Essentials of Topology with Applications Book

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.