Topology
Munkers, Topology
Topology is the first step in geometry.
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
15 The Product Topology on $X\times Y$
17 Closed Sets and Limit Points
21 The Metric Topology (continued)
Chapter 3 Connectedness and Compactness
24 Connected Subspaces of the Real Line
25 Components and Local Connectedness
27 Compact Subspaces of the Real Line
Supplementary Exercises Nets
Chapter 4 Countability and Separation Axioms
34 The Urysohn Metrization Theorem
35 The Tietze Extension Theorem
Supplementary Exercises Review of the Basics
Chapter 5 The Tychonoff Theorem
37 The Tychonoff Theorem
38 The Stone-tech 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 Metnc Spaces
*44 A Space-Filling Curve
45 Compactness in Metric Spaces
46 Pointwise and Compact Convergence
47 Ascolis Theorem
Chapter 8 Baire Spaces and Dimension Theory
48 Baire Spaces
49 A Nowhere-Differentiable Function
50 Introduction to Dimension Theory
Exercises: Locally Euclidean Spaces
Chapter 9 The Fundamental Group
Section 51 Homotopy of Paths
Section 52 The Fundamental Group
Section 53 Covering Spaces
Section 54 The Fundamental Group of the Circle
Section 55 Retractions and Fixed Points
Section 56 The Fundamental Theorem of Algebra
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 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
ssuppLementary Exercises: Topological Properties and $\pi_1$
Chapter 14 Applications to Group Theory.
83 Covering Spaces of a Graph
84 The Fundamental Group of a Graph
85 Subgroups of Free Groups
Reference
OPTIMIZACIÓN ENTERA Y DINÁMICA - James Mukres
Munkres Topology First Part Solution