دانلود کتاب 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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