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Aspects of Topology, 2nd Ed.
by C. Christenson and W. Voxman
Table of Contents
Preface . v
Chapter 0: Preliminaries . 1
A. Basic Notation
B. Cartesian Products and Relations
C. Functions
D. The Axiom of Choice and Some of its Variants
E. Cardinal Numbers and Ordinal Numbers
F. Minkowski's Inequality
Chapter 1: The Basic Constructs . 11
A. Topological Spaces and Continuity
B. Further Examples of Topological Spaces
C. Metric Spaces: A Preview
D. Building New Spaces from Old Ones
E. A Potpourri of Fundamental Concepts
F. Continuity
G. Homeomorphisms
Problems
Chapter 2: Connectedness and Compactness . 47
A. Connectedness: General Results
B. Connectedness: Slightly Deeper Results
C. Path Connectedness
D. Components
E. Locally Connected Spaces
F. Simple Chains
G. Compactness
H. Two Characterizations of Compactness
I. Compactness: Local, Countable, and Sequential
J. The One Point Compactification
Problems
Chapter 3: Metric Spaces . 77
A. Compactness in a Metric Setting
B. Complete Metric Spaces
C. Convex and Hausdorff Metrics
Problems
Chapter 4: Separation Properties . 97
A. Normal Spaces
B. Uryson's Lemma; Tietze's Extension Theorem
C. Further Results Concerning Normal Spaces
D. The Separation Axioms
Problems
Chapter 5: 1-Manifolds and some Plane Theorems . 121
A. 1-Manifolds
B. On the Contractibility of \sS1
C. The Jordan Curve Theorem
D. The Schönflies Theorem
E. The Annulus Theorem
Problems
Chapter 6: The Product Topology and Inverse Systems . 151
A. The Product Topology Revisited
B. Inverse Systems: the Preliminaries
C. Compact Metric Spaces are Continuous Images of the Cantor Set
D. The Dyadic Solenoid
Problems
Chapter 7: Functions Spaces, Weak Topologies, and Hilbert Space
. 175
A. The Point-Open Topology
B. The Compact-Open Topology
C. The Weak Topology: I
D. The Weak Topology: II
E. Hilbert Space
Problems
Chapter 8: Quotient Spaces . 193
A. The Quotient Topology
B. Identifications
C. Identification Maps
D. The Strong Topology and k-Spaces
E. CW-Complexes
F. Upper Semicontinuous Decompositions: An Introduction
Problems
Chapter 9: Continua . 221
A. The Arc
B. Peano Continua
C. Chainable Continua
D. Decomposable and Indecomposable Continua
E. The Pseudo-Arc
Problems
Chapter 10: Paracompactness and Metrizability . 249
A. Paracompactness
B. Partitions of Unity
C. A Sampling of Metrizability Theory
D. Moore Spaces
E. Completion of Metric Spaces
Problems
Chapter 11: Nets and Filters . 271
A. On the Failings of Sequences
B. Nets
C. Filters
D. Nets vs. Filters
Problems
Chapter 12: The Algebraization of Topology . 283
A. The Fundamental Group (The First Homotopy Group)
B. Elementary Properties of the Fundamental Group
C. Continuous Functions and Homomorphisms
D. Categories and Functors
E. The Seifert-van Kampen Theorem
F. Direct Limits
Problems
Chapter 13: Covering Spaces . 317
A. The Lifting Theorems
B. p1(\sS1,s) =
\intr (A Diversion)
C. Regular Covering Spaces
D. Map Liftings
E. Covering Morphism and Translations
F. Universal Covering Spaces
Problems
Chapter 14: Some Elements of Simplicial Theory . 341
A. The Polyhedral Category
B. Basic Constructions
C. Simplicial Approximation Theorems
D. Edge Path Groups: How to compute p1(|K|,v0)
for any Complex
Problems
Chapter 15: Further Applications of Homotopy . 375
A. The Extension Problem (Revisited)
B. The Separation Problem
C. Unicoherence
D. p1(\sE3,G), for
a Polygonal Graph G in \sE3
E. Wild Sets Do Exist
Problems
Chapter 16: 2-Manifolds . 401
A. The Triangulation of 2-Manifolds
B. The Classification of Compact 2-Manifolds
C. A Characterization of \sE2
Problems
Chapter 17: An Introduction to n-Manifolds . 419
A. Some Preliminaries
B. n-Annuli
C. Cellular Sets
D. The Generalized Schönflies Theorem
E. Compact n-Manifolds
Problems
Chapter 18: Dimension Theory . 439
A. Little Inductive Dimension
B. Big Inductive and Covering Dimension
C. Some Final Results
Problems
Appendix: . 459
A. Free Groups
B. Group Presentations
C. Homomorphisms and Tietze's Theorem
D. Direct Limits of Groups
Bibliography . 471
Index . 481
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