1st Edition

Essentials of Topology with Applications

By Steven G. Krantz Copyright 2009
    420 Pages 156 B/W Illustrations
    by Chapman & Hall

    420 Pages 156 B/W Illustrations
    by Chapman & Hall

    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.


    What Is Topology?

    First Definitions


    The Separation Axioms






    Totally Disconnected Spaces

    The Cantor Set

    Metric Spaces


    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


    Quotient Topologies


    Morse Theory

    Proper Mappings


    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


    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


    Fundamental Ideas of Graph Theory

    Application to the Königsberg Bridge Problem

    Coloring Problems

    The Traveling Salesman Problem

    Dynamical Systems


    Planar Autonomous Systems

    Lagrange’s Equations

    Appendix 1: Principles of Logic


    "And" and "Or"


    "If - Then"

    Contrapositive, Converse, and "Iff"


    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





    Vector Spaces

    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.