Topology

Contents
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 Finitesets
		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 rcl
	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