TOPOLOGICAL ANALYSIS
BY
Gordon Thomas Whyburn
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1958
Preface While the title "Topological Analysis" could embrace a wide range of subject matter, including all phases of analysis related to topology or
derivable by topological methods, the material to be presented here will be centered largely around results obtainable with the aid of the circulation index of a mapping and properties resulting from openness of a mapping. This choice of topics has been governed largely by the special interests and tastes of the author, but it is hoped these may be shared to an appreciable extent by others. Organization of some of this material was begun in 1952-1953 while
the author held a Faculty Fellowship supported by the Fund for the Advancement of Education of the Ford Foundation. It was further aided by the author's participation in the University of Michigan conference on Functions of a Complex Variable in the summer of 1953, and some of the later work, especially that in Chapters V, VIII and X involving new results, was supported in part by a grant from the National Science Foundation (G 1132). The generous provision of a private study,
offering freedom from interruption, by the Alderman Library of the University of Virginia during the past two sessions has greatly assisted in bringing the book to conclusion. To all of these the author takes pleasure
in recording here an expression of his sincere gratitude. Thanks are due also to the editors of the Princeton Mathematical Series for their interest
in the completed book and to the Princeton University Press for its careful and sympathetic handling of the task of printing and publishing it. Charlottesville, Virginia February, 1957
G. T. W$YBURN
Introduction Topological analysis consists of those basic theorems of analysis, especially of the functions of a complex variable, which are essentially topological in character, developed and proved entirely by topological and pseudo-topological methods. This includes results of the analysis type, theorems about functions or mappings from one space onto another
or about real or complex valued functions in particular, which are topological or pseudo-topological in character and which are obtainable largely by topological methods. Thus in a word we have analysis theorems and topological proofs. In this program a minimum use is made of all such machinery and tools of analysis as derivatives, integrals, and power series; indeed, these remain largely undefined and undeveloped. The real objective here is the promotion, encouragement, and stimulation of the interaction between topology and analysis to the benefit of both. Certainly many new and important developments in topology, some of recent discovery, owe their origin directly to facts emerging from studies of the topological character of analytic results. Also, the opinion may be ventured that basic recent developments in analysis are due in
considerable measure to the better understanding of the fundamental nature of the classical situations provided by topological concepts, results and methods. Of course, the topological character of many of the classical
results of analysis has been recognized since Riemann and Poincar6, and even before them. Indeed, the very fact of this character and its recognition is in large part responsible for the origins and developments of the field of topology itself. However, the full depth of the penetration of topological nature into analytical results was surely not realized until the fairly recent past. Contributions of fundamental concepts and results in this type of work have been made during the past twenty-five years by a large number of mathematicians. Among these should be mentioned: (1) Stoilow [1],
the originator of the interior or open mapping, who early recognized lightness and openness as the two fundamental topological properties of the class of all non-constant analytic functions; (2) Eilenberg [1] and Kuratowski [1, 2] who introduced and used an exponential representation for a mapping and related it to properties of sets in a plane; Vii
viii
INTRODUCTION
(3) Marston Morse who in his book (see Morse [1]) and in joint studies with Heins [1] and with Jenkins [1] analyzed invariance of topological indices
of a function under admissible deformations of curves in the complex plane and also conjugate nets and transverse families of curves in the plane,
which has greatly clarified the action of the mapping generated by an analytic function and opened the way toward admissible simplifying assumptions; (4) the Nevanlinna [1) brothers and L. Ahlfors [1) whose outstanding work on exceptional values of analytic functions led to conclusions partly topological in character which contain the suggestion of new connections with topology still awaiting development; and (5) Ursell and Eggleston [1] as well as Titus [1] and Young who have contri-
buted elementary proofs for the lightness and openness of analytic mappings using novel methods which have stimulated considerable further effort using these methods in the same area as well as for mappings in a more general topological setting. My own work on this subject began
around 1936 as a result of reading some of Stoilow's early papers, and has been published in an extended sequence of papers spanning the interval of nearly twenty years to the present. On two occasions, however,
summaries of some of my results have been given and these are to be found in Memoirs No. 1 of the American Mathematical Society series, entitled Open mappings on locally compact spaces and as Lecture No. 1 in the University of Michigan's recently published Lectures on functions of a complex variable.
The portion of topology which is used is surprisingly small and is entirely set-theoretic in character. This is developed in Chapters I-III and consists, in brief, of (1) introductory material on compact sets, continua, and locally connected continua in separable metric spaces; (2) a discussion of continuity of transformations and of the extensibility of a uniformly continuous mapping to the closure of its domain, and a proof of the basic theorem characterizing the locally connected continuum as the image of the interval under a mapping; (3) the most basic theorems
of plane topology, that is, the Jordan curve theorem, the Phragm5n. Brouwer theorem, and the p' ine- separation theorem, which permits the separation of disconnected parts of compact sets in the plane by simple closed curves. The last group, as well as much of the first two, could be largely avoided by leaning on polygonal approximations to general curves. However, it is felt that while this would effectively reduce the topological base on which the discussion rests at present, the use of
arbitrary simple arcs and simple closed curves is more natural in the theory of functions of a complex variable. The analytical background, developed in Chapter IV, comprises only the simple properties of the complex number system and the complex
INTRODUCTION
ix
plane, limits and continuity of functions, and the definition of the derivative together with its representation in terms of the partial derivatives of the real and imaginary parts of the function, but nothing beyond this for the complex derivative other than differentiability of rational combinations of differentiable functions. The mean-value theorem for real functions is presupposed. Also, the definition and the simpler properties of the exponential and logarithmic function of the complex variable are introduced and used. For example, use is made of the fact that the logarithms of a complex number are distributed vertically in the plane 21r units apart and the fact that the logarithm has continuous branches, but of nothing at all about the form of the logarithm or its derivative in terms of power series. Integration is never defined. Only a short bibliography is included, listing some of the more closely related sources from which material has been drawn. A few citations to the bibliography are made at the ends of some of the chapters. These are meant to guide the reader to other results of similar nature and also to enable him to trace the original sources of the ideas and results through references contained in the works cited. Neither the bibliography nor the citations made to it are meant to be in any sense complete nor to indicate priority of authorship or originality of ideas or results for anyone, whether cited or not.
Table of Contents Preface
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Introduction
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5 5 6 7 9
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14 15 16
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Chapter I. Introductory Topology 1. Operations with sets . . . . . . . 2. Metric spaces . . . . . . . . . 3. Open and closed sets. Limit points . . . . . . . 4. Separability. Countable basis . . . . . . . . . 5. Compact sets 6. Diameters and distances . . . . . . 7. Superior and inferior limits. Convergence . 8. Connected sets. Well-chained sets . . . 9. Limit theorem. Applications . . . . . 10. Continua . . . . . . . . . . . 11. Irreducible continua. Reduction theorem . . . . . . 12. Locally connected sets . 13. Property S. Uniformly locally connected sets
Chapter II. Mappings 1. Continuity . . . . . . . . . . . 2. Complete spaces. Extension of transformations 3. Mapping theorems . . . . . . . . . 4. Arcwise connectedness. Accessibility . . . 5. Simple closed curves . . . . . . . .
Chapter III. Plane Topology 1. Jordan curve theorem . . . . . . . . 2. Phragmen-Brouwer theorem. Torhorst theorem 3. Plane separation theorem. Applications . . 4. Subdivisions . . . . . . . . . . .
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Chapter IV. Complex Numbers. Functions of a Complex Variable 1. The complex number system . . . . . . . . . 2. Functions of a complex variable. Limits. Continuity . . . . 3. Derivatives . . . . . . . . . . . 4. Differentiability conditions. Cauchy-Riemann equations . 5. The exponential and related functions . . . . . .
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20 22 24 26 28 29 32 34 36 41
49 51
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TABLE OF CONTENTS
xii
Chapter V. Topological Index 1. Exponential representation. Indices . . . . . . 2. Traversals of simple arcs and simple closed curves . . 3. Index invariance . . . . . . . . . . . 4. Traversals of region boundaries and region subdivisions 5. Homotopy. Index invariance . . . . . . . .
Chapter VI. Differentiable Functions 1. Index near a non-zero of the derivative . . . 2. Measure of the image of the zeros of the derivative
3. Index
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4. Lightness of differentiable functions 5. Openness of differentiable functions 6. Applications . . . . . . .
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Chapter VII. Open Mappings 1. General theorems. Property S and local connectedness 2. Extension of openness . . . . . . . . . . 3. The scattered inverse property . . . . . . . 4. Open mappings on simple cells and manifolds . . . 5. Local topological analysis . . . . . . . . . 6. The derivative function . . . . . . . . .
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Chapter VIII. Degree, zeros. Rouch6 theorem 1. Degree. Compact mappings . . . . . . . . 2. Degree and index . . . . . . . . . . . 3. Zeros and poles . . . . . . . . . . . 4. Rouch6 and Hurwitz theorems . . . . . . . 5. Reduced differentiability assumptions. Concept of a pole Chapter IX. Global Analysis 1. Action on 2-manifolds . . . . . . . . . . 2. Differentiable functions . . . . . . . . . 3. Orientability . . . . . . . . . . . . 4. Degree and index . . . . . . . . . . . Chapter X. Sequences 1. Lightness of limit . 2. Uniform openness . 3. Hurwitz theorem . 4. Quasi-open mappings
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110 110
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TOPOLOGICAL ANALYSIS
I.
Introductory Topology
1. Operations with sets. We shall have occasion to use sets of points
and sets or collections of point sets of various sorts. Capital letters will be used to designate sets and, in general, small letters A, B, C, stand for points. a e A means " a is an element of the set A" or "a is a point of A" if A is a point set. a non e A means that a is not an element of A. If A and B are seta,
A = B means that every point in the set A is also a point in the set B, and conversely every point in B is also in A.
A c B-read "A is a subset of B" or "A is contained in B"-means that every point of A is a point of B. A
B means B a A, "A contains B." A = B is equivalent to
AcBandBaA.
A + B (sum or union of A and B) means the set of all points belonging either to A or to B. In general, if [G] is a collection of sets :EG is the set of all points x such that z belongs to at least one element (or set) of the collection [G].
(intersection or product) means the set of all points belonging to both A and B. For any collection of sets [G], HG is the set of all points x such that x belongs to every set of [G].
A - B is by definition the set of all points which belong to A but not to B. If [G] is a collection of sets, any collection of sets each of which is an element of the collection [G] is called a subcollection of [G].
Real and complex numbers and their properties will be used freely. A set or collection whose elements can be put into (1-1) correspondence with a subset of the set of all positive integers will be called countable, or enumerable. If such a correspondence is established and the elements arranged in order of ascending integers, e.g., al, a2, a3, , the resulting arranged set is called a sequence. The empty or vacuous set is designated by 0. Two sets A and B are said to be disjoint if their intersection is empty, i.e., 0.
2. Metric spaces. By a metric space is meant a class of elements, or points, in which a distance function or metric is defined, i.e., to each 3
[CH". I
INTRODUCTORY TOPOLOGY
4
pair of elements x, y of S a non-negative real number p(x, y) is associated satisfying the conditions:
p(x, y) = 0 if and only if x =y,
(1) (2) (3)
p(x, y) = p(y, x) (symmetry), p(x, y) + p(y, z) > p(x, z) (triangle inequality). The following examples of metric spaces are of fundamental import-
ance. (i) The real number system R in which the distance function is defined as
x,yeR. p(x, y) = Ix - yj, (ii) Euclidean n-space R" with the ordinary distance function
P(x, y) _
(X, _' yi)2,
x = (x1, x2, ... , x"), y = (y1, y2,
, y"),
;,ytaR".
Hereafter we assume all our spaces are metric. For any set X and real positive number r, V,(X) denotes the set of all points p with p(p, x) < r for some x e X. The set V,(X) will be called the spherical neighborhood of X with radius r.
8. Open and closed sets. Limit points. A set G in a space S is said to be open provided that for each x e U there exists an r > 0 such that V,(X) C U. A set F is said to be closed provided its complement S - F
is open. A point p is said to be a limit point of a set of points X provided every
open set containing p contains at least one point of X distinct from p. For any set X, by the closure` of X is meant the set consisting of X together with all of its limit points. The following statements are easily proven and are left as exercises for the reader: (3.1) (a) If a eet is open, its complement is closed.
(b) If a Bet is closed, its complement is open. (3.2) (a) The union of any collection of open sets is open. (b) The intersection of any collection of closed sets is closed.
(3.3) (a) The intersection of any finite number of open sets is open. (b) The union of any finite number of closed sets is closed.
(3.4) A point p is a limit point of a set X if and only if for each e > 0 there exists a point x e X different from p such that p(x, p) < E. (3.5) A set F is closed if and only if it contains all of its limit point8,
i.e., P = F.
(3.8) The closure of any set whatever is closed, i.e., ,Q set X.
= I for any
§ 5]
COMPACT SETS
6
4. Separability. Countable basis. A metric space S is separable provided some countable subset P = > pti of S is dense in S in the sense
that every point of S either belongs to P or is a limit point of P, i.e., P = S. The open sets in S are said to have a countable basis provided there exists a sequence R1, R2,
-
- of open sets in S such that every open
set in S is the union of a subsequence of these sets R, in other words, if U is any open set and x s U there exists an m such that x e R. U. Such a sequence Rl, R$, . will be called a basis or a fundamental sequence of open sets in S. (4.1) Every separable metric space has a countable basis of open sets.
Proof. Let P = pr + p2 +- - - be dense in S. Then the collection [VV(p,)] for n = 1, 2, - - - and for all rational positive numbers r is a countable collection of open sets in S and it forms a basis in S. For let
p e 0 where 0 is open. Then p e V2r(p) c 0 for some rational r. There exists an n such that p e V,(p), and we have p e V,.(p,) r- V2,-(P) (-- G. (4.2) LrxDELOF THEOREM. Every collection U of open sets in a separable metric space contains a countable subcollection whose union is identical with the union of all seta in the whole collection.
be the be a basis in S. Let R.,, Proof. For let B1, R2, subsequence of all basis sets such that R,, lies in at least one element Gi of U, i.e., an R becomes an
if and only if it lies in some element of
U. Then the union of all the G is the same as the union of all 0 e U because if x e 0 a U, there is an n such that x s R 0 and hence R C 0;. is an R., so that x e 5. Compact sets. Henceforth it is assumed that all spaces used are separable and metric. A set K is compact provided every infinite subset of K has at least one limit point in K. A set M is conditionally compact provided every infinite subset of M has at least one limit point (which may or may not be in M). A set L is locally compact provided that for
each x e L there exists an. open set U containing x such that U-L is compact. If L is the whole space, this obviously is the same as saying that there exists a conditionally compact open set containing p. A real valued function f (x) defined on a set X in a metric space is upper semi-continuous (u.s.c.) at xe e X provided that for any e > 0 there exists an open set U containing x0 such that f (x) < f (xe) + E for all x e U. (6.1) Any real u.s.c. function on a compact set K i8 bounded above on K, i.e., there exists a constant M such that f(x) < M for all x e K.
For if not, there exists a sequence of distinct points x1, x2, x3, .
.
in K such that n for each n. By compactness of K, 7'x has at least one limit point xo in K. But any open set U containing xu contains an x with f(xo) + 1, contrary to u.s.c. off at x0.
INTRODUCTORY TOPOLOGY
6
[CHAP. I
(5.2) BOREL THEOREM. A Bet K is compact if and only if every collection
U of open sets covering K contain a finite subcollection also covering K.
Proof. To prove the "only if" part, we note first that by (4.2) U of elements whose contains a countable subcollection G1, 02, G3, union contains K. For each x e K, we define f (x) = n where n is the least integer such that x e G,,. Then f is u.s.c. on K because xa E G implies
f(x) S n for all x E G,,. Thus by (5.1) there is an integer m such that f(x) S m for all x e K. In other words, K e > G,,. For the "if" part, we suppose K non-compact. Then there exists an infinite set I of distinct points in K having no limit point in K. Then for each x e K there exists an open set G,, containing x but containing at most one point of the set I. If a finite number of sets G. covered K, there could
be only a finite number of points in I. Thus this covering property implies compactness of K. (5.3) (a) Every compact set is closed.
(b) Every closed conditionally compact set is compact.
(c) If X is conditionally compact, I is compact.
The proofs of these statements are simple and are left as exercises. (5.4) If K1 n K2 K3 . is a monotone decreasing sequence of non-empty compact Bets, the intersection P = H 'K. of all these sets is non-empty.
For if P is empty, the sequence of open sets S - K1, S - K2, covers K1. Thus by (5.2), for some n, K1 c S - K,,, contrary to K a K1.
6. Diameter and distanoes. As an immediate consequence of the definition we have (6.1) Any distance function p(x, y) is continuous.
That is, for any two points a and b and any e > 0 there exist neigh. borhoods U. and Ub of a and b, respectively, such that for x e Ua, y e Ub jp(a, b) - p(x, y)l < e.
(i)
To prove this we have only to take Ua and Ub so that for x e Ua, y e Ub p(a, x) < e/2, p(b, y) < E/2. This gives by the triangle inequality (ii)
p(x, a) + p(a, b) + p(b, y) < p(a, b) +
E,
p(a, b) S p(a, x) + p(x, y) + p(y, b) < p(x, y) +
e,
p(x, y) or
p(a, b) - e < p(x, y) < p(a, b) + e, which is equivalent to (1).
SUPERIOR AND INFERIOR LIMITS
§ 7]
7
DExnuTIows. By the diameter 8(N) of any set N is meant the least upper bound, finite or infinite, of the aggregate [p(x, y)] where x, y e N. By the distance p(X, Y) between the two sets X and Y is meant the greatest lower bound of the aggregate [p(x, y)] for x e X, y e Y.
Obviously 8(N) = 6(N) and p(X, Y) = p(f, $) for any sets N, X, Y.
(6.2) If N is compact there exist points x, y e N such that p(x, y) _ 8(N) < oo. be a sequence of pairs of points of N For let (x1, yl), (x=, y2), 8(N), finite or infinite. Since N is compact, such that lira p(x,,, contains a subsequence which converges to a the sequence x1, x2, x3, point x in the sense that every open set containing x contains almost all points of the subsequence, i.e., all but a finite number. We may suppose the notation adjusted so that x -+ x. Similarly the sequence [y.], after the adjustment for [xe] has been made, contains a subsequence converging
to a point y. Again we can adjust the notation so that x - x, y -r Y. Since lira p(x,,, 6(N), it results from (6.1) that p(x, y) _ 6(N) < oo. (6.3) If X and Y are disjoint compact sets, there exist points x e X and y e Y such that P(x, Y) = P(X, Y) > 0.
To prove this we choose a sequence of pairs of points [(x,,, yn)] as in (6.2) so that x a X, y a Y, x - x e X, y, -o- y e Y, and lira p(x,,, p(X, Y). The continuity of p gives p(X, Y) = p(x, y) and since
x0y,P(x,y)>0. 7. Superior and inferior limits. Convergence. Let 0 be any infinite collection of point sets, not necessarily different. The set of all points x of our space S such that every neighborhood of x contains points of infinitely many sets of G is called the superior limit or limit superior of G and is written lira sup G. The set of all points y such that every neighborhood of y contains points of all but a finite number of the sets of G is called the inferior limit or limit inferior of G and is written lira inf G.
If for a given system G, lira sup G = lira inf G, then the system
(collection, or sequence) G is said to be convergent and we write lira G
= lim sup G = lim inf G.
Under these conditions we say that G
converges to the limit lira G.
For example, let G be the collection of all positive integers. Then lim sup G = lim inf G = lim G = 0. Thus G is convergent and has a vacuous limit. Again, let G be the system of sets 1, where L. is the straight line interval joining the points [(-1)"(1 - (1/n)), 0]
INTRODUCTORY TOPOLOGY
8
[CHAP. I
and [(-1)"(1 - (1/n)), 1). Then lim sup G is the sum of the interval from (-1, 0) to (-1, 1) and the one from (1, 0) to (1, 1), lim inf G = 0, and thus lim 0 does not exist. From the definitions, we have at once for any system G lim inf G c lim sup G.
(i)
Furthermore, lim inf G and lim sup 0 are always cloned point seta. For if x is a limit point of lim inf G, then any neighborhood V of x contains a point y of lim inf G; and since V is a neighborhood also of y, then V
contains points of all save a finite number of the sets of 0 and thus x belongs to lim inf G. Similarly if x is a limit point of lim sup G, any neighborhood V of x contains a point z of lim sup G and thus V, a neighborhood of z, contains points of infinitely many of the sets of G. Therefore x belongs to lim sup G. It is a consequence also of our definitions that any system G having a vacuous lim sup is convergent and has a vacuous limit. In this case G is necessarily a countable system, and its elements may be ordered into a sequence A 1, A2, As, " ' . If G* is any infinite subcollection of a collection G, we have at once
lim inf G c lim inf G* C lim sup G* c lim sup G.
(ii)
Therefore if G is a convergent system, then every infinite subcollection G* of G is convergent and has the same limit as G. (7.1) THEOREM. Every infinite sequence of sets contains a convergent subsequence.
Proof. Let [As] be any infinite sequence of sets and let us set up the following array of sequences:
[All
A', A',
[A2] = As, [As] = A1,
A2,
As,
A',..
A3, ... As, ...
In this array, the first sequence [A;] is identical with the given sequence [A{]; and in general for each n, the sequence [Ai +1] is obtained from the sequence [A,] in the following manner. If the sequence [A"] contains any infinite subsequence whose elements occur in the same order as in
[A{ ] and whose limit superior has no point in the set R, where R is the nth set in the fundamental sequence of open sets R1, R2, B31 , then we pick out one such subsequence of [Ai ] and call it [A; +']. If, on the other hand, the limit superior of every infinite subsequence of [Ai] has a point in R, then we take for [A;+'] exactly the sequence [A?].
§ 81
CONNECTED SETS
9
We shall now prove that the diagonal sequence [A%] in the above array is convergent. Clearly it is an infinite subsequence of [Ai] = [AV], S'
since for k # j, Ak # Al. Suppose, on the contrary, that it is not convergent and thus that there exists a point x belonging to lim sup [All but not to lim inf [A']. Then there exists a neighborhood V of x, which we may take = R, for some m, and an infinite subsequence [A;-] of [An] all elements of which are contained in the complement of R.. Now the sequence [A ;] for n > m is an infinite subsequence of [AM] and therefore so also is [A*') for i > m. Thus [A; `] does contain an infinite subsequence, namely [AA], i > m, whose limit superior has no point in R.. Therefore by the method of choice of [A; +1], lim sup 0. But [AA], for n > m, is a subsequence of (A'+'] and hence lim sup [AA]-R,,, = 0, which is absurd since x belongs to both B. and lim sup [An]. Thus the supposition that the sequence [AA] is not convergent leads to a contradiction. (7.2) THEOBSM. If [Aij is a sequence of eels whose limit superior is L and the sum of whose elements is a conditionally compact set, then for
each e > 0 there exists an m such that for every n > m, A. C VE(L). Proof. Suppose this is not so. Then there exists an e > 0 and an infinite sequence of points p1. p,, p3, - such that for each i, pi belongs to A,, but not to VE(L) and such that ni 0 n, for i # j. But since >.Ai is conditionally compact, there exists a point p which either is a limit point of the set px + p2 + pe + - or is identical with pi for infinitely many i's. In either case p must belong to L = lim sup [A']. But clearly this is impossible, since VE(L) contains no one of the -
points pi. CORoou a.iY. If [Ail is convergent and has limit L and if _71 Ai is conditionally compact, then for each t there exists an m such that for every n > m, P(An, L) < e.
8. Connected gets. Well-chained sets. A set of points M is said to be connected provided that however it be expressed as the sum of two disjoint non-vacuous sets M1 and M2, at least one of these sets will contain a
limit point of the other. In other words, a set M is connected if it is not the sum of two sets A and B such that A-B = A-B = 0. Two such sets A and B are said to be mutually separated, i.e., two sets A and B are mutually separated provided they are mutually exclusive (= disjoint) and neither of them contains a limit point of the other. Any division of a set M of the form M = A -{- B where A and B are non-vacuous mutually separated sets is called a separation of M. It follows at once from the definition that the sum of any number of connected sets whose product does not vanish is connected. Also
10
INTRODUCTORY TOPOLOGY
[CHAP. I
it follows that if M is connected, so also is any set Mo such that M Mo c X. For if there were a separation Mo = A + B, M would be wholly would be a in A or wholly in B, since otherwise M = M-A +
separation of M. This is impossible, because if M C A, every point of B would be a limit point of A and similarly if M a B. It follows from this in particular that the closure of any connected set is connected.
By a component of a set M is meant a maximal connected subset of M, i.e., a connected subset of M which is not contained in any other
connected subset of M. Thus for any point a of M, the component of M containing a consists of a together with all points of M in connected subsets of M containing a. If E is any set of points, any subset F of E is called a closed subset of E and is said to be "closed in E" or "closed relative to E" provided that no point of E - F is a limit point of F. A subset O of E is called an open subset of E and is said to be "open in E" provided that no point of 0 is a limit point of E - O. Now it is seen at once that a connected set may be defined as a set M no proper subset of which is both open and closed in M.
For if M is not connected and M = A + B is a separation, A is both open and closed in M; and on the other hand, if a proper subset X of M is both open and closed in M, M = % + (M - X) is a separation of M. Also any component of a set M is closed in M. DEYn mox. If a and b are points, then by an a-chain of points joining a and b is meant a finite sequence of points.
a=x1,x2,x3,...,xn=b such that the distance between any two successive points in this sequence is less than c. A set of points M is said to be well-chained provided that
for every e > 0, any two points a and b can be joined by an c-chain of points all lying in the set M. (8.1) Law. If A is any subset of any set M and a is any positive number, the set M. of all points of M which can be joined to A by an a-chain of points of M is both open and closed in M.
To prove this lemma, set Mb = M - M0. Then MQ cannot contain a limit point x of Mb; for if so then some point z of Mb is at a distance less than a from x and if [a = xl, xz, . , xn = x] is an a-chain in M from a to x, a e A, clearly [a = x1, xE, , x,,, z] is an c-chain in M from a to z, contrary to the fact that z does not belong to M. . Thus M, is open in M.. Likewise no point z of Mb is a limit point of M,; for if so
then some point x of Ma is at a distance less than e from z; and if [a = x1, x1, , xn = x] is an a-chain in M from a to x, a e A, then [a = x1, x2, , x,,, z] is an a-chain in M from a to z, contrary to the
LIMIT THEOREM
§ 9]
11
fact that z does not belong to Ma. Thus M. is closed in M and the lemma is proved. (8.2) Every connected Bet is well-chained.
Suppose on the contrary that some connected set M contains two points a and b which, for some e > 0, cannot be joined by an a-chain of points of M. But then by the lemma the set Ma of all points which can be so joined to b is both open and closed in M, which clearly contradicts the fact that M is connected. Now it is not true, conversely, that every well-chained set is connected.
For clearly the set B of all rational points on the unit interval (0, 1) is well-chained but not connected. However, if a Bet K is compact and well-chained, it is connected, a fact which will be deduced a little later from a more general proposition.
(8.3) If N is any connected subset of a connected act M such that M - N is disconnected, then for any separation M - N = Ml + Ma, Ml + N and M2 + N are connected. For if there existed a separation Mi + N = A + B, N would have to lie wholly either in A or B, say in A, since otherwise N = N-A + N-B
would be a separation of N. This would give B a MI. Whence M = (M: + A) + B would be a separation of M, contrary to the con. nectedness of M. Similarly M. + N is connected. 9. Limit theorem. Applications. (9.1) THEOREM. If [A J is an infinite sequence of sets such that (a) 7A{ is conditionally compact, (b) for each i, any pair of points of Ai can be joined in A{ by an e{-chain and c, -- 0 with l /i, (c) lim inf [A t] # 0, then lim sup [A1] is connected.
Proof. Let L = lim sup [A{], l = lim inf [A1]. Since L is closed and contained in the compact set JA;, it follows that L is compact. Thus if
L is not connected and we have a separation L = A + B, A and B are compact disjoint sets at least one of which, say A, intersects 1. Thus by (6.2) p(A, B) = 4d > 0. This gives P[Vd(A), Vd(B)] > d.
By (7.2) there exists an integer N such that for n > N, A c Vd(L) _ Vd(A) + Vd(B); and since l-A # 0 we may suppose also A,; Vd(A) # 0.
Thus there exists an integer k > N such that ek < d, At- Vd(A) 0 0 0 AkV4(B). Clearly this is impossible, since if x is the last point in Vd(A) of an e,,-chain in At from a point of Vd(A) to a point of Vd(B) and y is the Vd(B) we have p(x, y) > d > ek. successor of x, then since y (9.11) If [A1] is a convergent sequence satisfying (a) and (b) of (9.1), Jim [A J is connected.
INTRODUCTORY TOPOLOGY
12
(OHM'.I
(9.12) If [A{] is a sequence of connected seta satisfying (a) and (c) of (9.1), lim sup [A8] is connected.
(9.2) If two points a and b of a compact set K can be joined in K by an a-chain for every e > 0, they lie together in the same component of K.
Proof. For each i, let Ai be the set of all points of K which can be joined to a by an l/i chain in K. Then since > Al a K, condition (a) of (9.1) is satisfied. Condition (b) is satisfied by choosing ei = l/i; and since lim inf [A1] a + b, (c) is satisfied. Accordingly L = lim sup [A,] is connected. Since a + b e L e K, clearly our conclusion follows.
It is of interest to note that as here defined the set L actually will be a component of K. (9.21) Every compact well-chained set is connected.
(9.22) Every interval of real numbers is connected. (Hence the real number space R is connected.) (9.3) If A and B are disjoint closed subsets of a compact set K such that no component of K intersects both A and B, there exists a separation
K = K. + Kb, where K, and Kb are disjoint compact Bets containing A and B, respectively.
Proof. We first show that there exists an e > 0 such that no a-chain in K joins a point of A to a point of B. If this is not so, then for each i there exists an l/i-chain A{ in K joining a point at of A to a point bi of B. Since, by (7.1), the sequence [A t] contains a convergent subsequence, there is no loss of generality in supposing the whole sequence converges.
Clearly if we take et = l/i, condition, (b) of (9.1) is satisfied; and since JAj C K, (a) is satisfied. Accordingly, by (9.11), L = lim [A{] is connected. But since A,A at, A,-B b{, and A and B are compact, we have L-A # 0 0 L-B, contrary to the hypothesis that no component of K intersects both A and B. Thus an e satisfying the above statement exists.
Now let K. be the set of all points of K which can be joined to some
point of A an c-chain in K and let Kb = K - K,. Then A e K B c Kb. By (8.1), K. is both open and closed in K. Accordingly K. is open and closed in K. Since K is compact, K. and Kb are compact. (9.4) If Mi Ma = M3 ... is a monotone decreasing sequence of nonvacuous, compact connected Bets, IlM1 is a non-vacuous compact connected set.
For we have only to note that under these conditions lI W Mi = lim [M{] = lim sup M1 # 0, and > M1 = M,. Accordingly our conclusion follows from (9.12).
10. Continua. A compact connected set will be called a continuum. A locally compact connected set will be called a generalized continuum. If G is an open set, the set 0 - 0 will be called the boundary or frontier
§ 10]
13
CONTINUA
of 0 and will be denoted by Fr(G). Since Fr(0) = 0-(S - 0), where S is the whole space, the boundary of every open set is closed.
(10.1) If N is a generalized continuum and 0 is an open set such that N-G is non-vacuous and different from N and N-( is compact, every component of N-c3 intersects Fr(G).
For suppose some component A of N-0 fails to intersect Fr(G). Let K = N-0, B = N-Fr(G). Since K is compact, and A and B are closed subsets of K [B is non-vacuous because otherwise we would have the where S is the entire space], separation N = N-G + we may apply (9.3) and obtain a separation K = K. + Kb where N K, A, Kb B. But then Ka c U, Kb N - K) would be a separation since K. and K. are compact and Y--K-(7 C Fr(0). This contradicts the connectedness of N.
If the non-degenerate continuum K is a subset of a set M, then K will be called a continuum of convergence of M provided there exists in M a sequence of mutually exclusive continua K1, K2, K3, - - no one of which contains a point of K and which converges to K as a limit, i.e.,
lim[Ki]=K. For example, let M = Q + 1i K{, where Q is the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) and, for each i, K. is the straight line interval from (1/i, 0) to (1/i, 1) and let K be the interval from (0, 0) to (0, 1). Then K is a continuum of convergence of M, for K = lim [K,]. It is to be noted that this continuum M is not locally connected, that is, it contains points x, e.g., the point (0, 1/2), in every neighborhood of which there are points of
M which cannot be joined to x by a connected subset of M of diameter less than some positive number given in advance. It is in connection with the study of the property of local connectedness that we find the
principal applications for the notion of continuum of convergence. Indeed we shall show presently that any non-locally connected continuum always has continua of convergence of a particular type.
DErzxrriox. A point set M is said to be locally connected at a point p of M if for every e > 0 a 6 > 0 exists such that every point x of M whose distance from p is less than 6 lies together with p in a connected subset of M of diameter less than e ; or, in other words, if for each neigh-
borhood U of p a neighborhood V of p exists such that every point of lies in the component of M-U containing p. A set M which is locally
connected at every one of its points is said to be locally connected. (10.2) THEOREM. If the generalized continuum M is not locally connected at one of its points p, then there exists a spherical neighborhood R with center p and an infinite sequence of distinct components N1, N21 N31
of M-b converging to a limit continuum N which contains p and has no point in common with any of the continua N1, N2, N31 -
.
INTRODUCTORY TOPOLOGY
14
[CHAP.I
Proof. Since M is not locally connected at p, there exists some
spherical neighborhood R with center p and radius e such that M-A is compact and for every positive b, V,(p) contains points of M which do containing p. Let x1 be such a not belong to the component C of point lying in V,,/2(p) and let Cl be the component of M-8 containing x1. Since Cl is closed and does not contain p, there exists a point xy in M. VE14(p)
which does not belong to Cl + C. Let C2 be the component of M-h containing x2. Likewise since Cl + Cz is closed and does not contain p, there exists a point x$ in M- V,,8(p) which does not belong to.C + C1 +
C.
Let C$ be the component of M-E containing x3, and so on. Continuing this process indefinitely, we obtain a sequence C1, C!, CS, - of distinct 0] comCZ + - - - + [since for each n, C x, and ponents of M- 1t whose inferior limit contains p. Now, by (7.1) the sequence [C{] contains a convergent subsequence
[C.4] with limit N which necessarily contains p. For each i, set N, = C. Now, by § 9, N is connected and hence is a continuum. Furthermore N c C, because N = p. Therefore N-Y Nf = 0. This completes the proof.
(10.3) Every set M which contains a generalized continuum which is not locally connected has a continuum of convergence.
(10.4) If the generalized continuum M is not locally connected at a point p, then there exists a subcontinuum H of M containing p and such that M is not locally connected at any point of H. For if e is the radius of R and if H denotes the component of N- V,12(p)
containing p as in (10.2), then it is clear that M is not locally connected at any point of H, and H is a continuum containing more than one point because it contains p and by (10.1) it must contain at least one point of Fr[VE12(p)l
11. Irreducible continua. Reduction theorem. A set of points H is said to be irreducible with respect to a given property P provided the set
H has property P but no non-empty closed proper subset of H has property P. A set M which is irreducible with respect to the property of being a continuum containing two points a and b (or more generally a closed set K) is called an irreducible continuum from a to b (or about K) or a continuum irreducible between a and b. This means, of course, that M is a continuum containing both a and b but no proper subcontinuum of M can contain both a and b. In this section we shall prove that an
arbitrary continuum M contains an irreducible continuum between any two points a and b of M or, indeed, about any given closed subset K of M. This will follow easily from a general theorem known as the Brouwer Reduction Theorem, which we now proceed to establish.
§ 12]
LOCALLY CONNECTED SETS
Is
A property P is said to be inducible (or inductive) provided that when each set of a monotone decreasing sequence A1, As, As, - - - of
compact sets has property P, so also does their product A = II; A,. For example, the property of being non-vacuous is inducible, as was shown in (5.4). Also the property of being connected is inducible, as was proved in (9.4). (11.1) BROVwER REDUCTION Tsxoazm. If P is an inducible property, then any non-vacuous compact set K having property P contains a non-vacuous closed subset which is irreducible with respect to property P. Proof. Let R1, R1, - - - be a fundamental sequence of neighborhoods
in the spaoe. Let n1 be the least integer such that K contains a nonvacuous closed subset A 1 having property P and not intersecting RN. Let n2 be the least integer greater than n1 such that Al contains a closed
non-vacuous subset A s having property P and not intersecting Similarly, let n3 be the least integer greater than n2 such that A. contains a closed non-vacuous subset A. having property P and not intersecting RV and so on. Continuing this process indefinitely we obtain an infinite monotone decreasing sequence of compact sets A1, A2, A3, . each having property P. Now by (5.4), if A denotes IIA then A is non-vacuous and
compact; and since P is inducible, it follows that A has property P. Furthermore, A is irreducible with respect to property P. For if some proper closed subset B of A had property P, there would exist an integer
k such that Rk A # 0 but Rk B = 0. But since
0 for all i's
we have k ,-e n, for all i's, whereas Rk B = 0 would give K = n, for some i S k by the definition of the integers n,. (Note. We have assumed throughout that K itself is not already irreducible relative to property P.) (11.2) If K is any closed subset of a continuum, M, then M contains an irreducible subcontinuum about K. To we this we have only to note that the property of being a subcontinuum of M containing K is inducible.
12. Locally connected seta. It will be recalled that a set M is locally connected provided it is locally connected at each of its points, i.e., for each p e M and each neighborhood U of p there exists a neighborhood V of p such that M V lies in a single component of M- U (see § 10). A connected open subset of a set M will be called a region in M. (12.1) A set M is locally connected if and only if each component of an arbitrary open subset of M is itself open in M. (12.2) A act M is locally connected if and only if each point of M is contained in arbitrarily small regions in M.
The proofs of these propositions result at once from the definition of local connectedness and are left as exercises.
INTRODUCTORY TOPOLOGY
16
[CsAP.I
(12.3) THEOREM. Every connected open subset (or region) of a locally connected generalized continuum is itself a locally connected generalized continuum.
For let B be a region in a locally connected generalized continuum M and let p e R. There exists a neighborhood U of p so that M 17 is Thus B is connected and locally compact and such that compact. Finally, by local connectedness of M, there exists, for any E > 0, a region Q in M with p e Q c R and b(Q) < e. Since clearly Q is also a region in B, it follows by (12.2) that R is locally connected.
13. Property S. Uniformly locally connected sets. A point set M is said to have property S provided that for each E > 0, M is the sum of a finite number of connected sets each of diameter less than E. (13.1) If M has property S, it is locally connected.
For let x be any point of M and a any positive number. Let M = M1 + M2 + + M,,, where 6(M8) < e/2. Let K be the sum of all those sets Mi which either contain x or have x for a limit point. Then clearly K is connected and b(K) < e. Thus since x is not a limit point of M - K it follows at once that M is locally connected at x. (13.2) If M has property S, so also does every ad Mo such that M c Mo C M.
+ M. For let e be any positive number and let M = M1 + M2 + where Mi is connected and of diameter less than e for 1 < i < n. Then Mo = M2 M0 + ' + M.-Mo and clearly is connected (since M, a a M,) and of diameter less than e. The two facts just established yield at once the following proposition: (13.3) If M has property S, then every set Mo such that M e Mo a . is locally connected.
DEJntmox. If N is any subset of a metric space D and e is any positive number, we shall denote by TE(N) the set of all points x of D which can be joined to N by a chain of connected subsets L1, L5, , L. of D such that for each i, b(Li) < e/21, 0 0, L,, n x, and any two successive sets (links) Li and Li+1 have at least one common point. Such a chain will be called a chain of type T. or simply type T. (13.4) THEOREM. If the metric space D has property S, then every TE(N) has property S. For let 6 be any positive number, and let us choose an integer k such that _7r a/2i < 614. Let E be the set of all points in TE(N) which can
be joined to N (i.e., to points of N) by a chain of type T which has at most k links. Let us express D as the sum of a finite number of connected sets each of diameter less than E/2k+1; and of these sets, let Q1, Q%,
.
, Q.
be the ones which contain at least one point of E. Then we have
§ 13]
PROPERTY S
17
E c 2iQi, Now for each i, Qi c TE(N); for Q; contains a point x of E, and x can be joined to N by a chain L1, L21- , L, of type T having k , L Q;] links or less; and since 8(Q;) < -E/2k+1, therefore [L1, Ls,
is a chain of type T, and hence Q; c TE(N). For each i (1 S i 5 n) let W, be the set of all points of TE(N) which can be joined to some point
of Q1 by a connected subset of TE(N) of diameter less than 8/4. Then for each i, W{ is a connected subset of TE(N) of diameter less than b. It remains only to show that TE(N) c :Ei W;. To this end let x be any , L. be a chain of type T joining x to point of TE(N) and let L1, L2, N. Obviously we need consider only the case in which x does not belong
to E, and in this case m > k. Then since Lk c E, it follows that for some j, Lx Q, # 0; and since k 'e/2i < 6/4, it follows that 8( 'L,) S 2k 8(L{) < 6/4. Hence >k L; is a connected subset of TE(N) of diameter < 814 which joins x to a point of Q5. Therefore x c Wj, and our theorem is proved. (13.41) CoBou &1tY. Any metric apace D having property S is the sum of a finite number of arbitrarily small connected subsets each having property S. Furthermore these subsets may be chosen either as open sets or as closed sets.
For let 8 be any positive number, let e = 6/3, and let D = 71!Di, where each D{ is connected and of diameter less than e. Then, for each i, TE(Di) is connected and of diameter less than 6 and clearly D -_ :E; TE(D;).
Now the sets [TE(Di)] themselves are open; and since it is true that if a
set E has property S, so does every set E. such that E C E0 c 2, it follows that the sets [TE(D,)] have property S, and of course they are closed.
From this corollary it follows that in any metric space having property S there exists a monotone decreasing fine subdivision into connected sets.
In other words, we can subdivide such a space into a finite number of connected sets each having property S and being of diameter less than 1; then we can subdivide each of these sets into a finite number of connected sets of diameter less than 1/2, and so on indefinitely. (13.42) COROLLARY. Any point p of a metric space D having property S is contained in an arbitrarily small connected open set (region) which has property S.
To see this we have only to take N = p, and then the set TE(P) is the desired region. Since TE(p) also has property S and hence is locally
connected (because any set having property S is locally connected), we have shown that p is contained in an arbitrarily small region whose closure is locally connected. (13.43) Every locally connected generalized continuum has property S locally, i.e., each p e M is contained in an arbitrarily small region in M having property S.
INTRODUCTORY TOPOLOGY
18
[CHAP. I
For let p e M and let e > 0 be chosen so that M VE(p) = K is compact. Then TE(p) has property S. This is proved by the same argument as given
for (13.4), substituting p for N and changing the first part of the third sentence to read: "By the Borel Theorem we can cover K by a finite number of regions in M each of diameter less than e/2,-+'."
DEFINrrION. A set M is said to be uniformly locally connected if for each e > 0, a 6e > 0 exists such that every two points x and y of M whose distance apart is less than 6e lie together in a connected subset of M of diameter less than E' Obviously any uniformly locally connected set is also locally connected. In case the set is compact, the converse is also true, that is: (13.5) Every compact locally connected 8etM is uniformly locaUy connected.
For if not then for some e > 0 it is true that, for every positive 1/n integer n, some two points x and y of M exist with p(x,,, but which he together in no connected subset of M of diameter less contains a convergent than e. Since M is compact, the sequence subsequence [xn,] with limit point p in M. Clearly the sequence y,) < 1/ne < 1/i. But since M is locally also converges to p, since connected at p, there exists a 6 such that VE(p) lies in a region R of diameter less than e. And for n{ sufficiently large, R, contrary to*the definition of x,, and Y,,,' We proceed now to show that the property of being uniformly locally
connected is stronger for conditionally compact sets than property S. In the first place it is seen at once that, for example, if C is a circle and p is a point of C, then the set C - p has property S but is not uniformly locally connected. Thus there exist sets having property S but which are not uniformly locally connected. (13.6) Every conditionally compact and uniformly locally connected set M has property S. To prove this, let t be any positive number, let 6 be a number greater
than 0 such that every two points x and y with p(x, y) < 6 lie together in a connected subset of M of diameter less than e/3, and let P = pl +
ps + -
be a countable set of points dense in M (i.e., such that P
M).
For each n, let R. be the set of all points of M which lie together with p in a connected subset of M of diameter less than E/3. Then for each n, R. is connected and
e. We now show that for some k, M = :E1 R,,.
If this is not so, then an infinite sequence [p,, J of the points of P exist such that for each i, p , is not contained in I i'-1 R,,. Since M is con. ditionally compact, [p.,] has a limit point p. But then some two points, say and r), are such that p,.) < 6 and hence p,, c
c
i'-1R,,, contrary to the definition of the sequence
for some k, M = 2IR,,, and hence M has property S.
Therefore,
§ 13]
PROPERTY S
19
The two propositions just established yield at once the following characterization of locally connected continua: (13.7) In order that a continuum M should be locally connected it is necessary and sufficient that M should have property S. The condition is necessary, because by (13.5) every locally connected
continuum is uniformly locally connected and hence, by (13.6), has property S. It is sufficient, because by (13.1) any set having property S is locally connected. In concluding this section we note
(13.8) The spaces R"(n > 0), (see § 2) are connected and uniformly locally connected.
This results at once from the facts that these spaces are "convex" and that an interval is connected by (9.22). To exhibit the convexity we have only to note that if x = (xl, x2, ' .. , x"), y = (Vi, yt, ' ' ' , y")
are two points of R", every point of the interval [x, + A(y, - xi)], (0 < A S 1), belongs to the space. Since any such interval is obviously equivalent to the real number interval 0 S A < 1, our result follows.
H. Mappings 1. Continuity. If A and B are sets, any law which assigns to each point x e A a unique pointf(x) e B is called a (single-valued) tran8formation
of A into B. If, in addition, for each point y of B there is at least one
x E A such that f (x) = y, f is said to be a transformation of A onto B and we indicate this by writing f(A) = B. Further, if for each y e B there is one and only one x e A with f (z) = y, f is said to be one-to-one, written 1-1 or (1-1). In this case if for each y e B we set f-1(y) = x we have a single-valued "inverse" f-1 for f.
Let f (x) be a transformation of A into B. For any subset A 1 of A, f(A1) denotes the set of all points y e B such that for some x e A1, f(x) = y and is called the image or the transform of Al under f. For any subset B1 of B, f-1,(B1) denotes the set of all x e A such that f(x) e B1 and is called the inverse of B1 under f.
A transformation f of A into B is said to be continuous at a point x e A provided any one of the following four equivalent conditions is satisfied: (i) For any neighborhood U off (x) there exists a neighborhood V of x
such that f(A-V) c U. (ii) For any e > 0 a 6 > 0 exists such that if p e A and p(p, x) < then p(f(p), f(x)) < e. (iii) If (xt) is any sequence of points in A with x; -+ x, then f (x;)
(iv) If x is a limit point of a subset M of A, f(x) is either a point or a limit point of f(M), i.e., if x e R, f(x) e f(M). The proof that these conditions are equivalent is left as an exercise. If f (x) is continuous at all points of A 1, A 1 A, it is said to be continuous or A1. That f (z) is continuous on the whole set A on which it is defined is indicated merely by the statement that "f is continuous".
If for the (1-1) transformation f(A) = B, both f and its inverse f-1 are continuous, f is called a topological transformation or a homeomorphism.
If such a transformation exists for two sets A and B, these sets are said to be homeontorphic.
(1.1) The image under any continuous transformation of a Bet ie {wed}. Let A be compact and let f(A) = B be a mapping. If [G] is any open 20
CONTINUITY
§ 1]
21
covering of B, then [f-1(G)] is an open [see (1.3) below] covering of A and
hence reduces to a finite covering, say A c f-'(G1) + f-1(G2) + + + G. so that B is compact. f-l(G,,). This gives B -- G1 + G2 +
Now if B is not connected, B = B1 + B2 where B1 and B. are disjoint and open. This gives the separation A = f-'(B1) + f-1(Bs) where f-'(B1) and f-1(B2) are disjoint and open so that A is likewise disconnected.
DErzxinox. A transformation f is said to be uniformly continuous on
a set A provided that for any e > 0 a b > 0 exists such that if x and y are any two points of A with p(x, y) < b, p(f(x), f(y)) < e. (1.2) Any transformation f which is continuous on a compact set A is uniformly continuous on A.
Suppose that f is continuous on A but not uniformly continuous. and in A such that, for each n, p(x,,, 1/n, but p[ f(x ), f(y )] > e; and Then for some e > 0 there exist two sequences of points
since A is compact we can suppose them so chosen that x e A. But then also x and thus by continuity [ f so that, for n sufficiently large, to the definition of and [y ].
x where f (x) and
e, contrary
(1.3) In order that a transformation f(A) = B be continuous it is necessary and sufficient that for every {=,,d} subset K of B, f-1(K) be in A.
To prove the necessity, let K be any closed subset of B and let x be any limit point of f-1(K). Then since by continuity, f(x) is a point or a limit point of K and K is closed, we have f(x) e K so that x e f-1(K). Thus f-1(K) is closed.
To show the sufficiency, let us suppose the condition is satisfied but that f is not continuous at some point z e A. Then there exists a sequence [x{] - x in A and some neighborhood Y off (x) in B such that for infinitely
many ni's, e (B - V). But B - V is closed in B whereas .f--'(B - V) a :Ez,, and thus f-1(B - V) is not closed since x is a limit point of it. This contradiction proves f continuous. (1.31) If A, B and C are sets and f1(A) = B and f2(B) = C are continuous transformations, the transformation f(A) = fsf1(A) =f2U1(A)] _ C is continuous.
For let K be any closed subset of C. Then f2 1(K) is closed in B since fs is continuous, and fj 1[f2 1(K)] = f-1(K) is closed in A since fl is continuous. Hence f is continuous. (1.4) If A is compact, f is continuous and (1-1) and f (A) = B, then f -I is continuous and thus f is a homeomorphism sending A into B.
Suppose, on the contrary, that ff1 is not continuous at some point y e B, where y = f(x). Then there exists in B a sequence [yt] - y and a
MAPPINGS
22
[Caer. II
neighborhood U of x in A such that if y, = f(x{), then infinitely many of
the points [x"t] of [x{] are in A - U. But since A is compact, the sequence [x",] has at least one limit point z and since z r- A - U, x; thus f(z) 0 y, whereas [f(x"t)] --> y. This contradicts the continuity of f, and thus our theorem follows. Alternate proof. We have f-1(B) = A. Let K be any closed set in A. Then since f is (1-1) we have z
(f-1)-1(K) = f (K)
and f(K) is compact by (1.1). Thus (f-1)-1(K) is closed and hence by (1.3) f-1 is continuous. (1.5) If A has property S and f (A) = B is uniformly continuous, B has property S.
For let e > 0 be given. By uniform continuity of f there exists a a > 0 such that any set in A of diameter less than a maps into a set in B of diameter less than e. Thus if A = JA, where A{ is connected and 8(A{) < 8, we have B = i f(Aj) where each set f(A{) is connected, by (1.1), and of diameter less than e. (1.51) The image of a locally connected continuum under any continuous transformation is itself a locally connected continuum.
For (1.2) gives uniform continuity in this situation. Hence this result follows from (1.5) and [I, (13.7)].
2. Complete spade, Extension of trandormations. A sequence of points x1, x=, -
is called a fundamental sequence or a Cauchy sequence
provided that for any e > 0 an integer N exists such that if m, n > N, P(xm, <'6-
A metric space or set D is said to be complete provided every funda-
mental sequence in D converges to a point in D. A space or set is topologically complete provided it is homeomorphic with some complete space. We have immediately (2.1) Every compact set is complete. Every closed subset of a complete set is complete.
Since the real number system R is complete, it results at once that is a for every n > 0, the Euclidean n-space R" is complete. For if p1, p81 fundamental sequence in R", pi = (xi, x2, . , x;,), the sequences (x1), , (x") are fundamental sequences in R. Hence they converge to (x2), limits x1, x2, , x", respectively, in R; and accordingly the sequence p1, pa, - - - converges to the point p = (x1, x2, , x") in R". DEFnNTTON. For any metric space or set X, by the complete enclosure
% of $ will be meant the space whose points are the fundamental
COMPLETE SPACES
§ 21
23
sequences in X, any point x e X being identified with the sequence (x, x, x, ) e X and two fundamental sequences (x1, x2, ) and
point
(yi, ya, ' ' being the same or equivalent and determine the same provided (x1, y1, x2, ys, ) is a fundamental sequence, and with the distance function P(x, y)
where x=
= lim p(x.,, y,.), y=(y1,Y2,...)eX.
(x1,x2,...) eX,
It results at once that I is a metric space and that X is embedded in X ieornetrically, i.e., if x, y e X, then p(x, y) = p(xl, y1) where xi = (x x x ) a X, y' = (y, y, y, .. ') a X. Furthermore, X is dense in X, that is, any point of is a point or a limit point of X or, in the space k, we have l = X. For clearly if x = (x1, x2, ) e X where xi a X, we have x = Jim x{ where x{ = (xa xt, ' ' -) e k, because p(x, x;) = lim, p(xn, xj) and for n and i sufficiently large p(x,,, x,) is arbitrarily small by the Cauchy condition. Finally, the apace X is complete. For let x1, x2, x3,
be a funda-
mental sequence in I where x = (x" j), x7 e X and where 6(1x) < 11n. Then the sequence (x*) is a fundamental sequence. For given e > 0, an N exists so that for in, n > N, p(xm, xA) = lim p(xk, x7) < e/3. k-. co
Thus if N is chosen also so that 1/N < E/3, then for m, n > N we have p(xk, xk) < E/3 for a definite k sufficiently large. Whence p( m, S p(xm, xk) + p(xk , xk) + p(xk, 4) < E/3 + e/3 + e/3 = E.
Accordingly p = (xn) is a point of X. Furthermore, since p(xk, p) = lim,,,,.,,, p(xn, xn) < e, we have xk --, p so that X is complete. Thus we have proved (2.2) THEOREM. Any metric space X can be isometrically imbedded in a complete space % (called the complete enclosure of X) in which X is dense.
(2.21) A metric space X is complete if and only if X = X.
We next prove an extension theorem for uniformly continuous transformations. (2.3) If the transformation f is defined and uniformly continuous on set D and if f(D) is a subset of a complete space C, the definition off may be extended to all limit points of D in one and only one way so that the extended transformation is uniformly continuous on 1) and is identical withf on D. with Proof. For each x e 1) let us select from D some sequence x -x, agreeing that if x e D, is simply the sequence x, x, x,
.
24
MAPPINGS
[CHAP. II
Then, by uniform continuity, for any e > 0 an integer N exists so that for m, n > N, p[f(xm), f(x,)] < e. Accordingly, [f(xj)] is a fundamental sequence, and since C is complete, [ f (xr)] converges to some point of C which we define to bef(x). is any other sequence in D with y -* x, then since the Now if also converges to x, it follows in the same way sequence x1, y1, x2, y2, converges to f(x) so that that the sequence f(x1), f(yl), f(x5), f(y1), surely f (y.) -f(2:)- Hence &) is independent of the particular sequence (xn)
Now to prove uniform continuity off on 1S, let e > 0 and let 6 > 0 be given by the uniform continuity off on D. Let x, y e 1) with p(x, y) < 6, and let (x1) and (y{) be the sequences selected above for x and y, respectively. Since x{ -. x, yj --* y, for n sufficiently large we have p(x,,, 6 and hence p[f(x ), fly.)] < e. Accordingly f is uniquely determined and uniformly continuous on 13. (The uniqueness results at once from continuity, since D is dense in D.) 3. Mapping theorems. A continuous transformation will be called a mapping. Whenever the term mapping is used, it is understood that continuity is assumed. (3.1) THEOREM. If E ie any locally connected continuum, there exiat8 a mapping of the interval onto E.
Proof. Let E be expressed as the union of a finite collection 0 of locally connected continua each of diameter <1 and let a and b be two distinct points of E. The elements of G may be arranged into a chain
C:Ei,E2,...En so that a e Ei, b e E,1, and 0 0 for i = 1, 2, n - 1. For in the first place a and b can be joined by a chain C' of sets of G such that
successive links intersect. To see this we have only to note that the union F of all points b' e E which are so joinable to a is the union of a subcollection of G; and since then F would intersect some additional element H of G if there were any not in F, by connectedness of E, whereas
H would have to be in F since its points clearly are chainable to a, it follows that F = E. Thus there is a chain C' from a to b. If C' includes k links of G and k < n, some link H of G not in C' intersects a link Ef of C'; and if we replace El in C' by the chain Ell, If, E,', clearly we get a chain from a to b including k + 1 sets of G. By repetition of this process we obtain a chain C from a to b including all sets of G as links. Since a link of C may be repeated any desired finite number of times, we may suppose the number of links n in C is a power of 2, say n = 2'1.
MAPPING THEOREMS
§ 3]
25
Then we may select a chain of points
a=4xl,..x2°1=b where
xi z E:'E:+1
Next for each i = 1, 2,
for i = 1, 2, ..., (2°1 - 1).
, 2°1 we express Es as the union of a finite
collection of locally connected continua each of diameter <j and arrange
these in a chain from x,'_1 to xi with successive links intersecting as before. Further we may suppose the number of links in each of these chains is a power of 2 and indeed the same power of 2, say 2°" for all I so that our chain has the form i 2"+1 + 1 2°s+ 42"',
,
Then if we choose a point in the intersection of each pair of successive links we obtain a chain of points !1
4-1 = (i-1)2°s' 22(i-1)2°s+L, xi 2°s = xi Next we express the sets E;, i = 1, 2, , 2°1+°l, as the union of a finite collection of locally connected continua each of diameter <1/22, arrange these in a chain of 2°+ links from xi_1 to xj2 and select the points xi for (i - 1) 2°s S j S as before. Continue this process indefinitely. For each k we thus have a chain of points
a = 4, xi, ... ,
x2vl
+ t2 ... + Vk = b.
Now let t be any dyadic rational number on I = (0, 1) and define f(t) = xk where t = i/2v1+ .. ' +°t. Then since the points zk are so chosen
that x;-1, it follows that f(t) is independent of the particular representation of t and thus f is single valued. Further, f is uniformly continuous on the set D of dyadic rationals on I. For if E > 0 let k be chosen so that 1/2k < E/2 and let 6 = 1/2°1+ +°r = 1/2°. Then if t1, t2 e D and It1- t2I < 6, t1 and t2 lie between successive values (j - 1)/29, j/2", (j + 1)/2° of tin D. Let us suppose t1 lies between the first two of these values. Then xj_1, xf e E and by the method of selection of the points xi and definition of f, we have f(t1) e Similarly f(t2) e Ef+l if t2 is between j/2° and j + 1/2°. In any case f(t1) and f(t2) both lie in EE + Ei+1 and as this latter set is of diameter <E, it follows that f is uniformly continuous on D. Finally to obtain a mapping of I onto E we have only to extend f to I. Then f(I) = E since f(D) is dense in E. DxazxrrioN. A mapping f(A) = B is said to be monotone provided that each y e B, f-1(y) is a continuum.
MAPPINGS
26
[CH". H
(3.2) THEOREM. Any non-degenerate monotone image of an interval is homeomorphic with the interval.
Proof. Let f (J) = E be monotone, where J = (0, 1) and E is non. degenerate. Let 1(1/2) be the closure of the interval J - f-lf(0) - f-'f(1) and let x(1/2) be its mid point. Similarly let 1(1/22) and I(3/22) be the closures of the left and right intervals remaining on the deletion of f-lf[x(1/2)] from 1(1/2) and let x(1/22) and x(3/22) be their respective mid points. Likewise 1(1/23), 1(3/23), I(5/23), 1(7/23), are the closures of the intervals into which 1(1/22) and 1(3/22) are divided by removing
f-lf[x(1/22)] and f-If [z(3/22)] ordered from left to right, and soon indefinitely. In this way we define a collection of intervals I (m/2") and their mid points x(m/2") for all dyadic rational numbers m/2", 0s m S 2", so that the length of I(m/2") is <' 1/2"-1. Now for any dyadic rational m/2" on J we define h(m/2") = f[x(in/2")]. We next show that h is uniformly continuous. Let e > 0 be given. By uniform continuity off there exists a b > 0 such that for any interval H
in J of length <6, f(H) is of diameter <E/2. Let n be chosen so that 1/2"-1 < 6. Then if tl and t2 are points of the set D of dyadic rationals with It, - t2I < 1/2", there is at least one point t = j/2" such that for each i (i = 1, 2), t is an end point of an interval Ti of the nth dyadic subdivision of J containing t;. If S1 and S2 are the corresponding intervals to T1 and T. in the set I(m/2"}1), since each is of length <6 we have
h(ti) + h(t) c f(Sj), i= 1, 2, and 6[ f(S1) +f(S2)] < E since 6[f(Si)] < E/2, i = 1, 2. Accordingly, p[h(t1), h(t2)] < E and h is uniformly continuous on D.
Let h be extended continuously to 1) = J. Then h(J) = f(J) = E, because for each n the union of the intervals I (m/2") maps onto E under f, so that the images of the mid points of all these intervals for all n, i.e.,
the set h(D), is dense in E. Finally, h is (1-1). For if xl and x2 are distinct points of J, there exists a point t = j/2" between x1 and x2 with h(t) 0 h(xl). Then if H1 and H. are the closed intervals into which J is divided by f--1h(t) where xl e H1, f(H1) f(H2) = h(t) by monotoneity off and thus h(t) since h(Hi) f(Hi) by definition of h. Accordingly h(x1) non e h(HZ) so that h(x1) # h(x2). Thus h(J) = E is a homeomorphism.
4. Arewise oonneetedness. Accessibility. A set T homeomorphic with a straight line interval is called a simple arc. If a and b are the points of such an are T which correspond to the end points of the interval under the homeomorphism, then a and b are called the end points of the simple are T and T is said to join a and b. The are T is written ab, and
the set T - (a + b) is written ab or (ab).
ARC WISE CONNECTEDNESS
§ 41
27
A boundary point p of a region R is said to be accessible from Rprovided that for any q e R, R + p contains a simple are pq joining p and q.
(4.1) THEOREM. Every two points a and b of a locally connected continuum M can be joined in M by a simple are.
Proof. Let f(I) = M be a continuous mapping of the unit interval I onto M. Let us say that a closed subset F of I has property P provided f(F) a + b and if (xy) is any maximal segment of I - F, then f(x) _ fly)-
Then property P is inducible. For let F, be closed and have property
. Let F = n F, and let (xy) be a maximal P and F1 D Fp F3 open segment of I - F. Then there exists a sequence (x1y1), (x2yq). such that (xy) = of maximal segments complementary to F1, F2, I(x,y,), x, -. x, y, -, y. Since f(x,) = f(y,), clearly this gives f(x) = f(y). Thus by the Brouwer Reduction Theorem, since I has property P, there exists a closed subset A of I which has property P irreducibly. We shall show that f(A) = T is a simple arc in M from a to b.
To do this we note first that if f (x) = f (y) for two distinct points xy) would x and y of A, then xyA = x + y because clearly A have property P. Thus if we define h(t) = f(t) for t e A and h(t) = f(x) = fly) for t e [complementary segment zy to A in 1], we obtain a mapping of I onto T which is monotone because for each p e T, h-1(p) is either a single point of A or a closed interval xy with xyA = x + y. Accordingly T is a simple are from a to b by (3.2). (4.11) CoxouY. Every locally connected generalized continuum L is arcwise connected.
For if a, b e L, there exists a locally connected continuum M in L containing both a and b. To see this we let M. be the set of all points of L
which lie together with a in a locally connected continuum in L. Then Ma is open by 1, (13.43), since each point y of L is interior to a locally connected continuum K. in L. Also Ma is closed since if y e Ma, K contains a point x of M. and K + [a locally connected continuum in L joining x and a] is such a continuum joining y and a. (4.2) THEOREM. In order that a continuum T be a simple arc from a to b where a, b e T it is necessary and sufficient that every point of T - a - b separate a and b in T.
Proof. The necessity is immediate since any simple are is homeomorphic with an interval. To prove the sufficiency we show first that T is locally connected. If not then T contains an infinite sequence of disjoint continua K1, K2, K3, converging to a non-degenerate con. tinuum K where 0 for all n. But then let x, y and z be distinct points on K, each distinct from a and from b. One of these three points must separate the remaining two in T. For let T = Ta(x) + Tb(x) be a
28
MAPPINGS
(Cawr. II
x. Then if both y and z division of T into continua where lie in a single one of these continua, say in Tb(x), let T = Ta(y) + Tb(y) be a similar division of T. Then if z e Tb(y), y separates x and z because x e Ta(x) c Ta(y); and if z E Ta(y), z separates x and y because if T = Ta(t) + Tb(z) we have x e Ta(x) c Ta(z), y e T5(y) a Tb(z). Hence one of these
points, say z, separates the other two, x and y, in T. But then since x + y e K, for all n sufficiently large we have K.-T6(z) # 0 0 K,, Tb(z) so that z e K,,, contrary to the fact that the K. are disjoint. Thus T is locally connected. Then a and b can be joined by a simple arc N in T; and clearly N = T since no point of T - N could separate a and b in T. (4.3) THEOREM. If a region R in a locally connected generalized continuum has property S, every boundary point p of R is aceesaible from R.
Proof. Since for any e > 0 any set having property S is the union of a finite number of connected open sets having property S and being of diameter < e, there is a region Rl in R of diameter < 1 having property S and having p on its boundary. Likewise in RI there is a region R2 of diameter <1/2 having property S and having p on its boundary; and in R. there is a region Rs of diameter <1/3, having property S and having p on its boundary and so on indefinitely. For each n let p,, be a point of R and let T,, be a simple are in R. from p,, to pi+1. Then if T = p -=0 i T,,, T is a locally connected continuum in R + p, because and .fin p + A Ti. Thus T, and hence also R + p, contains a simple are joining pi and p. 5. Simple closed curves. A set J is called a simple closed curve provided
J is homeomorphic with the set consisting of the points on a circle. It results at once that a set J is a simple closed curve if and only if J is the union of two simple arcs axb and ayb having just their end points a and b in common. Further, if J is any simple closed curve, then for any two distinct points a and f on J, J is the union of two simple arcs a x f and a y f intersecting in just a + P. The proofs for these statements are simple and are left as exercises.
III. Plane Topology 1. Jordan curve theorem. Use will be made of the geometry results that every simple polygon P in the plane separates the plane into just two regions and is the boundary of each of these regions; and also that if P meets a segment ab in a single interior point o of ab which is not a vertex of P, then ab crosses P so that P separates a and b in the plane. In this chapter all sets considered are assumed to lie in a plane ir. (1.1) No simple arc separates the plane.
For if an are 0 separates the plane 7r, then if Rl is a component of it - a14, Q is a component of 1r - (Rl + a$l), where al and fl, are the first and last points of Fr(R1) on a#, and B is the component of ?r - (Q + ab) containing B1, where a and b are the first and last points of Fr(Q) on al/l, then ab separates a, and Q and R and components of
it - ab. Let Ca and Cb be disjoint circles with disjoint interiors centered at a and b respectively and let x and y be points on C. and C. respectively
so that ax - x and by - y lie within C. and Ca respectively. Let Ca and Cb be circles with centers a and b respectively within Ca and C. and so that Ca'-a xb = 0. Choosing b as origin, and using polar coordinates, let T (O) be the triangle with vertices (r, 0), (r, 0 + 27r/3), and (r, 0 - 27r/3) where r is the radius of C. Let Cb be a circle, with center b, small enough that it is within T(0) for every 0; let z be the first point of ba on Cb in the order b, a and let Jb be a circle with center b which is within Cb and such that a b and Q, there exist broken line segments albs and a2b2 in R and Q respectively such that sib{ Ca = ai, bi, i = 1, 2. Then albs + chord ala2 of C. -r a2bz + chord b2b1 of Jb = P is a simple polygon not intersecting xz of ab. Now for a suitable choice of 0 (indeed for all save a finite number of 0's) we have that T(0) = T is a triangle no vertex of which is on P and which itself contains no vertex of P. Let W denote the region obtained by adding
to the interior of T the complementary region I to P which contains xz and let C denote its boundary. Then C is a graph made up of a finite number of vertices and non-intersecting edges. Any vertex of C which is a vertex of either P or T is not on the other one of these sets and hence 29
PLANE TOPOLOGY
30
[CHAP. III
is incident to exactly two edges of C; and any vertex v of C which is a vertex of neither P nor T is on an edge e of P and edge t of T. Thus we have
exactly four segments of T + P meeting in v, and of these one is within T and one within I so that exactly two are on C. Thus each vertex of C is on exactly two edges of C. Hence if G denotes the graph obtained by deleting the open segment ala$ from C, G has exactly two odd vertices a1 and a2 and therefore these lie in the same component L of G. (Note: the number of odd vertices in any finite graph G must be even. Whence, the number of odd vertices on any component L of G must be even, as L itself is a graph. Accordingly if 0 has exactly two odd vertices, they lie in the same component of G.) Since xb C W we have L c albs + T + asbz cz T + R + Q we have Lax = 0. Accordingly L-ab = 0, contrary to the fact that a1 e R and a.. E Q and L joins a1 and a2. This contradiction establishes our theorem. (1.11) CoRou Y. If Q is any component of the complement of a simple closed curve J in the plane, then Fr(Q) = J.
For otherwise some simple are ab on J contains Fr(Q) and thus separates the plane.
DEFnnrr1ON. Any simple closed curve containing a straight line segment will be called a semi-polygon. (1.2) AuxuaeaY THEOREM. Any semi-polygon separates the plane 7T into exactly two regions.
Proof. Let S be any semi-polygon containing a straight segment a circle a sob of a line L and an arc axb with with center o which neither contains nor encloses any point of axb, and which therefore meets S in only two points u and v of aob. Let CI and C2 be the two arc segments of C - (a + b). Then no broken line in ir - S joins points of both C1 and C.. For suppose such a line K joins a point p1 of C1 and a point P2 of C. and lies otherwise without C. Then K + the line segment P1P2 is a polygon P meeting S in only one point,
and that point is within C and hence on sob. But P separates a and b and hence must meet axb. Therefore rr - S is not connected. Now by (1.2), S must be the boundary of each component of A - S. Thus each component of rr - S must meet C - (a + b) and hence must contain either C1 or C2. Therefore there are just two such components. (1.3) JORDAN CURVE THEOREM. Every simple closed curve J divides the plane rr into just two regions and is the boundary of each of these regions.
Proof. There exists a straight line segment ab meeting J in exactly the two points a and b. Let axlb and axzb be the two arcs of J from a to b, and denote by Sl and S2, respectively, the semi-polygons ab + ax1b
and ab + axtb. By (1.3), it - S1 and ,r - Sz are composed of two regions each, say B1 + D1 and R2 + D2, respectively, where D1
ax=b
JORDAN CURVE THEOREM
§ 1]
31
and D2 ax1b. Now since IT - (S1 + S2) = R1 + R= + D,-D., we and we thus have a division of have it - J = (R1 + B2 + ab) + 0, because Dl x2 and rr - J into two non-vacuous sets. x2 is a boundary point of D2). These two sets are also mutually separated.
For D,Da is open, as is also R1 + R2; and no point of ab is a limit point of
For if so then there exists a straight segment po where
o e ab and po - o c
and if C is a circle with center o and
not enclosing any point of p + J, then C meets ab in two points u and v and each of the three sets B1, R2, and
must contain components
of C - (u + v); but there are only two such components and hence we have (B1 + R2)-(D1-D2) # 0, which is impossible. Therefore D,-D. and R1 + R2 + ab are mutually separated, and thus 7r - J is not connected. Finally, DI-D2 must be connected. For by (1.11) any component of D1D2 has all of J for its boundary. Let G be one component of
Then there exists a broken line L = a1b1 such that L -- (a1 + b1) c G, and so that a1 and b1 belong to axlb of J and 0. Then L + a1a + ab + bbl = S is a semi-polygon lying in G + J + ab. If H and K are the two components of ,r - S, then because S contains points of both G and ab. Thus if 00 there were a second component G1 of we would have 0. Accordingly is connected and our theorem is proven. DzFIxrrioR. By a 0-curve will be meant a continuum which is the sum of three simple arcs axb, ayb, azb intersecting by pairs in just their end points. As a direct consequence of (1.3) we get (1.4) A 0-curve separates the plane 7r into just three regions.
Proof. Let 0 = axb + ayb + azb, axb + ayb - J1, axb + azb = J2, and ayb + azb = J3. For each i (i = 1, 2, 3) let R; be the complementary region of J1 which does not contain the segment 0 - J1. Then rr - 0 =
Rl + R2 + R3. For if, on the contrary, D,-D2-D30- 0, where D1 = or - J1 - B1, then any component D of D has interior points of all three area of 0 on its boundary. For any point q in D can be joined to x, say, by an are qx a D3; and if r is the first point of axb on qx in the order q, x, we have qr - r c D so that r e Fr(D). Now D + x' + y'
contains an are x'y', and R1 + x" + y" contains an arc z"y" where x' x" a axb Y, ' y"e ayb. Let J= x'y' + x'x" + x"y" + y"y and ,r J = U + V where U z> a. Then V must contain a point p of J1 since it contains points of both R1 and IT - R1. Say p c axb. But then since p + a c axb c J2 = F(R2), J must contain a point of R2, contrary to the fact that J c D + 0 + R1. Note. If axb, ayb, azb are the edges and R1, R2, R3 the regions of
PLANE TOPOLOGY
32
[CHAP. III
a 0-curve 0, and if we understand that "an edge is on a region" means it is on the boundary of that region and "a region is on an edge" means it has the edge as a part of its boundary we get at once the following dual conclusions: (1.5) I. Each edge of 0 is on exactly two regions. I'. Each region of 0 is on exactly two edges.
II. Each pair of edges of 0 is on exactly one region. II'. Each pair of regions of 0 is on exactly one edge.
2. Phragm8n-Bronwer Theorem. Torhorst Theorem. It is understood
throughout this section that all sets considered lie in a plane which is usually denoted by ir. If K is a closed set, by a complementary domain
of K is meant a component of the complement of K, i.e., of a
- K.
(2.1) PEmAOMfN-BxovwER THEOREM. The boundary of every bounded
complementary domain R of a closed generalized continuum N is itself a continuum.
Proof. For el < 1 take a hexagonal division D, of the plane of norm less than es.' Let H1 be the sum of all the subdivisions in D1 lying wholly in R. Let Kl be any component of Hl. Assuming that has been constructed, let e,, = p(K,,_1, Fr(R)), and construct a hexagonal divison D. of the plane of norm less than min (E,,, 1/n). Let H. be the sum of all subdivisions of D which lie wholly in R, and let K be the component of H,, containing
for all n, and let F =
Make this construction
F is a simple polygon. To show this we note first
that since each vertex e of D on F is on three hexagons of D. either one or two of which are in K,,, then in any case just two edges of F meet at e. Hence if al and a2 are the end points of an edge iala2 of F,,, then a1 and a2 are joined in the graph G = F. - alas by a broken line L, as they are in the same component of G, and alai + L is a simple polygon P in F,,. As P e R and B is bounded, N is wholly in the exterior
of P. Thus P together with its interior I lies in K,,. If K,, # P + I then some hexagon of D in K,, would be exterior to P but contain an edge e of P; and this is impossible as then e would be interior to K,, and thus not on F,,. Accordingly K = P + I so that F = P. But, by construction, for a sufficiently large n we have Hk K,,. Thus Fr(R) = lim F,,, so that Fr(R) is a continuum, as was to be proved. 1 That is, the plane is represented as the union of a sequence of congruent regular hexagons plus their interiors each of diameter <e,, so that the intersection of any two of these is either empty or a common side of both.
§ 2]
PHRAGMIEN-BROUWER THEOREM
33
(2.11) Coaouay. Any compact set which is the common boundary of two domains is a continuum.
Since one of the domains D1 and D2, say D2, is bounded, we have only to note that D2 is a complementary domain of Dl. (2.12) If the boundary F of a complementary domain of a generalized continuum is bounded, F i8 a continuum. (2.2) TORHORST THEOREM. The boundary F of every complementary domain R of a locally connected continuum M is itself a locally connected continuum.
Proof. For if F is not locally connected, then it contains a continuum of convergence K = lim (K ), where Km K = 0, m n, and for some 8e for every n. Now since M is locally connected e > 0 we have and thus has property S, we have M = >i M,, where each M, is a locally connected continuum of diameter less than e. Now each K,, must meet each set of some pair A, B of sets (M,) which have no common points (A-B = 0). Since there are only a finite number of disjoint pairs A, B in (M,), it follows that for some such pair A, B of sets M; there exist three
of the sets (K,,), say K1, K2, K3, each of which meets both A and B. Now take or > 0 such that 8a < min [p(K1, K2), p(K2, K3), p(K3, K1)] and consider the regions Ra(K,) = I R, (x), x e K{ in M (i = 1, 2, 3) [R0(x) is the component of Va(x,)-M containing x]. These clearly are disjoint and contain arcs a,b,, respectively, such that a;b,-A = a,, a,b, B =
bi (i = 1, 2, 3). The set A contains an are a1a2 and an arc a3a where a3a-a1a2 = a (a may be a3) and similarly B contains an are b1b2 and an arc bib where bib-bibs = b. Then albs + albs + aab3 + a1a2 + a3a + blb2 + b3a
= ax1b + ax2b + ax3b
= 0, a 0-curve,
where x; is on a,b, and p(x A) > or and p(x B) > a (i = 1, 2, 3). But since x, lies in Ra(K,) it can therefore be joined to some point of K, by a region of diameter less than a and hence not meeting 0 - ax,b. But then if rr - 0 = R1 + R2 + R3, where R3 B, we have that Fr(R3) meets all three edges of 0, which is impossible. (2.21) If the boundary B of a complementary domain of a locally connected generalized continuum is bounded, B i8 a locally connected continuum.
DEFINITIONS. A point p of a connected set M is cut point of M provided M - p is not connected. A connected set having no cut point is said to be cyclic. A locally connected continuum containing no simple closed curve is called an acyclic curve. (2.3) LEMMA. No acyclic curve separates the plane.
PLANE TOPOLOGY
34
[CHAP. III
Proof. For if so, then some acyclic curve A is the common boundary of two domains D1 and D8. But since A separates the plane it cannot be
an are, so that it must contain a triod ao + bo + co. Now construct a 0-curve 0 = a'ob' + a'xb' + a'yb', where a'ob' a ao + bo, a'xb' a D1 + ao + bo, a'yb' c D2 + ao + bo. By (1.4), it - 0 = R1 + R2 + R3. One of these, say R1, contains co - o. Then if Fr(R1) = a'ob' + a'xb', then D2 R1 = 0; but also D2 -(7r - R1) # 0, since DE a'yb'. This is impossible, since D.-Fr(R1) = 0. (2.4) If M is a cyelicly connected locally connected generalized continuum,
the boundary B of any complementary domain R of M is a simple closed curve provided B is bounded.
Proof. Since, if B :t- M, B separates the plane and M is cyclicly connected, it follows that in any case B contains a simple closed curve C.
Then B = C. For, if B - C contains a point p, let px be an are in B with x and let py be an are in M - x with (py)-C = y. Then px + py contains an arc xqy, where q lies in B. Let xay and xby be the
arcs of C from x to y and set 0 = xqy + xay + xby, rr - 0 = R1 + R$ + RS, where R. B. But since Fr(R) z) q + a + b, whereas Fr(R1) = two of the arcs zqy, xay, xby, we have a contradiction; and hence B is a simple closed curve. (2.41) If N is a locally connected continuum in a plane 7r and a and b
are points of IT lying in different complementary domains Ra and Rb, respectively, of N, there exists a simple closed curve J in N separating a and b in ?r.
For either R. or Rb, say Rb, is bounded. Then if M
lib, it results
at once that M is a cyclic locally connected continuum. Thus the boundary J of the complementary domain of M containing a is a simple closed curve and clearly J separates a and b. 8. Plane separation theorem. Applications. (3.1) SEPARATION THEOREM. If A is compact, B is a closed set with
B-
T totally disconnected and a, b are points of A
-
and respectively, and a is any positive number, then there exists
a simple closed curve J which separates a and b and is such that B) c and every point of J is at a distance less than a from some point of A.
-
Proof. For each point x of A let C. be the circle with center x and radius less than min [e/2, 112 p(x, B)], and let I. be the interior of this circle. Let H1 be the set of all points x of A with p(x, B) 1, and for each n > 1, let H. be the set of all points x of A with 1/n S p(x, B) S 1/(n - 1). Since for each n, H,, is compact, it follows that for every n there exists a finite number of the sets (I.) with centers in H,, whose
PLANE SEPARATION THEOREM
§ 3]
35
sum K,, = :Em=1 I," , covers H,,. Set K = :E' K,,, and let Q be the com-
ponent of K which contains a. Then Q has no out point, for every point of Q is an inner point of a circle lying in Q. Thus Q also has no cut point. Q is locally and B) c Also is totally disconnected and any point p of 0 not connected, since
is a point or a limit point of only a finite number of sets
in
lying in Q. Thus Q is a cyclicly connected, locally connected continuum Let J be the boun. c containing a in its interior and such that dary of the complementary domain of Q containing b, Then, by (2.4), J is a simple closed curve. Clearly J separates a and b, and since J C Fr(Q), we have
B) c
B) a
Finally, since every point of Fr(Q) is at a distance less than e/2 from some point of A we have the same property for J. (3.11) CoRotrARY (ZoBEITI THEOREM). If K is a component of a
compact set M and a is any positive number, then there exists a simple closed curve J which encloses L and is such that of J is at a distance lees thane from some point of K.
0, and every point
For, by [I, (9.3)], there exists a separation of M into two mutually separated sets A and Be, where A K, and every point of A is at a distance less than e/2 from some point of K. Let r be a ray emerging from some point b such that p(r, K) > 2e. Set B = B0 + b, and apply (3.1) obtaining the curve J every point of which is at a distance less than e/2 from A. Then J cannot enclose b, since 0; and every point of J is at a distance less than e/2 + e/2 = e from some point of K. DEFnTmow. A point p is called a regular point of a set K provided that for any e > 0 there exists an open set U of diameter < e containing p and whose boundary intersects K in only finite number of points.
If for each e > 0 such a U can be chosen so that
contains
S n points, p is said to be of order < n in K. If p is of order -<,' n but not of order S n - 1, then p is of order n in K. (3.2) THEOREM. If p is a regular point of a compact set K, then for each e > 0 there exists a simple closed curve J enclosing p with 8(J) < e and such that
pinK.
is finite and of power less than or equal to the order of
This follows directly from (3.1) by taking A = K U, B = K- (7T - U). (3.3) If the boundary of a region B is a simple closed curve C, then R is uniformly locally connected.
Proof. For if not, then there exists a d > 0 and two sequences and Y: yl, y2, of points in R each converging to one and the same point p of C and such that for no n do x,, and y lie together %: x1, x5,
36
PLANE TOPOLOGY
[CHAP. III
in any connected subset of R of diameter less than d. Take e = d/16. Then since p is a point of order 2 of C, by (3.2), there exists a simple a + b, two points. closed curve J enclosing p, with d(J) < E and Now J - (a + b) =axb + ayb, where axb and ayb are arc segments one of which, say axb, lies within C and the other without C. Now for some n, say n = k, xk and yk both lie within J. But then if xku and ykv are u and (ykv)-J = v, we have u + v C axb and arcs in R with hence xku + axb + ykv is a connected subset of R of diameter less than d containing xk and yk, contrary to the definitions of X and Y. (3.31) COROLLARY. If p is any point on a simple closed curve C, then for each e > 0 there exists an arc apb of C and an arc axb such that axb is within (without) C and d(apb + axb) <.F. (3.32) COROLLARY. If o is any inner point of an are aob and c is any positive number, then there exists an E-simple closed curve J (i.e., 8(J) < E) enclosing o and meeting aob in exactly two points u and v, where we have
ueao,veob. (3.33) CoB.oLLARY. The interior of every simple closed curve J has property S. The exterior E of J has property S locally, (i.e., the intersection
of E with the interior of a square enclosing J has property S). Whence, every point of J is accessible from both the interior and the exterior of J.
4. Subdivisions. By a closed 2-cell in the plane will be meant a set consisting of a simple closed curve plus its interior. In general, any set homeomorphic with such a set will be called a closed 2-cell whether in a plane or not. It follows from (4.4) below that all closed 2-cells are homeomorphic with each other, so that a closed 2-cell could be defined as a set
homeomorphic with a circle plus its interior. By a subdivision of a set X is meant a representation of X as a union X = I X,, of some of its subsets Xa to be specified and meeting further specified conditions. (4.1) SuBnmsIow ThEOREM. Let A be a closed 2-cell with interior R and boundary C. For any e > 0 there exists a subdivision of A into a finite number of 2-cells of diameter <E such that the intersection of any two is either empty, a single point or a simple arc. Proof. Choose points pl, p2, - - - , pn on C cyclically ordered on C and such that the diameter of the arcs p1p2, pep, ' ' ' , PnP1 are all < e/18. Let x1y1, x2y2, - ' , x,,yn be disjoint linear segments of length , n, xty;-C = xi E p pi+1, < E/9 chosen so that for i = 1, 2, -
xiyi - xi c R. Now by the proof of the Phragmen-Brouwer Theorem there exists a simple polygon P in R (obtained from a hexagonal subdivision H of the plane of mesh <e/18) whose interior I contains all the points yl, y2 ... , Y.
SUBDIVISIONS
§ 4]
37
together with all points of R at a distance >e/18 from C. For each i, let zi be the first point of P on x;y; in the order x;, y;. Let z;z;+l denote that one of the two arcs of P from z; to z;+1 which together with the area x;z;, xixi+l and x;+lz; forms a simple closed curve C, not enclosing the
other are of P from z; to z;+1. Now since for each i, C - x;x,+1 and P - z;z,+1 lie together on the same side (outside) of the simple closed curve C; = x;xi+1 + xi+ lzi+1 + zi+lz, + z;x; so that the points z;, z2, ... , Zn
are cyclically ordered on P, it follows that for each i, 6(C; + I;) < e where I; is the interior of C;. For if x e I;, x lies in a hexagon h1 of H contained in P + E where E is the exterior of P; and since p(x, C) < e/18 there exists a chain of hexagons of H: hl, h2, - , hk such that h; and 0 h,+1 have a common side, j = 1, 2, . , k - 1, and such that but hi-C = 0 for j < k and 8[:Ei h;j < E/9. No h, can lie in P + I, because if so then 2i h, would lie in P + I as it is connected and does not intersect C, whereas hl is in P + E. Hence the union Q of the interiors of the h, together with the open segments (sides) common to two successive hexagons h1, ht+1, j = 1, 2, - , k - 1 cannot intersect P and, since it is connected,
Jih; must intersect x;z; + x;+lzi+1 +xixi+l = K. Since 8(Q + K) < e/3, we have x e VE,s(xtx;+l) for any x e I. Whence 8(C; + I;) = r (I;) < e. Thus the hexagons of H in P + I together with the cells C; + I;, i = 1, 2, - - - , n constitute the desired subdivision of A, except that possibly some hexagon of H in P + I may have a nonits closure
connected intersection with z;z;+l for some i. Thus for each i we further
subdivide C; + I; by taking a finite number of disjoint broken line segments in C; + I; each having one end in z;z;+l and the other in x;xi+1 and so that each new cell thus obtained in C; + I; intersects P in an arc which either lies on a single edge of P or on two consecutive intersecting edges of P. These new cells together with the hexagons of H in P + I now meet the intersection requirement of our theorem. (4.11) Coxou uy. The interior of any simple closed curve is uniformly locally connected.
This results at once from the fact that any two cells intersecting C have an arc in common or else their intersection is empty.
Note. By replacing each z; which is a vertex of the subdivision of P + I by a point z; very near z, on P and changing x;z; to a broken segment x;t;zs very near x;z;, we obtain an E-subdivision of A having the property that the intersection of each pair of 2-cells is either empty or an are segment. A subdivision of the plane or of a region of the plane having this property will be called simple. It will be noted that hexagonal subdivisions have this property of simplicity. Further, a subdivision of
the sort described in (4.1) is simple if and only if each point p of its 1-dimensional structure 0 (i.e., the complex of the boundaries of all its
PLANE TOPOLOGY
38
[CHAP. III
2-cells) is of order <3 i.e., has at most 3 segments of G meeting in R. Now since the same procedure as in the above proof can be applied to the exterior of C, we have the (4.2) THEOREM. Given any simple closed curve C in the plane or, any
e > 0 and any 8uiciently fine hexagonal or simple subdivision H of it, there exists a simple E-subdivision H' of it whose 1-dimensional structure includes C and which is identical with H outside VE(C).
Now let R be any elementary region in the plane it, i.e., a region bounded by a finite number of disjoint simple closed curves C1, C8, - , C.. Then if we take a less than J the minimum distance between any two of the C, and take H sufficiently fine, m-fold repeated application of (4.2) yields a simple -E-subdivision H' of it whose 1-dimensional structure includes all the C,, j = 1, - , m. The cells of H' lying in R constitute a simple c-subdivision of R. Here we have (4.3) THEOREM. If R is any bounded elementary region, for any e > 0 -
there exists a Simple E-subdivision of R. (4.31) Every elementary region i8 uniformly locally connected.
DEFINITION. If A and B are closed 2-cells, or bounded elementary regions, then simple subdivisions S. and Sb of A and B respectively are
said to be isomorphic or Similar provided there exists a similarity correspondence between them. By a similarity correspondence is meant a 1-1 relationship between S. and Sb, say h(SQ) = h(Sb), which maps
the graph G. consisting of the union of the boundaries of the 2-cells of S. topologically onto the graph Gb made up of the union of the 2-cells of Sb and, in addition, establishes a 1-1 relation between the 2-cells of
S. and those of Sb under which boundaries are preserved, that is, if Ca is a 2-cell of Sa and Cb is its correspondent under h, then h maps the boundary of Ca homeomorphically onto the boundary of Cb.
(4.4) THEOREM. If J is any simple closed curve in the plane and h(J) = C i8 any homeomorphism of J onto a circle C, then h can be extended
to give a homeomorphi8m of J plu8 its interior onto C plus its interior.
Proof. Let R and I denote the interiors of J and C respectively. We first show
(f) For any e > 0 there exist Simple E-8ubdivi8ions S and S' of J + R and C + I respectively which correspond under a similarity correspondence which coincides with h on J.
To show this, let So be a simple E-subdivision of J + R as given by (4.1). We next construct a similar subdivision of C + I. This is done by regarding So as being constructed by a finite sequence of steps each consisting of the addition of a spanning are to the previously con-
structed boundary graph. That is, we have the set J to begin with. Next we add a simple are al to J having just its ends on J and lying in
§ 4]
SUBDIVISIONS
39
J + B; then we add an arc a2 spanning J + al, i.e. having just its ends in J + al, and lying in J + B, and so on. A finite sequence, say n, of such steps can be taken in such a way that the final graph J + : i a; = Go is identical with the union of the boundaries of the 2-cells in S. Now, going over to C + I, let x1 and y1 be the ends of al and let Y1 be an are (line segment!) in C + I joining h(x1) and h(z2) ; and let h be extended to a
homeomorphism from J + al to C + P1 and to a 1-1 correspondence
between the 2 regions of R - al and the two of I - Y1 preserving boundaries. Thus if x2 and Y2 are the ends of a2, then h(x2) and h(y2) are
on the boundary of one region Q in I - N1. Hence we construct, similarly, an arc (line segment!) fl, from h(x2) to h(y2) and lying except for
its ends in this region Q, and let h be extended to a homeomorphism from J + a1 + a2 to C + N1 + #2 and to a 1-I correspondence between
the regions of R - a1 - a2 and those of I - N1 - Y2 preserving boun-
daries. Continuing in this way for n steps we obtain the desired subdivision So of C + I similar to So. Now since So may fail to be an e-subdivision, we next take a simple e-subdivision S' of C + I which is obtained by adding a finite number of area (line segments!) to So. [This is possible by (4.1)]. Then we make
corresponding additions to the boundary graph in S. by the procedure outlined above, thus obtaining a simple e-subdivision S of J + R which corresponds to S' under a similarity correspondence coinciding with h on J. This establishes (t). To prove the theorem we note that repeated application of (t) yields infinite sequences of subdivisions Sl, S2, and S,, S2, - of J + R and C + I respectively such that (1) for each n, S and S;, are simple 1/n-subdivisions of J + B and C + I respectively corresponding under a similarity correspondence which is identical with h on J and which maps the boundary graph on (i.e. the union of the boundaries of the cells of S,,) topologically onto the corresponding graph G; for S;, and and are refinements of S,, and S;, respectively in the sense that G D 0,',. Thus h is extended topologically to G for each n. We next show that as thus extended, h is uniformly continuous on G = i G,,. To this end let e > 0 be given and let us then choose n so that 2/n < e. With n thus fixed, let 8 = min p(A, B) where A and B are an arbitrary pair of non-intersecting 2-cells of the subdivision 5,,. Then if (2) for each n,
x, y e 0 and p(x, y) < 8, x and y lie together either in a single 2-cell A of S,, or in two intersecting 2-cells A and B of 5,,. Thus by the method of extension of h, h(x) and h(y) lie either in a single cell A' or in the union
of 2 intersecting cells A' + B' in S so that p[h(x), h(y)] < 6(A' + B') < §/n < e. Thus h is uniformly continuous on G. By the same type of
40
PLANE TOPOLOGY
[Casr. III
argument, h-1 is uniformly continuous on G' _ 2 i G. Accordingly, since G and 0' are dense in J + R and C + I respectively, it follows by II, (2.3), that h extends to a homeomorphism of J + R onto C + I. (4.41) COROLLARY. A set A is a closed 2-cell if and only if it is homeomorphic with a circle plug its interior. (4.42) CoROLLARY. If A and B are arbitrary closed 2-cells (in a plane of the edge of one onto the edge of the other can or not), any be extended to a homeomorphism of A onto B.
REFERENCES
The material in Chapters I-III is closely related to that in the first part of Whyburn [1]. The reader is referred to this book for appropriate references to the original on other related sources. See also Moore [1). In connection with § 4 of Chapter III see also Kerikjbrtd [1].
IV. Complex Numbers. Functions of a Complex Variable 1. The complex number system. We recall that we are assuming as known all usual properties of the real numbers. The real number system is adequate for many purposes (e.g., for linear measurement) but not for all, for example, for solving simple equations. The equation
x=+1=0 has no solution in the real number system since the left hand side is positive for any real x. Thus we are led to define a more inclusive system, called the complex number system. This is readily accomplished by using the real numbers
and their properties which we assume known; and we proceed now to outline the procedure for doing this. (a) Definition. A complex number is any ordered pair of real numbers. If a and b are real, we denote (temporarily) the complex number defined by the pair a, b by (a, b). Two such numbers (a, b) and (c, d) are equal
if and only if a = c and b = d. Note that in general the number (a, b) is different from the number (b, a). Thus (1, 2) is not the same as (2, 1).
If a is real, the complex number (a, 0) is to be identified with the real number a. In other words every real number a automatically becomes a complex number by identifying it with the pair (a, 0). Of course we must be careful in defining rules of combination of complex numbers to make sure that these agree, in case the numbers happen to
be real, with the same operations already defined for real numbers. (b) Sum and product. If (a, b) and (c, d) are complex numbers (a, b, c, and d real), their sum is defined to be the complex number (a + c, b + d) and their product the complex number (ac - bd, ad + bc). Thus we write
(a, b)+(c,d)=(a+c,b+d) (a, b)(c, d) _ (ac - bd, ad + be).
We note that these definitions are consistent with the same operations on real numbers. For if a and c are real, we have
a+c=(a,0)+(c,0)=(a+c,0)=a+c ac
(a, 0) (c, 0) = (ac, 0) = ac. 41
COMPLEX NUMBERS
42
[CHAP. IV
Also, already we can show that in our new number system the equation x2 + 1 = 0 has the solution x = (0, 1). For
(0,1)2+1 =(0,1)(0,1)+(1,0)
=(--1,0) + (1,0)= (0,0)=0 Similarly we could show also that (0, -1) is a solution. (c) The number i. The form a + ib. The complex number (0, 1) has special significance and is denoted by i. If (a, b) is any complex number whatever,
a+ib=(a,0)+(0,1)(b,0)_(a,0)+(0,b)=(a,b). Thus any complex number (a, b) is expressible in the form a + ib. This form will be used in what follows in preference to the more cumbersome
and formal (a, b). Any complex number of the form (a, 0) or a + i0 is a real number; and one of the forms (0, b) or 0 + ib is said to be pure imaginary. For any complex number (a, b) = a + ib, a and b are called the real part and the imaginary part respectively of a + ib. Thus if
a = a+ib we write R(a) = a, I(a) = b. We note further that
i8=(0,1)(0,1)=(-1,0)=-1. Thus i is a square root of -1. Similarly (0, -1), which will be -1, is also a square root of - 1.
(d) Rules of combination. From the definition of addition and multiplication of complex numbers we obtain at once the Commutative laws:
aP = fix, Associative laws:
(a+li)+y=a+(f+Y) W )y = a(i y)
Distributive law:
a(f +Y)= a4+ay. In each case, a, f and y are arbitrary complex numbers. The proofs here are left as exercises for bbe reader.
THE COMPLEX NUMBER SYSTEM
§ 1]
43
(e) Conjugates. Neutral elements and inverses. If z = (a, b) = a + ib = a + bi is any complex number, the number z = (a, -b) = a + (-b)i = a - bi is called the conjugate of z. (By commutativity it is clearly immaterial whether we write a + ib or a + bi.) The complex numbers 0 and 1 [i.e., (0, 0) and (1, 0)] are neutral
elements with respect to addition and multiplication respectively,
i.e.,if a=a+ib a+0= (a, b) + (0,0) = (a, b) =a (a, b)(1, 0) = (a, b) = a.
Also these neutral elements are unique. For if
a+E=a we have
a=a+ib,
=8+it,
(a+ib)+(8+it)=(a+8)-l-i(b+t)=a+ib.
Thus
a+8=a, Whence
b+t=b
=0.
8=t=0
so that Similarly, we show that 1 is the only complex number which is neutral with respect to multiplication by showing that for a # 0, cx-P = a implies = 1. (Exercise for the reader.) For any complex number at = a + bit the negative of a, -a (addition inverse) is defined to be -a + (-b)i; and for at 0, the reciprocal of at, 1/a (multiplication inverse) is defined to be
a + (-b)i
a a2 + b2
=
-
a2 + b2
It results at once from these definitions that a + (-a) = 0 and a(l /a) = 1. Further these inverses are unique. Suppose, for example,
aa'=1
0#a=a+bit'
a'=a'+b'i.
Then (aa' - bb') + (ab' + a'b)i = 1 = 1 + Oi. Whence aa' - bb' = 1, ab' + a'b = 0. Solving these two equations simultaneously gives
a'=a2+ab2,
-b b'=a2+ 62
We note finally that if a and fi are complex numbers, a# = 0 implies either at = 0 or = 0. For if at 0, multiplying a = 0 by 1/a gives ot
(X
(f) Subtraction and division are now defined in terms of the inverses under addition and multiplication respectively. Thus if at = a + bit
[CHAP. rv
COMPLEX NUMBERS
44
= c + di we define fi - a to be fi + (-a) and, for a # 0, fl/a to be so that
fi-a=(c-a)+(d-b)i -ac+bd ad - bci
a#0.
az+bz + az+bz These latter numbers are readily seen to be solutions, respectively, a
of the equations
z+a=# and
az = P Further, these solutions are unique. The reader should prove this. By adding and subtracting the equations
a=a+bi i=a - bi we get the relations
a +a=2a=2R(a)
(x -Ft =2b=2I(a), for the real and imaginary parts of a in terms of a and its conjugate. It is now easy also to show that every complex number has two square roots. For let
a+bi=(x+iy)z=xz-yz+2xyti;
whence
xz - yz=a 2xy = b. Solving these latter two equations simultaneously gives
x=f
a+
az+bz, 2
yaz+bz-a
2
The numbers under the outer radicals are always positive or 0 so that
x and y aie real. However the signs must be paired so as to satisfy 2xy = b, that is, the signs must agree if b > 0 and must differ if b < 0. It now follows that any quadratic equation with complex coefficients has solutions given by the usual quadratic formula. (g) The complex plane. It is natural to represent the complex numbers x + yi by the points (x, y) in a cartesian plane, that is we make the point with coordinates (x, y) in the plane correspond to the complex number x + yi. When used for this purpose the plane is called a complex plane. Clearly the relationship between the points in the complex plane and the set of all complex numbers is one to one. The real numbers are represented
§ 1]
THE COMPLEX NUMBER SYSTEM
45
by the points on the x-axis and the pure imaginary numbers by points on the y-axis. For this reason the x-axis in the complex plane is referred to as the real axis or axis of reals and the y-axis as the imaginary axis. Addition and subtraction of complex numbers when transferred to the complex plane become ordinary vector addition and subtraction where the complex number x + iy is interpreted as the vector with components x and y. (h) The polar form. Using polar coordinates (p, 0) in the complex plane, the number x + iy takes the form p(cos 0 + i sin 0) since
x= pcos0
and
y= psin0.
This is called the polar form of the number x + iy. We have
p=
0 = arctan y x
and these are called the modulus (or absolute value) and amplitude respectively of x + iy. We shall always take p to be a non-negative number and will designate it also by Ix + iyl. Thus as we use it p = IzI has a unique value for each complex number z. The amplitude, on the other hand, is many valued. For x = 0, by convention we take 0 = ±'rr/2 + 2k7r, k = 0, ±1, ±2, , the sign in front of 1/2 being chosen so as to agree with that of y for y 0 0; and for both z and y = 0, i.e., for x + iy = 0, 0 is not defined. Since p is restricted to non-negative values it is necessary to restrict also the choice of 0 among the values of aretan y/x. This is accomplished by taking for 0 these values of arctan
y/x and only those of the form 00 ± 2k7r (k = 0, 1 , 2, - - ) where 05 00 < 2or and where 00 = 0
when x > 0 and y = 0
0<00<1/2
when x > 0 and y > 0
00 = 1/2
when x = 0 and y > 0
1f2<00<1 0= 1
when x < 0 and y > 0
1<00<3w/2
when x < 0 and y < 0
00 = 31/2
when x=Dandy<0 when x>Oandy<0.
31/2 < 00 < 27r
when z < 0 and y = 0
COMPLEX NUMBERS
46
[CHAP. IV
It may be noted that although apparently geometric, the modulus and amplitude of a complex number can be made purely analytic concepts. This is clear for p; and to accomplish it for 0 it is only necessary to note
that the trigonometric and inverse trigonometric functions of real arguments are definable analytically either as infinite series or as definite integrals. Multiplication and division of complex numbers is exhibited lucidly in terms of the polar forms of the numbers. For if z1 = pl(cos 01 + i sin 01), zZ = p,(cos 02 + i sin 0=), we have zlzy = p1p,[(cos 01 cos 0E - sin 01 sin 0E) + i(cos 01 sin 02 + sin 01 cos 01)
= PiP,1° (01 + 02) + i sin (01 + 0a)]. Similarly, for z$
0,
Thus to multiply we multiply moduli and add the amplitudes and to divide we divide moduli and subtract the amplitude of the divisor from that of the dividend. In particular the reciprocal of a (non-zero) complex number with modulus p and amplitude 0 is the number with modulus l/p and amplitude -0, i.e., if z = p(cos 0 + i sin 0) we have 1
1
z=P[cos(-0)+ssin(-0)]
=1 [cos 0 - i sin 0],
since sin (-0) = -sin 0.
P
(i) Powers and roots. From the above rule for multiplying complex numbers it follows by a simple induction argument that if z = p(cos 0 + i sin 0) and n is any positive integer, then z" = p"(cos no + i sin n0).
The same relation holds for negative integers n as is seen by using the rule for division. In particular for p = 1 we have De Moivre's Theorem: (cos 0 + i sin 0)" = cos no + i sin no.
Now let n be any positive integer. We proceed to find the nth roots
of a complex number a # 0. We set z" = a and seek solutions for
z.
THE COMPLEX NUMBER SYSTEM
§ 1]
47
Let z = p(cos 0 + i sin 0). Then at = z" = p"(cos nO + i sin n0). Thus if a = r[cos 4 + i sin 0] we have
r= p", whence
p=
(k=0,
4,=nO+2kir
=
0
n
, n - 1 and only these give distinct and clearly the values k = 0, 1, 2, values of z. Thus we have found exactly the n nth roots z
cos
(t + 2k7r) n
n
+ i sin
(+)] n
n
for the number at = r[cos 0 + i sin 4,]. (j) Absolute values. Inequalities. It was shown in (h) above that for any two complex numbers
x1=x1+iy1,
zE=x2 +iy2,
IZIZ21= I=11 Iz81 I
zl z81
$ # 0.
iIzll
=
zal
These relations are readily deduced algebraically directly from the definitions of product and quotient. We proceed to show further, the relations
(*)
121 + zt) S Iz1I + Izzl
Izl+z,I>_ I=11 or, combining, (*)
,Z21,
-
Iz1I - Iz,I S Izl + za1 S Iz11 + Iz2I
Since for a and b real, (a - b)2 > 0 so that as + b$ > 2ab, we have x2jy2 + xzy > 2xlx?yly2
Whence
z2,4+y2ly2+2xlxyly2S44+AY2+x2iy2+zV, or (i)
(xlxs + YIY2)2 S (x1 + yi) (x2 + y2).
Thus x1X2 + YIY2 S
V-(;[+ y21) (x2 + y2)
or
(ii)
2x1x2 + 2yly$ S 2
(xi + yi) (x2 + y2)
,
ors
[CHAP. IV
COMPLEX NUMBERS
48
so that
(iii) (x2,+4+ 2x122) + ( + y2 + 2yiy2)
Sx2l +y2l +202,)(x2+y2)+4 +y22
(xl + X2)2 + (Yi + y2)2 < (V xi + yl + V'02 + y2)=
(iv)
Accordingly
-+y2 +
(xi + xE)Q + (yi + y2)2 <
(v)
2
+ Y2
or
i21+2211211+IZ21. Now since it results also from (i) that XiX2 + Yiys > -
(
+
y2 (X2
+ y2)
we get (ii) and (iii) with the inequality and the sign of the radical reversed
and (iv) with the inequality reversed and a minus sign between the radicals. This gives (v) similarly altered or -Iz21
Of course this is also deducible directly from the first relation in (') since this latter gives
lzl+zZ-Z21=1211 or
I21+7-21>=1211-1-221=!211-I221.
Now for any single complex number z = x + yi, since
Vii -+y2
1x1,
we have )z)
1x1,
121
4(lx1 + lyl)
1yl so that
Iz)
.
Also by (") above
121=1x+iyl< 1x1--1iy1 =1xl+ly1
'
Combining we get
(t)
4(Ixl + lyl) <- lzl < 1x1 + Iyl
The fraction j in (f) can be replaced by JV2-. The reader may find it interesting to prove this algebraically. (k) The metric. It results at once from what has been shown above
FUNCTIONS OF A COMPLEX VARIABLE
§ 2]
49
that if we define a metric in the set of complex numbers as p(zl, z2) = Izl - z21, we have a metric space. For this number vanishes if and only if z1 = z2 and is symmetric; and for three complex numbers z1, z8, z3, we have Izl - Z$l + IZa - 231
f
Izl - Z3l
by (*) above so that the triangle inequality holds. Further, this metric is identical with the ordinary cartesian distance between the points z1 and zE in the complex plane for every pair of complex numbers zl and z$. (1) The complex number systems as a topological space. The topology
of the complex number system as given by the metric just defined is identical with that of a euclidean plane because the correspondence between the complex numbers and the points in the complex plane is 1-1 and continuous both ways. The latter results from the fact that the metric IZ1 - z$I in the complex number system is equal to the distance between the corresponding points in the complex plane for each pair of complex numbers z1, z2.
Thus, in particular, the complex number space is separable, metric, connected, locally connected, and locally compact. Every bounded closed set is compact and the space is complete. The completeness of the complex number system is readily deduced directly from the fact that a sequence a1, z$, z3, . of complex numbers converges to a limit z = x + yi, lim zn = z, if and only if both xn -+ x and y --s y where zn = x, 1+ iyn. For by relation (t) in (j)
Ox. - xI + Iy,, - yI) 5 IZn - ZI Ixn - xl + l yn - yI so that z - z -* 0 if and only if both x,, - x --*- 0 and yn - y -+ 0. It follows similarly that a sequence z1, z2,
of complex numbers
are Cauchy sequences is a Cauchy sequence, since the
is a Cauchy sequence if and only if both (xn) and
where zn = xn + iyn. Thus if z1, z2, . real number system is complete there exist real numbers x and y such that xn -+ x and yn -+ y and this gives zn --+ z = x + yi.
2. Functions of a complex variable. Limits. Continuity. Any rule or law f which assigns to each complex number z in a set Z of complex numbers one or more definite complex numbers f(z) as values is called a function of the complex variable z. Thus a function f(z) is a transformation
of the set Z on which it is defined into the complex number system. The function is single valued if it assigns just one value f(z) to each z in the set Z. Unless otherwise specified all our functions are supposed to be single valued. When we deal with multiple valued functions such as Vz- or arg z we will nearly always limit our considerations to a single
COMPLEX NUMBERS
50
[Cser.IV
branch of these functions on which they are single valued. It will be convenient to write w = f (z) for a function and to consider that the values w of f lie in a separate complex number system. Thus we may think of the values assumed by z as lying in one complex plane, the z-plane, and the values of w as being in another, the w-plane. Of course, these planes are alike in every essential respect and may actually be the same.
Now for any z = x + iy where f is defined let us write w = f (z)
_
u + iv. Then clearly u and v are real valued functions of the real variables
x and y. Thus we have w = f(z) = u(x, y) + iv(x, y) Suppose now that f(z) is defined for all z in a neighborhood of zo though not necessarily defined at zo. Then we define lim..,s. f(z) = wo in the usual way to mean that for any e > 0 a 6 > 0 exists such that f (z) - wol < E
provided Iz - zol < 6 and z # zo.
It follows at once that limy,Z, f (z) = wo = uo + ivo if and only if we have both the simultaneous limits
lim u(x, y) = so and lim v(x, y) = vo. z-.x,
Y-Y,
Y-.Ya
For by the inequality (t) in §1, (i), l f(z) - wol will be <E provided I u(x, y) - uoI and I v(x, y) --vol are each <E/2 and conversely each of these will be <E provided I f (z) - wol < E/2.
If f (z) is defined at zo, f (z) is said to be continuous at zo provided lim, , f(z) = f(zo). Clearly this is equivalent to the statement that f is
continuous at zo when considered as a transformation of its range of definition into the w-plane. Also by what was just shown it results that f (z) is continuous at zo if and only if each of the real functions u(x, y) and v(x, y) is continuous in (x, y) simultaneously at the point (xo, yo) The usual theorems on limits and continuity now follow without further proof from the corresponding results for real functions, because we can reduce all such questions to the same questions concerning the pair of real functions u(x, y) and v(x, y). Thus, for example, lim:.,, f(z) = F and lim,=. g(z) = 0 give lim U (z) (z)
± g(z)] = F ± 0,
z--.z,
lim f (z)g(z) = FU z"'zo
and, for G :p6 0, lim
f x)
r'-z' g(z)
- FQ
Likewise any function continuous on a closed and bounded (compact)
DERIVATIVES
§ 3]
51
set K is bounded and uniformly continuous on K; and if f (z) is continuous at zo and g(w) is continuous at wo = f(zo), then 9f(z) is continuous at zo. These results also follow from the corresponding results for continuous transformations.
8. Derivatives. Let w =f(z) be defined throughout a neighborhood of zo and let Az be a complex variable. If lira f (zo + Az) - f(zo) Az
At--O
exists and has a definite finite value of f'(zo), f is said to be differentiable
at zo and to have f'(zo) as its derivative at z = zo. For example w = zs is differentiable and has derivative 2zo for any zo whatever, since hm
(zo + Az)2 - zo
A
= limyo (2zo + Az) = 2zo. &Z--0
On the other hand the function w =f is nowhere differentiable. For we have (for any zo)
Ax - iAy
Aw
Ax+iAy'
Az
and by taking Az real (Ay = 0) this ratio can be made to approach 1 whereas by choosing As pure imaginary (Ax = 0) it will approach -1 as Az -, 0. Thus the limit fails to exist. Using the limit theorems referred to in §2 it now follows as in the case of real functions that the sum, difference product and quotient of two differentiable functions are differentiable and that the same relations exist as in the real case. That is, if f(2) and g(z) are differentiable,
d(f f g)
df
dg
dz±dz
dz
dg
d(fg)
d_f
dz = f dz + 9 dz df
dg
dffg_gdz
fdz
dz
=
g2
(where g(z) * 0).
Thus, in particular, all polynomials in z are everywhere differentiable, since obviously w = z is everywhere differentiable. Also if z and 8 are complex variables and if w = f(8) is differentiable
[Ca& . IV
COMPLEX NUMBERS
62
at a = Eo and 8 = g(z) is differentiable at z = za then the composite function w = fg(z) is differentiable at zo and we have dw
dw ds
dz
da dz
For let
Aw = BAs + e(AS)A8, where 1iw Qs - dw
for ,6 # 0
da
E(0) = 0.
Then
_
+ c(A8)
for all Ds.
Whence dw
az=
lim
Ow
dw d8
= o0zdadz
4. Differentiability conditions. Canchy-Riemann equations. Suppose the function w = f (z) = u(x, y) + iv(x, y) has a derivative at the point z. Then since Aw/Ai has a limit as dz -+ 0 and this limit is independent
of the way in which dz -+ 0, let us first take Az to be real. Then Az = Ax and (a) f ' (z) = hm
Az-o
-S
Ow
u(x + Ax, y) - u(x, y)
Jim = AZO
AX
+ i Jim
v(x + Ax, y) - v(x, y) Ax
az-.o
au
av
ax
tax
Similarly if we now let Az be pure imaginary, so that Az = i Ay, we have
(b) f'(z) = lim u(x, y + Ay) - u(x, y) + Jim v(x, y + Ay) - v(x, y) tidy av
1 au
av
AV-0
au
=ay+say=ay-iay
AY
THE EXPONENTIAL AND RELATED FUNCTIONS
§ 5]
53
Since the two forms (a) and (b) for f'(z) must be identical we have au av au_av and
ax - ay
ax
ay
These equations which necessarily hold when f(z) has a derivative are known as the Cauchy-Riemann equations.
We now show, conversely, assuming that the functions u(x, y) and v(x, y) have continuous first partial derivatives throughout a neighborhood of a point z = x + iy, the holding of the Cauchy-Riemann equations is also sufficient to insure differentiability of the function w = u + iv at that point.
In view of the continuity assumption, the Mean Value Theorem for real functions enables us to write, for any Az of sufficiently small modulus,
Au =
Ax +
Ay + E(1 x, Ay)Ax
27(Ax, Ay)Ay
AV = axL x + ayAy + Y(Ax, Ay) Ax + MAX, Ay)Ay
where the indicated partial derivatives are evaluated at the point (x, y) and where each of the functions e, 71, y, 6 has the limit 0 as (Ax, Ay)
-
(0, 0). Now replacing aulay by -av/ax and av/ay by au/ax (by the Cauchy-Riemann equations) and combining we get
Aw=AU +iAv=
au
(AX+iAy)+ia(AX+iAy)
+(E+iy)Ax +(i+ia)Ay, so that
Aw
au+iav+(E+iY)AX+(77
Az
ax
ax
Az
+ib)Ay
Az
Since IAx/AzI and IAy/Azl are each < 1 by §1, (j), each of the last two terms has the limit 0 as Az -+ 0 and thus Aw/Az has the limit au/ax + i(av/ax) as Az --* 0 so that f'(z) exists.
5. The exponential and related functions. define the exponential function
For any z = x + iy we
e==e (eosy+isiny) Note that if z = x is real, e= reduces to the ordinary real value ex; and if z = iy is pure imaginary we have err =cosy + i sin y
.
COMPLEX NUMBERS
64
[CHwr. IV
Thus ez = ezeiY = ex+rY for any z. For any z ,
lezl=ex, argez=y. If we write z in polar form
z=p(cos0+isin0), then since eis = cos 0 + i sin 0, this also takes the form z = peie which is frequently a most convenient representation of a complex number. It is now easy to show that the complex exponential function e° has properties analogous to the real exponential. For example eslez, = ezl+z, es ez.
= esl-z,
follow from the above together with the multiplication and division rules developed in §1, (h). For ezlez, = e etViez,esY, = axlex,ety,eiY, = e(zl+z,lei(YI+Y,) _ ezl+z,,
and similarly for the second relation. Also, however, we note that for any z, ez+2,i = ez++(Y+2A)
= ez[eos (y + 27r) + i sin (y + 27r)] = es.
Hence the function es is periodic of period &rri.
The exponential function ez is everywhere differentiable. For if
f(z)=z=u+iv=ezcosy+iezsiny,wehave au
_ax-exeosy=avay,
au av ay= -e'siny= -ax'
so that the Cauchy-Riemann equations hold and all first partial derivatives
are continuous. Further
so that des f dz = es.
Using the exponential function, the trigonometric functions are readily defined
sin z =
eiz - e-iz 2i
,
cos z =
eis + e-is 2
tan z = sin z/cos z, and so on. The usual properties of these functions
§ bl
THE EXPONENTIAL AND RELATED FUNCTIONS
65
are now easily developed. The reader should show, for example, such relations as sin2 z + cos2 z = 1
d (sin z) = cos z . For any complex number z 0 0 we define
log z= logp+i0 where z = peie and where log p represents the ordinary real logarithm of the positive number p. The logarithm function thus is infinitely many valued, since different values of 0 = arg z give different values of log z. For any z # 0 the values of log z are distributed vertically as points in the complex plane and are 27r units apart. For any one value w0 of log z the whole set of values of log z could be represented in the form wo + 2k7ri, k = 0, ±1, ±2, - . We have at once, for w = log z, e10 = elm v+ie = er° n.ie = peie
= Z.
Thus log z is the inverse function to the exponential function. If for any fixed value 00 of 0 we restrict 0 to values 00 0 S 00 + 27r,
we get only one value of log z for any z # 0 and thus we get a single valued branch of log z. It is readily verified that any branch of log z satisfies the Cauchy-Riemann equations at any z 0 0 and thus is differenti. able on its whole range of definition. Also, d log z
1
dz
z
as is readily shown by writing
x# 0
+ i arctan y
log z= log
X
or
x
log z = log N/i2 + y2 + i arccot -
for
y#0
Y
and computing
au
av
x
iy
1(z)=ax+iax=xs+y$-X2 +y2
1
z
V. Topological Index 1. Exponential representations. Indices.
Let O(x) be any mapping of a metric space x into the complex plane Z. Let (X) = E and let p e Z - E. If there exists a continuous complex valued function u(x) on X such that
eb(x)_4i(x)-p, xeX, (t) we will call (t) an admissible exponential representation of #(z) - p on X, and u(x) will be called a continuous branch of the logarithm of #(x) - p on X. (1.1) If there exists a ray pq with origin p and lying in Z - E, then #(x) has an admissible representation (t) on X. For any admissible u(x) and any a, b e X, the difference in the imaginary parts of u(a) and u(b) is numerically <27r. Accordingly fi(b) = 4(a) implies u(b) = u(a). Further, u(a) may be any preassigned value whatever of log [#(a) - p].
For if u(a = log l¢(a) - pl + i0,, we have only to define u(x) = log 10(x) - pl + i(Oa -{- 0i) where 0y is the signed angle, -21r < 0z < 27r, from the ray po(a) to the ray p4(x) not containing the ray pg.
(1.2) If X is an interval ab, a ray, a line, or a plane, #(x) - p always has an admissible exponential representation (t) on X. Further, u(x) is uniquely determined up to an additive constant.
For let a = xo, x1, x2, , x,, = b be a subdivision of the interval ab so that #(xjxi+1) lies inside a circle not enclosing p for each i = 0, 1,
, n - 1. Then by (1.1), O(x) - p has a representation (t) on
xox1; and again it has a representation (t) on x1x2 where the new u(x1) agrees with the u(x1) in the representation on xox1. Likewise there is a representation (t) on x2x agreeing at xZ with the previous one and so on
to x,-Ix,. Thus we obtain a continuous branch u(x) of log (#(x) - p) on ab. In case X is a ray or a line, clearly a continuation of the same procedure gives a continuous branch u(x) of log [O(x) - p] on X. Now to see that this is unique up to an additive constant, we have only to note that e1(r) = e°(d) on a connected set I implies u(x) - v(x) = 2k7ri on I for some fixed integer k, because u(x) - v(x) is continuous on I and is always a value of log 1. Now let X be a plane, which we take as a complex plane with cartesian 66
EXPONENTIAL REPRESENTATION
§ 1]
67
coordinates (8, t) so that x = 8 + it. As just shown above, 4(x) - p has an admissible exponential representation on the real axis t = 0, giving a continuous branch uo(s) of log [0(x) - p]. Further, for each so, there is a continuous branch u, (t) of log (0(x) - p] on the vertical line s = 8o so chosen that u,.(0) = uo(so). For each point x e X we define
u(x) = u(8, t) = u,(t),
where x = 8 + it.
Then e"(-) = #(x) - p for each x e X and we shall show that u(x) is continuous. To this end we first prove the
Ls. If on a rectangle R: ao S 8 (i)
larg [#(xi)
(ii)
a', to < t S t',
- P] - arg Wxa) - p11 < rr,
x1, xs e R
u(x) is continuous on the base t = to, 8o 5 s S 8' of R,
then u(x) is continuous at aU points of R.
Since by (i) there is a ray from p not meeting O(R), it follows by (1.1) that there is a continuous branch v(x) of log [#(x) - p] on B, and we suppose this chosen so that v(xo) = u(xo) where xo = 8o + ito. It
then follows that u(x) = v(x) on R so that u(x) is continuous on R. For let w(x) = v(x) - u(x), x e B; and define g(r) = w(8o + r, to),
for O< r-< a -80, g(r) = w(8, to + r - 8 + 80),
for 8 - 8o S r S t - to + 8 - so = q. Then since u(x), and therefore w(x), is continuous on the base of B and is also continuous in t on the vertical line through (8, to) and (a, t) where u(x) is identical with u,(t),
it follows that g(r) is continuous in r on the interval 0 s r S q. Now since e(ID) = e"(e) = (x) - p for all x e B so that ef0(x) = ev(:)-°(x) = 1 on B, we have eo(r) = 1 on (0, q). Thus g(r) is of the form 2k7ri where k is an integer, so that g(r) must be constant on (0, q) as this interval is connected. Thus g(r) = 0 on (0, q) since g(0) = w(ao, to) = v(xo) v(xo) = 0.
This proves the lemma; and it now follows readily that u(x) is continuous on any rectangle Q with a base on the a-axis and thus is continuous throughout the whole plane. For by uniform continuity of q,(x) on Q, Q can be divided by horizontal and vertical lines into a finite
number of subrectangles such that (i) holds on each of them. Then since each point xl in Q can be joined to the base of Q on the s-axis by a chain Q1, Q=, - - , Q of these subrectangles each having a base in common with its predecessor and where Q1 has a base on the s-axis and xl a Q,,,
[CHAP. V
TOPOLOGICAL INDEX
58
repeated application of the lemma to this chain gives continuity of u(x) on Q and thus at x1. Thus eu(x) = #(x) - p is an admissible exponential representation on X; and uniqueness follows as before. (1.21) COROLLARY. The same conclusions as on (1.2) hold in case
X is any region of the form -oo < 8 < co, a < t S b or of the form 8 < oo, a S t < b in an (8, t) plane, or where X is any rectangle c in a plane. Further, the property thus defined on X is a topological in-
variant, 8o that the same conclusions hold on any set X' homeomorphic with a set X on which they hold. We now limit our considerations to the case in which X is an interval or a simple arc ab.
DEFrxrriow. For any admissible exponential representation eu(z) of O(x) - p on ab we define µ.e(#, p) = u(b) - u(a) . When no confusion is likely to result, some or all of the symbols ab,
0 and p in the expression u b(o, p) may be omitted. We have immediately the following: Notes.
(i) The index u is independent of the particular u(x) used in the exponential representation.
(ii) For any O(x) and any p e Z - E µ(4), p) = µ(9 -- p, 0).
(iii) For any factorization #(x) - p = 41(x)-42(x), we have µ* p) tu(01, 0) + J"42, 0).
(iv) If 1(z) is any non-singular linear transformation of the z.plane into itself, µ[14, l(p)] = µ(l, p). Note (ii) is a trivial consequence of the definition and (i) is a restate. ment of part of (1.2). To see (iii), we take representations 4(x) - p = e"cz),
01(x) = e.(*),
02(x) = eu'(z)-
Then e«,(:)+V,(:) = 01(x)02(x) = eU(X)
gives
µ(0l, 0) + 1u(02, 0) = u1(b) - ul(a) + u2(b) - u2(a)
= u(b) - u(a) = µ(#, p), by (1.2).
Note (iv) is a direct consequence of (ii) and (iii). For let l(z) _ 0. Then 10 = ao + fl, l(p) = ap + f and (ii) gives
ax + tg where a and P are constants, a
µ0,1(p)) = u(ao + A, ap + jS) = µ((Xo -- ap, 0) = µ(a4, ap);
EXPONENTIAL REPRESENTATION
§ 1]
59
and since obviously the index of any constant' function about any point different from that constant value is 0, (iii) gives
u(a4, ap) = 14100 - p), 0] = µ(a, 0) + u(#, p) = µ* p).
(1.3) If 0(b) _ 0(a), then (1/27ri)y(o, p) i8 integer valued and con-
tinuous in p. Thug it is constant in each component of Z -
E.
For if g(x) = p - p1/0(x) - p, x e ab, we have
0(x) - pi = (O(x) - p)[g(x) - (-1)]. Thus by notes (ii) and (iii) µ(0, P1) = µ(O, p) + µ(g, -1).
Also if pl is any point inside the circle with center p and radius p(p, E), g(E) lies within the circle Izi = 1 and does not intersect the ray of the negative real axis from -1 to oo, so that u(g, -1) = 0 by (1.1). (1.4) If 0(a) = fi(b), then µ(¢, p) = 0 for all points p in the unbounded component Q of Z - E. For there exists a point p1 e Q which is the origin of a ray lying wholly in Q. Thus by (1.1) and (1.3), 4u(0, pl) = 0 = µ(O, p). (1.5) For any homeomorphism h(ab) = a# which preserves sense [i.e., h(a) = a], we have µab* P) = JAp(Oh-1, P)
For if 01(x) = Oh-1(x) and u1(x) = uh-1(x), x e xj9, where e"(=) _ O(x) - p, x e ab, we have euA-1(:)
_ Oh-1(x) - p = 41(x) - p,
x e 00,
and
u10) - ul(a) = uh-'(fl) - uh-1(a) = u(b) - u(a). (1.6) Given #(x) = f4(x), x e ab, where and f are continuous on ab and C(ab) respectively. If t; (a) = fi(b), then (a) may be taken as an arbitrarily given point of H = d(ab) without affecting the value of udb* p).
For let q e H
- (a) and let at eWe suppose, as we may by
(1.5), that ab is an interval and let # = b - a + a. Then if e* z) = (x) - p is an admissible representation on ab, we define 01(x) = /(x), for at < x < b
#1(x)=O(x-b+a),forb<x
u1(x) = u(x - b + a) + u(b) - u(a) for b < x < .
TOPOLOGICAL INDEX
60
[CHAP. V
Then 1(z) and u1(x) are continuous on m fl and we have e" ltm = e t:) = O(x)
for aSxS - p = 01(x) - p eultx = ea(x-D+a) +u(b)-u(a) = elite)-ubta) euts-b-ra) = O(x - b
b
+ a)
= 01(x) - P, for b< x S P, since e" (b)-°(a) = 1; and
u1(9) - ul(a) = u(a) + u(b) - u(a) - u(a) = u(b) - u(a). 2. Traversals of simple arcs and simple closed curves.
Let t be a simple arc and let f(x) be any continuous function from t to the complex plane. If we understand by a traversal of t a homeo. morphism h of an interval ab onto t, then since as shown in §1, ,u b(fA, p) =
p), it follows that µb(f h, p) can be computed most simply directly from t and that its value depends only on the sense a, f or f, at on tin which it is taken, where a and f denote the end points of t. Thus we have µab(fh, p) == µ t(A p) =
µp.(.f, p)
when h(a) = at. Accordingly, indexes ,u for functions taken over simple arcs t will be indicated directly on t with the sense shown by the order
of the end points a and f of t. In case f is the identity function on t, f will be omitted and we write simply yo(p) or when p is fixed in a discussion.
We note further that for any subdivision at = x0, z1, x2, of tit is obvious that
,x=f
w-1
p)
µae(f, p) = 0
Let J be a simple closed curve and let f be a continuous function from
J to the complex plane. If we understand by a traversal of J a mapping C of an interval or simple arc ab onto J with (a) = C(b) but with C-1(y) unique for y e J - C(a), it is readily shown that µb(f , p) depends only on the sense in which C traverses J. That is, if at, f and y are distinct points on J, then u depends on whether, as x moves from a to b on ab, C(x) takes on in turn an even or an odd permutation of the order a, P, y of these values. Further, the value of p is independent of the starting point C(a). Indeed we have (2.1) For any two sense agreeing traversals C and bl of a simple closed curve J we have
MA, P) = PVC" P).
For let C and C1 be sense agreeing traversals of J. Let = 01 = Al. Then supposing, as we may by (1.6), that C(a) = i l(a1) (where
TRAVERSALS OP SIMPLE ARCS
§ 2]
61
41 maps albs onto J), h(x) = 1 1 4(x) for x a, b, h(a) = a1, h(b) = b1 isypa sense preserving homeomorphism of ab onto a1b1. Whenoe ¢h-1 fCC-1 S1= fS1 = 01; and by (1.5), Fa * P) = 1Aa,b,(O1, P), which is our desired conclusion.
Thus for any two traversals i; and 41 ofJ we have µ(fC, p) = f µ(f ,. p)
the sign being + or - according as and S1 agree or disagree in sense. Thus if a, f and y are any three distinct points of J and a#, #y, ya denote the arcs of J containing just two of these points, then for any traversal C of J we have
µ(fC, P) = ±[µ.#(f, P) + u ( f, p) + ,u (f, p)l (*) p) for the index taken over J in the sense Accordingly we write
giving the + sign in (*), i.e., when an even permutation of the order at, A, y of these values is taken on in turn as x traverses the parameter p) for the opposite sense. range ab, and Now if C is a circle with center p and radius r, it follows from (2.1) that any traversal of C agreeing in sense with that defined by Z(x) - p = 62w1z+,Og r, 0
S x < 1,
x real,
has index µ(C, p) = 21ri and all other traversals of C have index -21ri as they agree in sense with 4(x) - p. (Here we are taking the function f to be the identity mapping f (z) = z and the bar indicates the complex conjugate.) A similar conclusion holds in the case of any simple closed curve as we now show (2.2) For any traversal C of a simple closed curve J in the complex plane and any point p within J we have I 2Tri
µ
P)
=
+1.
Proof. Let apb be the interval of the horizontal line through p which contains p and has only its ends in common with J and where a is the right end of apb. Let C be a circle with center p and lying entirely within J and let u and v be the points of ap and pb which are on C. Let a and t be interior points on the upper and lower semi-circles respectively of C with ends u and v. Let x and y be interior points on the arcs of J from a to b such that p is without the simple closed curves
J1=au+usv+vb+bxa and
Jy=au+utv+vb+bya
TOPOLOGICAL INDEX
82
(CAP. V
Now since by (1.4), for any traversals and for the identity function we have IAJ,(p) = IAJ,(p) = 0,
we obtain by direct computation of indexes over the arcs concerned µaxb = fUau + jU. + ft'ob
Ybw = Pbv + Yotr + F.Adding we get, since end terms cancel, +.r = luao,bt+a = lua¢b + ,bva = ua,v + lpvt« = fA+c = 2irI
Similarly, of course, µ_,r =
µ_C = -27ri. DErn,TTzON. If J is a simple closed curve in the complex plane,
by a positive traversal of J is meant a traversal
of J for which
(1/2m)µ(4, p) = 1 where p is any point within J. It is now clear from (2.1) and (2.2) that if f is any continuous complex valued function defined on J, then all positive traversals of J give the same value p j(f, q) of µ(f, q) for anyq not inf(J) and similarly all negative traversals give this index the same value p_.r(f, q) and that
IA-.r(f, q) _ -µ,,(f, q). Thus we can compute the index µ directly from J; and if a, fl, y are distinct points of J, we have ±p.r(f, q) = paArac (f, q) _ /Lp(f, q) + µ6,(f, q) + 1a (f, q), the sign being + or - according as the order a, ,8, y determines the
positive or the negative sense on J. If J and J' are simple closed curves in a plane and a, b, c and a', b', c'
respectively are triples of distinct points on J and J', the senses determined by the orders a, b, c on J and a', b', c' on J' are said to agree provided that if p and p' are points interior to J and J' respectively then (P) In other words, the senses a, b, c and a', b', c' agree or disagree according as the signs of 1/27ri times the corresponding circulation indices agree or disagree.
Now let J' be within J. We then can take p = p' since p' is within both J' and J. In this case we show (2.3) The senses a, b, c and a', b', c' and J and J' agree if and only if there exist disjoint simple area aa', bb', cc' lying except for their ends in the annular region between J and X.
§ 2]
TRAVERSALS OF SIMPLE ARCS
63
Proof. Suppose such arcs exist. Then since p is without each of the simple closed curves
J, = ab + bb'+b'a'+a'a
J0=be+cc'+c'b'+b'b Jb = ca + aa' + a'c' + c'c, we have 0 = µ r (p) + ur,(p) + u,,(P). Thus by direct computation with cancellation,
0 = f4db + /5b, + luca + 14W.' + µc'b' + hilt' = PAP) - UJ'(P) This gives our assertion that a, b, c agrees with a', b', c'. Now suppose the senses a, b, c and a', b', c' on J and J' respectively
agree, but that area aa', bb', cc' do not exist as asserted. Then there do exist disjoint area aa' and bb' lying except for their ends in the annular region between J and X. Now c can be joined to a point c" on c' - a' - b'
by an arc cc" lying in A except for its ends and not intersecting aa' or bb'; and since c' cannot be so joined, c" must be on the arc a'b' of J' and not on a'c'b' of X. But then by the first part of this proof, the senses a, b, c and a', 6', c" would have to agree. Clearly this is impossible because a', b', c" and a', 6', c' are opposite senses on J' whereas they both
agree with a, b, c on J. This contradiction shows that area aa', bb', cc' must exist as asserted. (2.31) CoBOLLARY. Let J and J' be simple closed curves bounding
an annular region A in a plane and let a, b, c and a', b', c' be triples of distinct points on J and X. If the senses a, b, c and a', b', c' agree so also will the senses h(a). h(b), h(c) and h(a'), h(b'), h(c') on h(J) and h(J') agree where h is any homeomorphism of A + J + J' into a plane. Thus "agreement in sense" is a topological invariant in this situation. DEFINr oN. If two 2-cells A and A' with edges J and J' respectively
lie together in the interior of a 2-cell E with boundary C, we may now define traversals C and 1' of J and J' as agreeing in sense provided each agrees with one and the same traversal of C. This definition will be complete as soon as we define "agreement in sense" for the case where A' lies entirely interior to A, which we now proceed to do. In any situation in which two simple closed curves J and J' are disjoint and constitute the boundary of a "cylinder" surface D, i.e. a set homeomorphic with a closed circular ring, senses a, b, c and a', b', c' of J and J' are said to agree provided there exist disjoint simple arcs aa', bb', cc' lying except for their ends interior to the surface D. From (2.3) we have at once (2.32) If A is a 2-cell, "agreement in sense" for two Simple closed curves interior to A is invariant under any homeomorphism on A.
[CHAP. V
TOPOLOGICAL INDEX
64
(2.4) If J = xay + xby and J1 = xay + xcy are simple closed curves in the plane or on a 2-cell having an are zay in common, then (i) If the interior of J contains the interior of J1, then positive traversals of J and Jl agree in sense on the common arc xay. (ii) If the interiors of J and Jl are disjoint, positive traversals of J and J1 are opposite in sense on the common arc xay.
Proof. By (2.32) there is no loss in generality in assuming that J and J1 lie in a plane. Statement (i) is a direct consequence of (2.3) because a circle C may be taken interior to both J and J1 and disjoint area xx', aa', yy' in the annular region between J1 and C will exist with x', a', y' e C. Then if yaxby is the positive traversal of J, yaxcy must be the positive traversal of J1 since y, a, x agrees in sense in both cases with the sense y', a', x' on C. To prove (ii) let p and pl be points interior to J and J1 respectively. We may suppose the positive traversal of J is in the order yaxby. Then xbycx is the positive traversal of the simple closed curve K = xby + xey; and since K also encloses pl, then xbycx must be the positive traversal of J1 by (i). As xay and yax are opposite, we have established (ii).
DExn4rrioN. A homeomorphism h(x) of a plane X into a plane Y is positive (negative) provided that if fi(t) is a positive traversal of a simple closed curve C in X, then hi(t) is a positive (reap. negative) traversal of l(C) in Y. In other words, h(x) is positive provided (*)
1uo[hC, h(p)] = µaM p]
for any C,
and p.
As an immediate consequence of (2.3) we have (2.5) TaEOREM. Given a homeomorphism h(x) of X into Y (planes). If there exists one simple closed curve C in X and a traversal 4 of C such that (*) holds, then (*) holds for arbitrary C, t; and p so that h is a positive homeomorphi3m.
Also as a consequence of Note (iv) of § 1 just following the definition of the index u, we have
(2.8) Any non-singular linear tran8formation of a complex plane into itself is a positive homeomorphi8m.
Finally, we record the immediate result.
(2.7) If hl(X) = Y and h2(Y) = Z are positive homeomorphisms, so also is h(X) = Z where h(x) = h2hl(x). For since h1 is a positive traversal of h1(C) in Y, hah14 (= ht;) is a positive traversal of h$h1(C) [=h(C)] in Z because h2 is positive.
3. Index invariance.
(3.1) THEOREM. If #(x) and 1(x) are mappings of an are or interval
INDEX INVARIANCE
§ 3]
65
ab into Z satisfying I#1(x) - #(x)I < O(x) - pl on ab and if #(a) and /(b) = #1(b), then
O1(a)
Ilab(#, P) = I0ab(011 P)
Proof. If we take a representation eu(x) _ 4(x) - p,
x e ab
,
then
u(x) = log 10(z) - pl + i9(x).
Since for each x the circle with center #(x) - p and radius I#1(x) - O(x)l neither contains nor encloses the origin, there is a unique 01(x) satisfying etao 141(z) - Al
+iel(x) = #1(x) - P
9(x) - 7r/2 < 91(x) < 9(x) + a/2 If we set ui(x) = log l#1(x) - pI + i61(x), x e ab, then u1(x) is continuous and we have (iii)
u(b) - u1(b) = i[9(b) - 01(b)],
since
#(b) _ 01(b)
(iv)
u(a) - ul(a) = i[O(a) - 91(a)],
since
¢(a) _ 01(a).
Subtracting (iv) from (iii) we get (v)
IAb(O, P) - paa(01, p) = i{[9(b) - 01(b)] - [9(a) - 91(a)]} = 21ri(kb - ka),
where kb and ka are integers. But also by (ii) I9(b) - 91(b)l and I0(a) -91(a)
are each
(3.11) Co$oLLARY. If O(x) and 41(x) are mappings of ab into Z satisfying I#1(x) - Owl < lo(x) 01(a) = 41(b), then
- pi on ab and if #(a) = #(b) and
/6b(#, P) = µab(#1, P)-
(3.12) CoaoLLABY. If f(z) and fi(z) are mappings of a simple dosed curve C into Z and if l fl(z) - f(z)I < If(z) - pi on C, then
µc(f, P) _ /c(f1, P) For if C is a traversal of C and 0 = f , 01 = f1 C, (3.11) gives Iib(fS, p)
= Isb(fS, p) which is identical with our desired conclusion by (3.1). (3.13) CoBoLLAEY. As a function of f, uc(f, p) is constant on each component of the mapping space (Z - p)C, i.e., the space of all mappings of a simple closed curve 0 into Z - p.
TOPOLOGICAL INDEX
66
[CHwr. V
(3.14) Similarly, as a function of f, µab(f, p) is constant on each component of the space of all mappings of the simple are ab into Z - p which agree on the end points of ab.
(3.2) Let f(x) be a mapping of a separable metric space X into Z and let ab and albs be simple arcs in X. If there exists a homeomorphism h(ab) = albs with h(a) = a1 and satisfying If(x) - fh(x)I < If(x) - pj,
x e ab,
and f(a) = fh(a) = f(a1) = a,
f(b) = fh(b) = f(b1)
then
µab(f, p) = µa1b1(f, P).
Proof. By (3.1) we have µab(f, P) = µab(fh, p)+
and since h is a sense preserving homeomorphism of ab onto albs µab(fh, P) = µa1b1(f, p) by (1.5).
By iteration of (3.2) we get
(3.21) CoEora BY. Given a mapping f (x) of X into Z as in (3.2) and two points a, P in Z - p. For any sequence of simple arcs albs, a$bs, joining f-1((x) and f-1(f) [i.e. a{ a f-1(a), bi E f'-'(#)] for which sense pre. serving homeomorphisms h{(a{bi) = at +1bc +1, i = 1, 2, exist satisfying
1f(x) -fhi (x)I < 1f(x) - pl,
x e sbs,
we have
µa,bl(1, P) = µa b, (f, P)
(3.3) Given a mapping f(x) of a separable metric space X into Z - p and two points a, # e Z - p. If H is any continuous family of simple arcs
H=[h(8,t)],
0<8< 1,0
where h(0, t) = at, h(1, t) = P for all t and h(8, t) continuous in (8, t), for any two arcs t1 and t2 whatever in H we have
4u (f, P) = µt,(f, P) Similarly for simple closed curves in X, corresponding to (3.2) we have (3.4) Let f(x) be a mapping of a separable metric space X into Z and let C and C1 be simple closed curves in X. If there exists a sense preserving homeomorphism h(C) = C1 satisfying
f(x) -fh(x)l < If(x) - pl, then
µ0(.f, P) = µ0,(f, P). Corresponding to (3.4) we have
x e C,
TRAVERSALS OF REGION BOUNDARIES
§ 41
67
(3.5) For any mapping f of X into Z - p, µc(f, p) is constant on any continuous family H = [h(s, t)] of simple closed curves in X.
4. Traversals of region boundaries and region subdivisions.
Let R be an elementary region, i.e., a bounded plane region whose boundary consists of a finite number of disjoint simple closed curve
+ J and is thus , J,,, where J0 encloses J1 + J2 + the outer boundary of R. By a positive traversal of Fr(R) is meant , J,, which is positive on J0 and a collection of traversals of J0, J,,. It is clear from the above that if f is any con, negative on J1, and , tinuous function defined on Fr(R), and 1; = (%a,
J0, Ji,
o,
,
r') are arbitrary positive traversals of Fr(R) then F4M, P) _
P) = A(A ', P),
µa,b,(A., P) =
i.o.w., i(ft;, p) is independent of the particular positive traversals.
Now let f be a continuous function defined on a graph 0 which includes Fr(R) and lies in R and subdivides R into a finite number of regions R1, , R. each bounded by a simple closed curve C{ of G and where the intersection of any two of the C, is either a vertex or an edge of G. A set A is a graph provided it is the sum of a finite set V of points, called vertices, and a finite number of open arcs al, a$, , Mn, called edges, so that the two end points of each edge a are distinct and belong to V.
Next let
= (ii,
,
,,,) be a traversal of 0 obtained by taking
positive traversals of the C. This is called a positive traversal of such a G. Then each edge of G which lies interior to R lies on exactly two of the C, and by (2.3), (ii), above is traversed in opposite senses by l;. Further, any edge of G lying on J0 lies on just one C; and is traversed positively by by (2.3), (i), since R{ lies interior to R; and similarly an edge of G lying on a J{ with i 1 is traversed negatively just once by 4, since the RJ is exterior to Jt, where R, is the interior of the C, containing the edge. Hence we have the (4.1) LEMMA. For any positive traversal of a subdivision G of the elementary region B and any continuous complex valued function defined on G we have in
Ur, (fr, P) _ 2Pe'(,fl, P) = #a(fi, P)
F/Pr(R) (X P) _ 0
1
where p s Z - f(G) and where, in general, px(o, p) refers to the index computed over those intervals mapping into X under the traversal C. Further FiFr(R)(fC, p) is independent of the particular traversal .
68
(CHAP. V
TOPOLOGICAL INDEX
In view of the last statement together with the fact that µF,18) is also independent of the particular subdivision G, the 4 will usually be omitted and we shall write /AF,(R)(f, p). (4.2) THEOaEM. Let R be an elementary region with boundary C and let f be any mapping of It onto a set E in the complex plane Z. Then
for any point p in Z - E, µC(f, p) = 0.
Proof. Let G be an admissible subdivision of R with mesh small enough that for each region R{ into which R is divided by 0 we have aU(R{)] < p(p, E).
Then since p is in the unbounded component of f( {) for each i, we have MRi)(fC, p) = 0 for each i, where % is any positive traversalof 0; and thus 4u (f, p) = GpF(R,)(& p) = 0 by the lemma (4.1). (4.21) CoaOLJARY. If is a traversal of Fr(R) which is positive on each of the curves J{, 0 < i < n, n
µr.(& P) =
:SµJ,(A, p).
1
(4.22) CoRouAEY. If B is an elementary region with boundary C and f is any mapping of C into Z - p which admits an extension over R into Z - p, then i c(f, p) = 0. 5. Homotopy. Index invariance.
Two mappings f (x) and g(x) of one metric space X into another one Y are said to be homotopic (on X relative to Y) provided there exists a function f (x, t) defined for each x e X and t e (0, 1) and with values in Y which is continuous in (x, t) everywhere and satisfies h(x, 0) = f(x),
h(x, 1) - g(x)
for all x e X.
In other words, f and g are homotopic if they can be joined by a continuous family of mappings of X into Y or provided there is a continuous function
h of the cartesian product space X X I into Y, where I = (0, 1), which agrees with f(x) on X X (0) and with g(x) on X X (1). We shall write f = g to mean "f and g are homotopic". Now in case the space X is compact so also is X X I ; and if f ^ g, uniform continuity of h(x, t) yields that for any to E I and e > 0 a b > 0 exists such that p[h(x, t), h(x, t0)] < E
69
HOMOTOPY
§ 5]
for all x e X provided it - teI < a. Accordingly, by (3.12) and (3.13) we have
(5.1) THEOREM. If f (x) and g(x) are mappings of a simple closed curve C into Z - p and f and g are homotopic (on C rel. to Z - p), then z0(f' P) = iC(g, P).
Our next objective will be to show the invariance of the index u when the image plane is subjected to an arbitrary positive homeomorphism.
Some further developments on homotopy are needed as preliminary steps toward this objective. We next prove (where (0) denotes the origin in each case):
(5.2) Any mapping f(x) of a (punctured) complex plane X - (0) into another such punctured plane W - (0) is homotopic [on X - (0) rel. W - (0)] to the mapping w = x" for some integer n = 0, t1, f2,--Proof.
Let Z be another complex plane with 8- and t-axes so that
z = s + it, and let Ze be the region - 00 < e < 00, -Tr S t S IT of Z. The function x = es then maps Ze onto X - (0) continuously. Accordingly f(es) is a mapping of Z. into W - (0). Hence by (1.2) this mapping has an admissible exponential representation (i)
f(es) =
eu(z),
where u(z) is continuous and maps Zo into X. Now since e'+M = e'-*t for every real 8 so that f(e'+r:) = f(e'-"'), we have by (i) u(8 + iri) - u(s - Tri) = 2irin(8), (ii) where n(s) is an integer for each 8. But since n(8) is continuous in 8 on the whole s-axis and this axis is connected, n(8) must be a constant n for all 8. Define:
w(x) = u(z) - nz, where x e X - (0) and x = es. Then w(x) is continuous and is single valued by (ii), even though there are two values of z, namely s + rri and 8 - iri, for each x on the negative real axis of X (here s = log IxI). Thus by (i) and x" = e"z we have f(x) = eu(z) = xneu(z)-nz = xnew(z)
for each x in X - (0). Hence we have only to define h(x 0) = x"e(1-8)w(z)
0<0<1,
to obtain a continuous function in (x, 0) on X X I to W - (0) reducing to f(x) for 0 = 0 and to x" for 0 = 1. Thus f(x) x" on X - (0) rel.
W - (0).
TOPOLOGICAL INDEX
70
[CHAP. V
(5.21) COROLLARY. If f is a homeomorphism of X - (0) into W - (0),
then f is homotopic either to the identity mapping x or to the reciprocal mapping 1/x. For if we define 4(t) = f(e"),
-7T < t < IT,
then is a simple traversal of the simple closed curve f(C) where C is the circle Ixl = 1. Thus since (0), the origin in W, is necessarily within the simple closed curve f(C), by (2.2) and (ii) above, we have, (taking s = 0 and recalling that n(8) is a constant n) ±27ri =
(0)] = u(7r) - u(-7r) = 2niri. Accordingly, n = ±1 and the corollary is proven. (5.22) If f is a positive (negative) homeomorphism, f i8 homotopic to the identity (reciprocal).
(5.23) If f is any homeomorphism of the plane X into the plane W, then for any p e X we have either f(x) - f(p) = x - p or f(x) - f(p) c[x - p]-1 according as f is a positive or a negative homeomorphism.
(5.3) THEOREM. If a mapping h(y) of a metric space Y into itself is homotopic to the identity mapping, then for any mapping f(x) of a metric space X into Y, we have f ; hf (on X rel. Y).
Proof. By hypothesis there exists a continuous function g(y, t), y e Y, 0:!5,' t < 1, such that g(y, 0) = y, g(y, 1) = h(y) for each y e Y. Then if we define
k(x, t) = g[f(x), t]
for x e X, 0 < t < 1,
k(x, t) is continuous in (x, t) and k(x, 0) = g[ f(x), 0] = f(x), k(x, 1) = g[ f(x), 1] = hf(x), for all x e X.
Whence f ^r hf on X. Clearly the same method of proof gives (5.31) If h(y) and l(y) are mappings of Y into Z and h ` l on Y rel. Z, then for any mapping #(X) = Y we have
ho = 10
(on X rel. Y) where X, Y and Z are arbitrary metric spaces. (5.4) THEOREM. If f is any mapping of a simple closed curve C into the z-plane Z and h i8 any positive homeomorphism of Z into itself, then
foranypeZ-f(C), (#)
luc(f, p) = ,c[hf, h(p)].
HOMOTOPY
§ 61
71
Proof. By the notes in § 1 following the definition of index and by
virtue of the fact that f
hf clearly implies f - p ^r hf - p, we may
assume that p = 0 (the origin). In case h(0) = 0 our conclusion results at once from (5.1), because by (5.22) h(z) is homotopic to the identity z and thus by (5.3) f and hf are homotopic on C rel. Z - (0). If h(0) 0 we define, k(z) = h(z) - h(0). Then since k = lh where l is the linear transformation 1(z) = z - h(0), it follows by (2.6) and (2.7) that k(z) is likewise a positive homeomorphism of Z into itself; and since k(0) = 0, the case just treated gives
µc(f, 0) = µc(kf, 0) = µc(lhf, 0).
(t)
Thus by Note (iv) of § 1, since 1-1 is a linear transformation,
,uc(lhf, 0) = µc(hf,1-1(0)) _ 14c[hf, h(0)]
This combined with (t) gives (*) for p = 0 and hence the theorem is proven. REFERENCES
In connection with the material in this chapter, particularly that dealing with the topological or circulation index, and for references to other related sources see Eilenberg [1], Kuratowski [1, 2], Morse [1], Morse and Heins [1] and Whyburn [2, 3].
VI. Differentiable Functions 1. Index near a non-zero of the derivative. (1.1) Let f(z) be a function continuous in a region R and differentiable at a point zo in R. If f'(zo) # 0, for any sufficiently small simple closed curve C enclosing zo we have (1/2zri)u (fC, wo) = 1, where wo = f(zo) and where C is any positive traversal of C.
For we can write
(*) f(z) - wo = (z - zo)'ff'(zo) - e(z)] where lim,--so e(z) = 0. If C is taken sufficiently small it will lie inside R and neither sideey of will vanish on C. Then by Note (iii) of V, § 1, µC(fC, w0) = PC(Z - Zo, 0) + JAc[f'(zo) - C(Z), 0)
= RC(z, zo) + µC[-E(z), f'(zo)]
The first term on the right is 27ri by V, (2.2); and the second term is zero for C sufficiently small by V, (1.4), since e(C) lies in an arbitrarily small neighborhood of 0.
2. Measure of the image of the zeros of the derivative. (2.1) Lx&m!k. Let f(z) be differentiable on a subset E of a closed square D of side a. If a is any positive number, E is representable in the form
E where
E{,
8(Ei)2 S 2a2 and where each Ei contains a point zi such that z
f (z )
- zt
r)
-f'(z{) < e
is satisfied for all z e Ei.
Proof. For each n > 0 let D be the subdivision of D into 4" equal closed squares. Now for each z' e E there exists an n > 0 and a set Ee which is the union of 1, 2 or 4 squares of D" each containing z' and such that z' is interior to Ee and z' ZZ' )-f'(z)I <e for all zeE,.
x f()-
(
72
INDEX
§ 31
73
By the Lindelof Theorem, E is contained in the union of a countable number of these sets Ez., say E e Ezi. We now arrange all those squares (if any) of Dl which appear in any of the (E_,) in a sequence : , E., respectively, letting , S., intersecting E in El, Es1 Si, S2 , ; z in each case so that z{ e Ej be the corresponding points ,z z1, zy, and E, a Ez, Similarly, all squares of D$ which appear in any of the (Es;) but which are not contained in Jii Si are arranged in a sequence respectively and the corresponding points z +1, , z,, similarly chosen. Next all squares Ez, but not contained in 1i' S, are ordered of D. appearing in any of
S intersecting E in
EAI+$, -
-
-
S +v ... , S intersecting E in E. +i, dehnitely. TLien clearly E _ areas. Also for each i, z{ e E, and
,
z; _ Ej and
B,, , and so on ina(Ei)= S 2a= by
f(z) -Ax{) - f'(zi) < e for z e Ei since Er c Ezi where zj = z
x - zi
(2.2) If f'(z) = 0 for all z e E, then the 2-dimensional measure of f(E) is 0.
Proof. Let a > 0 be arbitrary. Choose e < 1/2a V and apply lemma (2.1). Then for z e Ei we/ have If(Z)
-f(Z4)12 < E= IZ - Ztls
6U(Ee)]2 S l.u.b. 4 If(z) -f(zr)I2 S 4c26(Ei)s S 4E2
8(E{)$ <
as
2.2a2 = 8.
Since f(E) = ,2f(Ei), this gives our conclusion. (2.21) CoBOL ARY. The image of the set of all zeros of the derivative of a differentiable function f(z) can contain no open set in the w-plane.
8. Index. (3.1) ThEomm. Let f (z) be continuous inside and on a simple closed curve C with interior R and differentiable on the inverse of a dense open
subset Be of E = f(R + C). For any p s E - f(C) and any positive traversal C of C, we have 2rri 1AC(A, P) > 0.
Proof. Let Q0 be the component of E -f(C) containing p. As Qe is open in E, Qo Ee contains a region Q in E and f-'(Q) is open in B. Since then there is a component of f-'(Q) on which f is not constant, there is a point ze a f-'(Q) where f'(ze) # 0 (an easy consequence of the
(CHAP. VI
DIFFERENTIABLE FUNCTIONS
74
mean value theorem for real functions). Then from (1.1) it follows that if J is a sufficiently small circle lying with its interior I in f-1 (Q) enclosing zo we will have 1
we) = 1
for any positive traversal to of J, where wo = f(zo). Then by V, (1.4), wo lies in a bounded component D of W - f (J); and thus by V, (1.3) and (4.2) every point of D lies in f (I) and hence in Q. Thus Q contains a square plus its interior. Accordingly by (2.2), Q must contain a point q such that f'(z) does not vanish on the set f -1(q). Then f -1(q) must consist of a finite number , C. be circles enclosing q1, , q,. in R. Let C1, C21 of points.g1, q2, , q,,, respectively and lying within C and such that no one C, intersects any other or the interior of any other C, and small enough so that [see (1.1)],
)µc, (ft, q) = 1
(1/2
for any positive traversal t; of C,. Now by V, (4.21), since f does not take the value q in the elementary region between C and C,, . , Cm, we have 1
1
AM µc(fR, q) _
2
pc,(ft, q) = m > 0-
Whence I 1ri
2-µ0(f2;,p)=m>0byV,(1.3),since peQ.
(3.11) COEOLLARY. If f(z) is continuous inside and on a simple closed
curve C and differentiable at all points of the interior R of C lying in the inverse of an open dense subset of f(R), then f(R + C) consists of f(C) together with a collection of the bounded complementary domains of f(C) in the plane W. Thus in particular if, f(z)I < M on C then I f(z)I S M in R. (3.2) Let w = f (z) be continuous in a region R and differentiable at all points of the inverse Re of an open dense subset of f(R) and let C be a simple closed curve lying together with its interior I in Ro. If zo is a point of
I such that f'(zo) = 0 and f(z) 0 f(zo) = wo at all z e C + I, 1
2rri
FC(f, wo) > 1.
Proof. We define a function g(z) for z e C + I as follows:
g(z) _ [f(z) - wo](z - zo)-1, g(zo) = 0.
if z
zo
1 41
LIGHTNESS OF DIFFERENTIABLE FUNCTIONS
75
Then g is continuous in C + I, since g(z) -- f'(zo) = 0 as z --o. z., and differentiable in C + I - zo. Thus g is differentiable at all inverse points for all values which g takes in C + I except the origin w = 0, and g(z) = 0 only for z = zo. Accordingly, by (3.1), we have (*)
2rri µc(9'
0) > 0.
Now the relation
f(z) - WO = (z - zo)9(z)
is satisfied identically for z e C + I. Whence, by Notes (ii) and (iii) of Chapter V, § 1, we have
(t)
µc(f, wo) = µc(z, z0) + µc(9, 0).
Since ac(z, zo) = 21ri by V, (2.2), (t) and (*) give
27n
1AC(f, wo) = 1 +
-; µc(9, 0) > 1.
(3.21) COROLLARY. If f(z) is differentiable in a region R, f'(zo) = 0
for a point zo in R which is an isolated point of f-'f(zo) if and only if (l/2in)µc[f, f(zo)] > 1 for all sufficiently small circles C centered at zo.
This results as once from (3.2) together with (1.1). 4. Lightness of differentiable functions. and differentiable in a region R (4.1) THEOREM. If f(z) is then f is light, i.e., non-constant on any non-degenerate continuum in R.
Proof. Suppose on the contrary that f(z) = a for all points z on some non-degenerate continuum in R. Then if Ro is the subset of R on which f(s) # a, Fr(R0) contains a non-degenerate continuum M lying wholly within R. Let zo e M, and let N be a subcontinuum of M of diameter or < ?a' p[zo, Fr(R)] which is irreducible between zo and z1. Let x be a point on N with p(zo, x) = 4p(zo, zl) let y be a point of R. with p(y, x) <
Jp(zo, x), and let C, be the circle with center y and radius 4p(zo, x). Then since x is within C and both zo and zl are without C,,, C, must contain at least two points s and t on N. For otherwise the part of N without and on C would be a proper subcontinuum of N containing both zo and zl. Now since y e Ro and B0 is open, there exists a point z' of Ro within C, and on the perpendicular bisector of the segment 8t, so that a circle C
with center z' passes through both s and t, and so that the angle sz't
DIFFERENTIABLE FUNCTIONS
76
[CHAP. VI
is a rational multiple of 27r, say 48z 't = 21rp/q where p and q are integers prime to each other. Next consider the function a-1
9(z) = 11 {f[(z T-0
- z')es*{nr/a + z'] - a).
Since
I(z - z')es*tyrla + z' - zol < Iz - z'I + Iz' - zol S Iz - z'I + }izo - ztl < Iz - z'I + AP[zo, Fr(R)], g(z) is differentiable inside the circle S with center z' and radius 5a. Now since g(z) = 0 for z s N whereas g(z') 0 0, there exists a region Q containing N and lying within the circle with center z' and radius 2a
and such that lg(z)I < d = JIg(z')I for z z Q. Let 8t be a simple are (polygonal line segment) in Q joining 8 and t and let h(x) be a homeomorphism of the interval (0, 1) onto st with h(0) = 8. For any x on the interval (1, q) we define h(x) = [h(x - r) - z']e;w'r91a + z' for r < x S r + 1, (r = 1, , q - 1). Then h maps the interval I = (0, q) onto a set E and Ig(z)I < 6 on E since g(E) = g(st). However we note that if e,.t:)
= h(x) - z',
0S XS 1,
is a representation of h(x) - z' on (0, 1), then e (W-r) +
h(x)
- z' for r < x
r + 1, r = 0, 1,
,q-
1
P(,r+i)(h, z') = q[u(1) - u(0)] 0. Since h(q) = 8 = h(0), it follows by V, (1.4), that z' is in a bounded complementary domain of E. Thus z' is within a simple closed curve (polygon) J contained in E. But since Ig(z)I < a on J, this must hold also inside J by (3.11), contrary to the fact that Ig(z')` = 26. This Accordingly u(o,a) (h, z')
contradiction establishes our theorem. 5. Openness of differentiable functions.
A mapping f(A) = B c B0, where A, B, and R. are topological spaces, is open (or interior) provided the image of every open set in A is open in B and is strongly open (or strongly interior) provided the image of every open set in A is open in Bo. (5.1) ThEousm. I f f (z) is differentiable in a region R, f is strongly open in R.
Proof. Let U be any open set in R, let wo E f(U) and zo e U f-1(wo). Since by § 4, f is light in R, there exists a simple closed curve C lying together with its interior I in R and such that f (z) # wo on C. By (3.11), f (C + I) and therefore also f (U), contains the complementary domain of f(C) to which wo belongs. Accordingly f(U) is open in the tv-plane. This establishes our theorem.
APPLICATIONS
§6]
77
6. Applications.
Let w = f (z) be analytic and non-constant in an open set R of the complex plane. We define m(z) = if (z)I, z e R, and call m(z) the modulua function. Obviously it is continuous. As an immediate consequence of the strong interiority of f (z) on R we have THEOREM. The mapping m(z) of R into the non-negative real axis i8 strongly interior. This yields at once the following for the most part standard results. (1) For any a e R and any neighborhood V of a in R, there exists z e V
such that m(z) > m(a); and if m(a) # 0, there exists zl s V such that m(zi) < m(a). (2) m(z) has no relative weak maximum points and no relative weak minimum points other than zeros. Note. For real valued functions on R the absence of relative maximum
and minimum points implies strong interiority. (3) Mexnimi MODULUS THEOREM. If f (z) is analytic in a bounded open set G and continuous on Fr(G) and I f (z)1 < M on Fr(G), then I f (z)I <
MforzeG. For otherwise, by (1), m(z) exceeds M for some z e G. Then m(a) _ max2ea m(z) for some a s G, which is impossible by (1). (4) If f(z) is analytic in a bounded open set G complementary to a level
curve L :I f(z)I = k and continuous on Fr(G) where Fr(G) C L, f(z) has at least one zero in G.
For by (3), I f (z)I < k on G. Thus m(a) = mina m(z) for some a e G; and by (1), m(a) must be 0. (5) Every (non-constant) polynomial has at least one zero.
For let wo 0 0 be a value of a polynomial P(z). The open set G defined by IP(z)I < Iwcl is bounded, since P(z) -- oo as z -). oo, and Fr(G) is contained in the level curve IP(z)I = Iwe.I Thus the result follows from (4).
(6) Let B be any bounded region with connected boundary B and let f(z) be analytic on R and continuous on B. Then either m(R) C m(B) or else f has at least one zero in R.
For the set m(B) is a continuum and hence is an interval a S x of the non-negative real axis; and if it does not contain m(R), we have a > m(ze) > 0 for some ze e R by (3). Thus m(a) = minuIt m(z) for some a s R and m(a) = 0 by (1). REFERENCES
1-5. Ursell and Eggleston [1], Stoilow [1], Titus [1] and Young, Whyburn [2, 4]. § 6. Whyburn [4]. Compare also, Morse and Jenkins [I].
VII. Open Mappings 1. General Theorems. Property 8 and local connectednesa. (1.1) THEOREM. If A is locally compact, separable and metric and
the mapping f(A) into B is strongly open on f-'(K) into K, where K is a given continuum in B, then any compact component of f-'(K) maps onto K.
Proof. Let H be a compact component off-'(K). Evidently f (H) C K.
Now let V be any neighborhood of H. Choose an open set U n H so that U is compact, U is contained in V, and Fr(U) -J'-'(K) = 0. Let
f-1(K)-U = W. Then W is open in f-'(K); and f(W) is open in K, since f is open on f-'(K). Also W is compact and hence f(W) is compact.
Thus f(W) is closed in K. Therefore f(W) = K. Whence, f(H) = K. for if f (H) were a proper subset of K, so also would be f (W) for V sufficiently near H. (1.11) If in addition B is locally compact and locally connected, then for any region R in B such that f is strongly open on f-' (B) into R, each conditionally compact component of f-'(R) maps onto R. For if Q is such a component and x e R, y E f (Q) and xy is an arc in
B, Q must contain a compact component of f-'(xy) which therefore maps onto xy. Thus f(Q) x. (1.2) Let f(A) = B be open where A is a locally connected generalized continuum. If K is a region in B or a continuum in B having an interior point, then only a finite number of components of f-'(K) can lie in any given compact set H in A.
For let y be an interior point of K. Each point of Hf-'(y) is interior to a unique component of f-1(K). Since Hf-'(y) is compact, it therefore lies in and intersects just a finite number of such components of f-'(K). But by (1.1) and (1.11) every component of f-'(K) in H must intersect Hf-'(y) since it must be conditionally compact. Thus there are only a finite number of components of f-'(K) in H. (1.3) THEOREM. Let f (A) = B be light where A and B are locally compact, separable, and metric. Then for any compact subsets K of B and H of A and any e > 0, there exists a d > 0, such that for any 5.8et B in B intersecting K, each component of f-'(R) intersecting H is conditionally compact and of diameter <e.
For suppose that for some K, H and e no such 6 exists. Then there . of subsets of B each intersecting K
would exist a sequence R1, R2,
78
§13
GENERAL THEOREMS
79
of components of 1/n and a sequence Q1, Q2, respectively such that for each n, Q. intersects H f-1(R1), f-'(R2), and is of diameter > e. Since K and H are compact, we may suppose the chosen so that it converges to a single point p e K sequence B1, R2, chosen so that it converges to a limit set L and also suppose Q1, Q2, which intersects H and thus is not empty. But by continuity of f, we must have L e f-1(p) and thus each component of L would reduce to a single point. On the other hand, since 6(QA) > e for each n and each Q. is connected and intersects H it readily follows from 1, § 9, that any component of L intersecting H is non-degenerate. This contradiction
and with
establishes (1.3). (1.4) THEOREM. Let f(A) = B be open and light. (A and B locally connected generalized continua.) Then for any conditionally compact, locally compact subset N of B having property S and whose interior (rel. B) is dense in N, the sum of all components of f-1(N) contained in any compact set H in A has property S.
Proof. Let Q be the sum of all components of f-1(N) contained in H and let e > 0 be given. Then using K = 19, apply (1.3) and determine 6 > 0 so that any component of the inverse of a 6-subset of N lying in H is of diameter < e. Since N has property S, k
(i)
N = > Nr
where each N1 is a region in N of diameter <6. For each i, the collection (Ni) of components of f- 1(N,) lying in H must be finite. To show this we show first that for each such component Ni, we have (ii)
f(i) = Ns.
If this were not so, then since N4 is a region in N, we could find an arc ab in N1 such that f(Ni) contains a but not b. But this is impossible since, by (1.1), the component of f-1(ab) containing a point z e N; f-1(a) lies in N and would map onto ab so that f(NN) b. Thus (ii) is established. Now since the interior (rel. B) of N is dense in N, N{ contains a region R in B. Then if y e R, and Y = Hf-1(y), Y is compact and each point of Y lies in just one component R1 of f-1(R). Further, each component of f-1(R) which lies in H must intersect Y since it maps onto R. Since these components are open and disjoint and cover Y, they must be finite in number; and since each component Ni in H of f-1(N1) contains a component of f-1(R), the collection (N) is finite as asserted. k Now clearly
QN,;
and since b(Ni) < E for each i and j, Q has property S.
OPEN MAPPINGS
80
[CHAP. VII
(1.5) THEOREM. If A i8 a focally connected generalized continuum,
f(A) = B i8 light and open and Y is any locally connected generalized continuum in B whose interior is den8e in Y, then f-'( Y) is locally connected.
Proof. Let x e f 1(Y), y = f(x). Let e > 0 be chosen that V$E(x) is compact. Then with H = Ve(x), K = y determine B from (1.3) so that any component of f-1[Vs(y)] intersecting His of diameter <e. Then let R be a region about y in Y- V5(y) which has property S. Then by (1.4), the sum Q of all components of f-1(R) lying in V$E(x) has property S. Also any component of f--1(R) intersecting VE(x) is in Q; and since f-1(R)
is open in f-1(Y) it follows that x is not a limit point of f-1(Y) - Q. Thus f-1(Y) is locally connected at x since Q has property S. DEFrnm o . A point p e S is an isolated point of S provided there
exists an open set U containing p such that (U - p)-8 = 0. It is clear that the method of proof in (1.4) and (1.5) gives (1.6) THEOREM. If A is locally compact, separable and metric, f(A) = B is open and for each y e B, every point off-1(y) is an isolated point of f-1(y),
then for any locally connected generalized continuum Y in B, f -'(Y) is locally connected.
(1.7) THEOREM. Let f(A) = B be open and light (A and B are locally connected generalized continua) and let K be any continuum in B. Then any compact component H of f-'(K) is interior to a locally connected continuum N in A on which f is open.
Proof. Since K is compact, for any 6 > 0 there exists a region R in B having property8 and satisfying K c R c V5(K). For 6 sufficiently small, the component N of f -'(R) containing H will be compact. Since R is locally connected, it follows by (1.5) that so is f-1(R). Hence N is open in f-1(R) and thus is a locally connected continuum on which f is open; and H lies in the interior of N. (1.8) THEOREM. Let f(A) = B be light and open where A is a locally connected generalized continuum, let K be any continuum in B and let H be any compact component of f-1(K). Then H lies in a conditionally compact region Q such that f is open on Q. Further, if e > 0 there exist
regions W in B, K c W e VE(K), and U in A with f [Fr(U)]- W = 0 and such that if R is any region in B about W satisfying R J[Fr(U)] = 0, f is open on the closure of the component Q of f-1(R) containing H. Proof. Let W be chosen as a region such that W is a locally connected
continuum and the component E of f-1(W) containing R ,is compact. Then let U be chosen so that U is compact, U E and U f -1(W) = E, the latter being possible because f-1(W) is locally connected, by (1.5),
so that E is both open and closed in f-1(W). Now let R satisfy the
EXTENSION OF OPENNESS
§ 2]
81
conditions in the above statement. Then Q lies in U and contains E, since
f(Q) C B so that
0. Hence f(Q) = R and f(Q) = R. Now
f-1(R)-U = Q. For if f-1(R)-U contained a point x not in Q, x would be a
limit point of f-1(R) and hence there would be a component Q1 1 Q of f-'(R) in U. This is impossible, since f(Q1) would necessarily contain W, whereas E is the only component of f-1(W) in U and E c Q. Accordingly, Q is open in f -'(A) and hence f is open on Q. (1.81) Conor tray. If R i8 chosen so as to have property S, Q will have property S.
If A and B are manifolds or closed regions, a mapping of A on B which maps the edge (or boundary) of A on the edge of B and the interior of A on the interior of B will be said to be a normal mapping of A on B. Thus if A and B are manifolds, the edge of A maps on the edge of B and the ordinary part of A maps on the ordinary part of B.
If f(X) = Y is a mapping, a region (or open set) R in X is said to be normal (rel. f) provided the set S = f(R) is open in Y and the mapping
f IR is normal in the sense just defined. If in addition the mapping fIR
is an open mapping, R will be said to be binormal (rel. f ). The following statements follow readily:
(i) For an open mapping f(8) = Y, a conditionally compact region B in X is binormal if and only if f 11? is open and f [Fr(R)] c Fr[f(R)]. (ii) If R is a binormal region (rel. f ), the mappings f I R and f I Fr(R) are open.
(1.9) THSoBx t. Let f(A) = B be light and open, where A and B are locally connected generalized continua. If K is a continuum in B, H is a compact component of f -'(K) and a > 0, there exists a binormal region Q having Property S and satisfying H c Q c V0(H). Proof. The Proof of (1.8) gives this result. For if a is small enough, E and U of (1.8) will lie in V0(H). Then if B is taken so as to contain W, have property S and not intersect f[Fr(U)], the component Q of f-1(R) containing E will meet our requirements. 2. Extendon of Opeiweee. That a mapping may be open on a region B and also open on the boundary of B and yet fail to be open on the closure
of B is shown by the mapping of a 2-cell A into the projective plane P by identifying diametrically opposite points on the edge of A. Here the mapping is topological inside A and equivalent to w = zz on the edge of A but fails to be open on A. (2.1) TaEoRXM. Let A and B be locally connected generalized continua,
let f(A) = B be a light mapping and let G and H = f(G) be open Bets in A and B respectively with boundaries C and 8 respectively such that the
[CHAP. VII
OPEN MAPPINGS
82
mappings f(G) = H and f(C) = S are open. Then if for each q eS there is connected, exists an arbitrarily small open set N q Such that the mapping f(O) = H is open.
For let x e C and let U be any open subset of 0 containing x and such that U is compact and Fr(U) f-1(q) = 0 where f(x) = q. Let N be an open set in B containing q and such that N f [Fr(U)) = 0 and so that f is open on R = H-N is connected. Since Thus if Q is any component of f -'(R) lying in U, by (1.11) we have f(Q) = R; and there must be at least one such Q since f(x) = q and f-1(R) Fr(U) = 0. Thus f(U) R; and since q is interior to f(U-C) by openness of the mapping f(C) = S, it follows that q is interior rel. H to f(G + C).
(2.11) CoBouaY. If A and B are 2-cells with edges C and S and the light mapping f (A) = B maps C onto S openly and the interior of C on the interior of S openly, then f is open.
(2.2) RpmARR. If a mapping f(A) = B is open on a conditionally compact open set Q and is topological on Q - Q and f (Q) f (Q - Q) = 0, then f is open on Q.
For if x e Fr(Q) and y = f (x), then for any sequence y1, y2, f(Q) converging to y, any convergent sequence x1, x2,
-
must converge to x since it converges to a point of reduces to x.
-
in
with x, e Q f -1(yi)
and this set
(2.3) THEOREM. If A and B are locally connected generalized continua, f is a light mapping of A into B, Bs is a closed non-dense subset of B such that B0 separates no region in B and f is strongly open on A - A 0 where A0 = f'-'(BO), then f is -strongly open on A.
Proof. Let x e Aa, let y = f (x) and let U be an open set in A containing x so chosen that it is conditionally compact and so that its boundary C does not intersect f-1(y). Then if Q0 is the component of B - f(C) containing y, clearly it suffices to show that (*)
f(U) Qo By hypothesis the set Q = Qo - Q0-B0 = Qo(B - Bo) is connocted
and thus is a region in B. Let Ro be the component of f -'(Qe) containing
x. Then Re c U since f(R0) c Q0 and Qo f(C) = 0 so that R be any component of f -'(Q) lying in Rc.
Then since R is condi-
tionally compact, because R c Rc c U, and f is strongly open at all points of R (since
Whence, f(U) D
0), it follows by (1.11) that
f(B) = Q and f(P) = Q. f(. - C) Q - f(C) = Q0, which is (*).
§3l
THE SCATTERED INVERSE PROPERTY
83
(2.31) CoRoLI..uw. If A and B are 2-manifolds, f is a light mapping of A into B and B0 is a closed totally disconnected Bet in B such that f is strongly open on A - f -'(Bo), then f is strongly open on A. 3. The scattered inverse property. DxFrxrrioN. A separable metric space M is said to be a 2-dimensional
manifold provided that for any x e M there exists a neighborhood U
of x such that 17 is a 2-cell, say 0 = R + J where J is the edge or boundary curve of U and R is the interior. If x e R, it is an ordinary or regular point of M whereas if x e J, it is an edge point (or boundary or singular point) of M. If there are no edge points the manifold is said to be an ordinary manifold or a "closed" manifold, whereas if there are edge points it is a manifold with boundary. If f denotes the set of all edge points of a manifold M, since each point of f is a point of order 2 of P and each component of P is open in f while f itself is a closed set, it results that each component of f is either a simple closed curve or an open curve (topological line or open segment). Thus if M is compact, # consists of a finite number (possibly 0) of simple closed curves.
(3.1) If f(A) = B is light and open, where A and B are 2-manifolds, then f has Scattered point inverses, i.e. for each y e B, each point of f-1(y) is an isolated point of f-1(y). (3.11) T zoR . The mapping generated by a non-constant diferen. tiable function w = f(z) has scattered point inverses.
Proof. Suppose on the contrary that for some value wo of f, f-'(wo) contains a limit point p of itself. Let U be a neighborhood of p such that U is compact and Fr(U) f-1(wo) = 0 and so that U lies on. an open 2-cell E c A. Let N1, N=, N3 be continua in B chosen so that each is the closure of an S-region (actually it may be taken as a curved triangle), N1-N, = N1-N3 = N= N3 = we and so that (N1 + N2 + N3) f [Fr(U)] = 0. Let
R{ = Nj - wo, i = 1, 2, 3. Then R{ has property S and thus so also does any component of f-1(Ri) lying in U by § 1. Now by (1.2) there are only a finite number of components of f-'(Rd
in U, i = 1, 2, 3. Also each such component maps onto R, by (1.11). Thus every point of U f-1(wo) is on the boundary of at least one such component of f-1(R,), i = 1, 2, 3; and since U f t(wo) is an infinite set, there exist components Q1 ,Q$, Q3 of f-'(R1), f-'(R2), f-'(R3) respectively and an infinite subset I of U f-1(wo) such that I =
Let x, y and z be distinct points of I. Since Qt has property S, each of these points is accessible from Q1, i = 1, 2, 3. Thus Q1 + z + y
and Q, + x + y contain arcs xay and xby respectively and Q1 + z and QE + z contain arcs az and bz respectively so that az-xay = a, b. Thus axb + ayb + azb = 0, a 0 curve. But then Q3 must
OPEN MAPPINGS
84
[CRAP. VII
lie in a single region complementary to 0 in the 2-cell E, contrary to the fact that x + y + z c Fr(Q). Second proof of (3.11). (Proof using the topological index.)
Let p be any value of f. Then since f is light, so that any point x e f-1(p) lies within a simple closed curve not intersecting f-1(p), our theorem follows at once from (3.12) Under the conditions of (3.11), for any simple closed curve C in Z and any p e W - f (C) there are at most m = (1 /27ri) pc(f, p) points of f-1(p) within C.
To prove this we note first that if Q is the component of W - f(C) containing p, then it was shown in the proof of VI, (3.1) that for each q e Q such thatf'(z) # 0 for all z e f-1(q), f -1(q) has exactly m points within C. Now if there were m + 1 distinct points pl, P2' - - , respectively
of f -1(p) within C we could choose disjoint open sets U, containing q, and so that f(EUj) Q. Then since f is open fl' f(U,) would contain a point q of the above type which is impossible, as then each U,, i = 1, 2, , m + 1 would contain a point of f-1(q) within C whereas there are just m such points. If a function f(z) is non-constant and differentiable in a region R of Z, for any value of wo of f in R, by the scattered inverse property all points of f -1(wo) are isolated points of f -1(wo). Thus by VI, (3.21), we have (3.2) THaoaEM. The zeros of f'(z) are identical with the points zo in R such that (1/27ri) UC(f, wo) > 1 for all sufficiently small circles C enclosing zo, where wo = f(ze) Now if we define, for each zo in R k(zo) = 211ri
µc(f, wo),
where wo = f(zo),
for any circle C containing and enclosing no other inverse of wo except zo, then k(zo) is an integer valued function of zo alone and all its values are 1. Clearly its value does not depend on the circle C. A little later on k(zo) will be called the local degree of f (z) at zo. We next show
(3.3) The function k(z) is upper semi-continuous. Thus the zeros of f'(z) form a closed set.
To prove this we have only to take zl near enough to zo that f(zl) and f (zo) lie together in the same component of f (I) - f (C), where I
is the interior of C. Then ,u0(f, wo) = yc(f, wl), where wl = f(zl); and if z2, C2,
, z, are the remaining points, if any, of f-1(wl) in I and C1, , C, are disjoint circles in I centered at z1, , z, respectively
and with disjoint interiors, by V, (4.21), we have
(2iri)-luo(f, w1) _ I (21ri)-1pc,(f, w1).
THE SCATTERED INVERSE PROPERTY
§ 3)
86
Since each term on the right is >0 by (3.1), this gives k(zo)
=
(2iri)-1pc(f, wo) = (21ri)-1(f, w1) > (27ri)-lµc,(f, w1) = k(z1)
and this establishes the upper semi-continuity of k(z). This completes the proof, since the second statement of (3.3) follows from the first. DErnrn7zox. A mapping f(X) = Y is said to be locally topological at x e X (ar to be a local homeomorphism at x) provided there exists an open set U in X containing x such that f (U) is open in Y and the mapping f I U is a homeomorphism of U onto f (U ).
(3.4) TsxORFM. If the function w = f(z) is differentiable in a region R of Z, then f is locally topological at zo e R if and only if f'(zo)
Proof. Suppose first that f'(zo) 0 0. Then by VI, (1.1),
0. (1/21ri)
µc(f, wo) = 1 for any sufficiently small circle C in B enclosing zo, where wo = f (zo). Let C be taken small enough that this holds and also so that wo on C and within C except at zo. Then if Ro is the component f(z) of W - f(C) containing wo and Q is the component of f-1(Bo) containing zo, Q lies inside C and f(Q) = Bo. Further, for q e Ro, (1/21ri)µc(f, q) = (1/27ri)(f, we) = 1. Thus by (3.12) there is one and only one point of f -1(q) inside C and this belongs to q. Accordingly the mapping f (Q) = Bo is topological.
Next let the mapping w = f (z) be locally topological at zo a B. Let U be an open set containing zo such that f maps U topologically onto f(U). Then if C is any circle enclosing zo and lying together with its interior in U, since f is topological on C it follows that f4 is a traversal of C' = f(C) whenever C is a traversal of C. Accordingly, by V, (2.2), (1/21ri) p0(f f , wo) = ±1 for any traversal of C because wo is within C'. Hence we must have f'(zo) # 0, since by (3.2) f'(zo) = 0 would imply (1/27ri)ua(f, wo) = (1/21ri) pc(ft;, wo) > 1 for C sufficiently small and for a choice of C which makesff a positive traversal of C'. (3.6) Tu oB.n . Let f(A) = B be open and have completely scattered point inverses, where A and B are locally compact separable and metric.
Then A is the union A = :An of a sequence of compact Beta such that f IA. is topological for each n.
Proof. Let (U,,) be a countable basis of open sets in A, so chosen that U. is compact for each n. For each n, let F be the set of all x e U,, such that g 1 g,,(x) = x where g denotes the mapping fIU,,. Then F. is closed in U. by openness of f. Accordingly each F is the union of a countable sequence of compact sets and thus we can write ) FR = 2A where each A is compact and lies in some F,,,. Thus f IA n is topological for each n. Finally, )2A = A, because if x e A, there exists an m such
that x e U. and Um f -lf (x) = x and hence so that x e F. c 2F. _ IA,,.
OPEN MAPPINGS
86
[CHAP. VII
(3.51) COROLLARY. If K is any closed set in A and V is any open subset of f (K), then V contains an open subset U which is homeomorphic with a subset of K.
For let K. denote the set K-A for each n. Then since V a and each f(K,,) is compact, some f(K,,) contains an open subset U of V. Then K f-1(U) maps topologically onto U under f. (3.52) CosoLLARY. If A and B are 2-manifolds, (or n-manifolds), then if K is non-dense in A, f(K) is non-dense in B. [Here "non-dense" means "contains no open set"].
4. Open mappings on simple cells and manifoWL Two mappings f(A) = B and O(X) = Y are said to be topologically equivalent provided
there exist homeomorphisms h1(X) = A and h2(B) = Y satisfying for all x e X.
#(x) = h2fh1(x)
(4.1) If a and f are simple arcs, any normal open mapping of a onto is topological.
For if some interior point y e f had more than one inverse point, there would exist a component Q of a - f-1(y) which contained neither end point of a, whereas by (1.11), f (Q) must contain an end point of fi since it is a component of f - y. (4.2) If A and B are 2-cell8 with edges J and C respectively, any normal light open mapping f of A on B which is topological on J is topological on A.
Proof. Let y be any interior point of B, and let ayb = P be a simple are in B with C-ayb = a + b. Since each component of f-'(ayb) maps onto ayb and thus must itself contain the unique inverse points, a' = f-'(a) and b' = f-1(b), it follows with the aid of (1.6) that f-'(ayb) is a locally
connected continuum. Accordingly it contains a simple arc a'b' = a. Since each of the two components R1 and B2 of B - f contains a component of C - (a + b) and each component of A - f-'(f) must map onto B1 or B2, there can be only two components Q1 and Q2 of A
- f-'(fl)
as each such component must contain one of the two components of J - J f 1(j9) = J - (a' + b'). Then since every point of f-'(fi) must be a limit point of both Q1 and Q2 while Q1 and Q2 are separated by a, it follows that a = f-'(#). Accordingly the mapping f(a) = fl is normal and open and thus by (4.1) is topological; and since f-1(y) C a, f-1(y) is a single point for each y e B. (4.3) Any open mapping f(J) = C of one simple closed curve J on another one C is topologically equivalent to the mapping w = zk on Izl = 1 for some positive integer k.
Proof. Let a e C. Since each component of J - P1 (a) maps onto C - a, the number of such components and therefore the number of
1 41
ON SIMPLE CELLS AND MANIFOLDS
87
points in f-1(a) must be finite, say k. Then if b e C - a, since each component of J - f-1(a) contains a point of f-1(b) and conversely each component of J - f-1(b) contains a point of f-1 (a), f-1(b) contains exactly
k points and, further, each component of J - f-1(a) maps topologically
onto C - a. Also these mappings are in the same sense. For let the , ak on C in either sense; and let points of f-1(a) be ordered a1, a2, bi be the point off-1(b) on aai+1 (ak+1 = a1). Then a1b1 maps topologically
onto one of the arcs of C from a to b, say axb; b1a2 maps topologically onto the other say bya; a2b2 maps onto axb; b2a3 onto bya, and so on to bkal which maps topologically onto bya. Clearly this shows that f is topologically equivalent to w = zk on Izl = 1. (4.4) Let f(A) = B be a normal light open mapping of one 2-cell A onto another one B. If there exi8ts an interior point q of B with a unique inverse point p in A, then on A the mapping f i8 topologically equivalent to the mapping w = zk on Izl
1 for Some positive integer k.
Proof. Since the edge J of A maps openly onto the edge C of B, the mapping f(J) = C is topologically equivalent to w = zk on I=I = 1 for some k. Thus if a and b are distinct points of C the inverse points of a and b are cyclically ordered a1, b1, a2, b2, - - , ak, bk, a1 on J so that a1b1 maps onto one are a8b of C, b1a2 onto the other arc bta from a to b
and so on. Let a = aqb be a simple arc in B with a-C = a + b. Then f-1((x) is a locally connected continuum, since each of its components
must contain p. Thus f-1(x) contains simple arcs asp and b;p, i = 1, 2, , k. Let W = Jia1p + b{p. Now since a;p - p and a,p - p are separated in A by b{p + b,p and likewise b;p - p and b,p - p are separated in A by a,p + afp, it follows that each pair of the arcs {a,p}, {b{p} intersect in just the point p. Let H1 and H2 be the components of B - aqb containing a8b - (a + b) and atb - (a + b) respectively. Since every component of A - f-1((X) maps onto either H1 or H2 and thus contains a component of J - f-1(a) f-1(b), the components of A - f-1(a) are 2k in number and they may be ordered S1, T1, S2, T2, -
, Sk, Tk , where S1 contains a1b1 - (a1 + b1), T1 contains b1a2 - (b1 + a2), and so on. Since every point of f-1(a)
must be a boundary point of one of the components S1, - - , Sk and also of one of the components T1, - , Tk, whereas every St is separated in A from every T, by the set W, it follows that W = f-1(a). Thus W maps openly onto at. Now consider the mapping of the 2-cell S1 onto the 2-cell H1. The edge J1= a1p + pb1 + b1a1 of S1 maps onto the edge aqb + bsa = C1 of
H1 and the interior S1 - (albs - a1 - b1) of S1 maps openly onto the interior H1- (aeb - a - b) of H. Further, J1 maps topologically onto C1, because albs maps topologically onto asb by (4.3), and a1p and b,p
OPEN MAPPINGS
88
[CHAP. VII
map topologically onto aq and bq respectively by (4.1). (The latter holding because a1, p and b1 are the only inverse points of a, q and b respectively on alp + b1p.) Accordingly by (4.2), Sl maps topologically onto H1. Similarly T 1 maps topologically onto F!2, S2 topologically onto
Hl, and so on so that on A f is topologically equivalent to w = zk on Izl < 1 is asserted. 5. Local topological analysis.
(5.1) Let A and B be 2-manifolds and f(A) = B be light and open. For any ordinary point q E B any p e f-1(q), there exists a closed 2-cell neighborhood E of p and an integer k such that, on E, f is topologically equiva-
lenttow=zkonIZI < 1. Proof. Let b e B - q. Since f is light, by (3.1) there exists a 2-cell neighborhood W of q with interior R and edge C such that the component
E of f-1(W) containing p lies inside a 2-cell P on A which contains no point of f-1(b) and no point of f-1(q) other than p and E = P f-1(W). Let Q be the component of f-1(R) containing p. Now since f-1(C) is locally connected, by § 1, and separates p from the edge of P, it contains a simple closed curve J and Q is inside J relative to P. However since every
point of Ef-1(C) must be a limit point of Q (as no other component of R intersects P) and also must be a limit point of f-'(B - W), it follows that
J = E f 1(C), because if any point interior to J were in f-1(B - W), a component of f-1(B - W) interior to J would map onto B - W by § 1, contrary to the fact that P f-1(b) = 0. Thus E is a closed 2-cell with edge J and interior Q, and since E f -1(q) =
p and f maps E normally and openly onto W, our theorem now follows directly from (4.4). (5.11) CoxoL ARY. With conditions as in (5.1), given any finite sequence Pv P2, ' ' , P of distinct points off-1(q), there exists a closed 2-cell neighborhood W of q such that the components E1, E2, -, E. of f-1(W) containing Pit P2' . ' ' , Pn respectively are disjoint and each is a closed 2-cell which maps onto W by a power map, i.e., a mapping topologically equivalent to w = zk on lzl < 1 for some k.
(5.2) If in addition f is a compact mapping and if PI + P2 + + Pn = f-1(q), then El + E2 + ....+ En = f-1(W). For by compactness of f, f-1(W) is compact; and since thus each component of f-'(W) maps onto W and hence contains a p,, our result follows. (For definition of compact mapping, see §1 of Chapter VIII below.) (5.3) THEOREM. The mapping generated by a non-constant differentiable function w = f (z) in a region R is locally equivalent to a power mapping.
Thus in particular it is established by topological methods that any differentiable function in a region R acts locally on the region precisely
THE DERIVATIVE FUNCTION
§ 6]
89
like a function which can be developed in a power series. Hence it has the action of an "analytical function" from the topological viewpoint. Indeed the same is true for any light strongly open mapping from R to the w-plane, by (5.1). This remarkable result was first formulated (in a somewhat
different way) by Stollow and represents one of the major results of Topological Analysis.
6. The derivative unction. Assuming a function w = f (z) is differen. tiable in a region R of Z, we summarize here the properties of the derivative function f'(z) which have been established above along with some of their immediate consequences.
By (3.4) the zeros of f'(z) coincide with the points at which f fails to be locally topological. By (5.1) such points are necessarily isolated. Thus we have: (6.1) The zeros of f'(z) form a countable closed set each point of which i8 an isolated point of the set. Now if we let a be any value whatever of f'(z) and define the function
O(z) = f (z) - az,
z e R,
0'(z) = f(z) - a,
z e R,
then since
so that 4'(z) vanishes if and only if f'(z) = a, we have (6.2) The function w = f'(z) has closed and Scattered point inverses. This should be compared with (3.1) above. In particular note that this shows that the transformation generated by f'(z) is light; and although we are not yet able to show continuity of f' by topological methods, at least we have shown in this direction that the inverse of a single point is closed. Openness of f' remains inaccessible at present by these methods. REFERENCES
§ 1, 2, See Whyburn [1, 4] and Stollow [1, 2] and further work cited in 4, 5. these sources. Result (4.2) was proven by Stollow in his paper [2]. § 3. Whyburn [1], Plunkett [1].
VIII. Degree, Zeros.
Rouche Theorem
1. Degree. Compact mappings.
We recall that a mapping f of A into B is compact provided that for every compact set K in B, f -1(K) is compact or, equivalently, that f is closed and the inverse of each point of B is compact. If k is a positive integer, a mapping f(A) = B is said to be of degree k provided that for each q e B the sum of the multiplicities of the points in f -1(q) is k, where pi e f -1(q) has multiplicity k{ provided that on a 2-cell neighborhood of pi, f is topologically equivalent to w = zk on IzI < 1. (1.1) THxoxxs. If A and B are 2-manifolds without edges (B connected,
A not necessarily connected), a light open mapping f(A) = B has a finite degree if and only if it is compact.
Proof. Suppose f is compact. Then by VII, (5.2) it follows that for each y e B there exists a neighborhood W of y and an integer k(y) such that every point p in W - y has exactly k(y) inverse points. Indeed k(y) is merely the local degree at y or the sum of the multiplicities at the points of f -1(y), i.e., the sum of the integers k{, i = 1, 2, - , n, where f is topologically equivalent to w = zk,, Izi < 1 on the set E,. If we consider k(y) as a function on B, it is therefore continuous and thus must be constant, say k(y) = k, on B as B is connected. Thus f is of degree k.
On the other hand suppose f is of degree k. Let K be any compact set in B. By VII, (5.11) each q e K is interior to a set W such that there are disjoint components E1, E2, - - , E. of f-1(W) containing p1, - - , p respectively where f -1(q) = p1 + P2 + ' - - + p and such that the Ei map onto W{ by power mappings of degree k, with I;_1 ki = k. Since each point of W - q then has k distinct inverse points in Ji E1, it follows that f -I(W) = I1 E{ and thus that f -1(W) is compact. Since K is covered by a finite number of such sets W, it follows that f-1(K) is compact. (1.11) CoRou s i Y. If f (A) = B is a light open mapping of degree k, the points y e B having less than k distinct inver8e8 form a completely scattered set D. On the get A - f-1(D), f is a local homeomorphism. (1.2) THxoREM. Let A and B be separable metric spaces, let f be any mapping of A into B and let R be any set in B. For any non-empty conditionally compact component Q of f-1(R) or any conditionally compact 90
DEGREE AND INDEX
§ 2]
91
set in f-1(R) which is closed relative to f-1(R), the mapping of Q into B by f is a compact mapping.
Proof. Denote f IQ by g and let K be any compact set in R. Then since f-1(K) is closed and Q is compact, Qf 1(K) is compact. But Q:f-1(K) = Qf-'(K) = g-'(K) since (Q - Q)f-'(K) c (Q - Q)f-1(R) =0.
(1.21) Coiamm&uY. If A and B are locally connected generalized continua, f is strongly open on f-'(R) where B is a region in B, and Q is a conditionally compact component of f-1(B), then f (Q) = R and fI Q:Q --3-. B is compact. For since Q is open in A, f (Q) is both open and closed in R and thus is
equal to R. (1.22) COROLLARY. Let w = f(z) be a light and strongly open mapping
of the z-plane Z into the w-plane W and let R be any region whatever in
W. If U is any conditionally compact open set in Z with Uf-1(R) # 0 whose boundary does not intersect f-'(R), then on S = Uf-1(R) f is a compact mapping of S into R and thus is of degree k for some integer k. Each of the finitely many non-empty components of S maps onto R under f. (1.23) COROLLARY. On any bounded normal region, f is compact and thus is of degree k for some k.
2. Degree and index. (2.1) THEOREM. Let w = f(z) be light and strongly open on a region of Z containing the simple closed curve C and its interior I, let p be any
point of f(I) - f(C) and let R be the component of W - f(C) containing p. Suppose, further, that f is differentiable at all points of f-'(R0) for some non-empty open subset Ra of R. Then for any positive traversal 4 of C,
(I/2iri) pc(f, p) = degree of f on If-'(R) = sum of the multiplicities of the points I f-1(p).
Proof. By (1.22) f is compact and hence of finite degree, say k, on
If-1(R); and for any point q e R, k is the sum of the multiplicities of the points of f-1(q) lying in I. Now let q be chosen in R so that f'(z)
exists and is ¢ 0 for all z e If-1(q) and at the same time so that this set 1 f-1(z) contains exactly k distinct points z1, z2, , zk, this being possible as shown in Chapter VI and in § 1 above. Let C1, C21 , Ck be disjoint simple closed curves each lying within C and enclosing the points z1, , zk respectively and such that the interiors of the curves are disjoint. Further, by VI, (1.1) we may supppose the C, chosen so that if C is any traversal of the graph G consisting of C, C1, C21 , Cr, then 1
2rri
c,(f , q)
Then by the lemma in V, (4.1) we have k
µc(& p) = /Ac(& q) =
IC,(& q) = 2kiri, 1
DEGREE, ZEROS
92
[CHAP. vm
the first equality resulting from V, (1.3) because p and q are in the same component B of W - f (C). 3. Zeros and poles. A point p is called#a pole of a function w = f (z) provided f is finite and
differentiable at all points U - p for some open set U containing p and lim, f(z) = oo. The function, w = f(z) is said to be meromorphic in a region R provided it is differentiable at all points of R except for poles. (3.1) THEOREM. If f(z) is meromorphic on a region containing the simple closed curve C and ite interior and is finite and #p on C then {i)
2rri
pc(f, p) = N - P,
where N = no. of p places inside C, P = no. of poles inside C.
Proof. Let C. and C. be disjoint simple closed curves inside C en. closing all p-places and all poles respectively off inside C and such that their interiors are disjoint. Then since f is finite and p on the closed elementary region B bounded by C, C. and C., (ii)
f'c(f, P) = µc,{f, P) + f'C0(f, P)Also since f is finite on C, plus its interior and : p on C,, by (2.1) we have (lu}
Uc,(f, p) = N.
Now consider the function
(z) = p + f(z)1
,
for f finite
P (z) = p, when z is a pole off. This function is finite, continuous, light and strongly open on C, plus its interior. The strong openness at points of 0-1(p) follows from VII, (2.3), taking B as the w-plane and B. = p in that theorem. Also 0 is differentiable at all points of C. plus its interior except for the poles of f. Thus again by (2.1) we have (iv)
2ri µc.(0, p) = number of p-places of 0 inside C. = P.
Now if
1
-P f-P
Whence, using (iv), (v)
IAO.(f, p) _ fAcp) = -21ri P.
Then (v), (iii) and (ii) combine to give (i).
§ 6]
REDUCED DIFFERENTIABILITY ASSUMPTIONS
93
4. Rouchd and Hurwitz Theorems. (4.1) Roucn 's THEOREM. If f (z) and g(z) are continuous and
differentiable within and on a simple closed curve C and Ig(z)I < I f (z)I on C, then f(z) and f(z) + g(z) have the same number of zeros within C. This is an immediate consequence of (2.1) or (3.1) together with V,
(3.12). We need only take p as the origin and let fl(z) = f(z) + g(z). (4.2) Huawrrz's TsE0REM. Let the sequence of functions fn(z), each continuous and differentiable in a region R, converge uniformly in R to a differentiable function f (z) not identically zero. Then if e R is an M -fold zero (i.e. a zero of multiplicity m) of f(z), every sufficiently small neighborhood D of C contains exactly m zeros of fn(z) for n > N(D).
For let C be a circle in B containing no zero of f (z) and whose interior
D lies in R and contains no zero of f (z) other than . Since f (t) # 0 in C, there exists an integer N(D) such that if n > N(D), I fn(z) - f(z)I < I f(z) I for all z c C. Then by (4.1), f(z) and fn(z) have the same number, m, of zeros inside C.
5. Reduced differentiability assumptions. Concept of a pole.
We now show that even under certain reduced differentiability assumptions for a function in a region of Z, lightness and openness of the mapping into W generated by the function can still be established. This is of significance in connection with the behavior of the function
on the boundary of regions in which it is differentiable and also in connection with the notion of a pole. We need first a result concerned with the preservation of dimensionality or non-density of a set under the sort of mapping we are studying. A set K will be said to be non-dense in a space X provided K contains no open set in the space X. To say that
K is dense in X, however, means that every point of X is a point or a limit point of K, i.e., k X. (6.1) LEMMA. Let f1(z), f2(z), - , fg(z) be continuous functions in the closure of a region R of Z such that for each n < q, f(z) is differentiable
everywhere in a region R n R - Kn, where Kn is a compact non-dense set in R on which fn is constant and has the value a. For each k < q and , r, of k distinct positive integers
, rk, z)
Let K = BK,. Then for each k < q, the union of the images of K under all the mappings g(r1, r8, - , r,, z) for j < k is non-dense.
Proof. (By induction on k) For k = 1, let H,, be the image set of K under the mapping g(n, z). Then to prove that >jH is non-dense in W it clearly suffices to show that H,, is non-dense for each n S q. Now
94
DEGREE, ZEROS
[CHAP. VIII
g(n, z) = aQ-If (z) and thus g(n, z) is differentiable everywhere on R and is constant and equal to a° on K,,. Since then K - K is closed and non-dense in B,, - K,,, it follows by VII, (3.52) that H,, - as is non-dense in W. Thus so also is H,,. We now suppose our conclusion valid for all k < m and consider the case k = m < q. Then the union N of the images of K under all the mappings g(r1, r=, . , r f, z) for j < m is non-dense in W. Let (1)
h(z) = g(rv r2, ... , r,,,, z) = a°_mf,. (z)f,. (z) ... f.,,(z)
f,(z). Then since the number of such functions g is finite, clearly it suffices to show that the image of K under each one of them is non-dense, because be any one of the g-functions having m factors from the set f I(z),
,
each such image is compact. Thus we show that h(K) is non-dense. Now since Jr, is differentiable everywhere on R,,, it follows that h(z) is differentiable everywhere on the set r l' R,. = Q. Thus since K-Q = K - :Ei` K,,, is closed in Q and non-dense in Q, it follows by VII, (3.52) that h(K-Q) is non-dense in W. Also since for any n < m, f,*(z) is constant on K,,, and takes the value a everywhere on K,,, we have (2)
h(z) = a°-"`+If,,(z)f.,(z) ... f,.
(z):f.,,..,(z) ...f.,,(z)
for all z e K,* Whence, h(K,*) c N because the right hand member of (2) is one of the functions g with only m - 1 of the factors from the set fl, fs, ' - - , fQ and N contains the images of K under all such mappings. Thus since K - K-Q c 11 K,., we have h(K) = h(K-Q)
+ h(> K,.) c h(K.Q) + N;
and since each of the sets on the right is non-dense in W, so also is h(K) because both N and h(K) are compact. (5.12) COROLLARY. (Case of k = q). In particular, if g(z) denotes the function f1(z)f2(z) - - f,(z), then g(K) is non-dense in W. (5.2) THEOREM. Let w = f (z) be non-constant and continuous in a
region B of Z and differentiable everywhere on B - H where H is a closed non-dense subset of R on which f is constant and has the value a. Then (which contains H). H is totally disconnected, as is also
Proof. Suppose on the contrary that f(z) = a for all z on some nondegenerate continuum in R. Then if Ro is the subset of R on which f(z) _* a, Fr(R0) contains a non-degenerate continuum M lying within
R. Let ze e M and let N be a subcontinuum of M of diameter a < ]IF p[ze, Fr(R)] which is irreducible between za and a point zI : ze. Let to be a point on N with Ize - wl = #Izo - zll, let y be a point of Re
REDUCED DIFFERENTIABILITY ASSUMPTIONS
§ 5]
96
with l y - wl < }Izo - wl and let C, be the circle with center y and
radius 4Izo - wl. Then since w is within C, and both zo and zl are without C,,, C must contain at least two points 8 and t of N, (by irreducibility of N).
Now since y e Ro and Re is open, there exists a z' a R. within C,, and on the perpendicular bisector of the segment 8t so that the angle az't is a rational multiple of 27r, say /8z't = 2irp jq where p and q are integers prime to each other. Next consider the function a-1
g(z) = 11 {f[(z - z )e2rivla + z'] - a}. .-o Since
I(z-z')es.tr'IQ +z'-xellz-z'l +Iz'-zel< Iz - iI + IIzo - zll < Iz-z'I+$a, and a < }p[zo, Fr(R)], it follows that g(z) is continuous on the interior
Do of the circle Co with center z' and radius z a, since Do c R. Let C be the circle with center z' and radius 2a and let D be its interior. Let KO, = and let K1= H-D. For each r = 1, 2, ,q-1 and each z e Do let h,(z) = (z - z')e-2*'p'I4 + z' and set K°+1 = h,(K°), K,+1= h,(KI) = K° Then if for each n = 1, 2, - , q we define fn(z) = f [(z -
z' ] - a,
then fn(z) is continuous on Do and thus on 1), differentiable in a region Do - K° D - Kn, K. is non-dense and, on K,,, fn(z) is constant and
has the the value 0. Accordingly by the lemma and its corollary it follows that g(Jg Kn) [which in this case is the same as g(K1)] is non-dense. Thus every open set in g(D) contains a 2-cell A such that g is differentiable
at all points of because A can be taken in g(D) - gQi making D-g-1(A) c: D - >JK,, and g is differentiable at all points of D - BBKA.
Now since g(z) = 0 for all z e N whereas g(z') z 0, there exists a region Q lying in D and containing N such that (*)
Ig(z)l < 6 = ilg(zll
for all z e Q. Let 8t be a broken line segment in Q joining 8 and t and let h(x) be a homeomorphism of the interval (0, 1) onto 8t with h(0) = 8. For any x on the interval (1, q) let us define h(x) = [h(x - r) - z']e2*"PIQ + z', for
r<x
98
DEGREE, ZEROS
[Caer. VIII
Then h maps the interval I = (0, q) onto a set E and Ig(z)l < a for z e E because g(E) = g(st). However we note that if
en(z) = h(x) - z',
0 < x < 1,
is a representation of h(x) - z' on (0, 1), then eu(x-r) + 2,irpla = h(x) for
- z',
rsxsr+1,
Accordingly, µ(e,Q)(h, z') _ :E°ol #(r.r+1)(h, z') = q[u(1) - u(0)] # 0. Since
h(q) = 8 = h(0), it follows by V, (1.4), that z' is in a bounded comple. mentary domain of E. Thus z' is within a simple closed curve J (simple polygon) contained in E. But since Ig(z)I < 6 on J this must hold also within J by VI, (3.11), because J and its interior lie in D and, as shown above, every open set in g(D) contains a 2-cell A such that g is differenti-
able at all points of D-g-1(A). But this is contrary to the fact that Ig(z')l = 26 by (*). This contradiction establishes our theorem. (5.21) CoEoLLAnY. If w = f (z) is non-constant and continuous in a
region R and differentiable on B - fl(a) where a is some value of f in B, a e f(R), then f is light and strongly open in R. Since f would automatically be differentiable at any possible interior
points of f-1(a), that f is light results from the theorem. To see that f is strongly open we need only apply Corollary (3.11) of VI, (3.1). For let U be any open set in B, let wo ef(U), zo e R fl(wo) let C be a simple closed curve lying together with its interior I in U so that I zo and C f '(we) = 0. Then, applying the corollary just mentioned, we have that f(C + I), and therefore f(U), contains the component of W - f(C) containing we. Thus wo is interior to f (U ). (5.22) CoRoLLARY. Let w = f(z) be differentiable everywhere in R - K where R is a region and K is a closed non-dense subset of R. Further suppose that for each zo e K we have
lim f(z) = 00. Then K is a scattered set and the mapping w of R into the complex sphere is light and strongly open.
For let R. be the region B - f-1(0). Then let us define
g(z)=f(
z)
for zeRo--K
=0 for
z e K.
§5l
REDUCED DIFFERENTIABILITY ASSUMPTIONS
97
Then g(z) is continuous in R. and differentiable everywhere in R. - K; and on K, g is constant and has the value 0. Thus by (5.2), K is totally disconnected and, by (5.21), g(z) is light and strongly open in Re. Accordingly K must be a scattered set; and as a mapping of B into the complex sphere
f(z) =
1
on
g(z)
:0_K
on
=0
on
R - Ro
and thus f is light. Also f is strongly open even on K by (2.3) of VII because f is strongly open on R - K and we have only to take B as the complex sphere and Be as co on B and apply the theorem quoted.
Note. Thus all points of K are topologically precisely like poles of f. Hence the concept of a pole is approachable topologically. No reference to expansions in series need be made. A pole is simply a point where f (x) -+ oo and such that in some neighborhood of this point there are only points of this same type and points where f is finite and differentiable. (5.3) THEOREM. Let w = f(z) be non-constant and differentiable
in a region R. Then if f is continuous at all points of a continuum K of Fr(R) and constant on K, then K is a continuum of condensation of Fr(R). For suppose on the contrary that some p e K is not a limit point of
Fr(R) - K. Then there exists a circle C with center p and interior I For any I) K. Let H = such that z s I - PA let us define g(z) = a, where a is the value of f on K, and for z e I..R let g(z) = f(z). Then in the region I, g(z) is differentiable everywhere except on H. Since H is closed in I and is non-dense and since g is constant on H, it would follow from Theorem 1 that H was totally disconnected; but since K is a continuum this clearly is impossible. Accordingly our theorem follows. (5.31) COROLLARY. If w = f(z) is non-constant and differentiable in the interior of a simple closed curve C (or in an elementary region with boundary C), there exists no are on C on which f is continuous and constant. Thus if f is continuous on C it must be light on C.
IX. Global Analysis 1. Action on 2-manifolds
By a (straight) 0-, 1- or 2-dimensional simplex is meant a point, line
segment or triangle plus its interior respectively. The end points of a 1-simplex are called its vertices or 0-sides. The sides of a triangle are the 1-sides of the 2-simplex defined by the triangle and the vertices of the triangle are its 0-sides or simply its vertices. A set homeomorphic with
a simplex is called a curved simplex, with dimensionality and sides determined in obvious manner from the homeomorphism. A complex is a set K of simplexes (straight or curved) such that (1) any side of a simplex of K is itself a simplex of K, (2) the intersection of any two simplexes of K is either empty or a side of both simplexes and (3) any point on a simplex of K has a neighborhood which intersects only a finite number of simplexes of K. A simplicial subdivision of a set M is a repre-
sentation of M as the union of a set of simplexes (straight or curved) which form a complex in the sense just defined. In our considerations, the set M will be a 2-dimensional manifold. If K is a finite 2-dimensional complex (i.e., has only a finite number of simplexes), the Euler character-
istic x(K) is the number ao - al + as, where a1 is the number of jsimplexes in K, j = 0, 1, 2. If M is a compact set, any simplicial subdivision of M is a finite complex K and we set x(M) = x(K). It can be verified that x(M) is independent of the particular subdivision K used. For a light open mapping f(A) = B, A and B 2-manifolds, a point y s B is a singular point if f fails to be locally topological at some point of f-1(y). (1.1) TsaoRzm. Let A and B be compact 2-manifolds and let f(A) = B be light, open and normal. Then
kx(B) - X(A) = kr - n, where k is the degree of f, r is the number of singular points off on B and n is the number of inverse points in A of these r singular points.
Proof. Let at and f be the edges of A and B respectively. Then f must be locally topological at all points of a. For the mapping f : a -* # is open; and since both a and f consist of finitely many simple closed curves
this mapping is locally topological. Let A. and Bo be closed manifolds 98
ACTION ON 2-MANIFOLDS
§ 11
99
obtained from A and B respectively by fitting disjoint open 2-cells into each edge curve so that the open cell together with the edge into which it fits forms a closed 2-Dell; and then extend f to Ae by mapping each of the new open 2-cells E in A0 onto the new one R in Be fitting into the edge which is the image of the edge of E by a power mapping agreeing with f on the edge of E. The thus extended mapping of A 0 onto Be is light and open and by VIII, (1.1), is of the same finite degree k as originally on A - a. Hence by VIII, (1.11), some points of f have exactly k distinct inverse points and this holds for all points of # as the mapping f : a -+ f4 is locally topological. Accordingly, f must be locally topological at each point of a.
Now let Q = q1 + qi +
+ q, be all the singular points of f on
B, i.e., the points (if any) of B having at least one inverse point at which
f is not locally topological. By what was just shown it follows that r and n are finite. For each i< r, let f-1(q;) = pil + pig + + p,,,,. By VII, § 5, there exist disjoint closed 2-cell neighborhoods W1, W2, - - - , r, f-1(Wi) consists of W, of q1, - - - , q, respectively so that for each i
n, disjoint 2-cells W,1, Wiq, .. , W,,,, so that W contains p in its
interior and maps onto W, by a power mapping. For each i S r, let Si be a simple arc in W, containing q, and having just its end points on the edge of W. Let T,, = W f -1(S,), j = 1, - - - , n, and let T = :E7 T i,.
There exists a number e > 0 such that for any connected set E in A - T of diameter <e, f is topological on E. For if not, there would exist a sequence El, E2, - - - of connected sets with 0 and which converges to a point p of A and such that for each n some two points of
R. have the same image under f. Since f is locally topological on A - ff1 (Q), we must have p = p,, for some i and j. But then for n sufficiently large, E. lies interior to W,, and does not intersect Ti,, whereas f is topological on the closure of each component of W,i - Til. Next, since f is light, there exists a d > 0 such that for any set N in B of diameter
GLOBAL ANALYSIS
100
[CH". IX
Let am and am(m = 0, 1, 2) be the number of m-simplexes in K
and H respectively and let n = nt = the total number of points in f-1(Q). Then since f is of degree k and is locally topological on A - f-1 (Q), we have
a2 = 02 a1 = kf1
ao = k(flo - r) + n. Whence,
a2-a1+ao=k(#2-#1+flo)-kr+n-
or kX(B) - X(A) = kr - n. REMARK. Given a local homeomorphismf(X) = Y. If C is any simple closed curve in Y, every component of f -1(C) is either a simple closed curve or an open curve (topological line). This results at once from the fact that these two types of sets are the only ones locally homeomorphic with an open segment. This leads at once to (1.2) If A0 and B0 are 2-manifold8 without edges and f(A0) = B0 is light and open and B is any 2-manifold on B0 whose edge contains no singular point of f, then any compact component A of f-1(B) is a 2-manifold which maps nor ally, lightly and openly onto B and the relation
kX(B) - x(A) = kr - n holds. Proof. For if Q is a component of f-1(R) in A where R is the ordinary part of B, then Q is bounded by a finite number of simple closed curves in f-1(f) where f is the edge of B. Thus Q is a manifold; and Q must be equal to A because each edge point of Q is on the boundary of a region of
A0 - Q which maps into B. - B, so that there can be no other component of f-'(B) in A except Q. S. Differentiable lanctiona
(2.1) THnomcm. Let W = f(z) be non-constant and differentiable in a region D. For any 2-cell A in D which is a component of the inverse of B = f(A), the mapping f of A onto the 2-cell B is light, open, and normal; no singular point of f is on the edge of A and we have (i)
kr-n=k-l
Proof. Since f-1(B) is locally connected (see Chapter VII) and has A as a component, there exists a component E of f-'[f(D) - B] exterior to A having the entire edge C of A on its boundary. Let I be the interior of A. Then since every point of f (C) is a boundary point of each of the disjoint regions f(I) and f(E), f(I)f(C) = 0 so that C = f is open on C, because by VII, § 1, it is open on A; and if J = f(C) then J is a simple closed curve. Let R be the interior of J. Then since I is conditionally compact, it follows by VIII, (1.21), that f(I) = R. Hence
ORIRNTABILITY
§ 3]
101
B = R + J is a 2-cell and the mapping f(A) = B is normal, light and open. Also, if C contained a singular point p of f, then by VII, (5.1), there would be a set of at least four arcs in f-'(J) each pair of which intersect in just the point p. This clearly is impossible. Finally, since x(B) = x(A) = 1, the relation (i) now is a direct consequence of § 1 above.
(2.11) CoRoi.LiaY. If D is simply connected and B is any 2-cell in f(D) whose edge contains no singular point of f, any compact component A of f-1(B) is a 2-cell and (i) holds.
Proof. For by (1.2), A is a 2-manifold. Hence A contains a simple
closed curve J with interior I such that A c J + I e D. If there were a component Q of D - A in I, f(Q) would be all of f(D) - B so that f(J + I) = f(D), contrary to the fact that f(D) cannot be compact. Accordingly, A = J + I. (2.12) COROLLA a.Y. If f vanishes k times (or takes any of its values k times) in a 2-cell A in D which is a component of the inverse of its image, then f'(z) vanishes on a set in A the total of whose "deficiencies" is k - 1.
Proof. For let P = p1 + p2 + ... + p, be the singular points
off in B. Then if for each j < r, the points off-'(pi) on A are denoted by p,.1, p, 2. , , ' , p, , , and the local degree of f at p!'. is Ic,,, (i.e., f maps a neighborhood of p,,. onto a neighborhood of p, by a power mapping of the deficiency d,,, = k,,. - 1 does not vanish at some of the
degree
pl,,,. We suppose d,,,, # 0 for n < m1 and d,,,, = 0 for n > mi. Then since, for each j < r,
kl1 +ks.+...+k!*, =k, we have
(kt,i-1)+(kt.-1)+....+(k,.,,,-1)=k-n,, or
Thus
f
1n
mu
nj
n-l
n-i
dl,,,
= k - n!, since d,,,, = 0 for n > m!.
di,n=kr-(nl+n2+...+n,)=kr-n=k-Iby(i) 1
Accordingly, the deficiency sum over the set P
p,,, equals
k - 1; and since f fails to be locally topological at all points of P, f'(z) - 0 on P. 8. Orientability
An orientation of a straight or curved triangle A with vertices a, b, and c is determined by an ordering of its vertices. For example (a, b, c)
102
GLOBAL ANALYSIS
[CHA?. IX
and (b, a, c) determine (opposite) orientations of A. Now an orientation (a, b, c) of a triangle A, straight or curved, induces orientations (a, b), (b, c), (c, a) onto the sides or segments of the sides of A. Two orientations of the same triangle are the same provided the orderings of vertices in one can be changed to the ordering in the other by an even number of interchanges. Thus (a, b, c) and (b, c, a) are the same whereas (b, a, c) and (a, b, c) are opposite. Thus each triangle, or 2-cell, A has exactly two distinct orientations and these correspond precisely to the two senses of traversals of its boundary. A traversal of the boundary of A determines a sense a, b, c of permutation of the vertices of A and thus corresponds uniquely to an orientation of A and conversely. Two oriented triangles
or 2-cells lying together in a plane or on a 2-cell are said to agree in orientation provided the corresponding traversals of their boundaries agree in sense as defined in Chapter V, § 2. Thus as shown there, if a 2-cell A' with edge J' lies interior to a 2-cell A with edge J and a, b, c and a', b', c' are triples of distinct points on J and J' respectively, the orientations (a, b, c) of A and (a', b', c') of A' as simplexes will agree if and only if there exist disjoint simple arcs aa', bb', cc' lying except for their ends in the annular region of A between J and X. Also if two oriented
triangles or 2-cells (a, b, c) and (x, y, z) otherwise disjoint and lying in a plane or a 2-cell are so situated that a side xy of (x, y, z) lies in a side
ab of (a, b, c) and the corresponding traversals abc and xyz of their boundaries agree in sense on xy, then the orientations (a, b, c) and (x, y, z) will be the same or opposite according as A xyz lies except for xy entirely inside or entirely outside A abc. A 2-dimensional manifold M is said to be orientable provided orientations can be so assigned to the curved triangles in a triangulation of M, i.e., a simplicial subdivision of M, that any pair of triangles with a common side will be "coherently" oriented in that the common side obtains opposite
induced orientations from the two triangles. If such an assignment of orientations is not possible, M is said to be non-orientable. It is readily seen that if M is orientable it has exactly two different orientations and
these are said to be opposite. It is true, though not obvious, that the property of being orientable is independent of any particular triangulation of M and in fact dependent only on the topological structure of M itself.
If one triangulation of M can be oriented, then so can any other triangulation of M. This in particular is a direct consequence of the main theorem of this section now to be established.
(3.1) TSsoIxM. Let f(A) = B be light, open and normal, where A and B are 2-manifolds. Then if B is orientable, so also is A.
Proof. Let T. and Tb be triangulations of A and B respectively and let Tb be oriented. For each triangle A in Ta we assign an orientation
ORIENTABILITY
§ 3]
103
as follows. Let 6 be a triangle interior to A which does not intersect the set W of all points x of A such that f fails to be locally topological at some
point of f-i(x) together with all edge points of A and which is small enough that 6 maps topologically onto its image y under f and so that y lies inside a triangle P of Tb. Then if a, b, and c are the vertices of a, we give 6 such an orientation (a, b, c) that [ f(a), f(b), f(c)) is the positive
orientation of y in B, i.e., this orientation in y agrees with the given orientation of r in Tb. We then assign an orientation to A so as to agree with that just given to 6. Thus we assign an orientation to each triangle A in Ta. Now to show that this yields an admissible orientation of A, we must show that any pair of triangles 01 and A2 in T. with a common side ab are coherently oriented in the sense that the induced orientations of ab from the orientations of the two triangles Al and A,
are opposite. To this end let 61 and 62 be the triangles in 01 and A. respectively which determine the orientations as defined above. Since A - W is connected and open and contains b1 and 62, it contains a chain of triangles.
81= a, a common side for i = 1, 2, - - , n - 1, and ai -a, is at most a common vertex for j > i + 1, (2) ai c 01 for i < k and ai OQ for i > k so that the common side 4 of at and ak+1 lies in ab, and (3) so that f is topological on the union of any pair ai + a,+1 of successive triangles in the chain and f(a{ -- a,+1) intersects at most one edge (i.e., 1-simplex of Tb and thus lies interior either to a single triangle of T. or to the union of two triangles of Tb. To obtain the
chain so as to satisfy (3) we note that there are only a finite set F of points interior to Al + A. which either belong to W or map into a vertex of Tb. Then if we take simple arcs alp and pa2 in 01 - F and A. - F respectively with p e ab and alp - p interior to A1, a$p - p interior to A2, any chain in Al + A2 satisfying (1) and (2) and of sufficiently small mesh will necessarily satisfy (3) provided each link of the chain intersects t = alp + pat, because f is locally topological at all points of t and f(t) contains no vertex of Tb. Thus we have only to construct the chain using the arc t as a guide with each link intersecting t and being of sufficiently small diameter. Now for each i, f(aE) lies interior to a 2-cell E which is either a single triangle of Tb or the union of two such triangles with a side in common. Thus in either case E is oriented by the given orientation of Tb. Now if for each i we assign such an orientation (a{, b;, c,) in a, that [f(ai), f(b,), f(cj)] agrees in E with the given orientation in Tb, it follows by property (3) that any two successive triangles a1 and ai+1 in the chain are
GLOBAL ANALYSIS
104
[CHAP. IX
coherently oriented because f(ai + ai+1) is a 2-cell interior to such a cell E on B with which both f(a;) and f(a;+1) agree in orientation. Thus ai for i < k has the same orientation as does a1= 81 and hence the same as A1; and a, for i > k agrees in orientation with a = a2 and thus with A,2.
Since a,8 therefore receives opposite induced orientations from
or,, and ar+,, the common side ab of Ol and 02 which contains a# likewise receives opposite induced orientations from Ol and A2. Accordingly Al and A. are coherently oriented. COROLLARY 1. Non-orientability is invariant under light open mappings of ordinary 2-manifolds onto ordinary 2-manifolds.
Note. This is not valid for manifolds with edges. For a projective plane can be mapped onto a 2-cell by a light open mapping, although of course the mapping cannot be normal.
Also it is not true that orientability is invariant under light open mappings on ordinary closed manifolds. For the sphere maps readily onto a projective plane by 2-1 local homeomorphism. CORoLLARY 2. Orientability i a topological invariant for 2-manifolds. If one triangulation of a given 2-manifold admits an orientation, so also does any other.
To get the first statement we have only to apply the above theorem to h-1, where h(A) = B is a homeomorphism. For the second, we let h be the identity and work with Ta and T, as in the above proof. 4. Degree and index We are now in position to extend to light open mappings the relation established in VIII, (2.1) between degree and index. We note first that a traversal of the edge C of a 2-cell E corresponds
uniquely to an orientation of E in a natural way. For if a, b and c are chosen on C to be vertices of the simplex E and if the traversal t; maps
the interval (0, 1) onto C so that a = 1;(t1), b = (t2), c = l;(t3) for tl < t2 < t3, then Y corresponds to or determines the orientation (a, b, c) of E. Conversely, an orientation of E corresponds uniquely to or determines a sense of traversal of the edge of E. Now if in a 2-manifold A, E1 and E2 are non-overlapping 2-cells with edges C, and C2, then traversals y, and 42 of C1 and C2 are said to agree in sense provided that there exists a simple chain of 2-simplexes
El = al, a2, ... , an = E2 such that a, and a{+1 have in common a set which is a side of both and
a common vertex for Ii - jl > 1 and so that one orientation of the complex made up of the simplexes in this chain agrees with both traversals i;1 and t 2.
§ 4]
DEGREE AND INDEX
105
(4.1) LEMMA. Let f (A) = B be a light open normal mapping of the 2-manifold A onto the orientable 2-manifold B. Let E be a circular disc on B with center p and let El and E2 be 2-cells on A With edges Cl and C= such that E1 E2 = 0 and each of which maps topologically onto E under f. Then
µ0,(f, p) = uc,(f, p) = +27ri for traversals of C1 and C2 which agree in sense.
Proof. If h1 and h2 are sense agreeing traversals of C1 and C2, then since f maps both C1 and C2 topologically onto the edge C of E, the mappings f h1 and fh2 are traversals of C each of index +27ri about p. Thus if we choose a definite orientation of B so that the orientation thus given to E agrees with the traversal fh1 of C. we have /2
(f, p) = 277i.
Now to see that also i c,(f, p) = 21ri, we let
E1= ab
a2,..., an= E2
be a simple chain of 2-simplexes in A oriented so as to agree with the traversals h1 and h2 of C1 and C2 and so chosen that f is topological on any pair of successive simplexes in this chain. Then since the mapping
fhl provides a traversal of the edge of E agreeing with the orientation of B, likewise f maps a2 onto f(a2) so that the resulting traversal of the edge of &2) a2) agrees with the orientation in B since a2 and al agree in orientation and have a common side. Similarly f maps a3 onto f (03) so that the resulting traversal of the edge of &3) a3) agrees with the orientation of f(a2) in B and hence with the orientation in B. Continuing in this way we arrive at the nth step at the conclusion that f maps an = E2 onto E so that the resulting traversal f h2 of C agrees with the orientation
on E and hence with the orientation of B. Accordingly the traversal fh2 of C agrees in sense with f h1 so that foes(!, p) = 2iri = fuc,(.f, P).
(4.2) THEOREM. Let w = f (z) be a light and strongly open mapping of a region A of the z-plane Z into the w-plane W, let C be a simple closed curve lying together with its interior I in A, let p be any point of f(I) and let R be the component of W - f (C) containing p. Then (1 /2 ri) uc(f, p) =
degree of f on I f -1(R) = sum of multiplicities of the points of I f -1(p). Proof. By VIII, (1.2), f is compact and hence of finite degree, say k, on I f-1(R); and for any point q E R, k is the sum of the multiplicities of the points of f-1(q) lying in R. Now let q be chosen in R so that f is locally topological at all points of 1f-1(q) and hence so that 1-f -I(q)
106
GLOBAL ANALYSIS
[Cawr. IX
, Z. Let C11 Cs, . , Ck consists of exactly k distinct points z1, zz, , zk respectively with interiors be simple closed curves enclosing z1, II, , I. such that for each n < k, On + In lies in I and maps topologi. cally under f onto a circular disk E of edge D and interior 0 containing , k). q and where the sets C. + I. are disjoint (n = 1, 2, Now for any positive traversal of the boundary C + C1 + Cs + + , Ck we have PC,(f, q) Ck of the elementary region between C and C1, 21ri, n = 1, 2, , k, by (4.1) above, since f is a homeomorphism of each
C. onto D. Thus by V, (4.1), we have It
suc(f, p) _ pc(f, q)
uc (f, q) =
the first equality resulting from V, (1.3), because p and q are both in R. EEFEEEBCE$ Whyburn [1], Stollow [1].
X. Sequences 1. THEOREM. If f(z) is non-constant in a region R and is the limit function of a uniformly convergent sequence [ f (z)] of differentiable Junctions
in R, then f i8 light.
Proof. Suppose, on the contrary, that f (z) = a for all points z on some non-degenerate continuum in R. Then if Ro is the subset of R on which f(z) :*a, Fr(R0) contains a non-degenerate continuum M lying wholly within R. Let zo e M and let N be a subcontinuum of M of diameter a < } p[zo, Fr(R)] which is irreducible betwet,n zo and a point zl # zo. Let w be a point on N with Izo - wl = U Izo - Z11, let
y be a point of R. with y - wl < JIzo - wl and let C be the circle with center y and radius '] zo - wI. Then since w is within C, and both zo and z1 are without C,,, C, must contain at least two points 8 and t of N.
For otherwise the part of N without and on C. would be a proper subcontinuum of N containing both zo and z1. Now since y e Rs and fRo is open, there exists a point z' of R. within C,, and on the perpendicular bisector of the segment 8t so that the angle sz't is a rational multiple of 2a, say 48z 't = 21rp/q where p and q are integers prime to each other. Next consider the functions
v-1 gw(z) = 1 [ ( fj(z -
z'] - a} n = l 2 .. .
r-0
since
I(z-z')et.trr/a +z' - zoI
Iz-z'I+Iz'-zol
Iz-z'I+1Izo-zil
within the circle C with center z' and radius 2a. Also the sequence [g.(z)] converges uniformly in R to the function
g(z) _e-, IT ( f[(z - z r-0
z'] - a
Thus since g(z) = 0 for all z e N whereas g(z') such that (*)
0, there exists an n
Ign(z)I < a = ilgn(z')I
for all z e N; and hence there exists a region Q lying within C and containing N such that (*) holds for all z e Q. Let 8t be a broken line 107
[CHer. X
SEQUENCES
108
segment in Q joining 8 and t and let h(x) be a homeomorphism of the interval (0, 1) onto st with h(O) = 8. For any x on the interval (1, q) let us define
h(x) = [h(x -- r) - z']e2°'rr/a + z',
for r < x< r + 1, (r = 1, 2,
,q-
Then h maps the interval
1).
I = (0, q) onto a set E and Ign(z)I < b for z e E because gn(E) = gn(at). However, we note that if
e"W = h(x) - z',
0
is a representation of h(x) - z' on (0, 1), then 04 - r)+2+hrp/q = h(x)
- z',
q - 1. Accordingly
_ 0. Since h(q) = 8 = h(0), it follows by V, (1.4), that z' is in a bounded complementary domain of E. Thus
for r < x < r - 1, r = 0, 1, GO-1 p(,. r + 1)(h,z') = q [u(1) - u(0)]
u(o q)(h,z')
z' is within a simple closed curve (simple polygon) J contained in E. But since Ign(z)I < 6 on J this must hold also within J by VI, (3.11), contrary to Ign(z')I = 26 by (*). This contradiction establishes our theorem.
2. Uniform openness
A sequence U .W] of open mappings of X into Y (separable metric spaces) is uniformly open on a subset A of X provided that for each e > 0 there exists an integer N such that if n > N then for any x e A Jn[VE(x)] = V1JN[.!n(x)]
(1)
(2.1) THEOREM. Let the sequence of strongly open mappings f,,: X -. Y converge uniformly to the light mapping f(x), the convergence being uniform
on each compact set in X, where X and Y are locally compact and locally connected. Then the sequence is uniformly open on each compact set in X and f is strongly open. Proof. First we show: Given e > 0, for each x e X, there exists an integer N(x) such that for
any n> N(x) (ii)
r
fn[V E(x)] = V 1,N(Z)W n(x)]
Since f is light we can choose a region U in X about x such that C' is compact, U e Vt,(x) and p[f(C), y] = 2d > 0 where C is the boundary of U and y = f(x). Next we choose h such that V2d(y) is compact and lies in the component R of Vd(y) containing y. Then by uniform convergence
on C + x we can choose an integer N(x)
1/6 and such that for any
UNIFORM OPENNESS
§ 2]
109
n > N(x), fn(x) a V6(y) and fn(C) e V4[f(C)] so that fn(C)-R= 0. Then if Qn is the component of Y - fn(C) containing y and U. is the component of f,-'(Q.) containing x, we have .fn[V((x)]
fn(U.) = Q. R
fn(U)
V2a(y)
Ve[fn\x)]
VI,N(x)[fn(x)]
which establishes (ii).
Next we show that if for each x e X, N(x) is the least integer such that (ii) holds, then as a function of x on X, N(x) is upper semi-continuous. To see this we note that by openness and continuity of fn there exists
a a > 0 and a'-< a, a' > 0, such that fn[VE
- a(x)] z V IIN(x)+aU n(x)]
and so that z' s V, (x) implies f,,(x') e V 1/N(z)+a[f l(x)]. Whence if x' e V,, (x), fn[VE(x )]
AlV,-a(17)1
V1IN(x)+a[fn(x)]
V11N(x)[fn(x') ,
so that N(x') < N(x). Thus N(x) is upper semi-continuous. Accordingly, on any compact set A in X, N(x) is bounded and thus N(x) < N, a constant, for x e A. This gives (i) which establishes uniform openness of the sequence (f.(x)] on A.
Finally to prove strong openness of f at any x e X, let U be any open set in X containing x, let e > 0 be determined so that V,,(x) is compact
and contained in U and, using (ii), find N(x) so that for n > N(x) (iii)
f,,[VE(x)] D V2e(yn)
V6(y),
where y = f(x), yn = fn(x), 26 = 1/N(x). This will imply (iv)
f(U) z f[VE(x)]
V6(y)
For otherwise there is a z e Va(y) and a a > 0 such that V0(z) f[VE(x)] = 0. Then uniform convergence on VE(x) gives an integer I > N(x) such that
for n > I p[fn(w), f(w)] < a for all w e V,,(x),
whereas by (iii) for each n > I there exists a zn a VE(x) with f,,(zn) = z so that p[ n(zn),f(zn)] = p[z,f(zn)] _> a,
since Va(z) f [VE(x)] = 0. Thus we have (iv) which gives strong openness of f.
SEQUENCES
110
[CHAP. X
3. Hurwitz Theorem A strengthened form of the Hurwitz Theorem, not requiring knowledge of the differentiability of the limit function, is now readily established. (3.1) THEOREM. (HuRwrrz). Let the sequence
of function ff(z),
each continuous and differentiable in a region R, converge uniformly in R to a function f(z) not identically zero. Then if e R is an m -fold zero of f (z), every sufficiently small neighborhood D of t; contain exactly m zeros of
f (z) for n > N(D). For let C be a circle in R containing no zero of f(z) and whose interior
D lies in R and contains no zero of f(z) other than Z. Since f(z) # 0 on C, there exists an integer N(D) such that if n > N(D), I f (z) - f(z)I < f(z)I for all z e C. Then by V, (3.12), 0) n > N(D). µc(f, 0) = Now by the theorem in §1 above, f is light and hence by (2.1) it is both light and open. Thus by I,X (4.2), µc(f, 0) and pc(f,,, 0) are 21ri times the numbers of zeros respectively of f and f,, inside C. Accordingly our conclusion follows from (*).
(*)
4. Quasi-open mappings. A mapping f (A) = B Be is said to be quasi-open provided that for any y e B and any open set U in A con. taining a compact component of f-1(y), y is interior relative to Be to f(U). Notes.
1. Every strongly open mapping is quasi-open. 2. Every light quasi-open mapping is strongly open. (4.1) THEOREM. On a locally connected generalized continuum A, a mapping f(A) = B c Be, where Be is locally connected, is quasi-open if and only if for each region R in Be, each conditionally compact component of f-1(R) maps onto R under f. Proof. Necessity. Let Q be a non-empty conditionally compact component of f-1(R). Let y ef(Q). Then Q contains a compact com-
ponent of f-1(y). For if x e Qf-1(y), the component of f-1(y) containing x lies in Q, is a closed set, and thus is compact since Q is conditionally compact. Accordingly, y is interior to f(Q). Then if f(Q) 0 R there is a point z e R - f(Q) which is a limit point of f(Q). Let [z{] converge to z, where zi a f(Q) and let x e Q f-1(z ). Since Q is conditionally compact,
we may suppose that [xn] converges to a point x e Q. Then f(x) lim f(xi) = Jim zi = z, and hence x e f-1(R). Then x e Q, since Q is a component of f-1(R). This implies that z e f(Q), which is a contradiction. Sufficiency. We must show f quasi-open. Let y e B and let X be
a compact component of Pl(y). Let U be an open set containing X.
QUASI-OPEN MAPPINGS
§ 4]
111
Let V be an open set containing X such that V c U, V is compact, and f-1(y)-Fr(V) = 0. Then y non E f[Fr(V)]. Choose a region R in B. so that y E R and R f [Fr(V )] = 0, which is possible since f [Fr(V )] is a compact 0, set not containing y. Since P is compact and since there is a conditionally compact component Q of f-1(R) contained in V. By hypothesis, f(Q) = R and hence y is interior to f(U). (4.2) THEOREM. If A and B0 are locally connected generalized continua, a mapping f(A) = B c Be is qua8i-open if and only if for each conditionally compact region R in A, f(R) consists of f[Fr(R)] together with a collection of Q, components of Be - f[Fr(R)] (or, equivalently,f(R)-Q 0 0 implies f(R) where Q is any component of Be - f [Fr(R)])
Suppose this condition is satisfied. Then if Ro is any region in B. and G is any non-empty conditionally compact component of f-'(R0), we take R = G. By our condition, f(O) consists of f [Fr(G)] together with certain components of Be - f[Fr(G)]. Sincef(G) c R. and Re f[Fr(G)] = 0, we have f (O) B. Thus f (G) = R. and by (4.1), f is quasi-open. On the other hand let f be quasi-open and let R be any conditionally compact region in A. Then if Q is a component of B0 - f[Fr(R)] which
intersects f(R), B will contain a conditionally compact component G of f-1(Q). By (4.1), G maps onto Q so that f(R) f(G) Q. Hence our condition is satisfied.
The condition in (4.2) is very close to the property established in (3.11) of Chapter VI. Indeed, in case the spaces A and B. are planes this latter property likewise characterizes quasi-openness of the mapping as we now show
(4.3) If X and Y are planes, a mapping f:X -* Y of X into Y i8 quasi-open if and only if for each elementary region R in X with boundary 0 vie have (*)
f(R + C) = f(C) + certain bounded components of Y - f(C).
That quasi-openness off implies (*) is a direct consequence of (4.2). To prove the reverse implication let y e Y, let K be a compact component
of f-1(y) and let U be any open set in X containing K. Now X - U is contained in a finite number of components of X - K, say in H. + H1 + - - - + H. where Ho is the unbounded component of X - K and H4 for i > 0 is a bounded component of X - K. Now for each i > 0 let Ki be a continuum lying in H{ and containing H{-(X - U)-clearly this may be constructed by covering Hi-(X - U) by a finite number of circular disks lying in Hr and joining these by arc segments in Hi. By the Plane Separation Theorem, (3.1) of Chapter III, there exists a simple closed curve Ct lying in H4 separating K and K, and not intersecting f-1(y); and since Hi is bounded for i > 0, C{ must enclose K{ and leave
SEQUENCES
112
[CxAr. X
K in its exterior. Similarly, if Jo is a large circle enclosing K, we can construct a simple closed curve Co lying in He and also within J., not
intersecting f-1(y) and separating K from (X - U)-Ho + (Jo + its exterior). Then Co must enclose K and thus encloses Ct for 0 < i S n. Thus if R denotes the elementary region bounded by
C = Co + Ci + ... + C,,, we have Cf-i(y) = 0 and also
R+Cc U, because X - U lies in the exterior of Co plus the union of the interiors
of C{ for i > 0. Thus f(U)' f(R + C); and since by (*) f(R + C) contains the component of Y - f(C) containing y, it follows that y is interior to f(U). Hence f is quasi-open. Methods of proof similar to those employed above suffice to establish the following characterization of quasi-openness of a mapping on even less restricted spaces. (4.4) THEOREM. If X and Y are locally compact separable metric spaces, a mapping f:X -3o. Y is quasi-open if and only if for each condition. ally compact open set U in X
Fr[f(U)] c f[Fr(U)].
(*)
To see that (*) implies quasi-openness of f, let 0 be any open set containing a compact component K of f-1(y) for y e Y. Choose an open set U containing K with l7' compact and contained in 0 and such that
Fr(U) f`1(y) = 0. Then (*) implies that y is interior to f(U) and thus also to f(0), because y belongs to f(U) but not to Fr[f(U)]. On the other hand quasi-openness of f implies (*). For suppose that
for some conditionally compact open set U in X there exists a point y in Fr[f(U)] which is not in f(C) where C = Fr(U). We first show that y ef(U). If not, there exists a sequence of points y1, y2, in f(U) converging to y. By conditional compactness of U there exists an infinite sequence of points x, , x,,, . - with x,,, e U f-1(y,,,) so that x,,, - . x e C. Then x is not in C because f(x) = y and y is not in f(C). Hence x e U so that y E f (U). Then by quasi-openness of f, y must be interior to f(U) because U clearly contains a component K of f-1(y) and K is compact by conditional compactness of U. This contradicts the fact that y e Fr[ f (U)].
We next prove a theorem on sequences of quasi-open mappings analogous to the Weierstrass double series theorem. This theorem includes as a special case the strong openness part of the conclusion in (2.1).
QUASI-OPEN MAPPINGS
§ 41
ltd
(4.5) THEOREM. If A and B0 are locally connected generalized continua
and the sequence of quasi-open mappings f (A) = B,, C Be converges to the mapping f(A) = B e B0, the convergence being uniform on each compact set in A, then f is quasi-open. Proof. Let y e B and let X be a non-empty compact component of f-1(y). Let W be an open set containing X. Choose an open set U contain. ing % such that U is compact, U c W, and such that f-1(y)-Fr(U) = 0. We show that y is interior to f (C). Define e so that 3e, = p(y, F), where F = f[Fr(U)]. Then e > 0 and we may choose 0 < e < e such that V,,(y) is contained in some component B of V,(y). Since the convergence is uniform on 0, there exists an integer N such that for n N we have f(x)] < e for all x E U. Then for n > N we have f (x) c VE(y) c: R and F. = f,[Fr(U)] c V,(F).
Then R-F = 0, and B is contained in the component R. of B. - F. which contains y. Then if U is the component of containing
%, we have U. c U. Then f, (U,,) = R R f (U)
VE(y) and hence R for n sufficiently large. Since 17 is compact, we obtain
f (17)
R. This completes the proof. REFERENCES
Whyburn [4, 5]. See also Polak [1].
Bibliography AnLFo1s, L. V. , Contributions [1] Development of the theory of conformal mapping, to the Theory of Riemann Surfaces, Princeton University Press, 1953 (Annals of Mathematics Studies, No. 30).
EaaiasTON, H. C. [1] (with Ursell, H. D.), On the lightness and strong interiority of analytic functions, J. London Math. Soc. 27 (1952), 260-271. EILENBEBO, S.
[1] Transformations continues en circonference et la topologie du plan, Fund. Math. 26 (1936), 61-112.
Hxn s, M. H. [1] (with Morse, M.), Deformation classes of meromorphic functions and their extensions to interior transformations, Acta Math. 79 (1947), 51-103. JENXSNS, J.
[1] (with Morse, M.), Conjugate nets on an open Riemann surface, from Lectures on Functions of a Complex Variable, University of Michigan Press, 1955.
KBR*xJ1sT6, B. v. [1] Topologie, Berlin, Springer, 1923. KUB.4TOWSKI, C.
[1] Homotopie et functions analytiques, Fund. Math. 33 (1945), 316-367. [2] Topologie II, Warszawa, Polska Akad. Nauk, 1950. MooB.E, R. L. [1] Foundations of Point Set Theory, New York, 1932 (American Mathematical Society, Colloquium Publications, vol. 13). MORSE, M.
[1] Topological Methods in the Theory of Functions of a Complex Variable, Princeton University Press, 1947. (Annals of Mathematics Studies, No. 15). [2] See Heins [1]. [3] See Jenkins [1]. 115
BIBLIOGRAPHY
116
(B:s.
NEVANraNNA, R.
[1] Eindeutige Analytische Funktionen, Berlin, Springer, 1936. PLUNSETr, R. L. [1]. A topological proof of a theorem of complex analysis, Proc. Nat. Acad. Sci., 42 (1956), 425-426.
PoLAx, A. I. [1] On functions taking no constant value on a non-degenerate connected set, Dokladi Akad. Nauk, USSR, 100 (1955), 213-215. STofWW, S.
[1] Principes Topologiques do la Theorie des Fonctions analytiques, Paris, (lauthier-Villars, 1938. [2] Sur un theorime topologique, Fund. Math., 13 (1929), 186-194.
Trrus, C. J. [1] (with Young, G. S.), A Jacobian condition for interiority, Michigan Math. J. 1 (1952), 89-94. UBSELL, H. D. [1] See Eggleston [1]. WHYBURN, G. T.
[1] Analytic Topology, New York, 1942 (American Mathematical Society, Colloquium Publications, vol. 28).
[2] Introductory topological analysis, from Lectures on Functions of a Complex Variable, University of Michigan Press, 1955.
[3] Topological analysis, Bull. Amer. Math. Soc., 62 (1956), 204-218. [4] Open mappings on locally compact spaces, Amer. Math. Soc. Mem., No. 1, New York, 1950.
[5] Sequence approximations to interior mappings, Ann. Soc. Polon. Math., 21 (1948), 147-152. YOUNG, G. S.
[1] See Titus [1].
Index accessibility, 27 acyclic curve, 33 admissible representation, 56 agreement in orientation of triangles, 102 agreement in sense of traversals, 63, 104 applications of modulus function, 77 we, 26 arcwise connectedness, 26 basis, 5
binormal mapping, 81 Borel Theorem, 6 boundary curve (edge), 83 boundary of a set, 12 boundary point, 83 branch, 56 Brouwer Reduction Theorem, 15 Cauchy-Riemann equations, 52 Cauchy sequence, 22 chain of type T, 18 characteristic, Euler, 98 circulation index, 58 "closed" manifold, 83 closed set, 4 closed 2-cell, 36 "coherently" oriented triangles, 102 compact mapping, 90 compact set, 5 complete enclosure, 22 complete space, 22 complex, 98 complex number system, 41 complex plane, 44 complementary domain, 32 component, 10 conditionally compact set, 5 connected set, 9 continuity of a function, 50 continuous branch, 58 continuous transformation, 20 continuum, 12; generalized, 12; irreducible, 14 continuum of convergence, 13 convergent sequence of sets, 7 countable set, 3
curve; acyclic, 33; boundary, 83; level, 77; open, 100; simple closed, 28 curved simplex, 98 cut point, 33 cyclicly connected set, 33 cyclic set, 34 deficiency of a function, 101 degree of a function, 90 dense, 5, 93 derivative, 51 diameter, 7 differentiability conditions, 52 disjoint sets, 3 distance between two sets, 7 distance function, 3 domain, complementary, 32
edge point, 83 edges of a graph, 67 elementary region, 38 enumerable set, 3 e-chain (epsilon chain), 10 Euler characteristic, 98 exponential function, 53 exponential representation, 58 extension of openness, 81 extension of transformations, 22 fine subdivision, 17 frontier, 12 function; exponential, 53;
logarithm,
55; meromorphic, 92; modulus, 77; power, 88; upper semi-continuous, 5 fundamental (Cauchy) sequence, 22 generalized continuum, 12 graph, 67 hexagonal subdivision, 32 homeomorphism, 20 homeomorphism, positive (negative), 64 homotopy, 68 Hurwitz Theorem, 93, 110 image, 20 index, 58 index invariance, 64, 68 117
INDEX
118
index near a non-zero of the derivative, 72
index, topological or circulation, 58 inducible (inductive) property, 14 inferior limit, 7 interior mapping, 76 invariance of agreement in sense, 63 invariance of index, 64, 88 inverse, 20
irreducible continuum, 14 isolated point, 80 isometric, 23 isomorphic (similar) simple subdivisions, 38
Jordan Curve Theorem, 29 level curve, 77 light mapping, 75
lightness of differentiable functions, 75 limit; inferior, 7; of a function, 50; point, 4; superior, 7; Theorem, 11 LindelOf Theorem, 5 local degree of a function, 84, 101
locally compact set, 5 locally connected set, 13 locally topological mapping (local homeomorphism), 85 local topological analysis, 88
logarithm function, 55 manifolds; ordinary or "closed," 83;
orientable and non-orientable, 102; 2-dimensional, 83; triangulation of 2-dimensional, 102; with boundary, 83 mappings; compact, 90; homotopic, 68; light, 75;
neighborhood, spherical, 4 non-; dense set, 93; orientable manifolds, 104 normal mapping, 81
one-to-one (1-1) transformation, 20 open (interior) mapping, 76 open mappings on simple cells and manifolds, 86 openness of differentiable functions, 76 openness, uniform, 108 open set, 4
order n, point of, 35 ordinary ("closed") manifold, 83 ordinary (regular) point, 83 a orientability and orientation, 101 orientable 2-dimensional manifolds, 102 orientation agreement of triangles, 102 Phragrn n-Brouwer Theorem, 32 Plane Separation Theorem, 34 point; cut, 33; edge (boundary or singular), 83; inverse of a, 20; isolated, 80; limit, 4; regular, 35 point inverse, scattered, 83 point of order n, 35
polar form (of a complex number), 45 pole of a function, 92 positive homeomorphism, 64 positive traversal, 62 power mapping, 88 powers and roots, 46 Property S, 16 punctured complex plane, 69 quasi-open mapping, 110
locally topological, 85;
modulus, 77; monotone, 25; normal and binormal, 81; open (interior), 76; power, 88; quasi-open, 110; strongly open (strongly interior), 76; topologically equivalent, 86 mapping theorems, 24 Maximum Modulus Theorem, 77 measure of the image of the zeros of the derivative, 72 meromorphic function, 92 metric, 3 metric spaces, 3 m-fold zero of a function, 93 modulus function (mapping), 77 monotone mapping, 25 mutually exclusive sets, 9
reduced- differentiability assumptions, 93 Reduction Theorem, Brouwer, 15 region, 15 regular (ordinary) point, 35
Rouchb Theorem, 93
scattered inverse property, 83 semi-polygon, 30
sense agreement of traversals, 63, 104 separability, 5 separation, 9 sequence; fundamental (Cauchy), 22; uniformly open, 108
sets; arcwise connected, 26; boundary of (frontier of), 12; closed, 4; compact,
complete, 22; complete enclosure components of, 10; conditionally compact, 5; connected, 9; convergent collection of, 7; countable 5;
of, 22;
negative homeomorphism, 64 negative traversal, 62
INDEX (enumerable), 3; cyclic, 33; cycliely connected, 34; dense, 5, 93; diameter of, 7; disjoint (mutually exclusive), 3; distance between, 7; locally compact, 5; locally connected, 15; nondense, 93; open, 4; operations on, 3; subdivision of, 38; uniformly locally connected, 16; well-chained, 10 similarity correspondence, 38 similar (isomorphic) simple subdivisions, 38
simple are, 28 simple closed curve, 28 simplex; curved, 98; sides of a, 98; straight 0., 1-, 2-dimensional, 98 singular point, 98 apace; complete, 22; metric, 3; topologically complete, 22 spherical neighborhood, 4 S-property, 16 strongly open (strongly interior) map-
ping, 76 subdivisions; fine, 17; hexagonal, 32;
isomorphic simple, 38; simple, 37; simplicial, 98 Subdivision Theorem, 36 superior limit, 7
9-curve (theta-curve), 31 topological index, 56 topologically complete space, 22 topologically equivalent mapping, 86
119
topological transformations, 20 Torhorst Theorem, 32 transform, 20 transformation, 20 traversal of arcs and simple closed curves, 60 traversal of region boundaries and region subdivisions, 67 traversal, positive, 82 traversals, agreement in sense of, 63, 104 triangles, "coherently" oriented, 102 triangulation of 2-dimensional manifold, 102
2-cell (two-cell), closed, 36
2-dimensional (two-dimensional) manifold, 83 Type T Chain, 18
uniformly continuous transformation, 21 uniformly locally connected set, 16 uniformly open sequence, 108 uniform openness, 108 upper semi-continuous function, 5 vertices of a graph, 67 vertices of a simplex, 98 well-chained set, 10
zero sides (0-sides) of a simplex, 98 Zoretti Theorem, 35
