Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. It presents the traditional concepts of topological space, open and closed sets, separation axioms, and more, along with applications of the ideas in Morse, manifold, homotopy, and homology theories.
After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. He also explores meaningful applications in a number of areas, including the traveling salesman problem, digital imaging, mathematical economics, and dynamical systems. The appendices offer background material on logic, set theory, the properties of real numbers, the axiom of choice, and basic algebraic structures.
Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond.
Author(s): Steven G. Krantz
Series: Textbooks in Mathematics
Edition: 1st
Publisher: Taylor & Francis Group/CRC
Year: 2009
Language: English
Pages: xvi,404
Cover
S Title
Series Editor
Essentials of Topology with Applications
Copyright
© 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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