A Guide to Topology

Steven G. Krantz was born in San Francisco, California and grew up in Redwood City, California. He received his undergraduate degree from the University of California at Santa Cruz and the Ph.D. from Princeton University. Krantz has held faculty positions at UCLA, Princeton University, Penn State University, and Washington University in St. Louis. He is currently Deputy Director of the American Institute of Mathematics. He has written 160 scholarly papers, over 50 books and is the holder of the Chauvenet Prize and the Beckenbach Book Award.

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.