Essentials of Topology with Applications / Edition 1

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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.

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Product Details

  • ISBN-13: 9781420089745
  • Publisher: Taylor & Francis
  • Publication date: 7/27/2009
  • Series: Textbooks in Mathematics Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 420
  • Product dimensions: 7.20 (w) x 10.10 (h) x 1.10 (d)

Meet the Author

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

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Table of Contents


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.

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