Topological analysis (Princeton mathematical series No. 23)

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Topological analysis (Princeton mathematical series No. 23)

TOPOLOGICAL ANALYSIS BY Gordon Thomas Whyburn PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1958 Preface While t

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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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vii

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3 3

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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

52 53

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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60 64

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67 68

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72 72 73

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75 76

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77

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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83 86 88 89

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92 93 93

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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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98 100

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104 107 108

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110 110

Bibliography

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115

Index

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117

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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, 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 - 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 0

00 = 1/2

when x = 0 and y > 0

1f2 -

(

+

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) 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 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 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)
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 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 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 0 such that for any connected set E in A - T of diameter 0 such that for any set N in B of diameter 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