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Brings Readers Up to Speed in This Important and Rapidly Growing Area
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.
Fundamentals
What Is Topology?
First Definitions
Mappings
The Separation Axioms
Compactness
Homeomorphisms
Connectedness
Path-Connectedness
Continua
Totally Disconnected Spaces
The Cantor Set
Metric Spaces
Metrizability
Baire’s Theorem
Lebesgue’s Lemma and Lebesgue Numbers
Advanced Properties of Topological Spaces
Basis and Sub-Basis
Product Spaces
Relative Topology
First Countable, Second Countable, and So Forth
Compactifications
Quotient Topologies
Uniformities
Morse Theory
Proper Mappings
Paracompactness
An Application to Digital Imaging
Basic Algebraic Topology
Homotopy Theory
Homology Theory
Covering Spaces
The Concept of Index
Mathematical Economics
Manifold Theory
Basic Concepts
The Definition
Moore–Smith Convergence and Nets
Introductory Remarks
Nets
Function Spaces
Preliminary Ideas
The Topology of Pointwise Convergence
The Compact-Open Topology
Uniform Convergence
Equicontinuity and the Ascoli–Arzela Theorem
The Weierstrass Approximation Theorem
Knot Theory
What Is a Knot?
The Alexander Polynomial
The Jones Polynomial
Graph Theory
Introduction
Fundamental Ideas of Graph Theory
Application to the Königsberg Bridge Problem
Coloring Problems
The Traveling Salesman Problem
Dynamical Systems
Flows
Planar Autonomous Systems
Lagrange’s Equations
Appendix 1: Principles of Logic
Truth
"And" and "Or"
"Not"
"If - Then"
Contrapositive, Converse, and "Iff"
Quantifiers
Truth and Provability
Appendix 2: Principles of Set Theory
Undefinable Terms
Elements of Set Theory
Venn Diagrams
Further Ideas in Elementary Set Theory
Indexing and Extended Set Operations
Countable and Uncountable Sets
Appendix 3: The Real Numbers
The Real Number System
Construction of the Real Numbers
Appendix 4: The Axiom of Choice and Its Implications
Well Ordering
The Continuum Hypothesis
Zorn’s Lemma
The Hausdorff Maximality Principle
The Banach–Tarski Paradox
Appendix 5: Ideas from Algebra
Groups
Rings
Fields
Modules
Vector Spaces
Solutions of Selected Exercises
Bibliography
Index
Exercises appear at the end of each chapter.
Biography
Steven G. Krantz is a professor in the Department of Mathematics at Washington University in St. Louis, Missouri, USA.
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