1st Edition

# An Illustrated Introduction to Topology and Homotopy

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**An Illustrated Introduction to Topology and Homotopy** explores the beauty of topology and homotopy theory in a direct and engaging manner while illustrating the power of the theory through many, often surprising, applications. This self-contained book takes a visual and rigorous approach that incorporates both extensive illustrations and full proofs.

The first part of the text covers basic topology, ranging from metric spaces and the axioms of topology through subspaces, product spaces, connectedness, compactness, and separation axioms to Urysohn’s lemma, Tietze’s theorems, and Stone-Čech compactification. Focusing on homotopy, the second part starts with the notions of ambient isotopy, homotopy, and the fundamental group. The book then covers basic combinatorial group theory, the Seifert-van Kampen theorem, knots, and low-dimensional manifolds. The last three chapters discuss the theory of covering spaces, the Borsuk-Ulam theorem, and applications in group theory, including various subgroup theorems.

Requiring only some familiarity with group theory, the text includes a large number of figures as well as various examples that show how the theory can be applied. Each section starts with brief historical notes that trace the growth of the subject and ends with a set of exercises.

**TOPOLOGY**

**Sets, Numbers, Cardinals, and Ordinals**

Sets and Numbers

Sets and Cardinal Numbers

Axiom of Choice and Equivalent Statements

**Metric Spaces: Definition, Examples, and Basics**Metric Spaces: Definition and Examples

Metric Spaces: Basics

**Topological Spaces: Definition and Examples**The Definition and Some Simple Examples

Some Basic Notions

Bases

Dense and Nowhere Dense Sets

Continuous Mappings

**Subspaces, Quotient Spaces, Manifolds, and CW-Complexes **Subspaces

Quotient Spaces

The Gluing Lemma, Topological Sums, and Some Special Quotient Spaces

Manifolds and CW-Complexes

**Products of Spaces**Finite Products of Spaces

Infinite Products of Spaces

Box Topology

**Connected Spaces and Path Connected Spaces **Connected Spaces: Definition and Basic Facts

Properties of Connected Spaces

Path Connected Spaces

Path Connected Spaces: More Properties and Related Matters

Locally Connected and Locally Path Connected Spaces

**Compactness and Related Matters **Compact Spaces: Definition

Properties of Compact Spaces

Compact, Lindelöf, and Countably Compact Spaces

Bolzano, Weierstrass, and Lebesgue

Compactification

Infinite Products of Spaces and Tychonoff Theorem

**Separation Properties**The Hierarchy of Separation Properties

Regular Spaces and Normal Spaces

Normal Spaces and Subspaces

**Urysohn, Tietze, and Stone-Čech **Urysohn Lemma

The Tietze Extension Theorem

Stone-Čech Compactification

**HOMOTOPYIsotopy and Homotopy **Isotopy and Ambient Isotopy

Homotopy

Homotopy and Paths

The Fundamental Group of a Space

**The Fundamental Group of a Circle and Applications **The Fundamental Group of a Circle

Brouwer Fixed Point Theorem and the Fundamental Theorem of Algebra

The Jordan Curve Theorem

**Combinatorial Group Theory**

Group Presentations

Free Groups, Tietze, Dehn

Free Products and Free Products with Amalgamation

**Seifert–van Kampen Theorem and Applications**

Seifert–van Kampen Theorem

Seifert–van Kampen Theorem: Examples

The Seifert–van Kampen Theorem and Knots

Torus Knots and Alexander’s Horned Sphere

Links

**On Classifying Manifolds and Related Topics **1-Manifolds

Compact 2-Manifolds: Preliminary Results

Compact 2-Manifolds: Classification

Regarding Classification of CW-Complexes and Higher Dimensional Manifolds

Higher Homotopy Groups: A Brief Overview

**Covering Spaces, Part 1 **Covering Spaces: Definition, Examples, and Preliminaries

Lifts of Paths

Lifts of Mappings

Covering Spaces and Homotopy

**Covering Spaces, Part 2 **Covering Spaces and Sheets

Covering Trans formations

Covering Spaces and Groups Acting Properly Discontinuously

Covering Spaces: Existence

The Borsuk–Ulam Theorem

**Applications in Group Theory**

Cayley Graphs and Covering Spaces

Topographs and Presentations

Subgroups of Free Groups

Two Subgroup Theorems

Bibliography

### Biography

Sasho Kalajdzievski

"… reflects interesting aspects and will find its readers."

, 1323

—Zentralblatt MATH"… an ideal college or university textbook and an invaluable addition to academic library mathematical studies reference collections."

—Library Bookwatch, May 2015