ISBN 9789380108117,Foundations of Topology

Foundations of Topology

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ISBN 9789380108117
Publisher

Jones & Bartlett Learning

Publication Year 2010
ISBN-13

ISBN 9789380108117

ISBN-10 9380108117
Binding

Paperback

Edition 2nd edition
Number of Pages 406 Pages
Language (English)
Subject

Educational: Mathematics & numeracy

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.


Table of 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
Embeddings



Chapter 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

 

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 Algebra


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