1. Fundamentals
1.1 What Is Topology?
1.2 First Definitions
1.3 Mappings
1.4 The Separation Axioms
1.5 Compactness
1.6 Homeomorphisms
1.7 Connectedness
1.8 Path-Connectedness
1.9 Continua
1.10 Totally Disconnected Spaces
1.11 The Cantor Set
1.12 Metric Spaces
1.13 Metrizability
1.14 Baire’s Theorem
1.15 Lebesgue’s Lemma and Lebesgue Numbers
2. Advanced Properties
2.1 Basis and Sub-Basis
2.2 Product Spaces
2.3 Relative Topology
2.4 First Countable and Second Countable
2.5 Compactifications
2.6 Quotient Topologies
2.7 Uniformities
2.8 Proper Mappings
2.9 Paracompactness
3. Basic Algebraic Topology
3.1 Homotopy Theory
3.2 Homology Theory
3.3 Covering Spaces
3.4 The Concept of Index
4. Manifold Theory
4.1 Basic Concepts
4.2 The Definition
5. Function Spaces
5.1 Preliminary Ideas
5.2 The Topology of Pointwise Convergence
5.3 The Compact-Open Topology
5.4 Uniform Convergence
5.5 Equicontinuity and the Ascoli-Arzela Theorem
5.6 The Weierstrass Approximation Theorem
6. Knot Theory
6.1 What Is a Knot?
6.2 The Alexander Polynomial
6.3 The Jones Polynomial
Solutions of Selected Exercises
Biography
Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He has previously taught at UCLA, Princeton University, and Penn State University. He earned his PhD from Princeton University in 1974. Prof. Krantz has directed 20 PhD students and 8 master's-degree students, and he published over 130 books and over 300 scholarly articles. He is the holder of the Chauvenet Prize, the Beckenbach Book Award, and the Kemper Prize, and he is a fellow of the American Mathematical Society.
