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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.