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    Topology for Analysis
    Topology for Analysis
    Topology for Analysis
    Ebook692 pages9 hours

    Topology for Analysis

    By Albert Wilansky

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    About this ebook

    Appropriate for both students and professionals, this volume starts with the first principles of topology and advances to general analysis. Three levels of examples and problems, ordered and numbered by degree of difficulty, illustrate important concepts. A 40-page appendix, featuring tables of theorems and counter examples, provides a valuable reference.
    From explorations of topological space, convergence, and separation axioms, the text proceeds to considerations of sup and weak topologies, products and quotients, compactness and compactification, and complete semimetric space. The concluding chapters explore metrization, topological groups, and function spaces. Each subject area is supplemented with examples, problems, and exercises that progress to increasingly rigorous levels. All examples and problems are classified as essential, optional, and advanced.
    LanguageEnglish
    PublisherDover Publications
    Release dateJan 18, 2013
    ISBN9780486150758
    Topology for Analysis

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      Indeholder "Preface", "Acknowledgements", "1. Introduction", " 1.1 Explanatory Notes", " 1.2 n-Space", " 1.3 Abstraction", "2. Topological Space", " 2.1 Topological space", " 2.2 Semimetric and metric space", " 2.3 Semimetric and metric topologies", " 2.4 Natural topologies and metrics", " 2.5 Notation and terminology", " 2.6 Base and subbase", "3. Convergence", " 3.1 Sequences", " 3.2 Filters", " 3.3 Partially ordered sets", " 3.4 Nets", " 3.5 Arithmetic of nets", "4. Separation Axioms", " 4.1 Separation by open sets", " 4.2 Continuity", " 4.3 Separation by continuous functions", "5. Topological Concepts", " 5.1 Topological properties", " 5.2 Connectedness", " 5.3 Separability", " 5.4 Compactness", "6. Sup, Weak, Product, and Quotient Topologies", " 6.1 Introduction", " 6.2 Sup topologies", " 6.3 Weak topologies", " 6.4 Products", " 6.5 Quotients", " 6.6 Continuity", " 6.7 Separation", "7. Compactness", " 7.1 Countable and sequential compactness", " 7.2 Compactness in semimetric space", " 7.3 Ultrafilters", " 7.4 Products", "8. Compactification", " 8.1 The one-point compactification", " 8.2 Embeddings", " 8.3 The Stone-Cech compactification", " 8.4 Compactifications", " 8.5 C- and C*-embedding", " 8.6 Realcompact spaces", "9. Complete Semimetric Space", " 9.1 Completeness", " 9.2 Completion", " 9.3 Baire category", "10. Metrization", " 10.1 Separable spaces", " 10.2 Local finiteness", " 10.3 Metrization", "11. Uniformity", " 11.1 Uniform space", " 11.2 Uniform continuity", " 11.3 Uniform concepts", " 11.4 Uniformization", " 11.5 Metrization and completion", "12. Topological Groups", " 12.1 Group topologies", " 12.2 Group concepts", " 12.3 Quotients", " 12.4 Topological vector spaces", "13. Function Spaces", " 13.1 The compact open topology", " 13.2 Topologies of uniform convergence", " 13.3 Equicontinuity", " 13.4 Weak compactness", "14. Miscellaneous Topics", " 14.1 Extremally disconnected spaces", " 14.2 The Gleason map", " 14.3 Categorical algebra", " 14.4 Paracompact spaces", " 14.5 Ordinal spaces", " 14.6 The Tychonoff plank", " 14.7 Completely regular and normal spaces", "Appendix, Tables of Theorems and Counterexamples", "Bibliography", "Index".

    Book preview

    Topology for Analysis - Albert Wilansky

    DOVER BOOKS ON MATHEMATICS

    HANDBOOK OF MATHEMATICAL FUNCTIONS, Milton Abramowitz and Irene A. Stegun. (0-486-61272-4)

    TENSOR ANALYSIS ON MANIFOLDS, Richard L. Bishop and Samuel I. Goldberg. (0-486-64039-6)

    VECTOR AND TENSOR ANALYSIS WITH APPLICATIONS, A. I. Borisenko and I. E. Tarapov. (0-486-63833-2)

    THE HISTORY OF THE CALCULUS AND ITS CONCEPTUAL DEVELOPMENT, Carl B. Boyer. (0-486-60509-4)

    THE QUALITATIVE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION, Fred Brauer and John A. Nohel. (0-486-65846-5)

    ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES, Richard P. Brent. (0-486-41998-3)

    PRINCIPLES OF STATISTICS, M. G. Bulmer. (0-486-63760-3)

    THE THEORY OF SPINORS, Élie Cartan. (0-486-64070-1)

    ADVANCED NUMBER THEORY, Harvey Cohn. (0-486-64023-X)

    STATISTICS MANUAL, Edwin L. Crow, Francis Davis, and Margaret Maxfield. (0-486-60599-X)

    FOURIER SERIES AND ORTHOGONAL FUNCTIONS, Harry F. Davis. (0-486-65973-9)

    COMPUTABILITY AND UNSOLVABILITY, Martin Davis. (0-486-61471-9)

    ASYMPTOTIC METHODS IN ANALYSIS, N. G. de Bruijn. (0-486-64221-6)

    PROBLEMS IN GROUP THEORY, John D. Dixon. (0-486-61574-X)

    THE MATHEMATICS OF GAMES OF STRATEGY, Melvin Dresher. (0-486-64216-X)

    APPLIED PARTIAL DIFFERENTIAL EQUATIONS, Paul DuChateau and David Zachmann. (0-486-41976-2)

    ASYMPTOTIC EXPANSIONS, A. Erdélyi. (0-486-60318-0)

    COMPLEX VARIABLES: HARMONIC AND ANALYTIC FUNCTIONS, Francis J. Flanigan. (0-486-61388-7)

    ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS, Kurt Gödel. (0-486-66980-7)

    A HISTORY OF GREEK MATHEMATICS, Sir Thomas Heath. (0-486-24073-8, 0-486-24074-6) Two-volume set

    PROBABILITY: ELEMENTS OF THE MATHEMATICAL THEORY, C. R. Heathcote. (0-486-41149-4)

    INTRODUCTION TO NUMERICAL ANALYSIS, Francis B. Hildebrand. (0-486-65363-3)

    METHODS OF APPLIED MATHEMATICS, Francis B. Hildebrand. (0-486-67002-3)

    TOPOLOGY, John G. Hocking and Gail S. Young. (0-486-65676-4)

    MATHEMATICS AND LOGIC, Mark Kac and Stanislaw M. Ulam. (0-486-67085-6)

    MATHEMATICAL FOUNDATIONS OF INFORMATION THEORY, A. I. Khinchin. (0-486-60434-9)

    ARITHMETIC REFRESHER, A. Albert Klaf. (0-486-21241-6)

    CALCULUS REFRESHER, A. Albert Klaf. (0-486-20370-0)

    PROBLEM BOOK IN THE THEORY OF FUNCTIONS, Konrad Knopp. (0-486-41451-5)

    INTRODUCTORY REAL ANALYSE, A. N. Kolmogorov and S. V. Fomin. (0-486-61226-0)

    SPECIAL FUNCTIONS AND THEIR APPLICATIONS, N. N. Lebedev. (0-486-60624-4)

    CHANCE, LUCK AND STATISTICS, Horace C. Levinson. (0-486-41997-5)

    TENSORS, DIFFERENTIAL FORMS, AND VARIATIONAL PRINCIPLES, David Lovelock and Hanno Rund. (0-486-65840-6)

    SURVEY OF MATRIX THEORY AND MATRIX INEQUALITIES, Marvin Marcus and Henryk Minc. (0-486-67102-X)

    ABSTRACT ALGEBRA AND SOLUTION BY RADICALS, John E. and Margaret W. Maxfield. (0-1186-67121-6)

    FUNDAMENTAL CONCEPTS OF ALGEBRA, Bruce E. Meserve. (0-486-61470-0)

    FUNDAMENTAL CONCEPTS OF GEOMETRY, Bruce E. Meserve. (0-486-63415-9)

    FIFTY CHALLENGING PROBLEMS IN PROBABILITY WITH SOLUTIONS, Frederick Mosteller. (0-486-65355-2)

    NUMBER THEORY AND ITS HISTORY, Oystein Ore. (0-486-65620-9)

    MATRICES AND TRANSFORMATIONS, Anthony J. Pettofrezzo. (0-486-63634-8)

    THE UMBRAL CALCULUS, Steven Roman. (0-486-44139-3)

    PROBABILITY THEORY: A CONCISE COURSE, Y. A. Rozanov. (0-486-63544-9)

    LINEAR ALGEBRA, Georgi E. Shilov. (0-486-63518-X)

    ESSENTIAL CALCULUS WITH APPLICATIONS, Richard A. Silverman. (0-486-66097-4)

    A CONCISE HISTORY OF MATHEMATICS, Dirk J. Struik. (0-486-60255-9)

    PROBLEMS IN PROBABILITY THEORY, MATHEMATICAL STATISTICS AND THEORY OF RANDOM FUNCTIONS, A. A. Sveshnikov. (0-486-63717-4)

    TENSOR CALCULUS, J. L. Synge and A. Schild. (0-486-63612-7)

    MODERN ALGEBRA: Two VOLUMES BOUND As ONE, B.L. Van der Waerden. (0-486-44281-0)

    CALCULUS OF VARIATIONS WITH APPLICATIONS TO PHYSICS AND ENGINEERING, Robert Weinstock. (0-486-63069-2)

    INTRODUCTION TO VECTOR AND TENSOR ANALYSIS, Robert C. Wrede. (0-486-61879-X)

    DISTRIBUTION THEORY AND TRANSFORM ANALYSIS, A. H. Zemanian. (0-486-65479-6)

    Copyright

    Copyright © 1970, 1998 by Albert Wilansky All rights reserved.

    Bibliographical Note

    This Dover edition, first published in 2008, is an unabridged republication of the work originally published by Ginn and Company, Waltham, Massachusetts, in 1970.

    Library of Congress Cataloging-in-Publication Data Wilansky, Albert.

    Topology for analysis / Albert Wilansky.

    p. cm.

    Originally published: Waltham, Mass. : Ginn, [1970].

    Includes bibliographical references and index.

    9780486150758

    1. Topology. I. Title.

    QA611.W53 2008

    514—dc22

    2008026255

    Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501

    To my mother, Esther (Sidel) Wilansky,

    although a widow with six children,

    she saw to the completion of my education.

    Preface

    This book is intended to serve the needs both of the beginning student and of the mature mathematician. Also it is intended as a reference and handbook. As an elementary text it begins with first principles and develops without haste all that part of topology which may be described as generalized analysis; for the mathematician, the subject is carried further in examples and problems. The handbook function of this text is carried out in an experimental concept: the Tables of Theorems and Counterexamples in the Appendix. For a discussion of the use of these tables, the reader is referred to the introduction immediately preceding the tables.

    are harder and more special. Each problem section is broken into three parts —the second and third parts are numbered from 100 and 200, respectively. The 100 problems are special and somewhat challenging; the 200 problems may be extremely difficult.

    Both nets and filters are explained and used throughout the text. All the standard counterexamples of the subject are presented; an almost successful effort was made to use the Stone–Cech compactification as a universal counterexample—thus the use of ordinal spaces is reduced to a minimum, almost zero.

    NOTE. I have followed the custom, now quite respectable, of not using more separation axioms than needed. Arguments developed for T2 spaces are adapted in several ways. One is by the device of using retractions extensively (see, for example, Sec. 6.7, Problem 102). Another consists of using closed graph instead of continuous functions (see, for example, Sec. 7.1, Problem 109).¹ Metrization theorems, usually proved by embedding in products, are replaced by semimetrization theorems, proved by means of weak topologies. Occasionally the hypothesis compact must be replaced by locally compact, an unrelated condition in non-Hausdorff spaces (see, for example, Lemma 13.1.1 and Theorem 13.1.1). This procedure is set forth in detail in my article "Life without T2," Amer. Math. Monthly, 77(1970), pp. 157–161.

    AXIOMATICS. One should beware of the disease called Axiomatics, which consists of wasting time wondering whether a, b, and c imply d, where a, b, c, and d are properties selected at random. We can do no better than quote Edwin Hewitt’s remark made at the 1955 Madison Summer Institute on Set Theoretic Topology and printed on p. 17 of its Proceedings: "I am happy to say that it (the disease of axiomatic topology) has been almost totally cured. Right now I don’t care a bit whether every beta capsule of type delta is also a T-spot of the second kind. However, every young mathematician must have an injection of the live virus; and every mathematician should be aware of a few basic implications such as regular, Lindelöf implies normal, and regular, second countable implies semimetrizable"; results of this basic nature are included in the text. For those who absolutely must know whether every locally compact, completely regular, separable space is σ-compact, a search in the Tables will reveal a counterexample.

    Acknowledgments

    It has always been my custom to write to everybody about everything. The kindness of the mathematical community in replying to my letters has been so great that I could not even begin to list the names of those people who made significant contributions to this book. In addition to this correspondence, I have incorporated hundreds of results from the mathematical literature. In some cases the authors are cited at appropriate places in the text. The greatest source of help and encouragement has been the Lehigh University Mathematics Department. My familiar notes on our bulletin board with the large letters HELP on top were invariably answered by the elegant examples of Jerry Rayna, or the erudite references and proofs of Bill Ruckle, to mention only two of my very helpful colleagues. Murray Kirch used my typescript in a course at SUNY Buffalo and made helpful criticisms. John W. Taylor made an enormous contribution! Unfortunately some of his remarks came too late to be used.

    The typing was done by Helen Farrell, Rosemarie Ehser, and Judy Arroyo; I thank these three ladies for their excellent work and their devotion to a tedious task.

    The administration of Lehigh University was at all times sympathetic and helpful. My deepest thanks go to all.

    In addition I want to express my great appreciation to the staff of the college division of Ginn and Company and most especially to Mrs. Claire Felts whose contributions to this book occur in every line.

    A.W.

    Table of Contents

    DOVER BOOKS ON MATHEMATICS

    Title Page

    Copyright Page

    Dedication

    Preface

    Acknowledgments

    Topology for Analysis

    1 - Introduction

    2 - Topological Space

    3 - Convergence

    4 - Separation Axioms

    5 - Topological Concepts

    6 - Sup, Weak, Product, and Quotient Topologies

    7 - Compactness

    8 - Compactification

    9 - Complete Semimetric Space

    10 - Metrization

    11 - Uniformity

    12 - Topological Groups

    13 - Function Spaces

    14 - Miscellaneous Topics

    Appendix

    Tables of Theorems and Counterexamples

    Bibliography

    Index

    Topology for Analysis

    1

    Introduction

    1.1 Explanatory Notes

    There are certain standard and all-pervasive notations and terminologies used by mathematicians. In addition, we use a few special notations with less currency.

    The following notations will be used:

    We draw the reader’s attention to the following:

    A B, read "A does not meet B" meaning A ∩ B = ∅.

    is used, it is assumed that a set X . When the presence of X is not clear from the context, the notation X \ A will be used.

    The words space, set, family, and collection are synonymous.

    When a space X has been designated the members of X will be called points.

    The words map, mapping, and "function are synonymous. If ƒ: X → Y we call X the domain, and Y the range of ƒ; "ƒ will sometimes be written as x → ƒ(x)."

    ITALICS. A word in italics is being used for the first time and is defined by the sentence in which it appears.

    PROOF BRACKETS. Part of a discussion enclosed in square brackets means the statement immediately preceding the brackets is being proved. As an example, suppose the text reads, "Since x if x = 0, it follows that cos x = 1 contradicting the hypothesis], we may cancel x from both sides. The reader should first absorb Since x is not zero, we may cancel x from both sides." He may then proceed with the text, or, if desired, return to the proof in brackets.

    are special and of limited interest.

    100 PROBLEMS. In each section Problems 101, 102, . . . are devoted to extending the text, and exposing interesting results beyond the scope of the book.

    200 PROBLEMS . Problems 201, 202, . . . may be extremely challenging.

    .

    BIBLIOGRAPHY.  References to the bibliography at the back of the book are indicated by [ ].

    THE EMPTY SET. We state our conventions concerning the empty set ∅. These cannot be proved since we do not set up our set theory formally. There is only one empty sent ∅ ; ∅ ⊂ A for all sets A. (This makes the statement "A ⊂ B for subsets of a space X," true if B = X ) Wherever some property of sets is tested for a set A by examining an arbitrary point of A, then this property is true of the empty set. For example, "all positive integers a, b, c satisfying an + bn = cn for some integer n {(a, b, c): an + bn = cn, a, b, c ∈ ω}.) The empty set is finite.

    .

    Problems

    1. If ƒ: X → Y, g: Y →Z , define g ƒ: X → Z by (g ƒ)(x) = g[ƒ(x)]. If ƒ, g are one-to-one and onto prove that (g ƒ)−1 = ƒg−1. (ƒ−1 means the map from Y to X satisfying ƒƒ = ix, ƒ ƒ−1 = iy where ix(x) = x for all x ∈ X.)

    2. Let {Sα: α ∈ A} be a family of subsets of a set X, and let ƒ: X → Y. Prove that

    ƒ[∪ {Sα: α ∈ A}] = ∪ {ƒ.[Sα]: α ∈ A};

    ƒ[∩ {Sα: α ∈ A}] ⊂ ∩ {ƒ[Sα]: α ∈ A};

    if ƒ is one-to-one, inclusion may be replaced by equality in (b), but not in general, even if A has only two members;

    if ƒ is one-to-one, ƒ] c {ƒ[S]}∼;

    if ƒ is onto, {ƒ[S]}∼ ⊂ ƒ];

    one-to-one cannot be omitted in (d), and onto cannot be omitted in (e).

    3. Let {Sα: α ∈ A} be a family of subsets of Y, and let ƒ: X → Y. Then (in contrast with Problem 2), ƒ−1[∪ Sα] = ∪ ƒ−1[Sα], ƒ−1[∩ Sα] = ∩ƒ−1[Sα], ƒ] = {ƒ−1[S]}∼.

    4. Let A, B be two collections of subsets of a set X with AB. Prove that ∩ {S: S ∈ A} ⊃ ∩ {S: S ∈ B}. What is the corresponding result for union?

    5. Let ƒ: X → Y and let S be a subset of X or Y. Prove that ƒ[ƒ−1[S]] ⊂S, ƒ−1[ƒ[S]] ⊃ S, ƒ[{ƒ−1[S]}∼] ∅S.

    6. What assumption about ƒ would produce equality in the first two parts of Problem 5?

    7. Let X, Y be sets. Let X x Y be the set of all ordered pairs (x, y) with x ∈ X, y ∈ Y. Show that R² = R x R.

    if and only if A ⊃ B.

    9. Prove the following formulas (given by the 19th-century mathematician, A. de Morgan).

    10. Let ƒ: X → X. Let A, B be disjoint subsets of X, and set G = A ∩ ƒ−1[B]. Show that ƒ[GG.

    11. Let ƒ: X → Y, g : Y → X. We say that g is a left inverse of ƒ, and ƒ is a right inverse of g if g ƒ is the identity on X. Show that f has a left inverse if and only if ƒ is one-to-one, and a right inverse if and only if ƒ is onto.

    1.2 n-Space

    The space R of real numbers will not be defined in this book. It will be assumed to allow the usual operations of arithmetic, to have its usual ordering (in short, it is a totally ordered field), and to have the property that every bounded set S has a least upper bound, written sup S. Such facts as x² ≥ 0 for all x ∈ R will be used without scruple.

    A set S is called countably infinite if it can be put in one-to-one correspondence with ω; that is, there exists ƒ: S → ω which is one-to-one and onto. A set is called countable if it is finite or countably infinite. A set which is not countable is called uncountable. We shall assume the existence of an uncountable set. (Some are shown in Problems 201, 202, etc., and R is also proved uncountable in Sec. 9.3, Problem 114.)

    We denote by Rn the set of all ordered n-tuples of real numbers, where n is a positive integer; R¹ is the same as R. For x, y ∈ Rn, say, x = (x1, x2, . . ., xn), y = (y1, y2, . . ., yn), we define ||x||, pronounced norm x, by the formula

    and x • y, pronounced x dot y, by the formula

    so that, in particular, x • x = ||x||².

    Note that ||x + y||² = (x + y)•(x + y) = x • x + 2x • y + y • y. Thus, for all x, y,

    2x • y = ||x + y||² − ||x||² − ||y||²,

    and, similarly

    −2x • y = ||x − y||² − ||x||² − ||y||².

    Since ||x ± y||² ≥ 0 we get ±2x • y ≤ ||x||² + ||y||²,hence,

    (1.2.1)

    Now let x, y be different from 0. (By 0 is meant the n-tuple (0, 0, . . . , 0).) Let x′ = x/||x||, y′ = y/||y||. Then ||x′|| = ||y′|| = 1 and so, by (1.2.1), we have |x′ • y′| ≤ 1, and so

    (1.2.2)

    Formula (1.2.2), called Cauchy’s inequality (named for the famous 19th-century mathematician, A. Cauchy), was proved for x, y different from 0, but obviously holds in this case also.

    We now have

    Taking positive square roots we obtain

    (1.2.3)

    If now we define d(x, yx − y , the familiar distance (familiar at least for n = 1, 2, 3), we have, for any x, y, z,

    This function d is called the Euclidean distance for Rn, also the Euclidean metric for Rn

    Problems on n-Space

    x − y x y |.

    2. Fix m, n with m > n and define f : Rn → Rm by (x1, x2, . . ., xn) → (x1, x2, . . ., xn, 0, 0, . . ., 0). Is f f(xx for all x ∈ Rn.

    3. Let a ∨ b (read: a sup b) and a ∧ b (read: a inf b) be the larger and smaller, respectively of a, b ∈ R. Prove that

    101. Show that R and R(x, y) → z where z = .x1y1x2y2x3yin decimal notation. Make xi, yi = 0 if possible. This map is one-to-one but not onto.]

    102. If there exists a one-to-one f : S → ω; then S must be countable.

    103. If there exists an onto f : ω → S; then S must be countable.

    where each Sn is an infinite set and {Sn} is a disjoint family.

    201. Prove that the set S of all sequences of 0’s and 1’s is not countable. (A sequence of 0’s and 1’s is a sequence {xn} with xn = 0 or 1 for all nLet f : ω → S. Then the characteristic function of {n ∈ ω : yn = 0 if y = f(n)} does not belong to f[ωDeduce that R is uncountable.

    202. The set of all subsets of a set S is denoted by 2S. Prove that 2ω

    203. If there is a one-to-one map from A to B and no one-to-one map from B to A, we write |B| > |A| (say: B has a larger cardinality than A). Show that |2S| > |S| for every set S.

    204. Show that S x S can be put into one-to-one correspondence with S if and only if S is an infinite set or is empty.

    205. Show that the set of all permutations of ω is uncountable. (Indeed there are exactly 2|ω| of them.)

    206. A closed interval in R cannot be the union of a disjoint countable family, with more than one member, of closed intervals.

    207. Show that there is room in R² for uncountably many disjoint L’s but only countably many disjoint TMR 21

    208. Given a countable set S ⊂ R, show that there exists t ∈ R such that x + t is irrational for every x Q − S is countable, hence there exists t ∈ R \ (Q − S

    209. Let s be the set of all sequences of real numbers, and Jω the set of all sequences of irrational numbers. Let A be a countable subset of s. Show that there exists t ∈ s such that x + t ∈ Jω for all x ∈ A. [For each n, let Sn be the set of all nth terms of members of A. Choose tn as in Problem 208, and let t = {tn

    1.3 Abstraction

    In the course of his mathematical education, the student encounters various structures. By a structure we mean a set together with some rules concerning its members and subsets. For example, the set of real numbers together with the operation of addition, the set of positive real numbers with multiplication, and the set of integers with addition, are three structures. In each of these three examples, there is a set and one operation. When we say that each of them is a group we are performing an abstraction; forgetting the differences between these examples, we consider only their similarities. If a remark is made to the effect that a group must have a certain property, this means that every group, in particular the three just mentioned, must have this property. The three examples, and any others, are called realizations, or examples, of the abstraction: group. Even more briefly, they are called groups.

    The study of the real numbers has led to many abstractions; for example, the concept of ring is suggested by the two operations + and ×, while the concepts of ordered system and lattice are suggested by the fact that real numbers may be compared in size. In every case, the abstraction has turned out to have realizations, other than the real number system, the study of which has been fruitful. For example, there exist finite groups.

    The abstraction topological space is suggested by those aspects of the real number system which are studied in advanced calculus courses, namely those which enter into discussions of such topics as continuity and convergence. When topological space is defined (Sec. 2.1, Definition 2), it will be pointed out that the set of real numbers is a topological space (Sec. 2.1, Example 7).

    2

    Topological Space

    2.1 Topological Space

    After reading Definition 1, below, the student may not recognize that, as promised in Section 1.3, we are speaking of things involved in discussions of continuity and convergence. The only remedy for such doubts is the actual pursuit, carried on in the text, of such topics in an arbitrary topological space, that is, using only ideas introduced in Definition 1.

    DEFINITION 1. Let X be a set. Let T be a collection of subsets of X satisfying

    (i) ø ∈ T;

    (ii) X ∈ T;

    (iii) If G1 ∈ T and G2 ∈ T, then G1 ∩G2 ∈ T;

    (iv) If ⊂T≠ ø, then U {G: G } ∈ T.

    Then T is called a topology for X.

    Thus a topology for X is a certain collection of subsets of X. By definition, the empty set and X itself are included. Condition (iii) says that the intersection of any two sets in T must be in T, hence the intersection of any finite collection of sets in T must be in T. Condition (iv) says that if any collection, no matter how numerous, of members of T be given, then the union of this collection is also a member of T. Briefly, a topology for X is a collection of subsets of X containing ø, X, and closed under finite intersections and arbitrary unions.

    To topologize a set X means to specify a topology for X.

    EXAMPLE 1. The indiscrete topology. Let X be a set. Let T = {ø, X}. Then T is a topology for X. conditions (i) and (ii) of Definition I clearly hold. Conditions (iii) and (iv) are also clear since 0 ∩ X = ø and ø ∪ X = X. This particular topology is called the indiscrete topology. Any set may be given the indiscrete topology. For example, let X = {1, 2, 3, 4}. Then T = {ø, {1, 2, 3, 4}} is the indiscrete topology for X.

    EXAMPLE 2. The discrete topology. Let X be a set. Let T be the collection of all subsets of X. Then T is a topology for X, called the discrete topology.

    EXAMPLE 3. The cofinite topology. Let X be a set. Let

    T = {G ⊂ X: X \ G is a finite set} ∪ {ø}.

    Thus T consists, except four ø, of complements of finite sets. To check Condition (iii) of Definition 1, let G1, G2 ∈ T. If either G1 or G2 is empty, G1 ∩ G2 = ø ∈ T; otherwise, X \ (G1 ∩G2) = (X \ G1) ∪ (X G⊂ T= {ø}, there is nothing to prove; otherwise, let ø ≠ G . Then T {S: S } ⊂T G{S: S } ∈ T.

    It should be noted that if X is a finite set, the cofinite topology is identical with the discrete topology.

    EXAMPLE 4. Let X = ω (the positive integers) and let T be the cofinite topology for X. For n = 1, 2, 3, . . ., let Gn = {1, n, n + 1, n + 2, . . .}. Then each Gn ∈ T {Gn: n = 1, 2,. . .} = {1} ∉ T. This illustrates the finite intersection part of the definition of topology.

    EXAMPLE 5. Let X be an infinite set and T the collection of all finite subsets of X, together with X itself. Then T is not a topology since a union of finite sets need not be finite.

    DEFINITION 2. A topological space is a set together with a topology for the set. The members of the topology are called open sets.

    Thus we refer to a topological space (X, T), in which X is a set, and T is a topology for it. For S ⊂ X, the sentences "S is open and S ∈ T" are synonymous. When T is understood from the context, the space will be denoted by X, rather than (X, T). Our aim is to develop a usage which will not refer to T. Thus we shall speak of a topological space X together with its open subsets.

    EXAMPLE 6. If X is given the indiscrete topology, it has only two open subsets, ø and X. (But see Example 8.) With the discrete topology every subset of X is open. With the cofinite topology, a nonempty subset is open if and only if its complement is finite.

    EXAMPLE 7. The Euclidean topology. Let X = R (the real numbers) and let the empty set be called open; also any set G is called open if for every x ∈ G, there exists ε > 0 such that (x − ε, x + ε) c G. This defines a topology for X To check Condition (iii) of Definition 1, let G1, G2 be open sets. If G1 ∩ G2 = 0 there is nothing to prove. If G1∩ G2 ≠ ø, let x ∈ G1 ∩ G2. There exist ε1 > 0, ε2 > 0 such that (x − εi, x + εi) ⊂ Gi, i = 1, 2. Let ε = min (ε1, ε2). Then ε > 0 and (x − ε, x + ε) ⊂ G1 ∩ G2. Thus G1 ∩Gbe a collection of open sets, and we may assume that its union is nonempty. Let x {G: G }. Then x ∈ S for some S . There exists ε > 0 such that (x − ε, x + ε) ⊂ S. Then (x − ε, x + ε{G: G

    EXAMPLE 8. In Definition 1, it is possible that X is the empty set. Then the only topology for X is {ø}. Needless to say, in any discussion we are always thinking of a nonempty set, and thus, on occasion, will make a statement which is not strictly true. Fcr example, the indiscrete topology for øhas only one member, not two as stated in Example 6. It seems hardly worth while to be sufficiently careful at all times to cover this case, but the student is warned that such an occasion may arise.

    If T, T′ are topologies for a set X, it may happen that T ⊃ T′. In this case we shall say that T is larger than T′, (including the possibility that T = T′), and T′ is smaller than T. Very commonly, "stronger and finer are used instead of larger; weaker aid coarser instead of smaller." Of course if T is both larger and smaller than T′, then they are equal.

    A set F in a topological space is called closed if its complement is open. (Use of the letters F—French fermé—and G—German Gebiet—for closed and open sets, respectively, has become traditional.)

    EXAMPLE 9. Let ω have the cofinite topology. Then each finite set is closed, indeed a proper subset is closed if and only if it is finite. The set of even integers is neither closed nor open; ω is both open and closed. (Two morals: A set can be both open and closed. Just because a set is not open, there is no reason to think that it is closed.)

    Problems

    1. [0, 1) is not open in R, with the Euclidean topology. Neither is it closed.

    2. Let S ⊂X. Let T = {ø, S, X}. Show that T is a topology.

    3. Find all sets for which the discrete and indiscrete topologies are the same.

    4. Let S1 ⊂ X, S2 ⊂ X. Let T = {ø, S1, S2, X}. Is T a topology?

    5. Let T = {ø} ∪ {R} ∪ {(a, ∞): a ∈ R},

    T′ = {ø} ∪ {R} ∪ {[a, ∞): a ∈ R}.

    Show that T is a topology for R, but T′ is not.

    6. Let X be a topological space. Show that Ø, X, every finite union and every intersection of closed sets, are all closed.

    101. Let (X, T) be a topological space and let T′ be the collection of closed sets. Is T′ a topology?

    102. If, in Problem 101, X is finite, show that T′ is a topology.

    103. In Example 3 replace finite by countable. Show that a topology is defined. It is called the cocountable topology.

    104. The cocountable topology on a countable space is discrete.

    201. Let k(n) be the number of different topologies which can be placed on a set with n members. Show that k(0) = k(1) = 1, k(2) = 4, k(3) = 29. (Note: k(4) = 355, k(5) = 6942, k(6) = 209527, k(7) = 9535241. See [Evans, Harary, and Lynn].)

    202. Let d(n) be the number of topologies T which can be placed on a set with n members, satisfying T = T′ (Problem 102). Show that d(0) = d(1), d(2) = 2, d(3) = 5.

    2.2 Semimetric and Metric Space

    DEFINITION 1. Semimetric and metric. Let X be a set and suppose given a real-valued function d of two variables, each selected from X. Thus d is a function from X x X to the real numbers. Any such function satisfying the following conditions is called a semimetric.

    (i) d(x, y) = d(y, x) ≥ 0,

    (ii) d(x, x) = 0,

    (iii) d(x, z) ≤ d(x, y) + d(y, z),

    for all x, y, z ∈ X. Condition (iii) is called the triangle inequality. A metric is a semimetric, d, which satisfies the condition d(x, y) > 0 if x ≠ y.

    DEFINITION 2. A semimetric (or metric) space is a set together with a semimetric (or metric) for the set.

    Thus the pair (X, d), in which X is a set and d a semimetric, is a semimetric space. We shall usually denote it by X, when d is understood from the context.

    EXAMPLE 1. The discrete metric. Let X be a set. For x, y ∈ X, define d(x, y) = 1 if x ≠ y, 0 if x = y. It is easily verified that this is a metric, for example, the triangle inequality may be checked for x, y, z by noting that it is trivial if x ≠ y and y ≠ z, if x = y, or if y = z .

    EXAMPLE 2. The indiscrete semimetric. Let X be a set. For x, y ∈ X, define d(x, y.

    EXAMPLE 3. The Euclidean distance for Rn was proved, in Section 1.2, to obey the triangle inequality. Since it obviously has the other metric properties, it is a metric, called the Euclidean metric, and so Rn is, with this definition, a metric space.

    EXAMPLE 4. For w = (x, y, z) ∈ R³ define p(w) = |x| + |y|. Then d(w, w′) = p(w − w’) defines a semimetric for RFor example, with x = (1, 2, 3), y = (1, 2, 4), d(x, y) = 0.

    EXAMPLE 5. Let C be the set of continuous real functions on the closed interval [0, 1]. For f ∈ C, g ∈ Cf = max{|f(x)|: 0 ≤ x ≤ 1} and d(f, gf − g . As in Section 1.2, the triangle inequality for d To establish (1.2.3), we have, for any x ∈ [0, 1], |f(x) + g(x)| ≤ |f(x)| + |g(xf g . Choosing x so as to maximize |f(x) + g(x)| gives the result.]

    The following definitions apply to any semimetric space X. For a ∈ X and r ∈ R, the cell of radius r and center a, written N(a, r), is {x: d(x, a) < r}; the disc of radius r and center a, written D(a, r) is {x: d(x, a) ≤ r}; and the circumference of radius r and center a, written C(a, r), is {x: d(x, a) = r}. The word sphere is often used in place of circumference, and when X = Rn+1 with the Euclidean metric, C(0, 1) is called the n-sphere, written Sn. Thus S1, the 1-sphere is the unit circumference in R², {(x, y): x² + y² = 1}.

    It is possible for a cell, disc, or circumference to be empty. For example, N(a, 0) is certainly empty, while if X is empty for any a= {a}. For A ⊂ X, B ⊂X, x ∈ X, we define d(x, A), the distance from x to A, to be

    inf{d(x, a): a ∈ A},

    and d(A, B), the distance from A to B to be

    inf{d(a, b): a ∈ A, b ∈ B}.

    (See Problems 7, 8, and 9). The diameter of a set A is defined to be sup{d(x, y): x ∈ A, y ∈ A} (perhaps infinite). A set with finite diameter is said to be metrically bounded.

    In the next result it is assumed that A is not empty.

    LEMMA 2.2.1. |d(x, A) − d(y, A)| ≤ d(x, y).

    We may assume that d(x, A) ≥ d( y, A) since d(x, y) = d(y, x). Let ε > 0 be given and choose a ∈ A with d(y, a) < d(y, A) + ε. Then

    Let X, Y be semimetric spaces and f: X → Y. We call f an isometry if it is one-to-one and satisfies d[f(x), f(x′)] = d(x, x′) for all x, x′ ∈ X. (The assumption one-to-one is redundant if X .) It is called an isometry of X onto Y if it is onto; if a particular isometry is not known to be onto, we shall sometimes write isometry (into), for emphasis. If there is an isometry of X onto Y we say that X, Y are isometric.

    EXAMPLE 6. Define f: R → R² by f(x) = (x, 0). Then f is an isometry of R into R². (R and R² are assumed to have the Euclidean metric, e.g., Example 3.)

    EXAMPLE 7. For ω = (x, y, z) ∈ R³, define p(ω) = |z|, d(ω, (ω′) = p(ω − ω′). Just as in Example 4, d is a semimetric for R³. Let R have the Euclidean metric. Define f : R³ →R by f(x, y, z) = z. Then f is distance preserving, that is d[f(ω), f(ω′)] = d(ω, ω’), but it is not an isometry since it is not one-to-one.

    Problems

    In this list, (X, d) is a semimetric space.

    1. Verify Examples 1 and 2.

    2. Prove that |d(x, a) − d(y, a)| ≤ d(x, y).

    3. Prove that |d(x, a) − d( y, b≤ d(x, y) + d(a, b).

    4. Prove the triangle inequality for d Start with Formula (1.2.3) with p

    5. Let X {a} for all a ∈ X. What is D(a, 1)? C(a, 1)?

    6. Prove that d(x, S) = d({x}, S).

    7. Prove that d(A, B) = inf{d(a, B): a ∈ A}.

    8. Prove that d(A, B) = d(B, A).

    10. Prove that a distance-preserving map from a metric space to a semimetric space is an isometry (into).

    101. Let X be a set and u: X x X → R satisfy u(x, y) = 0 if and only if x = y; and u(x, y) ≤ u(x, z) + u(y, z). Show that u is a metric.

    102. Give an example of a metric space which contains a nonempty disc whose diameter is less than its radius.

    Easy examples are certain subspaces of R², R

    104. [0, ∞) and { − 1 } ∪ [0, ∞) are each isometric into the other, but the spaces are not isometric. (Thus isometry of metric spaces does not enjoy the crisscross property.)

    2.3 Semimetric and Metric Topologies

    There is a standard way of defining a topology for a semimetric space (X, d). The empty set is declared to be open; moreover, if G ⊂ X, G is called open if, for every x ∈ G, there exists r > 0 such that N(x, r) ⊂ G. Before proving that the collection of open sets is a topology, consider an example.

    EXAMPLE 1. Let R have the Euclidean metric, that is, d(x,y) = |x − y|. Let S = (0, 1]. Then S is not an open set since N(1, r) ⊄ S for all r For example, 1 + (r/2) ∈ N(1, r) \ S.] However, (0, 1) is open since if 0 < a < 1, N(a, r) ⊂ (0, 1) whenever r = min {1 − a, a}.

    We now prove that a topology has been defined. (It is called the topology induced by the semimetric.) To see that X is open, we merely note that for any x ∈ X, N(x, 1) ⊂ X. (Indeed, N(x, r) ⊂ X for arbitrary r.) Next, let G1, G2 be open sets and let a ∈ G1 ∩ G2. For i = 1, 2, choose ri > 0 with N(a, ri) ⊂ Gi. Then N(a, r) ⊂ G1 ∩ G2 if r = min{r1, r2}.

    be a collection of open sets and let x ∈ U {G: G }. Then x ∈ S for some S , and so there exists r > 0 with N(x, r) c S. Then N(x, r) ⊂ ∪ {G: G }.

    EXAMPLE 2. The topology for Rn induced by the Euclidean metric (Sec. 2.2, Example 3) is called the Euclidean topology for Rn (see Problem 1).

    THEOREM 2.3.1. Cells in a semimetric space are open.

    Let N(x, r) be a cell. It is empty, hence open, if r ≤ 0, so we may assume r > 0. Let y ∈ N(x, r). Then N(y, s) ⊂ N(x, r) if s < r − d(y, x). Let a ∈ N(y, s). Then d(a, x) ≤ d(a, y) + d(y, x) ≤ s + d(y, x) < r. Thus a ∈ N(x, r

    It is natural to ask whether every topology can be obtained from a semimetric, and, if not, which ones can. (Those which can are called semimetrizable.) This question will be more fully treated in Chapter 10; however, it is possible to give a simple example (Example 3) of a topology which is not induced by any semimetric. The most instructive procedure is to see what properties a topology induced by a semimetric must have. The one we give here is only one of many possibilities.

    THEOREM 2.3.2. Let x, y be points in a semimetric space such that every open set containing x also contains y. Then every open set containing y also contains x.

    We first observe that d(x, yFor suppose on the contrary that d(x, y) = ε > 0. Then N(x, ε/2) is an open set containing x but not y.] Let G be an open set containing y. There exists r > 0 such that N(y, r) ⊂ G. Since d(x, y) = 0 < r, we have x ∈ N(y, r), hence x ∈ G.

    Now it is an easy matter to construct a topology not obeying the conclusion of the preceding theorem.

    EXAMPLE 3. Let X be a set, and x, y distinct points in X. Let T = {∅, {y}, X}. This is a topology for X. set. 2.1, Problem 2.] Every open set containing x contains y. There is only one.] But {y} contains y and not x. Hence T is not semimetrizable Theorem 2.3.2].

    REMARK ON USE OF ADJECTIVES. Adjectives are applied equally often to a topological space and to its topology. For example, "X is metrizable can be written instead of T (the topology of X) is metrizable." Later we shall speak of a connected space, a connected topology, a compact topology, and so on.

    DEFINITION 1. Let d, d′ be semimetrics for a set X. Let Td , Td′ be the induced topologies. We say that d is stronger than d′, and that d′ is weaker than d, if Td ⊃ Td′. If Td = Td′ we say that d, d′ are equivalent.

    Problems

    Show that the Euclidean topology for R as given in Example 2 is the same as that given in Sec. 2.1, Example 7.

    The discrete and indiscrete topologies are semimetrizable.

    Let d, d′ be semimetrics for X and suppose that there exists a number k such that kd(x, y) ≥ d′(x, y for all x, y. Prove that d is stronger than d′.

    If d is a semimetric, and there exists a weaker metric, then d is a metric.

    Problem 4 is false with stronger instead of weaker.

    Every disc is closed.

    101. Give an example of a nonempty disc which is an open set.

    102. Let d be a semimetric. Then d/(1 + d) and d 1 are semimetrics equivalent to d. (By d 1 is meant the function whose value at (x, y) is min{d(x, yFor the triangle inequality use d/(1 + d) = 1 − (1/(1 + d

    .

    104. Every open set in R, with the Euclidean topology is the union of a disjoint family of open intervals.

    105. Let X = {a, b, c, d, e} ; T = {∅, {a}, {b, c}, {d, e}, {a, b, c}, {a, d, e}, {b, c, d, e}, X}. Show that T is a semimetrizable topology for X.

    106. Suppose that the condition d(x, y) = d(y, x) is omitted from the definition of semimetric. Would Theorems 2.3.1 and 2.3.2 be true?

    Consider {1/n

    108. Let X be a noncountable set and d a metric which induces the discrete topology. Then X has a noncountable subset on which d ≥ εD, where D is the discrete metric, and ε With Sn = {x: N(x, 1/n) = {x}} for n = 1, 2,. . ., U Sn is noncountable and so some Sn

    109. With X, d, as in Problem 108, there exists ε > 0 such that X is not a countable union of sets of diameter < ε.

    201. The converse of Problem 3 is false; that is, d might be stronger than d′ and no such k exist.

    202. Let X be the set of all closed convex curves in Rdefine a metric on X?

    2.4 Natural Topologies and Metrics

    Certain spaces will be assigned topologies which will, throughout this book, be referred to as the natural topologies for these spaces. For example, the natural topology for R is the Euclidean topology; wherever R is mentioned, it will be assumed to have this topology. If for some reason we want to consider R with some other topology, the topology will be explicitly stated. The phrase "let G be an open subset of R " will mean "let G ∈ T, where T is the Euclidean topology for R"

    The natural topology for R" is the Euclidean topology. The natural topology for Z is the discrete topology. We hasten to add that this is also the Euclidean topology for Z. [For m, n ∈ Z let d(m, n) = |m − n|. Then N(n) = {n}, that is, for each n, {n} is an open set. Hence d induces the discrete topology.] The natural topology for Q, and J, will be that induced by the Euclidean metric.

    Similarly certain spaces will be assigned natural metrics. The natural metric for Rn, Z, Q, J is the Euclidean metric. In each case, the natural metric induces the natural topology.

    , Then X inherits from R the metric d(1/m, 1/n) = |1/m − 1/n|. This gives X the discrete topology [{1/m} is open for each m since it is (1/m − ε, 1/m + ε) ∩ X for suitably chosen ε > 0]. Similarly Z has the discrete topology. However Z also has the discrete metric as its natural metric, while the metric given for X is not the discrete metric.

    EXAMPLE 2. We shall usually identify R² and the complex plane, writing |z| for (x² + y²)¹/² if z = (x, y).

    2.5 Notation and Terminology

    Let x, N be, respectively, a point, and a set in a topological space. We say that N is a neighborhood of x if there exists an open set G with x ∈ G ⊂ N. We call N a deleted neighborhood of x if x ∉ N, and N ∪ {x} is a neighborhood of x. (Note that a neighborhood need not be open.)

    THEOREM 2.5.1. A set is open if and only if it is a neighborhood of each of

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