Introduction to Topology

Third Edition

Bert Mendelson

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486663524
• eBook ISBN: 9780486135090
• Edition: 3
• Topic: Mathematics
• Subtopic: Topology
• Index: Mathematics

Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The book's principal aim is to provide a simple, thorough survey of elementary topics in the study of collections of objects, or sets, that possess a mathematical structure.

The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean n-space and which lead to the ordinary topology.

Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness. Throughout the text, Dr. Mendelson, a former Professor of Mathematics at Smith College, has included many challenging and stimulating exercises to help students develop a solid grasp of the material presented.

CHAPTER 1

Theory of Sets

1 INTRODUCTION

As in any other branch of mathematics today, topology consists of the study of collections of objects that possess a mathematical structure. This remark should not be construed as an attempt to define mathematics, especially since the phrase “mathematical structure” is itself a vague term. We may, however, illustrate this point by an example.

The set of positive integers or natural numbers is a collection of objects N on which there is defined a function s, called the successor function, satisfying the conditions:

1. For each object x in N, there is one and only one object y in N such that y = s(x);

2. Given objects x and y in S such that s(x) = s(y), then x = y;

3. There is one and only one object in N, denoted by 1, which is not the successor of an object in N, i.e., 1 ≠ s(x) for each x in N;

4. Given a collection T of objects in N such that 1 is in T and for each x in T, s(x) is also in T, then T = N.

The four conditions enumerated above are referred to as Peano’s axioms for the natural numbers. The fourth condition is called the principle of mathematical induction. One defines addition of natural numbers in such a manner that s(x) = x + 1, for each x in N, which explains the use of the word “successor” for the function s. What is significant at the moment is the conception of the natural numbers as constituting a certain collection of objects N with an additional mathematical structure, namely the function s.

We shall describe a topological space in the same terms, that is, a collection of objects together with a specified structure. A topological spa...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• 1 Theory of Sets
• 2 Metric Spaces
• 3 Topological Spaces
• 4 Connectedness
• 5 Compactness
• Bibliography
• Index