Essentials of topology with applications

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Essentials of Topology with Applications

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     © 2009 by Taylor & Francis Group, LLC
     ISBN 978-1-4200-8975-2 (eBook - PDF)

dedicated To the memory of Paul Halmos.

Table of Contents


Preface

Chapter 1: 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
     Exercises

Chapter 2: Advanced Properties of Topological Spaces
     2.1 Basis and Sub-Basis
     2.2 Product Spaces
     2.3 Relative Topology
     2.4 First Countable, Second Countable, and So Forth
     2.5 Compactifications
     2.6 Quotient Topologies
     2.7 Uniformities
     2.8 Morse Theory
     2.9 Proper Mappings
     2.10 Paracompactness
     2.11 An Application to Digital Imaging
     Exercises

Chapter 3: Basic Algebraic Topology
     3.1 Homotopy Theory
     3.2 Homology Theory
          3.2.1 Fundamentals
          3.2.2 Singular Homology
          3.2.3 Relation to Homotopy
     3.3 Covering Spaces
     3.4 The Concept of Index
     3.5 Mathematical Economics
     Exercises

Chapter 4: Manifold Theory
     4.1 Basic Concepts
     4.2 The Definition
     Exercises

Chapter 5: Moore-Smith Convergence and Nets
     5.1 Introductory Remarks
     5.2 Nets
     Exercises

Chapter 6: Function Spaces
     6.1 Preliminary Ideas
     6.2 The Topology of Pointwise Convergence
     6.3 The Compact-Open Topology
     6.4 Uniform Convergence
     6.5 Equicontinuity and the Ascoli-Arzela Theorem
     6.6 TheWeierstrass Approximation Theorem
     Exercises

Chapter 7: Knot Theory
     7.1 What Is a Knot?
     7.2 The Alexander Polynomial
     7.3 The Jones Polynomial
          7.3.1 Knot Projections
          7.3.2 Reidemeister Moves
          7.3.3 Bracket Polynomials
          7.3.4 Creation of a New Polynomial Invariant
     Exercises

Chapter 8: Graph Theory
     8.1 Introduction
     8.2 Fundamental Ideas of Graph Theory
     8.3 Application to the K¨onigsberg Bridge Problem
     8.4 Coloring Problems
          8.4.1 Modern Developments
          8.4.2 Denouement
     8.5 The Traveling Salesman Problem
     Exercises

Chapter 9: Dynamical Systems
     9.1 Flows
          9.1.1 Dynamical Systems
          9.1.2 Stable and Unstable Fixed Points
          9.1.3 Linear Dynamics in the Plane
     9.2 Planar Autonomous Systems
          9.2.1 Ingredients of the Proof of Poincar´e-Bendixson
     9.3 Lagrange’s Equations
     Exercises


Appendices
     Appendix 1: Principles of Logic
          A1.1 Truth
          A1.2 “And” and “Or”
          A1.3 “Not”
          A1.4 “If-Then”
          A1.5 Contrapositive, Converse, and “Iff”
          A1.6 Quantifiers
          A1.7 Truth and Provability

     Appendix 2: Principles of Set Theory
          A2.1 Undefinable Terms
          A2.2 Elements of Set Theory
          A2.3 Venn Diagrams
          A2.4 Further Ideas in Elementary Set Theory
          A2.5 Indexing and Extended Set Operations
          A2.6 Countable and Uncountable Sets

     Appendix 3: The Real Numbers
          A3.1 The Real Number System
          A3.2 Construction of the Real Numbers

     Appendix 4: The Axiom of Choice and Its Implications
          A4.1 Well Ordering
          A4.2 The Continuum Hypothesis
          A4.3 Zorn’s Lemma
          A4.4 The Hausdorff Maximality Principle
          A4.5 The Banach-Tarski Paradox

     Appendix 5: Ideas from Algebra
          A5.1 Groups
          A5.2 Rings
          A5.3 Fields
          A5.4 Modules
          A5.5 Vector Spaces


Solutions of Selected Exercises


Bibliography

Index

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