Topology Without Tears꞉ Including a Graduate Course on Topological Groups

Description

"Topology has several different branches — general topology (also known as point- set topology), algebraic topology, differential topology and topological algebra — the first, general topology, being the door to the study of the others. I aim in this book to provide a thorough grounding in general topology." Contents 0. Introduction 0.1 Acknowledgments 0.2 Readers – Locations and Professions 0.3 Readers' Compliments 0.4 Helpful Hint on Hyperlinks 0.5 The Author 0.6 Credits for Images 1. Topological Spaces 1.1 Topology 1.2 Open Sets 1.3 Finite-Closed Topology 1.4 Postscript 2. The Euclidean Topology 2.1 Euclidean Topology 2.2 Basis for a Topology 2.3 Basis for a Given Topology 2.4 Postscript 3. Limit Points 3.1 Limit Points and Closure 3.2 Neighbourhoods 3.3 Connectedness 3.4 Postscript 4. Homeomorphisms 4.1 Subspaces 4.2 Homeomorphisms 4.3 Non-Homeomorphic Spaces 4.4 Postscript 5. Continuous Mappings 5.1 Continuous Mappings 5.2 Intermediate Value Theorem 5.3 Postscript 6. Metric Spaces 6.1 Metric Spaces 6.2 Convergence of Sequences 6.3 Completeness 6.4 Contraction Mappings 6.5 Baire Spaces 6.6 Postscript 7. Compactness 7.1 Compact Spaces 7.2 The Heine-Borel Theorem 7.3 Postscript 8. Finite Products 8.1 The Product Topology 8.2 Projections onto Factors of a Product 8.3 Tychonoff's Theorem for Finite Products 8.4 Products and Connectedness 8.5 Fundamental Theorem of Algebra 8.6 Postscript 9. Countable Products 9.1 The Cantor Set 9.2 The Product Topology 9.3 The Cantor Space and the Hilbert Cube 9.4 Urysohn's Theorem 9.5 Peano's Theorem 9.6 Postscript 9.7 Credits for Images 10. Tychonoff's Theorem 10.1 The Product Topology For All Products 10.2 Zorn's Lemma 10.3 Tychonoff's Theorem 10.4 Stone-Cech Compactification 10.5 Postscript 10.6 Credit for Images 11. Quotient Spaces 11.1 Quotient Spaces 11.2 Identification Spaces 11.3 Möbius Strip, Klein Bottle and Real Projective Space 11.4 Postscript 11.5 Credit for Images 12. The Stone–Weierstrass Theorem 12.1 The Weierstrass Approximation Theorem 12.2 The Stone-Weierstrass Theorem 12.3 Credit for Images Appendix 1: Infinite Sets Appendix 2: Topology Personalities Appendix 3: Chaos Theory and Dynamical Systems Appendix 4: Hausdorff Dimension Appendix 5: Topological Groups: A Graduate Course Appendix 6: Filters and Nets Bibliography Index