Essentials of Topology with Applications 1st Edition by Steven Krantz 1420089749 9781420089745
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Product details:
ISBN 10: 1420089749
ISBN 13: 9781420089745
Author: Steven G. Krantz
Brings Readers Up to Speed in This Important and Rapidly Growing AreaSupported 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
Table of contents:
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
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
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
4 Manifold Theory
4.1 Basic Concepts
4.2 The Definition
Exercises
5 Moore-Smith Convergence and Nets
5.1 Introductory Remarks
5.2 Nets
Exercises
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 The Weierstrass Approximation Theorem
Exercises
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
8 Graph Theory
8.1 Introduction
8.2 Fundamental Ideas of Graph Theory
8.3 Application to the Königsberg Bridge Problem
8.4 Coloring Problems
8.4.1 Modern Developments
8.4.2 Denouement
8.5 The Traveling Salesman Problem
Exercises
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 Poincaré-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
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Tags: Steven Krantz, Essentials, Topology, Applications