A Guide to Topology
Steven George Krantz · American Mathematical Society eBooks · 2009
Preface Part I. 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 Part II. Advanced Properties: 2.1 Basis and subbasis 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. Morse theory 2.9. Proper mappings 2.10. Paracompactness Part III. Moore-Smith Convergence and Nets: 3.1. Introductory remarks 3.2. Nets Part IV. Function Spaces: 4.1. Preliminary ideas 4.2. The topology of pointwise convergence 4.3. The compact-open topology 4.4. Uniform convergence 4.5. Equicontinuity and the Ascoli-Arzela theorem 4.6. The Weierstrass approximation theorem Table of notation Glossary Bibliography Index.