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.
Table of Contents
What Is Topology?
The Separation Axioms
Totally Disconnected Spaces
The Cantor Set
Lebesgue’s Lemma and Lebesgue Numbers
Advanced Properties of Topological Spaces
Basis and Sub-Basis
First Countable, Second Countable, and So Forth
An Application to Digital Imaging
Basic Algebraic Topology
The Concept of Index
Moore–Smith Convergence and Nets
The Topology of Pointwise Convergence
The Compact-Open Topology
Equicontinuity and the Ascoli–Arzela Theorem
The Weierstrass Approximation Theorem
What Is a Knot?
The Alexander Polynomial
The Jones Polynomial
Fundamental Ideas of Graph Theory
Application to the Königsberg Bridge Problem
The Traveling Salesman Problem
Planar Autonomous Systems
Appendix 1: Principles of Logic
"And" and "Or"
"If - Then"
Contrapositive, Converse, and "Iff"
Truth and Provability
Appendix 2: Principles of Set Theory
Elements of Set Theory
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
The Continuum Hypothesis
The Hausdorff Maximality Principle
The Banach–Tarski Paradox
Appendix 5: Ideas from Algebra
Solutions of Selected Exercises
Exercises appear at the end of each chapter.
Steven G. Krantz is a professor in the Department of Mathematics at Washington University in St. Louis, Missouri, USA.