An Illustrated Introduction to Topology and Homotopy

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





HOMOTOPY
Isotopy 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