General Topology

by Stephen Willard

Overview

Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: "continuous topology," represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and "geometric topology," covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures.

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ISBN-13:	9780486131788
Publisher:	Dover Publications
Publication date:	06/14/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
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Pages:	384
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General Topology

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By Stephen Willard

Dover Publications, Inc.

Copyright © 1998 Stephen Willard
All rights reserved.
ISBN: 978-0-486-13178-8

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Contents

Title Page,
Copyright Page,
Preface,
Chapter 1 - Set Theory and Metric Spaces,
Chapter 2 - Topological Spaces,
Chapter 3 - New Spaces from Old,
Chapter 4 - Convergence,
Chapter 5 - Separation and Countability,
Chapter 6 - Compactness,
Chapter 7 - Metrizable Spaces,
Chapter 8 - Connectedness,
Chapter 9 - Uniform Spaces,
Chapter 10 - Function Spaces,
Historical Notes,
Bibliography,
Index,


CHAPTER 1

Set Theory and Metric Spaces

1 Set theory

The material of this section is introduced primarily to serve as a review for those with some background in set theory and as an introduction to our notational conventions and terminology. The reader entirely unfamiliar with any aspect of set theory should not be content with the intuitive discussion given here, but should consult one of the standard references on the subject (see the notes).

Most of the material in this book is accessible to anyone who understands 1.1 through 1.8 below. It is recommended that the remainder of this section be skipped on first reading and referred to later as needed.

1.1 Sets. A set, family or collection is an aggregate of things (for example, numbers or functions or desks or people), called the elements or points of the set. If a is an element of the set A we write a [member of] A and if this is false we write a [??] A.

If A is a set and S is a statement which applies to some of the elements of A, the set of elements a of A for which S(a) is true is denoted {a [member of] A | S(a)}. Thus if N is the set of positive integers, the positive divisors of 6 form the set {a [member of] N |ab = 6 for some b [member of] N}. In the case of small sets, such as this one, it is easy to describe the set by listing its elements in brackets. Thus the set just given is the set {1, 2, 3, 6}.

This discussion is rather nave and leads to certain difficulties. Thus if P is the set of all sets, we can apparently form the set Q = {A [member of] P |A [??] A}, leading to the contradictory Q [member of] Q iff Q [??] Q. This is Russell's paradox (see Exercise 1A) and can be avoided (in our nave discussion) by agreeing that no aggregate shall be a set which would be an element of itself.

1.2 Elementary set calculus. If A and B are sets and every element of A is an element of B, we write A [subset] B or B [contains] A and say A is a subset of B or B contains A. The collection P(A) of all subsets of a given set A is itself a set, called the power set of A.

We say sets A and B are equal, A = B, when both A [subset] B and B [subset] A. Evidently, A and B are equal iff they have the same elements.

We write A - B to denote the set {a [member of] A|a [??] B} and (unlike some writers) use this notation even when B is not a subset of A, i.e., even when B [??] A. When we do have B [subset] A, A' B is called the complement of B in A.

The empty set, [empty set]; is the set having no elements. By the criterion for equality of sets, there is only one empty set and, by the criterion for containment, it is a subset of every other set.

Note that element and subset are different ideas. Thus, for example, x [member of] A iff {x} [subset] A.

A few sets will keep recurring and we will establish now a conventional notation for them.

R: the set of real numbers,

Rn Euclidean n-space,

N: the set of positive integers,

I: the closed interval [0, 1] in R,

Q: the set of rational numbers in R,

P: the set of irrational numbers in R,

Sn the n-sphere, {x [member of] Rn + 1 x| = 1}.

Eventually, each of these sets will be assumed to carry some "usual" structure (a metric, topology, uniformity or proximity) unless the contrary is noted. Additional less often used conventional notations will be introduced in the text. All can be found in the index.

1.3 Union and intersection. If Λ is a set and, for each λ [member of] Λ, Aλ is a set, the union of the sets Aλ is the set [union] λ [member of] ΛA,λ of all elements which belong to at least one Aλ. When no confusion about the indexing can result, we will write the union of the sets Aλ as simply [union] Aλ. The intersection of the sets Aλ is the set [intersection] λ [member of] ΛAλ, or simply Aλ, of all elements which belong to every Aλ. In case is the collection {A| λ [member of] []LAMBDA}, the union and intersection of the sets Aλ are sometimes denoted [union]A and [intersection]A, respectively.

When only finitely many sets A,1 ..., An are involved, the alternative notations A1 [union]···[union] An or [union]nk = 1 Ak are sometimes used for the union of the Ak, while A1 [intersection]? [intersection]An or [intersection]nk = 1 Ak sometimes denotes their intersection. When denumerably many sets A1, A2, ... are involved, their union will sometimes be denoted by A1 [union] A2 [union] ? or [union]∞k = 1, their intersection by Ak [intersection] A2 [intersection] ? or [union]∞k = 1 Ak.

We say A meets B iff A [intersection] B ≠. Otherwise, A and B are disjoint. In general, a family A of sets is pairwise disjoint iff whenever A,B [member of], A [intersection] B = [empty set].

For those who wish to test themselves on the concepts just introduced, here are a few easily proved facts:

a. A [subset] B iff A [union] B = B,

b. A [subset] B iff A [union] B = A,

c. If A is the empty collection of subsets of A, then [union] A = [empty set] and intersectionA = [empty set] .

d. A [union] B = A [union] (B - A).

e. A [intersection] (B [union] C) = (A [intersection] B) [union] C iff C [subset] A.

1.4 Theorem. If A is a set, Bλ [subset] Λ A for each λ [member of] [;AMBDA] and B [subset] A, then

a) [MATHEMATICAL EXPRESSION OMITTED]

b) [MATHEMATICAL EXPRESSION OMITTED]

c) [MATHEMATICAL EXPRESSION OMITTED]

d) [MATHEMATICAL EXPRESSION OMITTED]

Proof. a) If x [member of] A - ([union] Bλ), then x [member of] A and x [??] Bλ for any λ, so x [member of] A - Bλ for each λ; hence x [member of] [intersection] (A - Bλ). Conversely if x [member of] [intersection] (A - Bλ), then for each λ, x [member of] A and x [??] B; hence x [member of] A - [intersection] Bλ. Thus x [member of] A - ([union]Bλ iff x [member of] [intersection] (A - Bλ), so that

A - ([union] Bλ) = [intersection] (A - Bλ).

b) Similar to (a). See Exercise 1B.

c) If x [member of] B [intersection] ([union] Bλ then x [member of] B and [union [Bλ; thus x [member of] B and x [member of] Bλ0 for some λ0. Hence x [member of] [union] (B [intersection] Bλ). Conversely, if [union] (B [union] Bλ), then x [member of] B [MATHEMATICAL EXPRESSION OMITTED] for some λ0 [member of] Λ; thus x [member of] B and [MATHEMATICAL EXPRESSION OMITTED], so that x [member of] B and x [member of] [union] Bλ. Hence x [member of] Bλ. We have shown x [member of] B [intersection] ([union] Bλ) iff x [member of] [union] (B [intersection] Bλ); it follows that B [intersection] ([union] Bλ) = [union] (B [intersection] Bλ).

d) Similar to (c). See Exercise 1B.

1.5 Small Cartesian products. If x1 and y2 are distinct elements of some set, the two-element sets {x,x} and {x1, x2} are, by the criterion for set equality, the same. It is useful to have a device for reflecting priority as well as membership in this case, and it is provided by the notion of the ordered pair (x1, x2). By definition, ordered pairs (x1, x2) and (y1, y2) are equal iff x1 = y1 and x2 = y2. For a somewhat more formal approach to ordered pairs, see Exercise 1C.

Now if X1 and X2 are sets, the Cartesian product X1 x X2 of X1 and X2 is defined to be the set of all ordered pairs (x1x2) such that x1 [member of] X1 and x2 [member of] X2. This definition, for example, gives the plane as the set of all ordered pairs of real numbers. Other examples: S1 x I is a cylinder, S1 x S1 is a torus, R x Rn = Rn + 1.

Once defined for two sets, Cartesian products of any finite number of sets can be defined by induction; thus, the last example in the previous paragraph could be taken as the definition of Rn + 1.

For more about finite Cartesian products, and for a bridge between the definition given here and the definition provided in Section 8 for products of infinitely many sets, see Exercise 1D.

1.6 Functions. A function (or map) f from a set A to a set B, written f:A ->B, is a subset of A x B with the properties:

a. For each a [member of] A, there is some b [member of] B such that (a,b) [member of] f.

b. If (a, b) [member of] f and (a, c) [member of] f, then b = c.

More informally, we are requiring that each a [member of] A be paired with exactly one b [member of] B. The relationship (a, b) [member of] f is customarily written b = f(a) and functions are usually described by giving a rule for finding f(a) if a is known (rather than, for example, by giving some geometric or other description of the subset f of A x B). This reflects the common point of view, which is prone to regard a function not so much as a static subset of A x B as a "black box" which takes in elements of A and spits out elements of B.

When regarded as a set in its own right, the collection of functions from A to B is denoted BA.

If f:A ->B and C [subset] A, we define f(C) = {b [member of] B| b = f(a) for some a [member of] A}. If D [subset] B, we define f-1(D) = {a [member of] A |f(a) [member of] D}. Hence every function f:A ->B induces functions f: P(A) ->P(B) and f-1P(B) ->P(A). (and here we are following the unfortunate, but common, practice of denoting the elevation of f from A to P(A) by f also). The properties of these induced functions are investigated in Exercise 1H, which should be mandatory for anyone who cannot provide easily the answers to the questions it poses.

Note that if f:A ->B, then f-1(B) = A but it need not be true that f(A) = B. We call f(A) the image of f (or the image of A under f), calling B the range of f and A the domain of f. When f(A) = B, we say f is onto B. Note also that, for b [member of] B,f({b}) [which is always abbreviated f-1(b)] may consist of more than one point; in extreme cases, we may have f-1(b) = A. When such behavior is proscribed, f is called a one-one function. In addition to the usual requirements for a function, then, a one-one function f:A ->B must evidently obey the rule: a1 ≠ ab [??] f(a1) ≠ f(a2). In words, such a function takes distinct elements of A to distinct elements of B.

If f: A ->B and g: B ->C, then f and g determine together a natural function, their composition g° f: A ->C, defined by

(g ° f)(a) = g]f(a)], for a [member of] A.

More formally, (a, c) [member of] g° f iff for some b [member of] B, (a, b) [member of] f and (b,c) [member of] g. Less formally, put two black boxes end to end.

1.7 Special functions. A function f: N ->A is called a sequence in A. It can be described by giving an indexed list x1, x2, ... of its values at 1, 2, ... and this is often abbreviated (xn)n [member of] N or even simply (xn). Thus '(n) = 1/n, (1/n) n [member of] N and 1, 1/2. ...., 1/n, ... describe the same sequence in R.

A real-valued function on A is a function on A whose range is R. The collection RA of all real-valued functions on A inherits an algebraic structure from R since we can define addition, multiplication and scalar multiplication in RA as follows: given a [member of] A and r [member of] R,

(f + g)(a) = f(a) + g(a),

(fg)(a) = f(a)g(a),

(rf)(a) = r]f(a)].

For this and other reasons, the real-valued functions merit special attention in any branch of mathematics, and topology is no exception.

(Continues...)

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Excerpted from General Topology by Stephen Willard. Copyright © 1998 Stephen Willard. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents

1. Set Theory and Metric Spaces.
2. Topological Spaces.
3. New Spaces from Old.
4. Convergence.
5. Separation and Countability.
6. Compactness.
7. Metrizable Spaces.
8. Connectedness.
9. Uniform Spaces.
10. Function Spaces.
Historical Notes.
Bibliography.
Index.