Introduction to topology Mendelson, Bert

Table of Contents:

I Theory of Sets

01 Introduction
02 Sets and subsets
03 "Set operations: union, intersection, and complement"
04 Indexed families of sets
05 Products of sets
06 Functions
07 Relations
08 Composition of functions and diagrams
09 "Inverse functions, extensions, and restrictions"
10 Arbitrary products

II Metric Spaces

01 Introduction
02 Metric spaces
03 Continuity
04 Open balls and neighborhoods
05 Limits
06 Open sets and closed sets
07 Subspaces and equivalence of metric spaces
08 An infinite dimensional Euclidean space

III Topological Spaces

01 Introduction
02 Topological spaces
03 Neighborhoods and neighborhood spaces
04 "Closure, interior, boundary"
05 "Functions, continuity, homeomorphism"
06 Subspaces
07 Products
08 Identification topologies
09 Categories and functors

IV Connectedness

01 Introduction
02 Connectedness
03 Connectedness on the real line
04 Some applications of connectedness
05 Components and local connectedness
06 Path-connected topological spaces
07 Homotopic paths and the fundamental group
08 Simple connectedness

V Compactness
01 Introduction
02 Compact topological spaces
03 Compact subsets of the real line
04 Products of compact spaces
05 Compact metric spaces
06 Compactness and the Bolzano-Weierstrass property
07 Surfaces by identification

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.


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