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- ISBN-10:
- 0486434796
- ISBN-13:
- 9780486434797
- Pub. Date:
- 02/27/2004
- Publisher:
- Dover Publications

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## 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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## Product Details

ISBN-13: | 9780486434797 |
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Publisher: | Dover Publications |

Publication date: | 02/27/2004 |

Series: | Dover Books on Mathematics Series |

Pages: | 384 |

Sales rank: | 643,754 |

Product dimensions: | 6.14(w) x 9.21(h) x 0.80(d) |

## Read an Excerpt

#### General Topology

**By Stephen Willard**

**Dover Publications, Inc.**

**Copyright © 1998 Stephen Willard**

All rights reserved.

ISBN: 978-0-486-13178-8

All rights reserved.

ISBN: 978-0-486-13178-8

#### 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,

**R n** 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,**

**S n** the

*n*-sphere, {

*x*[member of]

**R**

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

When only finitely many sets *A,*1 ..., *A*n are involved, the alternative notations *A*1 [union]···[union] *An* or [union]*nk* = 1 *Ak* are sometimes used for the union of the *Ak*, while *A*1 [intersection]? [intersection]*An* or [intersection]*nk* = 1 *Ak* sometimes denotes their intersection. When denumerably many sets *A*1, *A*2, ... are involved, their union will sometimes be denoted by *A*1 [union] *A*2 [union] ? or [union]∞*k* = 1, their intersection by *Ak* [intersection] *A*2 [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, B*λ [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] *B*λ), then *x* [member of] *A* and *x* [??] *B*λ for any λ, so *x* [member of] *A* - *B*λ for each λ; hence *x* [member of] [intersection] (*A* - *B*λ). Conversely if *x* [member of] [intersection] (*A* - *B*λ), then for each λ, *x* [member of] *A* and *x* [??] *B;* hence *x* [member of] *A* - [intersection] *B*λ. Thus *x* [member of] A - ([union]*B*λ iff *x* [member of] [intersection] (*A* - *B*λ), so that

A - ([union] *B*λ) = [intersection] (A - *B*λ).

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

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

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

**1.5 Small Cartesian products.** If *x*1 and *y*2 are distinct elements of some set, the two-element sets {*x,**x*} and {*x*1, *x*2} 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* (*x*1, *x*2). By definition, ordered pairs (*x*1, *x*2) and (*y*1, *y*2) are equal iff *x*1 = *y*1 and *x*2 = *y*2. For a somewhat more formal approach to ordered pairs, see Exercise 1C.

Now if *X*1 and *X*2 are sets, the *Cartesian product X*1 x *X*2 of *X*1 and *X*2 is defined to be the set of all ordered pairs (*x*1*x*2) such that *x*1 [member of] *X*1 and *x*2 [member of] *X*2. This definition, for example, gives the plane as the set of all ordered pairs of real numbers. Other examples: **S**1 x **I** is a cylinder, **S**1 x **S**1 is a torus, **R** x **R**n = **R***n* + 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 **R***n* + 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*-1*P*(*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: *a*1 ≠ *a*b [??] *f*(*a*1) ≠ *f*(*a*2). 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 *x*1, *x*2, ... 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 **R***A* of all real-valued functions on *A* inherits an algebraic structure from **R** since we can define addition, multiplication and scalar multiplication in **R***A* 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...)*

Excerpted fromGeneral TopologybyStephen 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.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## 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.