Preface
A Note to the Reader
Part I GENERAL TOPOLOGY
Chapter 1 Set Theory and Logic
1 Fundamental Concepts
2 Functions
3 Relations
4 The Integers and the Real Numbers
5 Cartesian Products
6 Finite Sets
7 Countable and Uncountable Sets
8 The Principle of Recursive Definition
9 Infinite Sets and the Axiom of Choice
10 Well-Ordered Sets
11 The Maximum Principle
Supplementary Exercises: Well-Ordering
Chapter 2 Topological Spaces and Continuous Functions
12 Topological Spaces
13 Basis for a Topology
14 The Order Topology
15 The Product Topology on X x Y
16 The Subspace Topology
17 Closed Sets and Limit Points
18 Continuous Functions
19 The Product Topology
20 The Metric Topology
21 The Metric Topology (continued)
*22 The Quotient Topology
*Supplementary Exercises: Topological Groups
Chapter 3 Connectedness and Compactness
23 Connected Spaces
24 Connected Subspaces of the Real Line
*25 Components and Local Connectedness
26 Compact Spaces
27 Compact Subspaces of the Real Line
28 Limit Point Compactness
29 Local Compactness
*Supplementary Exercises: Nets
Chapter 4 Countability and Separation Axioms
30 The Countability Axioms
31 The Separation Axioms
32 Normal Spaces
33 The Urysohn Lemma
34 The Urysohn Metrization Theorem
*35 The Tietze Extension Theorem
*36 Imbeddings of Manifolds
*Supplementary Exercises: Review of the Basics
Chapter 5 The Tychonoff Theorem
37 The Tychonoff Theorem
38 The Stone-Cech Compactification
Chapter 6 Metrization Theorems and Paracompactness
39 Local Finiteness
40 The Nagata-Smirnov Metrization Theorem
41 Paracompactness
42 The Smirnov Metrization Theorem
Chapter 7 Complete Metric Spaces and Function Spaces
43 Complete Metric Spaces
*44 A Space-Filling Curve
45 Compactness in Metric Spaces
46 Pointwise and Compact Convergence
47 Ascoli's Theorem
Chapter 8 Baire Spaces and Dimension Theory
48 Baire Spaces
*49 A Nowhere-Differentiable Function
50 Introduction to Dimension Theory
*Supplementary Exercises: Locally Euclidean Spaces
Part II ALGEBRAIC TOPOLOGY
Chapter 9 The Fundamental Group
51 Homotopy of Paths
52 The Fundamental Group
53 Covering Spaces
54 The Fundamental Group of the Circle
55 Retractions and Fixed Points
*56 The Fundamental Theorem of Algebra
*57 The Borsuk-Ulam Theorem
58 Deformation Retracts and Homotopy Type
59 The Fundamental Group of Sn
60 Fundamental Groups of Some Surfaces
Chapter 10 Separation Theorems in the Plane
61 The Jordan Separation Theorem
*62 Invariance of Domain
63 The Jordan Curve Theorem
64 Imbedding Graphs in the Plane
65 The Winding Number of a Simple Closed Curve
66 The Cauchy Integral Formula
Chapter 11 The Seifert-van Kampen Theorem
67 Direct Sums of Abelian Groups
68 Free Products of Groups
69 Free Groups
70 The Seifert-van Kampen Theorem
71 The Fundamental Group of a Wedge of Circles
72 Adjoining a Two-cell
73 The Fundamental Groups of the Torus and the Dunce Cap
Chapter 12 Classification of Surfaces
74 Fundamental Groups of Surfaces
75 Homology of Surfaces
76 Cutting and Pasting
77 The Classification Theorem
78 Constructing Compact Surfaces
Chapter 13 Classification of Covering Spaces
79 Equivalence of Covering Spaces
80 The Universal Covering Space
*81 Covering Transformations
82 Existence of Covering Spaces
*Supplementary Exercises: Topological Properties and
Chapter 14 Applications to Group Theory
83 Covering Spaces of a Graph
84 The Fundamental Group of a Graph
85 Subgroups of Free Groups
Bibliography
Index