Synopses & Reviews
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. It provides a simple, thorough survey of elementary topics, starting with set theory and advancing to metric and topological spaces, connectedness, and compactness. 1975 edition.
Synopsis
Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introductionto 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."
Synopsis
An undergraduate introduction to the fundamentals of topology engagingly written, filled with helpful insights, complete with many stimulating and imaginative exercises to help students develop a solid grasp of the subject.
Synopsis
Concise undergraduate introduction to fundamentals of topology — clearly and engagingly written, and filled with stimulating, imaginative exercises. Topics include set theory, metric and topological spaces, connectedness, and compactness. 1975 edition.
Description
Includes bibliographical references (p. 201-202) and index.
Table of Contents
Preface
1 Theory of Sets
1 Introduction
2 Sets and subsets
3 "Set operations: union, intersection, and complement"
4 Indexed families of sets
5 Products of sets
6 Functions
7 Relations
8 Composition of functions and diagrams
9 "Inverse functions, extensions, and restrictions"
10 Arbitrary products
2 Metric Spaces
1 Introduction
2 Metric spaces
3 Continuity
4 Open balls and neighborhoods
5 Limits
6 Open sets and closed sets
7 Subspaces and equivalence of metric spaces
8 An infinite dimensional Euclidean space
3 Topological Spaces
1 Introduction
2 Topological spaces
3 Neighborhoods and neighborhood spaces
4 "Closure, interior, boundary"
5 "Functions, continuity, homeomorphism"
6 Subspaces
7 Products
8 Identification topologies
9 Categories and functors
4 Connectedness
1 Introduction
2 Connectedness
3 Connectedness on the real line
4 Some applications of connectedness
5 Components and local connectedness
6 Path-connected topological spaces
7 Homotopic paths and the fundamental group
8 Simple connectedness
5 Compactness
1 Introduction
2 Compact topological spaces
3 Compact subsets of the real line
4 Products of compact spaces
5 Compact metric spaces
6 Compactness and the Bolzano-Weierstrass property
7 Surfaces by identification
Bibliography
Index