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Description: Topology is a branch of pure mathematics that considers the abstract relationships found within geometry and analysis. Written with the mathematically mature student in mind, Foundations of Topology, Second Edition, provides a user-friendly, clear, and concise introduction to this fascinating area of mathematics. The author introduces topics that are well-motivated with thorough proofs, making them easy to follow. Historical comments are dispersed throughout the text, and exercises that vary in degree of difficulty are found at the end of each chapter. Foundations of Topology is an excellent text for teaching students how to develop the skills necessary for writing clear and precise proofs.
Key features:
•The text is organized in a flexible fashion, allowing instructors to teach topics in the order they desire for their specific course
• A useful background section on Set Theory is available as an appendix.
• Exercises of varying degrees of difficulty allow students to test themselves on the important mathematical concepts at hand.
Contents: Chapter 1: Topological Spaces • Metric spaces • Topological spaces: The definition and examples • Basis for a topology • Closed sets, closures and interiors of sets • Metric spaces revisited • Convergence • Continuous functions and homeomorphisms
Chapter 2: New Spaces from Old Ones • Subspaces • The product topology on X x Y • The product topology • The weak topology and the product topology • The uniform metric • Quotient spaces
Chapter 3: Connectedness • Connected spaces • Pathwise and local connectedness • Totally disconnected spaces
Chapter 4: Compactness • Compactness in metric spaces • Compact spaces • Local compactness and the relation between various forms of compactness
Chapter 5: The Separation and Countability Axioms • To’, T 1’ and T2’ Spaces • Regular and completely regular spaces • Normal and completely normal spaces • The countability axioms • Urysohn’s Lemma and the Tietze Extension Theorem • EmbeddingsChapter 6: Special Topics • Contraction mappings in metric spaces • Normal Linear spaces • The Frechet Derivative • Manifolds • Fractals • Compactification • The Alexander Subbase and the Tychonoff Theorems
Chapter 7: Metrizability and Paracompactness • Urysohn’s Metrization Theorem • Paracompactness • The Nagata- Smirnov Metrization Theorem
Chapter 8: The Fundamental Group and Covering Spaces • Homotopy of paths • The fundamental group • The fundamental group of the circle • Covering spaces • Applications and additional examples of fundamental groups
Chapter 9: Applications of Homotopy • Inessential maps • The fundamental theorem of algebra • Homotopic maps • The Jordan Curve Theorem
Appendix A Logic and Proofs • Appendix B Sets • Appendix C Functionss • Appendix D Indexing Sets and Cartesian Productss • Appendix E Equivalence Relations and Order Relationss • Appendix F Countable Setss • Appendix G Uncountable Setss • Appendix H Ordinal and Cardinal Numberss • Appendix I AlgebraISBN - 9789380108117
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Pages : 406
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