Essentials of Topology with Applications

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.

Steven G. Krantz is a professor in the Department of Mathematics at Washington University in St. Louis, Missouri, USA.