinite open cover Y of Y there exists afinite open cover Q of X such that 9 A A < f-'(Y).
33
Extensions of Mappings I
The following is more convenient in applications.
3.18. Theorem (Taimanov [ 19521). Let Y be a compact Hausdorf space. Then a continuous map f:A + Y is extendable over X i f CI, f -'(C)n CI, f -l(D) =
0
holds for any closed subsets C and D of Y with C n D =
0.
Proof. Let Y be the collection of all covers of Y which are refined by finite open covers. Then Y is a complete uniformity on Y compatible with the topology. Since every finite open cover {GI, . . . , G,,} of Y has an open star-refinement X , the cover A {&;I i = I , . . . , n } where Xi = {G;,St(X - G;,X ) } ,refines {G,, . . . , G,,}. Thus the collection of all the binary open covers of Y is a subbase for Y. Hence Theorem 3.18 follows 0 immediately from Theorem 3.9. 3.19. Remark. Any binary open cover {HI,H,} is refined by a binary cozero-set cover of Y. Hence in Theorem 3.18 it suffices to assume the condition merely for any two disjoint zero-sets C and D. 3.20. Corollary (Engelking [1964]). Let Y be a realcompact space. Then a continuous map f:A + Y is extendable over X if we have nCl f -'(9) = 0 for any countable collection 9 of zero-sets of Y with 09 = 0. Proof. For a completely regular Hausdorff space Y the collection of all countable normal covers of Y is a uniformity on Y compatible with the topology, and Y is called realcompact if this uniformity is complete. Since every countable normal cover is refined by a countable cozero-set cover (see Section 1 in Chapter 2), as an immediate consequence of Theorem 3.9 we have Corollary 3.20. 0 In concluding this section we shall prove a generalization of a theorem of Lavrentieff.
3.21. Theorem. Let ( X , 0 )and ( Y , Y ) be complete semi-uniform Hausdorf spaces, and let A and B be dense subspaces of X and Y respectively. Let f:A + B be a homeomorphism. Then there exist the largest subspaces A, of X and B, of Y such that A c Ao, B c Bo and f is extendable to a homeomorphism fo : A, 4 B,. In case each of 0 and Y has a subbase of cardinality < m, each of A. and Bo is the intersection of m open sets, where rn is an infinite cardinal number.
K. Morira
34
Proof. Let g :B + A be the inverse off, Let us considerf and g as continuous mapsf : A + Y and g: B + X, and apply Theorem 3.8 to these maps. Then, with the notation in Theorem 3.8 there exist continuous mapsf, :H( f ) + Y and g, : H(g) + X. Let us put A, = H ( f ) n f i - v w ) ,
Bo = H(g) n g ; W f ) ) .
Then g, ofi :A, + Xcoincides with the inclusion map iA:A + Xon A. To see this,letx E A.Thenfi(x) = f(x) E Bandg,(fi(x)) = g(f(x)) = x.SinceX is Hausdorff and A is dense. in A,, we have (8, ofi)(x) = x for x E A,. Similarly we have (fi og,)(y) = y for y E Bo. Since g,(fi(A,)) = A,, we have f i ( A , ) c g;'(A,) c g;l(H(f)). On the other hand, since A, c A-'(H(g)), it follows that f i ( A , ) c H(g). Therefore f i ( A , ) c H(g) n g;l(H(f)) = B,, that is,fi(A,) c B,. Similarly g,(B,) c A,. Now, let us Put f, = fiIA,:A, B,, go = g,IB,:B, + A,. +
As has been proved above g,(f,(x)) = x for x E A , and henceg,(f,(x)) = x for x E A,. Similarly we havefo(g,( y)) = y for y E B,. Thereforef, : A, + Bo is a homeomorphism onto. Suppose that A c C, B c D and f is extended to a homeomorphism cp : C + D . Let $ :D + C be the inverse of cp. Let us consider cp and $ as continuous maps cp : C + Y and $ : D -+ X respectively. Then cp : C + Y is an extension off: A + Y and hence by Theorem 3.8 we have C c H ( f ) and cp = fi I C. Similarly we have D c H(g) and $ = g, ID. Therefore, we have C = cp-'(D) c fi-'(H(g)), and consequently, C c H c f ) nf,-'(H(g)) = A,. Similarly D c B,. Thus, the first statement of the theorem is proved. Suppose that 0, = {el I a E R} (resp. Yo = {
H(f1 =
n{uJ%(f-l<<)) I Y E r>,
H(g) = n { u w - i w i a E and hence H( f ) (resp. H(g)) is the intersection of at most m open sets of X (resp. Y). This proves the second statement. 0
3.22. Corollary (Lavrentieff Theorem, Lavrentieff [1924]). Let X and Y be complete metric spaces, let A c X and B c Y, and let f :A + B be a homeomorphikm. Then f is extendable to a homeomorphismf,:A , + B, such that A c A, c CI,A, B c B, c CI,B and A, (resp. B,) is a G,-set o f X (resp.
Y).
Extensions of Mappings I
35
Proof. Since CI,A and C1,B are complete metric spaces, we can apply Theorem 3.21 to the case that A c CI,A and B c CI,B, and we see that there exist G,-sets A , and B, of C1,A and CI,B and that f is extendable to a homeomorphismf,: A , -,B,. Since C1,A and C1,B are G,-sets of X and Y respectively, A, and B, are also Gd-setsof X and Y respectively. 0
Appendix. A generalization of the Ascoli Theorem For topological spaces X and Y , let C ( X , Y ) be the space which consists of all continuous maps from X to Y and has the compact-open topology. In Kelley [I9551 the two types of generalizations of the Ascoli Theorem are given: Theorem A. Let X be a regular locally compact space, ( Y , Y) a Hausdorf ungorm space. Then a closed subset F of C(X, Y ) ' i scompact iy (a)
F is equicontinuous,
(b)
C1 F[x] is compact for each x
E
X , where F [ x ] = { f(x) If
E
F}.
Theorem B. Let X be a regular locally compact space and Y a regular Hausdorff space. Then a closed subset F of C ( X , Y ) is compact i f f (a)
F is evenly continuous,
(b)
CI F[x] is compact for each x
E
X.
Here a family F of maps from X to Y is called evenly continuous if for each X , each y E Y and each nbd G of y there exist a nbd U of x and a nbd H o f y s u c h t h a t f ~ F a n d f ( x ) ~ H i m p l y f ( U )c G. On the other hand, the definition of equicontinuity presupposes a compatible uniformity Y on Y :F c C(X, Y ) is called equicontinuous if for each X E XandeachV E Y thereexistsanbd Uofxsuchthatf(U) c St(f(x), V ) for all f E F. Since Y has such a uniformity iff Y is completely regular, there is a discrepancy between Theorems A and B. It was perceived first by Poppe [I9671 that a good remedy for this situation is provided by semi-uniformities. In this appendix we shall define the concept of equicontinuity for a semiuniforrh space ( Y , Y) in the same way as above and prove the following theorem, which was given by Poppe [I 9671 with a stronger definition of equicontinuity than ours.
x
E
36
K. Morita
A.l Theorem. Let X be a topological space and ( Y, Y) a semi-uniform space. Let F c C(X, Y ) .
If F is equicontinuous, then F is evenly continuous. If F is evenly continuous and CI F [ x ] is compact for each x E X ,
(a) (b)
then F is equicontinuous.
Proof. (a) Let x E X , y E Y and let G be any nbd of y. Then there exists Y E Y such that St*(y, Y ) c G (see Proposition 1.1 1). Since F is equicontinuous, there is a nbd U of x such thatf(U) c S t ( f ( x ) , Y ) .Let us put H = St(y, Y ) .Then, iff ( x ) E H, thenf ( U ) c StCf(x), Y ) c St2(y,Y ) c G. (b) Let x E X , and let Y be any open cover of Y. For each y E C1 F[x] there is G( y ) E Y with y E G( y). Since F is evenly continuous, there exist a nbd U, of x and a nbd H( y) ofy such that f (x) E H( y ) impliesf(U,) c G( y). Since C1 F [ x ]is compact, there exist y j E CI F[x],i = 1, . . . ,n ( n finite) such that CI F[x] c U { H ( y , ) l i= 1 , . . . , n } . Let US put U = n { U , l i = 1 , . . . , n } . Then U is a nbd of x . If f ( x ) E H ( y , ) , then f ( U ) c f(U,,) c G ( y J c
0
S t ( f ( x ) , Y).
A.2. Corollary. Theorem A holh if (Y, Y ) is a Hausdor- semi-uniform space. A direct proof of Corollary A.2 without appealing to Theorem B was given by Poppe [ 19671 for the special case of X being compact Hausdorff and by Rolicz [ 19751 generally. As for the assumption on X Bagley and Yang [I9661 proved that Theorems A and B hold if X is a Hausdorff k-space. Here a topological space Xis called a k-space if G c X is open whenever G n K is open in K for each compact subset K of X . Any locally compact space is a k-space, and for such a space Hausdorffness implies regularity but not conversely. Therefore, Theorems A and B are not contained in the above results of Bagley and Yang. Our next theorem extends these theorems and results simultaneously. A.3. Theorem. Let X be a k-space and ( Y , Y) a semi-uniform space. Let F c C ( X , Y ) . Then C1 F is compact iff
F is equicontinuous (resp. evenly continuous),
(a) (b)
’
CI F[x] is compact for each x
E
X.
Our proof rests upon the following lemma.
Extensions of Mappings I
31
A.4. Lemma. Let X be a k-space, Y a regular space and F a locally compact subset of C(X, Y ) . Then the evaluation map o :F x X + Y, which is dejined by o ( f ,x ) = f ( x ) , is continuous. Proof. (i) Let K be a compact subset of X . Then o I F x K is continuous. To see this, let x E K, f E F, y = f ( x ) . Let G be any nbd of y . Since Y is regular, there is a nbd H of y with C1 H c G. Let U be a nbd of x in K such that f ( U ) c H. Then f(C1,U) c Clf(U) c C1 H c G. Thus, if we put C = C1,U and M(C, G ) = { g E F l g ( C ) c G } , then C is compact, M(C, G ) is a nbd off in F and w(M(C, G ) x U ) c G . . (ii) Since Y is regular, so is C ( X , Y) (see Kelley [1955, Theorem 7.41) and hence Fis regular. The proof of Lemma 3 in Morita [1953], in which it suffices to assume local compactness and regularity for the space Y there, shows that G c F x Xis open whenever G n (F x K ) is open in F x K for each compact subset K of X (and hence F x X is a k-space). Hence it follows from (i) that o is continuous. 0
Proof of Theorem A.3. Let us first observe that if F in Lemma A.4 is compact, then F is equicontinuous. To see this, let x E X and let Y be any open cover of Y. Then for each f E F there are a nbd W ( f )off and a nbd 4)for some 5 E Y .Since F is compact, U’of x such that W ( f ) x U, c o-’( . . . , W ( f , ) } of F. Let us put there is a finite open cover {W(f,), U = n{U,li = 1 , . . . , n } . I f f € F, thenfe W ( f ; )for somef; and hence f ( U ) c f(v,) c v/, c S t ( f ( x ) , Y ) .Now, assume C1 F is compact. Then Cl F, and hence F, is equicontinuous and o(C1 F x { x } ) = (Cl F ) [ x ] is compact for each x E X by Lemma A.4. Since the closure of a compact subset of a regular space is compact, C1 F [ x ] is compact. This proves the necessity of (a) and (b). Conversely, assume (a) and (b). Let us consider the product space Y x = Il{Y, I x E X } where Y, = Y for each x E X; Y x consists of all maps f :X + Y with f ( x ) = p , ( f ) where p , : Y x + Y, is the projection. Let P be the closure of F in Y x . It suffices to prove: (1) E is compact, (2) P is evenly continuous and hence P c C(X, Y), (3) the subspace topology B of P c Y x coincides with the compact-open topology % on P, (4) P = C1 F as topological spaces. Assettion (1) follows from the relation F c n{Cl F [ x ]I x E X } by virtue of (b). By Theorem A.l, F is evenly continuous. Hence for x E X,y E Y and a closed nbd G of y there exist a nbd U of x and an open nbd H of y such
38
K. Morita
that if f~ F then f ( x ) E H implies f(U) c G, that is, F c Q where Q = p ; ' ( Y - H ) u n { p ; ' ( G ) l x ' E U}.SinceQisclosed,wehaveP c Q which implies the even continuity of P since Y is regular. Thus (2) is proved. If we define a map w : P x X + Y by w ( g , x ) = g ( x ) , then E c Q implies w(p;'(H) x V ) c G. Hence w is continuous. Therefore, for K c X compact and V c Y open, B(K, V) = { g E El { g } x K c w - ' ( V ) } is open in F. Since B(K, V) = { g E El g ( K ) c V } and {B(K, V) I K c X compact, V c Y open} is a subbase for V ,we have V c 8.Since 8 c V we have (3), and hence (4). 0
For various geiieralizations of the Ascoli theorem see Poppe [ 19741. References Bagley, R. and J. Yang [I9661 On k-spaces and function spaces, Proc. AMS 17, 703-705. Bourbaki, N. and J. Dieudonnt [I9391 Note de teratopologie 11, Revue Scientifque 77, 180-181. Dugundji, J. [ 19661 Topology, Boston. Eilenberg, S. and N. Steenrod [ 19521 Foundations of Algebraic Topology, Princeton.
Engelking, R. [I9641 Remarks on real-compact spaces, Fund. Math. 55, 303-308. [I9771 General Topology, Warsaw. Harris, D. [I9711 Structures in topology, Mem. AMS 115. Herrlich, H. [I9671 Fortsetzbarkeit statiger Abbildungen und Kompaktheitsgrad topologischer Raume, Math. Z . 96, 64-72. [I9741 A concept of nearness, General Topology Appl. 4, 191-212. [1974a] On the extendability of continuous functions, General Topology Appl. 4, 213-215. [1974b] Topological structures, Math. Centre Tracts 52, 59-122. Isbell, J. R. [I9641 CJni/brmSpaces, Providence. Kelley, J. L. [I9551 General Topology, New York. Morita, K. [I9511 On the simple extension of a space with respect to a uniformity, I-IV, Proc. Japan Acad. 27, 65-72, 130-137, 166-171, 632636. [I9531 On spaces having the weak topology with respect to closed coverings, Proc. Japan Acad. 29, 537-543. [I9811 Theory of Topological Spaces (Japanese), Tokyo.
Extensions of Mappings I
39
Nagata, J. [19851 Modern General Topology, Amsterdam. Poppe, H. [I9671 Ein Kompaktheitskriterium fur Abbildungsraume mit einer verallgemeinerten uniformen Struktur, Proceedings of the Second Prague Topology Symposium 1966. Praha 1967, pp. 284-289. [ 19741 Compactness in General Function Spaces, Berlin.
Know, W. [1967] uber die verallgemeinerten uniformen Strukturen von Morita und ihre Vervollstandigung, Proceedings of the Second Prague Topology Symposium 1966, Praha 1967, pp. 297-305. [1975] Lehrbuch der Topologie, Berlin. Rolicz, P. [1975] Generalizations of Ascoli’s theorems, Colloq. Math. 33, 213-218. Shanin, N. A. [I9431 On the theory of bicompact extensions of topological spaces, Doklady URSS 38, 154-156. Steiner, A. K. and E. F. Steiner [I9731 On semi-uniformities, Fund. Math. 83, 47-58. Taimanov, A. D. [I9521 On the extension of continuous mappings of topological spaces (in Russian), Mat. Sb. 31, 459-462. Tukey, J. W. [1940] Convergence and Uniformity in Topology, Princeton. Wooten, D. F. [I9731 On the extension of continuous functions, Fund. Math. 83, 59-65.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 2
EXTENSIONS OF MAPPINGS I1
Takao HOSHINA Institute of Mathematics, University of Tsukuba, Ibaraki 305, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. C*-, C-, P”- and Pembeddings. . . . . . . . . . . . . . . . . . . . . . . 3. Unions of C*-embedded subsets . . . . . . . . . . . . . . . . . . . . . . 4. C*-embedding in product spaces . . . . . . . . . . . . . . . . . . . . . . 5. Homotopy extension property . . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 42 51 62 69 75 78
Introduction
In this chapter we shall consider cases, such as C*-, C- or P-embedding, of extending continuous maps defined on a (not necessarily dense) subspace to the whole space. For a space Xand its subspace A we shall say that A is C*-embedded (resp. C-embedded) in X if every bounded real-valued (resp. real-valued) continuous function on A can be extended to a continuous map on X. These notions, which apparently come from the well-known Tietze Extension Theorem for normal spaces, were investigated with nice expositions by Gillman and Jerison [1960] and are now familiar to us. Another useful notion we mention is that of the P-embedding. The P-embedding was introduced by Shapiro [ 19661in connection with the extension of continuous pseudometrics defined on a subspace to the whole space. Afterwards it has been investigated in various studies, in consideration of several equivalent but distinct forms, some of which will be our main concern in this chapter. One of such forms, which will be adopted here as our
T. Hoshina
42
definition of the P-embedding, is described in terms of extending normal covers of a subspace to normal ones on the whole space. However, the P-embedding will also answer the purpose of this chapter because it will be characterized in terms of extending continuous maps from a subspace into Cech-complete AR spaces to the whole space. On the other hand, C*- and C-embeddings will also be characterized by using finite and countable normal open covers respectively in a form analogous to our .definition of the P-embedding. Thus, C*-, C- and P-embeddings can be approached by each of two methods: extension of continuous maps or extension of normal covers. The latter will be a good tool to develop our study. Section 1 is devoted to some preliminary notions and results. In Section 2 we shall prove some basic properties of C *-, C-, P" - and P-embeddings and their characterizations as stated above, by introducing the notion of P" -embedding which is an intermediate one between C-embedding and P-embedding. In Section 3, C*-embeddability of finite, or certain infinite, unions of C*-embedded subsets will be discussed. It will be found that, among other results, C*-embeddability of unions of merely two C*embedded subsets is the most fundamental. In Section 4, C*-embedding in product spaces will be discussed. We shall characterize C-, P"- or P-embeddings in terms of C*-embedding on product spaces. Results obtained in these sections are interesting in themselves, yet the more interesting lie in applications to the homotopy extension theorems. In the final section we shall completely describe and generalize the homotopy extension theorems which were originally studied by K. Borsuk and afterward by C. H. Dowker. Throughout this chapter, by a space we shall mean a topological space without any separation axiom, unless otherwise specified.
1. Preliminaries
We use the same notation and terminology as in Section 1 .O of Chapter I . N and I denote the set of all positive integers and the closed unit interval respectively. For a cover 9 of a space X we put
9* = {St(U, %)I
u E 9}, a"
= {St(.,
a)lx
E
X}.
Hence, 9*c Y for another cover Y of Xmeans that 9is a star-refinement of Y . If 9"< Y , 91 is said to be a A-refinement of Y . By a simple calculation we always have 9 c 42" c a* c (a!")".
Extensions of Mappings II
43
First let us recall a normal cover of a space. A sequence {%,,In E N} of open covers of a space Xis said to be normal if < a,, for each n E N. An open cover 9 of X is said to be normaI if there exists a normal sequence {Q,,} such that Q1refines Q. Let Q be a normal cover of a space X , and @ = {a,,I n E N} be a normal For any subset A of X we define sequence of open covers of X with Q, < 9. Int(A; @) = {x E XI St(x, Q,,) c A for some n } . Then Int(A; @) is an open subset of X. For, let x E Int(A; 0).Then St2(x,a,,+') c A for some n since < Q,. Hence we have c Int(A; a), which shows Int(A; @) is open. St(x, For each cover Q, of 0 let us put V,,= {Int(U; @)I U E Q,,}. Since < Q, , one can prove that Y,,is an open cover of X and for each n E N (1) Vn+l is a star-refinement of Y,,, (2) % + I < "IT, < QnThat is, {V,,,(nE N} is a normal sequence, and it'is denoted by @*. Let us denote by (X,@) the topological space obtained from X by taking {St(x, Q,,) In E N} as a nbd base at each point x of X. Let X / @ denote the quotient space obtained from (X, 0)by defining those two points x and y equivalent for which y E St(x, 9,,) for each n E N. Let cp be the composite of the identity map from X onto (X, @) and the quotient map cjj from ( X , @) onto X / @ . Then we have @-'(cjj(Int(A;@))) = Int(A; @) for any A c X. Therefore, cp is continuous. Taking also views of (1) and (2), {cp(V,,) I n E N} is a normal sequence of open covers of X / @ , which generates a uniformity of X / @ compatible with the topology. Thus, X/@is metrizable. These arguments and results can be seen in Morita [1970]. Finally, we note that cp-'(cp(%)) = % < Ql. 1.1. Theorem (Morita [1970]). An open cover % ' o f a space X is normal if there exist a metric space T, a continuous map cp :X + T and an open cover Y of T such that cp-'(V)reJnes 9.
Proof. The "only if" part immediately follows from the arguments above. Conversely, suppose that T, cp : X + T and V of the theorem exist. Then V is a normal cover of T since every open cover of a metric space is normal. Consequently, cp-'(*Y) is also a normal cover of X and so is Q. 0 A subset Z of a space X is called a zero-set if 2 can be written as Z = {x E Xlf(x) = 0} with some continuous mapf:X + I. A cozero-set is the complement of a zero-set. A cozero-set (resp. a zero-set) cover is a cover consisting of cozero-sets (resp. zero-sets).
T. Hoshina
44
As another characterization of normal covers we have the following theorem. 1.2. Theorem (Morita [1962, 19641). For an open cover 4 of a space X the following are equivalent. (a) 9 is normal. (b) 42 is refined by a local&Jinite cozero-set cover. (c) 9 is refined by a o-locallyfinite cozero-set cover. (d) 4 is refined by a o-discrete cozero-set cover. (e) 9 has apartition of unity subordinated to it, that is, a family (cpl I A E A} of continuous maps cpl : X -+ Zsuch that Zcpl(x) = 1for eachpoint x of Xand {{x E XI cpl(x) > 0} 1 rl E A} refines %. Here, Lp1(x) means sup{X:,,, cpl(x) 1 y is afinite subset of A}.
Proof. (a)*(b). Suppose 9 is normal. By Theorem 1.1 there exist a metric space T, a continuous map cp :X -+ T and an open cover V of T such that cp-'(V)refines 4. Since T is paracompact, we can take a locally finite open refinement W of V ;each member of W is a cozero-set since T is metric. Hence, cp-'(W)is also a locally finite cozero-set cover of X,and refines 9. (a)*(d). By replacing "locally finite" by "a-discrete" the same argument as in (a)*(b) can be applied to this case. (b)*(c), (d)*(c). These are obvious. (c)*(e). Suppose that there exists a cozero-set cover Y = UncN Vnsuch that V refines 4 and each V, = { V,, Ia E Q,} is locally finite. Let cpna :X + I be a continuous map such that V,, = {x E XI rp,,(x) > O}. Since Vnis locally finite, cp,(x) = cpna(x),x E X defines a continuous function cp, over X.If we put
then rp is also continuous over X.Since V covers X,for every point x E X we have ~ ( x > ) 0. Let us put +na(x) = Pna(x)/2"
E
X-
Then co
1 JInU(x) =
1 for X E X and
n = l asn,
Ka
Hence
= {x~XlICIna(x)> 01-
I a E Q,, n E N} is a partition of unity subordinated to 9.
Extensions of Mappings 11
45
(e)*(a). Let {cpl I 1 E A} be a partition of unity subordinated to 9. Let us consider a metric space M which consists of { xl I 1E A} with xl E Z such that C xl = 1 and has a metric d(x, y ) = C IxI - y l l . Define cp:X -+ M by cp(x) = {cpl(x)11 E A} for x E X. We show cp is continuous. Let xo E X and E > 0. Since C cpl(xo) = 1, for a finite subset y of A and a nbd U of xowe have
Then we have for x E U,
Let 9 be a collection of cozero-sets of a space X. Then it is well known that U 9 = U{U 1 U E 9}is also a cozero-set in case either Card 9 ( = cardinality of 9)< Noor 9 is locally finite. Hence, further in case 9 is a-locally finite, U 9 is also a cozero-set; this can be assured by the similar construction of cp such as in the proof of (c)=.(e) of Theorem 1.2. For collections of zero-sets the corresponding results need not be true (see Example 3.14 below). The following lemma is useful in discussing unions of zero-sets. 1.3. Lemma (Morita and Hoshina [1976]). Let {FaI a E R} be a collection of zero-sets of X . Assume that there exists a locally Jinite collection { G, I a E n} of cozero-sets of X with Fa c G, for each a E R. Then the union U{F,I a E R} is a zero-set.
Proof. Since Fais a zero-set and G, a cozero-set of X with Fa c G,, there exists a continuous map f, : X -,Z such that I
Fa
=
{XE
X l f , ( x ) = l},
G, =
{ X E
Xlf,(x) > O}.
T. Hoshina
46
Define g : X
-, Z by
g ( x ) = s u p { ~ ( x ) I uE R}
for x
Then g is continuous, and we have U{F,I u is, F, I a E f2} is a zero-set of X.
u{
E
E
f2}
X. = { x E Xlg(x) =
l}, that
0
In calculating normal covers the following theorem will be useful. 1.4. Theorem. For any normal cover 42 = { U, I u E R} of a space X , there exist a locally finite cozero-set cover { V ,1 u E R} and a zero-set cover {F, I u E R} of X such that F, c V , c U, for each a E f2. Proof. By Theorem 1.2 there exists a locally finite cozero-set cover Y of X refining 42. For each Y E Y choose uv E R so that V c UUv, and define a map s: Y -, R by s ( V ) = u,,. Let us put for each u E f2 V, = U{ V E V Is(V) = u } . Then V, is a cozero-set with V, c U, and { V ,I u E R} is locally finite and covers X. Hence, { V ,I u E R} is constructed. Next, by Theorem 1.2 there exists a locally finite cozero-set cover W such that W* < { V , l u ~ R } . I f w e p u t F , = X - S t ( X - V , , W ) f o r e a c h u ~ R , then one can prove that F, is a desired zero-set. 1.5. Lemma. Let {G, I u E R} be a locallyfinite collection of cozero-sets and
{ F, I u E R} be a collection of zero-sets of a space X such that F, c G, for u E R, then the collection
where r is the set of all finite subsets of R, is a locally finite cozero-set cover of X such that St(F,, W ) c G, for each u E R.
nuEy
Proof. For y E r let W(y) = G, n (X - UBdY F8). Then W(y) is a F, is a zero-set of X by Lemma 1.3. Let x be any point cozero-set since UBCy of X. Let y I and y2 be finite subsets of R such that x E G, iff a E y I (yl may be empty) and for a suitable nbd V of x, B 4 y2 implies V n G, = 8. Then x E W(y,),and y Q y z implies V n W(y) = 0. Hence W is a locally finite cozero-set cover. Finally, if F, n W(y) # 8, then u E y, that is, W(y) c G,. Hence, St(F,, W ) c G,. 0 Remark. In the lemma above we note that Card W < KO or Card W = Card Y according as Card Y < No or KO < Card Y, and that in case 9 is further a cover of X then so is Y and we have W A< Y.
Extensions of Mappings 11
47
1.6. Theorem. Any normal cover Q of a space X admits a normal sequence {Q,,} such that Ql < Q and either Card Q, < KOfor each n E N or Card Qn = Card Q for each n E N according as Card Q < KOor KO < Card Q. Proof. By Lemma 1.5 and its remark we can inductively construct a sequence {W,,}of open covers such that W: < W n pwhere l , Wo= 42, and either Card Wn< KOfor each n E N or Card W,, = Card Q for each n E N according as Card 42 < KOor KO < Card 9. Let Qn = W2,,for each n E N. Then {Q,,} is the desired normal sequence since we have = W2fn+I) < 0 (Ktn+I)IA < %:+I < Kn = Qn (n > 1).
Theorem 1.6 strengthens Theorem 1.1 as follows. 1.7. Theorem. For any normal cover 42 of a space X there exist a metric space T which is either compact or tech-complete with weight < Card Q according as Card 42 < KOor KO < Card Q, a continuous map cp : X + T and an open cover Y of T such that q-'(V)< 42. Proof. Let @ = {Q,,} be a normal sequence obtained in Theorem 1.6. Applying the same arguments as above to @, the metric space X/@, a normal sequence {Y,,} and a continuous map cp : X + X/@ are constructed so that refines Ql, Card = Card a,,, cp-'cp(Vn) = V,,,and Y = {cp(Vfl)} is a normal sequence of open covers of X / @which generates a uniformity of X / @ compatible with the topology. Let T be the completion of X/@with respect to Y. Regard cp as a map from X into T , and finally put Y = { T - ClT(X/@ - V')I V' E rp(Y1)).
Then T , cp : X + T and V satisfy all the required properties.
0
By a linear topological space we mean a real vector space L with the topology such that a : L x L + L and m : R x L + L are both continuous, where a(x, y ) = x + y , m(a, x ) = ax. A linear topological space L is locally convex if each point x of L has a nbd base consisting of convex nbds of x. It is well known that every Banach space is a locally convex linear topological space.
1.8. Theorem (Dugundji [ 195 11). Let L be a locally convex linear topological space. Let X be a metric space and A its closed subspace. Then every continuous map f : A + L is extended to a continuous map g : X + L so that g ( X ) c the convex hull of f ( A ) in L.
T. Hoshina
48
Proof. Let d be a metric on X. Let B ( x ; E ) = { y E XI d ( x , y ) < E } . Let x E X - A and let E, = d ( x , A ) . Then B ( x ; 38,) c X - A and W = { B ( x ; +e,)lx E X - A} covers X - A. Since X - A is paracompact, W is refined by a locally finite open cover 4 of X - A. First we shall prove: if U E 4 and U n B(a; E ) # 8 for some a E A, then 6 ( U ) (=diameter of U ) < 2 ~ For, . choose z E U n B(a; E ) . Let B ( x ; + e x ) E W containing U . Then 6(U) <
Ex
=
d ( x , A ) 6 d ( x , 2)
+ d(z, a) < $ E x + E.
Hence, we have 6 ( U ) < 2.5. Let U be any member of 9 with U # 0. Take a point xu E U . Then d(x,,, A) > 0, and so a point a, E A can be chosen so that d(x,,, a,,) < 2d(x,,, a). Then the following assertion holds: (*)
For each a E A and a nbd W of a in X , there exists a nbd V of a with V c W such that whenever U n V # 0 and U E 9 then U c Wanda,,€ W.
Suppose that Indeed, take E > 0 so that B(a; E ) c W. Put V = B(a; h~). U n V # 8 and U E 4. Then by the fact proved above 6 ( U ) < + E . Hence, U c B(a; * E ) . Consequently, U c Wand we have
d(a,,
4 < d(a,,, xu)
< 3d(x,,
+ d(x,,, a) < 2d(x,,, A ) + d(x,, 4
a) <
ZE,
that is, a,, E B(a; E ) c W. Hence (*) is proved. Since X is a metric space, each U E 9 can be written as U = { x E XI g,,(x) > 0 } for some continuous map g , :X + Z. Define K,, : X + Z by K,,(x) = g,,(x)/C,,.,,g,.(x) for x E X - A . Now let us define h :X + L by
x
E
A,
Clearly, h I A = f. We prove the continuity of h. Since X - A is open in X and 4 is locally finite, h is continuous at each point of X - A. We show h is continuous at each a E A. Let 0 be an arbitrary nbd of h(a) = f ( a ) . Since f is continuous and L is locally convex, there exist a nbd W of a in X and a convex nbd C off@) in L such thatf( W n A) c C c 0. For W find a nbd V of a satisfying (*). Then we have h( V) c C. To see this, it suffices to prove h(x) E 'C for each x E X ,- A. Then for this point x we have x E Q for at most finitely many U , , . . . , U, E 4. Then n V # 0, which implies a,, E W by (*). Therefore,f(a,,) E C, i = 1, . . . , n. Hence, h ( x ) E C since
Extensions of Mappings II
49
C is convex. Thus, we have h ( V ) c C c 0, which shows that h is continuous at a E A. By construction h ( X ) c the convex hull of f ( A ) . Thus, h is the desired extension off. 0 1.9. Theorem (Kuratowski-Wojdyslowski, see Hu [1965, p. 851). Let Y be a metric space of weight < m (m 2 No).Then there exist a Banach space L and an isometrical embedding cp : Y + L such that cp( Y ) is a closed subset of its convex hull Z in L and weight of Z < m.
Proof. Let C * ( Y ) denote the set of all bounded real-valued continuous maps defined on Y. Then it is known that C*( Y) is a Banach space endowed with the usual linear structure and a norm )If 1) by taking )If )I = supy, If ( y) I. Let d be a metric on Y. Choose p E Y, and p will be fixed throughout the proof. For a point a E Y definef, by f , ( Y ) = d(Y, 4 - d(Y, P ) f o r y E y* Then f, is continuous, and If,( y) I < d(a, p). Hence, f, E C*( Y). Define cp:Y+ C * ( Y ) by c p ( 4 =A,, Y . Since If,(yj - h ( y ) I = I d ( y , a ) d( y, b) I < d(a, b ) for each y E Y and If,(b) - fb(b)1 = d(a, b), we have
II c p ( 4 -
cpw I1
= SUP If,(y) - h(Y ) I =
4 2 ,
b).
YEY
Hence, cp is an isometrical embedding. We shall show that p ( Y ) is closed in its convex hull Z in C * ( Y )and weight of Z < m. To see cp( Y) is closed in Z, let g be any point of Z - cp( Y). Since Z is the convex hull of cp( Y), there exist al, . . . , a, E Y such that g =
AIL,+
*
n
- + A,f,
where li > 0 with
1 li = 1.
i= I
Since g # q ( Y ) implies 11 g - f,,11 > 0 for i = 1, . . . , n, 6 > 0 can be chosen so that 6 < 11 g - f,,11 for every i = 1, . . . , n. Let us put V = { f E 21llg - f I1 < a}. Then Vis an open nbd ofgin Z. We will show V n cp(Y) = 8. Suppose not and choose a point a E Y with cp(a) = f, E V. Then for every i = 1, . . . , n d(a, ail =
IILi - f a l l 2 llg
- L,ll - Ilg
- LII > 26
-
Consequently, we must have
Aid(a, ai) >
= i= I
(il i )6
a contradiction. Hence, V n cp(Y) =
= 6,
i= I
8,and cp(Y) is closed in Z.
6 = 6.
T. Hoshina
50
Next we show weight of Z < m. Since Z is metrizable, it suffices to prove that Z has a dense subset of cardinality < m. Since q( Y) is homeomorphic to Y, q( Y) has also weight < m, and has a dense subset D of Card D < m. Let H ( D ) be the convex hull of D in C*(Y). Then H ( D ) c Z . Let r be the set of all finite subsets of D.Denote by H(y) the convex hull of y in C*(Y) for a y = { g l , . . . ,g,} in r. Then H(y) is a finite union of closed simplexes with vertices g , , . . . , g, and hence it is separable. Therefore, H ( D ) has a dense subset of cardinality < m since Card < m and H ( D ) = U { H ( y )I y E r}.Hence, it remains to show that H ( D ) is dense in Z . To see this, let f E Z and E > 0. Then there exist fi, . . . ,fn E q( Y) such that
f
=
A,fi + . . . + A,f, where A, > 0 with
n
Izi = 1. i= I
Since D is dense in q(Y), there exist g , , . . . , g, in D such that llx - g,II < $ E for i = 1, . . . , n. Let g = Algl + * * * + Angn. Then g E H(y), where y = { g , , . . . , g , } , hence g E H ( D ) . Moreover, we have
Thus, H ( D ) is dense in Z . The theorem is proved.
0
Let X be a space, which is a subspace of another space Y. Then X is a retract (resp. neighborhood retract) of Y if there is a retraction from Y (resp. a neighborhood of X in Y = an open subset of Y containing X ) onto X . Let V be a class of spaces. Then a space Xis said to be an absolute retract or shortly AR (resp. absolute neighborhood retract or shortly ANR) for V if X belongs to V and whenever X is a closed subspace of a space Y in V X is a retract (resp. neighborhood retract) of Y. A space Yis said to be an absolute extensor or shortly AE (resp. absolute neighborhood extensor or shortly ANE) for V if Y belongs to 9 and for every continuous mapf: A --* Y from every closed subspace A of any space X in V into Y there exists a continuous extension off from X (resp. a neighborhood of A in X ) into Y. It is obvious that for any class V every AE (resp. ANE) for %? is an AR (resp. ANR) for V , and it is known that the converse also holds for many classes %? (see Hu [1965]); in particular the class A of all metric spaces is such one asis shown by the following. 1.10. Theorem. A space X i s an A R (resp. A N R ) f o r A iflit is an A E (resp. A N E ) f o r A.
Extensions of Mappings II
51
Proof. We shall prove an ANR for I is an ANE for A. The proof for the case of an AE space for 4 is similar and simpler. Let Y be a space which is an ANR for A. Let X be a metric space and A its closed subspace. Let f:A + Y be a continuous map. Then by Theorem 1.9 there exists a Banach space L such that Y is closed in its convex hull Z in L. Since Y is an ANR for .M and 2 metrizable, there exists a nbd V of Y in Z together with a retraction r : V + Y. On the other hand, by Theorem 1.8f: A + Y is extended to a continuous map g : X -+ L such that g ( X ) t the convex hull off(A). Hence g ( X ) c Z. Let U = g - ' ( V ) . Then U is a nbd of A in X , and we see the composite r o ( g 1 U ) :U -+ Y is clearly an extension off. Thus, Y is an ANE for A. 0 In the sequel of this chapter an AR (resp. ANR) for A will be called simply an AR (resp. ANR). By Theorems 1.8 and 1.10 every locally convex linear topological space is an AR if it is metrizable. 2. C*-, C-, P"- and P-embeddings
Let us begin with notions of C*- and C-embeddings and their basic properties. 2.1. Definition. Let X be a space and A its subspace. Then A is said to be C*-embedded (resp. C-embedded) in X if every bounded real-valued (resp. real-valued) continuous mapfon A is extended to a continuous map g over X. We note in the definition above that iffis bounded, that is, there exist a, b E R such that a < f(x) d b for any x E A , then the extension g can be taken so that a < g(x) 6 b for any x E X. For, let g' be a continuous extension off. Then define g by g(x) = min{b, max{g'(x), a } } for x E X . Let E and F be a pair of disjoint subsets of a space X . Then E and F a r e said to be completely separated in X (or E is said to be completely separated from F ) if there exists a continuous mapf: X -+ I that takes values 0 on E and 1 on F, that is,f(x) = 0 for x E E andf(x) = 1 for x E F. 2.2. Lemma. E and F are completely separated in X i f l there exist disjoint zero-sets Z , and Z , such that E c Z , and F c Z , .
Proof. ' If there exists a continuous mapf: X
+
I such thatf
=
0 on E and = l } are
f = 1 on F, then Z, = {x E Xlf(x) = 0} and Z , = {x E Xlf(x)
disjoint zero-sets and we have E c Z , and F c Z 2 .
52
T. Hoshina
Suppose conversely, then there exist continuous mapsf, g : X + I such that E c Z, = { X E X l f ( x ) = 0},F c Z, = { X E X l g ( x ) = 0 } andZ, n 2, = 8. Define h : X + I by h ( x ) = f ( x ) / (f ( x ) g ( x ) ) ,x E X . Then h is continuous and is equal to 0 on E and to 1 on F. 0
+
The following is fundamental (cf. Gillman and Jerison [1960]).
2.3. Lemma. Let A be a subspace of a space X . Then we have the following: (a) A is C*-embedded in X iffevery pair of completely separate subsets of A are completely separated also in X . (b) A is C-embedded in X i f A is C*-embedded in X and is completely separated from any zero-set of X disjoint from A . Proof. (a) Assume A is C*-embedded in X. Let E and F be a pair of subsets of A and f : A + I be continuous with f = 0 on E and f = 1 on F. If g : X -, I is an extension off, then clearly g equals 0 on E and 1 on F. Hence E and F are completely separated in X , which proves the “only if” part. To prove the “if” part, assume that the condition described in (a) is satisfied. Let f be a bounded real-valued continuous map on A . Let If I < c for some c > 0, where If I = sup{/f ( x ) 1 1 x E A } . Define r,, = $c($)” for each n E N. We shall define inductively for each n E N a real-valued continuous map f,, on A so that If,[ < 3rn. Define fi = f. Then clearly Ifi I = If I < c = 3r,. Assumef, is given. Let us put En
=
{ X E
Alf,(x)
<
-r,,},
Fn = { x E A I f , ( x )
> r,,}.
Then it is easy to see that Enand F,, are completely separated in A and so by assumption they are completely separated in X . Hence, there exists a continuous real-valued map gn on X such that g, = - r,, on E,, and r, on F,, and with I g,, I < r,, . Then we define
L+.l(X)
= f,(4
- gnw, x
E
A.
Thenf,,, is continuous on A and satisfies If,+,I < 2rn = 3r,,+,.Thereforef, can be defined inductively. Let us now put g ( x ) = X.“=,gn(x)for x E X . Since I gnI < r,, X,”=,g,,(x)converges uniformly on X . Hence g defines a continuous map on X . Finally, observe that X b , g i I A = (fi - fi) + (fi - h ) + . . . + (f,- A+,) = fi f,,,. Hence we have gl A = f, because for x E A ,-
(b) Assume A is C-embedded in X . Clearly A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Let g : X + I be continuous with
Extensions of Mappings I1
53
Z = { x E Xlg(x) = O}. Then f = l/g is defined on A and continuous. By assumption take h which is a continuous extension off over X . Then clearly gh = 0 on 2 and 1 on A. Hence, A and Z are completely separated in X. This proves the “only if” part. Suppose the converse, and let f be any real-valued continuous map on A. Define g : A + (-in, in) by g(x) = tan-’(f(x)), x E A. Regard g as g : A + [-in, i n ] and by assumption take a continuous extension h : X + [-+n,+n]ofg.SinceZ = { x ~ X I l h ( x ) = I +n}isazero-setofX and 2 n A = 8, by assumption there exists a continuous map cp : X + I such that cp = 1 on A and 0 on Z. Let us now define k on X by k = tan (cph). Then clearly k is a continuous extension off over X . 0 2.4. Corollary. Every C*-embedded zero-set is C-embedded.
Here we shall give some comments about when a subspace A can be C*-embedded in a space X. Familiar and easy facts are that every compact subspace in a Tychonoff (=completely regular and T,) space is C-embedded, and that every C*-embedded pseudo-compact subspace of a space is C-embedded. By Tieze’s Extension Theorem we have the following.
2.5. Theorem. For a space X the following are equivalent. (a) X is normal. (b) Every closed subspace is C*-embedded in X . (c) Every closed subspace is C-embedded in X . Let us denote flX the Stone-Cech compactification of a Tychonoff space X . flX is shown as the compactification in which Xis C*-embedded. A fact related to Chapter 1 is that flX is the completion of a Tychonoff space X with respect to the uniformity generated by all the finite normal covers of X . Also, many basic and important facts on flX have been studied until now (see for details Gillman and Jerison [1960] and Walker [1974]). In this text we shall use, without proof, the following fact: a subspace A is C*-embedded in a Tychonoff space X iff flA = CI,,A. Lemma 2.3 can be applied to another description of C*- and C-embeddings as follows: (a) was proved in Morita and Hoshina [I9751 and (b) was proved by Gantner [ 19681.
2.6. Theorem. For a subspace A of a space X the following hold.
T.Hoshina
54
(a) A is C*-embedded in X iyfor everyfinite normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 . (b) A is C-embedded in X i f f o r every countable normal cover 4 of A there exists a normal cover Y of X such that Y A A < 4 .
Proof. (a) To prove the “if” part, assume that the condition in (a) is satisfied. Let E and F be completely separated subsets of A. By Lemma 2.2 there exist disjoint zero-sets Z , and Z , of A such that E c Z, and F c Z,. Let 4 = {X - Z,, X - Z,}. Then 4 is a binary cozero-set cover, and hence a normal cover of A by Theorem 1.2. By assumption there exists a normal cover Y of X such that Y A A < 4. Since Y is normal, there exists a locally finite cozero-set cover W of X such that W * < Y . Let us now put Z;= X - U { W E W I W ~ Z ~ = ~ }1 , 2~. T=h e n e a c h o f Z ; , i = 1,2, is a zero-set of X , and we have Z, c Z,‘ and Z , c Z;. Since W * < Y and Y A A < 4, we have Z,’ n Z; = 0. Hence by Lemma 2.2 E and F are completely separated in X . Thus, A is C*-embedded in X by Lemma 2.3 (a). Conversely, assuming that A is C*-embedded in X , let 4 = { U,,. . . , U,,} be any finite normal cover of A. Then by Theorem 1.4 there exist a cozero-set K and a zero-set 4 of A for i = 1, . . . , n such that F, c K c 17and , A = Ui&. Since A - r/; is a zero-set of A and t;; n (A - K) = 0, by assumption and Lemma 2.3 (a), there exist disjoint zero-sets Zi, and Zi, of X for i = 1, . . . , n such that t;; c Zi, and A - I( c Zi,. Let us put
w,
=
x - z,,v z,, v
. . . v Znl,
K = X - Z i 2 , i = l , . . . , n. y ,j = 0, 1, . . . , n is a cozero-set of X
Then each and W = { ylj = 0, 1, . . . , n } covers X . Hence W is a normal cover of X . Since W, n A = 0 and K n A c K c U, for i = 1, . . . ,n, we have W A A < 4, which proves the “only if” part. (b) To prove the “if” part, assume that the condition in (b) is satisfied. Then by (a) A is C*-embedded in X . Let Z be a zero-set of X disjoint from A. Letf: X + I b e a continuous map such that Z = { x E XI f ( x ) = O } . For each n E N let us put U,= {x E XI f (x) > I / n } . Then U,, is a cozero-zet of X , and since Z n A = 0, 4 = {V,, n A In E N} covers A. Hence, by assumption, there exists a normal cover -‘9 of X such that Y A A < 42. By Theorem 1.4 there exist a locally finite cozero-set cover W = { W,I a E R} and a zero-set cover 9 = {F,I a E fZ} of X such that W refines Y and Fa c Wafor a E R. Let a E R. Since 9 A A < 4,there exists an n(a) E N such that F, n A c U,(,)n A. Let us put
E, = F, n { x E XI f ( x ) 2 l/n(a)}.
Extensions of Mappings 11
55
Since { x E X l f ( x ) 2 l/n(a)} is a zero-set of X , E, is also a zero-set of X. Moreover we have
E, c W, for each a E R, E, n 2 = 8 for each c1 E R and u{E,la E R}
3
(2.1) A. (2.2)
Since W is locally finite, by (2.1) and Lemma 1.3 Z’ = U { E , I a E a} is a zero-set of A’, and by (2.2) we have A c Z’ and Z’ n Z = 0.Hence, A and 2 are completely separated in X.Thus, A is C-embedded in X by Lemma 2.3 (b). This proves the “if” part. To prove the “only if” part, suppose that A is C-embedded in X.Let 4 be a countable normal cover of A. Let *Y = { V ,I n E N} be a countable cozeroset cover of A that refines 9. Let V , = { x E A Ifn(x) > 0} with some continuous mapf, : A + I. Since A is C*-embedded in X , there is a continuous extension g, off, over X . Let us put W, = {x E X ( g , ( x ) > O}. Then W, is a cozero-set of X with W, n A = V,. Note that A = U V , c W,. Therefore, if we put 2 = X W, I n E N}, then Z is a zero-set of X and we have 2 n A = 0. By assumption and Lemma 2.3 (b) we can take a zero-set Z’ of X such that A c Z’ and Z’ n Z = 8. Let us put W = {X - Z ’ , W,ln E N}. Then W is a countable cozero-set cover and, therefore, a normal cover of X,and we have W A A c 9,which proves the “only if” part. 0
u
u{
Theorem 2.6 leads us naturally to the following definitions of P“- and P-embeddings. Let m be an infinite cardinal number. 2.7. Definition. A subspace A of a space Xis said to be P”-embedded in X if for every normal cover 9 of A with Card 9 < m there exists a normal cover Y of X such that Y A A c 4. A is said to be P-embedded in X if A is PI”-embedded in X for any m.
In view of Theorem 2.6 Puo-embedding coincides with C-embedding. Therefore we have the following implications. P-embedding P”-embedding Puo-embeddingo C-embedding
C*-embedding
For the converses it will be shown later that there exist spaces, one of which contains a closed C*-embedded but not C-embedded subspace and the other one contains a closed C-embedded but not P-embedded subspace. Moreover, concerning extensions of mappings, the following theorem makes it clear how these embeddings relate to each other; (c) is due to Morita [1975a].
T. Hoshina
56
2.8. Theorem. Let A be a subspace of a space X . Then each of the following holds. (a) A is C*-embedded in X i f f any continuous map from A into any compact AR is continuously extended over X . (b) A is C-embedded in X iff any continuous map from A into any techcomplete separable AR is continuously extended over X . (c) A is P“-embedded in X for any continuous map from A into any techcomplete AR with weight < m is continuously extended over X . (d) A is P-embedded in X @any continuous map from A into any techcomplete AR is continuously extended over X .
Proof. (c) To prove the “if” part, assume that the condition in (c) is satisfied. Let 9 be any normal cover of A of Card 9 < m. By Theorem 1.7 there exist a complete metric space T of weight < m , a continuous map f : A + T and an open cover W of T such that f - ’ ( W ) refines 9. By Theorem 1.9 there exists a Banach space L such that T is isometrically embedded in L and closed in its convex hull Z, in L and the weight of Z, < m. Let Z = C1 Z,. Then Z is also convex with weight < m and it is complete as a closed subset of L. Therefore if we consider f as the map f:A + Z, by assumption and Theorem 1.9 there exists a continuous map g : X -P 2 such that g I A = f.Since T is a complete metric subspace of Z, T is closed in Z. Hence, there exists an open cover Y of Z such that Y A T = W . Since 2 is metrizable, Y is a normal cover of Z. Hence g - ’ ( Y ) is a normal cover of X , and Y A T = W and f - ‘ ( W ) < 9 imply that g - ’ ( Y ) A A refines 9. Hence, A is P”-embedded in X. This proves the “if” part. Conversely, suppose that A is P”-embedded in X . Let Y be a cechcomplete AR of weight < m. Then there exists a normal sequence { *w; I i E N} of open covers of Y such that {St( y , K )I i E N} is an nbd base at each point y of Y , the cardinality of K < m for each i E N, and such that Y is complete with respect to { K } . Let f : A + Y be a continuous map. T h e n f - ’ ( K ) is a normal cover of A with cardinality <m. Since A is P”-embedded in X , we can inductively construct a sequence {qI}of open covers of X such that for each i E N,
9, I is a star-refinement of 9, ,
(2.3)
911 A A refines f
(2.4)
-I(%).
Therefore 0 = {qII i E N} is a normal sequence of open covers of X . Let us now apply the arguments given in Section 1 to 0. Then we can construct a
Extensions of Mappings II
57
metric space S, a continuous map cp:X + S and a normal sequence {.Y; I i E N} of open covers of S such that
cp-’(c)refines for each i E N, n St(x, cp-l(c))= St(x, a,) for each x E X , m
I=
(2.5)
XI
I
{St(t,
I=
c)I i
I
(2.6)
N} is a nbd base at each point t of S.
(2.7) Let us define a map g : cp(A) + Y by g( y ) = f ( x ) , where y = cp(x), x E A. Then, by (2.4), (2.5) and (2.6) g is single-valued, and, since for each i E N we have that A cp(A) refines g-’(“w;), g is uniformly continuous when we regard cp(A) as a uniform subspace of S with the uniformity {.Y; 1 i E N} and Y as a uniform space with the uniformity {“w; I i E N). Since Y is complete with respect to {W,},g is extended to a continuous map g : Cl,cp(A) + Y. Since Y is an AR, there exists a continuous map h : S + Y such that h (Cl,cp(A) = g. Now consider the composite h0cp:X + Y. Then we see that h cp is an extension off. This proves the “only if” part of (c). (a) Since any closed interval [a, b] in (w is compact AR, the “if” part is obvious. Using Theorem 2.6 (a), the “only if” part can be proved similarly as (c); in the present case we further take “w; so that Card “w; < KO. (b) By putting m = KO,(b) follows from (c) and Theorem 2.6 (b). (d) This is a restatement of (c) for arbitrary cardinal m. 0 E
0
As was mentioned in the introduction, Shapiro [I9661 defined the notion of P-embedding, where P”-embedding was also defined, using terms of continuous pseudo-metrics. A pseudo-metric d on a space X is said to be continuous if as a function d : X x X + R, where X x X is a product space, it is continuous. d is said to be m-separable if the pseudo-metric space ( X , d ) has a dense set with cardinality < m. The following is easy to prove. 2.9. Lemma. If d is a (resp. an m-separable) continuous pseudo-metric on a space X , then there exist a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T such that d ( x , y ) = e(cp(x),cp(x)).Conversely, f there exists a metric space ( T , e) (resp. with weight < m) and a continuous map cp : X + T, then the pseudo-metric d on Xdefined by d ( x , y ) = e(cp(x), cp( y)), x, y E X is continuous (resp. and m-separable).
Now Shapiro’s definition of P”- and P-embeddings is as follows: a subspace A of a space Xis P”-embedded in Xiff for every rn-separable continuous pseudo-metric d on A there exists a continuous pseudo-metric d on X
T. Hoshino
58
such that dl ( A x A ) = d. P-embedding is defined to be P“-embedding for every m. We shall show that the definitions of Shapiro’s P”-embedding and ours are identical. 2.10. Theorem. Let A be a subspace of a space X . Then A is P“-embedded in X in the sense of Definition 2.7 i f l A is P”-embedded in X in the sense of Shapiro.
Proof. To prove the “only if” part, assume that A is P”-embedded in X in the sense of Defnition 2.7. Let d be an m-separable continuous pseudo-metric on A . Then by Lemma 2.9 there exist a metric space (T, e) with weight < m and a continuous map cp: A + T such that d ( x , y) = e(cp(x), cp( y)). By Theorem 1.9 ( T , e) is isometrically embedded in a Banach space B and the convex hull Z of ( T , e) in B has weight < m. Let Z’ be the closure of Z. By assumption and Theorem 2.8, as is shown in the proof of Theorem 2.8, cp is extended to a continuous map $: X -+ Z’, where we regard cp as the map cp : A + 2’. Now define a pseudo-metric d on X by 4 x 9 Y ) = G($(x), $ ( Y h
x9 Y
E
x,
where 6 is a metric on B. Then clearly d is continuous. Moreover, since + I A = cp, we have for x , y E A &x,
r)
$(A) = G(cp(4, d Y ) ) e(cp(4,cp!~)) = d(x, v),
= G(+(x), =
that is, dl ( A x A ) = d as desired. This proves the “only if” part. Conversely, suppose that A is P”-embedded in X in the sense of Shapiro. Let 42 be a normal cover of A with Card 9 < m. By Theorem 1.6 there exists a locally finite cozero-set cover Y of A such that Y A= {St(x, Y ) I x E A } refines 9. Let Y = { & 1 A E A}. For each A E A take a continuous map q i : A + Z such that 6 = { x ~ A I c p ~ (>x )0). Define + A : A+ I by + l ( x ) = cp,(x)/C,,,cp,(x). Then $i is continuous, X )(IA(x)= 1 for x E A, and 5 = { x E A I t,hi(x) > O } . Let M be the set which consists of { x AI A E A} with xi E Zsuch that Z xi = 1 and x , = 0 for all but a finite number of A’s, and define a metric e ( x , y) = C I x , - y , 1 on M. Define a map : A + M by $ ( x ) = {+,(x)l A E A} for x E A . Then, similarly as in the proof of (d)*(a) of Theorem 1.2, the map $ is continuous. Moreover, it should be noted that the metric space (M, e) has weight < m. Therefore, if we put for x , y E A , d(x, y ) = e ( $ ( x ) , +( y)), then by Lemma 2.9 d is an m-separable
+
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continuous pseudo-metric on A. By assumption there exists a continuous pseudo-metric d o n X such that d ( ( A x A) = d. Let us put 9Y = { B ( x ;4)lx E X}, where B(x; 4) = { y E X l d ( x , y ) < 3). In view of Lemma 2.9 and Theorem 1.1 it is easy to see that B is a normal cover of X. Now suppose that B ( x ; f) n A # 8. Choose a point xoE B ( x ; 4) n A . Let y be any point of B ( x ; f) n A. Then we have d(xo, y ) =
d(x0,
y)
< d(x0, X)
+ d(x, JJ) < 4 + +
= 1.
Hence e($(xo), $ ( Y ) ) = 4 x 0 , Y ) < 1.
(2.8)
Let yo = {A E Alx,, E q} and y = {A E h l y E K}. Then by (2.8) we must have yo n y # 8. Therefore, there is a 1 E A such that xo,y E K . That is, y E St(xo, V ) .Thus, we have
B(x; 4) n A c St(xo, Y ) . Since Y * < 4, we have B A A < 4. Hence A is P”-embedded in X in the sense of Definition 2.7, which proves the “if” part. 0 As for when a subspace of a space is P”-or P-embedded, one sees that every compact subspace in a Tychonoff space is P-embedded and that every C*-embedded pseudo-compact subspace of a space is P-embedded, since every locally finite cozero-set cover of a pseudo-compact subspace is nothing else a finite cozero-set cover. Analogous to Theorem 2.5, we shall obtain spaces in which every closed subspace is P”-embedded. Recall that a space X is said to be m-collectionwise normal if for every discrete collection {FaI a E R} of closed sets of X with Card R < m there exists a discrete collection (G, I a E R} of open sets of X such that Fa c G, for each a E R. Clearly, X is collectionwise normal iff X is m-collectionwise normal for every m.
2.11. Theorem (Dowker [1952]). A space X is m-collectionwise normal z f f every closed subspace is P”-embedded in X . Proof. The “if” part. Let 9 = {FaI a E R} be a discrete collection of closed subsets of X with Card R < m. Since each Fa is closed and open in the closed subspace A = U{FaI a E R}, 9can be regarded as a locally finite cozero-set cover and so a normal cover of A. By assumption there exists a normal cover V of X such that Y A A < 9. Yet take a locally finite open cover W such
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60
that W * < -Y-. Let G, = St(F,, W ) for a E R. Then we have F, c G,, and easily see that {G, I a E R} is discrete. Hence X is m-collectionwise normal. This proves the “if” part. For the proof of the “only if” part, we need a lemma; this lemma will be used also in Section 4. 2.12. Lemma (Morita and Hoshina [1976]). A subspace A of a space X is P” -embedded in X if A is C-embedded in X and for every discrete collection {G,l a E R} of cozero-sets of A with Card R Q m and every collection {F, I a E R} of zero-sets of A with F, c G, for each a E R, there exists a locally Jinite collection {HaI a E R} of cozero-sets of X such that Fa c Ha n A t G,
for a
E
R.
Proof. Suppose A is P”-embedded in X . Then A is C-embedded in X . Let {G, 1 a E R} and {F,l a E R} be collections described in the lemma. Let us put
Since UaGnF, is a zero-set of A by Lemma 1.3, 9 is a locally finite cozero-set cover of A with Card 9 Q m. Moreover, note that St(F,, 9)c G , for a E R. By assumption there exists a locally finite cozero-set cover -Y- of X such that -Y- A A refines 42. We then have for V E -Ya #
p, a,
E
R
* either
Y n Fa =
8 or
Y n FB =
8.
Therefore, if we put Ha = St(F,, “f) then H, is a cozero-set of X , and the above and the locally finiteness of -Y- imply that { H ,I a E R} is locally finite. Since F, c H, n A c St(F,, 9)c G, for a E R, the “only if” part is proved. Conversely, suppose that the condition in the lemma is satisfied. Let 9 be a normal cover of A with Card 9 Q m. Then by Theorem 1.2, 9 is refined by a a-discrete cozero-set cover -Y- = UlsN of A , where = { I a E a,} is discrete with Card R, Q m. By Theorem 1.2, Y is normal. Hence by and Theorem 1.4 there exists a zero-set F,, of A such that F,, c {F,, I a E Q,, i E N} covers A . By assumption there exists a locally finite collection { H I ,I a E R,} of cozero-sets of X such that F,, c HI, n A c for each a E R,. Let us put D = u{H,,Ia E R,, i E N}. Since {H,,la E R,, i E N} is a a-locally finite collection of cozero-sets of X , D is a cozero-set of X . Note that D 3 A . Since A is C-embedded in X , by Lemma 2.3 (b) there exists a cozero-set G of X such that G n A = 0,G v D = X . Hence if we put 2 = { G } u { H I ,I a E R,, i E N}, then the above arguments show that
<
<
v,
v,
vu
Extensions of Mappings II
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.% is a a-locally finite cozero-set cover and so a normal cover of X such that .% A A < 42. Thus, A is P”-embedded in X . 0
Proof of Theorem 2.11 (continued). The “only if” part. Assume that X is m-collectionwise normal. Let A be a closed subspace of X. By Theorem 2.5, A is C-embedded in X . Let (G, I a E R} and {F,I a E R} be a discrete collection of cozero-sets and a collection of zero-sets of A with Card R < m respectively such that F, c G,, a E R. Since A is closed, {F, 1 a E R} is discrete also in X . By assumption there exists a discrete collection {K,I a E Q} of cozero-sets of X such that F, c K, for a E R. On the other hand, since A is C*-embedded in X , there exists a cozero-set L, of X such that L, n A = G,. Finally, let H, = K, n L, for a E R. Then we see that ( H , 1 a E R} is a discrete collection of cozero-sets of X , and F, c H , n A c G, for a E R. Hence, by Lemma 2.12 A is P“-embedded in X . 0 2.13. Corollary. A space X is collectionwise normal iff every closed subset of X is P-embedded in X .
Remark. Bing’s Example G (Bing [ 19513) is a normal space but not collectionwise normal. By Theorem 2.5 and Corollary 2.13 such a space contains a closed C-embedded but not P-embedded subspace.
For the case Y in Theorem 2.8 (c) ranges over all (not necessarily Cechcomplete) AR spaces with weight < m, we have the following theorem. 2.14. Theorem (Morita [1975]). Assume that A is a P”-embeddedzero-set of a space X . Then every continuous map from A into any A R with weight < m is extended continuously over X .
Proof. Let Y be an AR with weight < m and f:A + Y a continuous map. Then the proof of the “only if” part of Theorem 2.8 (c) shows that there exist a metric space T of weight < m and continuous maps g : X + T and cp : g ( A ) + Y such that f = cp ( g I A ) . On the other hand, since A is a zeroset, there is a continuous map k :X + I such that A = { x E XI k ( x ) = O } . Consider the subspace S = ( ( g ( x ) ,k ( x ) )1 x E X ) of X x Z and define a map h : X + S by h ( x ) = ( g ( x ) , k ( x ) ) for x E X. Then h is continuous. Let $ : S + T be the projection map. Then it should be noted that h ( A ) is closed in S arid g = h holds. Since Y is an AR, the composite cp 0 (I) I h ( A ) ): h(A) + Y is extended to a continuous map q:S + Y. Then we see that h o q : X + Y is the desired extension off. 0 0
$ 0
T. Hoshina
62
Remark. Theorem 2.14 suggests us to characterize a subspace A of a space X such that every continuous map from A into any AR space with weight < m is continuously extended over X. Such a subspace A is said to be M" -embedded in X after Sennott [19781; M-embedding means M" -embedding for every m. Theorem 2.14 reads that every P"-embedded zero-set is M" -embedded. Characterizations of M" -embedding were given in Sennott [1978] (see also Hoshina [1977'J)and results analogous to Morita and Hoshina [1975] were also obtained there. As for spaces in which every closed subspace is M"-(or M-) embedded, it was shown in Hoshina [1977] that, as compared with Theorem 2.12, not even paracompact spaces are such one; indeed the Michael line (see Michael [1963])contains Q ( = rational numbers) as a closed subspace which is not MKo-embedded. A remarkable result due to Sennott [1978a] is that if X is a collectionwise normal P-space in the sense of Morita [I9641( = a collectionwisenormal space whose product with any metric space is normal), then every closed subspace is M-embedded in X (for an extended result, see WaSko [1984]). We note that the converse of this result is not true because the Rudin's Dowker space (Rudin [19711) is collectionwise normal but not countably paracompact, and in which every closed subspace is shown to be M-embedded.
3. Unions of C*embedded subsets In discussing C*-embeddabilityof a given subspace, we often meet whether any union of C*-embedded subspaces is again C*-embedded or not. But, it need not be C*-embedded even for two P-embedded subspaces: 3.1. Example. For any ordinal number a let a itself mean the space of all ordnals < a with the usual order topology. Let
X =
(0,
+ 1) x
A = w1 x (wl},
(0,
+ 1) -
B =
((01,
( 0 , )x
0,>},
wI,
where wIis the first uncountable ordinal. Each of A and B is a closed, C*-embedded countably compact subspace of X. Hence, they are Pembedded in X. But the union A u B is not C*-embedded in X. In this section we shall first discuss when the union A u B of two C*-embedded subspaces A, B is C*-embedded, and secondly give results related to certain unions of infinitely many C*-embedded ones. The following two theorems explain how A u B can be C*-embedded.
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3.2. Theorem (Morita and Hoshina [1975]). Let A and B be C*-embedded in a space X . Zf for anyfinite normal cover 9 of X the cover { X - A, X - B, U I U n A n B # 8, U E 9}is refined by a normal cover of X , then A u B is C*-embedded in X . Proof. Let Y be a finite normal cover of C = A v B. Take a finite normal cover X of C such that X * < Y. Since each of A and B is C*-embedded in X , by using Theorem 2.6 twice, we can take a finite normal cover 9 of X such that 9 A A and Q A B refine X . By assumption there exists a finite normal cover Y of X such that Y < {X - A, X - B, U ( U n A n B # 8, U E a}.Let W = 9 A V .Then W is also a finite normal cover of X . We show W A C < Y. Let U E 4,V E Y with U n V n C # 8. Incaseeither V c X - A or V c X - B, then our purpose is clear. Otherwise, there exists a U' E 9 such that V c U' with U' n A n B # 8.Then there exist HI,H2 E X such that U' n A c H I and U' n B c H 2 . Hence H I n H, # 8 and consequently we have U n V n C c U' n C c H , v H, c St(H,, X ) c G for some G E Y. Hence in any case we see W 2.6, C = A v B is C*-embedded in X .
A
C < Y. Thus, by Theorem 0
3.3. Theorem (Morita and Hoshina [1975]). Let A, B be subsets of a space X which are closed in A u B. Zf A v B is C*-embedded in X , then for any normalcover@ofXthecover{X - A,X - B , U I U n A n B # ~ , U E % } is refned by a normal cover of X . Proof. Let Q be a normal cover of X . Using Theorem 1.4, we can take a locally finite cozero-set cover V = { K I 1E A}, a cozero-set cover { H AI 1E A} such that Y < 9and for some continuous maph : X -P I for 1E A we have 0 if x h(x) =
1
E
HA,
ifxEX-K.
Now, suppose that HA - St(A n B, V ) # 8 for some 1E A. Then we have 5 n A n B = 8. Define a map g A :A u B + I by g,l A = & and g, I B = 1; then g, is single valued and continuous on A u B sincef, = 1 on A n B'and each of A and B is closed in A u B. By assumption there is a continuous extension h,: X + I of g,. Then h, is 0 on HA n A and 1 on HAn B. Let us define cozero-sets HA,and HA2by HA, = { x E H, I h,(x) < $},
T. Hoshina
64
Hi, = { x E Hj.I h,(x) >
i}.Finally, let us put
W = { H i i )i = I, 2; H , - St(A n B, V ) # u {V,lV, n A n B # & P E A } .
8,
E
A}
It is easy to see that W is a locally finite cozero-set (and so a normal) cover of X that refines {X - A, X - B, UI U n A n B # 8, U E 4 } ,completing the proof. 0 Remark. In Theorem 3.3, the assumption that A and B be closed in A u B is essential, even when both A and B are C*-embedded in X . Indeed, let X = wI + 1, A = wl, B = {wl}. Then A, B and A u B = X are C*embedded in X. But {X - A, X - B) cannot be refined by a normal cover; if otherwise B would be a zero-set of X , a contradiction. 3.4. Corollary. Let A and B be C*-embedded in X . Then A u B is C*embedded in X i f f A u B is C*-embedded in Cl(A v B) and for any finite normal cover 4 of X the open cover { X - C1 A, X - C1 B, UI U n C1 A n C1 B # 8, U E 4 } is normal.
Remark. In case X is Tychonoff, our condition that for any finite normal cover 4 of X {X - A, X - B, UI U n A n B # 8, U E 4 } is refined by a normal cover is equivalent to Cl,,(A n B) = Cl,,A n Cl,,B; this can be proved similarly as the proof of Theorem 2.35 of Chapter 1. The/following two theorems assert that for A u B to be C- or P"embedded the essential is C*-embeddability of A v B. 3.5. Theorem (Morita and Hoshina [1975]). Suppose A and B be Cembedded in X . Then A u B is C-embedded in X i f f A u B is C*-embedded in X.
Proof. We only have to prove the "if" part. Assume A u B is C*-embedded in X. By Theorem 2.3 (b) it suffices to prove that if Z is a zero-set of X with (A u B) n Z = 8 then A u B and Z are completely separated in X. Take such a zero-set Z. By assumption and Theorem 2.3 (b) there exist zero-sets Z, and'Z, of X such that Z, 3 A, Z, n Z = 8 and Z , B, Z , n Z = 8. Then Z' = Z, u Z , is a zero-set and we have A u B c Z', Z' u Z = 8. Hence A u B and Z are completely separated in X . 0 =)
Extensions of Mappings II
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3.6. Theorem (Morita and Hoshina [1975]). Suppose A and B be Pmembedded in X. Then the following are equivalent. (a)
A u B is P"-embedded in X .
(b)
A u B is C-embedded in X .
(c)
A u B is C*-embedded in X .
Proof. We shall prove (b)*(a) since (a)*(b) is obvious and (b)*(c) follows from Theorem 3.5. Assume A u B is C-embedded in X . Let 9 be a normal cover of A u B of Card 9 < m. Let Y be a locally finite cozero-set cover of A u B with Card Y < rn that refines @. By assumption, for every V E Y there is a cozero-set 9of X such that v n (A u B) = V.On the other hand, since each of A and B is P"-embedded in X , there exists a locally finite cozero-set cover .W of X such that .W A A < Y and .W A B < Y . For any H E .W with H n A # 0,choose V,, E Y so that H n A c V,, and put W,, = H n Let W-, = { W,l H E S,H n A # 0).Then W-,is a locally finite collection of cozero-setsof X and we have UW-, 3 A and W-, A ( A u B) refines Y .Similarly let us construct WBfor B. Let K = UW-,u UWB.Then K is a cozero-set which contains A u B. By assumption and Theorem 2.3 (b) there exists a cozero-set L of X such that L n ( A u B) = 8 and K u L = X . Now let us put W = {L} u WAu WB.Then the above shows that W is a locally finite cozero-set (and so a normal) cover of X such that W A ( A u B ) < Y < 9.Hence A u B is P"-embedded in X . 0
v,,.
3.7. Corollary. Suppose A and B be P-embedded in X . Then thefollowing are equivalent. (a)
A u B is P-embedded in X .
(b)
A u B is C-embedded in X.
(c)
A u B is C*-embedded in X
Remark. (1) By the same method one sees that Theorems 3.5, 3.6 and Corollary 3.7 are also valid for a finite union A, u . * . u A,. (2) A subspace A is z-embedded in a space X if any zero-set Z of A can be written as Z = Z' n A with some zero-set Z' of X (see Alo and Shapiro [1974]). C*-embedding implies z-embedding. It is not hard to see that A is C-embedded in X iff A is z-embedded in X and is completely separated from any zefo-set of X disjoint from A. In view of this, as R. L. Blair noted (cf. Morita and Hoshina [1975]), in Theorem 3.5 and 3.6 and Corollary 3.7 C*-embedding of A u B can be weakened to z-embedding.
T. Hoshina
66
We shall proceed to prove applications of results above, the first of which will be used also in Section 5. 3.8. Theorem (Morita and Hoshina [19751). Let A and B be C-embedded in X. If one of A and B is a zero-set of X,then A u B is C-embedded in X.
Proof. Let 9 be a finite normal cover of X. Let Y be a finite cozero-set cover of X that refines 9. Assume A is a zero-set. Then, if we put G = (X - A ) u U { V l V n A n B # 8, V E Y},Gisacozero-setcontaining B. Since B is C-embedded, there is a cozero-set H such that H n B = 8 and H u G = X.Thisimplies that {X - A, H, VI V n A n B # 8, V E Y }is a finite cozero-set (and so a normal) cover of X, which refines {X - A, X-B, V J V n A n B f 8 , V~Y}.HencebyTheorem3.2AuBis C*-embedded in X,and hence by Theorem 3.5 it is C-embedded in X. 0 3.9. Corollary. Let A and B be P"-embedded in'X. If one of A and B is a zero-set of X,then A u B is P"-embedded in X. Since a C*-embedded zero-set is C-embedded (2.4), we have the following corollary. 3.10. Corollary. Let A and B be C*-embedded zero-sets of X. Then A u B is C-embedded in X. For the case of C*-embedding, a theorem analogous to Theorem 3.8 does not hold.
+
3.11. Example. Let X = (aI 1) x BN - {a1} x (BN - N), A = a1x (BN - N ) and B = {a1} x N. Observe that BX = (aI 1) x BN, PA = (a1+ 1) x (BN - N ) = Cl,,A, BB = {a1}x /IN = Cl,,B. Hence A and B are C*-embedded in X. A is a countably compact zero-set of X,and hence A is a P-embedded zero-set of X. On the other hand, it is easy to see that A u B is not C*-embedded in X.
+
For the case of finite unions we have the following theorem. 3.12. Theorem. Let A l , . . . , A,, be C*-embedded (resp. C-embedded or P"-embedded) subsets of a space X. If any of A, u Aj (1 < i, j d n) is C*-embeddedinX,then A l u . . . u A,, is C*-embedded(resp. C-embeddedor P"-embedded) in X.
Extensions of Mappings 11
61
Proof. Let B, and B, be completely separated subsets of A = A , u . * u A,.
For each pair (i,j), 1 < i, j < n, by assumption and Lemma 2.3 (a) there exist disjoint zero-sets Z,(i,j ) and Z,(i, j ) of X such that Bk n ( A i u A j ) c Zk(i,j ) , k = 1, 2. Here, we may assume Z,(i, j ) = Z k ( j ,i ) . Let us put for 1 < i < n , k = 1,2,
Z k ( i ) = Z,(i, 1) n . . * n Z,(i, n). Then Z,(i) is a zero-set of X, and we have
Bk n A ,
c Z,(i),
Z , ( i )n Z,(j) =
k = 1, 2,
0,
1
< i, j < n.
-
Hence, if we put Z, = Z k ( l )u . u Zk(n),then Z, is a zero-set of X and we have that B, c Z,, k = 1,2, and 2, n Z, = 0.Thus, by Lemma 2.2, B, and B, are completely separated in X. Hence A is C*-embedded in X. The theorem has been proved for the case of C*-embedding. From this case together with (1) of the remark following Corollary 3.7 the other cases follow 0 easily. For infinite unions any corresponding results to those obtained above is not true, even for a countable union UieNAi such that each Ai is a one-point (and so P-embedded) set and {Ail i E N} is discrete.
3.13. Example. Let X and B be as in Example 3.11. Then B is a union of {(q, n)} and {{(q, n)} In E N} is discrete. But B is not C-embedded in X. 3.14. Example. Let X be the Niemytzki Space (= the space R, in Nagata [1984]), that is, X is the subset {(x, y) E R2I y > 0} of the Euclidean plane Rz, with the topology: nbds of ( x , y) with y > 0 are those usual ones in R2, and basic nbds of points z = (x, 0) are of the form {z} u { ( x ’ , y ’ ) E XI ( x - x’)* + ( E - y’)’ < E ’ } , E > 0. Then each point of X is a zero-set. Let A = { ( x , O ) I X E Q}, B = { ( x , O ) I X E P}, where Q = the set of rationals, P = the set of irrationals. Then B is a zero-set of X , while A is not. Consider disjoint subsets Q,and Q, of Q, each of which is dense in the subspace Q in R. Then Q,x (0) and Q, x {0} are obviously completely separated in A, but not in X. Hence A is not C*embedded in X. Similarly B is not C*-embedded in X.
T. Hoshina
68
3.15. Example. Let X be the space A given in Gillman and Jerison [1960, p. 971, that is, A = fiR - (/?N - N). Then each point of N is a zero-set of A and { { n } 1 n E N} is discrete. N is C*-embedded in A since fiA = flR and Cl,, N = Cl,, N = /?N.On the other hand, since A is pseudo-compact, N can not be C-embedded. We shall give a convenient notion for the union of a locally finite collection of C*-embedded subsets to be C*-embedded.
3.16. Definition (Morita [1980], Ohta [1977]). A collection d of subsets of a space X is said to be uniformly locallyfinite if there exists a normal cover Q of X such that each member of Q intersects at most finitely many members of d. 3.17. Theorem (Morita [1980], Ohta [1977]). A collection d = { A, 11 E A} of subsets of a space X i s uniformly locally finite fi there exist a cozero-set GI and a zero-set Z , of X for A E A such that A, c Z, c G, and { G II 1 E A} is locally finite. Proof. Suppose d is uniformly locally finite. Let 9 be a normal cover such as given in Definition 3.16. Take a locally finite cozero-set cover Y = {V.I a E R} and a zero-set cover 8 = {EuIa E R} of X such that Y * c Q and Eu c V,, a E R. Let us put GI = St(A,, Y ) ,
Z , = St(A,, 8).
Then G, is a cozero-set and by Lemma 1.3, Z, is a zero-set and we have A, c Z, c G,. Let I/ E Y .Then St(V, Y ) c U for some U E Q. Let y be a finite subset of A such that U n A, # 8 implies A E y. Then we have V n G, # 8 1 E y. Thus, {G,} is (uniformly) locally finite. Conversely, assume that G2 and Z,, 1 E A described in the theorem exist. For each finite y c A let W, = n l s y G , n ( X - U { Z , I A $ y } ) , and put W = { F l y is a finite subset of A}. Then by Lemma 1.5 W is a locally finite cozero-set (and so a normal) cover of 1.Since A , n 6 # 8 implies 1 E y , d is uniformly 0 locally finite. With the aid of uniformly local finiteness Theorem 3.12 is extended to the following theorem.
3.18. Theorem (Morita [1980]). Let GI = {A, I 1 E A} be a uniformly locally finite collection of C*-embedded (resp. C-embedded or P”-embedded) subsets of a space X . I f A n v A,, is C*-embedded in X for each A. p E A, then the union ud is C*-embedded (resp. C-embedded or P“-embedded) in X .
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69
Proof. Let 42 be a finite normal cover of U d . By assumption there exists a normal cover 9 = (G, I a E R} of X such that each a E R admits a finite subset y, of A with A, n G, # 8 = A E y,. We may assume B is a locally finite cozero-set cover. Let a E R. By assumption and Theorem 3.1 1 B, = U { A , l A E y,} is C*-embedded in X. Hence by Theorem 2.6 (a) there exists a locally finite cozero-set cover V, of X such that V; A B, refines 42. Let us put = ( V n Gal V E a E nt.
c,
Then V is a locally finite cozero-set (and so a normal) cover of X . Let a E R and V E Va.Then V n G, n ( U d ) = V n G, n B, c V n B,. Hence V A ( U d ) refines 42. Thus, U d is C*-embedded in X by Theorem 2.6 (a). Using Theorem 3.12, in view of Theorem 2.6 (b) and Definition 2.7, the other cases can be proved similarly. 0 3.19. Corollary. Let d be a ungormly locallyfinite collection of C*-embedded (resp. P"-embedded) zero-sets of X . Then U d ' i s a C*-embedded (resp. P'"-embedded)zero-set of X . Proof. By Corollary 3.10 Theorem 3.18 implies that U d is C*-embedded (resp. P"-embeeded) in X. On the other hand, by Lemma 1.4 and Theorem 3.17, U d is a zero-set. 0 4. C*-embedding in product spaces As we have learned in Section 2, C*-embedding (resp. C-embedding or P"-embedding) is precisely the notion of extending normal covers of cardinality < KO(resp. < KOor < m) of a subspace to normal ones of the whole space. On the other hand, we know that normal (resp. countably paracompact normal or m-paracompact normal) spaces are precisely those ones of which every open cover of cardinality < No (resp.
Theorem A (Dowker [1951]). equivalent.
For a Hausdorff space X the following are
(a)
X is countably paracompact normal.
(b)
X x Y is normal for any compact metric space Y
(c)
X x I is normal.
T. Hoshina
I0
Theorem B (Morita [1962]). For a Hausdorflspace X the following are equivalent. (a)
X is m-paracompact and normal.
(b)
X x Y is normalfor any compact Hausdor-space Y of weight
(c)
X x I" is normal.
< m.
Being motivated by these theorems, we shall consider in this section the problem of A x Y being C*-embedded in X x Y for a C*-embedded subspace A of a space X and a compact HausdorfT space Y. This problem was first studied by Alo and Sennott [1972] and afterward by Morita and Hoshina [ 19761 and Przymusinski [ 19781. First of all we note the following proposition. The proof is easy and has been omitted. 4.1. Proposition. For a subspace A of a space X and a space Y if A x Y is C*-embedded in X x Y , then A is C*-embedded in X . The following theorem, which was proved in Morita [1975], plays an essential role; for the proof see Chapter 3. 4.2. Theorem. Let X be a space and Y a compact Hausdorfspace. Let 9 be a normal cover of X x Y. Then there exist a normal cover Q = { U, I A E A} of X and a collection {V,I A E A} offinite open covers of Y such that the cover { V i x V J V E K , I E A } ~ ~Y rXe fXi n e s Y a n d C a r d Q <max{CardY, weight of Y } . Now the first theorem we shall prove is the following.
4.3. Theorem (Alo and Sennott [1972]). Let A be P"-embedded in a space X . Then A x Y is P"-embedded in X x Y for any compact Hausdorfspace Y with weight G m .
Proof. Let Y be a compact Hausdorff space with weight G m . Let Q be a
normal cover of A x Y of Card Y < m. By Theorem 4.2 there exist a normal cover 6 = { V, I A E A} of A with Card A < m and finite open covers VA, I E A of Y such that { V , x VI V E VA,I E A} refines Y. Here, we may assume that each set in VA is a cozero-set of Y since Y is normal. Now since
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A is P"-embedded in X , by using a technique as in the proof of Theorem 1.4 we can take a locally finite cozero-set cover W = { W, I 1 E A} of X such that W , n A c U,foreach1EI\.Let% = { W , x V I V E Y , , ~ E A } . T ~ ~ ~ ~ is a locally finite cozero-set cover of X x Y and we have % A ( A x Y ) refines 3. Hence A x Y is P"-embedded in X x Y. 0 In Section 2 we noted that every compact subspace in a Tychonoff space is P-embedded. But we obtain a stronger version with the following lemma. Using Theorem 4.2 the proof is straightforward and left to the reader.
4.4. Lemma. Let B be a compact subspace of a Tychonoff space Y. Thenfor any space X X x B is P-embedded in X x Y. The following lemma, which is another characterization of P"-embedding, is needed to prove Lemma 4.6 below. 4.5. Lemma (Morita and Hoshina [1976]). Let A be C*-embedded in a space X . Then A is P"-embedded in X iff for every normal cover Q of A with Card Q < m there exists a normal cover Y of X such that each set V n A , V E Y is contained in a union of ajinite number of sets in Q.
Proof. We only have to prove the "if" part. Let 3 = {G,Ia E R} be a normal cover of A with Card R < m. 3 may be assumed to be a locally finite cozero-set cover of A . Then there exist a cozero-set cover { H , I a E R} of A and a continuous map& : A -P Z for a E R such thatf, equals 0 on Ha and 1 on A - G,. Let g, :X + Z be an extension off, and put for a E R K,
= {XE
Xlg,(x)
=
0},
La
= {XE
X l g , ( x ) < l}.
Then K, is a zero-set and L, a cozero-set of X . On the other hand, for { H , I a E R}, there exists by assumption a locally finite cozero-set cover { U, I 1 E A} of Xwith the property in the lemma. For each 1 E A let us choose a finite subset y, of R so that U,n A c U { H ,I a E y l } . Then we have K, c La for a E R, H, c K, n A c La n A c G, for a E R, U, n A c U{K,Ia E y,}
(4.1) (4.2)
for 1 E A.
(4.3)
Let us put Y- = { u, n Lu I a E Y, 1 E A} u { u, n ( X 9
Uucyl
Ka) I 1 E A}.
Then by (4.1) Y is a locally finite cozero-set cover of X and by (4.2) and (4.3) Y A A refines 9.Hence A .is P"-embedded in X. 0
T. Hoshina
12
4.6. Lemma. Let A be a subspace of a space X and Y a non-discrete compact Hausdorff space. If A x Y is C*-embedded in X x Y, then A is C-embedded
in X .
Proof. Suppose A x Y is C*-embedded in X x Y. By Proposition 4.1 A is C*-embedded in X . Let Q = {G, I n E N} be a countable normal cover of A; assume each G, is a cozero-set of A. Since Y is nondiscrete Hausdorff, Y contains an infinite discrete subspace B = { y,ln E N}. Let us put
< n} I n E N}, (C1 B - { y i l i < n } ) I n E N}.
H , = U{G, x { yil i H, = U{G. x
Then each of HI and H , is a cozero-set of A x C1 B since each pointy. of B is isolated in C1 B, and it is easy to see that { H , , H , } covers A x c1 B. By Lemma 4.4, A x C1 B is C*-embedded in A x Y, and hence by assumption A x C1 B is C*-embedded in X x C1 B. Hence,. by Lemma 2.6 (a) and Theorem 4.2 there exist a locally finite cozero-set cover Q = { U, I I E A} of XandfiniteopencoversV,,I E AofC1 Bsuchthat{(U, n A) x V ( V E VA, I E A} < { H I ,H , } . Suppose that U, n A # 8. Since C1 B - B # 8, for some V E V, (C1 B - B) n V # 8. This implies (U, n A) x V c H , . Moreover, since Vcontains a y, of B, we have then U, n A c UiGnG,. Thus, by Lemma 4.5 A is C-embedded in X. 0 4.7. Corollary. A subspace A is C-embedded in a space X embedded in X x I.
iff
A x I is C*-
We shall now establish Theorem 4.9 below which is the main theorem in this section. Before proving this we state a lemma, where y denotes the initial ordinal with Card m. Let Y be a Tychonoff space of weight m. Thenfor each u < y there are subsets A , , B,, U, and V, of Y such that
4.8. Lemma.
(a)
A , and B, are zero-sets and A , c U, and B, c V,,
(b)
U, and V , are cozero-sets and disjoint,
(c)
Vg <
a, either A , Q U, or B, Q
V,.
In case it is only required that each of A, and B, be closed and each of U, and V, be open, Lemma 4.8 was proved by Starbird (cf. Rudin [1975]) (see, Chapter 4) and his proof can be modified easily so as to yield the lemma.
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Extensions of Mappings I1
4.9. Theorem (Morita and Hoshina [1976], Przymusinski [1978]). Let A be a subspace of a space X and Y a compact Hausdorfspace of weight m. Then A is P”-embedded in X i f f A x Y is C*-embedded in X x Y.
Proof. We shall prove the “if” part since the “only if” part follows readily from Theorem 4.3. Suppose that A x Y is C*-embedded in X x Y. We shall prove the theorem by applying Lemma 2.12. First let us note that A is C-embedded in X by Lemma 4.6. Let {G, I u < y} be a discrete collection of cozero-sets of A with Card y = m and {F,I u < y } a collection of zero-sets of A such that F, c G, for each u < y. Let A,, B,, U, and V , be subsets of Y with the properties described in Lemma 4.8. Let us put
Since F, x A, is a zero-set and G, x U, is a cozero-set of A x Y, F, x A , c G, x U, for tl < y and { G, x U , 1 u <’ y } is discrete, by Lemma 1.3, Z, is a zero-set of A x Y. This goes similar for Z , , and we see Z, and Z , are disjoint. Also, we have the disjoint zero-sets Z;
=
u { F , x B,[u< y } ,
< y}.
Z; = u { F , x ( Y -
By assumption and Theorems 2.6 (a) and 4.2 there exist a locally finite cozero-set cover A = { M ,I 1 E A} of X and finite open covers Mj,,I E A of Y such that { ( M , n A) x N ( N E Ni, 1E A} refines each of {A x Y - Z , , A x Y - Z , } and {A x Y - Z ; , A x Y - Z ; } . Then for a set Mj,of A we have (4.4) Mi n F, # 8 and Mi. n Fp # 8 =. u = /?. To see this, suppose /? < u. Then by Lemma 4.8 either A, - U p # 8 or B, - 5 # 8. For example, let A, - Up # 8. Then for some N in Mj, (A, - U p )n N # 8. We have then
((Mi n A ) x N ) n Z ,
= ((Mj.n A ) x
((Mi. n A ) x N ) n Z ,
3
N ) n (F, x A , ) #
8,
((M;.n A ) x N ) n (Fp x ( Y - U s ) #
0.
This contradicts that { ( M i n A ) x N ( N E Mi, I E A} refines {A x Y - Z , , A x Y - 2,). Thus, (4.4) is proved. Let us put H , = St(K,, A) for u < y. Then by (4.4) we see that { H , 1 u < y } is locally finite, and each Ha is a cozero-set of X containing F, . Since A is C*-embedded, there is a cozero-set c, of X such that G, n A = G, . Then { H , n G, I u < y } is a locally finite collection of cozerosets of X , and we have F, c H , n G, n A c G, for u < y. Therefore by Lemma 2.12 A is P” -embedded in X. 0
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T. Hoshina
4.10. Corollary. A subspace A is P"-embedded in a space X if A x I" is C*-embedded in X x I". Remark. It should be noted that Theorem 4.9 has been proved for a given compact Hausdorff space Y of weight m . As for normality of product spaces the corresponding result is not true. For, let A ( w , )be the one-point compactification of the discrete space of Card w , . Then weight of A ( w , ) is w , . It is known that wl x A ( w , ) is normal, but w , is not w,-paracompact. 4.11. Corollary (Starbird [1974]). Let Y be a compact Hausdorfspace with weight m. Then a space X i s m-collectionwise normal ifffor every closed set A of X A x Y is C*-embedded in X x Y . 4.12. Corollary (Rudin [1975]). If X x Y of a space X with a compact Hausdorflspace Y of weight m is normal, then X is m-collectionwise normal. In the results above we have been concerned with products with a compact factor. As for products with a metric factor Przymusinski [I9831 obtained several results. On such a product the following example is interesting, which asserts that the corresponding result to Theorem 4.3 does not hold.
4.13. Example (Morita [1977]). There exist a paracompact Hamdorff space X , a closed subspace A of X (hence it is P-embedded in X) and a metric space Y such that A x Y is not C*-embedded in X x Y. Indeed, let M be the Michael line given in Michael [1963]. Then as is proved there, Q is a closed set of M and M x P is not normal because Q x P and A(P) = {(x, x) I x E P} cannot be separated by open sets. Note that A(P) is a zero-set of M x P. Hence by Theorem 2.3(b) Q x P is not C-embedded in M x P. Let K be a non-discrete compact subset of P. Then by Theorem 4.9, Q x P x K is not C*-embedded in M x P x K, and by Lemma 4.4, Q x P x K is C*-embedded in Q x P x P. Hence Q x P x P is not C*-embedded in M x P x P. Since P x P = P, Q x P is not C*-embedded in M x P. Concerning products with a metric factor the following problem discussed in Przymusinski [1983] is still open.
4.14. Problem. Let Y be a non-discretemetric space. I f A x Y is C*-embedded in X x Y , must A x Y then be C-embedded in X x Y ? Finally, we shall discuss spaces X such that for any closed set A of X A x Y is C*-embedded in X x Y for any space Y.
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4.15. Theorem (Morita [1977]). Let X be a locally compact paracompact Hausdorfspace and A its closed subspace. Then A x Y is C*-embedded in X x Y for any space Y . Proof. By assumption on X there exist collections Y = {Galu E R} of cozero-sets, 9 = {F,I u E R} of zero-sets and V = {C, I u E f2} of compact subsets of X such that C, c F, c G,, u E R, V covers X and Q is locally finite. Let B, = (A n C,) x Y . Then by Lemma 4.4 for any u, B E R B, u B, = ( A n (C,u C,))x Y is C*-embedded in X x Y. Since B, c F, x Y c G, x Y for u E i2, by Theorem 3.17 {B,} is uniformly locally finite. Thus by Theorem 3.18 U { B , I u E f2} = A x Y is C*-embedded in X x Y . In case X being a metric space Theorem 4.15 is also valid; the proof is omitted because of its length.
4.16. Theorem (Michael, see Starbird [1974]). Let X be a metric space and A its closed subspace. Then A x Y is C*-embedded in X x Yfor any space Y . Remark. (1) As a common generalization of locally compact paracompact spaces and metric spaces, we know paracompact M-spaces (Morita [19641). WaSko [ 19841 showed by an example that in case Xis a paracompact M-space the same result as above need not hold. (2) Theorem 4.16 has been generalized by Fujii [1984] to the case X is an M 3 (= a stratifiable) space. 5. Homotopy extension property Let (X, A) be a pair of a space X and its subspace A, and let Y be another space. Then ( X , A) is said to have the homotopy extension property (abbreviated HEP) with respect to Y if every continuous map f:Xx{O}uAxZ+Y
can be continuously extended over X x Z. It is well known that the original Borsuk's homotopy extension theorem (cf. Hu [1965]) was generalized by Dowker [1956]:
Theorein A. Let Y be a tech-complete separable ANR space. Thenfor every closed subspace A of a countably paracompact normal space X ( X , A ) has the HEP with respect to Y.
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T. Hoshina
Theorem B. Let Y be an ANR. Then for every closed subspace A of a countably paracompact, collectionwise normal space X ( X , A ) has the HEP with respect Y , if either A is a Gs-set of X or Y is tech-complete. In this section, as an application of results obtained in preceding sections, we will show that Dowker's theorems above can be generalized to the case A being C-embedded or P-embedded in X , and conversely that the HEP of ( X , A ) with respect to an arbitrary Y as in theorems above implies C- or P-embeddability of A , under discussions based on Morita [1975a] and Morita and Hoshina [1975]. The following theorem is fundamental and gives a characterization for ( X , A) has the HEP with respect to an arbitrary (5ech-complete ANR of weight < m; (a)=>(d) was previously proved by Morita [1975a]. 5.1. Theorem (Morita and Hoshina [1975]). For a subspace A of a space X the following are equivalent. (a)
A is P"-embedded in X .
(b)
X x B u A x Y is €'"-embedded in X x Y for every compact Hausdorff space Y of weight < m and its closed subset B.
(c)
X x {0} u A x I is P"-embedded in X x I.
(d)
( X , A ) has the HEP with respect to any tech-complete ANR of weight < m .
Proof. (a)*(b). Assume (a). Then by Theorem 4.3 A x Y is P"-embedded in X x Y. By Lemma 4.4 X x B is P"-embedded in X x Y. Let % be a finite normal cover of X x Y. Let V be a finite cozero-set cover of X x Y that refines %. Let us put G = St(A x B, V ) .Then G is a cozero-set of X x Y. Letf : X x Y + I be a continuous map such that G = ((x, y ) If (x, y ) > O}. Definef ': X + I byf"(x) = inf{f(x, y ) I y E B } for x E X . Since the projection : X x B + Xis open and closed, one can prove thatf' is continuous. If we put H = { x E X l f ' ( x ) > 0}, then H is a cozero-set of X such that A x B c H x B c G. Since A is C-embedded in X , there is a cozero-set W of X such that A n W = 0, W u H = X . Since X x B is C-embedded in X x Y , there is a cozero-set L of X x Y such that L n ( X x B ) = 0, L u ((W x Y ) u G ) = X x Y. Now we have the cover { L , W x Y , V 1 V n (A x B ) # 0, V E Y } , which is a finite cozero-set cover of X x Y refining { ( X - A) x Y , X x ( Y - B), UI U n (A x B ) # 0,UE%}.Thus, byTheorem3.2X x B u A x Y
Extensions of Mappings 11
77
is C*-embedded in X x Y, and hence by Theorem 3.6 it is P"-embedded in x x Y. (b)*(c). This is obvious. (c)*(a). Assume (c). Let %! = { U ,12. E A) be a locally finite cozero-set cover of A with Card A < m. Let us put V = {U; x
(5,
l]lA E A} u { X x ( 0 ) u A x [0, 5)).
Then it is easy to see that V is a locally finite cozero-set cover of X x (0) u A x I . By (c) there exists a normal cover W of X x I such that W A ( X x ( 0 ) u A X I )< V . L e t u s s e t K = W A ( X x { I ) ) . T h e n K anormalcoverofX x {l}such that% A ( A x {I}) < V A ( A x (1)) = { U, x { 1) 11 E A}. This implies that A is P"-embedded in X . (d)*(c). This follows readily from Theorem 2.8 (c). (c)*(d). Assume (c). Let Y be a Cech-complete ANR of weight < m, and let f:( X x ( 0 ) ) u A x I + Y is any continuous map. By Theorem 1.9 there is a Banach space L, in which Y is embedded as a closed subset and the closed convex hull Z of Y in L has weight < m. Since Y is an ANR, there is an open subset U of Z with a retraction r : U + Y. By assumption and Theorem 2.8 (c) there exists an extension g : X x I + Z off. Since U is a cozero-set of Z , g - ' ( U ) is also a cozero-set of X x I containing A x I . Then as above there is a cozero-set H of X such that A x I c H x I t g - ' ( U ) . Since A is C-embedded in X by (c)-(a), there exists a continuous map cp : X x I such thatrp= 1 o n A a n d O o n X - H . D e f i n e h : X x I - , Z b y h(x, t ) = g ( x , cp(x)t), ( x , 1) E X x I .
Then h is continuous, and it is easy to see that h I ( X x { 0 } u A x I ) = f and h(X x I ) c U . Therefore, the composite ro h : X x I + Y is now a desired 0 extension off. 5.2. Corollary (Morita and Hoshina [1975], Morita [1975a]). For a subspace A of a space X the following are equivalent.
(a)
A is C-embedded in X .
(b)
X x B u A x Z is C-embedded in X x Z for every compact metric space Z and its closed subset B.
(c)
X x ( 0 ) u A x I is C-embedded in X x I .
(d)
( X , A ) has the HEP with respect to every separable tech-complete ANR.
78
T. Hoshina
Remark. (1) (a)*(b) of Corollary 5.2 is also implied by Theorem 3.8 since in this case X x B is a zero-set of X x Z. (2) (a)*(b) has been proved by Starbird [1975] in case Y is [0, 11 or (0, 1). 5.3. Corollary (Miednikov [1975]). Let A,, . . . , A, be closest subsets of a normal space X , and let B,, . . . , B,, be closed in I. Then A, x B, v ‘ . . v A, x B, is C-embedded in X x I. Proof. Since each Ai x Bi is C-embedded in X x I, by Theorem 3.12 the proof is sufficient for the case n = 2. Also, we may assume that X = A , u A , a n d I = B, v B,. Let C = A , x B, v A, x B,,andletfbea real-valued continuous map on C. Since C , = A, x B, v (A, n A,) x I is C-embedded in A , x I by Corollary 5.2,fJ C , is extended to g , over A , x I. Similarly f l C2 is extended to g , over A, x I, where C, = A2 x B2 u (A, n A2) x I. Then clearly the map g on X x Idefined by g = g, on A , x I and g = g, on A, x I is continuous and an extension off.
If, in addition, A is a zero-set of X , by using further the same method as in the proof of Theorem 2.14, the following theorem, generalizing Theorem B, can be proved similarly to Theorem 5 . I . 5.4. Theorem (Morita [1975a]). Let A be a P”-embedded zero-set of a space X . Then ( X , A) has the HEP with respect to every ANR of weight <m. Remark. Applying the methods in Morita [1975a] and Morita and Hoshina [1975] to the notion of M”-embedding, Sennott [1978] obtained that ( X , A) has the HEP with respect to every ANR of weight < m iff A is M”-embedded in X .
References AlO, R. A. and L. I. Sennott
[I9721 Collectionwise normality and the extensions of functions on product spaces, Fund. Maih.76, 231-243. Aid, R. A. and H.L. Shapiro [I9741 Normal Topological Spaces, Cambridge.
Bing, R.,H. [1951] Metrization of topological spaces, Canad. J . Math. 3, 175-186. Dowker, C. H. [I9511 On countably paracompact spaces, Canad. J . Math. 3, 219-224.
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[I9521 On a theorem of Hanner, A r k . f . Mar. 2, 307-313. [I9561 Homotopy extension theorems, Proc. London Math. SOC.6, 100-1 16. Dugundji, J. [1951] An extension of Tietze’s theorem, PaciJic J. Math . 1, 353-367. Fujii, S. [I9841 Ir-embedding and Dugundji extension theorem, MSc thesis, University of Tsukuba. Gantner, T. E. [I9681 Extensions of uniformly continuous pseudometrics, Trans. A M S 132, 147-1 57. Gillman, L. and M. Jerison [1960] Rings of Continuous Functions, New York. Hoshina, T. [I9771 Remarks on Sennott’s M-embedding, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 13, 284-289. Hu, S. T. [I9651 Theory of Retracts, Detroit. Michael, E. [I9631 The product of a normal space and a metric space neid not be normal, Bull. A M S 69, 375-376. Miednikov, L. E. [I9751 Embeddings of bicompacta into the Tychonoff cube and extension of mappings from subsets of products (in Russian), Dokl. Akad. Nauk SSSR 222, 1287-1290. Morita, K. [19621 Paracompactness and product spaces, Fund. Math. 50, 223-236. [1964] Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. I19701 Topological completions and M-spaces, kci. Rep. Tokyo Kyoiku Daigaku, Sect. A 10, 27 1-288. [I9751 Cech cohomology and covering dimension for topological spaces, Fund. Math. 87, 3 1-52. [1975a] On generalizations of Borsuk’s homotopy extension theorem, Fund. Math. 88, 1-6. [I9771 On the dimension of the product of topological spaces, Tsukuba J. Math. 1, 1-6. [1980] Dimension of general topological spaces, in: G. M. Reed, Ed., Surveys in General Topology, 297-336, New York. Morita, K. and T. Hoshina [1975] C-embedding and the homotopy extension property, General Topology Appl. 5,69-81. [1975a] Review of Morita and Hoshina [1975], Zentralbl. Math. 2%, #54014. [I9761 P-embedding and product spaces, Fund. Math. 93, 71-86. Nagata, J. [19851 Modern General Topology, Amsterdam. Ohta, H. [I9771 Topologically complete spaces and perfect maps, Tsukuba J. Math. 1, 77-89. Przymusinski, T. C. [19781 Collectionwise normality and extensions of continuous functions, Fund. Math. 98, 75-8 1. [I9831 Notes on extendability of continuous functions from products with a metric factor, Manuscript.
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Rudin, M. E. [I9711 A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. [I9751 The normality of products with one compact factor, General Topology Appl. 5,45-59. Sennott, L. 1. [I9781 On extending continuous functions into a metrizable AE, General Topology Appl. 8, 219-228. [1978a] Some remarks on M-embedding, Topology Proc. 3, 507-520. Shapiro, H.L. [I9661 Extensions of pseudometrics, Canad. J . Math. 18, 981-998. Starbird, M. (19741 The normality of products with a compact or a metric factor, PhD Thesis, University of Wisconsin. [I9751 The Borsuk homotopy extension theorem without the binormality condition, Fund. Math. 87,207-2 I 1. Walker, R. C . [I9741 The Stone-tech CompactiJication, Berlin. Waiko, A. [I9841 Extensions of functions defined on product spaces, Fund. Math. 124, 27-39.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 3
NORMALITY OF PRODUCT SPACES I
Masahiko ATSUJI Josai University, Sakado, Saitama, Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental results. . . . . . . . . . . . . . . . . . . . . . 2. The first of Morita’s three conjectures . . . . . . . . . . . . . . 3. The second and third of Morita’s three conjectures . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
81 82 92 109 116
Introduction
Unless explicitly stated to the contrary, the product of spaces is the topological product space of two spaces. As a separation axiom normality is not a special one, and it is the natural one next to regularity or complete regularity. However, the situation is quite different in the theory of product spaces. A product space satisfies a separation axiom T,, 0 < i < 3+, if and only if each factor space of it does. However, surprisingly, even a product of the unit closed interval I on the real line with a normal space is not necessarily normal (Rudin [1971]), so normality is found to be a very interesting property in the theory of product spaces. On the other hand, Dowker [1951] characterized a structure of a space X , countable paracompactness, by normality of X x I, and Tamano [1960] did the same: a characterization of paracompactness of a completely regular space Xby normality of the product of Xwith its Cech-Stone compactification. The works of Dowker and Tamano have stimulated a study of general topology and contributed to a development of the study, and we have had many beautiful and important results on product spaces since then. Thus we
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recognize a special importance of normality of product spaces in general topology, and we expect further possibilities for finding a characterization of various spaces in this direction. Among other topologists, especially K. Morita has done much in this area, and various interesting problems derived from or connected with his works have been researched and a further study is expected. So, in this chapter, we will concern ourselves mainly with Morita’s work, particularly with his three conjectures and related topics which are about the characterization of spaces by means of normality of product spaces. Section 1 deals with some, not all, fundamental results on the normality of product spaces with a compact factor. In the proofs of many propositions we use a pseudometric in order to make them simpler. The main contents of Sections 2 and 3 are clear by their titles, which also include various topics derived from Morita’s conjectures. Throughout this article, a space is a topological Hausdorff space unless otherwise specified, whose cover is open. Let A be a subset of a product space X x Y and y a point of Y. A[ y ] = { x E XI (x, y ) E A} is called a slice of A at y- 1 A 1 is the cardinality of a set A, w ( X ) is the weight of a space X.w and w,are the first countably infinite and the first uncountable ordinal numbers respectively; rc is a cardinal number or an initial ordinal of the cardinal, which often stands for the set of all ordinals smaller than K, and also for a topological space equipped with the order topology or some topology indicated. A space X is said to be rc-paracompact if any cover of X with power < rc admits a locally finite refinement. If X is rc-paracompact for any infinite IC, then Xis said to be paracompact; so if w ( X ) < rc, then rc-paracompactness is equivalent to paracompactness for X . o-paracompactness is nothing but a countable paracompactness. For undefined notions and symbols, for facts used without mention and also for related subjects the reader is referred to Engelking [1977] or Nagata [1985]. 1. Fundamental results 1.1. Theorem (Tukey [ 19401). A cover 4 of a space X is normal if and only ifthere exists a pseudometric don X such that { U ( x ;E ) 1 x E X } is a refinement of 4 for some E > 0, where U ( x ; E ) = { y E XI d(x, y ) < E } . 1.2. Definition. The pseudometric d stated in this theorem is called the pseudometric associated with the cover 4 , and the topology induced by d is denoted by Fd. The following theorem is a generalization of the well-known Stone’s theorem (Corollary 1.6 below).
Normality of Product Spaces I
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1.3. Theorem (Morita [1981, Theorem 29.101). Let .T and Y be topologies and @, n E o,are on a nonempty set X . Suppose that Y is a cover of (X,9) covers of (X,9’) with the property thatfor any x E G E 3 ’ there exists a number n with StS(x,@,,) c G. Then there exists a locallyfinite, with respect to 9, cover of (X,9) which refines 9.
Proof. Suppose the index set R of 9 = {G, I a E R} is well-ordered. For each point x in X we take the smallest a with x E G, which is denoted by a,. We let Bna = Ana - U{GpIB < a}, An, = {xlStS(x,@ n ) c Ga}, Bn = U{&ala
E Q},
HI, = St2(Bla,@I),
H,,, = St2(B,,,, @,) - U{St(Bj, % ) l j < n} for n > 1, where the bar
H
-
means the closure by the topology 9,
= { H , , , ~ ~ EaW ER , }.
Then we have StS(Bn,, q n ) c StS(An,, an) c G,, and p < a implies St5(Bn,,@,,) n B,, = 8, hence, for any point x E X and a member U of @, including x , we have U n St2(Bn,,@,,) = 8 or U n St2(Bn,,@,,) = 8. Therefore, for every n E o,{St2(Bn,,@,J I a E R} is locally finite with respect to 9. For each x E X there exists an n E o with StS(x,@,) c Gax,and x is in BnaX, hence
X
= U{BnInE a}.
For any x E X there exists a smallest number n E o such that x E St(B,, @,,). Since St(B,,, @,,) = U{St(Bna,@,,) 1 a E R} and {St(B,,, @,,) 1 a E R} is locally finite, there exists an a with
x
E
St(Bna, @ n ) c st2(Bna,a n ) ;
hence, if n = 1, then x x
E
E
HI,, and if n > 1, then
St2(B,,,, @ n ) - U{St(Bj, @j)Ij < n}
=
Hna.
Consequently, .W is a cover of (X,Y ) ,which refines Y. For x E X,there exists an n with x E B,,, and St(x, @,,) c St(B,,, @,,), so rn > n implies H,, n St(x, a,,) = 8. Since { q . a l j= 1, . . . ,n, a E R} is locally’finite, H is locally finite with respect to 9’.
Remark. If IR( 2 w , then tH I = IRI. 2t‘ is o-discrete.
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M.4tsuji
1.4. Corollary. Every normal cover of a space has a locallyfinite refinement. The converse of this corollary is valid in a normal space, namely, a locally jinite cover of a normal space is normal. We can omit the normality of the space if we replace “cover” with “cover consisting of cozero sets”, and then the condition “locally finite” can be weakened to “a-locally finite” (Morita [1964, Theorem 1.21;see Nagata [1985, VS.D]), where G is called a cozero set of a space X if there exists a real-valued continuous map f on X such that G = {x E XI f ( x ) # 0). Since a space is normal if and only if every finite cover is normal, we obtain the following assertion by these facts: a space is normal and K-paracompact if and only if any cover of the space with power at most K is normal. We can restate Corollary 1.4 as follows.
1.5. Corollary. Let Y be a normal cover of a space and let d be a pseudometric associated with 9. Then B has a locallyfinite refinement consisting of Fd-open sets. Since every paracompact space is normal, we have the following corollary. 1.6. Corollary (Stone [1948]). A space X i s paracompact ifand only ifevery cover of X is normal.
We also mention the following proposition (cf. Nagata [1985, Theorem V. I]): If X is a regular space, then X is paracompact i f and only if every cover of X has a a-locallyfinite refinement (Michael [1953]). 1.7. Corollary (Tukey [1940], Stone [1948]). compact.
A pseudometric space is para-
1.8. Lemma (Morita [1975]). Let (X,Y) be a space, and let (C, 9)be a compact space. Then for any normal cover Y = { G a l aE R} of ( X x C, F x 9) there exist a pseudometric d on X and a normal cover &‘ = {Ha 1 a E R} of ( X x C, F d x 9’) such that Ha c G, for all a E R.
Proof. Let e be a pseudometric on X x C associated with Y. For any points x and x’ in X we define d(x, x’) = sup{e((x, Y ) , (x’, Y ) ) I Y
E
C};
then d is a pseudometric on X and the topology Y d is coarser than Y. It is Let also eaSy to see that the topology Y d x 9’ on X x C is finer than YQ. &‘ = {Ha I a E R} be a cover of (X x C,Yo)with Ha c G,, a E R; then it is normal and a cover of (X x C, Y d x 9’). 0
Normality of Product Spaces I
85
Let us here present two useful lemmas the first of which is found in Nagata [1985, Section V.5.E]. 1.9. Lemma (Morita [1962]). Suppose that U{Vi:,lIE A} is a cover of a space X and { 11 E A} is a normal cover of X , and that { Wi n V I V E "v;} is a normal cover of the subspace Wi for each I E A; then { K, n V I V E 9;. , Iz E A} is a normal cover of X , and also is U-V;.
1.10. Lemma (Morita [1975]). Let ( X , 9 )be a space, let (C, 9)be a compact space with a base a,1931 < K, o < K, and let Y = {G,la E R}, I R I < K, be a cover of X x C. Then there exists a cover 4 = { U, 11 E A}, I A I < K, of X satisfying conditions (i) and (ii) below. (i) There exists a family {Vi11 E A} of finite covers of C satisfying the condition that Vi c for every I E A, { U, x V I V E Vi,I E A} is a cover of X x C and { U i x VI V E Vi,I E A } refines 9. (ii) Y is a normal cover of X x C if and only if@ is a normal cover of X .
Proof. (i): We can take a family { V, c 9 l I 6 E A}, I A I < K, of finite covers of C such that any finite cover of C is refined by some V,, 6 E A. Let Vd= {hiI i = 1, . . . , r,}, and let r, be the set of all maps f of { 1, . . . , r,} to R. Then I r, I < K. For 6 E A and f E r, we denote U(6,f ) = { x E XI { x } x 6;c G,,;,for each i < r 6 } . Since is compact, U(6,f ) is open in X. For any point x of X there is a finite subset y of R such that {x} x C c U{G,la E y } . Hence there is a 6 E A such that { { x }x V E V,} refines { G,1 a E y } . Therefore, 4 = p(6,f)ifE
r,, 6 E A}
is a cover of X , (U(6,f ) x VI V E V,,f E r,, 6 E A} covers X x C , and { U ( 6 ,f ) x VI V E 6 , f E r,, 6 E A} refines 9.Denote A = ((6, f ) I f E r,, 6 E A} and 9'&) = 6 ; then 1 A I < K. Thus condition (i) is satisfied. (ii): Suppose that Y is normal. By Lemma 1.8 there exist a pseudometric don X and a normal cover & = { H , I a E R} of (X x C, Ydx 9) such that H , c G,, a E R. We let M(6,f ) = { x E XI { x } x
6, c
H,(i) for each i
< r,}.
As above, ( M ( 6 ,f ) } is a cover of ( X , Yd), and 4 = ( U ( 6 ,f ) } is normal because of M(6, f ) c U(6,f ) . Conversely, suppose that 4% = { U ,I I E A} is normal. Then { U , x CI I E A} is a normal cover of X x C, and { U i x VI V E V i }is a normal cover of U , x C, and hence, { U , x V I V E Yj,,I E A} is a normal cover of X x C by Lemma 1.9, which refines Y by (i), and thus Y is normal.
M. Alsuji
86
1.11. Definition. Let p be a point of a space X. We denote by ~ ( pX) , the character of p , namely ~ ( pX, ) = min{ I Y I I Y is a local base for p } . Let p be an accumuiation point with ~ ( pX) , = K . If there exist a local base 9 = { K 1 1 < K ] for p and a set A with p E A - A satisfying the condition that, for any 1 < K, there exist a point x E A n V, and an index a, < K such that x does not belong to any V, with a, < a < K, then we call p a simple accumulation point for 9 and A (or briefly a simple accumulation point). If { A n K I 1 < K} is monotone decreasing, then p is a simple accumulation point for Y and A ; particularly, if p has a monotone decreasing local base 9, then p is a simple accumulation point for Y and X. Another example is this. Denote X = (0, 1) x (w l), p = (w,, w), V,, = (a, w , ] x (n, 01, A = {(a, w ) l a < w , ) , and B = {(a,, n)ln < w>. Order { a n l a c w , , n < w } lexicographically; then p is a simple accumulation point for Y = {&,,la < w , , n < w } and A, but not for Y and B.
+
+
9.12. Theorem. Suppose that X i s a space and C is a compact space including a simple accumuiation point p , with ~ ( p ,Y, ) = v for each v < K . I f X x C is normal, then X is normal and K-paracompact. Proof. We select a local base 9 = { V ,I a < K} for p , and a subset A, of C for which p Kis simple. We shall verify the theorem inductively. Xis obviously wrmal, and we assume that, for any infinite cardinal K’ < K, our theorem :s valid. Let 93, 193 I = p, be a base for C consisting of members of Y and and let 0 = { 0,I a < K ] be a cover spen sets contained in C - q , Y, E 9, of X. We denote
G, = u{O, x V,la <
K}
and G2
=
X X C - X X {p,};
then 9 = {GI, G,} is a cover of X x C , and by Lemma 1.10 there exist a normal cover Q = { U, I 1 E A}, I A I < p, of X and a family {V,I I E A} of finite covers of C with V, c & such I that { U, x V I V E VL, 1 E A} is a cover o f X x Cand{U, x P I V ~ V , , A ~ A } r e f i n e s Y . F o r a n y 1 ~ A t h e r e e x i s t s a V E Y,includingp,. Vis in 2,and we rewrite it V,, ,j, < K. We then have (*)
ui. x V,,
c GI,
and we can find an a, < (**)
U, c U(0,la
K
such that
< a,}.
87
Normality of Product Spaces I
In fact, if we cannot, then select a point y in VS; n A, and an index a,, < K such that y is not in Vy E Y for all y > a,, in K. There exists a point x in U, - u{O,Ia < a,,}; then the point (x, y ) of U, x F$, does not belong K } , which contrato both u{O, x V,la < a y ) and U{O, x KIa, < a dicts (*). Since X and $! are normal, $! is refined by a normal cover W = {W,16EA}ofXsuchthat % t U,forsomeAEA,and
-=
c U, c u{O,Ia
< aI}.
Since W,x C is normal and a1 < K, u(0.l a < a l } has a normal refinement on W, by the inductive assumption. Therefore, by Lemma 1.9,O has a normal refinement. 0 1.13. Theorem (Morita [1961, 19621). Let X be a space, and let K be an infinite cardinal number; then the following conditions are equivalent: (1) X is normal and K-paracompact. (2) The product space X x C is normal and K-paracompact for any compact space C with w ( C ) < K . (3) X x C is normalfor any compact space C with w ( C ) < K . (4) X x M" is normal for any compact metrizable space M . ( 5 ) X x I" is normal for I = [0, 11. (6) X x D" is normal for D = (0, l} c I. (7) X x (K 1) is normal'.
+
Proof. (1)=42): Let Y = {G, I a E R} be a cover of X x C with 1 R I < K ; then we construct a cover 92 = { U, I 1E A} of X stated in Lemma 1.10.9 is normal because of 1A1 < K, and Y is also normal by condition (ii) in the same lemma. (2)*(3): Trivial. (3)*(4): Obvious because M" is a compact space with w ( M " ) < K. (4)*(5): Obvious. (5)*(6): Obvious because D" is a closed subset of I". (6)*(7): Obvious because K + 1 is closedly embedded in D". (7)*(1): Obvious by Theorem 1.12. 0 Probing Theorem 1.13, Y. Katuta obtains a result which we introduce here without proof. Let K and 1 be infinite cardinal numbers. A space X is ' S o w mathematicians, e.g., Przymusinski [1984], say that the equivalence of (1) and (7) is Kunen's theorem. Some others, e.g., Rudin [1985], call it Morita's theorem, and she shows a proof of the theorem which is largely the same as the one of Lemma 2.5 in Morita [ 19621. Though the lemma does not explicitly assert the equivalence, it states an essential part of the proof.
88
M.A fsuji
K-collectionwise normal if, for every discrete family {F,I a < K} of closed subsets of X,there exists a family {C,I a < K } of disjoint open subsets of X with F, c G, for all a < K. A family .W of subsets of a space Y is said to separate points of Y if, for any two distinct points of Y, there exists a member of .W including only one of them. We denote by w( Y) the smallest cardinal A such that there exists a family 3Ep = U{.W’( fl < A} separating points of Y, where JE., is a family consisting of disjoint cozero sets of Y for each fl < A. For the one-point compactification Y of a discrete space, we have v ( Y) = w and w( Y) = I Y I; in general, it is known that for a compact space Y with I Y I 2 w it holds that t( Y) < w( Y ) < w( Y ) , where t( Y) is the tightness of Y (= the smallest cardinal number a such that, for every point y in Y and A c Y, if y E A then there exists a subset E of A with x E B and I E I < a). We denote by I @ ) the subspace of the product I Aof A copies of I consisting of points p such that at most one coordinate of p is not 0. D(A) is similarly defined by using the discrete D = (0, l} instead of I. D(A) is nothing but the one-point compactification of a discrete space with A points. Theorem 1.13 is the case K = A of the following theorem.
1.14. Theorem (Katuta [ 1977al). Thefollowing statements are equivalenrfor a space X : (1) X is K-paracompact and A-collectionwise normal. (2) The product X x C is normal for any compact space C with v(C) < K and w ( C ) < A. (3) X x I(A)l is normal. (4) X x D(A)l is normal. We can derive the following corollary from this theorem (cf. Corollary 2.23): A space X is u-paracompact and collectionwise normal if and only if X x C is normal for every compact space C with w(C) < K . In particular, X is countably paracompact and collectionwise normal if and only if X x C is normal for any compact space C with w(C) = w.
Remark. Ohta [ I985/86] improves Katuta’s theorem above by replacing w(C) in the statement (2) with a smaller cardinal u(C), which is defined by replacing “disjoint cozero sets” in the definition of v(C) by “boundedly pointfinite cozero sets”, where a family of subsets of C is called boundedly pointfinite if there exists a natural number n such that every point of C is included in at host n members of the family. Trivially, u(C) < w(C). A compact space C with u(C) < w is precisely a uniform Eberlein compact space shown in Benyamini and Starbird [1976](cf. Negrepontis [1984, Section 6]), namely,
Normality of Product Spaces I
89
Cis homeomorphic to a subset of a Hilbert space in its weak topology. In the note cited above Ohta also discusses many topics together with problems on cardinal functions of compact spaces in connection with Katuta’s theorem. Let us present another form of result giving normality of the product with a compact factor. 1.15. Theorem (Morita [1962]). Let X be a subspace of a compact C . Then X is paracompact if and only if X x C is normal.
Proof. The “only if” part is obvious by Theorem 1.13. Now, suppose that X x C is normal and Y = { G, I a E R} is a cover of X, then for every a E R there exists an open set Ha of C with G, = X n H a . Denote H = U { H , I a E R} and A = {(x, x) I x E X}, then A is a closed subset of X x C contained in X x H. For the cover A? = {X x H , ( X x C) - A} of X x C, by Lemma 1.10, we obtain a normal cover 9 = { U, I I E A } of X and a family {V,11 E A} of finite covers of C such that { U, x VI V E f,, I E A} covers X x C and {U, x PI V E V,, I E A} refines A?. If U, n # 8, then U, x P meets A, and so U, x c X x H. Hence the compact set U { U, n P # 0, V E .U;} is contained in H and contains U,, and so there exists a finite subset y of R such that U, c U { H , I a E y } . Thus we have U, c U{G, I a E y } , and Y has a locally finite refinement. 0 This theorem can be generalized a bit as follows: Let X be a subspace of a space Z , let C be a compact space, and let f be a continuous map on C to Z with f ( C ) 3 X . Then X is paracompact if and only if X x C is normal. However, this is an immediate consequence of the theorem above because the map ix x f:X x C -, X x f(C) is perfect, where ix is the identity map on X. An investigation of a role of maps in the theory of product spaces is of great interest, which is one of main subjects of Hoshina’s Chapter 4. The following classical results are easily deduced from some of the propositions above. 1.16. Corollary (Dieudonnt [1944]). The product space of a compact space and a paracompact space is paracompact.
1.17. Corollary (Dowker [1951]). A normal space X is countably paracompact if and only if X x I is normal.
Proof. Since D” is a closed subset of I,X x D” is normal if X x I is normal. 0
M. Atsuji
90
1.18. Corollary (Tamano [1960]). A completely regular space X i s paracompact if the product of X and its cech-Stone compactijication flX is normal.
The next theorem is one of the natural generalizations of Corollary 1.16. 1.19. Theorem (Morita [1963a]). If X is a paracompact and a-locally compact space, then X x Y is paracompact for any paracompact space Y.
Proof. Let Y be a cover of X x Y. Since X is a-locally compact and paracompact, X is covered by U { d i li E a},where di = { A , I a E Q i } is a locally finite family consisting of compact subsets of X,and we can construct, for each i E w ,a locally finite open family { L i ,1 a E Q }with Aia c L, for all a E 0,.We apply Lemma 1.10 to the space Ai, x Y, and we have a locally finite cover qiaof Y and a family {“v;,, 11 E Aim}of finite covers of Aia such that every member of Kdis contained in Li,; moreover, each product of a is contained in some member of Y. Since member of qi,and a member of KaA {Ux V I U E ~ ~ V ~ ~, ~, ~ , ~ ~ A , , a ~ R , } i s l o c a l l y f i n i t e f o r e a c h i ~ w , B is refined by a a-locally finite cover, and the proof is completed by the proposition stated after Corollary 1.6. 0 After this theorem of Morita’s was published, a lot of topologists have searched for further generalizations. One of the results obtained is the following theorem (a space is C-scattered if every nonempty closed subspace S has a point with a compact neighborhood in S). 1.20. Theorem (Telgarsky [1971]). If X is paracompact and a-C-scattered, then X x Y is paracompact for any paracompact space Y.
Theorem 1.19 develops to Theorem 3.6 and to Morita’s Conjecture I11 stated later on the one hand, and on the other hand it is followed by Theorem 1.20 and others which are also considered partial solutions of Tamano’s problem (Tamano [1962]): Characterize a space X with which the product X x Y is normal for any paracompact Y. Applying Corollaries 1.16 and 1.18, Tamano himself verified the following theorem. 1.21. Theorem (Tamano [1962]).
Thefollowing conditions on a space X are
equivalknt: (1) X x Y is normal for any paracompact space Y . (2) X x Y is paracompact for any paracompact space Y .
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91
Though Katuta [1971] gives a solution of Tamano’s problem, a simpler characterization is also desired. Telgarsky [19751 defines an interesting notion: topologicalgumes, which can also be applied effectively to our subject. The reader will find some of the results concerned in Yajima’s Chapter 13 in this volume. Let us conclude this section by presenting two more theorems for which we first mention a well-known proposition without proof. Let I be an infinite cardinal. We recall that a well-ordered set { y , I a < A} of points of a space Y is said to be afree sequence of length I in Y if { y , I /3 < a } n { y , I /3 2 a } = 0 for every a in I.
1.22. Lemma (Arhangel’ski [1971]). For a nondiscrete compact space C, it h o l h that t ( C ) = sup{I I there exists a free sequence of length A in C } . ‘1.23. Lemma (Przymusinski [1984]). For a nondiscrete compact space C, it holh that t ( C ) = sup{A I there exist a closed subset F of C and a continuous m a p f : F + A + l o f F o n t o A + I}. Proof. We denote by p the cardinal number of right-hand side of the equality in our lemma, and by { y , I a < I } a free sequence of length 1in C. We let C, = { y,IS 6 a } for each B < I , C, = n{C,l/3 < A}. The map f:C, -+ 1 I defined by f ( y ) = sup{BI y E C,} is clearly surjective and continuous because the C, are closed and open in C,, and so we have t ( C ) 6 p. If t ( C ) < p, then there exists a cardinal 1‘ > t ( C ) in p + 1 satisfying the condition stated in the definition of p. However, this is impossible because a continuous map on a compact space cannot increase a tightness. 0
+
1.24. Theorem (Katuta [1977b]). r f the product X x C of a space X with a compact space C is normal, then X is I-paracompact for every A < t ( C ) . Furthermore, if t ( C ) is not weakly inaccessible, then X is t(C)-paracompact. Proof. This elegant proof is taken from Przymusinski [ 19841. Suppose that A < t(C);then, by Lemma 1.23, there exists a v > I in t ( C ) 1 such that a closed set F c C is continuously mapped onto v + 1. Since X x ( v + 1) is a perfect image of the normal space X x F, it is also normal, and, by Theorem 1.13, X is A-paracompact. When t ( C ) is not a limit cardinal, then the above proved fact immediately implies that Xis t(C)-paracompact; when t ( C ) is singular, then the fact that X is I-paracompact for all I < t ( C ) implies that X is t(C)-paracompact. 0
+
M.Atsuji
92
Giving an example, Katuta also notes in his paper cited above that t ( C ) being not weakly inaccessible is essential.
1.25. Theorem (Nogura [1976]). For an infinite cardinal K and a compact space C, the product K + x C is normal ifand only i f t ( C ) < K , where K + is the successor cardinal for K .
Proof. The “only-if” part is immediately obtained by Theorem 1.24 because -paracompact. = { G , , G,} be a binary cover of K + x C. For any y in C there exists an a < K + such that the interval (a, K + ) is contained in GI[ y ] or G,[ y]. Suppose (a, K + ) c GI[ y]; then there exists an open neighborhood V of y in C with (a, K + ) x P c G I .In fact, if not, then, for any ?‘ni&?, a local base for y in C, there exist a fly > a in K + and a z y in P such that the point (fly, zv)is not in GI. Since t ( C ) < K, there exists a set Z c {zyl V E U }with 121 < K and y E 2. We have s ~ p { f lZ} ~ = l ~y ~< ~K + , a < y and (a, K + ) $ G, [ y], a contradiction. Since the set a + 1 is compact and open in K + , it is a small step to the conclusion that Y is normal. 0 K+
is not
K+
“If” part: Let Y
Citing Kombarov” 19721, Chiba [I9761 remarks that the sufficiency of the theorem above is valid for a paracompact space C ;and, giving an example, she shows that the necessity does not hold for C. Afterward, K. Chiba [I9801 notes that in Nogura’s theorem we can replace “compact space” by “paracompact k-space” because it holds that t( Y) = sup{t ( H ) I Y 3 H: compact} for a k-space Y.
2. The first of Morita’s three conjectures Morita [1977] publicly posed the following three problems. (I) If X x Y is normal for any normal space Y, is X discrete? (11) If X x Y is normal for any normal P-space Y, is X metrizable? (111) If X x Y is normal for any countably paracompact space Y , is X metrizable and o-locally compact? Throughout this article, the term “P-space” is used in Morita’s sense; namely, it is defined in Morita [1964] and differently in Gillman-Jerison [1960], and it is characterized, together with its normality, by the property that its product with every metric space is normal (cf. Nagata [1985, Theorem VI.241). Further details will be introduced in Section 3 (see also Hoshina’s Chapter 4).
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In this section we discuss Problem I and related results. Conjecturing affirmatively, Morita raised Problem I mainly from the fact that we knew only few kinds of cases where products of normal spaces were normal. As we shall see in this section and also in the next one, conditions around one point of a factor space of a product space frequently plays an important role in deciding a structure of another factor space by normality of the product. We have already caught a glimpse of such circumstances in Theorem 1.12. Apparently, it seems special, but actually the way of approach from the local structure of a factor space to the characterization of another space through normality of their product has been shown to be effective in many important results, from the classical Dowker’s Theorem on countable paracompactness to the latest Chiba, Przymusinski and Rudin’s proof of Morita’s conjectures. 2.1. Definition. Let p be a point of a space Y. We denote by Y ( p ) the local base for p . Let { A , I y E Z } be a family of subsets of a space X with index set Z c Y . We call the following subset of X an upper intersection of the family at p :
n
u
Y E Y ( P )y s Y n Z
~ , . 2
The following lemma is an immediate consequence of this definition.
2.2. Lemma. Let X and Y be spaces, and let { A , I y E Z } be a family of subsets of X with index set Z c Y . Thefamily has the empty upper intersection at a point p of Y ifand only ifevery point x of X has a neighborhood which does not meet any A, with index y E V n Z for some neighborhood V of p . 2.3. Remarks. (1) The upper intersection at p E Y of a family d = {A, t X J Y E2 c Y } is the slice A [ p ] of A at p , where A = U { A , x { y } I y E Z } and A is the closure of A in X x Y, so the upper intersection is an invariant of local buses for p . The upper intersection at p of the family is empty if and only if A is disjoint from X x { p } . Consequently, A is a closed subset of X x Y if and only if for each y E Y the upper intersection at y is A, ( A , is empty for y 4 2).Trivially, the upper intersection at p of a family contains the intersection of all closures of sets of the family, not always the same. *Thissubset is denoted by limp sup A, in Atsuji [1977]. The notation was cited from Nobeling [1954, Section 8.21. The subset for a countable family is treated by some mathematicians with other notations, for instance, Isiwata [1964], and Alexandroff and Hopf [1935, p. 11 I) in which it is called “oberer topologischen Limes”. See also Comment 2.29.
M.Atsuji
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(2) If I Z - VI < y for every basic neighborhood V of p in the lemma above, then the family has the empty upper intersection at p if and only if the family is, as it were, locally less than y ; especially if y = w, it is locally finite. 2.4. Theorem. Let X and Y be topological spaces, and let d = { A , 1 y E Z , } and 1= {ByI y E Z z }be families of subsets of X with index sets in Y such that the upper intersections of d and 1at every point of Y are disjoint. If X x Y k normal, then there exkt open families 9 = {D, I y E 2,} and 6 = {E, I y E Z 2 } such that (i) 0,3 A, for y E 2,and Ey 3 Byfor y E Z,, and (ii) the upper intersections of 9 and 6 at every point of Y are dkjoint.
Proof. Denote A = U { A p { Y } I Y E Z I } and B = U{B,X { V } I Y E Z * } ; then the closures A and B of A and B in X x Yare disjoint, so there exist open sets D and E in X x Y containing A and 1 respectively and having disjoint closures. { D [y ] 1 y E Z , } and { E [y ] I y E Z , } are the desired open families. 0 In connection with this theorem it is interesting to find conditions under which the upper intersection is the same with the usual intersection. In order to present one of the conditions, we need some terms. Suppose that an order < is defined on a subset Z of a space Y whose relative topology on Z is not necessarily the same as the one defined by the order. A family {A,I y E Z c Y} of subsets of a space X is said to be increasing (decreasing) if A,, c A, whenever y , < y2 ( y , 2 yz).When Z is linearly ordered, the increasing (decreasing) family is also said to be monotone increasing (monotone decreasing). 2.5. Definition. Let an order be defined on a subset Z of a space Y. Z is said
to befit if Z satisfies the following two conditions: (i) for any point p in Z - Z there exists a subset Zpof Z with p E 2, - Z, such that for each point y in 2,there exists a neighborhood V of p satisfying y’ 2 y for all points y’ in V n Z; (ii) any point q in Z has a neighborhood U such that y’ 2 q for all y’ in U n Z. A family of subsets of a space X with a fit index set is also called to befit. The set Zpstated in condition (i) is called afit part of Z for p . If we require
Normality of Product Spaces I
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for Z only condition (i) for some point p E 2 - Z, then we say that Z (and the family of subsets with the index set Z ) is primarilyfit at p , Suppose that Z is a subset of Y and p is an accumulation point of Z. An accumulation degree of p is defined by @(PI = minil AllP E A - { P I } .
Let p be an accumulation point of a space Y with an accumulation degree K. Then there exists a subset D of Y such that ID I = K and p E b - D. Well-order D = { y,I ct < K}; then for any point y, of D there exists a neighborhood V of p with { ys 1 /3 < a } n V = 0,and thus D is primarily fit at p. The sets A and B defined in the example of the simple accumulation point, stated just before Theorem 1.12, are both primarily fit at the point p given there. The following lemma is interesting in the sense that many results on normality of product spaces include conditions which imply the equality of the upper intersection and the intersection of a family of subsets playing central roles in the proof of the results, and the equality comes from the fitness of the family assured by the conditions. This situation will be seen in many places in this chapter, too.
X and Y be spaces and let an order be deJined on a subset Z c Y, and let (A,I y E Z c Y } be a decreasingjit family of subsets of X . Then the upper intersection of the family at p E 2 - Z is the same as the intersection of {A,1 y E Z,,} for any jit part Z, of Z for p ; and the upper intersection of the family at q E Z is A,. 2.6. Lemma. Let
Proof. It will be enough to verify the first assertion, for which it suffices to show that the upper intersection of the family is contained in the intersection. Take an arbitrary pointy, in Z,; then we can find a neighborhood V, ofp such that y 2 y , for all y in V, n Z. H =
n u n A,.
Vc.%’(p) y c V n Z
H
=
A,
= yeynZ
= A,,,
Y‘ZP
2.7. Corollary. Let X and Y be spaces and { A , I y fit family of closed subsets of X . If
E
Z c Y } be a decreasing
M.Atsuji
96
for each point p in z - Z and for a fit part Z, of Z for p , then
UP,
x {YHY E
z>
is a closed subset of X x Y. The proof is obvious by Remarks 2.3(1) and the lemma above. 2.8. Definition. Let Z c Y and Z be ordered, and let 3 be a family of
subsets of Z. If, for any decreasing closed family f = {F, I y E Z } in a space X with the property that every family {F, E 9 1y E Z’}, Z’ E 3,has the empty intersection, there exists an open family 59 = {G,I y E Z } such that Fy c G, for all y E Z, and n{GyI y E Z ’ } = 8 for every Z’ E 3,then X is said to have a property Q(Z, a). Especially, in case %” = {Z},the property 9 ( Z , 3)is renamed a property 9(Z).
2.9. Corollary. Let Z c Y be ordered, and let %” be a point finite family of subsets Z, of Z, u E R, such that Z, is a primarily fit set at a point p , with fit part Z, for every u E R. Z f X x Y is normal, then X has the property Q(Z, 3).
Proof. Suppose that 9 = {FYIy E Z } is a decreasing closed family in X with the property that the intersection of fa= {F, E f I y E Z , } is empty for each u E a. Since Z, is a primarily fit set at pa with fit part Z , , the upper intersection of faat pa is empty by Lemma 2.6. Applying Theorem 2.4 to the families fa and { B y [y E { p a } } , Bpa= X , we have an open family 8, = {E,I y E Z,} such that Ey 3 F, for y in Z, and the intersection of {E, 1 y E Z,} is empty. Each point y in Y is included in only finitely many Z , , , . . . ,Zan,and wedefine G, = n{EZIEz3 F,, z E Zui,1 < i < n}. Thus we have the property 9(2,9’) of X . 0 The following corollary will be used in the next section.
2.10. Corollary. Let Z , and Z , be disjoint subsets of a space Y which are similarly ordered and fit, where the similarity is established by a map f : Z, + Z , , and let z.,i = 0, 1, be a family of subsets Z,, of Z , , u E R, such that (i) the similarity between Z,, and Z , , is established by f for each u E R, (ii) Z,, n z,, # 8for every u in R, and (iii)for any point y in 2, - Z , there exists an u in R such that Zi, is afit part of Zifor y , i = 0, 1. Zf X x Y is normal, then X has the property 9(Zo,3,).
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Proof. Let 9, = {F, I y E Z , } be a decreasing closed family in X such that the intersection of 9, = {F, E 9 , l y E Z,,} is empty for each a in R. We for every z in ZI and 9, = {F, I z E Zl}; then 2FI is a define F, = decreasing closed family. The upper intersections of Soand 9, at any point y of Yare disjoint. Indeed, if y E Zi - Z i ,then the upper intersection of at y is empty by assumption (iii) and ()Fa= 0 and by Lemma 2.6. If y E Z i , then y 4 Z,, j-= 1 - i, so the upper intersection of 5. at y is empty in both cases of y 4 Z, and y E Ziby Lemma 2.6. Therefore, by Theorem 2.4 there exist open families 9 = {D,ly E Z , } and I = {E, I z E Z,} such that G, = 0,n E’(,,, 2 F, for y in Z,, and the upper intersections of 9 and 6 at any point of Y are disjoint. Let a E R and x E X , and let p be in Z,,, n Z,,. x does not belong to the upper intersection of either 9or 6 atp. Suppose that
e.
X #
n
= H;
v a ~ ( p z) E v n Z l
then
n $,
x 4 H 3
ZEZlpl
and x is not in EZfor some z quently, we have (-{G,ly
E
Z,,}
=
= f( y),
0
y E Z,, , hence x is not in
G,. Conse-
for every a in R.
It is obvious that we can express the property 9 ( Z ) in other words as follows: any increasing open cover {G, I z E Z} of X can be shrunk (Rudin [1985]); namely, there exists a cover { H , I z E Z} of X with R2c G, for all z E 2 ; the latter cover is called a shrinking of the former. In general, a space is said to be (A-)shrinking,A an infinite cardinal, if every cover (of cardinality < A) of the space can be shrunk. Obviously, a A-shrinking space is normal and countably paracompact. 2.11. Definition. Suppose that I Z I is an infinite cardinal K and Z u { p} has an order similar to K + 1. In this special case 9 ( Z ) is called a property ~ ( I c ) . A space with the property 9 ( ~ for) any infinite K is said to have a property 9.’ If an open family {G, I y E Z}, whose existence is required corresponding to a monotone decreasing closed family in the definition of ~ ( I c ) is, also required to be monotone decreasing, then the property 9 ( ~ (or) 9)is called a property W ( K )(or W)(cf. Zenor [1970]). ’The author of this chapter temporarily called the property 9 ( ~ a property ) B*(K)in Atsuji [1977]. The property D was named a weak I-property by Yasui [1972]. Now, following Rudin [1985], the author would like to use the reasonable new term.
98
M. Alsuji
Trivially, the property 9 ( ~ implies ) countable paracompactness, and K shrinking =$ 9 ( ~ 6)1 ( ~ ) . Let Y be a space including an accumulation point p with accumulation degree K. Then, as stated before, Y contains a linearly ordered subset 2 which is similar to K and primarily fit at p , so, by Corollary 2.9, we have a following corollary. 2.12. Corollary (Atsuji [1977]). Suppose that a space Y includes an accumulation point p with an accumulation degree K. r f X x Y is normal, then X has the property 9 ( ~ ) .
A normal space without the property 9 ( ~ is called ) a K-Dowker space. An existence of K-Dowker space for an arbitrary infinite K is given by Rudin [1978], and it will be introduced in Definition 2.24 and Theorem 2.28. Thus Morita’s conjecture I is now answered affirmatively by the existence of ic-Dowker space along with the corollary above:
2.13. Theorem. Morita’s Conjecture I holds, namely, i f X x Y is normal for any normal space Y, then X is a discrete space. This theorem shows us that the property 9 ( ~ is a) rather interesting notion in the study of product spaces, and we have so many results on the property. Let us mention here some of the known basic properties of the property 9 ( ~ ) and the property B(K). 2.14. Facts. (1) The property 9(o)is equivalent to the countable paracompactness in topological spaces (cf. Ishikawa [19551). Yasui [ 19721 gives, among others, the following two examples: (2) o,is the normal space with the property 9 and without the property 1; (3) The Sorgenfrey line has the property 1,but its square does not. It is not hard to see that (4) i f X is K-paracompact, then X has the property B(K)(cf. a n o r [1970]). Using an example of Navy [ 19811, Rudin [1983](and [1985]) shows that the converse of this fact is not true (see also Fleissner [1984]), namely, ( 5 ) the property B, and also shrinkabiliiy, of a normal space does not imply paracompactness. She furthermore notices in the same note that (6) I-paracompactness for all I < cf(K) of a normal space does not always imply the property 9 ( ~where ) , Cf(K) is a confinality of K (cf. Theorem 2.28).
Normality of Product Spaces I
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Ohta [1987] obtains the following result on this issue (for details, see Yasui’s Chapter 5, Section 2; the definition of “FCP” is given at the end of this list): (7) Let X be a regular space. X is paracompact if and only if jor every increasing cover Y = (G, I a < u} of X there exists an FCP increasing cover (H,I a c u} of X with R, c G, for each a < K. K. Chiba [1984] shows that (8) for any regular different cardinals K and I the property 9 ( u ) does not imply the property 9 ( A ) ; and (9) the property 9 ( u ) is not productive for any regular cardinal u. (10) (Ohta [1983]) If X is a normal space that is a closed continuous image of a space with the property 9,then X also has the property 9. However, a closed continuous map does not preserve the property 9 in Tychonoff spaces. (1 1) (Rudin [19831). Closed continuous maps do not necessarily preserve the property W in normal spaces. (12) (Yasui [1984]). Every cover of a topological’spaceby open subsets with perfectly normal closures can be shrunk. Particularly, every perfectly normal space has the property 9. (K. Chiba [1984]). See also Yasui [1988a] and Chiba-Przymusinski-Rudin [19861.
A family 9 of subsets of a space X is said to be finitely closure preserving (FCP)in X if for any subfamily 9’ of 9 and for any point x in X there exist a neighborhood U of x and a finite subfamily 9” of 9‘ such that U n (Up’) t Cl(U9”). We have many interesting results connected with this subject (cf., e.g., Yasui [1986, 1988b] and Chiba [1987, 19881). For instance, results on equivalence of the property 9 and countable paracompactness on Z-product spaces are given in Chiba [1982, 19851. Moreover, including an answer to Chiba’s question, Yajima [1986] shows results on the property 9 and shrinkability of Z-products. As for the relation between shrinking spaces and the normality of products we additionally cite Hoshina [1984] and Rudin [1985]. For further information on this subject the reader is referred to Yasui’s Chapter 5 in this volume. The next corollary is an immediateconsequence of Corollary 2.12 and Facts 2.14 (1). Finding conditions on X under which the converse of Corollary 2.15 holds is an interesting problem. As we shall see in Section 3, one of the conditions is the property 9(Z, 9’) for some 2 and 9’ (Theorems 3.1 and 3.3). Taking another point of view in Chapter 4 of this volume, Hoshina shows another form of these conditions and develops it to the case of a Lainev space M :a closed image of a metric space.
M. Atsuji
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2.15. Corollary. Zfthe product X x M of a space X and a nondiscrete metrizable space M is normal, then X is normal and countably paracompact.
Let us now consider the converse of Corollary 2.9. When spaces X and Y are neither compact nor metrizable, we know quite few cases of normality of X x Y, so in order to search conditions for normality, we cannot help fairly restricting the structure of one of the factor spaces as we shall see in the following theorem (cf. Theorem 2.13). 2.16. Theorem! Let X be a normal space, and let Y = Z u { p } be a space in which every point of Z is isolated. The product space X x Y is normal if and only if any closed disjoint subsets K and X x ( p } in the product space can be
separated by open sets.
Proof. It suffices to prove the “if” part. Let A and B be nonempty disjoint closed subsets in the product. We denote
B = UP, x
(4Iz E Z } u (B, x
(PI).
For any y in Y, the sets A, and By are disjoint closed subsets of X,and there exist disjoint open subsets C, and D, of X containing A, and By respectively. Since the upper intersection of {A, I z E Z } at p is contained in A, by Remark 2.3 (I), the upper intersection of the family {(A, - C,)lz E Z } at p is empty, and by our assumption we have an open family { E , J zE Z } with A, - C, c E, c C,for every z E Z whose upper intersection at p is empty. Any point t of B, has an open neighborhood U, c D, satisfying Qn Iz E = 8 for some neighborhood of p .
(u{&
v})
F = u{V, x v l t ~ B,}
and
G
= ~{E,X(Z}~ZEZ}U(C,XY)
are the disjoint open sets containing Bp x { p } and A respectively. In the same way we have disjoint open sets F‘ and G’ containing A, x { p } and B. Thus A and B are separated by the open sets
G n (U{C, x { z } I z
E
Z}) u
F‘
4This theorem is explicitly stated in Starbird [1974], but it is essentially the same as a fact proved in Theorem 1 in Alas [1971], and the ideas of proofs in both papers are also identical. Our way of proving the theorem will follow Alas.
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and G’ n (U{Q x { z } I z E Z } ) u F.
0
2.17. Corollary. Let Z be a directed set, let Y = Z u { p } be a space in which every point except p is isolated and p has a local base Y consisting of all sets of the form V, = {z‘ E Z I z’ 2 z } u { p } for z in Z , and let Y be pointfinite at every point of Z . The product space X x Y of a space X with Y is normal if and only i f X is a normal space with the property 9(Z).
Proof. The “only-if‘’ part follows from Corollary 2.9 because Z is primarily fit at p . “If” part: By the theorem above it suffices to show that disjoint closed subsets A and X x { p } of the product space can be separated by open sets. Define B, = { x E X l ( x } x V, n A # S}; then, since {BzI z E Z} is the decreasing closed family with empty intersection, by the property 9 ( Z ) , there exists an open family { C, I z E Z} such that n{c,I z E Z} = 8 and C, 3 1,for all z E Z. Since (X - C,) x V , n A = 8 and only finitely many V , include a given point in Z,
(*I
U{(X - C,) x
V,lz E
z>
is a clos.:d subset of X x Y disjoint from A and containing X x { p } .
c]
The assumption of this corollary that Y is point-finite on Z is used only to say that the set (a) in the proof is closed, so we can replace the assumption by any one which assures the closedness of the set, for instance, by the condition “Z is linearly ordered and the family { C, I z E Z} is monotone decreasing”. The following example of Ohta is situated at an interesting place in Morita’s Conjecture I1 and suggests a new conjecture concerning Morita’s one on which we shall touch in Section 3. 2.18. Example (Ohta [1981]). Let r be the family of all finite subsets of wI, r = w;“; then r is an ordered set with respect to set inclusion, and we define wI > a for all a E r. We denote Y = r u {aI} and define a topology on Y in such a way that wIhas basic neighborhoods { fl E Y I fl 2 a } for each a in r and ,every element in is isolated. Then, by Corollary 2.17, X x Y is normal if and only if X is a normal space with the property 9(r),namely, every increasing open cover {G, I a E r}of X has a shrinking.
I02
M.Atsuji
Let X be a class of spaces. We denote by N ( X )the class of those spaces x Y is normal for every Y in X . Theorem 2.13 shows that, for the class X of all normal spaces, N ( X )is the class of discrete spaces. In many other cases of characterizing N(.X),all members of the class X are contained in a class of compact spaces (cf., e.g., Theorem 1.13, Corollaries 2.21 and 2.23). The following theorem of Yasui gives a characterization of N ( X )for a class X which is almost disjoint from the compact class. We equip YO&) = K + 1 with a topology in which K has neighborhoods in the usual order topology of K + 1 and each element in K is isolated.
X such that X
2.19. Corollary (Yasui [1983]). The product X x YO(x) of a space X with &(K) is normal ifand only i f X is a normal space with the property W ( K ) .Thus a space X is a normal space with the property W if and only if X x Y0(x) is normal for every infinite K.
Proof. Noticing the remark after the proof of Corollary 2.17, we see immediately that the "if" part of the first assertion in this corollary follows from Corollary 2.17. The "only-if" part of the first assertion follows from the relation, derived from fitness of YO&),
for a monotone decreasing closed family {FaI c1 < K}, and from the fact that { G, I o! < K} is monotone decreasing. 0
If K is a limit cardinal with cf(rc) = w , then K has a countable local base in YO@),and by Corollaries 2.17 and 2.19 we have the next corollary. 2.20. Corollary. Let K be a limit cardinal with cf (K) = w , then the following are equivalent 1 (1) X is normal and countably paracompact. (2) X is a normal space with the property W ( K ) . (3) X x Y,(K)is normal. Combining Corollary 1.17 with a special case of the corollary above, we have the following corollary.
2.21. Corollary. The following are equivalent for a space X : (1) X is normal and countably paracompact. (2) X x Z is normal.
Normality of Product Spaces I
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(3) X x (w + 1) is normal. (4) X is a normal space with the property W ( o ) . ( 5 ) X is a normal space with the property 9 ( w ) . One more interesting application of Theorem 2.16 is the following Alas’ result, for which we need the next well-known Dowker-KatEtov’s theorem. 2.22. Lemma (cf. Nagata [1985, V.3.D]). Let K be an infinite cardinal. A normal space X is countably paracompact and K-collectionwise normal if and only i f , for any locallyfinite family ( H aI a < K} of closed subsets of X,there exists a locallyfinitefamily {G,I a < K} of open subsets of X with Ha c G, for each a < K.
Let YI(rc)= K + 1 be the one-point compactification of the discrete space Remembering Remarks 2.3, we recognize immediately that the following three statements are equivalent. For a family d =’ {A, I a < K} of subsets of a space X and the subset A = U { A , x { a } I a < K} of X x Y,(K), (1) d is locally finite in X ; (2) A n (X x {K}) = 0; (3) the upper intersection of d at K is empty. Thus we obtain easily the following from Theorem 2.16 and the lemma above (cf. Theorem 1.14). K.
2.23. Corollary (Alas [19711). X x Y I @ )is normal if and only if X is countably paracompact and K-collectionwise normal. Following Starbird [1974], we call a space X a test space for a topological property P if the product X x Y is normal if and only if Y has the property P,namely, N ( { X } ) = P. Summing up, in connection with test spaces, the results mentioned above from Section 1 and some questions, we have the following Table 1. Now, let us introduce the well-known example of a K-Dowker space given by Rudin [1985], which plays important roles in the theory of product spaces: for K = w ,in virtue of the existence of an w-Dowker space, we can conclude that normality does not imply countable paracompactness, settling a long unsolved problem; for an arbitrary K, the existence of K-Dowker space is an essential half of Theorem 2.13; furthermore, the K-Dowker space X, constructed below has a fairly strong property so that it serves to clear relationship between various properties of spaces (cf., e.g., Facts 2.14(6); Rudin
M.Atsuji
104
TABLE I Test space I" D" K +
I
I
Property
Refer to
normal and K-paracompact
Theorem 1. I3
ltollectionwise normal and K-paracompact
Theorem 1.14
normal and countably paracompact
Corollary 1. I7
normal and property g ( K ) countably paracompact and K-collectionwise normal normal and property W K K-shrinking
Corollary 2.19 Corollary 2.23
)
[1983, 1984, 19851). Though its proof is not simple, we shall show the details of X, because, besides the importance observed above, a nicer example of a K-Dowker space is wanted; cardinal functions of X, seem larger than necessary as Rudin herself feels (cf. Rudin [1985]). Let K be an infinite regular cardinal and IC+ the successor cardinal for K. For a cardinal I we say that a Hausdorff space X is I-ultraparacompact if every open cover with the cardinality 1of X has a disjoint open refinement. Trivially, a I-ultraparacompact space has the property 9 ( I ) and the property A?@),and it is I-shrinking, I-paracompact and normal. 2.24. Definition. Let {A, I a < K}be a monotone increasing family of regular cardinals with I, = A: and I,, > K + . X, is the subspace of the box product (A, 1) consisting of mapsf for which there exists a /3 < K such that K < cf(f(a)) < I, for all a < K.
+
We denote F =
n (I, + 1)
U
and C =
{YEFly(.)
< I, for all a <
K}.
Forfand g in F we definef < g (orf < g) iff(a) c g(a) ( o r f ( a ) < g(a)) foralla < ~ . F o r g< f i n F w e l e t B , / = {hEX,lg < h < f } , a n d i t i s a
Normality of Product Spaces I
105
basic open neighborhood off in X , iff is in X,. It is easy to see that X , is Hausdorfland has the property that the intersection of < K open subsets of X , is open. For the proof that X , is a K-Dowker space we need three lemmas, the first ), of which is the one for the proof that X , does not have the property 9 ( ~and the other two are for A-ultraparacompactnessof X,. Note that in the first half of the proof of Theorem 2.28, actually, we verify that X , has a monotone increasing cover {G, 1 a < K} which cannot have any, not necessarily open, shrinking. 2.25. Lemma. Let a < K and G, = { f E X , If (B) < A, for all B 2 a in K}, and let H be a subset of X , with G, 3 R ( = the closure of H in X,). There exists an f E C with {g E C n HI f < g} = 8.
Proof. Index the set of maps P = {p:(a + 1) -,A,lp(B) < A, for all fl < a} as { p, I u < A,} in such a way that eachp isp, for A,-many u’s. If our lemma is false, then, for each u < A,, making an extension ofp, to K so that it is in C, we can inductively find a g, E C n H for each u < 1, such that P,(B) < g,(B) for B G a and sup{g,(B)It < 4 < g,(B) for B > a in K. Then we definef E X , byf(B) = A, for B < a andf(B) = sup{gU(/3)Iu < A,} for B > a in K. f is in X , and not in G,, so there exists a g < f i n F with Bd n H = 8.For each B, a < B < K, there exists a u, < 1, with g(B) < g,,(B). There exists a uo < 2, such that uo > sup{o,la < fl < K} and p,(B) = g(B) for all B < a because of g I (a + 1) E P. We already have gu0 in C n H, which is also in Bg,, a contradiction. 0 Let A < cf(K) and { 17, I p < A} be a cover of X,, and let V be an open subset of X,. We define the top of V: t V = t E F by t(a) =
sup{f(a)lf E V >
for all a < K, and we denote for y <
V,
= {g E VI cf(g(a)) <
K
1, for all a <
K}.
2.26. Lemma. ZfA < cf(K) and cf(t(a)) > K for all a < K, then for every y < uthereexistanf, < tinFandap, < Asuchthat{gE V , l f , < g} c U,,?.
Proof. Suppose that our lemma is false. Then, for some y < K and for any f E F w i t h f < t a n d f o r a n y p < 1 , w e c a n f i n d a g E V, - U,withf < g.
106
M.Atsuji
We denote A = {a < KIcf(t(a)) Q A,,}.
For any a E A we choose a T, c t(a) which is cofinal with t(a) and I T,I = cf(t(a)). Since I,, = I; and I < K < I,, we can index Q = nUEA T, as{p6,,lb < h a n d p < 1)insuchawaythatpE Qisp6,,forA,,many6’sfor each fixed p. Define the order /3v < 8 p lexicographically. By the induction on the index, we choose gap E V, - U,,for all 6 < A,, and p < I as follows. Definefa,, E FbYf,,(a) = P6p(a) for a E A andfa&(a)= sup{g,v(a) I Bv < dp} for a E K - A. Since gayis in V,, we have g,,(.)
Q r(a)
and cf(g,v(a)) < A,, < cf(t(a))
for a E K - A, and sofa,,(a) < t(a) for a E K - A because the number of /3v is less than A,,, and hence,fa,, < t . By our assumption we can select gJp> fa,, in V , - Up. Finally, define g in F by g(a) = t(a) for a E A and g(a) = sup{g,,(a) 16 < A,,, p < I}for a E K - A; then it is in X, and in some Urn, and hence there exists an h < gin Fwith Bhg c Urn.Since ~r, < I < K , there exists a 6, < A,, with g,#,(a) > h(a) for a in K - A; and by our choice o f p we can find 6 > sup(8,l a E K - A} withp6,(a) = h(a) for a in A. For a in A we have h(a) = p6p,3(a) =
< g6m(.)
t(a) = d a ) ,
and so h < gJrn< g; hence we have g6rnE Bhgc Urn,a contradiction. 0 2.27. Lemma. Suppose I < cf(u). Let t be the top of an open subset V of X, which is covered by {U,, I p < I}.Ifcf(t(a)) > K for all a < K , then there exist anf < t i n F a n d a p < Isatisfying{gEVlf < g } c Up.
Proof. For every y < K we have& < t and 4 < I stated in Lemma 2.26. Define f by f(a) = sup{&(a)ly < K } for all a < K; then f < t . Since I < cf(K), there exists a p < I such that p = c ~ yfor K many y’s less than K. Supposef < g E V . Since V = U,<, V , by the definition of V , , g is in some V,, v < K, and there exists a y > v in K w i t h 4 = p, and s o g is in V, 3 V,. Since& < f < g, g is in Uy.= U,,by Lemma 2.26. 0 2.28. Theorem. space.
If I < cf(K),
then X, is a I-ultraparacompact Ic-Dowker
Proof. , Let us first show that X,does not have the property 9(ic).We define an open set for every a < K C, = {fE X,If(fi) < I, for all /3 2 a in K}.
Normality of Product Spaces I
107
Suppose that the monotone increasing cover { GaI a < K } of X , has a shrinking {HaI a < .}. By Lemma 2.25 there exists anf. E C for every a < K such that
(*I
{ g E C n HUIL < g> =
8.
We may consider j#?) > sup{f,(B) I a < y} for every y < Define g in F by g(B) = sup{fa(b)Ia < }.
+ K+
for B <
K
and
B<
IC.
K;
then we have g(B) < I,, cf(g(B)) = K + and g E C n X,, and hence, g is in C n Hu for some a < K, which contradicts (*). Let us next show that X,is I-ultraparacompact. Let 4 = {Up1 p < I}be a cover of X,.We shall construct an open disjoint refinement W of 9. For this purpose we first define, by the induction on 0 < K + , a cover W, of X, consisting of disjoint open sets with the property that 7 < 0 < K + and W E Woimply the existence of V E WTsuch that . (1) w c V; (2) if W c t Up for any p < I, then tw # t,; (3) if V c U,, for some p < 2, then W = V. We let Wo= { X , } , and assume that W7has already been defined for all z < Q, where 0 < Q < u+. (i) When Q is a limit, then W, = W,l W , E WT for z < 0 and W, 3 W , for e < T < Q } is a desired cover satisfying the three conditions above because the intersection of < K open sets is open. (ii) When Q = T 1, then for each V E W7we define “w; as follows. If V c U,,for some p < I , then we define W, = { V}. Otherwise, dividing into two cases, we shall define W, below as an open cover of V by disjoint subsets of V which do not have t , as their top if they are not contained in any Up. If we obtain such W, for every V E *w,, then W, = U{WvI V E WT}is a desired cover of X , satisfying the three conditions. Now suppose that V Up for all p < I, and denote t = t,. Case 1: cf(r(a)) < K for some a < K. Let R = {adld < cf(t(a))} be a monotone increasing closed family of ordinals cofinal with t(a), where a, = 0.Foreachy < cf(t(a))weletW, = { f ~Via, < f ( a ) < a,+,}.Since cf(f(a)) > K by f~ X , and cf(a,) < K for limit y’s and also R is closed, W, = { W,ly < cf(t(a))} is a desired cover of V. Case 2: cf(t(a)) > K for all a < K. By Lemma 2.27 there exist anf < t in F and’a p < I satisfying {g E Vlf < g} c U,,. For S c K we denote W, = {g E Vlf(a) < g(a) for a E S and g(a) < f(a) for a E K - S}. If S # K, then tw, # t, hence W, = { W, I S c K } is a desired cover of V.
{n7<,
+
+
M.Arsuji
I08
Now we define W = { W E W,ICJ<
IC+
and W c U,forsomep < A},
then this is a desired refinement of 4. In fact, to see that W covers X , , take f i n X , . For any CJ < IC+ there exists a W, E W , including5 If T < 6,then, by condition (l), there exists a W , containing W,. For each a < IC we have t,(a) 2 t,(a), where ts = tw,, and so there exists a T= < K + such that t J a ) = t,(a) for all CJ > in K + . For r0 = sup{z,)a < K} < K + and for CJ with T,, < CJ < IC+ and W, 3 f we have t,, = t,. By condition (2), W, c U, for some p < I, and W, belongs to W . Obviously, W is made up of disjoint open sets by its construction and condition (3). Trivially, W refines 4. 0
2.29. Comment. Though it is an essential notion in the theory of product spaces, an application of “upper intersection” is not common. It has occasionally appeared explicitly and played an important role in the literature on product spaces (cf., e.g., Frolik [1960]and Isiwata [1964]). On the other hand, the notion of upper intersection plays an interesting role in another area of general topology. From a point of view of nonstandard analysis, Bernstein [1970] first defined and studied an 9-compact space, 9 a free ultrafilter on o,which was defined by use of an 9-limit point of a sequence of points {x,}. A point x is defined to be an 9-limit point of the sequence if x is in the set
namely, in our terms, an 9-limit point is a point in the upper intersection of the sequence. An 9-limit point is, if it exists, of course uniquely determined in Hausdorff space. 9-compactness is situated between compactness and countable compactness, and it is infinitely productive. Ginsburg and Saks [1975] generalize the notions of Bernstein. They call a point x an 9-limit point, 9 E Po - o,of the sequence {A, I n E o}of subsets of a space if x is in
n oxA,,
FEf
nsF
namely, in the upper intersection of the sequence. They study countable compactness and pseudocompactness of product spaces by making use of generalized 9-compactness. We also have interesting works of J. E. Vaughan on, specifically product spaces of, countably compact spaces by use of 9-limit points (cf., e.g., Vaughan [1984]).
Normality of Product Spaces I
109
3. The second and third of Morita’s three conjectures Before discussing Morita’s Conjectures I1 and 111, we show a few results on product spaces with a metric factor. Suppose that Mis a metrizable space with a a-locally finite base W = UW,, such that for every n a locally finite cover W,, of M has a star refinement a,,+, , a diameter of every member of which is less than l / ( n 1) with respect to some metric on M. Let W be the family consisting of members of W which are not discrete subspaces of M, and let W; = W‘ n W n. Let us show that we can inductively select a point p BE B for each B E a’in such a way that { pB1 B E W’} consists of distinct points. Well-order 49; = {litna}for each n and define an order nu < m/3 lexicographically. Suppose that we have already selected a desired point in En, for all na < m/3. Since Emsincludes an accumulation point which has a neighborhood meeting only finitely many members of U{W,,I n < m ) , Emsincludes a point p,,, which is not in {pen,I na < m g } . Thus we have a set
+
zo
=
{PeIBEW’J
of distinct points. We define an order on Z , by pel For any accumulation point z of A4 we denote 2, =
{PB
E ZOlZ E
< p,,
for El
3
B,.
B E @’}
and
9,=
I z is an accumulation point in M j.
It is easily seen that if M is not discrete, then a space with the property 9 ( Z o , 9,is )countably paracompact. 3.1. Theorem. Let M be a nondiscrete metrizable space. Ifthe product space X x M of a space X with M is normal, then X is a normal space with the property W Z o ,9,).
Proof. First, we show that Z , is a fit set. Let z be in 2, - 2,. For any p,, E Z , there exist an no with B, E W;, and an n, 2 no with St*(z, ail)c B,. and a neighborhood V of z such that {pel B E We find B, 3 z in U,,<,,l Wi}n V = 8. For everyp, in B, n V it holdsp, 2 p,,, and hence Z , is a fit part of Z, for z. Similarly for z E Z,, and thus Z, is fit. Next, we select a set Z, = { q B E BI B E W’}in the same way with Z , such that Z, n Z, = 8.Z, and 2, are similarly ordered, and Z, and Z, satisfy all the conditions required in Corollary 2.10. 0
M . Atsuji
110
In order to prove the converse of the theorem above, we shall use a lemma which is a very special case of a theorem given by Morita [1962].
3.2. Lemma. The property of being countably paracompact normal space is hereditary with respect to Fa sets.
Proof. The proof is not hard, so we only sketch it. First, we verify inductively that a set A which is a countable union of normal closed subspaces F, of a space Xis also normal. Next, if Xis normal and countably paracompact, then F, x I are normal closed subspaces of A x I by Theorem 2.21, and A x I is normal by the fact verified above, hence A is normal and countably paracompact by Theorem 2.21 again. 0 3.3. Theorem (Morita [1964]). If X is a normal space with the property 9 ( Z o ,To), then the product X x M of X with a nondiscrete metrizable space M is normal and countably paracompact. Proof. Let 0 = (0, I i E o}be a cover of X x M. We denote for B E 93 and
k
E
w
GB,k - U { L I L is an open set with L x B c O k }
and GB
=
U{GB,kl
‘We let G,, = GB for B U{G,,lz
E
E
O>.
93‘; then {G,, JpBE Z , } is increasing, and
BEW}
=
X
for any accumulation point z in M. By the property 9(Z,,, Z,), there exists an open Fa set HPBof X for each B in A?’ such that c G,, and
F,,
~ { H , , ) Z BE E a’}
=
X,
for any accumulation point z. We denote HB = HpBfor B E a‘ and HB = X or 0 for R F W - W according to B consisting of one point or otherwise; then we have HB c GBfor all B E 93 and U { H B l z ~ B ~ 9= 3 }X
for any point z in M. Since X is normal and countably paracompact, HB is also, normal and countably paracompact by the lemma above, and {GB.k n HB I k E w } is a normal cover of HB, and the cover {(GB,&
HB)
E w,
Normality of Product Spaces I
111
of H , x B is also normal. Since 9 is a-locally finite, X = (H,x B1 B E a} is a a-locally finite cover of X x M , and so it is a normal cover because H, and B are the cozero sets. {(GB,kn H,) x B I k E w } is, as stated above, a and normal cover of H, x B which refines the cover {(H, x B) n ok 1 k E a}, 0 is a normal cover by Lemma 1.9. 0 The definition of the property 9 ( Z o To) , is motivated by Characterization Theorem 4.1. in Przymusihski [19841, and one of fundamental notions of the theorem is essentially a basic cover of X x M with a special refinement defined by Morita [ 1963b], which is also one of main tools throughout Hoshina’s Chapter 4. 3.4. Definition (Morita [1964]). A space Xis said to be a P-space if, for any cardinal number K 2 1 and for every decreasing family {F,I a E r}, r = do, of closed subsets of X , there exists a family {GoI a E r}of open subsets of X such that F, c G, for each d E r and n{F,,.1 n E w } = 0 implies n{G,,. In E w } = 0 for every f E K”.
The following two theorems will be verified in Hoshina’s Chapter 4. As the converse of the theorems, Morita formed his conjectures. It is pointed out in Morita [1977] that Morita’s Conjecture I11 is affirmatively answered if his Conjecture I1 is, because, as is easily seen, any normal P-space is countably paracompact. Furthermore, it is known that every perfectly normal space is a P-space.
3.5. Theorem (Morita [1964]). In order that the product space X x M is normal for any metrizable space M , it is necessary and suficient that X is a normal P-space. Morita’s Conjecture 11. Let X be a topological space. X is metrizable if and only if the product space X x Y is normal for any normal P-space Y.
3.6. Theorem (Morita [1963b]). Let M be a metrizable space. In order that the product space X x M is normal for any countably paracompact normal space X , it is necessary and suficient that M is a a-locally compact space. Morita’s Conjecture In. Let X be a topological space. X is metrizable and a-locally compact if and only if the product space X x Y is normal for any countably paracompact normal space Y.
112
M. Atsuji
Morita’s Conjectures I1 and I11 are not completely solved yet. They are affirmatively answered under the set-theoretical assumption V = L (ChibaPrzymusinski-Rudin [1986]). However, they may be solved in ZFC because the assumption is used only in constructing an example; or they may have a counter example in some model for set theory, and thus Morita’s Conjecture I1 may be undecidable as Chiba, Prszymusinski and Rudin conjecture (cf. ibid.). So at present, we like to begin by mentioning two results on the conjectures obtained in ZFC by T. and K. Chiba. Since many important propositions on Morita’s Conjectures have too long proofs to cite in this chapter, we shall state them without proof. But it will be worthy to note that proofs of almost all key results heavily depend, directly or indirectly, upon the well-known example “H” of Bing [1951] or its modification in such a way that, under some assumption, if X is not metrizable, then X x Y is not normal for some perfectly normal space Y , the example H of Bing. The first remarkable step to Morita’s Conjectures I1 and I11 is the following theorem. 3.7. Theorem (T. and K. Chiba [1974]). Let X be a separable space. The following are equivalent: (a) X is metrizable. (b) X x Y is normal for any normal P-space Y . (c) X x Y is normal for any perfectly normal space Y .
The second large step to the conjectures will be this following theorem. 3.8. Theorem (K. Chiba [1980]). Let X be a closed image of an M-space. Then X is metrizable if and only if X x Y is normal for any normal P-space Y .
In the rest of this section we introduce the main results from ChibaPrzymusinski-Rudin [19861, which include interesting results not only on Morita’s conjectures, but also on related topics on perfectly normal spaces. 3.9. Definition. A Bing-type example H ( K ) , IC a cardinal > o,is a normal space With the following properties: (1) H ( K )contains a closed and discrete subset D, I D I 2 IC, consisting of functionsf: T + o for some T ;
Normality of Product Spaces I
113
(2) basic neighborhoods of elementf E D have the form B( f , F ) , where I; is a finite subset of T and B ( f , F ) n B(g, G ) # 8 iff1 F n G = gl F n G. The following proposition is fundamental in the paper referred to above. 3.10. Proposition. Suppose that (Y, F)is a nonmetrizable topological space and K = 19-I. There is a perfectly normal Bing-type example H ( K ) such that if Y x H ( K ) is normal, then there exists a monotone increasing family { Y, I a < o1} of sub.reis of Y such that
In particular, Y has an uncountable tightness.
The following example is given by BeSlagiC and Rudin [1985] under the assumption V = L. 3.11. Example ( V = L). For every cardinal 1there exists a collectionwise normal P-space X , having a monotone increasing open cover { U, I a < wI} with the property that if C,, is a closed, in X,,subset of U, for each u < o, and /3 < 1,then U{CasIa <
0 1 ,
B<
A}
+
3.12. Theorem ( V = L). Morita’s Conjecture I1 holds, namely, a space X is metrizable if and only if X x Y is normal for any normal P-space Y.
Proof. It suffices, by Theorem 3.5, to prove the “if” part. Suppose that X x Y is normal for any normal P-space X , and Y is not metrizable; then there exists a monotone increasing family { Y,la < o,}of subsets of Y satisfying relation (*) in Proposition 3.10. Let I YI < A. Index Z = u{Y,la < ol}= { y , l B < l}.Foranyyswetakea = min(y(Y,3ys)and rewrite y a p for y , . We define an order ap < yd lexicographically; then { y,, I a < wI,/3 < A} is primarily fit at a point y in the left-hand side of relation (*) in Proposition 3.10. Let X , and {Vala < ol} be the space and its cover given in Example 3.1 1. We let La, = X , - U,,, where U,, is the U , corresponding to y,,; then {L,, I yaSE Z } is the monotone decreasing closed family with the empty upper intersection at y by Lemma 2.6. Since the product X , x Y is normal, by Corollary 2.9 there exists an open family {DapIy,, E Z } with the empty intersection such that L,, t D,,,which is impossible. 0
114
M.Aisuji
3.13. Theorem ( V = L). Morita’s Conjecture I11 holds, namely, a space X is metrizable and a-locally compact if and only if X x Y is normal for any normal and countably paracompact space Y .
Proof. Suppose that the product space of X with any normal and countably paracompact space is normal. Then, since a normal P-space is countably paracompact, the product of X with any normal P-space is normal, so X is metrizable by Theorem 3.12, and hence X is a-locally compact by Theorem 3.6. Making use of a previous notation, we can state Theorems 3.12 and 3.13 in other words as follows. Let X be the class of all normal P-spaces (respectively all countably paracompact normal spaces); then, under V = L, N ( X ) = A = the class of all metrizable spaces (respectivelyN ( X ) = the class of all a-locally compact metrizable spaces). ‘On the other hand, as we have seen in many places above, perfect normality and being a normal P-space are closely situated each other on the problem of metrizability of a space X whose product X x Y is normal for any space Y with the properties. Therefore, it will be natural to pose the problem of characterization of N ( 9 ) , 9 the class of all perfectly normal spaces. Trivially, N ( 9 )2 A; and it seems that N ( 9 )is nearly A; but not the same as we shall see below. This problem is not answered yet, so we shall here introduce a little more detail on perfect normality. Let us recall the space Y in Ohta’s Example 2.18. As we point out there, the product X x Y is normal if and only if X is a normal space with the property 9 ( ~ : in~particular, ); it follows from Facts 2.14(12) that X x Y is normal for every perfectly normal space X , namely, Y belongs to N ( 9 ) . Meanwhile Y is obviously a regular nonmetrizable a-discrete space. This observation leads to the inequality N ( 9 ) # A. (This inequality was first disclosed by T. and K. Chiba [1974] giving another example.) In the meantime, Ohta’s example shows us an exciting other aspect of Morita’s Conjecture 11. Namely, if there exists some model for set theory in which every normal P-space has the property 9 ( Z )for any ordered set Z with I Z I = o,, then Ohta’s example will be a counter example for Morita’s Conjecture 11, and thus, together with Theorem 3.13, the conjecture becomes undecidable. Now, denote by c ( X ) and /(A’) the Souslin number and the Lindelof number of a space X. The following proposition and corollaries are a considerable improvement of results presented in T. and K. Chiba [1974], and in K. Chiba [1974, 1976, 19801.
Normality of Product Spaces I
115
3.14. Proposition. Every space X in N ( 9 ) is stratiJiable and w ( X ) = c(X) =
&(a.
3.15. Lemma. Let 4. be in N ( 9 )for every i < a.Then the product space n,,, X, x Z is perfectly normal for every perfectly normal space Z.
Proof. It suffices to show by induction that nicn X, x Z is perfectly normal fiJr every n < o (cf., e.g., Przymusinski [1984, Theorem 6.21). Assume that S = niG, X, x Z is perfectly normal; then T = X,,, x S is normal. Since X,,, is stratifiable, it has a o-discrete network ( A , , l n < o},A, = {M,,,1 c1 E Q,} a closed discrete family. Denoting L,, = U{ VI V is an open subset of S with M,, x V c G) for a given open set G c T, and observing that {Mnax L,, I n < o,c1 E a,} covers G, we can easily check that T is 0 perfectly normal. The following two corollaries are immediately derived from this lemma; note that a perfectly normal space is hereditarily normal.
3.16. Corollary. If X is in N ( 9 ) ,then X"' x Z is perfectly normal for every perfectly normal Z. 3.17. Corollary. N ( 9 )is countably productive and hereditary. 3.18. Corollary. Every space X in N ( 9 ) which is locally ccc or locally Lindelof or locally p-space is metrizable.
Proof. X is locally metrizable and paracompact by Proposition 3.14, and 0 .thus it is metrizable. The next is an immediate consequence of Proposition 3.10.
3.19. Corollary. Every space in N ( 9 )with countable tightness is metrizable. 3.20. Corollary. Every k-space X in .N(9) is metrizable.
1 I6
M . Atsuji
Proof. Let C be a compact subset of X. C is in N ( 9 )by Corollary 3.17, so it is metrizable by Corollary 3.18, hence X is sequential because X is a k-space, and thus the tightness of X is countable. 0
References Alas, 0. T. [I9711 On a characterization of collectionwise normality, Canad. Math. Bull. 14(1), 13-15. Alexandrof, P. and H. Hopf [I9351 Topologie I (Springer, Berlin). Arhangel'skii, A. V. [197I] On bicompacta hereditarily satisfying the Souslin condition. Tightness and free sequences, Dokl. Akad. Nauk SSSR 199, 1227-1230; English translation in: Soviet Math. Dokl. 12, 1253-1257. Atsuji, M. [I9771 On normality of the product of two spaces, in: General Topology and its Relations to Modern Analysis and Algebra, Part B, I V ; Proc. Fourth Prague Topology Symp., Prague 1976, 25-27. Benyamini, Y. and T. Starbird [I9761 Embedding weakly compact sets into Hilbert space, Israel J . Math. 23, 137-141. Bernstein, A. R. [I9701 A new kind of compactness for topological spaces, Fund. Math. 66, 185-193. BeslagiC, A. and M. E. Rudin [I9851 Set-theoretic constructions of nonshrinking open covers, Topology Appl. 20, 167-177. Bing, R. H. [I9511 Metrization of topological spaces, Canad. J . Math. 3, 175-186. Chiba, K. [I9741 On products of normal spaces, Rep. Fac. Sci. Shizuoka Univ. 9, 1-11. [1976] Two remarks on the normality of product spaces, Rep. Fac. Sci. Shizuoka Univ. 11, 17-22. [I9801 On the normality of product spaces, Math. Japon. 25, 559-565. [I9821 On the weak P-property of %products, Math. Japon. 27, 737-746. [I9841 On the weak 9-property, Math. Japon. 29, 551-567. [1985] Remarks on the weak @property of Z-product, Questions Answers General Topology 3, 1-9. [I9871 On the %property of a-products, Math. Japon. 32, 5-10. [I9881 A note on the weak @-property, to appear. Chiba, T. and K. Chiba [I9741 A note on normality of product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A 12, 165-173. Chiba, K., J. C. Przymusinski and M. E. Rudin [I9861 Normality of product spaces and Morita's conjectures, Topology Appl. 22, 19-32. Dieudonne, J. [1944] Une gknkralization des espaces compacts, J. Math. Pures Appl. 23, 65-76.
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Dowker, C. H. [I9511 On countably paracompact spaces, Canad. J. Math. 3, 219-224. Engelking, R. [19771 General Topology (Polish Scientific Publishers, Warsaw). Fleissner, W. G. [ 19841 The normal Moore space conjecture and large cardinals, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 733-760. Frolik, Z. [I9601 The topological product of two pseudocompact spaces, Czech. Math. J. 10, 339-349. Gillman, L. and M.Jerison [I9601 Rings of Continuous Functions (Van Nostrand, New York). Ginsberg, J. and V. Saks [I9751 Some applications of ultrafilters in topology, Pacific J. Math. 57, 403-418. Hoshina, T. 119841 Shrinking and normal products, Questiom Answers General Topology 2, 83-91. Ishikawa, F. [I9551 On countably paracompact spaces, Proc. Japan Acad; 31, 686-687. Isiwata, T. [I9641 Some classes of countably compact spaces, Czech. Math. J. 14, 22-26. Katuta, Y. [I9711 On the normality of the product of a normal space with a paracompact space, General Topology Appl. 1, 295-319. [ 1977al Paracompactness, collectionwise normality and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku See. A 13, 165-172. [1977b] Characterizations of paracompactness by increasing covers and normality of product spaces, Tsukuba J. Math. 1, 27-43. Kombarov, A. P. [I9721 On the product of normal spaces. Uniformity on Z-products, Dokl. Akad. Nauk S S S R 205, 1033-1035; English translation in: Sovier Math. Dokl. 13, 1068-1071. Michael, E. [I9531 A note on paracompact spaces, Proc. A M S 4, 831-838. Morita, K. [I9611 Note on paracompactness, Proc. Japan Acad. 37, 1-3. [I9621 Paracompactness and product spaces, Fund. Math. 50, 223-236. [1963a] On the product of paracompact spaces, Proc. Japan Acad. 39,559-563. [1963b] Products of normal spaces with metric spaces 11, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 8, 87-92. [I9641 Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. (19751 Cech cohomology and covering dimension for topological spaces, Fund. Math. 87, 31-52. [I9771 Some problems on normality of products of spaces, in: General Topology and its Relations to Modern Analysis and Algebra, Part B, I V ; Proc. Fourth Prague Topology Symp., Prague 1976, 296-297. (198I] Theory of Topological Spaces (Japanese) (Iwanami, Tokyo). Nagata, J. [ 19851 Modern General Topology (North-Holland, Amsterdam).
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Navy, C. L. [19811 Nonparacompactness in para-Lindelof spaces, Thesis, Univ. of Wisconsin. Negrepontis, S. [I9841 Banach spaces and topology, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 1045-1 142. Nobeling, G. (19541 Grundlagen der Analytischen Topologie (Springer, Berlin). Nogura, T. [I9761 Tightness of compact Hausdofispaces and normality of product spaces, J. Math. SOC. Japan 28, 360-362. Ohta, H. [I9811 On normal, non-rectangular products, Quart. J. Marh. Oxford 32, 339-344. [I9831 On a part of problem 3 of Y. Yasui, Questions Answers General Topology 1, 142-143. [1985/86] Katuta’s questions, normality of products, Quesrions Answers General Topology 3, 11 1-123. [1987] Private Communication to Y.Yasui. Przymusinski, T. C. [1984] Products of normal spaces, in: K. Kunen and J. E. Vaughan, eds., Handbook of SerTheoretic Topology (North-Holland, Amsterdam) 78 1-826. Rudin, [I9711 [I9781 [I9831 [I9841
M. E. A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. K-Dowker spaces, Czech. Math. J. 28(103), 324-326. Yasui’s questions, Questions Answers General Topo/ogy 1, 122-127. Dowker spaces, in: K. Kunen and J. E. Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam) 761-780. [I9851 K-Dowker spaces, in: I. M. James and E. H. Kronheimer, eds., Aspect of Topology, London Math. SOC. Lecture Note Ser. 93 (Cambridge Univ. Press,Cambridge) 175-193.
Starbird, M. [I9741 The normality of products with a compact or a metric factor, Thesis, Univ. of Wisconsin. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. A M S 54, 977-982. Tamano, H. [I9601 On paracompactness, Pacijic J. Math. 10, 1043-1047. [1962] On compactifications, J. Marh. Kyoto Univ. 1, 161-193. Telgarsky, R. [I9711 C-scattered and paracompact spaces, Fund. Marh. 73, 59-74. [1975] Spaces defined by topological games, Fund. Math. 88, 193-223. Tukey, J. W. [I9401 Convergence and Uniformity in Topology, Ann. of Math. Studies 2 (Princeton Univ. Press, Princeton, NJ). Vaughan, J. E. (19841 Countably compact and sequentially compact spaces, in: K.Kunen and J. E. Vaughan, eds., Handbook of Ser-Theoretic Topology (North-Holland, Amsterdam) 569-602. Yajima, Y. [1986] The shrinking property of Z-products, Questions Answers General Topology 4, 85-96.
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Yasui, Y. [1972] On the gap between the refinements of the increasing open coverings, Proc. Japan Acad. 48,8690.
[1983] On the characterization of the a-property by the normality of product spaces, Topology Appl. 15, 323-326. [I9841 A question of G. Gruenhage and E. Michael concerning the shrinkable open covers, Questions Answers General Topology 2, 124-126. [I9861 Some characterizations of a 8-property, Tsukuba J. Math. 10, 243-247. [1988a] Some remarks on the shrinkable open covers, Math. Japon., to appear. [l988b] Another characterization of some refinement of a monotone increasing open covering, to appear. Zenor, P. [1970] A class of countably paracompact spaces, Proc. AMS 42, 258-262.
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K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 4
NORMALITY OF PRODUCT SPACES I1
Takao HOSHINA Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
Contents 1. Products of normal spaces with metric spaces. . . . . . . . . . . . . . . . . 2. Products of normal spaces with generalized metric spaces . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 140 158
In this chapter we shall begin with the normality of the product of a normal space with a metric space and discuss related results, and afterwards extend our concern to the case of the product of a normal space with one of certain generalized metric spaces. All spaces are assumed to be Hausdorff, and maps continuous and onto unless otherwise specified. N denotes the set of positive integers.
1. Products of normal spaces with metric spaces In general, the product X x Y of a normal space X and a metric space Y need not be normal. This was shown by Michael [I9631 by constructing a hereditarily paracompact space M , which is called the Michael line, such that M x P is not normal, where P is the space of irrationals (Nagata [1985, Example V.21). On the other hand, in [1964] Morita, by introducing the notion of P-spaces, established his theorem that for a product X x Y to be normal for any metric space Y, it is necessary and sufficient that Xis a normal P-space. Then the notion of P-spaces, as well as the method of his proof of this theorem, greatly influenced many later studies in the area of normality or paracompactness of product spaces. Then, in Morita [1963] the method
122
T. Hoshina
above was sharpened further to study the normality of the product of two given spaces, one of which is metric. On such a product the notion of basic covers was defined and played an essential role. In the present section, by introducing first the notion of basic covers as a central concept, we shall explain several results on the normality of products with a metrix factor and other related results. Let Y be a metric space. Then there exists a sequence {V,,} of locally finite open covers of Y such that each member of V,,has diameter < l/n. Hence, In E N} forms an nbd basis at each point y of Y. Let V,,= {St(y , f,,) {V,,IaER,,}. Letusput
W(aI,. . , , a,,) =
v,, n
* *
n V,,"
for a, E R,, . . . , a,, E a,,.
Let X be a space. Then, following Morita [1963], an open cover Y of X x Y is said to be a basic cover if Y has the form Y = {G(a,, . . . , a,,) x W(a,,. . . , a,,) I a, E R,,
v = 1 , . . . , n ; n N} ~ with open subsets G(a,, . . . , a,,) of X such that G(a,, . . . , a,,) c G(a,, . . . , a n , an+,)
f o r a , E R , , v = 1, . . . , n , n + 1.
(1
.a
Let Y be a basic cover in (1.1). If there exists a family {F(a,, . . . ,a,,)1 a, E Q, v = 1, . . . , n; n E N} of closed subsets of X such that (1.3) U{F(a,, . . . , a,,) x W a , , . . . , a,,) I a, E Q,,
v = 1, . . . , n ; n ~ N } = X x y , then we shall say that Y has a special refinement.
(1.4)
1.1. Lemma (Morita [1963]). Let X be a normal space. Thenfor a basic cover Y in (1.1) the following are equivalent: (a) Y has a special refinement. (b) There exists afamiry {F(a,, . . . ,a,,)I a, E R,, v = 1, . . . ,n; n E N} of open F;-subsets of X satisfying (1.3) and (1.4). (c) There exists a family {F(a,, . . . ,a,,) I a, E R,, v = 1, . . . ,n; n E N} of F,-subsets of X satisfying (1.3) and (1.4).
Normalily of Product Spaces II
I23
Proof. The implications (a)-(b) and (b)*(c) are obvious. Assume (c). Then there exist closed subsets C(a,, . . . , a,,; k) of X such that F(a,, . . . , a,,) = U?=,C(a,, . . . , a,,; k). If we put
< n, k
K(a,, . . . , a,,) = U{C(a,,. . . , ai;k)lj
Q n},
then K(al, . . . , a,,) is a closed subset and, clearly, K(al, . . . , a,,) c G(a,, . . . , a,,) by (1.2) and (1.3). Let (x, y ) be any point of X x Y. Then (x, y ) is contained in some F(a,, . . . , a,,) x W(a,,. . . , a,,), and x E C(a,, . . . , a,,; k) for some k. Let m = max{n, k}, and choose a , , + I E Q , , + l, . . . , a,EQ,sothatyE Fa,,j= n 1, . . . , m.Wethenhave
+
( x , y ) E K(al, . . . , a,) x WaI, . . . , a m ) *
Thus, {K(a,, . . . , a,,) x W(a,,. . . , a,,)Ia, E Q,, v covers X x Y. Hence, (a) holds.
=
1,
. . . , n;
n E N}
0
For results on normal covers we refer to the preceding chapter or often to Chapter 2: “Extensions of Mappings 11”.
1.2. Theorem (Morita [1963]). Let Xbe a normalspace and Y a metric space. Then a basic cover Y in (1.1) has a special rejbement ifl it is a normal cover. Proof. If Y has a special refinement, by Lemma 1 . 1 , there exist open F,-subsets F(a,,. . . , a,,) satisfying (1.3) and (1.4). Since Xis normal, each F(aI, . . . , a,,) is a cozero-set of X. Hence,
{F(a,, . . . , a,,) x W(a,, . . . , a,,)Ia, E R,,
#‘ =
v = 1,
. . . ,n ; n ~ N }
is a a-locally finite cozero-set cover refining Y. Hence, Y is normal. Conversely, suppose Y is normal. Then there exists a locally finite open cover A = {MA I A E A} of X x Y such that 2 = 13, E A} refines Y. Define M(a,, . . . , a,,; A) and F(al, . . . , a,,) by
{aL
Wa,,.. =
an;
A)
U{PI P open in X and P x W(a,, . . . , a,,) c MA},
..
9
an) = U{M(ai,
- .-
9
an; A l l
M(a,, . . . , a,,; A) c G(a,, . . . , a,,), A
E
A}.
Since M(a,, . . . , a,,; A) x W(a,,. . . , a,,) c MAand .A is locally finite, F(a,, . . . , a,,) is closed and, clearly, F(a,, . . . , a,,) c G(a,, . . . , a,,). To show (1.4), let (x, y ) be any point of X x Y. Then, for some A E A and
T. Hoshina
124
E R,, v = 1, . . . , n, we have (x, y ) E M,, HAc G(a,, . . . , a,,) x W(a,,. . . , a,,). Since (x, y ) E M Aand y E W(aI,. . . , a,,), we can choose an open nbd U of x and a, E R,, v = n + 1, . . . , m such that
a,
(x, Y ) E U x Wal, . . . , a,,
a,+l,
. . . , a,)
c MA.
We then have U c M(a,, . . . , a,,, a,,+,,
. . . , a,; A)
M(al, . . . , a,,, a,,+lr.. . , a,;
and
4
. . . , a,. a,,+l, . . . , a,) c M A . Hence (a1, . . . , a,,, a,+1, . . . , a,; 1) c G(a,, . . . , a,,); consequently, M(al, . . . , a,,, a.+l,. . . ,arn;4c F(aI,. . . , an, a,,+I,. . . ,a,). x W(aI,
Therefore,
Let Y be a metric space. Let A = { M AI I, E A} be an open cover of X x Y. For a, E R,, v = 1, . . . , n and 1 E A, we put G(aI, . . . an; 1) 9
= U{PlP open in X and
P x W ( a I ,. . . , a,,)
c
M A } , (1.5)
and
. . . , a,,; A) 11 E A}. {G(al,. . . , a,,) x W(a,,. . . , a,)la,
G(a,, . . . , a,) = U{G(a,,
(1.6)
E R,, Then it can be shown that v = 1, . . . , n; n E N} is a basic cover of X x Y and, moreover, satisfies the following property: for a:) E R,, v = 1, . . . , n, n E N,
n= I
u 00
+X
=
G(ay), . . . , a t ) ) .
n= I
If, in general, a basic cover Y in (1.1) satisfies (1.7), Y is said to be proper. We shall now prove the following theorem, which is due to Morita [1963] and will be fundamental in our discussions.
1.3. Theorem (Morita [ 19631). For a countably paracompact normal space X and a metric space Y the following are equivalent:
Normality of Product Spaces II
125
(a) X x Y is normal and countably paracompact. (b) Every basic cover of X x Y has a special refinement. (c) Every proper basic cover of X x Y has a special refinement.
Proof. (a)-(b): Since every basic cover Y is a a-locally finite open cover and since such a cover of a normal countably paracompact space is normal, by Theorem 1.2, Y has a special refinement. (b)*(c): This is obvious. (c)*(a): Assume (c). Let A = {Mk 1 k E N} be any countable open cover of X x Y. To prove (a), it suffices to show that 4 is normal. Define G(a,, . . . , a,,; k) and G(a,, . . . , a,,) as in (1.5), (1.6), where A = N. Then {G(a,, . . . , a,,) x W(a,, . . . , a,,)la, E R,, v = 1, . . . , n ; n E N} is a proper basic cover of X x Y. Therefore, by (c), there exist closed subsets F(a,, . . . , a,,) of X satisfying (1.3) and (1.4). Since Xis countably paracompact and normal and since F(a,, . . . , a,,) c G(a,, . . . , an), in view of ( I S ) , there exists a locally finite collection X ( a , , . . . , a,,) = {H(a,, . . . , a,,; k) I k E N} of cozero-sets of X such that H(a,, . . . , a,,; k) c G(a,, . . . , a,,; k)foreachk E N andF(a,, . . . , a,,) c U%(al, . . . , a,,). Let us put f l = {H(a,, . . . , a,,; k) x W(a,, . . . , a,,)Ia, E R,,
v = 1,. . . , n ; n , k ~ N } . Then X is a a-locally finite cozero-set cover of X x Y refining A since H(a,, . . . , a,,; k) x W ( aI ,. . . , a,,) c G(a,, . . . , a,,;
k) x W ( aI ,. . . , a,,)
c
Mk.
Thus, 4 is a normal cover of X x Y. Hence, (a) is proved.
0
The following theorem establishes the equivalence of normality and countable paracompactness for the product with a metric factor; the “if” part is due to Morita [1963a] (for his proof, see Ishii [1966]) and the “only-if” part due to Rudin and Starbird [1975]. We shall call this theorem Morita-RudinStarbird’s Theorem. 1.4. Theorem (Morita [1963a], Rudin and Starbird [1975]). Let X b e normal and countably paracompact and Y metric. Then X x Y is normal fi it is countably paracompact.
Proof. In either implication, it is sufficient to prove that X x Y satisfies Theorem 1.3(c).
T. Hoshina
126
To prove the "if" part, assume X x Y is countably paracompact. Let Y = {G(al,. . . , a,,) x W(a,, . . . , a,)la, E Q, v = 1, . . . , n; n E N} be a proper basic cover of X x Y. Let us put for each n E N G(al, . . . , v = 1,
. . . , n}.
Since { W(aI,. . . , a,) I a, E a,, v = 1, . . . , n} is locally finite, D,, is a closed subset of X x Y. Clearly, Dn2 D,,+,, and since Y is proper, we have I D,, = 8. Therefore, by assumption, there exist open subsets U,, of X x Y such that D,, c U,,, U,,,,c U,and flu,, = 8. Let us put, for a, E R,,v = 1, . . .,n,
n,"=
F(a,, . . . , a,,) = { x E XI({.}
x W(a I,. . . , a,,)) n
U,,= S } .
Then F ( a I , . . . , a,) is a closed subset of X, and (1.3) and (1.4) above are clearly satisfied. Hence, Y has a special refinement. To prove the "only-if" part we need a lemma, which is essentially due to Rudin and Starbird [1975]. 1.5. Lemma. Let 42 = { U,I 1 E A} be a a-locally finite collection of open sets of a space X such that each U, is not discrete. Then we can select two distinctpointsp, andq, from U,so that 1 it p implies { p,, q,} n {p,,, q,,} = 8.
Proof. Rewrite Q = UnaN %,, where %, = { V,,1 1 E A,,} is locally finite. Let A,, be well-ordered by < . For 1 E A,, and p E A,, define 1 < p by either n < m o r n = m,and 1 < p i n A,,. Let 1 E A, and assume, for each p < 1 with p E A,,, that p,, and q,, have been selected from U,,, so that the property of the lemma is true for p X and qr with 1' < 1. Let D = { p , , , q,,Ip < A}. Since D = {p,,, q,,;(pE A,,, n < m} v { p , , , q,,lp E A,,,, p < A } and since each %, is locally finite, D is discrete and closed. Hence, U, - D must be infinite and from it two distinct points p , and q, can be selected. Then the property of the lemma is satisfied for all p,,, q,, with p < 1. 0 h f of k r e m 1.4 (continued). ("only-if" part): Assume X x Y is normal. Define a subset of A, of a, x * - * x a,,for each n E N by A, = { ( a l , . . . , a,,) E 0, x . . * x
a,,l
W(aI,. . . , a,) is not discrete}.
Normality of Product Spaces II
127
By Lemma 1.5, for each (a,, . . . , a,,) E A,, there exist two distinct points p ( a l , . . . , a,,) and q(a,, . . . , a,,) of W ( a I ,. . . , a,,) such that (QI,
* *
3
an)
(PI, - *
9
+ (BI, . . -
Bm),
(a19. .
* { ~ ( a i ,-
Ir,) E A m
-
9
*
-
9
9
an),
a,,) E An, q(ai,
-.
7
a,,)}
. B,)} = 0. (1.8) Let Y = {G(a,, . . . ,a,,) x W(al,. . . ,a,,) I a, E Q, v = 1, . . . ,n; n E N} be a proper basic cover of X x Y. Let us put, for each n E N, n {P(Bl,
* *
I
Bm),
q(81,
*
*
9
An = U { ( X - G(a,, . . . , a,,)) x { p ( a , ,. . . , a,,)} I
(a,, *
* * 7
an) E A,,},
Bn = U{(X - G(a,, * (al, .
* * 9
an)) x {q(a,, *
* * 3
an)} I
- . , a,,) E An},
and put A = UA,,, B = UB,,. Since { W ( a l ,. . . , a,,)I(aI,. . . , a,,) E A,,} is locally finite, A,, is a closed subset of X x Y. Moreover, observe that A,, c D,,, where D,, = u { ( X G(a,, . . . , a,,)) x W(a,, . . . , a,,)la, E 9, n = 1, . . . , n}. And {D,,} is a decreasing sequence of closed sets with n,,D,, = 0 since Y is proper. Therefore, A is closed in X x Y. Similarly, B is closed in X x Y. Furthermore, from (1.8) it follows that A and B are disjoint. By assumption, there exist open subsets U and V of X x Y such that A c U , B c V and 0 n P = 0. Let us put, for (a,, . . . , a,,) E A,,, * * * 3
an)
= { X E Xl{x} x
E(aI, * =
and
* * 9
{XE
W ( a I , . . . , a,,) c X x Y - U},
an)
X l { x } x W ( a , , . . . , a,,) c X x Y - V},
Fl(a,, . . . , a,,) = D(a,, . . . , a,,) u E(a,, . . . , a,,).
Then D(a, , . . . , a,,) and E(a,, . . . , a,,) are closed, and so is Fl( a , , . . . , a,,). We have Fl(a,, . . . , a,,) c G(a,, . . . , a,) for ( a I ,. . . , a,,) E A,, and, for any (x, y) E X x Y with y nonisolated, there exists an (a,, . . . , a,,) E A,, such that (x, y) E Fl(a,, . . . , a,,) x W ( a I ,. . . , a,,). On the other hand, let I(Y) = { y E Y Iy is an isolated point}. Then, since I(Y ) is 'discrete, X x I( Y )is countably paracompact and normal. Applying Theorem 1.3(c) to the basic cover {G(a,, . . . , a,,) x ( W ( a l , . . . , a,,) n I(Y))Ia, E R,,v = 1,. . . ,n; n E N} o f X x I(Y),wecan takeclosed subsets
I28
T. Hoshina
F2(a1,. . . , a,,) of X such that
Finally, let us define F(a,, . . . , a,,) by
F ( a I , . . . , a,,) =
u Wa1, . . . ,
i= 1.2
F2(al,. . . , a,,)
an)
if ( a 1 ,. . . , a,,) E A,,, if ( a I , . . . , a,,) 4 A,,.
Then we have F(al, . . . , a,,) c G(a,, . . . , a,,), and U{F(al,. . . , a,,) x W ( a l , . . . , a,,)la, E R,, v = 1, . . . , n; n E N} = X x Y. Thus, Y has a special refinement. 0 Theorems 1.3 and 1.4 will play a central role to obtain further results. First we shall prove the theorem of Morita [ 19641mentioned in the introduction. Let m be a cardinal number. A space X is a P(m)-space if for any set R of indices with Card R (=cardinality of R) < m and for any collection {G(a,, . . . , a,,) I a, E R, v = 1, . . . , n; n E N} of open subsets of X such that G(al, . . . , a,,) c G(al, . . . , a,, a,,+I)for a, E R, v = 1, . . . , n, n 1, there exists a collection {F(a,, . . . , a,,)lavE R, v = 1, . . . , n; n E N} of closed subsets of X such that the conditions (a), (b) below are satisfied:
+
(a)
F(al, . . . , a,,) c G ( a I ,. . . , a,,) for a, E R, v = 1, . . . , n;
(b)
X = U,"=I F(a,, . . . , a,,) for any sequence (a,,) such that X = U,"=IG(al,. . . , a,).
A P-space is a P(m)-space for every m. A normal space X is a P(1)-space iff X is countably paracompact. By Morita [1964], for 2 < m < No,X is a P(2)-space iff it is a P(m)-space. If n < m, every P(m)-space is a P(n)-space. The theorem of Morita above now follows from the following. 1.6. Theorem (Morita [1964]). Xis a normal P(m)-space i f X x Y is normal for any metric space Y with weight w(Y) < m (m 2 No).
Proof. ("only-if part"): Assume that X is a normal P(m)-space. Then we first note that X is countably paracompact. Let Y be a metric space with w ( Y ) Z m. As before, Y has a sequence {V,,} of locally finite open covers V,,= { V,, I a E a,,} with Card R,, < m such that {St(y, V,)1 n E N} is a nbd basis at each point y of Y. Here, we may assume R = R,, for all n E N. Let
Normality of Product Spaces II
129
Y = {G(al, . . . , a,) x W(aI,. . . , a,)[ a, E R, v = 1, . . . , n ; n E N} bea proper basic cover of X x Y. Since G(a,, . . . , a,) c G(al, . . . , a,,, a,+1) for each a, E R, v = 1, . . . , n, n + 1 and X is a P(m)-space, there exist closed subsets F(al, . . . , a,,) of X such that (a) and (b) above are satisfied. Let (x, y ) E X x Y. Select V,,, E V,,for each n E N so that y E V,,,. Then y E W(al, . . . , a,), and since Y is proper, we have X = (J,"=IG(al, . . . , a,,). Hence, by (b), X = U =; ,F(aI, . . . , a,). Thus, for some a, E R, v = 1, . . . , n we have (x, y ) E F(al, . . . , a,) x W(aI, . . . , a,,), which shows that {F(a,, . . . , a,) x W(aI, . . . , a,) I a, E R, v = 1, . . . ,n; n E N} covers X x Y. Hence, in view of (a), Y has a special refinement. Thus, by Theorem 1.3, X x Y is normal. ("if" part): Assume the condition of the theorem is satisfied. Let {G(al, . . . , a,) I a, E R, v = 1, . . . , n; n E N} be any collection of open subsets of Xwith G(al, . . . , a,) c G ( a l , . . . , a,,, a,+1)with an index set R of Card R < m. Let B(R) denote the Baire space, that is, the set of all sequences a = (a,) of elements of R and a basic nbd of a = (a,) is defined by V(a,, . . . , a,) = = (pi) E B(R) I fli = al, . . . , 8, = a,}. B(R) is known to be a metric space with a metric d defined by
{a
Ilk if
d(a, 8) = for a = (ai), defined by
0
pi for i < k and
a, =
if ai =
8, for each
= (/Ii).B(R) has weight =
u
i
E
tlk
#
Bk,
N
m. Let S be the subspace of B(R)
m
S = { a = (a,) E B(R)l
G(al, . . . , a,) = X } .
n= I
Let B(al, . . . , a,,) = V(aI, . . . , a,,) n S. Note that n"k1V(c$), . . . , a(')) # 8 implies a?) = a!+') = = a?), i = 1, . . . , n. And since G(a,, . . . , a,) c G(al, . . . , a,,, a,,+,), we see that Q = {G(al, .
. . , a,)
x B(al,
. . . , a,)Ia,
E
R,
v = l,,..,n;nEN} forms a basic cover of X x S. Since X x S is normal, by Theorem 1.4, X x S is countably paracompact. Hence, by Theorem 1.3, there exist closed subsets F(al, . . . , a,) of X such that F ( ~ I* ,
9
-
a,) c G(al,
*
.
U{F(al, 01), x v = l,...,n;nEN} a
a
*
1
a
an),
.
an)lav E Q,
= XXS.
T. Hoshina
130
Then, clearly, (a) is satisfied. Suppose X = U:=IG(aI, . . . , a,,). Then a = (ai) E S. Let x be any point of X . Then ( x , a) E F(/l,, . . . , P,,) x B(P,, . . . , P,) for some PI, . . . , /I E,R. , Since a E B(/?,,. . . , B,,) implies PI = aI,. . . , P,, = a,,, we have x E F(al, . . . , a,,). Hence, X = U:=lF(al, . . . , a,,), which proves (b). Thus, X i s a P(m)-space. 0
Remark. The Michael line is not a P(2)-space. Answering a question in Morita [1964] for every m, n with KO G m < n Vaughan [1975] constructed an example of a P(m)-space which is not a P(n)-space. Since every perfectly normal space is a P-space (see the proof of (c)-(a) of Lemma 1. l), we have 1.7. Corollary (Morita [1964]). Let X be a perfectly normal space and Y a metric space. Then X x Y is normal. As another but useful one of P-spaces, we quote M-spaces defined also in Morita [1964], where M-spaces are characterized as those spaces which can be the pre-images of metric spaces under quasi-perfect maps, and hence they are both generalizations of metric spaces and countably compact spaces. As is known for paracompact spaces being M-spaces and p-spaces due to Arhangel’skii [1963] are identical. The effect of Theorem 1.3 works well in proving the following theorem, which characterizes metric spaces whose product with any countably paracompact normal space is normal. 1.8. Theorem (Morita [1963]). Let Y be a metric space. Then X x Y is normal for any countably paracompact normal space X iff Y is a-locally compact; that is, Y is the union of a countable number of locally compact closed subspaces.
u,?,
Proof. To prove the “if” part, let Y = Ci,where Ci is locally compact for each i E N. Let X be any countably paracompact normal space. Let B = (G(a,, . . . , a,,) x W(al,. . . , a,,)Ia, E R,, v = 1, . . . ,n ; n E N} be a basic cover of X x Y. Then for each i E N
%
= {G(al,. . . , a,) x
(W(al,. . . , a,,) n Ci)la,E R,,
v = 1, . . . , n ; n E N }
Normality of Product Spaces I1
131
is a basic cover of X x Ci. Since Ci is locally compact and metizable, X x Ci is normal. Therefore, by Theorem 1.3, there exist closed subsets &(al, . . . , a,,) of X such that
W I , . . , a,,) = W,, . . . , an), U{&(ai,
-
v = 1,
7
a,,) x
..., n
(w(al,-
-
7
a,,) n Ci)Iav E Q v ,
; n ~ N }= X x C i .
If we put F(al, . . . , a,,) = Ui",,&(a,, . . . , a,,), then F(a,, . . . , a,,) is an F,-subset of X and clearly (1.3) and (1.4) are satisfied. Hence, by Lemma 1.1, Y has a special refinement. Thus, X x Y is normal. To prove the "only-if"' part, assume to the contrary that Y is not o-locally compact. Then, by Stone [1962], there exists a complete metric space Z containing Y as its subspace, but not an F, in Z. Let us define a space 2, where 2 = 2 as a set and nbds of z E 2 are either those of z in 2 if z E Z - Y or {z} if z E Y. Then 2 is a paracompact Hausdorff space. Consider E = (2 - Y) x Y and F = {( y , y ) I y E Y } . Then E and F are closed in 2 x Y and disjoint. By Michael [1963], we see that E and Fcannot be separated by open sets. Consequently, 2 x Y is not normal. 0 The following is another useful application of Theorem 1.4. 1.9. Theorem (Rudin and Starbird [1975]). Let Y be a metric space and Z a compact space. Suppose that X x Y and X x Z are normal. Then X x Y x Z is normal.
Proof. We may assume Y is nondiscrete. Then, since X x Y is normal, X is countably paracompact (see the preceding chapter). Hence, by Theorem 1.4, X x Y is countably paracompact, and since Z is compact, (X x Y) x Z is countably paracompact. Since (X x Z) x Y = (Xx Y) x Z and X x Z is normal and countably paracompact, again by Theorem 1.4, (Xx Z) x Y = X x Y x Z is normal. 0 Let m be an infinite cardinal number. The following corollaries are proved by Theorem 1.9 above and Theorems 1.13 and 2.21 of the preceding chapter. 1.10. Corollary (Morita [1963]). Let Y be a metric space. Suppose X x Y is normal. If X is (respectively m-)paracompact, then X x Y is (respectively m-)paracompact.
132
T. Hoshina
1.11. Corollary (Okuyania [1967]). Let Y be a metric space. Suppose X x Y is normal. If X is (respectively m-)collectionwise normal, then X x Y is (respectively m-)collectionwise normal.
The above two results are easily extended to the following. 1.12. Corollary. Let Y be aparacompact M-space. Suppose X x Y is normal. is m-paracompact (respectively m-collectionwise normal), then X x Y is m-paracompact (respectively m-collectionwise normal ).
If X
Proof. By assumption, there exists a perfect map f:Y + T from Y onto a metric space T. Since 1, x f:X x Y + X x Tis perfect, X x Tis normal. In case T is discrete, Y is the discrete union of compact subspaces, and so the result is clear. In case T is nondiscrete, X is countably paracompact and X x T is m-paracompact (respectively m-collectionwise normal) by Corollaries 1.10 and 1.1 1. Since 1, x fis perfect, X x Y is m-paracompact (respectively m-collectionwise normal). 0 We now proceed to a further application of Theorems I .3 and 1.4. 1.13. Theorem (Rudin and Starbird [1975]). Let Z be a metric space. Suppose + Y is a closed map, then Y x Z is normal.
X x Z is normal. Iff: X
Proof. Assume X x Z is normal. We may assume that Z is nondiscrete. Hence, as before, X is countably paracompact and, by Theorem 1.4, X x Z is countably paracompact. Also note that Y is countably paracompact and normal. Let Y = {G(a,, . . . , a,) x W(a,,. . . , a,,)Ia, E Q, v = 1, . . . ,n; n E N} be any basic cover of Y x Z. Then { f - ' ( G ( a , , . . . , a,)) x W(a,, . . . , a,,)Ia,
v = I,
E Q,,
...,n ; n ~ N }
is a basic cover of X x Z. Hence, by Theorem 1.3, there exist closed subsets F(q,. . . , a,) of X such that F(al, . . . , a n ) c f - ' ( G ( a , , . . . , a,,)), U{F(a,,
.
* * 7
an) x W(aI, *
..
3
an)Iay
E
Qv,
v = 1, . . . , n ; n ~ N }= X x Z .
Theref6re, by considering the closed subset f ( F ( a , , . . . , an)) of Y, we see that Y has a special refinement. Thus, Y x Z is normal by Theorem 1.3. 0
Normality of Product Spaces I1
133
Theorem 1.13 is originally related to the following problem posed by Morita (cf. Morita [1961]): Let Z be a compact space. Let f : X -, Y be a closed map. Suppose X x Z is normal. Then must Y x Z be normal? Morita conjectured that the answer to this problem was affirmative, and indeed the positive answer was given afterward by Rudin [1975]. On the other hand, when drafting his paper [I9631 Morita further conjectured that in case Z was metrizable, the problem above is also affirmative and he noticed that the answer is yes if, in addition, X x Z is countably paracompact. Thus, Theorem 1.13 is a positive answer to this problem. The remaining part of this section will be devoted to Rudin’s result above as well as to results given recently by BeSlagiC [1986]. 1.14. Theorem (Rudin [1975]). Let Z be a compact space. Let f : X a closed map. X x Z is normal, then Y x Z is normal.
+
Y be
For the proof of this theorem we shall need several results, the first of which is proved in the preceding chapter. 1.15. Lemma (Morita [1975]). Let X be a space and Z a compact space with weight w ( Z ) < m. Let Y be a normal cover of X x Z. Then there exist a normal cover 42 = { U, I 1 E A} of X and a collection { 6. I 1E A} ofjinite open covers of Z such that { U, x V I V E V,, 1 E A} refines Y, where Card A < max{m, Card Y}. 1.16. Lemma (Starbird) (see Rudin [1975]). Suppose Z is a regular space. Then,for each a < w ( Z ) , there exist subsets A,, B,, U, and V , of Z such that (a) A, and B, are closed and A, c U, and B, c V,, (b) U, and V , are open and disjoint, (c) Vg < a, either A, Q U , or B, Q 5.
Proof. We define A,, B,, U, and V , by induction on a. Suppose that y < w ( Z ) and that A,, B,, U, and V , have been defined for all a < y . Since Card { U,},<, < w ( Z ) , { Ua}a
T.Hoshina
134
1.17. Lemma (Rudin [1975]). Let Z be a compact space. Suppose X x Z is normal; then X is w(Z)-collectionwise normal.
Proof. Let {FaI a < K} be a discrete collection of closed subsets of X,where K = w ( Z ) . Let A , , B,, U, and V , be subsets of Z defined in Lemma 1.16. Let us set D = U{F, x ( Z - U J l a < K } . C = U { F , x A,la < K } , Then C and D are disjoint closed subsets of X x Z since {Fa}is discrete, each Fais closed and .A, c U,. Since X x Z is normal, there exist disjoint open subsets G and H of X x Z such that C c G and D c H. Let us put for each a
Ha =
{X
E X ~ { Xx} ( Z
- U,)
c H}.
Since A, and 2 - U, are compact, G, and Ha are open subsets of X . Note that Ha x ( Z - U,) c H . G, x A, c G, Fa c G, n H a , Similarly, find disjoint open sets G' and H' of X x Y such that U { F , x B,la <
K} c
G',
U{F, x ( Z - V,)la <
K} c
H'
and define open subsets G: and H,' of X such that Fa c G: n H:,
G: x B, c G',
H,' x ( Z
-
V,)
c H'.
Let us now define for a < K
Ma
= G, n Ha n G: n H,'.
Then M ais an open subset of X and Fa c Ma.We show (Ma I a < K} is disjoint. Suppose not. Then, for some /? < a < K , Ma n M, # 8. Choose x E Ma n M,. By Lemma l.l6(c), either A, U, or B, b. Suppose A, - U, # 8 and y E A, - U,. Then
+
+
(x, y ) E (G, x A,) n (H, x ( X - Up))c G n H =
a contradiction. Similarly for the case B, - V, # must be disjoint.
8,
8. Thus, { M a l a <
K}
0
The following lemma is interesting and also necessary for proving Theorem 1.14; the proof however is highly involved and is therefore omitted.
Normality of Product Spaces 11
135
1.18. Lemma (Rudin [1975]). Let I be a cardinal > 2. Suppose that a space Y is u-collectionwise normal for all u < I , and Y is an open cover of Y of cardinality I which is refned by a hereditarily closure-preserving closed cover of Y. Then there exists a locally finite closed cover of Y that refnes Y .
We note that if, in particular, Y is I-collectionwisenormal, then the lemma is true. We shall now prove Theorem 1.14. Proof of Theorem 1.14. Assume that X x Z is normal. Z may be assumed to be nondiscrete. So Zcontains an infinite sequence {z,,} with an accumulation point z,. Then an F,-subset X x ({z,,} u { z o } ) is normal, and therefore X is countably paracompact. Hence, Y is normal and countably paracompact. Definecp:XxZ+ Y x Z b y c p = f x 1,. Let E and F be disjoint closed subsets of Y x Z. We put G = X x Z cp-'(E) and H = X x Z - q - ' ( F ) . Since X x Z is normal, the binary open cover {G, H} is normal. Hence, by lemma 1.15, there exist a normal open cover 4 = { U, I I E A} of X and a collection {Y,I I E A} of finite open covers of Z such that (0, x PI V E Y,, I E A} refines {G, H}, where Card A < w(Z). Since 4 is normal, there exists a locally finite closed cover 9 = {FAI I E A} of X such that FA c U, for each I E A. Let I E A and let us put V,, = { V E " Y ; ~ F ,V X c G } , "v,, = { V E Y , I F , X P c H } .
Since {FAx PI V E V,, I E A} refines {G, H}, we have Y, = Y,, u V,,. Let us put
W,, = { y E YI { y } x P c Y x Z - E, V E Y,,}, W,, = { Y E Y ~ { ~ ) XYPx C Z-F,VEY,~}. Then W,,is an open subset of Y x Z since P( V E Y,,)is compact and V,,is finite. Similarly for W,,. Observe that f (F,) c W,,n W,,. Therefore, if we put W,= W,, n W,,and W = { W,I I E A}, then W is an open cover of Y such that {W,x P I V E " V ; , I E A }refines { Y x Z - E , Y x Z - F } , and f ( 9 ) = { f(FJ 11 E A} refines W . (1*9) Since Xis w(Z)-collectionwise normal by Lemma 1.17, Y is also w(Z)-collectionwise normal. And sincef is closed,f(9) is a hereditarily closure-preserving closed 'cover refining W . By Lemma 1.18, W has a locally finite closed refinement { D , 11 E A} such that D, c W, for each I E A. Since Y is w(Z)collectionwisenormal and countably paracompact, there exists a locally finite
I36
T. Hoshina
open cover { M i I A E A} of Y such that Di c Mi and A?i c W, for each A E A. Therefore, if we put V = { M i x VI V E Yi,A
E
A},
V is a locally finite open cover of Y x Z such that {CI C E V} refines { Y x Z - E, Y x Z - F} by (1.9). Finally, let us put L = St(E, U). Then L is open in Y x Z , and we have E c L and t n F = 8 since V is locally finite. Thus, Y x Z is normal. 0
1.19. Corollary (Rudin and Starbird [1975]). Let T be a metric space and C a compact space. Let f : X + Y be a closed map. r f X x T x C is normal, then so is Y x T x C. Proof. Use Theorems 1.4, 1.7 and 1.14.
0
As is defined in Starbird [1974], let us denote by N (respectively 9)the class of all spaces Z having the following property: Let f:X -, Y be a closed map. r f X x Z is normal (respectively paracompact), then so is Y x Z . We have seen that every metric space, every compact space and every product of a compact space with a metric space are all in N ,and therefore it is easy to see that they are all in 8.On the other hand, it is known that every paracompact M-space is characterized as a closed subspace of a product of a metric space and a compact space (for the proof, see below). In view of this fact together with Corollary 1.19 it is natural to ask whether every paracompact M-space belongs to N,although we note that it belongs to 8 (Starbird [1974]). This question was asked in Rudin [1975a], but it is still unsolved. In connection with this BeSlagiC [ 19861 recently obtained the following result.
1.20. Theorem (BeSlagiC [1986]). Let Z be a paracompact M-space. Let f:X + Y be closed. r f X x Z is collectionwise normal, then so is Y x Z . To prove this theorem we need a lemma. Let Z be a paracompact M-space. Then there exists a perfect map g : Z + T from Z onto a metric space T. Let PZ be the Stone-tech compactification of Z . Define a map h : Z + T x PZ by h ( 4 = (g(4, 4, z E
z.
Then h is a homeomorphic embedding such that h ( Z ) is closed in T x PZ and g = p;h holds, where p r is the projection: T x PZ + T. The following lemma was proved by Starbird [I9741 (for the proof, see BeSlagiC [1986]); the proof is, however, too involved to write down.
Normality of Product Spaces I I
137
1.21. Lemma (Starbird [ 19741). Let Z be a closed subset of T x C, where T is metric, C is compact, and p T ( Z ) = T. r f X x T is normal, then X x Z is C*-embedded in X x T x C.
Here, p T is the projection: T x C + T. For the notion of C*-embedding, see Chapter 2: "Extensions of Mappings 11".
Proof of Theorem 1.20. The proof is a somewhat simple combination of those of Theorems 1.3 and 1.14. Assume that X x Z is collectionwise normal. By Corollary 1.1 1 , it suffices to prove that Y x Z is normal. Let g : Z + T be a perfect map, where T is metric. Since 1, x g :X x Z -+ X x T is perfect, X x T is normal. Here we may assume T is nondiscrete. For otherwise, Z would be a discrete union of compact subsets and the theorcm would be clear. Therefore, X is countably paracompact and, by Theorem 1.4, X x T is countably paracompact. Let h :Z -+ T x C be the embedding obtained above, where C = PZ. Let cp = f x l , : X x Z +
YXZ
and
I//
=
fx1TXl,:XXTXC+YXTXC,
Now let E and F be disjoint closed sets of Y x Z. Let us put G = Y x T x C - (1, x h)(E), H = Y x T x C - (1, x h)(F).
Since h ( Z ) is closed in T x C, G and H are open subsets of Y x T x C and {C, H } covers Y x T x C. Since cp-'(E) and cp-'(F) are closed and disjoint and X x Z i s normal, the open cover { X x z - c p - ' ( ~ ) , x x z - cp-'(F)} of x x Z is normal. Since X x 2 is homeomorphic to X x h ( Z ) , { X x h(Z) - (1, x h)(cp-'(E)h X x h ( Z ) - ( 1 , x Mcp-'(FN)
is also a normal cover of X x h ( Z ) . Since X x h ( Z ) is C*-embedded in X x T x C by Lemma 1.21, there exists a binary normal cover {G', H') of X x T x C such that G' n ( X x h ( Z ) ) c X x h ( Z ) - ( 1 , x h)(cp-'(E)),
H' n ( X x h ( Z ) ) c
x
x h ( Z ) - ( 1 , x h)(cp-'(F)).
Since X x h(Z) is closed in X x T x C, we may further put
G'
=
X x T x C - (1, x h)(cp-'(E)),
H'
=
X x T x C - (1, x h)(cp-'(F)).
T.Hoshina
138
Then observe that $-‘(G) =
G‘,
$ - ‘ ( H ) = H’.
Since {G’,H’} is normal, by Lemma 1.15 there exists a locally finite open cover 4’ = { U; I A E A} of X x T and a collection {V,11 E A} of finite cozero-set covers of C such that {U; x 91 V E b,A E A} refines {G’, H’}. Let A E A. Let us put VAl = { V E VAl U; x V,* = { V E V,IUi x
P c G’}, P c H’}.
Then V, = V,, u 95,. Let us put
U,, = {p E Y x T l { p } x U,, =
{PE
Y X Tl{p}x
V c G, V E V A I } , P c H,V E V , , , } .
Since P ( V E V,)is compact and VAis finite, UAI. and U,,, are open sets of Y x T. Observe that (fx lT)(U;) c U,, n UA2.Hence, if we put U, = U,, n U,,, then U, is an open subset of Y x T and we have that %! = { U, [ A E A} covers Y x Tand { U, x PI V E V,, A E A} refines {G, H}. Now for the metric space T, let us adopt the same notations used in the previous part of this section such as W ( a , , . . . , a,,), a, E R,, v = 1, . . . , n. Let us put
u(ai, * * * an; A ) = U{PI P open in Y and P x W ( a I ,. . . , a,,) c U,}, 3
U’(al, . . . , a,,; A) = U{PI P open in
X and P x W ( a I ,. . . , a,,) c U;},
and
M(a,, . . . , a,; =
4
U’(aI, . . . , a,,; A) n f - ’ ( U ( a , ,
. . . , a,,; A)),
(1.10)
and put
M(a,, . . . , a,,) = U { M ( a , ,. . . , a,,; A ) l A E A}.
(1.11)
We shall show that the collection A = { M ( a , , . , . , a,,) x W ( a I ,. . . , a,,)Ia, E R,,
v = 1,
...,n ; n E N }
is a ba’sic cover of X x T. Clearly, for a, E a,, v = 1, . . . , n, n + 1, M ( a I ,. . . , a,,) c M ( a , , . . . , a,,, a,,+,) holds. Hence, it remains to show that A covers X x T. Let ( x , t) be any point of X x T. Then ( x , t) E U; for
Normality of Product Spaces II
139
some A E A. Hence, there exist an open nbd P I of x and a, E R,, v = 1, . . . , n such that f E W ( a I ,. . . , a,,)and P I x W ( a I ,. . . , a,) c U;. On the other hand, (fx 1T)( U;) c U,, we have ( f ( x ) , t ) E U,. Hence, there exist an open nbd P2 off(x) and a, E a,,v = n + 1,. . . ,m such that t E W(a,, . . . , a,, a,+,, . . . , a,,,) and P2 x W(aI,. . . , a,,, a,,+,, . . . , a,,,) c U,. Then we have, by (l.W,
x
E
U’(a,, . . . , a,,; A) n f - ‘ ( U ( a , ,
c U’(aI,
. . an, an+,, . -
-
. . . , a,,
a,,+,, . . . , a,,,; A))
a,; A)
. . . , a,,, a,,+I, . . . , a,,,; A)) 1 M(a,, . . . , a,,, a,+,, . . . , a,,,; A).
nf-W(a,, =
Hence, A covers X x T, and A is a basic cover of X x T. Therefore, by Theorem 1.3, there exist closed subsets F(al, . . . , a,,) of X such that F(a,, . . . , a,,) c M ( a , , . . . , a,)
(1.12)
U{F(al,. . . , a,) x W a , , . . . , an)lav E Q,, v = 1, ...,n;
EN}
= X X
T.
(1.13)
Since 42’ is locally finite, so is { U’(al, . . . , a,,; A)l A E A}. Hence, {M(a,, . . . ,a,; A) 1 A E A} is locally finite. Therefore, by (1.1 l), (1.12) and the normality of X, there exists a closed subset F(al, . . . , a,,; A ) of X such that {F(al,. . . , a,; A) 11 E A} is locally finite, and F(al, . . . , a,,; A) c M(a,, . . . , a,; A) F(al, . . . , a,,) = U{F(a,, . . . , a,,; A)lA
(1.14) E
A}.
(1.15)
Then, sincefis closed, { f ( F ( a , ,. . . , a,,; A)) I A E A} is a hereditarily closurepreserving closed cover of the closed subset f ( F ( a , , . . . , a,,)) such that f ( F ( a l ,. . . , a,,; A)) c U(a,, . . . , a,; A ) by (1.10) and (1.14). Since Y is collectionwise normal, so i s f ( F ( a , , . . . , a,,)). Hence, applying the fact following Lemma 1.18 to the open cover {U(a,, . . . , a,,; A) n f ( F ( a , , . . . ,.,,))In E A} o f f ( F ( a , ,. . . , a,,)), we can take a locally finite closed cover {E(a,, . . . , a,,; A ) l A E A} of f ( F ( a , , . . . , a,,)) such that ,?(a,, . . . , a,,; A ) c U(a,, . . . , a,; A) for each A E A. Since Y is countably paracompact and collectionwise normal, there exists a locally finite collection {H(a,, . . . , a,,; A) I A E A} of cozero-sets of Y such that E(a,, . . . , a,; A) c H(a,, . . . , a,,; A) c U(a,, . . . , a,,; A).
140
T. Hoshina
Now we have constructed a collection 9 = { H ( a , ,. . . , a,,; A) x W ( a l ,. . . , a,,) x VI V E
6,
A E A ; ~ , E R , , v= I , . . . , n ; n ~ N } ,
each member of which is a cozero-set of Y x T x C, and it is easy to check that Y is a-locally finite, covers Y x T x C and refines {G, H}. Thus, 3 is a a-locally finite cozero-set cover, and so a normal cover of Y x T x C refining {G, H}. Hence, {G, H} is itself normal, and consequently, { Y x Z - E, Y x Z - F } is a normal cover of Y x Z . Thus, as is well known, E and F are separated by open sets. 0 A proof similar to the one above yields the following corollary.
1.22. Corollary (BeSlagiC [1986]). Let C be compact and M metric and let Z be closed in C x M . Suppose X is w(Z)-collectionwisenormal. Let f:X -, Y be closed. Then if X x Z is normal, so is Y x Z .
2. Products of normal spaces with generalized metric spaces In Section 1 we have discussed several normal products, where the notion of basic covers and Morita-Rudin-Starbird’s Theorem play an important and fundamental role. In general, for two normal countably paracompact spaces X and Y the countable paracompactness of X x Y need not imply its normality. For example, let X = w , and Y = wI + 1. As for the converse, it has been a still unsolved problem (see Rudin [1975a]), but under some set-theoretic assumptions counterexamples were constructed by Wage [ 19781 and recently by BeSlagiC [1985]. Therefore, in view of these facts, the MoritaRudin-Starbird’s Theorem is meaningful and of interest. In this section we shall find several conditions on X or Y for the product X x Y to allow equivalence between normality and countable paracompactness and we shall obtain some related normal products. First we note: since w , + 1 is compact, w , x ( 0 , + 1) shows that the above equivalence does not hold in general in case Y is compact. Hence, in view of Morita-Rudin-Starbird’s Theorem any possible approach to such an equivalence may lie in cases where at least one factor space is in a suitable class including metric spaces but not compact spaces. Along this direction we shall btgin our discussion with products whose one factor is a LaSnev space. A space which is the closed image of a metric space was characterized by LaSnev [1966]; such a space is simply called a LuSnev space. LaSnev spaces,
141
Normality of Product Spaces I1
which are a natural generalization of metric spaces, have been often discussed to study several “metric-like’’ properties. One of the successful studies of LaSnev spaces can be found in Leibo [1974], where the well-known KatEtovMorita coincidence theorem for metric spaces in dimension theory was extended to the case of an arbitrary LaSnev space. Let Y be a LaSnev space. Then, analogous to Theorem 1.3, we shall first obtain, after some preliminary results, a necessary and sufficient condition for a given X x Y to be normal and countably paracompact. Since Y is LaSnev, there exists a metric space Tand a closed mapf : T -+ Y . Then we first note that Y is paracompact and perfectly normal. Our method will be based on the following fact: Y can be expressed as Y = Yo u Ui, I u., where y. is discrete closed for each i 2 1, f - I ( y ) is compact for each y E Yo and Yo n (Ui,l KJ = 8. This was proved by LaSnev [1965] and is assured by the following (see Nagata [1985, Theorem VII.41). 2.1. Theorem. Let X be a normal M-space and f : X -+ Y a closed map. Then Y = Yo u Ui,l F , where is discrete closed for each i 3 1, and f - I ( y ) is countably compact for each y E Yo.
Since T is a metric space, there exists a sequence {%,} of locally finite open covers of T such that each member of 42, has diameter < l/n. Let %, = {U,,Ia E a,}. Let us put F,, = f ( u , , ) for c1 E R, and put 9, = {F,, I c( E O n } Since . @, is a locally finite closed cover of T andf:T -+ Y is closed, 9, is a hereditarily closure-preserving closed cover of Y . Moreover, the sequence (9 }, has the following property: Let y E Yo.Then g,is locally finite at y for each n E N, and for each open nbd W of y in Y there exists an n such that St(y, 9,c )W .
(2.1)
To see this, let y E Yo.Since @, is locally finite and f - ‘ ( y ) is compact, there exist an open subset G containing f - I ( y ) and a finite subset y of R, such that a $ y implies G n on,= 8. Then G # = Y - f ( T - G ) is a nbd o f y such that a $ y implies G X n F,,, = 8. Hence, 9, is locally finite at y . Let W be an open nbd of y . Since f - ’ ( y ) is compact, there exists an n such that l / n < e ( J - ’ ( y ) , T - f - ’ ( W ) ) ,where e is a metric on T . Then we have c W. St(f - I ( y ) , @,) c f - I ( W ) , which implies St( y , 9,) Now let us apply arguments given in Morita [1955]. Let R, be well-ordered. Let us define G,,, for each ct E R, by Gno = Int Fno,
G,, = Int F,, -
u Int
B
Fna.
142
T. Hoshina
Let 9,= {G,, I a E a,}. Then the following properties are satisfied: each G,, is open, and a #
B implies G,,
g, is locally finite at each point y
E
n Gns=
8,
Yo,
(2.2) (2.3)
r, = Int(U%'n)
(2.4)
In fact, G,, is open since 9, is hereditarily closure-preserving, and 9,is clearly disjoint. Property (2.3) follows from (2.1). To see (2.4), let y E Yo.We first prove y E Int(U{Int F,, I a E a,}).By (2.1) there exist an open nbd V of y in Y and a finite subset { a l , . . . , a,} of a, such that V c FnUI u * u F,,,,. Now, u F,,, u * * * u F,,,) is not empty. Then V, suppose that V, = V - (Int FnUI is a non-empty open set with V,c FnUl, that is, V, c Int F,,,, a contradiction. Similarly V c Int F,,, u Int Fnu2 u F,,, u * u FnUb, and finally we have V c Int F,,, u . . . u Int F,,,. Hence, y E Int(U{Int F,,la E a,}). Since U {Int F,, I a E Q,} = Ug,,, we have (2.4). Let us further put H,, = Int and H,= '{H,,,I a E a,}. Note that H,, = G,,, and that G,, n Gns= 8 implies H,, n Hns= 0.Hence, the above (2.2), (2.3) and (2.4) are also satisfied for H,. Let us put K(a,, . . . , a,) = HIUln . . . n Hnan for a, E Q, v = 1, . . . , n, and X, = {K(a,, . . . , a,,) I a, E Q,, v = 1 , . . . , n}. Then we also note that X, satisfies the following:
-
a
..
*
9
an)
+ (81,
* * * 9
a
Bn)
* K(a1, . . . > an) n K(B1, fin) = 0, 9,is hereditarily closure-preserving and is locally finite * * * 3
(2.2') (2.3')
at each point of Yo, Y, c Int(U2,).
(2.4')
For each point y E Yo, let us put L , ( y ) = Int St(y, 9,). By (2.3') and (2.4'), L , ( y ) is an open nbd of y. Let 9,= { L , ( y ) ( y E Yo}. Then the sequence {9, satisfies: }
&
c
UY,, for each n
E
N,
for each point y of Yoand each open nbd G of y in Y there exists an n such that St(y, 9,)c G .
(2.5) (2.6)
Clearly, (2.5) holds. For (2.6), let y E Yoand let G be an open nbd of y in Y. By (2.1) St(y, 5,)c G for some k. Then we have that L , ( y ) c G by (2.4') and that 2,refines F,. Again by (2.1), there exist an n > k such that St(y, 9,) c L , ( y ) . We shall show that St(y, 9,) c G. Suppose that
Normality of Product Spaces I1
143
s,),
y E L,,( y’). Then since L,( y’) c St( y’, we have y’ E St( y, 9,). Hence, y’ E Lk(y ) . Let y’ E K(&, . . . , B,). Then we have y E K ( f i I ,. . . , Bk) since ify $ K(fi,, . . . , Bk), we have K(a,, . . . , ak) n K(BI, . . . , Bk) = 0 for any K(a,, . . . , ah)with y E K(al, . . . ,ak), and hence, Lk(y ) n K(fiI,. . . , Bk) = 0, which contradicts y’ E L h ( y ) .Therefore we have St(y’, 2,) c St(y, 2k) and hence L,,(y’) c L k ( y ) . Thus we have proved St(y, Y,,)c L,(y). Hence, St(y, Y,,)c G. This proves (2.6). Now, from the fact above, we obtain the following theorem.
2.2. Theorem. There exists a normal sequence {V,}of locallyfinite open covers of Y such that for each point y of &{St( y, V,,)I n E N} is a local nbd base at y in the whole space Y.
Proof. Let 9,,be obtained above. Let us put H,, = U9,,. Since Y is perfectly normal, there exist open subsets Hnkof Y such that H,, = Ukm,,Hnkand A,,, c H,,, for k E N. Let us put, for n, k E N, d n k
= { L n H n k + l I L ~ I P , l u { y%- k } -
Then A,,,is an open cover of Y. Rewrite newly these Ank as A,,(n E N). Then by (2.5) and (2.6) we can easily see that for each point y of Yo {St(y, A,,)I n E N} is a local nbd base at y in Y . Using paracompactness of Y, take inductively a locally finite open cover V,,for each n E N such that V,, refines A, and V,,+ I is a star-refinement of V,. Then { V,,}has the desired 0 properties of the theorem. Now let {V,,}be the sequence of covers of Y obtained in the theorem KCl, above.LetV,, = {V,,IaER,,}.LetusputW(a,, . . . , a,,) = q a , n . . . n f o r a , E n , , v = 1, . . . , n. Let X be a space. Then a collection Y of open subsets of X x Y is said to be a basic semicover if Y has the form Y = {G(a,, . . . , a,) x W ( aI ,. . . , a,)Ia,
E
...,n ; n ~ N } with open subsets G(a,, . . . , a,,) of X and satisfies G(a,, . . . , an) c G h , . . . , a,, a,+l) f o r a , E n , , v = 1, . . . , n , n + 1 v = 1,
and
I
U{G(a,, . . . , an) x
W E , , . . - , a,) I a, E a,,
v = 1, . . . , n ; n E N }
I>
X x Yo.
a,, (2.7)
T. Hoshina
144
Let E be a closed subset of X x Y contained in X x Yo. Then the basic semicover Y in (2.7) is said to have a special refinement relative to E if there exist closed subsets F(a,, . . . , a,) of X such that F(a,, . . . , a,)
= G(al, . . . , a,,),
(2.10)
and U{F(a,,. . . , a , ) x W a , ,. . . , a , ) l a , ~ Q , , v = I,
...,n
; n ~ N 13 } E.
(2.1 1)
The following lemma can be proved similarly to Lemma 1.1, 2.3. Lemma. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo. For a basic semicover Y in (2.7) the following are equivalent : (a) Y has a special refinement relative to E. (b) There exisrs a family {F(a,, . . . , a,) I a, E R,,v = I , . . . ,n; n E N} of open F,-subsets of X satisfying (2.10) and (2.1 I). (c) There exists a family {F(a,, . . . , a,) I a, E R,,v = I , . . . , n; n E N} of F,-subsets of X satisfying (2.10) and (2.1 1).
The following is analogous to Theorem 1.2. 2.4. Theorem. Let X be a normal space. Let E be a closed subset of X x Y contained in X x Yo.Then a basic semicover 3 ’ in (2.7) has a special reJinement relative to E i f there exists a a-locally finite collection Y of open subsets of X x Y such that E c U Y and 2 refines 9.
Proof. To prove the “only-if‘’ part, using Lemma 2.3(b), take a family {F(a,,. . . , a , ) ( a , E R,, v = 1, . . . , n; n E N} of open Fu-subsets of X satisfying (2.10) and (2.1 1). Then F(a,, . . . , a,) is a cozero-set of X and so is F(a,, . . . , a,) x W ( a , ,. . . , a,) of X x Y. Let L k ( a I ,. . . , a,) (k E N) be an open set of X x Y such that F(a,, . . . , a,) x W(a,, . . . , a,) = U k L k ( a l ,.. . ,a,) and &(alr . . . , a,) c LL+,(cq,. . . , a , ) for k c N. Since { W(a,, . . . , a,)lav E R,,v = 1, . . . , n } is locally finite, {L,(a,, . . . , a,)l a, E R,, v = I , . . . , n; n, k E N} is the desired a-locally finite collection of open subsets of X x Y. Conversely, let 9 = U Y k be a a-locally finite collection of open subsets of X x Y described in the theorem, where Yk= {Lkil1E A k } is locally finite. Put for a, E R,, v = 1, . . . , n, k E N and 1 E A Lk(a,, . . . an; 1) = U { P I P open in X a n d P x W(a,, . . . , a,) c Lki}, 9
Normality of Product Spaces II
145
and
Fk(al, . . . , a,,)
=
U{&(alr . . . , an; 41
L,(a,,. . . , a,,; 1) c G(a,, . . . , a,,), A E h k } . Since Ykis locally finite, Fk(a,, . . . , a,) is closed in X and Fk(aI,. . . , a,) c G(a,, . . . ,a,,). Let F(a,, . . . , a,,) = U k F k ( a l., . . ,a,,). Then F(a,,. . . , a,,) is an F,-subset of X and, in a similar way as in the proof of Theorem I .2, it can be shown that {F(a,, . . . , a,,) I a, E R,, v = I , . . . , n; n E N} satisfies (2.10) and (2.11). Hence, by Lemma 2.3, 9 has a special refinement relative 0 to E. Here, for later discussions let us consider several weak normality properties. A space X has property (6) if, for any open subsets U,, n E N, and any closed subset B such that 0, 0, n B = 8, U,, and B are separated by open subsets of X.A subset A of a space X is a regular Gb-set if A is written as A = Q, = Onwith some open subsets Unof X.Clearly, every zero-set is regular G b . According to Mack [1970] X is &normal if, for every pair of disjoint closed subsets one of which is regular G b , there are disjoint open subsets containing them. Normality implies property (6) and property (6) implies 6-normality. By Mack [ 19701, countable paracompactness implies &normality. This result is slightly refined as follows.
n,,
n,,
2.5. Lemma.
n,,
Every countably paracompact space has property (6).
Proof. Suppose Xis countably paracompact. Let U,, (n E N) be open subsets and Baclosedsubsetwith n B = 8. Then {X - On,X - Bin E N} is a countable open cover of X. Let V" be its locally finite open refinement. Consider G = St(B, V"). Then it is easy to check n,Un n G = 8. 0
n,,0,
Next for a product space X x Y we consider the following two properties: (*) For any point y of Y and any closed subset F of X x Y with F n (X x { y } ) = 8, Fand X x { y } are separated by open subsets of X x Y. (**) For any point y of Y and any open subsets U,, (n E N) of X x Y with X x { y } c U,,U,,, there exist open subsets V, of X x Y such that X x { y } c U,,V,,and c U,,foreachn~N.
<
2.6. Lemma. Each of the following holds. (a) Let Y be perfectly normal. I f X x Y is 6-normal, then it has property (*). (b) Property (**) always implies property (*). The converse is true if X is normal and countably paracompact.
T. Hoshina
146
Proof. (a) Since Y is perfectly normal, { y} is a zero-set of Y. Hence, X x { y ) is also a zero-set of X x Y. Thus, property (*) easily follows from &normality. (b) The first statement is clear. To prove the converse, assume X x Y has property (*) and Xis normal and countably paracompact. Let Un(n E N) be open subsets of X x Y with X x { y } c U,U,,. Since X x { y} is countably paracompact, there exist closed subsets F, such that X x { y } = UnFnand Fn c U, for each n E N. Since Xis normal, there exist disjoint open subsets G and H of X such that
Fn c G x { Y } ,
(XX{ Y } ) n (XX Y -
un)c
H x {Y}.
Then U, u (H x Y) is an open subset of X x Y containing X x { y}. By (*), take an open subset K of X x Y such that X x { y } c K and R c Unu (H x Y). Let V , = K n (G x Y). Then, clearly, we have F, c V , and c U,,. Thus, V , are the desired open subsets of X x Y.
0 Returning to the LaSnev space Y, we shall now prove the following theorem which is mentioned above. A basic semicover Y in (2.7) of X x Y is said to be proper if for a?) E a,, v = 1, . . . , n ; n ~ N
fi w(ap), . . . , a t ) ) n Y, z 8
n=
I
x
u ~ ( a p ). ,. . , W
=
at)).
n= I
2.7. Theorem. Let X be a countably paracompact normal space. Let Y be a Lainev space. Then the following statements are equivalent: (a) X x Y is normal and countably paracompact. (b) X x Y has property (*a) and any basic semicover has a special rejinement relative to any closed subset of X x Y contained in X x Yo. (c) X x Y has property (**) and any proper basic semicover of X x Y has a special refinement relative to any closed subset of X x Y contained in X x Yo. Proof. (a)=(b): Suppose X x Y is normal and countably paracompact. Clearly, X x Y has property (**). Let Y be a basic semicover in (2.7) and E a closed subset of X x Y with E c X x Yo.Then 9’= 9 u { X x Y - E} is a a-locally finite open cover. By (a), there exists a locally finite open cover X of X x Y such that 3 refines 9‘.Let 2’ = { H E Z 1 H n E # 0).Then A?’ is a a-locally finite open collection such that E c US’ and 3’ refines 9.Thus, by Theorem 2.4, 9 has a special refinement relative to E.
Normality of Product Spaces I1
147
(b)*(c): This is obvious. (c)*(a): Assume (c). Let A = { M k I k E N} be a countable open cover of X x Y . To show (a), it suffices to prove that there exists a o-locally finite open cover 2 ' of X x Y such that 2 refines A. Let y be any point of Uir I Then by property (**) there exist open subsets u y k O f X X Y such that X X { y } C U k u y k and U y k C h f k for each k E N.Let i > 1. Since Y is paracompact and Y, is discrete closed, there exists an open nbd V, for each y E such that { V , l y E y } is discrete. Let us put Hyk= v y k n ( X x V,) for each y E UialY,, and put
x.
r
Then we see that X I is a o-discrete open collection satisfying
(2.12) On the other hand, let us put, for a, E R,, v W E , ,
=
=
1, . . . , n and k E M,
. . . , a,,; k ) U { P I P open in Xand P x W ( a l , . . . , a,,) c
M k } ,
and put M(a,, . . . , a,,)
=
U(M(a,, . . . , a,,; k ) l k
E
(2.13)
N}.
Then it is easy to see that 9 = { M ( a , ,. . . , a,,) x W(a,, . . . , a,,) I a, E R,,
v = 1,
...,n ; n € N }
is a proper basic semicover of X x Y. Now apply (c) to 9 and the closed set E = X x Y - UH,which is contained in X x Yoby (2.12). Then there exist closed subsets F(a,,. . . , a,,) of X such that W a , ,. . . , a,,)
= M(a,, . . . , a,,),
U{F(al, . . . , an) x W a l , . . . , a,,)la,
v = 1, . . . , n ;IIEN}
2
(2.14) E Q,,
E.
(2.15)
Since X is countably paracompact and normal, by (2.13) and (2.14) there exists an open subset H ( a l , . . . , a,,; k) of X such that
F(a,,. . . , a,,) c U { H ( a , ,. . . , a,,; k)lk E N}, H(a,, . . . , a,,; k) c M(a,, . . . , a,,; k ) for each k
E
N.
(2.16)
T. Hoshina
148
Then we have H(a,, . . . , a,,; k ) x W ( a l ,. . . , a,,) c M k ,and therefore
Hz = { H ( a , , . . . , a,,; k ) x W ( a l ,. . . , a,,)Ia, E R,, v = 1, . . . , n ; n E N , k E N ) is a o-locally finite collection of open subsets of X x Y such that in view of (2.15) and (2.16) E c UXZand $z refines A. Now, H = Sl u H2is a o-locally finite open cover of X x Y such that 2 refines A. Thus, (a) holds.
0 With the aid of Theorem 2.7 we shall obtain several results on the normality of products with a LaSnev factor. The following theorem establishes the equivalence of normality and countable paracompactness of products with a LaSnev factor. 2.8. Theorem (Hoshina [ 19841). Let X be a normal and countably paracompact space and Y a LaSnev space. Then X x Y is’normal iff it is countably paracompact.
Proof. In either implication, it is sufficient to prove that X x Y satisfies (c) of Theorem 2.7; property (**) obviously holds and thus, only the latter condition of (c) is needed to prove. To prove the “if” part, assume X x Y is countably paracompact. Let 9 = { G ( a l ,. . . , a,,) x W ( a l ,. . . , a,,)[a, E R,, v = I , . . . , n ; n E N} be a proper basic semicover of X x Y and let E be a closed subset of X x Y contained in X x Yo.We shall show 9 has a special refinement relative to E. Let us put, for each n E N,
Ffl = U{(X - G(a1, v = 1,.
...
9
a,,)) x w(a1,
. . 1
9
a,,)Ia,
E Q,,
. .,n}.
Then F,, 3 F n + l ,and since { W(a,,. . . , a,,) I a, E Q,, v = 1, . . . , n } is locally finite, F,, is a closed subset of X x Y . Moreover, since S is proper, we have F, n ( X x Y,) = 8, n
(ui2,
that is, n,,F,, c X x q). Since X x Y is countably paracompact and X x Y, is a zero-set of X x Y which is disjoint from E, by Lemma 2.5, there exists an open subset U, of X x Y such that E c U,. and q n ( X x V , ) = 8. Then we have
Normality of Produci Spaces I1
149
ni,,
the latter implies q.n n,F, = 0. Thus, by Lemma 2.5 again, there exists an open subset K of X x Y such that
n F, = K,
RnE
=
8.
n
Then F,, - K is closed and {F,, - K l n E N} is decreasing and has empty intersection. Hence, since X x Y is countably paracompact, there exists I n E N} of X x Y such that n an increasing open cover H = {H,, (F,- K ) = 0 for each n E N. Let us put for a, E R,, v = 1, . . . , n
f‘(al, . . . , a,)
= {x E
XI { x } x W ( a I ,. . . , a,)
c
17,- K}.
Then F ( a I , . . . , a,) is a closed subset of X,and we have F(a,, . . . , a,) c G(a,, . . . , a,,). Moroever, one can easily check that
U { F ( a l ,. . . , a,) x W ( a , , . . . , a,)la, E a,, V =
1, . . . , n ; EN}
E.
3
Thus, Y has a special refinement relative to E. To prove the “only-if’’ part, assume X x Y is normal. Let us define a subset A, of Q, x . . . x R, for each n E N by
A, = {(a,, . . . , a,,) E R, x . . * x Q,I W ( a I ,. . . , a,) is not discretej. Then, by Lemma 1.5, for each (a,, . . . , a,j E A,, two distinct points u ( a l , . . . , a,) and v(a,, . . . , a,) can be selected !rom W ( a , , . . . an.! such that (,al.. . . , a,) E A,, (8,. . . . , 8,) E A m , ~
( a , , . . . , a,> # n {u(B,, . .
(B,, . . . . Em) = {#(aI, . . . . Z,J.
. B,,),
vtfi,, .
lj,c\t =
via,, . .
8.
. rn i -... ,
17’:
Let 3 = {Gi,a,, . . . , r,\r x Wta,, . . . . -,)la, E O,, v , . . . . n; r. &, be a proper basic semicover of A’ x Y and iet A be a ciosed subsei of .:. :.. ; contained in X x Y i . We shall show that % has a speciai reiinemeni remiv.. to A . Let us put, for n E N, -1
C;,
‘=
iJ((X - G(a,. . . . , a . , ) ~ x cu(a,. . . . , a,>ji jbi,
. . . , a,)
E
Anj,
T. Hoshina
I50
D,, = U { ( X - G(a1, -
* * 9
a,,)) x
{4%,. ,%)}I * *
(a,, . * . , a,,) E A,,} and put C = UC,, and D = UD,,. Since { W ( a l ,. . . , a,,)I ( a l , . . . , a,,) E A,,} is locally finite, C,, is a closed subset of X x Y. Moreover, we note that Cn c Fn =
U { ( X - G(al, . . . , a,,)) x W ( a l , . . . , a,,)Ia, E
v
= 1,.
9,
. . ,n},
and {F,,} is a decreasing sequence of closed subsets with OF,, n ( X x Yo) = 8 since Y is proper. Therefore we have C n ( X x Yo)= C n (X x Yo).Similarly, D n ( X x Yo) = b n ( X x Yo).And since C n D = 8 by (2.17), we have C n b n ( X x Yo) = C n D n ( X x Yo) = 8;
x).
that is, C n b c X x (UiLI Since A c X x Yoand X x Y is normal, there exist open subsets U,V and W of X x Y such that
C n b c W and P n A C - W c U ,
=
b - W c V
8, and
OnV=8.
Let us put, for ( a I ,. . . , a,,) E A,,,
D(al, . . . , a,,) = { x E XI { x } x W ( a I ,. . . , a,,) c
X x Y - (V u W ) } ,
E(a1, . . . a,,) = { x E X l { x } x W ( a l , . . . , a,,) c X x Y - ( V u W ) } 9
and
Fl(al,. . . , a,,) = D(a,, . . . , a,,) u E(a,, . . . , a,,).
Then Fl(a,, . . . , a,,) is closed. We have that
Fl(al,. . . , a,,)
c
G(al, . . . , a,,) for ( a I , . . . , a,,) E A,,,
and that, for any point (x, y ) E A such that y is a nonisolated point of Y , there exists an (a1,. . . , a,,) E A,,such that (x, y ) E Fl(al,. . . , a,,) x W(a,, . . . , a,,). On the other hand, let I( Yo) = { y E YoI y is an isolated point of Y } . Then X x I(Yo)is normal and countably paracompact and
iG(a1, * . a,,) x (W(a1, ' v = 1, . . . , n ; n E N } 3
* 9
a,,) n I(Y0))la"E
a,,
151
Normality of Produci Spaces 11
is a basic cover of X x I( Y,)in the sense of Section 1. Hence there exist closed subsets F2(a,,. . . , a,,) of X such that
Fz(aI, ... U{F*(a,,
*
a,,)
..
7
= G b l , . . . , a,,), a,,) x (Wa,, *
v = 1, . . . , n; n
Define
F(aI, . . . , a,,) =
E
*
9
a,,)
-f7
4 Y O ) ) l ~ "E
Q,,
N} = X x I ( & ) .
u eh,. . . ,
a n ) if (at,
i= 1.2
Fz(a1,
* * *
, @,I
. . . , a,,) E A,,,
if (al, . . . , a,) $ A,,.
We have then
Using Theorem 2.8 the same proof of Theorem 1.6 yields the following: in case Y is nondiscrete Lainev, Y contains an infinite convergent sequence, and so normality of X x Y implies countable paracompactness of X. 2.9. Theorem (Hoshina [1984]). Let Y be Lainev and Z compact. If X x Y and X x Z are normal. then so is X x Y x Z . 2.10. Corollary (Hoshina [ 19841). Let X be m-paracompact (respectively m-collectionwise normal) and Y Lainev. I f X x Y is normal, then it is m-paracompact (respectively m-collectionwise normal ).
Next we shall consider conditions on X under which the product X x Y with a Lagnev space Y is normal. In view of Theorem 1.6, a necessary condition is that Xis a normal P-space. However, that is not sufficient as is shown by the following example; detailed constructions of X and Y are already given in the preceding chapter and are omitted. A a-space is a regular space with a a-locally finite net (Okuyama [1967]). It is known that every normal a-space is perfectly normal and hence a P-space. 2.11. Example (T. Chiba and K. Chiba 1197411). A normal a-space X and a Lainev space Y such that X x Y is not normal.
. ,
.. . . I
152
T. Hoshina
Indeed, any necessary and sufficient condition on X has not been obtained yet. The following result may be a help to seek such a one.
2.12. Theorem. Let Y be a Lainev space. Let X be a normal P-space, more precisely, a countably paracompact normal space such that X x Yo is normal. Then X x Y is normal iflit satisjies (*). Proof. We have to prove the “if” part only. Suppose that X x Y has property (*) or, equivalently by Lemma 2.6, has property (**). Let Y = {G(a,, . . . , a,) x W(a,, . . . , a,)la, E R,, v = 1, . . . , n ; n E N} be a basic semicover of X x Y and E a closed subset of X x Y contained in X x Yo.Observe that Yo is metrizable and the collection {G(a,, . . . , a,) x (W(a,, . . . , a,) n Yo)I a, E R,, v = 1, . . . , n; n E N} is a basic cover of the product X x Yo.Since X x Yois normal, by Theorems 1.3 and 1.4, there exist closed subsets F ( q , . . . , a,) of X such that F(al, . . . , a,)
= G(a,, . . . , 4, U{F(a,, . . . , a,) x ( W a , ,. . . , a,) n yo)t a, E a,, v = 1, . . . , n ; n ~ N }= X x Y , .
Since E c X x Yo, the above obviously shows that Y has a special refinement relative to E. Thus, by Theorem 2.7, X x Y is normal. 2.13. Theorem (Chiba [1974]). Let X be a locally countably compact normal P-space and Y a Lainev space. Then X x Y is normal.
Proof. By Theorem 2.12 we only have to prove that X x Y has property (*). Let y E Y . Let F be any closed subset of X x Y such that F n (X x { y}) = 8. We shall find an open subset L of X x Y such that X x { y } c L and EnF=8. Let T be a metric space and f : T + Y a closed map. Let cp = 1, x f : X x T + X x Y . Then cp-‘(F) and X x f - I ( y ) are disjoint closed subsets of X x T, and since X x T is normal, there exists an open subset G of X x T such that X x f - I ( y ) c G and G n cp-’(F) = 8. Let x E X and take an open nbd U, of x such that D,is countably compact. Consider the projection p : 0, x T -+ T. Then p is a closed map since uxis countably compact and T is first countable. Hence, if we put V, = T - p(Dx x T - G ) , then V, is an open subset of T and we have f -I( y) c V, and U, x V , c G . Since f is closed, there is an open nbd W, of y such that f -‘(W,) c V,. Let us put K = U { U , x W,IXE X } .
Normality of Product Spaces 11
Then K is an open subset of X x cp-'(K) n cp-'(F) = 8 since cp-'(K) open subset H such that X x f - ' ( y ) way as above choose an open nbd U, x f - ' ( W i ) c H. Let us put
L = U{U, x
W,'IXE
153
Y and we have X x {y} c K and c G. Again by normality of X x T, c H and I? c cp-'(K). By the same W,' of y for each x E X such that
X}.
Then L is an open subset of X x Y with X x { y } c L and cp-'(L) c H. We will show E c cp(cp-'(K)).Suppose (x, z) .$ cp(cp-'(K)).Then ({x} x f - ' ( z ) ) n cp-'(K) = 8. Therefore, if we put A = {x' E X l z E W,.}, we must have x .$ U{ U,. I x' E A}. Take an open nbd 0, of x such that 0, is countably compact and 0,n U,, 1 x' E A}) = 8; the latter implies (0, x f -'( y)) n cp--'(K) = $4. Since A c cp-'(K), we have (0,x f-'(y ) ) n A = 8. Since 0, is countably compact, similarly as above we can find an open nbd Wofy such that (0, x f - ' ( W ) ) n A = 8. Since cp-'(L) c H, we have (0, x W) n L = 8. Thus, (x, y ) .$ E. Since cp-'(K) n cp-'(F) = 8, we have t n F = 8 as required. 0
(u{
2.14. Proposition. A space X is paracompact any regular space Y .
fi
X x Y has property (*)for
Proof. The "if" part of this proposition is essentially proved in Theorem 1.12 of the preceding chapter. The "only-if" part is easily proved. 0 With this proposition as well as with Theorem 2.12 we find that the product of a paracompact P-space and a LaSnev space is normal, and by Corollary 2.10 it is paracompact. But this is implied by a more general result below. In Nagami [I9691 the notion of X-spaces was defined. As is proved there, if Y is a X-space, then there exists a sequence {8,}of locally finite closed covers of Y, where 8,is written as 8,= {E(a,,. . . , a,)Ia, E R, v = 1, . . . ,n} for each n E N with a suitable set R of indices, and a cover {C(y) 1 y E Y} of countably compact closed subsets with y E C( y) for each y E Y such that
&,, . . . , a,)
= U{E(a,,
. . . , a,,
a,+I)Ia,+I E R},
for each y E Y there exists a sequence a', . . . , a,, . . . E R such that C( y) c V with V open implies C ( y ) c E(a,, . . . , a,) c V for some n.
(2.18)
(2.19)
It is known that all cr-spaces and all M-spaces are Zspaces. Since every LaSnev space is a a-space, it is also a Z-space.
T. Hoshina
154
2.15. Theorem (Nagami [1969]). Let X be a paracompact P-space and Y a paracompact X-space. Then X x Y is paracompact. Proof. Take {&,} and {C(y) I y E Y} as above. Let 9 be any open cover of X x Y. Let a" E R, v = 1, . . . ,n. Consider a pair ( V , U )of an open subset U of X and a finite collection C of cozero-sets of Y such that E(a,, . . . , a,) c U V and each U x V ( V E U )is contained in a set of 9.Denote all such pairs 3, E A@,, . . . , a,) and put W(a,,. . . , a,) = {Ui x VI V E f , , by (U,, 1E A(a,, . . . , a,)}. Let us put W = U { W ( a , , . . . , a,) 1 a, E R, v = 1, . . . , n; n E N}. Since Y is paracompact, each C( y) is compact. Hence, in view of (2.19), one sees that W is an open cover of X x Y and, by definition, W refines 9. F o r a , E R , v = 1 , . . . , nweputG(a,,..., a,)= U { U i . ( l ~ A (, a. .I. , a,)). Then G(a,, . . . , a,) is an open set of X , and since W(a,, . . . , a,) c W ( a I ,. . . , a,, a,,,) by (2.18), we have G(a,, . . . ,a,) c G(a,, . . . , a,, a,+,). Since X is a P-space, there exist closed subsets F(a,, . . . , a,) of X such that
c.),
F@,, . . . , a,) c G h , . . . , a,,),
u
G(a,, . . . , a,)
ns N
=
X
=-
u F(a,,. . . ,
(2.20) a,)
=
X.
(2.21)
nEN
Here we may assume F(a,,. . . , a,) c F(al, . . . , a,, a,+,). Since X is paracompact, in view of (2.20), there exists a locally finite cozero-set collection &'(aI, . . . ,a,) = {H(a,, . . . , an;L)11 E A(a,, . . . ,a,)} of X such that H(a,, . . . , a,; 1) c U,, 1E A and F(a,,. . . , a,) c U X ( a l , . . . , a,). On the other hand, by paracompactness of Y, take a locally finite cozero-set collection {L(a,, . . . , a,) I a, E R, v = 1 , . . . , n} of Y such that F(a,, . . . , a,,) c L(a,, . . . , a,). Let us now put
. . . , a,; 1) x ( V n L(al, . . . , a,))l V E C , 1E A(a,, . . . , a,); a, E R, v = 1, . . . , n; n E N}.
= {H(a,,
Then it is easy to check that &' is a a-locally finite cozero-set cover of X x Y and refines 9.Thus X x Y is paracompact. 0
Remarks. (a) Besides locally countable compactness or paracompactness any other (sufficient) condition on a normal P-space X,under which X x Y is normal for a LaSnev space Y, has not been found yet. In contrast to Corollary 1.7 and Example 2.11, perfect normality of X is not sufficient. In Chiba [1974] it is asked whether X x Y of a normal M-space and a LaSnev space Y is normal.
Normality of Product Spaces II
155
(b) In Hoshina [ 19871a necessary and sufficient condition on Xis obtained under which X x Y is normal for any closed image Y of a locally compact metric space. It is unknown to the author whether every LaSnev space belongs to the class JV due to Starbird [1974] (see Section 1); however, the following asserts that every Lahev space belongs to the class 8.
2.16. Theorem. Let Y be a Lainev space. Let f : X X x Y is paracompact, then so is X' x Y .
+
X' be a closed map. If
Proof. Let Y = {G(a,, . . . , a,) x W(a,,. . . , a,)lav E R,, v = 1, . . . , n; n E N} be any basic semicover of X' x Y and let E be any closed subset of X' x Y contained in X' x Yo.Then {f-'(G(al, . . . , a,)) x W(al, . . . , a,)lav
E
Q,,
v = 1, . . . , n ; n ~ N } is a basic semicover of X x Y and F = (fx 1, , - I ( E ) is closed in X x Y and contained in X x Yo. Since X x Y is normal, there exist closed subsets F(a,, . . . , a,) of X such that
F(a1, . . . , a,,) c f -'(G(al,
. . . , an)),
U{F(al, . . . , a,) x Wal, . . . , a,)lav E a,, v = 1, . . . , n ; n ~ N }3 F.
Then, sincefis closed,f (F(a,, . . . , a,,)) is a closed subset of X' and we have
f ( F ( a , , . . . , a,))
c G(a,,
...
a,,),
U{f(F(a,,..* 9 a n ) ) x W(al,...,an)Iav~Rv, v = 1, . . . , n ; n ~ N }2 E.
Thus, B has a special refinement relative to E. Observe X' x Y has property (*) by Proposition 2.14 since X' is paracompact. Therefore, by Theorem 2.7, X' x Y is normal. From Corollary 2.9 it follows that X' x Y is paracompact.
0 In case Y is an F,-metrizable space, Theorem 2.8 is also valid (Hoshina [1984a]);a space is F,-metrizable (Gruenhage [1980]) (= o-metric in the sense
T. Hoshina
156
of Nagami [1971]) if it is paracompact and is a countable union of closed metrizable subspaces. Lahev spaces and F,-metrizable spaces are mutually independent notions, but both included in the class of paracompact a-spaces. The author does not know whether Theorem 2.8 holds for paracompact o-spaces Y (cf. Hoshina [1983]). If, in addition, we assume several conditions on X besides normality and countable paracompactness, Theorem 2.8 may be true for other spaces Y. Indeed, in Hoshina [I9841 Theorem 2.8 was proved in case X is a normal P-space and Y a paracompact a-space; hence, the question above is affirmative if X is further assumed to be a P-space. As for the case Y being a closed image of a normal M-space the following theorem has been obtained. 2.17. Theorem (BeSlagiC and Chiba [1987]). Let X be a paracompact Jirst countable P-space and Y a closed image of a normal M-space. Then X x Y is normal fi it is countably paracompact. 2.18. Lemma (Hanai [1961]). Let f : X + Y be closed and Y paracompact. Suppose that for each y E Y every pair of disjoint closed subsets o f f ' ( y ) are separated by open subsets of X . Then X is normal. Proof. Let E and F be arbitrary disjoint closed subsets of X . Let y E Y . By assumption of the lemma, there exists an open subset U,' of X such that E n f"-'( y ) c U,'and ob'n (F n J ' ( y ) , = 8. Let F' = n F. Then F is closed and F' n f ' ( y ) = 0. Sincefis closed and Y is reguiar, there exists an open nbd 0 of y such that F' ' '(0i.r= 0. Put C: = U , f '((31. Then U, is open and we have E n J 'r y ) z L; and 6,n F = 8. Similarlq, we can take an open subset r ; oi' X such that J -'(yt - L', c 1: ano n E = 8. Then we note
u,'
~5
i
'( y )
y
[I,
-t
'b
15 n f ' ( y ) c O,, G , n F = QI,
-
F n f ' ( y >c V , ,
B ? E
= $9.
Since fis closed, there is an open nbd W , o f y such that f - I ( ) c U, LJ 1:. Then W ; I y t Y IS an open cover of L . l a ~ de iocally finite open cover [ L , , l yF Y ) such tha. 2, c El for each J' E t . N o w let us pui h = Ij(L', nf ' ( L , , ) , y E I.~).Theniriseasytocnec~thaiL: n H = @and c7 n I. = 8. Hence, x is normal c
Normality of Product Spaces I1
157
Proof of Theorem 2.17. By assumption there exist a normal M-space 2 and a closed map f:2 + Y. We first prove that X x 2 is normal (Chiba [1974]). Since Z is an M-space, there exists a quasi-perfect map g : 2 -+ T from Z onto a metric space T. Consider t,b = 1, x g : X x Z -+ X x T. Then it is not hard to see that t,b is a closed map because Xis first countable and g is quasi-perfect. By Corollary 1.10, X x T is paracompact. And since X is normal, every pair of disjoint closed subsets of t,b-'(x, y ) = {x} x g - ' ( y ) are obviously separated by open subsets of X x 2. Hence, by the lemma above, X x 2 is normal. Also we note X x Z is countably paracompact since I) is quasi-perfect and X x T is paracompact. Secondly we note that since a normal M-space is collectionwise normal, so is Y and, by Theorem 2.1, Y can be written as Y = Yo u Uf91 where is closed and discrete for each i 2 1 andf-'(y) is countably compact for' each y E Yo. Now, to prove the "only-if" part, assume X x Y is normal. Let 9 = { U, I n E N} be a countable open cover of X x Y with U, c U,, I . Since X is paracompact, by Proposition 2.14 and Lemma 2.6, X x Y has property (w). Therefore, by applying property (w) to each point y E Ufpl and using collectionwisenormality of Y, similarly as in the proof of (c)*(a) of Theorem 2.7, we can take a o-discrete collection V = UVnof open subsets, where V,, is discrete, such that X x (UfaI t U V and Q refines 9.Let us put F = X x Y - U V . Then Fis closed and contained in X x Yo. On the other hand, let cp = l X x f : X x 2 + X x Y. Then, since { q r 1 ( U n ) 1 E n N} is an open cover of X x 2 and X x Z is countably paracompact, there exists a closed cover {E,,ln E N} of X x 2 such that En c cp-'(U,) for each n E N. Now, since Xis first countable andf-'( y ) is countably compact for y E Yo, we note that the following holds for each n E N:
x,
x
x)
V(En) n F c
un.
Let 0,= U{ 7 1 Y E V,,}. Then the above arguments show that {D,,, cp(E,,) n F ( n E N} is a countable closed cover of X x Y that refines 9.Therefore, X x Y is countably paracompact since X x Y is normal. Conversely, assume X x Y is countably paracompact. Let E and F be any disjoint closed subset of X x Y. Then, similarly as above, there exists a a-discrete collection V = U k V kof open subsets of X x Y , where Vkis discrete, such that X x (UfrfY,)c U V and for each V E 9" either 7 n El = 8 or P n E2 = 0 holds. Let us put, f o r k E N a n d j = 1, 2,
q,k =
U{VEV l Y n
4 z S}.
158
T. Hoshina
Then we have, for k
ol,& n E2
E
N and j = 1, 2,
=
0
and El n 0 2 , k =
0.
On the other hand, since X x Z is normal, there exist disjoint open subsets GIand G2 of X x 2 such that cp-'(Ej) c Gj ( j = 1, 2). Let 4 = X x Y cp(X x 2 - Gj)( j = 1, 2). Then H I and H2 are open and disjoint, and by the same reason as above, we have
E, n (Xx 6 ) c Hj, j = 1,2. Let us put Lo = X x Y - R2and Lk = X x Y - i 7 2 , & , k 2 1. Then L,, are open and El - U k U 1 . k c L, (n = 0, 1, 2, . . .). Moreover, we have nnaOLn n E2 = 0.Hence, by Lemma 2.6, there exists an open subset Ul,oof
ol,o
X x Y such that El - U k a l Ul,&c U,,oand n E2 = 0.Similarly there exists an open subset of X x Y such that E2 - U k a I U2.k c U2,0and El n 022.0 = 0.Now we have chosen open subsets q,,, n = 0, 1, 2, . . . ; j = 1, 2 of X x Y such that
ul,,nE2 =
0,
EInO2,, =
0,
n=0,1,2,...
.
Thus, as is well known, E, and E2 are separated by open subsets of X x Y. Hence, X x Y is normal. 17 References Arhangel'skii, A. V. [1963] On a class of spaces containing all metric and all locally bicompact spaces, Soviet Math. Dokl. 4, 751-754. BeilagiC, A. [I9861 Normality in products, Topology Appl. 22, 71-82. (19851 A Dowker product, Trans. AMS, to appear. BeSlagiC, A. and K. Chiba [1987] Normality of product spaces, Preprint. Chiba, K. 119741 On products of normal spaces, Rep. Fac. Sci. Shizuoka Wniv. 9, 1-1 1. Chiba, T. and K. Chiba [1974] A note on normality of product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A . 12, 55-63.
Normality of Product Spaces 11
159
Gruenhage, G. [I9801 On the M3 =. M ,question, Topology Proc. 5, 77-104. Hanai, S. [I9611 Inverse. images of closed mappings, 11, Proc. Japan Acad. 37, 302-304. Hoshina, T. [I9831 Normality and countable paracompactness of product spaces, Questions Answers General Topology 1, 54-56. [I9841 Products of normal spaces with LaSnev spaces, Fund. Math. 124, 143-153. [1984a] Shrinking and normal products, Questions Answers General Topology 2, 83-91. [ 19871 Normality of product spaces, Questions Answers General Topology (Proc. Soviet-Japan Joint Symp. Japan) 5, 139-142. Ishii, T. [I9661 On product spaces and product mappings, J. Math. SOC.Japan 18, 166-181. LaSnev, N. [I9651 Continuous decompositions and closed mappings of metric spaces, Soviet Math. Dokl. 6, 1504-1506. [I9661 Closed images of metric spaces, Soviet Math. Dokl. 7, 1219-1221. Leibo, I. M. [I9741 On the equality of dimensions for closed images of metric spaces, Soviet Math. Dokl. 15, 835-839. Mack, J. E. [19701 Countable paracompactness and weak normality properties, Trans. A M S 148,265-272.
Michael, E. [1963] The product of a normal space and a metric space need not be normal, Bull. A M S 69, 375-376. Morita, K. [I9551 A condition for the metrizability of topological spaces and for n-dimensionality, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A 5, 33-36. [I9611 Note on paracompactness, Proc. Japan Acad. 37, 1-3. [I9631 Products of normal spaces with metric spaces 11, Sci. Rep. Tokyo Kyoiku Daiguku, Sec. A 8, 87-92. [1963a] Note on products of normal spaces with metric spaces, Unpublished manuscript. [I9641 Products of normal spaces with metric spaces, Math. Ann. 154, 365-382. [I9751 t e c h cohomology and covering dimension for topological spaces, Fund. Math. 87, . 31-52. Nagami, K. [I9691 Z-spaces, Fund. Math. 65, 169-192. [I9711 Dimension for u-metric spaces, J. Math. SOC.Japan 23, 123-129. Nagata, J. [1985] Modern General Topology (North-Holland, Amsterdam). Okuyama, A. [I9671 Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 8,236-254. Rudin, M. E. [I9751 The normality of products with one compact factor, General Topology Appl. 5,45-59. [1975a] Lectures on Set Theoretic Topology, Reg. Conf. Ser. in Math., 23.
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Rudin, M. E. and M. Starbird [1975] Products with a metric factor, General Topology Appl. 5, 235-248. Starbird, M. [I9741 The normality of products with a compact or a metric factor, Ph.D. Thesis, Univ. of Wisconsin. Stone, A. H. [I9621 Absolute F,-spaces, Proc. AMS 13, 495499. Vaughan, J. E. [I9751 Nonnormal products of o,-metrizable spaces, Proc. A M S 51, 203-208. Wage, M. L. [I9781 The dimension of product spaces, Proc. Natl. Acad. Sci. USA 75, 46714672.
K. Morita, J. Nagata, Eds., Topics in General Topology 0 Elsevier Science Publishers B.V. (1989)
CHAPTER 5
GENERALIZED PARACOMPACTNESS
Yoshikazu YASUI Department of Mathematics, Osaka Kyoiku University, Tennoji, Osaka, 543 Japan
Contents
1.
2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . Characterizations of paracompactness . . Characterizations of submetacompactness Characterizations of metacompactness . . Characterizations of subparacompactness Shrinking properties . . . . . . . . . . Examples. . . . . . . . . . . . . . . References . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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161 162 164 175 178 184 186 191 198
Introduction
The purpose of this chapter is to give some classes containing paracompact spaces and a rather detailed account of the characterizations of their classes. The class of paracompact spaces simultaneously contains two important classes: metric spaces and compact spaces. The concept of paracompactness was introduced by Dieudonni [1944] as a class which contains a class of compact spaces, and an important theorem (that is, every metric space is paracompact) was shown by Stone in 1948. After that, paracompactness and its associated concepts have become popular among topologists and analysts. In the last forty years, due to the introduction of locally finite or closurepreserving properties, many covering properties which are weaker than the property of paracompactness have been considered. In this chapter we consider some classes (subparacompact spaces, metacompact spaces, submetacompact spaces and shrinking spaces etc.) which
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seem to be very efficient and natural properties for studying the generalizations of paracompactness. This chapter can be read as a sequel to Nagata [1985, Chapter 51 but to make it self-containedI have recapitulated some of the main results announced in this book in some different format. A research had produced many results of various kinds concerning the covering properties. Many covering properties (for example paracompactness, metacompactness and subparacompactness etc.) are defined in terms of any open cover and so one of the purposes of this chapter is to characterize the covering properties in terms of any directed or any monotone increasing open cover (in Sections 2,3,4,5 and 6). In Section 7, we shall discuss the relations between the properties appearing in this chapter. The reader should be aware of several recent books on covering properties and applications. Among these are: Van Douwen [1980], Engelking [1977] and Nagata [1985]. This chapter lays emphasis on the relations among the characterizations of paracompact spaces and other covering properties. Although not very much knowledge is required to read this chapter, the reader is advised to consult Nagata [ 19851for some preliminary concepts and notions.
1. Preliminaries
Throughout this chapter, we assume the term space refers to a Hausdorff topological space. A mapping is a continuous onto function and the set { 1, 2, . . .} of the natural numbers is denoted by N. An ordinal is a set of all smaller ordinals and as a topological space, will have the order topology. Cardinal numbers are initial ordinals. The symbols w, w , and o2denote the first infinite ordinal, the first uncountable ordinal and the second uncountable ordinal, respectively. For a set A, the cardinality of A is denoted by I A I, and all the finite subsets of A are denoted by [A]. For a collection of sets 4, the family consisting of all finite unions of sets from 4 is denoted by 4'. For collections of sets 4 and Y and a set A, 4 A A denotes the collection ( U n A 1 U E 4 }and Q A Y the collection { U n V (U E 4 , V E Y } .If each element of 4 is contained in some element of Y , we shall say that 4 is a partial refinement of Y . If, moreover, 4 and Y are covers of X,then 4 is a refinement of Y . For a collection Q of subsets of a space X and a subset A of X,St(A, 4) denotes the set U{U I U n A # 8, U E 4}(if A = {x}, St({x}, 4) is simply denoted by St(x, 4)).The closure of A is denoted by C1 A and the collection {Cl U I U E Q} is denoted by C1Q.
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We say that a collection 4 is directed if %‘is a partial refinement of 4 ; that ! which is, if for any finite subcollection Y of 9, there is some element U of & contains UV. A collection 4 = { U,l a < t}, where t is an ordinal, is monotone increasing if U, c Upfor any a < fl < t. A collection Q of subsets of a space Xis said to be discrete (locallyfinite, locally countable) if for every x E X there is a nbd (=neighborhood) 0 of x such that the cardinality of {U E 4 1 0 n U # S} is at most one (resp. finite, countable). The collection Q is closure-preserving (resp. interior-preserving) if for any subcollection Y of 42, U(C1 VI V E V }is closed (resp. n(Int VI V E V }is open), where Int V denotes the interior of V. It is seen that a collection 4 is interior-preserving if and only if (X - U I U E 4 } is closure-preserving. A collection 4 of subsets of a set X is point-jinite (resp. point-countable) if each point x E X is contained in at most finitely many (resp. countably many) members of 4. A collection 4 is o-discrete (resp. 0-locallyfinite, etc.) if 4 can be expressed as 4 = U{%$ In E N} where each anis discrete (resp. locally finite, etc.) For later use we observe the following properties (they are easily verified and so we omit the proofs).
1.1. Lemma. Let 9 be a closure-preserving collection of closed subsets of a space X and A a closed subset of X. Then the collection 9 A A is closurepreserving .
1.2. Lemma. Let 4 be an interior-preserving collection of open subsets of a space X . Then the collection ( U Y 1 Y c 4 }(that is, all unions of all subcollections of 4 ) is also interior-preserving. The following properties are very useful later:
1.3. Lemma. Let 4 be a collection of open subsets of a space X and Y a point-finite open cover of X . Zf we let F ( U ) = { x E XISt(x, Y ) c U } for each U E 4, then the collection 9 = { F ( U )I U E 4 } is a closure-preserving collection of closed subsets of X. Proof. Let { U, I a E A } be any subcollection of 9 and any x # U{F(U,} 1 aEA}.SinceYispoint-finite,welet{V,, . . . , I / , } ~ ~ { V E Y J V}.Ifwe XE put Ai = { a E A I J( - U, # S} for each i = 1, . . . , n, then n (U{F(U,)( a E A i } ) = 8 for each i = 1, . . . , n and A = U { A i l i = 1, . . . , n}. So yl i = 1, . . . , n } is an open nbd of x which has the empty intersection with U(F(U,) I a E A}, that is, U{F(U,) I a E A } is closed. 0
n{
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2. Characterizations of paracompactness As stated above, we shall consider some classes of spaces as generalizations of paracompact spaces, and therefore it is unavoidable to study the characterizations of paracompact spaces. Morita [19621 introduced the notion of r-paracompactness in order to characterize the normality of the product space with a space and I' (where r is an infinite cardinal number and I' means the product space of r copies of Z = [0, 11). 2.1. Definition. Let r be an infinite cardinal. A space Xis t-paracompact if any open cover of X with cardinality < t admits a locally finite open cover as its refinement. If 7 = w, the notion of o-paracompactness is that of countable paracompactness. It is easy to see that the notion of a,-paracompactness is different from that of a,+,-paracompactness. In fact, for any ordinal a, the linearly ordered space a,+,is normal o,-paracompact but not w,+ ,-paracompact. r-paracompact spaces have many properties that are analogous to those of paracompact spaces. We will start with by stating the following characterizations of r-paracompactness. 2.2. Theorem (Morita [1962]). Let X be a space and r an infinite cardinal. Then the following are equivalent: (1)
X is r-paracompact.
(2)
X is countably paracompact and every open cover of X with cardinality < t has a a-locally finite open refinement.
Proof. (1)=42) Obvious. (2)~(1)Let9beanyopencoverofXwith(9~ < r a n d y = u{-V;,lnE N} a a-locally finite open refinement of 9, where each Ynis locally finite. If V , = uVnfor each n E N, then { V,,1 n E N} is a countable open cover of X , and therefore we have a locally finite open cover { W, I n E N} of X such that W, c V,, for each n E N. Then { W, n V J V E V,;n E N} is a locally finite open refinement of 02.Therefore Xis t-paracompact. It is interesting to be able to characterize the compactness of a space in terms of the existence of finite subcovers for all monotone increasing open covers of a space (Alexandrov and Urysohn [1929]). So we shall show the following characterizations of t-paracompact spaces in terms of monotone increasing (or directed) open covers:
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2.3. Theorem. Let X be a space and T an infinite cardinal. Then the following are equivalent: (1)
(2) (3)
X is T-paracompact. Every monotone increasing open cover of X with cardinality < T has a locally finite open refinement. Every monotone increasing open cover of X with cardinality < 7 has a 0-locallyfinite open cover Y of Xsuch that CI Y is its refinement.
Proof. (1)=>(2)Obvious. (2)*(3) Let 4 = { U, I a < t} be any monotone increasing open cover of X and Y a locally finite open refinement of 42, where we may express Y as {Val. < 7 } such that V, c U , for each a < z. If we let W, = u { O ( O is open in X , 0 n V, = 8 for any fl > a}, then W = { W,I a < t} is a monotone increasing open cover of X. Since C1 % n 5 = 8 for any p > a, we have: CI w, c x - U { y p > a }
= U{V,lB < a> = U{U,IP < a }
=
u,.
By (2), we have a locally finite open refinement d of W , and so CI d is a refinement of 4. (3)*(1) We shall prove this by transfinite induction on T. To begin with, we shall show that X is countably paracompact. Let 4 = {U,,ln E N} be any countable open cover of X . If we let 0,= U{& (i = 1, 2, . . . , n} for each n E N, then there exists a a-locally finite open cover Y such that CI Y is a refinement of { 0 , ln E N}, where we may express that Y = U{Y,,I n E N} and each Y,,= { V,,iI i E N} is locally finite with C1(V,,i) c Oifor each i E N. For each n, we put W, = K,j(i, j < n } . Then it is easily seen that { W,,1 n E N} is a monotone increasing open cover of X such that C1 W, c 0, foreachne N . L e t d = {Ul, U,, - CI W . - l l n 2 2}.Thenitisseenthatd is a locally finite open refinement of Q, and so Xis countably paracompact. Assuming that Xis A-paracompact for any Iz < T , we shall show that Xis r-paracompact. For this purpose let { U, I a < T } be any open cover of X . If 0,= U{U,l/3 < a } for each a .c T , then 8 = {O,(a < T ) has a a-locally finite open cover Y of X as in (3). Since X is countably paracompact, it is easily seen that there exists a locally finite open cover W = { W, I a < T} of X such that C1 W, c 0,. For each a < t (by the induction hypothesis), we have a locally finite open cover 9,of CI W, such that 9,is a partial refinement of {U,lB < a } , where local finiteness and openness of 9,mean in CI W,. Lastly we let
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d, = 9,A W,. Then dais a locally finite (in X) collection of open subsets of X which is a cover of W, and a partial refinement of 4 and therefore U{da 1 a < z} is a locally finite open refinement of 4. 0 2.4. Theorem. Let X be a space and z an infinite cardinal. Then the following are equivalent:
(1)
X is z-paracompact.
(2)
Every directed open cover of X with cardinality finite open refinement.
< z has a locally
(3)
Every directed open cover of X with cardinality finite closed refinement.
has a locally
Proof. (1)=.(2) Obvious. (2)*(3) Let 9 be a directed open cover of X with 1 9 I < z. Express 9 as 9 = { U, I a E D}, where D is directed and if a < /I (a, /I E D), then U, c U,. From the statement (2), we can take a locally finite open cover V = { V,I a E D } of X such that V , c U, for each a E D. For each a E D, we let W , = X U{Cl 5 I /I 4: a}. Since { W, I a E D } is a directed collection of open subsets of X such that C1 W, c 17, for each a E D, we shall show that { W, I a E D } is a cover of X.Let x E X.Then there is a finite subset (a,, . . . , a,,} of D such that X E n{Cl V,li = 1 , . . . , n} - U{Cl V,la # ai for any i = 1 , . . . , n } . Since D is directed, we have some a0 E D such that a. > ai for any i < n, and therefore x E Wq. From (2), we have a locally finite open refinement d of { W, I a E D } , Then the covering C1 d is a locally finite closed refinement. (3)=4) It suffices to show that (3) implies (2) by Theorem 2.3. We will use the method of the proof of Lemma 1 of Michael [1953]. Let { U, l a E D } be any directed open cover of X with ID1 < z and 9 a locally finite closed refinement of { U, 1 a E D}, where we may express 9 as 9 = {F, I a E D } such that Fa c U, for each a E D. For each A E [D], we let V ( A ) = X U{F, 1 a 4 A}. Since { V ( A )I A E [ D ] }is a directed open cover with cardinality < z, there is a locally finite closed refinement &' where we may express &' as &' = { H ( A )I A E [D]} such that H ( A ) c V ( A ) for each A E [D]. Let W, = U, - U{H(A))H(A) n Fa = 8} for each a E D. Since W = { W, I a E D } is an open cover of X such that W, c Uafor any a E D, it suffices to prove the local finiteness of W .For any x E X , we can choose an open nbd 0 of x such that {A E [ D ] ( On H ( A ) # S} is finite, and so we denote its finite set by { A l , . . . , A n } .For any a 4 U{Al, . . . , A n } ,we have H(AJ n F, c V ( A i ) n Fa = 8 and so H ( A i ) n W , = 8 for each i < n. Hence
Generalized Paracompactness
0 n W , c ( U { H ( A i )I i < n } ) n W , = elements of {W,laE U { A i l i < n } } .
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8.Therefore 0 meets at most many 0
2.5. Remark. In the proof of (2)*(3) in the above Theorem 2.4, we have proved that: (2)’ Every monotone increasing open cover 4 of X with cardinality < r has a locally finite open cover Y of X such that C1 Y is a refinement of 4, and therefore we have the following corollary. 2.6. Corollary. Let X be a space and r an infinite cardinal. Then thefollowing are equivalent: (1)
X is r-paracompact.
(2)
Every monotone increasing open cover 4 of X with I % I < t has a locallyfinite open cover Y of X such that C1 Y is a refinement of 4.
(3)
Every directed open cover 4 of X with 14 I < r has a locallyfinite open cover Y of X such that C1 Y is a refinement of 4.
If
T
= o in Theorem 2.3, the following well-known theorem holds.
2.7. Corollary (Ishikawa [ 19551). A space X i s countably paracompact ifand only iffor every countable increasing open cover { UnI n E N} of X,there exists an open cover { V ,I n E N} of X such that C1 V. c U,, for each n E N. As is well known the paracompactness can be characterized in terms of a
cushioned refinement (see Michael [1953]). Katuta [1977] characterized a t-paracompactness in the same terms which are used in order to obtain some results with respect to the normality of product spaces. We shall show the Katuta’s characterization. Let 4 and Y be collections of subsets of a space X:Then we say that Y is cushioned in 4 if we can assign to each V E Y a U( V) E 4 such that for every subcollection Y ’ of Y , Cl(UY’) c U { U ( V )I V E Y ’ } . 2.8. Theorem (Katuta [1977]). Let X be a normal space and t an infinite cardinal. Then the following are equivalent: ( I ) X is r-paracompact. (2) For every monotone increasing open cover 4 = { U, I a < A} of X with A < t,there exists a sequence { Y,I n E N} of collections of open sets of X such for that Y = U{YnI n E N} is a refinement of 4 and Ynis cushioned in Yn+, any n E N.
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Proof. (1)=+(2) Let 9 = {V,la < A} be any monotone increasing open cover of X with rl < T . Since Xis r-paracompact, there is a locally finite open refinement V of 9. Express V = { V,I a < A}. Since X is normal, there is a closed cover {FaI a < A } of X such that F, c V, for any a < A, and furthermore, for each a < A there is a sequence { & , I nE N} of open subsets of X such that F, c V,, c C1 V,, c V,n+l c V, for each n E N. If for each n E N, we let V, = { V,, I a < A}, then a sequence {V,I n } of open covers of X satisfies the statement (2). (2)=41) Let 9 = { V , I a < r } be any monotone increasing open cover of X. If we show that there exists a a-locally finite open cover V of X such that C1 Y is a refinement of 42, it follows X is r-paracompact by Theorem 2.3. At first we have some open refinement V = {V,1 n E N} of 9 such that V, is cushioned in V,+lfor each n E N, where we may assume that for each n E N, V, is indexed by T (that is, V, = { V,, I a < r } ) and V, is cushioned in V,+lwith respect to a < t (that is, for any A c r , Cl(U{ VnulaE A}) c u{V,,+lalaE A}). If for each n E N and each a < z, we let W,, = V,, Cl{U{V,+IsIp< a}),thenW = { W , , l n ~ N ( , < a r}isanopencoverofX such that C1 W is a refinement of 9. Therefore, to complete the proof, we shall show that { W,, I a < r } is locally finite in X. Fix n E N and x E X. If x is in U{V,+lula< r}, let a be the 1st of < T ~ X EV,+I,,}. Then V,,+lu n W,,, = 8 for any /3 < z with a < p and x$Cl(U{V,,IB < a}). Therefore an open set V,+lu - Cl(U{ V,,, I p < a}) is a nbd of x which intersects at most one member W,, of { WnsI fl < r } . I f x is not in U{ V,,+,, I a < r } , then 0 = X - Cl(U{ V,, I a < r } ) is a nbd of x which intersects no members of { W,,I a < r } . It completes the proof that { W,, I a < r } is discrete and therefore locally 0 finite (for any n).
{a
The above proof (1)*(2) of Theorem 2.8 shows that { W,, I CL < r } is discrete for each n E N , and therefore a normal space X is r-paracompact iff every monotone increasing open cover { U,I a < A} of X with A < T has a a-discrete open cover W such that CI W is a refinement of { U, I a < A}. In fact we have the following proposition. 2.9. Theorem (Katuta [1977]). Let X be a normal space and r an infinite cardinal. Then the following are equivalent: (I)
X is r-paracompact.
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169
(2)
For every monotone increasing open cover { U,I u < r} of X , there exists a 0-discrete open cover W of X such that Cl W is a refinementof {U,lu < T } .
(3)
For every open cover { U, I u < r} of X . there exists a a-discrete open cover W of X such that CI W is a refinement of { U, 1 a < r}.
Proof. It suffices to show that (1)*(3). Let 9 = { U, I u < r} be any open cover of X with cardinality < T and { V,Iu < r} a locally finite open refinec U, for each a < r. Since X is normal, there are a ment of 9 such that closed cover {F,I u < r} of X and a sequence { { W, I u < r} I n E N} of open covers of Xsuch that Fa c W, c CI W, c W,+ c V ,for any u < 'I. For each n E N and each u < r , we put On, = W,, - Cl(U{ Wn+lsI fl < a}). We leave the exercise of showing that if we put 0, = {On,I u < r} for each n E N, then U { U , l n E N} is an open cover of X such that CI (U{0,,,lnE N}) is a refinement of 9. To see that each 0, is discrete, let x be any point of X and a ( x ) the 1st of { a l x E W,,+l,}. For each u > a(x), it holds that Wn+lil(r)n On, = 8. For each u < u ( x ) , it holds that (X - Cl(U{ WnaIB < ~ ( x ) } ) n ) On, = 8. Therefore Wn+Iu(x) n (X - Cl(U { WnaI fl < ~ ( x ) } )is ) a nbd of x which intersects at most one member On,(x) of (0,. I a < r}. 0
,
2.10. Remark. (1) As mentioned above, a r-paracompactness was introduced by Morita [1962] in order to characterize the normality of a product space of its space and Z'. As is well known, the product space of a paracompact space and a metric space may fail to be normal; then, what is the necessary and sufficient condition for a normal space X in order that the product space of X with every space belonging to some class 9 of spaces be normal? We shall recall this problem in cases that its necessary and sufficient conditions are covering properties. A space Xis countably paracompact and normal iff X x (0,l/n} is normal, where (0, l/n} is the subspace of real line (Alas [1971]) iff X x Z is normal, where I is the closed unit interval (Dowker [1951]). A space X is countably paracompact and collectionwise normal iff X x u(X,) is normal, where u(X,) is the one-point compactification of a discrete space X , with I X , 1 = I X I , iff X x D" is normal, where D is two-point space with discrete topology (Alas [1971], Atsuji [1989]). (2) From Corollary 2.6 and Theorem 2.3, we have the following characterization of r-paracompactness:
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A space X is r-paracompact iff for every directed open cover Q of X with I Q I < T, there exists a a-locallyfinite open cover Y of X such that CI Y is
a refinement of Q. Since a space X is paracompact iff X is r-paracompact for any infinite cardinal T,the following theorem except (6) follows from the above theorems etc. 2.11. Theorem (Michael [1953, 1957, 19591, Mack [1967] and Junnila Let X be a regular space. Then the following are equivalent:
X is paracompact. Every open cover of X has a a-locally finite open refinement. Every monotone increasing open cover of X has a locallyfinite open refinement . Every monotone increasing open cover of X has a locallyfinite open cover whose closure is as a refinement. Every monotone increasing open cover of X has a a-locally finite open cover whose closure is as a refinement. Every monotone increasing open cover of X has a locally finite closed refinement. Every directed open cover of X has a locallyfinite open rejinement. Every directed open cover of X has a locallyfinite closed refinement. Every directed open cover of X has a locallyfinite open cover whose closure is as a refinement. Every directed open cover of X has a a-locally Jinite open cover whose closure is as a refinement.
Proof. It suffices to show that (6) implies (1). In order to show that X is paracompact, we shall show that by transfinite induction on 1:
(Pi)Every open cover Q with cardinality 1 of X has a locally finite closed refinement. Since it is easily seen that Xis countably paracompact, assuming that for some infinite cardinal T (T > w), every open cover Y of X with I Y I < r has a locally finite closed refinement and 43 any open cover of X with IQ I = r. If for each a < r, we let V , = U{V, 1 /3 < a}, then { V,I a < r} is a monotone
Generalized Paracompactness
171
increasing open cover of X and so there exists a locally finite closed cover { F , ( a < T} of Xsuch that F, c V, for any a c T. Since for each A < T, { U, I fi c A} A FA is an open cover of FA whose cardinality C T , there exists a locally finite closed refinement FA of { U, I fi < A } A FA(and hence .!FAis a locally finite collection of closed subsets of X).Hence U{.!FA I A} is a locally finite closed refinement of { U, I a c T} and so X is paracompact by the following theorem. The following characterizations are well known and so we give them without proof in order to show some contrast with another covering property.
2.12. Theorem (Michael [1957]). Let X be a regular space. Then the following are equivalent: (1)
X is paracompact.
(2)
Every open cover of X has a closure-preserving open refinement.
(3)
Every open cover of X has a closure-preserving refinement.
(4)
Every open cover of X has a a-closure-preserving open refinement.
2.13. Theorem (Michael [ 19591). Let X be a regular space. Then the following are equivalent: (1)
X
(2)
Every open cover of X has a cushioned open refinement.
(3)
Every open cover of X has a cushioned refinement.
(4)
Every open cover of X has a a-cushioned open refinement.
is paracompact.
2.14. Definition. Let d and LiJ be collections of subsets of a space X and x E X . We say that d is pointwise (local) W-refinementof 93 at x if there exists some W E [B] (and furthermore a nbd U of x) such that {A E d I x E A } ( { A E d I U n A # 8)) is a partial refinement of W . If we can choose %? to consist of a single element of W, then we say that d is a pointwise (local) star-refinement of W at x. If U d = UB and d is a pointwise (local) W-refinement of A? at each point of X , then we say that d is a pointwise (local) W-refinement of 9. These concepts were introduced by Worrell [1966b, 19681. Recall that a space Xis said to be fully normal if every open cover 42 of X has a pointwise star-refinement at any point x E X . It is known that a space
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X is paracompact iff X is fully normal (Stone [1948]). So we shall give a characterization of paracompactness which has a close connection with them of another covering property.
2.15. Theorem (Arhangel’skii [1961]). Let X be a regular space. Then the following are equivalent: ( I ) X is paracompact. (2) For every open cover 4 of X , there exists a sequence {4,, I n E N} of open covers of X such that for each x there is some n such that 4,,is a local star-refinement of 4 at x . The above sequence {%,, I n} of open covers of X is said to be locally starring in 4 .
Proof. (1)=42) It is clear because paracompactness implies full normality. (2)=41) Let 4 = {UrnI a E A } be an open cover of X and a sequence {q,, In E N} of open covers of X which is locally starring in 4. For each a E A and each n E N, we let C(rn,n) = U{O 1 0is open in X , St(0, 4,,)c V , } .Then it is seen that {CCrn,,,, I a E A , n E N} is an open cover of X and (C(rn,n) I a E A} is cushioned in 4 for each n E N. Therefore Xis paracompact by Theorem 2.13.
0 It is one of the important properties of paracompactness that all metric spaces are paracompact spaces. Stone was the first to prove directly the paracompactness of a metric space when he proved that every open cover of a metric space has a locally finite and a-discrete open refinement (Stone [1948]). After that, Michael [I9531 proved that if every open cover of a regular space X has a o-locally finite open refinement, then Xis paracompact and hence every metric space is paracompact because a regular space X is metrizable iff X has a a-discrete open base (Bing [1951]), or, iff X has a a-locally finite open base (Nagata [1950] and Smirnov [1952]). Furthermore, we may prove it by use of Arhangel’skii’s Theorem 2.15, that is, if for an open 0 is open, diam(0) < l/n and 0 c U for cover 4 of X , we let 4,,= (01 some U E 4 } ,then {@,,I n} is a locally starring in 4. Quite recently, Ohta [1987a] introduced a new class which contains a class of metric spaces and is contained in a class of M,-spaces (a regular space is said to be an MI-space if it has a a-closure-preserving open base (Ceder [1961]). The Ohta’s class is hereditary and countably productive (it is an open question whether a class of M,-spaces is closed hereditary or not).
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I73
In his paper, he introduced a new concept of a collection of subsets, which is called FCP, and he proved a new characterization of a paracompactness in terms of FCP.
2.16. Definition. A collection 9 of subsets of a space X is called to be finitely closure-preserving (FCP) in X (Ohta [1987a]) if for any subcollection 9’ of 9and any point x E X , there are a nbd 0 of x and a finite subcollection 9“ of 9‘such that 0 n (UP’) c Cl(U9”). It is easy to see that a local finiteness implies the FCP property and FCP property does a closure-preservingproperty. The following lemma is useful to prove Theorem 2.18. The proof is left to the reader. 2.17. Lemma (Ohta [1987b]). r f {Vala< 7 } is a monotone increasing collection of subsets of a space X , then the following are equivalent: (1)
(U,(a <
(2)
For any a. < T and any x E X , there are a nbd 0 of x and some a , < aosuch that 0 n (U{U,lB < a o } ) c C1 Ua,.
T}
is FCPin X .
2.18. Theorem (Ohta [1987b]). Let X be a regular space. Then the following are equivalent: (1) X is paracompacr. ( 2 ) For every monotone increasing open cover 43 = { iJaI a < T } of X , there exists a monotone increasing and FCP o-Den cover { V ,1 a < T} of X such that C1 V , c C,.for each a < T . Proof.
I‘ 1:*(2) Obvious. i 2 ) 2 ( 1 ) Let 42 = {Cu 1 ct .< T’. be any monotone increasing open cwer 01‘ = V ,I a < r } a monotone increasing and FCD open zovx of A’ A’. and such that C1 1.: c LT- for any o! < 7 If fot each Q : : ae let F = C1 V , - J{Cl C/,:o < a1.then.F = !F,ln < 7 , isacoverofXsuzh that C1 Fa c i‘,for any a < r. To finish the Droof it suffices to show thai F,I CI < T ; is locally finite by Theoreni 2.1 i . For t h ~ spurpose let x be any point of X-,a(Of the first of { a I x E 6 [ and H , = &, . Then we have thar ti,>
is an open nbd of x and (0;
Hc, n F,
=
8 for any y > ~ ( 0 )
By Lemma 2.17, there are an open nbd H , of x and some a( 1 ) < a(0) sucn thatH, n (U{l$Ib < a(O)}) c Cl(l&). Foranypwitha(1) . /3 < a(O),we
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have H I n l$ c H I n C1 c H I n Cl(U{yly < < /3}) = 8. Hence (C1 l$ - U(C1 (1)
H I n F, =
8
for any /3 with a(1) <
/3
/3}>
and so H I n
< a(0).
By repeated use of Lemma 2.17, there are an open nbd H,, of x and a sequence {a(n)ln}such that a(1) > a(2) > a(3) > . . . and:
(n)
H,, n FB =
8 for any /3 with a(n) < /3 < a(n -
1).
n{&.
Therefore a(n) = 0 for some n. If we let H = I i < n } , then H i s a nbd of x which does not meet the elements of 9 i = 0, 1 , . . . , n}. Hence we complete the proof that 9 is locally finite. 0 The last characterizations of a paracompactness are obtained in terms of an interior-preservingcollection. They do not appear frequently but we want to show them in order to show the contrast with other covering properties. 2.19. Theorem (Junnila [1979a]). Let X be a space. Then the following are equivalent : (1) X is paracompact. (2)Every interior-preserving directed open cover of X has an interior-preserving open local star-refinement. ( 3 ) Every interior-preserving directed open cover of X has a a-closurecovers X . preserving closed refinement 9 such that { Int F I F E 9) (4)Every directed open cover of X has a closure-preserving closed refinement 9 such that { Int F I F E 9} covers X .
Proof. (1)*(4) and (4)*(3) Obvious. (3)*(2) The first step is to note that X is countably paracompact by use of Theorem 2.3. Let 4 = {U,,In} be an increasing open cover of X. Then there is a a-closure-preserving closed refinement 9 of 4 such that {Int F I F E 9} covers X because a monotone increasing open cover is intenorpreserving and directed. In this place we may express 9 as 9 = U{9,, I n } where for each n, 9,, = {F,,,lm E N} such that F- c V, for any m and FnI c Fn2 c . . . . Then it is clear that {U,, - U { F h p 1 I i< n - 1}1n E N} is a locally finite open refinement of Q. To actually see that X satisfies the condition (2),let Q be any interior-preserving directed open cover of X and so there is a a-closure-preserving closed refinement 9 = U{9,,ln}(where each F, is, closure-preserving) such that { Int F I F E F}covers X. If for each n, we let C,, = u{Int F I F E F,, then },{C,,I n } is a countable open cover of X and so we have a locally finite open cover { V ,I n } of X with V , c C,, for
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175
each n E N. Since for each n, H,, = {Cl V , n F I F E Sn} is a cover of V ,and a closure-preserving collection of subsets of X by Lemma 1.1 and {CI V ,I n} is locally finite. U { S f l1 n } is a closure-preserving closed refinement of 4 such that { Int H 1 H E Hn, n} covers X. Therefore 4!l has an interior-preserving open local star-refinement by Lemma 4.2 which will be proved later on. (2)=41) By Lemma 4.2 and the preceding part of (3)*(2), Xis countably paracompact and so by Theorem 2.2, it suffices to show that every monotone increasing open cover of X has a o-locally finite open refinement. This condition follows from Lemmas 4.2 and 4.4.
3. Characterizations of submetacompactness A class of O-refinable spaces (= submetacompact spaces) was introduced by Worrell, Jr. and Wicke [1965]. This class is useful in the theory of covering properties as well as in the theory of generalizations of metric spaces. “8-rejinable” is called to be “submetacornpact” by Junnila [1978]. Recall the following definition of submetacompact spaces.
3.1. Definition. A topological space X is said to be submetacompact if for every open cover 4 of X , there exists a sequence {4,,I n} of open refinements of 9 such that for each x E X there is some n such that ord(x, %,) is finite, where ord(x, 4,,)denotes the cardinality of { U E 4,,I x E U } . Furthermore, the above sequence {a,,1 n> is called to be &sequence or 8-rejinernent of 4. Secondly we shall define a concept which is useful for the submetacompactness, subparacompactness and others.
3.2. Definition. Let 4 be a cover of a space X,{4”I n } a sequence of covers is a pointwise (resp. local) of X and x a point of X. A sequence {4,,11} W-refiningsequencefor 9 at x if there exists some n such that %, is a pointwise (resp. local) W-refinement of 9 at x. If a sequence {en I n} is a pointwise (resp. local) W-refining sequence for 4 at x for any x E X, then {%,,In} is said to be a pointwise (resp. local) W-rejining sequencefor 4 (see Worrell [1966a]). In the above definition, we can strengthen by adding that each a,,is a refinement of 4. By Worrell[1967], the next characterization of &sequence (which is useful to submetacompact spaces and subparacompact spaces, etc.) was given in terms of a pointwise W-refining sequence. We shall give it without proof.
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3.3. Proposition (Worrell [1967]). For an open cover 4 of a space X , the following are equivalent: (1) 4 has a &sequence of open refinements. (2) There exists a sequence {4,I n} of open refinements of 4 such that for each x there is a sequence { t(n)I n} of natural numbers such that 4tr(n+l) is a pointwise W-refinement of a,(,)at x for any n E N. By use of Proposition 3.3, we can prove the following theorem which is worth contrasting with Theorem 2.15.
3.4. Theorem (Worrell [19671). A space X is submetacompact if and only if every open cover of X has a pointwise W-refining sequence by open covers of x. Proof. “If” part: Let 4 be any open cover of X . By hypothesis we have a pointwise W-refining sequence W , for 4. By repeated use of hypothesis, we have a countable collection W, of open covers of X for each n E N such that for each W E W,, there exists a countable subcollection W‘ of W, such that W’ is a pointwise W-refining sequence for W . Then {W,In} satisfies the condition (2) of Proposition 3.3, and so 4 has a &refinement. “Only if” part: We can prove similarly by use of Proposition 3.3. 0
In Theorem 2.1 1, we can characterize paracompactness in terms of monotone increasingcovers, that is, it is enough to find the locally finite refinement for any monotone increasing open cover only. A similar statement is true for submetacompactness as follows:
3.5. Theorem (Junnila [1978]). A space X is submetacompact if and only if every monotone increasing open cover of X has a &refinement of open covers. Proof. It suffices to prove only the “if” part. This proof is based on Junnila [1980]. For each cardinal K , we set (P,) as follows:
(P,):
Every open cover of X with cardinality K has a pointwise W-refining sequence by open covers.
We shall prove the statements (P,) by the transfinite induction on K. Since it is clear for finite cardinal IC,we assume that for infinite cardinal K, (PI) holds for any il < K. Let 4 be any open cover of X with cardinal K and so we may express 4 as 4 = { 17I, a < K}. If for each a < K, we let V. = U{U,1 B < a}, then { V ,I a < IC} has a &sequence {V,I n } of open covers of X . For each n,
Generalized Paracompactness
I77
we let H,, = {x I ord(x, Vn) is finite} and a(x, n) the 1st of {a I St(x, Vn)c V,} foreachx E H,,. Ifforeacha < Kandeachqlet P,,,= ~ { V .V,l E V Q V,}, then a,,, = { U, I fl < a } u {Pan}is an open cover of X with cardinality < K and so a,, has a &sequence { Wa,,kI k} of open covers of X . For each n, k E N and each x E H,,, we can select some member W,,(x) of W,(x,n),,kwith x E W,k(X) and we set Gnk(x)= Wn,(x)n . . n W,k(X) n (n{vlXE VE~,,}).IfWelet%,,k = {Gnk(X)IXEHn}U {{Xlord(x,~,,)2 k + l}}, then the sequence {9,,, I n, k } of open covers of X will be a pointwise W-refining sequence of%. For any point x of X , we select some n with x E H,,. If we let A = {the 1st of {a I V c K} 1 x E V E V,,}, then A is finite. For each a E A, there exist some k(a) E N and some finite subcollection W ( a ) of a,,, such that { W I x E W E W&,} is a partial refinement of W(a). Let R = U{R(a) A 4 I a E A } and k = max{ord(x, V,,), k(a) I a E A}. Then it is seen that {G 1 x E G E gnk}is a partial refinement of a finite subcollection 9l of 4.
17 As the last characterization of submetacompactness, we shall have it, as well as that of paracompactness, in terms of refinements for interior-preserving open covers of a space (see Theorem 2.19) or a-closure-preservingclosed refinements (see Theorem 2.12).
3.6. Theorem (Junnila [1974]). For a space X , the following are equivalent: (1)
X is submetacornpact.
(2)
Every interior-preserving directed open cover of X has a a-closurepreserving closed rejinement .
(3)
Every directed open cover of X has a a-closure-preserving closed rejinement.
= { U, I a E A} be any directed open cover of X and {an In} a &sequence of open covers of X for 9. If for each n, k E N, let Fnk= {XIord(x, 4,,)4 k} and @,,k = { U E %,,I U n Fnk# s}, then {Fnk I n, k } is a closed cover of X and @,,k is a cover of &k . Furthermore, ord(x, %,,k) 4 k for any X E F n k . 1 n } ) be an enumeration of a collection {a,,,I n, k } Let {V,,I n} (resp. { Y,, (resp. {FnkIn,k}) indexed by N such that V,,covers H,, for any n. Then W = { V - U{H, I i < n} I V E V,,}is a point-finite open refinement of %. If for each a E A, we let F, = {x E XISt(x, W ) c U , } , then {Fala E A} is a closed cover of X by the direction of 4. Therefore {F, I a E A} is a closurepreserving closed refinement of 4.
Proof. (1)*(3) Let 4
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Y. Yusui
(3)*(2) Obvious. (2)=4 1) Since a monotone increasing open cover is interior-preserving, it suffices to show that every interior-preserving open cover 9 has a 8-sequence of open refinements from Theorem 3.5. Since Qfis an interior-preserving directed cover of X by Lemma 1.2. 9'has a a-closure-preserving closed refinement 9, where we express 9as 9 = U { F nI n} and each 9" is closurepreserving. If for each n~ N and each X E X , we let K,x= n { U E 9 l x E U } ~ { F 9E" I x $ F}, then K,xis an open nbd of x and Vn= { <.xIx E X} is interior-preserving by Lemma 1.2. Also we can see that {Vn 1 n} is a pointwise Since each Vnis an interior-preserving open cover, W-refining sequence of 9. we have a pointwise W-refining sequence of Yn,each of which is interiorpreserving. By repeated use of the above argument, we have a countable collection V of interior-preserving open refinements of 4 such that for any x E X and Y E V, there exists some W E V which is a pointwise W-refinement of V at x. Hence 9 has a 8-sequence of open refinements. This completes the proof.
3.7. Remark. Bennett and Lutzer [I9721 introduced the generalization of submetacompact spaces, which is called to be weakly O-refinable spaces, that is: A space Xis weakly 8-rejinable if for every open cover 9 of X there exists a sequence {a"1 n} of collections of open subsets of X such that (1) each a,, is a partial refinement of 9 and (2) for each x E X , we have some n with 1 < ord(x, 9,, <)o. Normal and weakly 8-refinable spaces are not aiways 8-refinable (see Example 6.1). But in a class of perfect spaces (that is, every open set is Fg),weak 8-refinability is equivalent to O-refinability (Bennett and Lutzer [1972]). Though weak 8-refinable spaces have many interesting and important properties, we shall omit to discuss them (refer to Bennett and Lutzer [1972], and De Caux [1976], etc.).
4. Characterizations of metacompactness Metacompactness was introduced by Arens and Dugundji [ 19501. They showed that a countably compact space is compact iff it is metacompact. On the other hand, Bing [ 19511 introduced the pointwise paracompact spaces (which are equivalent to metacompact spaces) and he showed that metacompact developable spaces are not necessarily paracompact. One of the important and interesting results on metacompact spaces is the theorem that every
179
Generalized Paracompactness
collectionwise normal metacompact space is paracompact, which was proved independently by Michael [1955] and Nagami [1955]. The techniques of the proof used in that papers are useful for the study of the covering and other properties. In this section we shall give the characterizations of metacompact spaces. A metacompactness is often called a weak paracompactness. Let us begin with the definition of metacompactness. 4.1. Definition. A space X is metacompact if every open cover of X has a point-finite open refinement. By the definition of submetacompact space, it is clear that every metacompact space is submetacompact (the converse is not true; see Section 6). Before giving the characterizations of metacornpact spaces, we shall show the next lemma, which is powerful for the study of covering properties. This lemma, essentially due to Worrell [1966a], is a weak form of a result from Junnila [1979a]. Our proof is due to Burke [1984]. 4.2. Lemma (Junnila [1979a]). Zf{%, I n E N} is a sequence of open covers of a space X such that @,+ I is a pointwise (resp. local) W-rejinement of 42, for any has a u-point-finite (resp. a-locally finite) open refinement. n E N, then Proof. We shall prove the lemma only for pointwise W-refinement. We express as {U,la < T}. If for any U E u{%,ln = 1, 2 , . . .}, we let 6 ( U ) = the 1st of { a < T ~ cU U,} and W, = { W E % , I ~ ( W )= 6 ( U ) if U E and W c U } for each n = 2, 3, . . . . To show that U{W,I n = 2, 3, . . .} is a cover of X , let x be any point of X. Since for each we let a, = sup{6( U )I x E U E a,}. n > 1, %, is a pointwise W-refinement of Then it holds that a2 2 uj 2 * . . and so there is some: k > 1 such that a k = U k + l = U k + 2 = * * * . Since %k+2 is a pointwise W-refinement of % k + l , of @ k + l such that { U E %k+2 I x E U } is a there is a finite subcollection partial refinement of where we may assume that x E U for any with 6(Uo) = a k . For UE From a k + l = a k , we have some U, E any U E %k with U, c U , we have ak = 6(Uo) < 6 ( U ) < ak and so 6(Uo) = 6 ( U ) , that is, U,, is a member of wk+l.Therefore W, I n > l} is a cover of X . If we let V,, = U { W E W,16(W) = a } , for each n > 1 and eacha < T , V ,= {V,,Ia}foreachn > 1 a n d V = U { V , l n > 1 } , t h e n V is an open refinement of In this place it suffices to show that Y, is point-finite for each n > 1. To show that, for any n > 1 and any x E X , we define A = { a < T 1 x E V,,}. For each a E A, we select some W, E W, such that x E W, and 6(W,) = a.
u{
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180
Since Wnis a pointwise W-refinement of %,+, , there is a finite subcollection 9 of 4 - ,such that { W,Ict E A} is a partial refinement of 9. Hence, if W, c U with U E 9, then ct = 6( W,) = 6 ( U ) E 6(9)which is finite, and so
0
A is finite.
The next characterizations of metacompact spaces are considered the modifications of paracompact spaces (see Theorem 2.15) and submetacompact spaces (see Theorem 3.4). 4.3. Theorem (Worrell [1966a], Junnila [1979b]). For a space X , the following
are equivalent: (1)
X is metacornpact.
(2)
Every open cover % of X has a point-finite refinement Y such that x E Int(St(x, Y ) )for any x E X .
(3)
Every open cover of X has an open pointwise W-refinement.
Such a cover “fof (2) is called to be a semi-open cover Proof. (1)=-(2) Obvious. be any open cover of X and I - a point-finite semi-open (2)*(3) Let refinement. For each I/ E I’ we select any member U ( V ) of @ with V c U( V ) .Iffor each x E X , let Yx, = Int(St(x, Y 7 ) n U( V )I x E V E $“}), then t i r , is an open nbd of x. It suffices to show that -w^ = { Qx)IxE X is a pointwise W-refinement of %. Let x be any point of X and I”, . . . . V” 1 be all members of %* which contain Let 1; be any member of -lI whicri contains x, then x E b , , z Sti p, I 1 and so ix, y : c I/‘ for some < n. and SO T{,, z h/(V”i. So { 1, v ) i x e I;,) E W ) is a partial refinemen1 of : LT(V‘) I i ( 3 b g1 1 This Drool IS due (0 Burke [1Q84].Let % be any open co\?r oL. X By ixnima 4.2. we niiw 9 a-point-finite open refinement Y’ = { *, I n : a1 %, where eacb 7 ” IS point-finite. If for each n. we let I., = (11 I/ I i E , L! . . . LJ 1 then there is an open pointwise W-refinement % of { 1;,.Jn, by repeated us: of Lemma 4.2. Since { V , , J d is increasing. E, = ( x E XI Sttx, %”) c 1 : : is a crosed set which is contained in Y :For each nl and {&,Ina I: A cover of A . If for eacri re, we define C, by el, = i l - LJ;€~IK < n\ C E r n i . then [ J ( U J n }is a point-finite open refinement of 9. J
(n{
),
2
821
~
-
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181
In order to discuss the next characterizations of metacompactness, it is necessary to prove the following lemma which is needed for the discussion of the interior- and closure-preservations and pointwise W-refinements, etc., which were used in the proof of the previous Theorem 2.19. 4.4. Lemma (Junnila [ 1979a, 1979b1). For an interior-preserving open cover Q of a space X,the following are equivalent:
(1)
4' has a closure-preserving closed rejinement (resp. furthermore whose interior is a cover of X ) .
(2)
4 has an interior-preserving open pointwise (resp.local) W-rejinement.
(3)
4' has an interior-preserving open pointwise (resp. local) starrefinement.
Proof. The following proof is due to Burke [1984]. (1)=42) Let 9be a closure-preserving closed refinement of 9'.If for each X E X , we define P,' as n { U l x E U E @ } - u { F l x $ F , F ~ 9 } then , { V,I x E X } is an interior-preserving open cover of X by Lemma 1.2. To see that { V, I x} is a pointwise W-refinement of 4, for each x E X we select any Fo E 9 with x E Fo and a finite subcollection of 4 such that Fo c U{U 1 U E 4 0 }Then . { V, I x E K.} is a partial refinement of @., ( 2 ) 4 1) Let Y be an interior-preserving open pointwise W-refinement of 4. If for each G E Q', let P ( G ) = { x I St(x, Y ) c G}, then { P ( G )1 G E 4'} is a closed refinement of a', and so we shall show that { P ( G ) JG E 4'} is closure-preversing. Let { G , 1 a E A } be any subcollection of Q', where we can express G , as G , = UQu for some finite subcollection Qu of Q (for each a). In order to show that U{P(G,)(a E A } is closed, let x be any point of X - U { P ( G , )I a}. Since Y is a pointwise W-refinement of 4, there exists a finite subcollection { U , , . . . , U,,}of 4 such that { V I x E V E Y }is a partial refinement of { U , , . . . , U,,}, where we may assume that x E U , for any i = 1, 2, . . . , n. For each u, there is some V , E Y such that x E V , and V, c t G, and so, there is some i ( a ) < n with V, c U,(,). Therefore 0= Ui(,)I a} is an open nbd of x such that 0 n P(G,) = 8 for any a, that is, U { P ( G , )I a} is closed. We shall leave the other parts of this lemma to the reader. (The proof is similar to the above.) 0
n{
The next characterizations of metacompact spaces are modifications of Theorems 2.11 (3) and 2.12 (3) for paracompact spaces and Theorems 3.5 and 3.6 for submetacompact spaces.
Y. Yasui
I82
4.5. Theorem (Junnila [1979a], Sconyers [1970]). For a space X , the following are equivalent:
(I)
X is metacompact.
(2)
Every directed open cover of X has a closure-preserving closed refinement.
(3)
For any open cover 9 of X , 9 ‘has a closure-preserving closed refinement .
(4)
Every monotone increasing open cover of X has a point-finite open refinement.
Proof. (1)=42) Let 9 be any directed open cover of X and express 42 as 9 = { U, I a E D},where D is directed. Furthermore, let Y be a point-finite If for each U E 4 ,we define W(V )as {x I St(x, Y ) c V } , open refinement of 9. then it is clear that { W ( U )I U E 9}is a closed cover of X such that W ( U ) c U for any U E 9, and so it suffices to show that for any D‘ c D, U{ W(U,)la E D’)is closed. For this prupose, let x be any point of X - U{ W ( U , )I a E D’}. Since *Ir is point-finite and St(x, Y ) U , for any a E D’,there is a finite subcollection { V,, . . . , V , } of Y with x E for each i < n such that for each a E D’ we have some i(a) < n with K(,) Q U,, which means that I&) n W(V,) = 8. Therefore V = Fl i = I , . . . , n} is an open nbd of x such that V n (U{ W(U,)I a E D’})= 8. (2)0(3) Obvious. (2)*(4) Let 9 be any monotone increasing open cover of X . By Lemma 4.4, there is a sequence {9, = 9, a2,423, . . .} of open covers of Xsuch that 9,,+, is an interior-preserving open pointwise W-refinement of 9,for any n. On the other hand, by Lemma 4.2, there is a a-point-finite open refinement V = U{V, I n } of 9,where each Y, is point-finite. If for each n, we let V , = U{UYkI k < n } , then { V ,I n} has a closure-preserving closed refinement IF, I n } by condition ( 2 ) such that F, c V , for any n. Then it is clear n = 1, 2, . . .} is a point-finite open that { V - U{eI i < n - l} I V E Yn, refinement of 9. (4)=4 1) We shall prove this by transfinite induction on the cardinality of a given open cover of X . Since Xis countably metacompact, we assume that for some uncountable cardinal T, every open cover of X whose cardinality is less than T has a point-finite open refinement. Let 4 be an open cover of X with cardinal T and so express it as 42 = { U, 1 a c r}. If V, = U( U, I /I< a } for each a < ‘I, then { V , I a c r} has a point-finite open refinement W = {W,la c z} such that W, t V , for any a c z. For each a, we let
+
n{
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9, = {U,lfl < a } u {U{W,lfl > a}}.Then9uisanopencoverofXwith cardinality < T and so there is a point-finite open refinement F, of Ya.Let Fu'= {TE FulT c U, for some j3 < a } for each a < T and d = { W, n TI T E Fu',a < T}. Then it is easily seen that d is a point-finite open refinement of 43. This completes the proof. 0
The next characterizations of metacompact spaces which were obtained by Junnila [1979a] are modifications of Theorems 2.19 and 3.6. Theorem 4.6 is useful to obtain the following analogue of Remark 2.10 (see Fletcher and Lindgren [1972]):A completely regular space Xis metacompact iff every open cover of X x B ( X ) has an interior-preserving open refinement, where B ( X ) is the Stone-(5ech's compactification of X . 4.6. Theorem (Junnila [1979a1). For a space X , the following are equivalent:
(1)
X is metacompact.
(2)
Every interior-preserving directed open cover of X has a closurepreserving closed refinement.
(3)
Every interior-preserving directed open cover of X has an interiorpreserving open pointwise star-refinement.
Proof. (1)*(2) Theorem 4.5. (2)*(3) Theorem 4.4. (3)*(1) We shall show this by use of Theorem 4.5. Let 4 = { Vula < T} be any monotone increasing open cover of X and an interior-preserving open pointwise star-refinement. Since 4; is interior-preserving and directed, there is an interior-preserving open pointwise star-refinement a2of 42;. By repeated use of Lemma 4.4, we have a sequence {a,,1 n } of open covers of X such that is a pointwise W-refinement of 4, and therefore we have a o-point-finite open refinement of 4 by Lemma 4.2. If for each n, we let W, = U{U"w;(i < n } , then there is an open pointwise star-refinement 9 of { W, I n } . If for each n, we let F, = {x I St(x, 9') c W.}, then it is easily seen that { W - Fn-lI W E W,; n = 1, 2, . . .} is a point-finite open refinement of 43, (where we assume that Fo = 0). 0 As is well known, spaces with the closure-preserving covers by compact sets are not necessarily paracompact (Potoczny [1972]), but it is easily seen that such spaces are metacornpact by the above Theorem 4.6.
4.7. Corollary (Katuta [1974], Potoczny and Junnila [1975]). I f a space X has a closure-preserving cover by compact subsets, then X is metacornpact.
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5. Characterizations of subparacompactness Subparacompactness is one of the important and interesting properties in general topology. Its property was introduced by McAuley who called it as F,-screenability (McAuley [ 19581). He showed that every semi-metric space is F, -screenable and every collectionwise normal F,-screenable space is paracompact. On the other hand, Arhangel’skii [I9661 studied a class of spaces with a a-discrete network (afterwards, every space of this class was named a a-space by Okuyama [1967]). To begin with, we shall give the definition of subparacompactness:
5.1. Definition. A space Xis said to be subparacompact if every open cover of it has a a-discrete closed refinement. This definition will be considered as a generalization of paracompactness by Theorem 2.1 1 (2). Since the following theorem is well known (Burke and Stoltenburg [1969], Coban [I9691 and Junnila [1978]), its proof can be found in many papers (Burke [I9841 and Nagata [1985], etc.) and will therefore be omitted.
5.2. Theorem (Burke, [1969], Junnila [ 19781). The following are equivalent for a space X : (1)
X is subparacompact.
(2)
Every open cover of X has a a-locallyfnite closed refinement.
(3)
Every open cover of X has a a-closure-preserving closed refinement.
(4)
Every open cover of X has a a-cushioned refnement.
From the above Theorem 5.2, we have the following corollary. 5.3. Corollary. r f a space X has a countable closed cover by paracompact sets, then X is subparacompact. We have the following characterizations of subparacompectness if we take a side view of Theorem 2.15. The second condition of Theorem 5.4 was introduced by Arhangel’skii [ 19661 who called it a-paracompact.
5.4. Theorem (Burke [1969], Burke and Stoltenburg [1969], Coban [1969]). The following are equivalent for a space X :
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(1) X is subparacompact. (2) For every open cover 4 of X there is a sequence {%,,In E N} of open covers of Xsuch thatfor any x E X there is some n E N such that 9, is pointwise star-rejinement of 4 at x. (3) Every open cover 4 of X has a sequence {%,, I n E N} of open refinements of 4 such that for each x E X , there is some n E N with ord(x, 4,,)= 1. The above sequence {@,,I n } of the condition (2) is called to be a pointwise starring in 4.
Proof. (1)-(3) and (3)-(2) Obvious. (2)=4 1) Let 4 = { U, I a E A } be any open cover of X and a sequence {%,, 1 n } of open covers of X which is pointwise starring in 4. For each a E A and n, we let U,, = {x I St(x, a,,) c U,}.Then it is seen that F,, = { U,, I a } is cushioned in 4 and U { F nI n} is a closed refinement of 4. 0 From Theorem 5.4, we have the following corollary.
5.5. Corollary (McAuley [ 19581). Every Moore space is subparacompact. It is easily seen that paracompactness is stronger than subparacompactness. For example Niemytzki space (see Engelking [1977] or Nagata [1985]) is subparacompact (by Corollary 5.3 or 5.5) and is not paracompact (by nonnormality of its space). In Section 6, we shall give a Moore space which is not metacompact (Example 6.8). Finally, we shall have a characterization for subparacompactness analogous to Theorems 2.19, 3.6 and 4.6. 5.6. Theorem (Junnila [ 19781). A space X is subparacompact if and only if every interior-preserving open cover of it has a a-closure-preserving closed refinement.
Proof. The “only if” part is obvious and so we shall prove the “if” part. By Theorem 3.6, Xis submetacompact. In order to show that Xis subparacompact, it suffices to show that for any discrete collection {F,I a E A } of closed sets of X , there exists a mutually disjoint collection {G, I a E A } of G,-subsets of X such that F, c G, for any a E A (see Theorem 7.15). If for any a, we let Ha = X - U{FBI fl E A , fl # a}, then J f = {HaI a } is an interior-preserving open cover of X and so Z has a a-closure-preserving closed refinement 9.
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Therefore, we may express 9 as 9 = U ( 9 , I n } such that each 9, is closurepreserving. If for each a E A and n E N, we let Can= X U{FE 9 , l F n Fa = then Can is an open set which contains Fa. Therefore, to complete the proof, it is sufficient to show that n{G.. I n } n (n{G,, I n } ) = 0 for any a, /? E A with a # /?.Let a, /? E A with a # /?, and any x E X. Since 9 is a cover of X,there are some i E N and some F‘ E q.with x E F‘. Here we may assume F’ n Fa = 0 without loss of generality, which means x E Cai 3 (-){Gun1 n } . Therefore (-){Can In } n
a},
(n{GB,In>>=
0.
6. Shrinking properties Normality of a space X is that for every binary open cover { U , , U z }of X , there is a binary closed cover { F , , F2} of X such that F, c U, for each i. Generally speaking, it is well known that a space is normal iff every pointfinite open cover { UaI a E A } of the space has a closed refinement {FaI a } such that Fa c Ua for each a E A iff every point-finite open cover { U, 1 a E A } of the space has an open refinement { V,1 a } such that CI V, c Uafor each a E A. One of the purposes of this section is to generalize the above properties. 6.1. Definition. An open cover { U a l aE A } of a space X is said to be shrinkable if there is an open cover { V ,I a E A } of X such that CI V, c Uafor each a E A . A space Xis shrinking if every open cover of the space is shrinkable.
In the above definition, we can say that a space X is shrinking iff for any open cover { U, I a E A } of X , there is a closed cover {FaI a E A } of Xsuch that F, c U, for any a E A. Every paracompact space is shrinking and by Corollary 2.7, Xis countably paracompact iff every countable increasing open cover of X is shrinkable. Furthermore, Chiba [ 19841 proved that every perfectly normal space or subparacompact and normal space is shrinking. Afterwards, Chiba’s result was generalized as follows. 6.2. Theorem (Yasui [1984]). A submetacompact space is shrinking if and only if it is normal. Proof. We shall prove “if” part: Let 42 = { UaI a E A } be any open cover of a normal and submetacompact space X and { VnI n E N} a &refinement of 42, where we may assume that Vn= {<.la E A } with c Ua for any
<,
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E A and any n E N. If for each n, we let Y, = { y I ord( y , V,)is finite}, then by a lemma of Yasui [1984], there exist open subsets Hnni( i E N) of X such that C1 Hnni c V,, for any i and any a, and Y, c u{HnaiIa,i } . Therefore, {Ifnni I a, n, i } is a cover of X. Therefore, every open cover of X is shrinkable 0 by Beilagik [ 19861.
a
Fleischman [ 19701 proved that every linearly ordered space is shrinking, and therefore, because every subspace of a linearly ordered space is a closed subspace for some linearly ordered space (see Nagata [1985]), we have the following theorem. 6.3. Theorem (Fleischman [ 19701). Every subspace of a linearly ordered space is shrinking.
6.4. Remark. There are several results for the shrinking properties of C-products. Let us recall the C-product which was introduced by Corson [1959]. Let {X,lA E A} be a collection of spaces, and X = n X , the product space of { X i I A} and f = (fi), E X. If we let C = { X I {A Jx, # A } is at most countable}, then the subspace C of X is called to be the C-product of spaces X , (A E A) with the base point f E X . Rudin [1983a] and Donne [I9851 showed the generalization of Rudin’s Theorem, that is: A C-product of paracompact M-spaces is shrinking iff it is normal. And quite recently Yajima [I9861 proved that a C-product of strong C-spaces is shrinking iff it is normal, where a space X is a strong C-space (Nagami [1969]) if there exist a cover X of X by compact sets and a o-locally finite collection 9of closed subsets of Xsuch that for any K E X and any open set U with K c U , there is some F E 9 with K c F c U . As mentioned above, a space is countably paracompact iff every countable increasing open cover of it is shrinkable, and so we shall consider the following definition. 6.5. Definition. A space X has a property 9 (resp. a property 93) if for every monotone increasing open cover { U, I a < T} of X , there is an open cover { V,la < T} of X such that C1 V , c U, for any a (resp. furthermore { V ,I a < T} is monotone increasing). A property was introduced by Zenor [ 19701who showed that a separable regular space is Lindelof iff it has a property 93 and every countably compact space with a property 33 is compact. On the other hand, a property 9 was introduced by Yasui [I9721 who called it a weak %property. In Remark 2.10, we noted that a normal space Xis countably paracompact iff X x l i s normal.
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Y. Yasui
Afterward, the issue of the existence of normal but not countably paracompact spaces arose and so any normal but not countably paracompact space soon become known as a Dowker space. Dowker space was constructed without using any special set-theoretic assumptions by Rudin [ 19711. Afterward, a generalization of a Dowker space was given as K-Dowker space, that is, a space which is normal and there exists a monotone increasing open cover of it which is not shrinkable (see Rudin [I9781 and Atsuji [1976]). From the above circumstances we shall give a new name as a property 9. Therefore a normal space X has a property 9 iff X is not a K-Dowker space for any K. We have the following trivial implications. 6.6. Proposition. Paracompactness able paracompactness.
=-property 93 =-property 9 * count-
Of the above implications, none of the converses hold (see Rudin [ l983b, 1984, 19851 and Yasui [1972]). With respect to a property W,we have the following characterizations which are modifications of those of paracompactness (see Theorem 2.12). 6.7 Theorem (Tani and Yasui [1972]). For a space X , the following are equivalent
(I)
X has a property W.
(2)
Every monotone increasing open cover of X has a cushioned open refinement.
(3)
Every monotone increasing open cover of X has a a-cushioned open refinement.
Proof. (1)=42) and (2)*(3) Almost obvious and so we shall omit. (3)*( 1) Let 4 = { 17I, a < T} be any monotone increasing open cover of X and V = Vn1 n E N} a a-cushioned refinement of Q, where each “v;, is cushioned in Q and we may express it as V, = {V,,la < T } and Cl(U{ V,. I a E A } ) c U{U, I a E A } for any subset A of z and any n. Since X is countably paracompact by the condition (3), there is a locally finite open cover { W, I n E N} of X such that W, c V,, I a } for any n. If for each a, we let 0, = U{ W, n (U{ KsI fl < a } ) In}, then {O,l a } is a monotone increasing open cover of X such that CI 0, c U, for any a. 0
u{
u{
As a characterization of a property W with use of terms of any open cover (not necessarily monotone increasing), we recently have the following.
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6.8. Theorem (Yasui [1986]). For a space X , the following are equivalent: (1) X has a property W . (2) For any monotone increasing open cover { U, I a < t} of X , there exists an open cover { V ,I a < t} of X such that (a) V , c U, for each a < t and (b) for each x E X , there exist a nbd 0 of x and some a. < T such that 0 n ( U { V , l a 2 a,}) = 0. (3) Every infinite open cover 4 of X has an open refinement Y such that each x E X h a s a nbd 0 of x such that the cardinality of { V E Y l O n V # $4) is less than 1 % I. Proof. (1)0(2) See Yasui [1986]. (3)*( 2) Obvious. (1)*(3) Let 4 be any open cover of X and Q express as { U, 1 a < t},where T is a minimal ordinal number whose cardinality is equal to I4 1. By Yasui [1986], there is an open cover Y = {GSl/3 < a; a < T} of X such that (i) KS c U, for any a, B with /3 < a, and (ii) each x of X has a nbd 0 and some a, such that 0 n (U{V,sl/3 < a ; a, < a } ) = 0.Then {V,,lB < a, a < t} is a desired cover. 0
With respect to Remark 2.10, it is well known that a regular space X is paracompact iff X x Y is normal for any compact space Y iff X x I' is normal for any cardinal t iff X x /3(X)is normal, where, B ( X ) is the StoneCech's compactification of X (Morita [1962], Tamano [1960, 19621, and Atsuji [ 19891). For the case of a property 9,there is no compact space Y such that X has a property 9 iff X x Y is normal (Rudin [1983b, 19851). But we have the following characterization: 6.9. Theorem (Yasui [1983]). For a normal space X , X has a property W if and only if X x I, is normalfor any cardinal t,where I, is the set of all ordinals < t with the topology { U c I, 1 t $ U or I, - U has a cardinality less than T}.
Proof. Suppose that X x I, is normal and {U,la -= t} is a monotone increasing open cover of X . Define H = U { ( X - U,) x { a } I a < t} and K = X x {t}. Since H and K are closed and disjoint in X x I,, there are disjoint open sets W and V of X x I, with H c W and K c V. For each a < t, define V, = { x i 0 x { B l a < /3 < t} c Vfor some nbd 0 of x}. Clearly V , is open and C1 V, c U,. Furthermore, { V,I a < t} is a monotone increasing cover of X . Conversely, suppose that Xis a normal space with a property a,and H and K are disjoint closed subsets of X x I,. By Starbird [1974], we may assume
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that H = X x { t } . For each a < 7 , let U, = U { U l U is open in X and (U x {/I1 a < /I < 7 } ) n K = 0}. Since { U, I a } is a monotone increasing open cover of X , there is a monotone increasing open cover { V , ( a < T} of XwithCIV=c U,.IfweletW= U{W,x{/IIa < / I } l a < ~ } , t h e n W i s an open set containing H . It is easy to check that K n C1 W = 8; thus X x Z, is normal. 0 When we concern with Theorems 2.3,2.18, 3.5 and 4.5, etc., which are the characterizations of covering properties in terms of monotone increasing covers, we have the following question: “If every monotone increasing open cover of X is shrinkable, then is every open cover of X shrinkable?”. The answer is “No” by BeilagiE and Rudin [1985] (see Theorem 7.5). The next is a sufficient condition in order that a space is shrinking, and such a space is Bing’s example C etc., for example. We shall omit the proof. 6.10. Proposition (Yasui [1985]). Let {A, B } be a disjoint cover of a normal space X such that each x of A is isolated in X and B is a discrete subspace of X . Then X is shrinking.
Next to the shrinking, we shall consider the para-Lindelof spaces, where a space is para-Lindelof if every open cover of it has a locally countable open refinement. The purpose of dealing with the para-Lindelof spaces is to introduce a Navy’s space (Navy [1981]). Navy had constructed a space which is a para-Lindelof, locally compact and normal space which is not collectionwise normal (and so not paracompact). We shall give a characterization of para-Lindelof spaces which is a modification of Theorem 4.3. 6.11. Theorem (Burke [1980]). A space X is para-Lindelif ifand only ifevery open cover of Xhas a locally countable rejinement Y such that x E Int(St(x, Y ) ) for any x E X .
Proof. It is sufficient to prove “if” part. Let 9 be any open cover of X and Y a locally countable refinement of 9 such that x E Int(St(x, Y ) )for any x . Furthermore, let W be an open cover each member of which meets at most countably many elements of Y and d a locally countable refinement of W such that x E Int(St(x, d)) for any x. If for each V E Y ,let us select some U ( V ) of Q with V c U ( V ) , and let G ( V ) = Int(St(V, d ) )n V ( V ) ,then Y = {C(V )I V E Y }is a locally countable open refinement of Q. 0
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7. Examples Throughout the above sections we discussed, for the most part, the characterizations of the covering properties. Implications among their properties are shown in the following diagram: subparacompact
I
dco~:t 1I para-Lindelof
+ countably paracompact 1 property 9
submetacompact
\
+ normal \
shrinking
\property 9 1
/
countably paracompact All the implications will be induced by the definitionsor their characterizations with some exceptions. In this section, we shall give the non-trivial implications, the examples in which the implications are not reversible and the sufficient conditions under which two different classes are coincident. At first we shall mainly give the subspaces of Bing's example G (Bing [1951]). For completeness, let us review Bing's example G: Let P be uncountable, d a collection of all subsets of P,D = (0, l } and F the product spaces of 19 I-copies of D , that is, F = {flf: 22 + D}. For each x E P, leth E F a s follows:f,(Q) = 1 ifp E Q andfp(Q) = 0 otherwise. and p E P, let We let a subset Fo as { f , I p -P}. ~ For each 43 U ( p , 43) = {YE F(f(R) = f , ( R ) for any R E W}.Let us describe a nbd basis 9(f) o f f € F as follows:
7.1. Example (Burke [1974]). not submetacompact.
If 1 P 1 > 2", then F (= Bing's example G ) is
Proof. Let% = { U ( p , { p } ) l p P~} u { F - F0}and{9,,!,n}anysequence of open refinements of 9, where we may assume that for each n, %, = { U ( P ,W ( , n ) )I P E P} u {{f}If€F - Fo} for some W ( p . n ) E [$I* For each p E P, let A,i = { R E W ( p , n ) I n E N, P E R } , Ap2 = { R E W ( p , n ) I n E N, P 4 R } and A, = API u A,, (hence A, = U{W,,,,, In}), Then there exists a subset P' of P such that API n A,, = 8 for any p, q E P' ( p # q ) and 1 P' 1 > 2" by
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Theorem I. 1 of Burke [ 19741. Now we define & E F as follows: S,(R) = 1 if and only if R E U { A , , J pE P’}.Then& E U(p, W(p,nl)for anyp E P’,that is, ord(fo, 9,,)2 I P’I > a. 0 If we let K = Fo u { f F~l f - ’ ( l ) is at most countable} be a subspace of F ( = Bing’s example G), then K is not metacompact by the similar proof of the above Example 7.1 (Burke [1974]) and it was shown by Hodel that K is meta-Lindelof, that is, every open cover of K has a point-countable open refinement. Later we shall show that every subspace of F ( = Bing’s example G ) is shrinking and so F is shrinking and not metacompact. The second example shows that subparacompact spaces and metacompact spaces are not coincident.
7.2. Example (Burke [1974]). Let F be a Bing’s example G with I PI > 2” and Y = { f Flf(R) ~ = 1 for at most two many elements of one-point sets R} a subspace of F. Then Y is normal and metacompact, but not subparacompact.
Proof. Every open cover 9 of Y has a refinement Y which is expressed as { W p , W,,)) n YIP E p } u {{f}I f € y - Fo} with {PI E g(,). Then ord(f(Y)) < 3 for anyfE Y, which shows the metacompactness of Y. To see that Y is not subparacompact, it suffices to show that 9 = { V ( p , {p}) ( pE Y} u { Y - Fo} does not have a &refinement. Let {a,,In} be any sequence of open refinements of 9. Without loss of generality, we may assume that 9,,= {U(p, W,,,,,) n Y l p E P} u {{f}I f € Y U(WP, g ( p , n ) ) Ip E P}},where { P} E W ( p , n ) for any P E P. For each P E P, wedefine A,, = { R E W ~ , , , , ) I ~ E N , ~ E R {} R, A EW ~ ~( ,=, , ) I ~ E N , ~ # R } and A, = A,, u A p 2 .Then there is a subset P‘ of P such that A,, n Aq2 = 0 for anyp, q E P‘ ( p # q) and 1 P‘ I > 2” by Theorem 1.1 of Burke [I 9741. We select two distinct pointsp and q of P’and define& E Y as follows:&(R) = 1 iff R E A,, u A q , . Then it is seen that ord(&, a,,)= 2 for any n, that is, Y is not subparacompact by Theorem 5.4. 0 The existence of a subparacompact space which is not metacompact is well known (for example Niemytzki’s plane (see Nagata [ 19851)). The third example is normal and submetacompact but not metacompact or subparacompact.
7.3. Example (Burke [1974]). Let 1 PI > 2”, {p,,ln E N} be a sequence of distinct points in P,9o= {{p}Ip E P},9, = { { q ,p , , . . . ,pn}I q E P} and
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193
X, = { f Flf-'(l) ~ n 9,, is at most finite} u {fp,, . . . ,fp,}. Then a subspace X = u { X ,I n E N} of F (= Bing's example G ) is normal and submetacompact but not metacompact or subparacompact.
Proof. Any open cover of X, has an open refinement $'- such that V = {U(P, W,,,)) n XIP E PI u {{fllf~ X - U { W , W(,JIP E PI}, where we may assume that {p}, {p, p , , . . . ,p,} E W,p, for any p E P. Since V is point-finite, each X , is closed and metacompact and so X = u{XnIn}is submetacompact. On the other hand, the space Y of Example 7.2 is a closed subspace of X, and so X is not subparacompact. We shall show that an open cover % = { U ( p , { p } ) n X ( p E P} u {X - F,} does not have a point-finite open refinement. Let W be any open refinement of@, where we may assume that W = { U ( p , W,,,) n X l p E P} u {{f}I f € X - F,}. Since, for each p E P, there is some n, E N such that W(,, n i 2 n,}) = 0,there are some subset P' of P and some n E N such that 1 P ' J > 2" and W,,, n i 2 n}) = 0 for any p E P'. If for each p E P', we let A,, = { R E W(,,Ip E R } and AP2 = W,,- A,,, then there is a subset P" of P' such that I P" 1 > 2" and A,, n Aq2 = 0 for any p, q E P" with p # q by Theorem 1.1 of Burke [1974]. DefinefE F as follows: f ( R ) = 1 iff R E IJ{ApI Ip E P"}.ThenfE X n V ( p , W,,,) for any p E P", which shows that W is not point-finite at f. 0
(u{QiI
(u{$]
7.4. Example. Every subspace of the space F (=Bing's example G ) is shrinking. If 1 PI is uncountable, then F does not have a property 93.
Proof. Since every subspace of F is shrinking by Proposition 6.10, we shall show that F does not have a property 93. Without loss of generality, we may assume that P = o,and so F,, = {f,I a < o,}. If for each a < wI, we let Ha = { f a l a < < ol},then { H a l a < ol} is a monotone decreasing closed collection with the empty intersection. Let { U, I a < o,} be any monotone decreasing open collection with Ha c U, for any a < 0,. If for we select some W,,, E [9] such that V(a,R,,,) c U,, then by the each a < a,, Sanin's lemma (see Juhasz [1970]), there exist some n, some subset P' of P and some W' = { R , , . . . , R,} E [9] such that I W,,,I = n for any a E P', I P' I = w , and {W,,, - 9' I a E I"} is mutually disjoint. Furthermore, we have a subset P" of P' with cardinality o,such that P" c Ri or P" n Ri = 0 for any i < m. Then we can define g c F as follows: g ( R ) = 1 if R = Ri 3 P" for some i, g ( R ) = f , ( R ) if R E W,,, - W' for some a E P" and g ( R ) = 0 otherwise. Then it is seen that g E U(a, W,,,) c U, for any a E P".By the monotone decrease of { U, I a < a,}, g E n { V, I a < o,}.
0
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Y. Yasui
The above example shows that a property W is stronger than a property 9. Another example of such a space is o,(Yasui [1972]). In the shrinking properties, the natural questions arise: If every monotone increasing open cover of X is shrinkable, then is every open cover of X shrinkable? It is answered negatively under some set-axiom as follows. Let ic be an infinite cardinal and E = { a E ~'(cofinalityof a = ic}. O + + ( E ) holds in L. We omit the proof.
7.5. Example ( O + + ( E )(Beilagii: ) and Rudin [1985]). There is a space A such that A is T2,ic-ultraparacompact and collectionwise normal and every monotone increasing open cover of A has a clopen shrinking, but there is an open cover having no closed shrinking. Rudin [1983b and 19841 showed that a Navy's space S does not have a property a.It was already known by Navy [1981]that S is para-Lindelof and countably paracompact but not paracompact (Theorem 6.8):
7.6. Example (Navy [1981], Rudin [1983b and 19851). A Navy space S has a property W but is not paracompact. To see that a property 9 is stronger than the countable paracompactness, we choose an increasing sequence of regular cardinals {A, Ia < ic] for which A: = A,, where ic is an uncountable cardinal. Then Xis the subset of the box product 0 {(A, + 1) I a < rc} consisting of precisely those functions f for which there is a B < K such that K
7.7. Example (Rudin [1978]). Let ic be an uncountable cardinal. Then a ic-Dowker space X is normal and countably paracompact and it has a monotone increasing open cover indexed by ic which is not shrinkable (and so it does not have a property 9). In Example 7.2, we described the space which is metacompact but not subparacompact. It is seen that the space is not locally compact, whose property is useful for the study of covering properties and so we shall give the existence of a space which is locally compact and metacompact but not subparacompact. The proof is left to the reader.
+
+
7.8. Example (Burke [1964]). Let X = (a2 1) x (a2 1) - {(a2, 02>}. LetL = { a } x (w2 + 1)andK = (02 + 1) x {a}.Topologized Ysuch that
Generalized Paracompactness
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the points in w2 x w2are isolated, the points (a, w 2 )have a nbd basis by sets of the form La - A (where A is finite) and the points (02, a ) have a similar nbd basis, with Ka replacing La.Then Xis locally compact and metacompact but not subparacompact. The above space X has some subspace which is interesting with respect to covering properties as follows:
7.9. Example (Burke [1964]). Let Y = (w2 + 1) x (w u { w 2 } )- {(w,, w 2 ) } be a subspace of Example 7.8's space X.Then Y is locally compact, metacompact and subparacompact but not paracompact.
Proof. Since Y is a closed set of X (=space of Example 7.8), Y is locally compact and metacompact. Also Y can be expressed as a union of countable members of closed paracompact subspaces and so Y is subparacompact by Corollary 5.3. It is easily seen that Y is not normal and so not paracompact. 0 Every Moore space is subparacompact by Theorem 3.4, but the next example shows that we cannot have the fact with subparacompactness replacing metacompactness. The proof is easy. 7.10. Example (Gillman and Jerison [1960]). Let d be a maximal collection of countably infinite subsets of N such that d is almost disjoint and Y(N) = d u N topologized as follows: the points of N are isolated in Y(N) and the points A of d have a nbd basis by sets of the form {A} u ( A - F) (where F is finite). Then the Moore space Y(N) is not metacompact. The above examples show that classes of paracompact spaces, subparacompact spaces, metacompact spaces and submetacompact spaces do not coincide with each other and so we shall discuss the conditions under which the above classes coincide. For example collectionwise normality is well known as one of such conditions and so we shall have the following without proof: 7.11. Theorem (Michael [ 19551, Nagami [1955], McAuley [1958], Worrell and Wicke [1965]). For a collectionwise normal space X, the following are equivalent : (1) X is paracompact. ( 2 ) X is subparacompact. (3) X is metacornpact. (4) X is submetacornpact.
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Next we shall consider the question: under what conditions is subparacompactness equivalent to metacompactness? For this purpose we shall define collectionwise &normality and prove some lemmas.
7.12. Definition. A space Xis said to be collectionwise 6-normal if for any discrete collection { F, 1 a E A} of closed subsets of X , there exists a mutually disjoint G,-sets’ collection {G,la E A } such that F, c G, for any a E A (Kramer [ 19711, Junnila [ 19801).
7.13. Lemma (Junnila [ 19811). Let X be collectionwise &normal, { F, 1 a E A } a discrete closed collection and { U,I a E A } an open collection with F, c U,for any a E A. Then there exist a a-discrete closed collection 9 and a Gs-set H such that U{F,I a E A } c H c U S and 9 is a partial refinement of { un I a E A } . Proof. Let {G, I a E A } be a mutually disjoint collection of G,-sets such that Fa c G, = n{G,,,In E N} c U,,where each G,,, is open and U,,3 G,, 3 G,, 3 . . . with G,, n F, = 8 for any fl # a (for any a E A). If we let G,, = U{G,, I a } for each n, then there are G,-sets H, = n{Hnk I k E N} and J,, = n{J,,, I k E N} (where all H,,, and Jnkare open) such that U{F,I a } c H,, and X - C,,c J,,. Let %,,k = {G,,Ia} u {Jnk}, c,,,= {xlord(x, grin) = I } and 9 , k = {G,,, n G,,, I a E A} for each n, k E N. Finally we let 9 = 1 n, k} and H = n{H,,1 n E N}, then 9 and H are desired.
u{&
7.14. Lemma (Junnila [1980]). Let Q be an open cover of a collectionwise &normal space X and X(42) = { x E XI ord(x, 42) isfinite}. Then there exists a a-discrete closed collection 9 such that X(Q) c UP and 9 is a partial refinement of Q. Proof. Let X , = {xlord(x, Q) < n} for each n. Then X(Q) = U{X,,In}. By repeated use of Lemma 7.13, we have a-discrete closed collections 9, and is a partial refinement of Q, for G,-sets H,, such that X , c H,, c U 9 , and 9,, each n. Then U{9,,ln}is desired. 0 Under the above lemma, the following theorem is easily seen: 7.15. Theorem (Boone [1973]). A space X is subparacompact if and only if X is collectionwise &normal and submetacompact.
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Next we shall consider a necessary and sufficient condition under which submetacompactness is equivalent to metacompactness. Smith and Krajewski [I9711 gave such a condition for the first time and a few years later, Boone [1973] gave a condition weaker than that used by Smith-Krajewski. To see this we shall prove the following fact which will be useful in a Boone’s theorem: 7.16. Proposition (Gittings [ 19741). Every submetacornpact space is countably metacompact.
Proof. Let { U, I n E N} be any countable open cover of a submetacompact space X and {@n,ln}a &refinement of it. Let V, = U,and V , = U, n St(X - ( U , u . . . u Un-l), A { @ i l i < n - I } ) for any n 2 2. Then { V ,I n E N} is a point-finite open refinement. 7.17. Theorem (Boone [1973]). A space X i s metacornpact ifand only i f X is submetacompact and every discrete collection of closed subsets of X has a point-finite open expansion.
Proof. The “only-if” part is easily seen and so we shall only prove the “if” part. Let {%,In} be a &sequence of an open cover 9 = { U, I a E A}. If for n, k E N, let x , k = { X I ord(x, a,,) < k } , then 4, is closed and X = U{XnkI n, k } . By repeated use of the given condition, there exist point-finite open collections “v;,, for n, k such that “v;,k is a partial refinement of 9 and a cover of Xnk - x n k - l . By Proposition 7.16, we have a point-finite open cover { W,k I n, k } of X such that W,, c u v n k for any n, k. Then it is seen that { W,k n VI V E V , k ; n , k E N} is a point-finite open refinement of %. The above Theorems 7.15 and 7. I7 characterize subparacompactness and metacompactness in terms of an expansion. In the case of paracompactness, the following theorem is known (the proof is left to the reader). 7.18. Theorem (Krajewski [1971]). A space X is paracompact if and only if X is submetacornpact and every locally finite collection of closed subsets of X has a locally finite open expansion.
Studying whether or not a topological property preserves under some class of mappings is one of the important and interesting problems. It is obvious that all classes of spaces appearing in this chapter do not preserve under the continuous mappings and so we shall consider the class of closed mappings.
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The next theorem follows easily from the definitions or the characterizations (Theorems 2.12 (2), 3.6 (3), 4.5 (2) and 5.2 (3)) because the closure-preserving property preserves under any closed mapping. 7.19. Theorem (Michael [1957], Worrell [1966b], Burke [1964], Junnila [1978]). Let f be a closed mapping from a space X onto a space Y . Zf X is paracompact (resp.metacompact, subparacompact, submetacompact, shrinking), then Y is so, respectively. Though paracompactness preserves under closed mappings, it is known that t-paracompactness does not (Zenor [1969]). But the domain space which was constructed by Zenor does not satisfy the complete regularity. Recently, Ohta constructed such an example in the class of Tychonoff spaces as follows. 7.20. Example (Ohta [1983]). There exists a closed mapping from a
Tychonoff space X with a property 9 onto a Tychonoff space Y which is not countably paracompact (and so does not have a property 9).
Proof. Let S be the subspace of all P-points of w 2 , 2 = {(a, 8) I /3 E S , /3 < a < w 2 } a subspace of the product space (a2+ 1) x S and X = Z x o.Then 2 has a property 9 (and so countably paracompact). Define Y by a quotient space obtained from X by collapsing the set {((02 /3),,n) I n < o}to a point j? for each fi E S. Since each is a P-point, it is easily seen that the quotient mapping f:X + Y is closed. Let us set F,, = f ({((/!I, /3), m ) I /3 E S , m < n } )for each n. Then {F, I n} is a sequence of closed subsets of Y with empty intersection. If G,,is an open set of Y containing F,,for each n, then C1 C,intersects {j? I /3 E S}at a cofinal subset, and so n{Cl G,,In} # 8.By Corollary 2.8, Y is not countably paracompact (and so does not have a property 9). 0
B
Though the countable paracompactness and the property 9 do not preserve under closed mappings, it is clear that in a class of normal spaces, their properties preserve under them.
References Alas, 0. T. [1971] On a characterization of collectionwise normality, Canad. Math. Bull. 14, 13-15. Alexandroff, P. S. and P. Urysohn [1929] Memoire sur les espaces topologiquescompacts. Verh. Akud. Wetensch. Amsterdam 14, 1-96.
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Arens, R. and J. Dugundji [1950] Remark on the concept of compactness, Portugal Math. 9, 141-143. Arhangel’skii, A. V. [I9611 New criteria for paracompactness and metrizability of an arbitrary T , space, Dokl. Akad. Nauk SSSR 141, 13-15. [I9661 Mappings and spaces, Russian Math. Surveys 21, 115-162. Atsuji, M. (19761 On normality of the product of two spaces, General Topology IV, Proc. Fourth Prague Topology Confer., Prague, B25-27. [I9891 Normality of product spaces I, Chapter I11 of this volume. Bennett, H. R. and D. J. Lutzer [I9721 A note on weak 0-refinability, Gen. Topology Appl. 2, 49-54. BeSlagiC, A. (19861 Normality in products, Topology Appl. 22, 71-82. BeSlagiC, A. and M. E. Rudin (19851 Set theoretic constructions of non-shrinking open covers, Topology Appl. 20, 167-177. Bing, R. H. [I9511 Metrization of topological spaces, Canad. J . Math. 3, 175-186. Boone, J. R. [I9731 A characterization of metacompactness in the class of O-refinablespaces, Gen. Topology Appl. 3, 253-264. Burke, D. K. [I9691 On subparacompact spaces, Proc. Amer. Math. SOC.23, 655-663. [1974] A note on R. H. Bing’s example G, Topology Confer. VPZ, Lecture Notes Math. 375 (Springer, New York) 47-52. [ 19801 Para-Lindelof spaces and closed mappings, Topology Proc. 5, 47-57. [I9841 Covering properties, in: K.Kunen and J. E. Vaughan, Eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam). Burke, D. K. and R. A. Stoltenberg [I9691 A note on p-spaces and Moore spaces, Pacific J. Math. 30,601-608.
De Caux, P. [I9761 A collectionwise normal weakly O-refinable Dowker space which is neither irreducible nor realcompact, Topology Proc. 1, 67-77. Ceder, J. [I9611 Some generalizations of metric spaces, Pacific J. Math. 11, 105-125. Chiba, K. [I9841 On the weak E-property, Math. Japon. 29, 551-567. Coban, M. M. [I9691 On u-paracompact spaces, Vestnik Moskov Univ. Ser. I, Math. Mech., 20-27. Corson, H. H. [I9591 Normality in subsets of product spaces, Amer. J. Math. 81, 785-796. Dieudonn6, J. [I9441 Une generalisation des espaces compacts, J. Math. Pures Appl. 23, 65-76.
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Donne, A. L. [1985] Shrinking property in Z-products of paracompact p-spaces, Topology Appl. 19.95-101. van Douwen, E. K. [I9801 Covering and separation properties of box products, in: G. M. Reed, Ed., Surveys in General Topofogy (Academic Press, New York) 55-129. Dowker, C.H. [1951] On countably paracompact spaces, Canad. J. Math. 3, 219-224. Engelking, R. [1977] General Topology (Polish Scientific Publishers, Warszawa). Fleischman, W. M. (19701 On coverings of linearly ordered spaces, Washington State University Topology Confer., March 1970, 52-55. Fletcher, P and W. F. Lindgren [1972] Transitive quasi-uniformities, J. Marh. Anal. Appl. 39, 397405. Gillman, J. and M. Jerison [1960] Rings of Continuous Functions (Van Nostrand, Princeton, NJ). Gittings, R. F. [1974] Some results on weak covering conditions, Canad. J. Marh. 26, 1152-1 156. Hodel, R. E. A note on separability and meta-Lindel6f spaces, unpublished. Ishikawa, F. [1955] On countably paracompact spaces, Proc. Japan Acad. 31, 686-687. JuhAsz, I. [I9701 Cardinal Funcrions in Topology (Math. Centrum, Amsterdam). Junnila, H. J. K. [ 19781 On submetacompactness, Topofogy Proc. 3, 375-405. [ 1979al Metacompactness, paracompactness and interior-preserving open covers, Trans. Amer. Math. SOC.249, 373-385. [ 1979bl Paracompactness, metacompactness and semi-open covers, Proc. Amer. Marh. SOC.73, 244-248. [1980] Three covering properties, in: G. M. Reed, Ed., Surveys in General Topology (Academic Press, New York) 195-245. KatEtov, M. [1958] Extension of locally finite coverings (in Russian), Colloq. Marh. 6, 145-151. Katuta, Y. [I9741 On spaces which admit closure preserving covers by compact sets, Proc. Japan Acad. 50, 826-828. [1977] Characterizations of paracompactness by increasing covers and normality of product spaces, Tsukuba J. Marh. 1, 27-43. Krajewski, L. [1971] Expanding locally finite collections, Canad. J . Marh. 23, 58-68. Kramer, T.R. [1973] A note on countably subparacompact spaces, Pacific J. Math. 46, 209-213. Mack, J. [1967] Directed covers and paracompact spaces, Canad. J. Math. 19,649-654.
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McAuley, L. F. [19581 A note on complete collectionwise normality and paracompactness, Proc. Amer. Math. SOC. 9, 796-799. Michael, E. [I9531 A note on paracompact spaces, Proc. Amer. Marh. Soc. 4, 831-838. [I9551 Point-finite and locally finite coverings, Canad. J. Math. 7, 275-279. [1957] Another note on paracompact spaces, Proc. Amer. Math. SOC.8, 822-828. [I9591 Yet another note on paracompact spaces, Proc. Amer. Math. SOC.10, 309-314. Morita, K. [19621 Paracompactness and product spaces, Fund. Marh. 50, 222-236. Nagami, K. [19551 Paracompactness and screenability, Nagoya Math. J. 8, 83-88. [1969] Z-spaces, Fund. Math. 65, 169-192. Nagata, J. [I9501 On a necessary and sufficient condition of metrization, J. Insr. Polytech., Osaka City Univ. 8, 93-100. [I9851 Modern General Topology (North-Holland, Amsterdam). Navy, K. [19811 Paralindelofness and paracompactness, Thesis, University of Wisconsin, Madison. Ohta, H. [I9831 On a part of problem 3 of Y. Yasui, Questions Answers Gen. Topology 1, 142-143. [ 1987al Well-behaved subclasses of M , -spaces, in preparation. [1987b] Letter. Okuyama, A. [1967] Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku, See. A 9, 236-254. Potoczny, H. B. [1972] A non-paracompact space which admits a closure-preserving cover of compact sets, Proc. Amer. Math. SOC.32, 309-31 1. Potoczny, H. B. and H. J. K. Junnila [I9751 Closure-preserving families and metacompactness, Proc. Amer. Math. Soc. 53,523-529. Rudin, M. E. [1971] A normal space X for which X x I is not normal, Fund. Math. 73, 179-186. [I9781 K-Dowker spaces, Czech. Math. J. 28(103) 324-326. [1983a] The shrinking property, Canad. Math. Bull. 26, 385-388. [1983b] Yasui’s questions, Questions Answers Gen. Topology 1, 122-127. [I9841 Dowker spaces, in: K. Kunen and J. E. Vaughan, Eds., Handbook of Set-theoretic Topology (North-Holland, Amsterdam). (19851 rc-Dowker spaces, in London Math. SOC.Lecture Note Series 93 (Cambridge University Press, Cambridge) 175-195. Sconyers, W. B. I19701 Metacompact spaces and well-ordered open coverings, Notice Amer. Marh. SOC.18,230. Smirnov, Yu. M. [1951] On metrization of topological spaces, Amer. Math. SOC.Transl. Ser. l(8) 63-77. Smith, J. C. and L. Krajewski [I9711 Expandability and collectionwise normality, Trans. Amer. Math. SOC.160, 437-451.
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Starbird, M. [I9741 The normality of products with a compact or metric factor, Ph.D. Thesis, University of Wisconsin. Stone, A. H. [I9481 Paracompactness and product spaces, Bull. Amer. Math. SOC.54,977-982. Tamano, H. (19601 On paracompactness, Pacific J. Math. 10, 1043-1047. [I9621 On compactification, J. Math. Kyoto Univ. 1, 161-193. Tani, T. and Y. Yasui [I9721 On the topological spaces with the 93-property, Proc. Japan Acad. 48, 81-85. Worrell, J. M. W., Jr. [1966a] A characterization of metacompact spaces, Portugal Math. 25, 171-174. [1966b] The closed continuous images of metacompact topological spaces, Portugal Math. 25, 175-179. [I9671 Some properties of full normalcy and their relations to Cech completeness, Notices Amer. Math. SOC.14, 555. [I9681 Paracompactness as a relaxation of full normalcy, Notices Amer. Math. SOC.15, 661. Worrell, J. M. W., Jr. and H. H. Wicke [I9651 Characterizations of developable spaces, Cunad. J. Math. 17, 820-830. Yajima, Y. [I9861 The shrinking property of 8-products, Questions Answers Gen. Topology 4, 85-96. Yasui, Y. [I9721 On the gaps between the refinements of the increasing open coverings, Proc. Japan Acad. 48, 8 6 9 0 . [I9831 On the characterization of the B-property by the normality of product spaces, Topology A&. 15, 323-326. [I9841 A note on shrinkable open coverings, Questions Answers Gen. Topology 2, 143-146. [I9851 Some remarks on the shrinkable open covers, Math. Japon. 30, 127-131. [ 19861 Some characterizations of a 93-property, Tsukuba J. Math. 10, 243-247. &nor, P. [ 19691 On countable paracompactness and normality, Prace Math. 13, 23-32. 42, 258-262. [I9701 A class of countably paracompact spaces, Proc. Amer. Math. SOC.
K. Morita, J. Nagata, Eds., Topics in General Topology 0Elsevier Science Publishers B.V. (1989)
CHAPTER 6
THE TYCHONOFF FUNCTOR AND RELATED TOPICS
Tadashi ISHII Chiba Institute of Technology, Narashino, 275 Japan
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Tychonoff functor. . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Product spaces and the Tychonoff functor . . . . . . . . . . . . . . . . . . 3. w-Compact spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A space X such that r ( X x Y ) = r ( X ) x T ( Y ) for any space Y . . . . . . . . . 5. A space X such that r(X x Y ) = T ( X ) x r ( Y ) for any k-space Y . . . . . . . . 6. Products of w-compact spaces . . . . . . . . . . . . . . . . . . . . . . . I. w-Paracompact spaces and the Tychonoff functor . . . . . . . . . . . . . . . 8. A generalization of Tamano's theorem. . . . . . . . . . . . . . . . . . . . 9. Rectangular products and w-paracompact spaces . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
203 204 201 209 215 218 22 1 224 230 234 241
Introduction It seems that the term of the Tychonoff functor was first used by Morita [1975]. The origin of the Tychonoff functor is the complete regularization of general topological spaces. It is now recognized, from the categorical viewpoint, as a covariant functor from the category of topological spaces and continuous maps into itself. The Tychonoff functor t is the reflector from the category above to the full subcategory of Tychonoff spaces. The purpose of this chapter is to present the basic theories of the Tychonoff functor and related subjects, for example, w-compact spaces (due to Ishii [ 1980a]), w-paracompact spaces (due to Ishii [ 19841) and those applications to rectangular products in the sense of Pasynkov [1975]. The reader should be aware of several recent articles containing results on the Tychonoff functor and related subjects. Some of them are: Pupier [1969],
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T. Ishii
Morita [1975, 19801, Oka [1978], Hoshina and Morita [1980], Ishii [1980a, 1980b, 19841, Pasynkov [1975, 19801. The starting point is a work of Pupier [I9691 who showed that if X is a locally compact Hausdorff space, then t ( X x Y) = t(X)x t ( Y ) for any topological space Y. After that, Morita proved that for topological spaces X and Y, t ( X x Y ) = r ( X ) x t( Y) if and only if any cozero-set of X x Y is a union of cozero-set rectangles of X x Y (cf. Hoshina and Morita [1980], Morita [1980]), and gave another proof of Pupier’s result. Further, by using Morita’s result above, Oka [I9781 proved that for a Tychonoff space X the converse of Pupier’s result is valid. For the case of X being a general topological space, Ishii [1980al found the necessary and sufficient condition for X to satisfy the condition that z ( X x Y ) = z ( X ) x t ( Y ) for any topological space Y by introducing the concept of w-compact spaces. The concept of w-paracompact spaces follows naturally that of w-compact spaces and is a generalization of paracompact Hausdorff spaces at the same time. It is interesting that every w-paracompact space X satisfies t ( X x Y) = t ( X ) x t( Y) for any Tychonoff space Y. A characterization of w-paracompact spaces in terms of product spaces is investigated in Section 8, which parallels to Tamano’s theorem for paracompact Hausdorff spaces. For this purpose the concept of w-normal spaces is introduced. Applications of w-paracompact spaces to rectangular products are stated in Section 9, which contains some generalizations of Hoshina and Morita’s theorems [1980]. For some facts, the proofs of which are omitted, the reader is referred to Hoshina’s chapter on extension of mappings I1 and Atsuji’s chapter on normality of product spaces I. All spaces in this chapter are assumed to be general topological spaces, and for regular spaces the T,-axiom is not assumed, unless otherwise specified. By a Tychonoflspace we mean a completely regular T,-space. Letters N, I and R denote the set of positive integers, the closed unit interval and the real line respectively. The cardinality of a set S is denoted by card S or I S I. The reader is referred to the texts of Nagata [I9851 and Engelking [1977] for standard terminologies and theorems in general topology.
1. The Tychonoff functor
Let us recall that a subset A of a space X is called a cozero-set if A = { x I f ( x ) > 0) for some continuous mapf: X -,I, and the complement of a cozero-set is called a zero-set.
The Tychonoff Functor and Related Topics
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For a space X , let I" be the set of all continuous maps cp : X + I and consider a continuous map t x :X + P ( X ) from X to the product space P ( X ) = n{Z, I cp E I"} defined by tx(x) = (cp(x))E P ( X ) , where I, = I for any rp E I". Let us put t ( X ) = t x ( X ) c P ( X ) . Then for a continuous map f : X + Y we have a continuous map t ( f ) : t(X) + t( Y ) by defining t( f ) ( t ) to be the point of t( Y )whose $-coordinate is the $ ofcoordinate o f t E t ( X ) , where $ E I " , and the diagram Xf.
Y
z ( X ) .o!,T ( Y ) is commutative. Thus t is a covariant functor from the category of spaces and continuous maps into itself. Obviously t(X) is a Tychonoff space, and as is easily seen, t X :X + t(X) is a homeomorphism if X is a Tychonoff space. Hence we call t the Tychonof functor. It is the reflector from the category above to the full subcategory of Tychonoff spaces. 1.1. Proposition. Any continuous map f from a space X into a Tychonof space Y is factorized through r ( X ) such that f = go tXfor some continuous map g : z ( X ) + Y , where g is determined uniquely by$
1.2. Proposition. For any cozero-set G of a space X , tx(G) is a cozero-set of z ( X ) with t;'(t,(G)) = G. Proof. Let G be a cozero-set of X such that G = { x ) g ( x )> 0) for some continuous map g : X + I. Since by Proposition 1.1 there is a continuous map h : t(X) + Z with g = h 0 tx, it holds that ti'({tE
z ( X ) ( h ( t )> 0 } )
=
G,
which implies that z,(G) = { t E t(X) I h(t) > 0 } and tx'(tx(C)) = G. Thus t,(G ) has the required properties. 1.3. Definition. A subset A of a space X is called t-open if A is a union of cozero-sets of X and the complement of a z-open set is called a t-closed set.
Clearly the union and the finite intersection of r-open sets are r-open and hence the intersection and the finite union of t-closed sets are t-closed. A subset P of a space Xis r-open if and only if P = s;'(Q) for some open set Q of t ( X ) .
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T. Ishii
1.4. Proposition. Let A be a subset of a space X . If for any cozero-set U of A there exists a z-open set V of X such that U = V n A , then ?,(A) = ? ( A ) , that is, ?,(A) is homeomorphic to ? ( A ) . Proof. By Proposition 1 . 1 , the restriction map T, I A :A + ?,(A) of z, to A is factorized through ? ( A ) such that zxI A = g 0 zA for some continuous map g : ? ( A ) --+ ?,(A). We shall verify that g is a homeomorphism. Sinceg is onto, it sufficesto see that g is a one-to-one open map. Let t E ?,(A) and u E g - ' ( t ) . Suppose there is a point x, E A such that x, E (?I, A ) - ' ( t ) - z;'(u). Then there is a cozero-set U of A such that x, E U and z;'(u) E X - U, since ?;I@) is z-closed in A. By assumption we can take a z-open set V of X with U = V n A. Since z;'(z,(V)> = V and t E z,(V), we have (zxl A)-'(t) c U , which is a contradiction. Therefore g is one-to-one. To see that g is an open map, let G be a cozero-set of ? ( A ) . By assumption, for a cozero-set t;I(G) of A there exists a z-open set W of X with z;'(G) = W n A. Since r x ( W ) is an open set of T ( X ) such that t x ' ( ~ , ( W ) )= W, it is easily seen that g(G) = z,(W) n ?,(A), which shows that g(G) is an open set of ?,(A). Thus g is a homeomorphism of ? ( A ) onto ?,(A). 0 A subset S of a space X is called z-embedded in X if any zero-set of S is represented in the form S n 2 for some zero-set 2 of X and is called C*-embedded in X if every continuous map f : S 3 I has a continuous extension 3:X --t I. Clearly, a C*-embedded subset of a space X is z-embedded in X . 1.5. Corollary. I f a subset A of a space X is z-embedded (or C*-embedded) in X , then we have ?,(A) = ? ( A ) .
1.6. Definition. A collection 9 of subsets of a space X is called r-locally finite in X if for any point x of X there exists a cozero-set neighborhood of x which intersects only a finite number of members of 9. A collection 9 of subsets of a space X is z-locally finite if and only if z x ( F ) = { ? , ( A ) / A E F}is locally finite in z ( X ) .
1.7. Proposition. For any locallyfinite cozero-set cover 4 of a space X , there exists a locallyfinite cozero-set cover V of t ( X ) such that z ; ' ( V ) refines %, where t;'(V) = {t;'(V)I V E Y } .
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The Tychonof Functor and Related Topics
In other words, any locally finite cozero-set cover of a space X admits a T-locally finite cozero-set refinement. To prove the proposition, we need the following lemmas.
1.8. Lemma. Let {FaI a E R} be a collection of zero-sets of a space X . Ifthere exists a locally finite collection { U, I a E Q} of cozero-sets of X such that FaE U, for each a E 0,then U{F, I a E a} is a zero-set of X . 1.9. Lemma. For any locallyjnite cozero-set cover { U, Ia E R} of a space X , there is a locally finite zero-set cover { F, 1 a E R} of X such that F. c U,for each a E R. As for the proofs of these lemmas, see Chapter 2, Lemma 1.3 and Theorem 1.4.
Proof of Proposition 1.7. Let 4 = { UaI a E R} be a locally finite cozero-set cover of a space X. Then by Lemma 1.9 there exists a zero-set cover { F, I a E R} of X such that F, c U, for a E R. For each a E R, take a continuous mapf,:X + I such that F, = f.-'(l) and U, = { x l f . ( x ) > o} and let
V,
= {.If.(.> >
f}
and K, = { x I f . < x >2
$1.
Then V, is a cozero-set and K, a zero-set in X such that F, c V , c K, c U, and X = U { V , l a E R } . Let us put U ( x ) = X - U { K , ~ X E X K- , } for each point x of X. Since U {K, I x E X - Ka} is a zero-set of X by Lemma 1.8, U ( x ) is a cozero-set of X , and {a I U ( x ) n V, # S } is clearly a finite set. Therefore Y = {T( V , )I a E R} is a locally finite cozero-set cover of ?(A'), and it is obvious that T; ' ( V )refines 4 , completing the proof. 0 We notice that Proposition 1.7 also follows from Proposition 1.1 and the following result: for any normal open cover 4 of a space X there exists a continuous map f : X + T of X onto some metrizable space T and a normal open cover V of T such that f - ' ( Y )refines 4 (see Chapter 2, Theorem 1.1 and Morita [1970]). 2. Product spaces and the Tychonoff functor Let { X , I a E R} be a family of spaces and let us put X = ll X , . For brevity we denote by T, the natural map T~~ : X , -, ? ( X u )for each a E R. By Proposition 1.1, for the product map f = n ~ ,X: + n r ( X , ) there is a continuous X )nt(X,) such that f = g o T x . map g = T ( ~ ~ T , ) : T (+
T. Ishii
208
2.1. Proposition. For any family of spaces { X uI a n 7 ( X a )is one-ro-one and onto, where X = H X , .
E
R}, the map g : r ( X ) +
Proof. Clearly the map g is onto. To show that g is one-to-one, assume that for a point t = (1,) E nt(X,)there are two different points u and v of r ( X ) such that g(u) = g(a) = t. Let x E r;'(u) and y E r ;' (v ). Then there is a continuous map h : X + I such that h ( x ) # h( y ) . Hence if we put x = ( x u ) and y = (y,) in X , it is shown that
h(x, x for some
(ZU)U#,)
# h(Y, x
(Zu).+,)
B E R and some point (z,),,,
. . . , E Q we have
h((x,))
= K Y ,
E
n{X,Ia # p } . If not, for ao, a,,
x (XU),#,)
= h ( ~ ux, Y,, x ( x a ) a # , , u , )
=
* *
3
so that by the axiom of choice (or the well-ordering of R and transfinite induction) we have h((x,)) = h(( y,)), which is a contradiction. Therefore, it holds that 7,(xS) # 7,( yo), implying that (r,(x,)) # (t,( y,)) in n t ( X , ) . Since (g07x)((xu))= ( 7 u ( x u ) ) and ( g 0 7 , ) ( ( ~ a ) = ) ( 7 a ( ~ u ) ) 9 we have ( g 0 7 x ) ((x,)) # (g 7*)(( y m ) )that , is, g(u) # g(a), which is a contradiction. Thus g is one-to-one. 0 0
If the map g = z(lI7,) : 7 ( r I X , ) + n 7 ( X m )above is homeomorphic, we put 7(nX,) = rI7(XU).
2.2. Definition. Let {Xu I a E Q} be a family of spaces and U,, a cozero-set of X U i ,where ai E Q, i = 1, . . . , n. Then a cozero-set U,, x . . . x Uunx n{X,I a # a ' , . . . , a,,} of nX, is called a cozero-set rectangle. 2.3. Theorem. For any family of spaces {XuI a E R}, thefollowing conditions are equivalent. (1) 7(nX,) = nT(X,). ( 2 ) For any cozero-set G of X = nX,and for any point x = ( x u )E G there exists a cozero-set rectangle A = U,, x * * * x Uunx n{X,I a # a ' , . . . , a,,} in X such that x E A c G. Proof. (1)+(2). Let G be a cozero-set of X with x = (x,) E G. Since 7,(G) is a cozero-set of 7 ( X ) with r;'(t,(G))= G by Proposition 1.2 and the map g = r(n7,): 7 ( n X , ) + n r ( X , ) is a homeomorphism by the assumption,
The TychonoffFunctor and Related Topics
209
( g 0 7,)(G) is an open set of n r ( X , ) containing g(7,(x)), and hence there is a cozero-set rectangle B = V,, x . . x V," x n { r ( X , )1 a # a,, . . . , a,} of the Tychonoff space l I t ( X , ) such that g(z,(x)) = (lI7,)(x) E B c (go 7,)(G). Therefore, letting U,, = r;'(V,,), i = 1, . . . , n, then A = U,, x . . . x U,. x l I { X aI a # a,, . . . , a,} is a cozero-set rectangle of X with x E A c G . (2)+(1). To show that the map g = r(nz,) : 7 ( X ) + n r ( X , ) is a homeomorphism, it suffices to verify that g is an open map, that is, for any cozero-set G of X , ( g 7,)(G) is open in n z ( X , > .Now let G be any cozero-set of X and let z E ( g 0 7,)(G) and x E ( g 0 z X ) - ' ( z ) . Since x E G , by the assumption there exists a cozero-set rectangle A = U,, x . . . x Uanx n{X,I a # a,, . . . , a,} of X such that x E A c G . Let us put V,, = 7,,(U,,),i = 1, . . . , n. Then B = V,, x . . . x 5, x II{r,(X,)la# a ' , . . . , a,} is a cozero-set rectangle of n z , ( X , ) such that z E B c ( g o T,)(G), from which it follows that (gor,)(G) is open in H7,(X,). 0 0
As a corollary of Theorem 2.3, we have the following result (Hoshina and Morita [ 19801). 2.4. Theorem. For two spaces X and Y the following conditions are equivalent. (1) 7 ( X x Y ) = 7 ( X ) x 7 ( Y ) . (2) For any cozero-set G of X x Y and f o r any point ( x , y ) E G there exists a cozero-set rectangle U x V of X x Y such that ( x , y ) E U x V c G .
3. w-Compact spaces
In this section we introduce the concept of w-compactness and study its basic properties. For a subset A of a space X , the 7-closure of A , C1,A in notation, means the set of points x in X such that any cozero-set neighborhood of x intersects A. If A = Cl,A, A is 7-closed in X . 3.1. Definition. A space Xis called w-compact if any open cover { U, I a E Q} of X contains a finite subfamily { U , , , . . . , U,"} such that X = Cl,(U,, u * . . u U,").
A Hausdorff space X is called H-closed if X is a closed subspace of every Hausdorff space in which it is contained, and it is known that a Hausdorff space X is H-closed if and only if every open cover { U, I a E Q} of X contains
T. Ishii
210
-
a finite subfamily { U , , , . . . , U,,} such that Oulu u Dan= X (see, for example, Engelking [1977]). Hence very H-closed space is w-compact as well as every compact space. However there exists a w-compact space which is not a H-closed space.
1x1
3.2. Example. There exists a regular TI space X with > 1 such that every continuous map f : X + R is constant (Novak [1948], Hewitt [1946], Herrlich [1965]). Such a space Xis not H-closed but w-compact. In fact, if we assume that X is H-closed, X is regular H-closed and hence it is compact, contradicting that any continuous map f : X + R is constant. It is obvious that X is w-compact. 3.3. Proposition. For a space X , the following conditions are equivalent. ( I ) X is w-compact. ( 2 ) Zfa family { A , } of closed sets of X is closed under thejinite intersection and each A , contains a nonempty cozero-set of X , then n A , # 8. ( 3 ) Zfa family {P.} of z-open sets of X has thejinite intersection property, then OFa # 8. Proof. It is obvious that (2) is equivalent to (3). (l)+(3). Let {PaI a E R} be a family of z-open sets of X with the finite intersection property. Suppose that OF, = 8.Then by (I), for an open cover {X - pala E R} of X,there exists a finite subset { a I , . . . , a,} of R such that
X
= Cl,(U{X - F , , ( i = I ,
. . . ,n } ) .
But this is a contradiction, since n{P,,I i = I , . . . , n } is a nonempty z-open set. Therefore we have nFa # 8. (3)+(1). Let { U, 1 a E R} be any open cover of X. Suppose that, for any finite subset { a l , . . . , a,} of R, X - Cl,(u{ Ua,l i = 1, . . . , n } ) # 8. Putting P ( a l , . . . , a,) = X - Cl,(u{ Uu,l i = 1,
. . . , n}),
the family {P(a,, . . . , a,) 1 aI,. . . , an E R, n E N} of z-open sets of X has the finite intersection property. Hence by (3) we have
n{ml,. . . , a,) l a 1 ,. . . , a, contradicting X =
u{U,I a
E
E
n, n E N}
#
8,
R}. Thus X is w-compact.
3.4. Proposition. Zfa space X is w-compact, then t(X) is compact.
0
21 I
The TychonoffFunctor and Related Topics
Proof. Let { V ,I a E n}be any open cover of r ( X ) . For each point t of ?(A/), take a(t) E R with t E K(,) and an open set W; of r ( X ) such that t E W; c q t K(,).Since Xis w-compact, for an open cover { r j l ( 1 t E r ( X ) } of X , there exists a finite subset {tl, . . . , t , ) of r ( X ) such that
w)
X = C I T ( U { T i 1 ( ~ , )=l i 1, . . . , n}) = U{cl&i'(W;,))p = 1,
. . . , n}.
Noting that CIT(~i'(W;)) = r i l ( q ) for t E ?(A'), we have r ( X ) i = 1, . . , n) c U{ V,(,,)I i = 1, . . . , n}. Thus r ( X ) is compact.
=
U{q,:,l
0
3.5. Proposition. There exists a regular Hausdorfspace X such that X is not w-compact but z ( X ) is compact. Proof. We prove the proposition by using an example (due to Tychonoff [1929]) of a regular Hausdorff space which is not Tychonoff. Let wo be the first countable ordinal and wIthe first uncountable ordinal. We denote by W(wi 1) the space of all ordinals a 6 mi with the interval topology, where i = 1, 2. Let us put
+
+ 1) x W ( 0 , + 1)
s
= W(w1
P
= {(a, w o ) l a < ol} and
-
(01, oo),
Q = {(a1, n)ln c w o } .
For each n c coo, let S, be the copy of the space S and p, a homeomorphism of S onto S,. In the topological sum US, of {Sn},we identify a point ~ 2 m I -( P ) with C P , ~ ( Pfor ) P E P and a point ~ 2 r n ( q )with ( ~ 2 m + l ( q ) for 4 E Q. By this identification we have a quotient space Y , which is locally compact Hausdorff. Let X be a space obtained by adding a new point 5 to Y and introducing the topology in X as follows: the base at 5 is the sets Y - U{p,(S)l 1 < j < n}, n < wo,and the base at x # 5 is the same as in Y. Then Xis a regular Hausdorff space which is not Tychonoff. Further it is shown that z ( X ) is compact but X is not w-compact. In fact, noting that for any continuous map g : W ( o , + 1) + I there exists < wIsuch that g(a) is constant for each a 2 (see, for example, Engelking [1977]), it is easily seen that for any continuous map f : X 4 I there is Po c w1 such thatfis constant on the set U~=I{pn((a, w,))l a 2 Po}. Hence every cozero-set of X containing has to include a set of the form
u(X where
\j
I=
I
Yu
pi(^) u i = I pi(T(y, ki)),
T(y, k ) = {(a, n) E S 1 a 2 y, n >, k } for y c w 1 and k <
0,.
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T. Zshii
Therefore any cozero-set cover of X has a finite subcover and so z ( X ) is compact. To see that X is not w-compact, we consider the family n < o,,} of closed sets of X , which is closed under the {cp,(T(tl,n))l tl < o,, finite intersection and each cp,(T(a,n)) contains an isolated point (that is, a cozero-set), while ncp,(T(a, n)) = 8. Consequently, X is not w-compact.
0 3.6. Proposition. A z-closed subset of a w-compact space is not always w-compact. We shall prove the proposition by an example. 3.7. Example. We introduce a topology e in the closed unit interval I as follows: the base at x # 0 is the &-neighborhoodsU,(x), E > 0, and the base at x = 0 is the sets U,(O) - A, E > 0, where A = { l / n l n E N}. Then the space X = (I,e) is H-closed and hence it is w-compact. But the z-closed subset S = A u {0} of X is not w-compact, because it is a countable discrete subspace of X . 3.8. Proposition. Let X be a w-compact space. Then every z-open set P of X has the w-compact closure.
3.9. Lemma. Let U be a cozero-set of a space X . Then every cozero-set V in U is also a cozero-set of X . Proof. Let f : X -+ I be a continuous map such that U = { X I f ( x ) > 0} and g : U -+ I a continuous map such that V = {x E U l g ( x ) > O}. Let us define a map h : X -+ I as follows: h(x) = f ( x ) g ( x )
for x
E
V,
and h(x) = 0 for x
E
X - V.
Then it is easily seen that h is continuous over X and V = {x1 h(x) > 0}, showing that V is a cozero-set of X . 0 Proof of Proposition3.8. Let P be a t-open set of a w-compact space X . Let { Q,} be a family of z-open sets of P with the finite intersection property. Then by Lemma 3.9, P n Q, is a non-empty z-open set of X for each a. Since the family { P n Q , } has the finite intersection property and Xis w-compact, we have n P n Q, # 8 and hence n Q , # 8. Thus P is w-compact.
The Tychonoff Functor and Related Topics
213
3.10. Proposition. The continuous image of a w-compact space is also
w-compact.
The above proposition is easily seen, and so the proof is left to the reader. A mapf : X + Y of a space X to another space Y is called a 2-map (Frolik [1961]) if it is continuous and f ( Z ) is closed for any zero-set Z of X . As for 2-maps, see also Isiwata [1967, 19691 and Noble [1969a, 1969bl. 3.11. Theorem. For a space X , the following conditions are equivalent.
(1)
X is w-compact.
(2)
The projection nY: X x Y
(3)
The projection K : X x Y + Y is a Z-mapfor any Hausdorflspace Y with at most one nonisolated point.
+
Y is a Z-map for any space Y .
Proof. (1)+(2). Let X be a w-compact space and Y an arbitrary space. Let 2 be a zero-set of X x Y such that Z = f - ' ( O ) for some continuous map f : X x Y + I. Suppose that n y ( Z )is not closed in Y . Then there is a point yo of Y such that yo E n Y ( Z ) - n y ( Z ) .Sincef (x, y o ) > 0 for x E X , we have inf{f (x, y o )I x E X } = a > 0; this follows from the compactness of 7 ( X ) . For each point (x, y o ) E X x Y , take an open neighborhood U, x V, of ( x , y o ) in X x Y such that f ( u , v ) > $ a for (u, v ) E U, x V,. Since { U, I x E X } covers the w-compact space X , there exists a finite number of points x,, . . . , x, of X such that X = U{Cl,U,,I i = 1, . . . , n } . Putting V = n{VJi = 1 , . . . , n } , we have Cl,(U{U,, x V l i
=
1 , . . . ,n } ) n 2 =
0,
because f ( u , v ) 2 + a for (u, v ) E Cl,(U,, x V ) , i = 1, . . . , n.
Hence it holds that ( X x V ) n Z = 0, contradicting the fact that yo E n Y ( Z ) .Therefore n y ( Z )is closed in Y . (2)+(3). This is obvious. (3)+(1). Suppose X is not w-compact, that is, there exists a family 9 = {PaI a E R} of 7-open sets of X with the finite intersection property and n { P uI tl E R} = 0,where we may assume without losing generality that 9is closed under the finite intersection. We introduce an order in R such that a < B means P, 3 Psand construct a space Y = R u { t}by adding a new point 5 to R and introducing a topology in Y as follows: each point of R is open and the base at is the sets U, = { /? E R I a < j} u { t},a E R. Then
214
T. Ishii
Y is a Hausdorff space with only one nonisolated point. Now for each a E R, take a cozero-set G, and a nonempty zero-set Fa in X such that F, c G, c P,. Then there is a continuous map g,:X + I such that g,(x) = 1 for x E X - G, and g,(x) = 0 for x E Fa. Let us define a map g : X x Y4Iinsuchawaythat g(z) = g,(x)
for z
=
(x, a) E X x a,
a E R,
and g(z) = 1 for z = (x, <) E X x
5.
Then it is easily seen that g is continuous. Hence if we put Z = g-'(0), it is a zero-set of X x Y and n y ( Z )is not closed in Y,since n y ( Z ) = Y - {<}. Thus X is w-compact. 0 Finally, we shall describe a theorem that is more precise than Theorem 3.11.
3.12. Theorem. For a space X the following conditions are equivalent. (1)
X is w-compact.
(2)
For any space Y and for any zero-set Z of X x Y, n y ( Z ) is a zero-set of Y.
(3)
For any Hausdorff space Y with at most one nonisolated point and for any zero-set Z of X x Y, n y ( Z )is a zero-set of Y.
The theorem above is an immediate consequence of the following lemma and Theorem 3.1 1.
3.13. Lemma. Let X be a w-compact space and Y an arbitrary space. Then for any continuous m a p f : X x Y + Y, the map g : Y + I defined by g(Y)
=
inf{f(x, Y ) l X E G Y Y ) }
is continuous. Proof. Let yo be any point of Y and E any positive number. For any point (x, y o ) E X x Y, take an open neighborhood U, x V, of (x, y o ) in X x Y such that If(u, v ) - f(x, yo)I c a e for (u, w ) E U, x V,. Then for an open cover { U' I x E X} of X, there is a finite number of points xI,. . . ,x, of Xsuch that X = U{Cl, Ux,I i = 1, . . . , n}, since X is w-compact. Let us put V = n{V,,li = 1, . . . , n}. Then it holds that If(x, y ) - f(x, y o )I
< 38
for x E X and y E V .
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The TychonoffFunetor and Related Topics
In fact, any point x of X is contained in some C1, Uxk,1 for y E V we have If(x,Y )
- f ( ~ k ,
.YO)(
< $8
< k < n, and hence
and If(x, yo)
-f(xk,
yo)[
< $&,
which implies that I f ( x , y ) - f ( x , y o ) ! < $ 8 . Consequently, from y E V it follows that 1 g( y ) - g( y o )I < E , showing the continuity of g : Y + I. 0
3.14. Remark. As for an open Z-map, Frolik [1961] proved the following result: Let f : X + Y be an open 2-map of a Tychonoff space X onto a Tychonoff space Y. If g:X4 R is continuous and if g,(y) = inf{g(x)l x E f - I ( y)}(g'( y ) = sup{g(x) I x E f - I ( y ) } ) is a real number for every point y E Y , then gi: Y 4 R (resp. g" : Y + R) is continuous. However his result is valid for the case of X and Y being general spaces. Hence by use of this result, Lemma 3.13 is also derived as a corollary of Theorem 3.11 (1
)+(a.
4. A space X such that r(X x Y ) = z ( X ) x r ( Y ) for any space Y The following theorem gives a characterization of a space X such that z ( X x Y ) = z ( X ) x z ( Y ) for any space (or any regular T I space) Y .
4.1. Theorem. For a space X the following conditions are equivalent. (1)
z ( X x Y ) = z ( X ) x z ( Y ) f o r any space Y.
(2)
t(X x Y ) = r ( X ) x r ( Y ) f o r any regular T , space Y.
(3)
For each point x of X , there exists a cozero-set neighborhood U of x such that 0 is w-compact.
For the proof of the theorem, we need two lemmas. 4.2. Lemma. Let U be a cozero-set of a space X whose closure is not wcompact. Then there exists a Hausdorffspace Y with only one nonisolatedpoint yo and a continuous map f 1 X x Y + I such that f ( z ) = 1 for z
E
( X x yo) u ( ( X - U )x Y ) ,
and f-'(o)n(Uxy>Z
B foryEY-yo.
The above lemma is shown in a similar way as Theorem 3.11 (3)+(1).
T. lshii
216
4.3. Lemma. There exists a regular TI k-space X containing two diferent points a and b such that g(a) = g(b)for every continuous map g : X + R.
Proof. Let X , P,Q , Snand qnbe the same as in the proof of Proposition 3.5, where n runs through all integers. In the topological sum US, of {Snln = 0, f 1, . . .} we identify a point q z n - , ( pwith ) qzn(p)forp E P and for q E Q . By this identification we have a a point qzn(q)with q2n+1(q) quotient space Y, which is locally compact Hausdorff. Let X be a space obtained by adding two different new points a and b to Y and introducing a topology in X as follows: the bases at a and b are the sets
respectively, and the base at y E X - a u b is the same as in Y. Then X is a regular TI space and the k-ness of X follows from the facts that each point y E X - a u b has a compact neighborhood and that each of two points a and b has a countable local base. Moreover it holds that g(a) = g(b) for every continuous map g : X --+ R, since any cozero-set of X including the point a has to contain a set of the form
where = {(a, n) E
T(y, k)
SI a 2 y , n 2
k} for y < o,and k <
Thus the space X has the desired properties.
w,,.
0
The fact that the space Xabove is a k-space will be used in the next section. Proof of Theorem 4.1. (1)+(2) is obvious. (2)+(3). Suppose that there exists a point x, of Xsuch that the closure of any cozero-set neighborhood of x, is not w-compact. Let { U,1 ct E Iz} be a base of cozero-set neighborhoods of x,. By Lemma 4.2, for each a E Iz there exists a Hausdorff space Y, with only one nonisolated point y, and a continuous map f,: X x Y, --t Z such that f,(z)
=
1
for z E ( X x y,) u ( ( X - U,) x Ye),
and
f,-'(O)
n (U, x y )
z0
for y
E
y, - y,.
Let 2 be a regular T Ispace containing two different points a and b such that g(a) = g(b) for every continuous map g : Z -+ R; such a space exists by
The Tychonoff Functor and Related Topics
217
Lemma 4.3. Let 2,be the copy of Z for each a E R and cp, a homeomorphism of Z onto 2,. In the topological sum Y , u Z , , we identify a point 6, = cp,(b) with y,. By this identification we obtain a quotient space T, which is regular T I .Let us define a space Y as a quotient space of the topological sum (JT, obtained by identifying every point a, = cp,(a) E Z , , a E R; Y is clearly regular T I .Let q: T, - Y be the quotient map and put 5 = q(a,) for every a E R. Then it is shown that there is a cozero-set G of X x Y including (xo. 5 ) for which there is no cozero-set rectangle U x V with (xo, 5 ) E U x V c G . To see this, we define a map g: X x Y -+ I in such a way that
u
g(z) = 1
for z E X x ( Y - u ( Y , - y,)),
and g(z) =
f , ( z ) for z
E
X x y, y
E
Y,
- y,.
Then g is a continuous map and G = {(x, y ) I g(x, y ) > 0} is a cozero-set of X x Y with (xo,5 ) E G . Let U x V be any cozero-set rectangle of X x Y containing (x,,, 5 ) and take /? E R such that U, c U . Since 5 E V , we have 6, = cp,(b) E Vfor each a E R and hence V n ( Y , - y,) # 8. On the other hand, we have fL'(0)n (Us x y ) # 8 for y E Y, - y,, so that g-'(0) n (Us x V ) # 8. Therefore U, x V is not contained in C,which implies that z ( X x Y ) # 7 ( X ) x z ( Y ) by Theorem 2.4. Thus (3) follows from (2). (3)-+(1). Let Y be an arbitrary space. Let us take a cozero-set G of X x Y and a point (xo,y o ) E G . If we put G ( y o ) = { x E X I (x, y o ) E C}, it is a cozero-set of X with xoE G( yo).As is easily seen by Proposition 3.8, there is a cozero-set neighborhood U of xo whose closure is w-compact and is contained in G ( y o ) . Since 6 x Y - G is a zero-set in 6 x Y , x y ( 6 x Y - G ) is a zero-set of Y by Theorem 3.12. Hence, if we put V = Y - x y ( 6 x Y - G ) , it is a cozero-set of Y with yo E V and so we have (xo,y o ) E U x V c G . Thus it holds that z ( X x Y ) = z ( X ) x z( Y ) by Theorem 2.4, completing the 0 proof. As a corollary of Theorem 4.1, we have the following. 4.4. Corollary. Let X be a Tychonoffspace. Then thefollowing conditions are
equivalent. ( 1 ) X is locally compact. (2) z ( X x Y ) = t ( X ) x z ( Y ) f o r any space Y .
Corollary4.4 (1)+(2) and (2)+(1) are due to Pupier [I9691 and Oka [1978], respectively.
T. Ishii
218
5. A space X such that r ( X x Y ) = r ( X ) x r( Y ) for any k-space Y
We shall begin with a characterization theorem of pseudocompact spaces in terms of product spaces. A space X is called pseudocompact if every continuous map f:X + Iw is bounded. As is easily seen, a space Xis pseudocompact if and only if for any decreasing sequence { U , } of nonempty # 0. cozero-sets of X we have
no,,
5.1. Theorem. For u space X the following conditions are equivalent.
(I)
X is pseudocompact.
(2)
The projection n,: X x Y k-space Y.
(3)
The projection n,: X x Y Y = W ( 0 , 1).
+
+
Y is a 2-map for any HausdorfT
+
Y is a Z-map for the space
is obvious. Let X be a pseudocompact space and Y any Hausdorff k-space. Let Z be a zero-set of X x Y such that Z = f -'(O) for some continuous map f : X x Y + I. To prove that n r ( 2 ) is closed in Y , it suffices to show that n,(Z) n K is closed for any compact set K of Y. Hence we may assume that Y is itself a compact Hausdorff space. For such a space Y, we have t(X x Y) = t(X) x T( Y) by Corollary 4.4, so that there exists a continuous map g : r ( X x Y) -+ I such that f = g o t x x , . Let us put 2, = g - ' ( 0 ) and denote by i , the identity map of Y onto itself. Since (rx x i , ) - ' ( Z I )= Z , it remains to show that n , ( Z , ) is closed in Y. Noting that the product z(X)x Y of a pseudocompact space r ( X ) with a compact space Y is pseudocompact (see, for example, Engelking [ 1977]), by Glicksberg's theorem (Glicksberg [1959]; see also Nagata [1985]) we have / I ( t ( X ) x Y ) = / I ( t ( X ) ) x Y , which implies that every continuous map h :t(X)x Y + I has a continuous extension over B ( r ( X ) ) x Y, where P ( r ( X ) )is the Stone-Cech compactification of t(X). Therefore the map g has a continuous extension 2 : / I ( t ( X ) ) x Y + I. If we put Z , = g-'(O), then we have n,(Z,) = n,(Z,) and n y ( Z 2 )is closed in Y by Theorem 3.1 1. Thus ny(ZI)(and hence n , ( Z ) ) is closed in Y. (3)+(1). Suppose Xis not pseudocompact. Then there exists a decreasing sequence { U,,} of nonempty cozero-sets of X such that non= 0. Take a nonempty zero-set Z,, of X contained in U, and a continuous mapf, : X + I such that 2, = f - ' ( 0 ) and X - U,, = f - ' ( l ) . Let us define a map f : X x W(o,+ 1) + I as follows:
Proof. (2)+(3) (1)+(2).
f ( x , n)
= fn(x)
for x E X and n < o,,,
The Tychonoff Functor and Related Topics
219
and f ( x , o,,)= 1 for x
E
X.
Then it is easy to see thatfis continuous. Further, letting Z = f -'(O), n y ( Z ) is not closed in Y = W(w, 1). Thus (3) implies (I), completing the proof. 0
+
As a more precise result than Theorem 5.1, we have the following theorem.
5.2. Theorem. For a space X the following conditions are equivalent. (I)
X is pseudocompact.
(2)
For any Hausdorf k-space Y and for any zero-set Z of X x Y , ny(Z) is a zero-set of Y.
(3)
For the space Y = W ( o , + 1) andfor any zero-set Z o f X x Y , ar(Z) is a zero-set of Y.
The above theorem is an immediate consequence of Theorem 5.1 and the following lemma. 5.3. Lemma. Let X be a pseudocompact space and Y a Hausdorf k-space. Then for any continuous map f : X x Y + I , the map g : Y + I defined by g ( y ) = inf{f(x,y)JxEn;'(y)} iscontinuous.
Proof. Since the lemma is reduced to the case of Y being a compact Hausdorff space and X being a pseudocompact Tychonoff space, it is derived from Glicksberg's theorem and Lemma 3.13. 0 We notice that Lemma 5.3 is also derived from Theorem 5.1 and Frolik's theorem mentioned in Remark 3.14. Now we prove a theorem characterizing a space X such that 7 ( X x Y ) = r ( X ) x 7 ( Y ) for any Hausdorff k-space Y.
5.4. Theorem. For a space X the following conditions are equivalent. (1)
7(X
x Y ) = 7 ( X ) x r ( Y ) f o r any Hausdor-k-spaces Y .
(2)
7(X
x Y)=
(3)
For each point x of X there exists a cozero-set neighborhood U of x such that 0 is pseudocompact.
7(X)x
7 ( Y )for any regular T , k-space Y .
T . Ishii
220
To prove the theorem, the following lemmas are needed.
u
5.5. Lemma. Let U be a cozero-set of a space X such that is not pseudocompact. Then there exists a continuous map f : X x W(w, + 1) -+ Isuch that f(z)
=
1 for z E ( X x
0,)
u ( ( X - U ) x W ( o , + l)),
and f - ' ( O ) n (U x n) #
0
for n <
0,.
Since the above lemma is shown by the similar way as in the proof of Theorem 5.1 (3)-+(1), the proof is left to the reader.
5.6. Lemma. Let U be a cozero-set of a space X such that 0 is pseudocompact. If V is a cozero-set of X contained in U, then is also pseudocompact.
v
Proof. Let {Gn>be a decreasing sequence of non-empty cozero-sets of p. Let us put H,, = G,, n V for each n. Then {H,,}is a decreasing sequence of nonempty cozero-sets of X by Lemma 3.9. Since D is pseudocompact and H,, c U for each n, we have nClOH,, # 0 and hence nClVG,, # 0,showing that is pseudocompact.
v
Proof of Theorem 5.4. (1)+(2) is obvious. (2)-+(3).Suppose there is a point xo of X such that any cozero-set neighborhood of x, does not have the pseudocompact closure. Let { U, I a E R} be a base of cozero-set neighborhoods of x,. Since 0, is not pseudocompact for each a E R, by Lemma 5.5 there exists a continuous map f , : X x Y, + I, where Y, = W(w, + 1) for each a E R, such that f , ( z ) = 1 for z
E
(X x
0,)
u ((X- U , ) x Ya),
and
f,-'(O) n (u, x n) z 0 for n < w,. Let Z be a regular T, k-space mentioned in Lemma 4.3 which contains two different points a and b such that g(a) = g(h) for every continuous map g : Z + R. Let Z , be the copy of the space Z and cp, : Z Z , a homeomorphism from Z onto Z , , a E R. Let us denoted by T, the space obtained from the topological sum Y, u 2, by identifying h, = cp,(b) with w, E Y,. We now define the space Y as a quotient space of the topological sum U{T,(a E R} obtained by identifying every a,, = q,(a), a E R. Let q be the quotient map and put 5 = q(a,) for each a E R. Since the topological sum Y, u Z , is a regular T , k-space, so is the quotient space T,, and hence the -+
The Tychonoff Functor and Related Topics
space Y is a regular T I k-space as well as the topological sum Let g : X x Y + I be a map defined by g ( z ) = 1 for z
E
X x U(Y - (Y,
-
22 1
u{T , I a
E
Q}.
w,)),
and g ( z ) = f , ( z ) for z E X x n , n
E
Y, -
0,.
Clearly g is continuous, and G = { ( x , y ) l g ( x , y ) > 0 } is a cozero-set of X x Y with (x,, 5 ) E G. Further it is easily seen that there is no cozero-set rectangle U x V of X x Y such that (x,, 5 ) E U x V c G, showing that r ( X x Y ) # t ( X ) x r ( Y ) . Thus ( 2 ) implies (3). (3)+( 1). This is derived from Lemma 5.6 and Theorem 5.1 by the similar way as in the proof of Theorem 4.1 (3)+(1) and so the proof is left to the 0 reader. Theorem 5.4 yields the following result due to Pupier [I9691 as a corollary.
5.7. Corollary. I f X is a space such that r ( X )is locally compact, then we have r(X x Y ) = r ( X ) x r ( Y )f o r any Hausdorfk-space Y . Proof. Let r ( X ) be locally compact. By Theorem 5.4 it suffices to see that each point x of X has a cozero-set neighborhood U such that 0is pseudocompact. Let x E X and t = t x ( x ) .Then there exists a cozero-set neighborhood V of t E r ( X ) such that P is compact. Let us put U = t x ’ ( V ) .Then U is a cozero-set of X with x E U and C1, U = z;’( F), and further every continuous map g :X + R is bounded on 0as well as C1, U,since g = h t xfor some continuous map h : z ( X ) + R. From this it easily follows that 0 is pseudocompact. 0 0
6. Products of w-compact spaces
This section is mainly concerned with the products of w-compact spaces. 6.1. Theorem. Let { X uI a E Q} be a family of w-compact spaces. Then the product IIX, is also w-compact and it holds that z ( n X , ) = llz(X,). 6.2. Lemma. Let X,, i = 1, . . . ,n, be any spaces and {P;, I A E A} a maximal family of z-open sets of the product X = II{X,.u,li = 1, . . . , n } with thefinite intersection property. Z f x , E n{Clx,ni(Pj,)I1 E A} for i = 1, . . . , n, then (XI,
where x i :X
3
. . . , x,)
E
npi I 1
X, is the projection.
E
A},
222
T. Ishii
Proof. We shall prove the lemma by induction for positive integers n. In case n = I , it is obviously valid. Suppose the lemma is true for n = k - 1, k > 2. To show that it is valid for n = k, let A = { P i 1 I E A} be a maximal family of r-open sets of X = II{xI i = 1, . . . , k} with the finite intersection and xi E n{Cl,ni(P,)II E A} for i = I , . . . , k. Let us denote by K the projection from X to X’ = ll(X,l i = 1, . . . , k - 1). Then it is easy to see that IT(&) = {.(Pi) 1 I E A} is a maximal family of t-open sets of X’ with the finite intersection property and xi E n{Cl,n,:(7z(Pi)) I I E A} for i = 1, . . . , k - I , where nj is the projection from X’ to X,. Hence by the assumption for the case of n = k - I , we have (XI,
...,
E
n{ci,.A(P,) I I E A}.
To show that X
= (XI,
.
* *
xk) E n{p$
E
A},
let us take an arbitrary open neighborhood UI x * . . x Uk of x in X. Then it is obvious that Qj.= ((17, x . . x Uk-1) x Xk) n Pi # 8 for each I E A. Since A is closed under the finite intersection, {Qil I E A} has the finite intersection property and so does { z k ( Q i ) l I E A}. Hence for each I E A, nk(Qi) u n , ( A ) has the finite intersection property. As is easily seen, nk(Qi) is a 7-open set of xk for I E A. Therefore we have ITk(Qi) E K k ( A ) for I E A, since q ( A )is a maximal family of t-open sets of Xk with the finite intersection property. Consequently, we have xk E Cl,Q, for each I E A, from which it follows that +
(ul x
’ ’ ’
x Uk-1 x
uk)
n Pi #
8
for 1 E A. Thus it holds that ( x I ,. . . , xk) E n{pi A E A}, completing the induction.
Proof of Theorem 6.1. Let X,, a E R, be w-com: act spaces and 9 any family of t-open sets of the product X = l l X , with the finite intersection property. Then by Zorn’s lemma there exists a maximal family A = { P i I I E A} of r-open sets of X having the finite intersection property and Let us denote by IT, the projection from X to X,. Since for each containing 9. a E R, X, is a w-compact space and {ta(PA) I A E A} is a family of t-open sets of X, with the finite intersection property, there exists a point x, of X, such that x, E n{Cl,n,(P,) I I E A}. Let us put x, = ( x , ) E X . Then it remains to n;’(U,) be an arbitrary show that x, E n { P i l I E A}. To see this, let open neighborhood of x o , where U, is an open neighborhood of xz, in X,, for each i. Let us put X’ = ll{X,,I i = 1, . . , , n} and let A be the projection
The Tychonoff Functor and Related Topics
223
from X to X’. Since {n(Pi)1 I E A} is a maximal family of z-open sets of X’ with the finite intersection property, by Lemma 6.2 we have (Xnp
f
..
9
E fI{C1,,n(P,)
X,J
II
E
A},
which implies that / n
\
that is,
n n,‘(Ui) n Pi # 8 n
for I
E
A.
i= I
Thus x, E (){Pi 11 E A}, showing the w-compactness of X = n X , . We next prove the latter half of the theorem. Denoting by z, for brevity, the natural map zx, of a space X, to z(X,), for the product map nz, : X --t n z ( X , ) there exists a continuous mapg :z ( X ) + n z ( X , ) such that nz,= g o z x . By Proposition 2.1, the map g is one-to-one and onto. Since X = nX, is w-compact, z ( X ) is compact by Proposition 3.4, and hence the map g is a homeomorphism of z ( X ) onto nz(X,). Thus we have r(IIX,) = n z ( X , ) , completing the proof. 0 We next describe an application of Theorem 6.1. The following theorem is reduced to Mibu’s theorem (Mibu [1944]) for the case of X,, a E R, being compact Hausdorff spaces.
6.3. Theorem. Let { X,I a E R} be a family of w-compact spaces. Thenfor any continuous map f:nX, -, Z, there exists a countable subset Q, of R and a continuous map g : n ( X , I a E Q,} + Z such that f = g 0 IC, where IC :nX, -, n{X,I a E Q,} is the projection. Proof. Let us put X = n X , and let f:X -, Z be any continuous map. Then there exists a continuous map g :z ( X ) + Z such that f = g z x . Since X,, a E R, are w-compact, we have z ( X ) = n z ( X , ) by Theorem 6.1 and z(X,) are compact. Hence by Mibu’s theorem there exists a countable subset Q, of R and a continuous map h : n { r ( X , ) I a E Q,} + Zsuch that g = h 0 n’, where n‘ : n z ( X , ) -, n { z ( X , ) I a E Q,} is the projection. If we denote by n the projection of X to X , = n { X , I a E Q,}, we obtain 0
f = g o t , = (hoa’)ot, = =
h 0 (zx0
0
Z)
h o ( ~ ’ o ~ , )
= (h 0 t x o0)71.
Therefore the map g = h , ,z :n { X , 1 o! E R,} that f = g n, completing the proof. 0
0
+
Z is a continuous map such
0
224
T. Ishii
7. w-Paracompact spaces and the Tychonoff functor In this section we introduce the concept of w-paracompact spaces, which is a generalization of both w-compact spaces and paracompact Hausdorff spaces, and study the basic properties of those spaces and the relationship between those spaces and the Tychonoff functor. Recall that any normal open cover of a space admits a locally finite cozero-set refinement and that any a-locally finite cozero-set cover of a space is a normal open cover (as for the proofs of these facts, see Chapter 2, Theorem 1.2). 7.1. Definition. A space X is w-paracompact if for any open cover { U ,I 1 E A} of X there exists a locally finite cozero-set cover { V ,I a E R} of X such that for each a E R there is a finite subset {,u(l), . . . , p(n)} of A such that V , c C17((J{U,(i)li= 1 , . . . , n}). A family { Aj,11 E A} of subsets of a space X is called weakly Cauchy (due to Corson [ 19581) if for any normal open cover { U, I a E R} of X there is b E R such that { U s } u { Aj,1 il E A} has the finite intersection property. The following proposition is a natural extension of Corson’s theorem that a Tychonoff space X is paracompact if and only if for any weakly Cauchy family { Aj , 1 1 E A} of subsets of X we have A ,I 1 E A} # 0.
n{
7.2. Proposition. A space X is w-paracompact if and only if for any weakly I 1 E A} of T-open sets of X we have n { F j ,1 1 E A} # 0. Cauchy family {P;.
Proof. Necessity. Suppose there is a weakly Cauchy family {Pj,I 1 E A} of z-open sets of X with n { F j I, I E A} = 0. If we put U, = X - Fj, for any 1 E A, { U ,I il E A} is an open cover of X with (Cl, U , ) n Pi = 8. Since X is w-paracompact, there exists a locally finite cozero-set cover { V , I a E R} of X such that for each a E R there is a finite subset { ~ ( l )., . . , p(m)} of A with
V , c CIT((J{U,(i)li= 1 , . . . , m}). Take /? E R such that { 5 ) u {Pj.11 E A } has the finite intersection property. Then, since V, c CIT(lJ{Uv(i)l i = 1, . . . , k}) for some v(l), . . . , v(k) E A, we have V, n n{Py(i)li= 1 , . . . , k} = 0, which is a contradiction. This proves the “only-if”-part. Sufficiency. Let { Uj.11 E A} be any open cover of X . In case X = U{Cl, U,(i)I i = 1 , . . . , n} for some A( l), . . . ,1(n) E A, it suffices to take the
225
The Tychonoff Functor and Related Topics
space X itself as a locally finite cozero-set cover { V ,1 a E f2} of X satisfying the required condition. In the case that X - u{Cl, Uj,(,) I i = 1, . . . , n } # 0 for any finite subset {A(l), . . . , A(n)} of A, if we let G(A(l), . . . , A(n)) = X - U{ClrUj.(i)li= 1,
. . . ,n } ,
and
9
= {G(A(l),
. . . , A(n))I L(l), . . . , A(n) E A, n E N},
then 9 is a family of r-open sets of X with the finite intersection property, while n{G(A(f), . . . , A(n))lA(l), . . . , A(n) E A, n E N} = 0. Hence 9 is not weakly Cauchy by our assumption, so that there is a locally finite cozero-set cover { V , I a E f2} of X such that each V , is disjoint from some Therefore for each a E R there are p( I), finite intersection of elements of 9. . . . , p ( k ) E A such that V, n G(p(l), . . . , p ( k ) ) = 0,showing that
V , c Cl,(u{Up(i)li= 1, . . . , k}). Thus X is w-paracompact, completing the proof.
0
7.3. Proposition. I f a space X is w-paracompact, then t ( X ) is paracompact. But the converse is not true in general. More precisely, even when r ( X ) is compact, X is not always w-paracompact.
To show this proposition, the following lemmas are needed. 1.4. Lemma.
For a space X , the following conditions are equivalent.
(1)
r ( X ) is paracompact.
(2)
For any weakly Cauchy family { P,.I A have (7{C1,Pj,(AE A} # 0.
E
A} of r-open sets of X , we
Since the above lemma is shown by the similar way as in the proof of Proposition 7.2, the proof is left to the reader. The following lemma is easily seen.
7.5. Lemma. I f a space X is w-paracompact and r ( X ) is compact, then X is w-compact. Proof of Proposition 7.3. The first part is an immediate consequence of Lemma 7.4. To prove the second part, let X be a regular Hausdorff space
226
T. Ishii
which is not w-compact but r ( X ) is compact, as is shown in Proposition 3.5. Then by Lemma 7.5, X can not be w-paracompact, which completes the proof. 17
By Proposition 7.3,any w-paracompact Tychonoff space is paracompact. However, even for regular Hausdorff spaces, w-paracompactness does not imply paracompactness. In fact, let X be a regular Hausdorff space with I XI > 1 such that every continuous mapf :X + R is constant. Such a space Xis w-compact and hence is w-paracompact, but it can not be paracompact, since it is not even a Tychonoff space. 7.6. Proposition. Let X be a w-paracompact space. Then,for any r-open set P of X , .p is w-paracompact. However a r-closed set (and hence a closed set) of X is not w-paracompact in general.
To prove the second part, we use a result due to Stone [I9481 as a lemma. 7.7. Lemma.
The product NK'is not normal.
Proof. We only sketch the proof of the lemma. Let A,, i = 1,2, be the set consisting of all { x n }E N" such that for any positive integerj # i we have xi. = j for at most one 2. Then A , and A, are closed and disjoint, and further it is shown that those sets are not separated by disjoint open sets. Thus NK1 is not normal. 0 Proof of Proposition 7.6. The first part of the proposition is easily seen, so that the proof is left to the reader. To show the second part, let X be a w-compact space mentioned in Example 3.7. Then by Theorem 6.1 the product XI is w-compact and hence is w-paracompact. But the r-closed subset (A u {O})K1 is not w-paracompact, is not normal and hence is not since A u {0}is homeomorphic to N and NK1 paracompact. This completes the proof. 0 7.8. Proposition. For a space X the following conditions are equivalent. (1)
X is w-paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(2)
X is w-paracompact and r ( X ) is locally compact.
The Tychonoff Functor and Related Topics
221
(3)
r ( X ) is paracompact and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(4)
There exists a locallyjnite cozero-set cover { U, I a E Q} of X such that 0, is w-compact for any a E R.
Proof. (1)+(2). Let x E X and t = r x ( x ) .Let U be a cozero-set neighborhood of x such that 0 is w-compact. Then r x ( U ) is a cozero-set of r ( X ) by ( )is compact by Proposition 3.10, which implies that Proposition 1.2 and t X 0 r x ( U ) is an open neighborhood o f t whose closure is compact. (2)+(3). We shall prove only the latter half of (3). Let x E X and t = r x ( x ) .Since r ( X ) is locally compact, there is a cozero-set neighborhood V of t such that P is compact. Let U = t i ’ (V ) and let {Pi I I E A} be any family of ?-open sets of 0 with the finite intersection property. Then Pi n U, I E A, are nonempty r-open sets of X by Lemma 3.9 and the family of those sets has the finite intersection property. Hence {rx(Pj.n U ) l I E A} is a family of open sets of t ( X ) with the finite intersection property and rx(Pi n U ) c V c V for each 1 E A. Since P is compact, we have n { r x ( P i n U ) ( 1E A} # 8, that is, n{Cl,(P,. n U ) l I E A } # 8, from which it follows that { P i n U I I E A} is a weakly Cauchy family of r-open sets of X. Therefore by the w-paracompactness of X we have n{-lL E A} # 8, showing that n{pi,llE A} # 8. Thus 0 is w-compact. (3)+(4). This is obvious. (4)+(l). We prove only that X is w-paracompact. Let { U, 1 a E Q} be a locally finite cozero-set cover of X such that each 0, is w-compact and let { I 1 E A} be any open cover of X. Since { 0, n 11 E A} is an open cover of a w-compact space 0,, there are p(I), . . . , p ( k ) E A such that
v,
v.
0,
=
CITE(U{O,n
c ( ii)=~ I , . . . , k } ) ,
where CITameans the .r-closure in 0,. Noting that CITaAc CIJ subset A of X , we have
for any
17~c CIr(u(OEn c(i)li= 1, . . . , k}) c Clz(U{<(i)li = 1,
. . . ,k } ) .
Consequently X is w-paracompact, completing the proof.
0
The following theorem gives the necessary and sufficient conditions for a space X to be w-paracompact in terms of the Tychonoff functor.
228
T . Ishii
7.9. Theorem. For a space X the following conditions are equivalent. (1)
X is w-paracompact.
(2)
T ( X ) is paracompact and r ( X x Y ) = r ( X ) x r ( Y ) for any Tychonofl space Y.
(3)
r ( X ) is paracompact and r ( X x Y ) = r ( X ) x T ( Y )for any Hausdorflspace Y with at most one nonisolated point.
The proof of Theorem 7.9 is based on the following result. 7.10. Theorem. For a space X the following conditions are equivalent. (1)
r ( X x Y ) = t ( X ) x r ( Y )for any Tychonoflspace Y.
(2)
r ( X x Y ) = t ( X ) x T ( Y )for any Hausdorffspace Y with at most one nonisolated point.
(3)
For any family {Pj,11 E A} of r-open sets of X with the finite intersection property which is closed under the finite intersection, n { C l ,Pj.11 E A} # 8 implies n{pj,11 E A} # 8.
Proof. (1)-(2) is obvious. (2)+(3). Suppose there is a family {Pi.I 1 E A} of r-open sets of X with the finite intersection property which is closed under the finite intersection and forwhich n{C\Pill E A} # 0but n{FilA E A} = 0. Letx, E n { C / P i ) I E A}. We denote by 9 = { Qa I a E R} a family of subsets of X each of which is in the form Pi. n U ( x o ) ,1 E A, where U ( x , ) is any cozero-set neighborhood of xo.Then 9is a family of r-open sets of X with the finite intersection property and I a E R} = 8. We now define an order in R such that a < b means Qs c Q, and construct a space Y = R u { <} by adding a new point to R and introducing a topology in Y as follows: each point of R is open and the u {<}, a E R. Then Y is a Hausdorff base at is the sets { B E R l a < /I} space with only one nonisolated point. For each a E R, take a nonempty zero-set F, and a cozero-set G, in X such that Fa c G, c Q,, and let g,: X + Z be a continuous map such that g i ' ( 0 ) = Faand g;'(l) = X - G,. Let us define a map g : X x Y + I as follows:
n{e,
r
<
g ( x , a ) = g,(x) and g(x,
r)
= 1
for (x, a ) E X x R,
for (x, 5 ) E X x
<.
Then g is clearly continuous. Further, if we put G = {(x, y ) Ig(x, y ) > 0 } , then G is a cozero-set of X x Y with (xo,5 ) E G and we have U x V - G # 0
The Tychonoff Functor and Related Topics
229
for any cozero-set rectangle U x V of X x Y containing (xo,0,contradicting (2) by Theorem 2.4. Thus (2) implies (3). (3)+(1). Suppose there exists a Tychonoff space Y such that r ( X x Y ) # r ( X ) x r ( Y ) . Then by Theorem 2.4 there is a cozero-set G of X x Y and a point (xo, y o ) E G such that any cozero-set rectangle of X x Y containing (xo, y o )is not contained in G. Let F be a zero-set and H a cozero-set in X x Y such that (xo,y o ) E H c F c G. Let us put P ( U , V) = n,(U x V - F) for any cozero-set rectangle U x Vof X x Y with (xo,y o ) E U x V , where nx is the projection of X x Y to X. Then it is easily seen that the sets P ( U , V ) are r-open in X and x,, E Cl,P(U, V). Since the family of such sets P ( U , V) has the finite intersection property and is closed under the finite intersection, by (3) there is a point x, of X such that xI E P ( U , V) for every cozero-set rectangle U x Vcontaining (xo, yo). We notice that (xl, y ) E H; otherwise, x, E X - P ( U , V) for any cozero-set neighborhood U of xo such that 0 c H ( y o ) = { x E XI (x, y o ) E H} and any cozero-set neighborhood V of yo. Hence there exists an open neighborhood A x B of (xI , y o ) contained in H , where we may assume that B is a cozero-set of Y , since Y is a Tychonoff space. But for any cozero-set neighborhood U of x,, the set P(U, B) does not contain any point of A , that is, P(U, B) n A = 8. Therefore we have xI E X - P ( U , B), which is a contradiction. Thus (3) implies (l), completing the proof. 0
Proof of Theorem 7.9. (1)+(2). Let X be a w-paracompact space. Since r ( X ) is paracompact by Proposition 7.3, we prove only the latter half of (2). Let {P;, I I E A} be a family of r-open sets of X with the finite intersection property and n{Cl,P, I I E A} # 8, which is closed under the finite intersection. Then it is clear that such a family is weakly Cauchy, so that we have n{Pil 3, E A} # 8 by the w-paracompactness of X. Therefore by Theorem 7.10 it holds that r ( X x Y ) = r ( X ) x r ( Y ) for any Tychonoff space Y. (2)+(3). This is obvious. (3)+(1). Assume that X satisfies the condition (3). Let {P;. I I E A} be any weakly Cauchy family of z-open sets of X , where we may assume without loss of generality that the family is closed under the finite intersection. Since r ( X ) is paracompact and { r x ( P j ,1)I E A} is a weakly Cauchy family of open sets of r ( X ) , we have n{r,(PJ 13, E A} # 8 by Corson’s theorem, which implies that n{Cl,PiI I E A} # 8. Hence by the latter half of (3) and Theorem 7.10, we have n { P , 11 E A} # 8. Thus Xis w-paracompact, completing the proof.
0
T. Ishii
230
8. A generalization of Tamano’s theorem
In this section we establish a characterization theorem of w-paracompact spaces in terms of product spaces, which parallels Tamano’s theorem (Tamano [19601) for paracompact spaces. For this purpose we introduce the concept of w-normal spaces.
8.1. Definition. A space Xis called w-normal if for any disjoint sets A and B of X such that A is z-closed and B is either z-closed or the closure of a z-open set, there exists a r-open set U of X with B c U c 0 c X - A. Let A and B be the disjoint sets of a w-normal space X with the properties mentioned above. If we put GI = X - A, GI is a z-open setand by the above definition there is a r-open set G(0) such that B c G(0) c G(0) c G I .Since G(0) c G I , there is a z-open set G ( $ ) such that (al)
-
-
G(0) c G ( $ ) and G O ) c G I .
Further, we can take z-open sets G(1/2’) and G(3/22) such that
-
(a’)
G(O) c G(1/2’), G(1/2’) c G O ) ,
Go) c G(3/2’), G(3/2’) By induction for n such that (a,)
E
c GI.
N, we can define r-open sets G(k/2”), k = 0, 1, . . . , 2 ” ,
0 < k < k‘
< 2“ * G(k/2“) c G(k’/2“).
Hence, as in the proof of Urysohn’s lemma (Urysohn [1925]), we can define a map f:X -+ Z such that f(x) = inf{k/2”Ix E G(k/2”)} for x
E
GI,
and f(x) = 1 for
X E
A = X - GI.
Then it is shown that f is a continuous map such that f(x) = 0 for x E B and f(x) = 1
f o r x e A.
Thus we obtain the following proposition.
8.2. Proposition. For a space X the following conditions are equivalent.
(1)
X is w-normal.
The Tychonoff Functor and Related Topics
23 1
(2)
z ( X ) is normal and for any z-open sets P and Q of X such that 0 c P, there is a t-open set U of X such that 0 c U c 0 c P.
(3)
Any disjoint sets A and B of X with the same properties as in Dejinition 8.1 are completely separated, that is, there exists a continuous map f : X + I such that f ( x ) = 0 for x E B and f ( x ) = 1 for x E A .
8.3. Proposition. normal.
There exists a non-w-normal space X such that r ( X ) is
Proof. Let X be a space mentioned in the proof of Proposition 3.5, that is, a regular Hausdorff space not being Tychonoff constructed by Tychonoff [1929]. As is shown in the proof of Proposition 3.5, Xis not w-compact but t ( X ) is compact. To see that Xis not w-normal, let us consider the one point set {t}which is z-closed and a z-open set q I ( S - (P u Q)). Then the point t is not contained in q , ( S ) = q l ( S - (Pu Q)) but there does not exist a continuous map f : X --t I such that f ( < ) = 1 and f ( x ) = 0 for x E q , ( S ) . Thus X is not w-normal, while t(X)is normal. 0 8.4. Proposition. Every w-paracompact space is w-normal.
Proof. Let X be a w-paracompact space. Since r ( X ) is paracompact by Proposition 7.3, it is normal. Let P and Q be t-open sets of X such that Q c P. For each point x of &, there is a cozero-set neighborhood C ( x ) of x such that Cl,G(x) c P, and hence we can consider an open cover $ = {G(x),X - Q I X E Q } OfX. In case Y has a finite subcover, we have
&
c U { G ( x i )I i = 1, c
. . . , n}
CI,(U{G(xi)li = 1, . . . , n } ) c P
for some points xI, . . . , x, E Q. Hence if we put U = U { G ( x i )I i = 1, . . . , n}, then U is a cozero-set of X such that Q c U c D c P, since
0
c
c1,u.
In case Y has no finite subcover, since Xis w-paracompact, there exists a locally finite cozero-set cover $2 = { U, I a E R} such that each U, is contained in the t-closure of some finite union of members of 9. By Proposition 1.7, Q admits a z-locally finite cozero-set refinement, so that we may assume that Q is itself z-locally finite. Let us put R’ = {u E R I U, n Q # @} and W, = U, n Q for tl E R’. Since { W, I u E R’} is a locally finite collection of
232
T. Ishii
r-open sets ofXand Q = U{ W,l u E a},we have Q = U{ R l u E a}.Further, for each u E Q’, W, is contained in the t-closure of some finite union L, of members of {G(x)JxE Q}. Let us put V, = U,, n L, for u E 0’. Since { V,I u E Q’} is r-locally finite in X and W, c C1, V,, we have
Q
= U { ~ ~ U € Rc ’ }U{Cl,V,~UEQ’} = Cl,(U{ V,lu
E
Q’}) c P .
By the normality of r ( X ) , there exists a cozero-set U of X such that cl,(U{V,~uER’})c so that
0c
uc
0
c
P,
U c 0 c P. Thus X is w-normal by Proposition 8.2.
0
8.5. Theorem. The product of a w-paracompact space with a w-compact space is w-paracompact.
Proof. Let X be a w-paracompact space and Y a w-compact space. Then by Theorem 4.1 we have r ( X x Y) = t ( X ) x r ( Y ) . Since r ( X ) is paracompact and t ( Y ) is compact, r ( X ) x r ( Y ) (=?(A’ x Y)) is paracompact (see, for example, Nagata [1985]). On the other hand, by Theorems 4.1 and 7.9, for any Tychonoff space Z we have T((X X Y ) X
z)
=
T(Y
X
(xX z)) =
= r(Y)x
r(X)x
z
T(Y)X
= r(Xx
T(xX z ) Y) x
z.
Thus X x Y is w-paracompact by Theorem 7.9, completing the proof.
[7
By the above theorem and Proposition 8.4 the product of a w-paracompact space with any w-compact space is always w-normal. The following theorem shows that the converse is also true. 8.6. Theorem. If for a space X the product X x /3( Y ) is w-normal for any Hausdorfspace Y with at most one nonisolatedpoint, then Xis w-paracompact.
To prove the theorem, we need a lemma which is due to Morita [ 19751 for the case of Y being compact Hausdorff. 8.7. Lemma. Let X be a space and Y a compact Hausdor-(or w-compact) space. Then for any normal open cover Y of X x Y , there exists a normal open cover 9 = { U, 1 u E Q} of X such that for a suitable collection {V,I u E Q} of jinite cozero-set covers of Y, the collection ( U , x V I V E V,,u E Q} is an open rejinement of 8.
233
The Tychonof Functor and Relaied Topics
As for the proof of Morita’s theorem for the case of Y being compact Hausdorff, see the chapter: Normality of product spaces I(Section 1). For the case of Y being w-compact, the lemma is a consequence of Morita’s theorem, Theorem 4.1 and Proposition 1.7, which will be used in the next section. Proof of Theorem 8.6. Let 9 = { P i I 1 E A} be any weakly Cauchy family of t-open sets of X . By Proposition 7.2, it suffices to show that (){pi 11 E A} # 8, where we may assume without losing the generality that 9 is closed under the finite intersection. Now suppose the contrary. Let us introduce an order in A such that 1 < p means Pj: 3 P, and, as in the proof of Theorem 7.10, construct a Hausdorff space Y = A u {<} with only one nonisolated point 5, where 5 is a new point and the base at 5 is the sets (1 E A 11 2 p } u { <}, p E A. Since each A E A is isolated in Y, so is in B( Y); this follows from the fact that Y is C*-embedded in B ( Y ) . Hence U{Pi x , I11 E A} is a t-open set of X x B( Y ) and its closure A = CL,,,,(U{P,. x 111E A>>
is disjoint from a t-closed set B = X x <. Therefore, by the w-normality of X x B ( Y ) , the binary open cover 92 = {X x B ( Y ) - A , X’x B ( Y ) - B} is normal, so that by Lemma 8.7 there is a locally finite cozero-set cover 4‘2 = { U, I a E Q} of Xsuch that, for a suitable collection {Vm I c1 E Q} of finite ( the collection { U, x V I V E Y,, a E Q} is an open cozero-set covers of /IY), refinement of Y. Since 9is weakly Cauchy, there is /? E Cl such that { U p >u 9 has the finite intersection property. Let V be an element of Y, with E V . Then we have (Us x V ) n B # 8, which implies that (Us x V ) n A = 8. But this contradicts the fact that { U p } u 9 has the finite intersection property. Thus we have n{Fj.1 1 E A} # 8,completing the proof. 0
<
Theorems 8.5 and 8.6 yield the following lemmas. 8.8. Corollary. For a space X the following conditions are equivalent.
(I)
X is w-paracompact.
(2)
X x Y is w-normalfor any w-compact space Y
(3)
X x /J(Y) is w-normalfor any Hausdor-space Y with at most one nonisolated point.
8.9. Corollary. For a Tychonoj” space X the following conditions are equivalent.
(1)
X is paracompact.
T. Ishii
234
(2)
X x Y is w-normalfor any w-compact space Y.
(3)
X x B ( Y ) is normal for any Hausdorfspace Y with at most one nonisolated point.
9. Rectangular products and w-paracompact spaces
The concept of rectangular products was introduced by Pasynkov [ 19751 as a generalization of Fproducts due to Nagata [1967]. 9.1. Definition. The product X x Y of two spaces X and Y is called rect-
angular if any finite cozero-set cover of X x Y is refined by a a-locally finite open cover of X x Y which consists of cozero-set rectangles.
As for rectangular products, Pasynkov [1975] proved an interesting result that, if X x Y is a rectangular product of Tychonoff spaces X and Y, then the inequality dim(X x Y) f dim X
+ dim Y
holds, where dim X means the covering dimension of a space X (as for the covering dimension, see, for example, Nagata [19831). After that, Hoshina and Morita [1980] improved his result to the case of X and Y being general spaces by showing the following theorem.
9.2. Theorem. The product X x Y of spaces X and Y is rectangular if and only i f r ( X x Y ) = 7 ( X ) x r ( Y ) and 7 ( X ) x r ( Y ) is rectangular. Proof. Since the “if”-part easily follows from Proposition 1.2, we prove the “only-if”-part. Assume that X x Y is rectangular. To show that 7 ( X x Y) = 7 ( X ) x 7 ( Y ) , let G be a cozero-set of X x Y with ( x o , y o ) E G. Taking a zero-set F of X x Y such that (x,,, y o ) c F c G, Y = {G,X - F} is a binary cozero-set cover of X x Y. Hence there exists a a-locally finite open refinement W of Y consisting of cozero-set rectangles. Let U x V be a cozero-set rectangle of W with ( x o , y o ) E U x V. Then we have U x V c G, so that 7 ( X x Y) = r ( X ) x r ( Y ) by Theorem 2.4. We next prove that 7 ( X ) x 7( Y) is rectangular. Let Q = { Ql i = 1, . . . , n} be a finite cozero-set cover of 7 ( X ) x 7 ( Y ) ( = T ( X x Y)). Since r i ; y(4Y) = y ( U , )I i = 1, . . . , n} is a finite cozero-set cover of X x Y, by the rectangularity of X x Y it is refined by a o-locally finite open cover -Ir = U/”=,5
{T~A
235
The Tychonofl Funcror and Related Topics
<
of X x Y consisting of cozero-set rectangles, where for each j , = { U,, x y a l a E Q j } is a locally finite collection of cozero-set rectangles of X x Y. If we take a nonempty cozero-set rectangle G,,, x Hi,, and a zero-set rectangle F,, x Kja in X x Y such that Gj, x Hja c
4ax
Kju c
qax ya,
then it is shown that {q,x 4 a I a E Q j } is t-locally finite in X x Y. In fact, noting that, for any subset Sofa,, (J{F,, x 4, I a E S } is a zero-set of X x Y by Lemma 1.8, the set W(z) = X x Y -
U{F,, x qalz E X
x Y -
F,,
x Kja,a E
a,},
which is defined for each point z E X x Y, is a cozero-set neighborhood of z and intersects only finitely many members of {Gja x HjuI a E Q j } . Thus {Gj, x HjaI a E a,} is t-locally finite in X x Y. Now, for each j and a, let us take a sequence of nonempty cosero-set rectangles {Gjakx HjakI k E N} and a sequence of zero-set rectangles (4.k X 4 . k I k E N} in X X Y such that Gjak
4ak
4 a k
4 a k
qa
vu,
and
qaX ya
= u{Gjak X
4 . k
Ik E
N}.
Then by the above argument, q k = {Gjmkx HjukI a E a,}is r-locally finite in X x Y and hence u { q k lj, k E N} is a o-t-locally finite open refinement of ti:,,(a).Thus admits a o-locally finite open refinement y($k) lj, k E N} consisting of cozero-set rectangles t,(Gjak) x tY(Hjak),a E aj,j,k E N, of t ( X ) x t ( Y ) , completing the proof. 0
u{t,,
9.3. Theorem. A space X i s w-paracompact ifand only i f X x Y is rectangular for any Hausdorflspace Y with at most one nonisolated point. Proof. Necessity. Let X be a w-paracompact space and Y a Hausdorff space with at most one nonisolated point. Then we can show that any cozero-set cover of X x Y admits a locally finite open refinement consisting of cozeroset rectangles, which implies that X x Y is rectangular. Let Y = { G; I 1E A} be any cozero-set cover of X x Y. In case Y has no nonisolated point, we have nothing to prove, and so we may assume that Y has only one nonisolated point 5. For each point (x, 5 ) E X x 5, take 1(x) E A with (x, 5 ) E Gi(,). Since r ( X x Y) = r ( X ) x Y by Theorem 7.9, there exists a cozero-set rectangle U, x V, such that (x,
5 ) E ux
x
v, = Gl(,,.
236
T. Ishii
By the paracompactness of T ( X )the cozero-set cover { U, I x E X} of Xadmits a locally finite cozero-set refinement { H , I a E R}. If for each a E Q we take x(a) E X such that H , c Uxca,.then YP = {Ha x I a E R} is a locally finite collection of cozero-set rectangles of X x Y covering X x <. Let {F, I a E R} be a locally finite zero-set cover of X such that F, c H , for any a E R. Since each V, is open and closed in Y , {F, x K(,)I a E R} is a locally finite collection of zero-set rectangles such that F, x P‘,(,) c Ha x <(,), and hence by Lemma 1.8, F = u { F , x K(.) I a E R} is a zero-set of X x Y. For each point y # in Y , let us put W ( y ) = (X x Y - F ) n (X x y). Since W( y) is a cozero-set of a w-paracompact space X x y, W( y) is also w-paracompact by Proposition 7.6. Therefore there exists a locally finite cozero-set cover dy= {A, I y E r v }of the space W(y ) which refines a cozero-set cover {G, n W ( y ) l AE A} of W ( y ) , and 91y = {A, n W ( y ) ( yE r,,}is a locally finite collection of cozero-sets of X x y covering W ( y ) . Finally let us put
<
3-
= Jf
u
U{qY z
r, y
E
Y}.
Then it holds that 3- is a locally finite refinement of 9 consisting of cozero-set rectangles of X x Y. In fact, it is clear that X refines Y and X is locally finite at every point (x, y) E X x y , y # 5. For each point (x, 5 ) E X x 5, take an open neighborhood N ( x ) of x which intersects only finitely many members of { H , I a E R}, say H a , , . . . , Ha,, and put V
=
n{V,(,,)Ii= 1 , . . . , n}.
Then it is easily seen that an open neighborhood N ( x ) x Vof (x, 5 ) intersects only finitely many members of X . This proves the “only-if”-part. Suficiency. Assume that X x Y is rectangular for any Hausdorff space Y with at most one nonisolated point. Then by Theorem 9.2 it holds that r ( X x Y ) = 7 ( X ) x Y. Hence by Theorem 7.8 it remains to show that 7 ( X ) is paracompact. Now let 92 = { U i 12 E A } be any cozero-set cover of X , where we may assume that 92 has no finite subcover. For each point x of X , let us take 2(x) E A with x E UiCx,. Then there is a zero-set Fi(,) and a cozero-set Ei(,) in X such that x E Ei(,, c FicX,c Uicx,.Let r be the set of all nonempty finite subsets of X , and for y = {x,, . . . , x,} E r let us put
V,
=
U{Ui(.r,)li= 1 , . . . ,n},
K,
=
u { F i ( x , ) p= 1, . . . , n}.
Then V, is a cozero-set and K, a zero-set in X with 4 c V,. We introduce an order in r such that y < 6 means y c 6 and construct a Hausdorff space
The Tychonoff Functor and Relaled Topics
231
Y = r u { <} with only one nonisolated point 5 , where 5 is a new point and the base at is the sets W, = {a E rib 2 y} u {t},y E r. n ( ( X - 4) x y) = 8 for x E X and y E I?, { ( X - 4) x Since (E,,,, x y I y E r}is a discrete collection of cozero-sets of X x Y. Hence by Lemma 1.8, Z = U { ( X - 5 ) x y I y E r} is a zero-set of X x Y. Letting P = U { ( X - K , ) x y I y E r},by the rectangularity of X x Y , a binary cozero-set cover {P,X x Y - Z } admits a a-locally finite open refinement Q = Up,9?, consisting of cozero-set rectangles, where 9?, = { L , x MmjI LY E SZ,} is a locally finite collection of cozero-set rectangles. Let us put
<
9
= {L,l5€Maj,LYERj,~E~}.
Then 9 is a a-locally finite cozero-set cover of X and each L, E 9 is contained in some V, = U{Uj.(x,)li = 1, . . . , n } ; this follows from the fact that ( L , x MaJ)n 2 = 8 for L, E 2’. Therefore, if we consider a cozero-set cover WQJ = { L , n U,,(v,) I i = 1, . . . , n } of L,, then W = U {WaJI LY E a,,j E N} is a a-locally finite cozero-set refinement of 42. Thus T ( X )is paracompact, completing the proof. 0 As a corollary of Theorem 9.2, we obtain a result due to Hoshina and
Morita [1980]. 9.4. Corollary. A Tychonof space X is paracompact if and only if X x Y is
rectangular f o r any Hausdor-space Y with at most one nonisolated point.
The following theorem gives a characterization of a space X such that X x Y is rectangular for any space (or any Tychonoff space) Y. To prove it, we make use of a method employed by Ohta [1978] in proving that, for a Tychonoff space X , X x Y is C-embedded in X x p ( Y ) for every Tychonoff space Y if and only if X is locally compact, where p ( Y ) is the topological completion of a Tychonoff space Y , that is, the completion of Y with respect to the finest uniformity consisting of all normal open covers of Y. A subspace S of a space X i s called C-embedded in X if every continuous map f : S -, R has a continuous extension3: X -, R (as for the uniformity, see, for example, Nagata [1985]). Recall that for any normal open cover 42 of a Tychonoff space Y there is a normal open cover Y of p( Y ) such that Y n Y = { V n YI V E Y } refines 42; equivalently, any normal open cover of Y is extended to a normal open cover of ,u( Y ) .
T. Ishii
238
9.5. Theorem. For a space X the following conditions are equivalent. (1)
X is w-paracompact (or t ( X ) is paracompact) and each point x of X has a cozero-set neighborhood whose closure is w-compact.
(2)
X x Y is rectangular for any space Y.
(3)
X x Y is rectangular for any Tychonofspace Y.
To prove the above theorem, we need the following lemmas.
9.6. Lemma. I f a space X is w-compact, then X x Y is rectangular for any space Y. More precisely, any normal open cover of X x Y admits a locallyfinite open refinement consisting of cozero-set rectangles of X x Y. The above lemma is an immediate consequence of Lemma 8.7. The following lemma is concerned with a regular cardinal. An initial ordinal I is called regular if there is no a < I which is cofinal with I , and a cardinal n is called regular if the initial ordinal I with I I I = n is regular (see, for example, Engelking [1977]). For a regular ordinal a,,we consider the space W(w,) = { I l I < w,} with the interval topology.
9.7. Lemma. Let w, be a regular ordinal. Then the space W(w,) has the following properties: (1) Any locally Jinite open cover (JiY of W(w,) has a finite subcover { U , , . . . , U,,},and there is yo < w, such that { y E W(w,) I y 2 y o } c U, for some k , 1 < k < n. (2) For every continuous map f:W(o,) + Z there is y , < w, such that f(r) = f ( y 1 ) f o r any Y 2 YI. Since the lemma above is similarly proved as in the case of W ( w , ) ,the proof is left to the reader.
Proof of Theorem 9.5. (1)+(2). We first notice that in (1) the w-paracompactness of X is replaced by the paracompactness of t(X)by Proposition 7.8. Let X be a space satisfying the condition (1). Then by Proposition 7.8 there exists a locally finite cozero-set cover { G , I a E R} of X such that G, is w-compact for each a E R. Let Y be an arbitrary space and %! = { U,I i = 1, . . . , n } a finite cozero-set cover of X x Y. For each a E R and i = 1 , . . . , n, let Hi,= n (G,x Y ) .Then Xu = {Z&I i = 1, . . . , n} is a finite cozero-set cover of G, x Y. Hence by Lemma 9.6 there exists a locally
239
The TychonoffFunctor and Related Topics
finite open cover X, consisting of cozero-set rectangles of G, x Y which refines X,. If P x Q E X,, then P n G, is a cozero-set of X by Lemma 3.9. Therefore X, n (G, x Y) = {(P x Q)n (G, x Y))P x Q E is a locally finite collection of cozero-set rectangles of X x Y which covers G, x Y, so that U{Xan (C, x Y) I a E a}is a locally finite open cover of X x Y consisting of cozero-set rectangles of X x Y which refines 42. Thus X x Y is rectangular. (2)+(3). This is obvious. (3)+(1). Assume that X x Y is rectangular for any Tychonoff space Y. Then by Theorem 9.3, X is w-paracompact. Now suppose that there is a point x, of X such that for any cozero-set neighborhood U of x,, 0 is not w-compact. Let { U, I I E A] be a base of cozero-set neighborhoods of xo. Then by Lemma 4.2, for any I E A there exists a Hausdorff space Y, with only one nonisolated point 5, and a continuous map hi : X x Y, + Z such that
x}
(X x t i ) u ( ( X
h,(z) = 1 for z
E
h,'(O) n ( V , x p )
z8
-
U , ) x Y,),
and forp E Yi
- ti,
where each Y, is an ordered set and the base at ti is the sets { q E Yi - t i 1 4 2 P} u ti},^^ Yi - ti* Let n be a regular cardinal greater than sup{I Y, 1 11 E A} and w, the initial ordinal of n. For each I E A, put Z ( I , 1) = W ( o , 1) x Y, and let Z ( I , 2) be the copies of W ( o , + 1) x W(wo + 1). By identifying a point (y, 5,) E Z(A, 1) with a point ( y , wo)E Z(A, 2) for each y < w,, we obtain a quotient space Z, and a quotient map cpi from the topological sum Z ( I , 1) u Z(A, 2) to Z,. Let Z be a quotient space obtained from the spaces Z,, I E A, by identifyingevery point cpi(wa,B ) E Z,, I E A, for each B 6 wo, where (a,, B) E Z ( I , 2), and let i,b: UZ, -, Z be a quotient map. Put e, = cp,, z,, = ei(o,,ti) for A E A and Y = Z - z,,. Then Y is clearly a Tychonoff space and it is shown that Z c p(Y). To see this, it suffices to show that for any locally finite cozero-set cover Y of Y there exists a locally finite cozero-set cover W of 2 such that W n Y = { w n Y l W E W } r e f i n e s V . L e V = {V,IoEX}beanylocally finite cozero-set cover of Y. For I E A, y < a,,p E Y;. - ti.and n < wo,let us define a subset
+
$ 0
A ( 5 71 P) = ((6, q ) l y < 6 G
ma,
P < 4 G ti} -
(ma,
ti)
of Z ( I , 1) and a subset
B(I, Y, n)
=
((7,
P)IY < 6 G
ma,
n < B G wo} -
(ma,
wo>
T. Ishii
240
of Z(A, 2). We denote by C(A, y(A), p(A), n(A)) the set Qi(A(A,y(A), p(A)) u B(A, y(A), n(A)) in Y. Then by Lemma 9.7 and the fact that n > sup{ 1 Y, I 11 E A}, for a fixed A, E A there is VboE V such that C@O, Y ( A 0 h P@O),
n(A0))
KO
for some y(&) < o,, p(Ao) E Y;, - tC and n(A,) < w,, from which it follows that for any A E A, there is y ( A ) < o,and p(A) E Y, - 5;. such that c ( A 7
Y ( 4 , P ( 4 , n(A0)) = KO.
W, = U{C@,y ( 4 , p ( 4 , n(AoNIA E A} and W , = V, n (Y - W,), Z. Then W, is an open and closed set of Y contained in V,,, so that V' = { W,, W,la E Z} is a locally finite cozero-set refinement of V . If we consider the set Po= W, u {z,}, @, is open and closed in Z and W = {I&, W, 1 0 E Z) is a locally finite cozero-set cover of 2 such that W n Y refines V , showing that Z c p(Y). Now let us define a map h : X x Y -, Z such that Let
CJ E
h ( x , v)
=
hi(&
h(x, y )
=
I
and
4 for Y
for y
E
=
@;.(Y, 4 E e;,(Z(A,1))
e,(Z(A,2))
-
-
zo,
z,.
Then it is obvious that h is continuous. Hence we can consider a binary cozero-set cover {G, H } of X x Y such that G
=
{( x , Y ) l h ( X , Y ) > 01 and H
=
I<.,
y)lh(x, y ) <
f}.
Since by the assumption X x Y is rectangular, there is a a-locally finite open cover 9 = UE,gi of X x Y consisting of cozero-set rectangles which refines the binary cover {G, H } of X x Y, where gi = {Gsi x Hail6 E Ai} is locally finite in X x Y. Since S ( x o ) = {HaiIx,E Gai, 6 E Ai, i E N} is a normal open cover of Y, there is a normal open cover S of p ( Y ) such that S n Y refines S ( x o ) . For each 6 and i, let H$
=
~ ( y-) Clp(y)(Y - Hai).
Then we have (x,,, z o ) E GEkx HZ for some k E N and E E A k .Take a cozeroset neighborhood Up of x , with Up c GEkr p E A. Then the open set H 2 of p( Y) with zo E HZ contains a set C ( p , y(p), p ( p ) , n(p)) for some y ( p ) < o,, p ( p ) E Y,, - 5, and n ( p ) < w,. Since h ( x , y ) = 1 for any y = e,,(y, 5,) E Q p ( Z ( p ,l)), we have GEkx H e k c G. On the other hand, it holds that h-'(Oj n (upx e P ( A ( p ,YW P(P)))
+ 0,
which is a contradiction. This completes the proof.
0
The Tychonoff Functor and Related Topics
24 1
As a corollary of Theorem 9.5, we obtain a theorem of Hoshina and Morita [1980].
9.8. Corollary. For a Tychonof space X the following conditions are equivalent. ( I ) X is locally compact and paracompact. (2) X x Y is rectangular for any space Y . ( 3 ) X x Y is rectangular for any Tychonofspace Y . 9.9. Remark. Concerning the relationship between rectangular products and topological completions, Hoshina and Morita [ 19801 proved that (a) for any Tychonoff spaces X and Y , X x Y is rectangular if and only if p ( X x Y ) = p(X)x p ( Y ) and p ( X ) x p ( Y ) is rectangular. On the other hand, Ohta [I9781 proved that (b) for a Tychonoff space X , p ( X x Y ) = p(X)x p( Y ) for any Tychonoff space Y if and only if X is locally compact. Using (a), (b) and Corollary 9.4, Hoshina and Morita [I9801 obtained Corollary 9.8. For furt
