Skip to main content**By Tai-Danae Bradley, Tyler Bryson, and John Terilla**

# Chapters

Topology

A Categorical Approach

MIT Press 2020

CC BY-NC-ND

**A graduate-level textbook that presents basic topology from the perspective of category theory.**

^{Click on the chapter titles to download pdfs of each chapter.}

0.1 Basic Topology

0.2 Basic Category Theory

0.2.1 Categories

0.2.2 Functors

0.2.3 Natural Transformations and the Yoneda Lemma

0.3 Basic Set Theory

0.3.1 Functions

0.3.2 The Empty Set and OnePoint Set

0.3.3 Products and Coproducts in Set

0.3.4 Products and Coproducts in Any Category

0.3.5 Exponentiation in Set

0.3.6 Partially Ordered Sets

Exercises

1.1 Examples and Terminology

1.1.1 Examples of Spaces

1.1.2 Examples of Continuous Functions

1.2 The Subspace Topology

1.2.1 The First Characterization

1.2.2 The Second Characterization

1.3 The Quotient Topology

1.3.1 The First Characterization

1.3.2 The Second Characterization

1.4 The Product Topology

1.4.1 The First Characterization

1.4.2 The Second Characterization

1.5 The Coproduct Topology

1.5.1 The First Characterization

1.5.2 The Second Characterization

1.6 Homotopy and the Homotopy Category

Exercises

**2 Connectedness and Compactness**

2.1 Connectedness

2.1.1 Definitions, Theorems, and Examples

2.1.2 The Functor

π_{0}2.1.3 Constructions and Connectedness

2.1.4 Local (Path) Connectedness

2.2 Hausdorff Spaces

2.3 Compactness

2.3.1 Definitions, Theorems, and Examples

2.3.2 Constructions and Compactness

2.3.3 Local Compactness

Exercises

**3 Limits of Sequences and Filters**

3.1 Closure and Interior

3.2 Sequences

3.3 Filters and Convergence

3.4 Tychonoff’s Theorem

3.4.1 Ultrafilters and Compactness

3.4.2 A Proof of Tychonoff’s Theorem

3.4.3 A Little Set Theory

Exercises

**4 Categorical Limits and Colimits**

4.1 Diagrams Are Functors

4.2 Limits and Colimits

4.3 Examples

4.3.1 Terminal and Initial Objects

4.3.2 Products and Coproducts

4.3.3 Pullbacks and Pushouts

4.3.4 Inverse and Direct Limits

4.3.5 Equalizers and Coequalizers

4.4 Completeness and Cocompleteness

Exercises

**5 Adjunctions and the Compact-Open Topology**

5.1 Adjunctions

5.1.1 The Unit and Counit of an Adjunction

5.2 Free-Forgetful Adjunction in Algebra

5.3 The Forgetful Functor

U: Top —> Set and Its Adjoints5.4 Adjoint Functor Theorems

5.5 Compactifications

5.5.1 The One-Point Compactification

5.5.2 The Stone-Cech Compactification

5.6 The Exponential Topology

5.6.1 The Compact-Open Topology

5.6.2 The Theorems of Ascoli and Arzela

5.6.3 Enrich the Product-Hom Adjunction in Top

5.7 Compactly Generated Weakly Hausdorff Spaces

Exercises

**6 Paths, Loops, Cylinders, Suspensions, . . .**

6.1 Cylinder-Free Path Adjunction

6.2 The Fundamental Groupoid and Fundamental Group

6.3 The Categories of Pairs and Pointed Spaces

6.4 The Smash-Hom Adjunction

6.5 The Suspension-Loop Adjunction

6.6 Fibrations and Based Path Spaces

6.6.1 Mapping Path Space and Mapping Cylinder

6.6.2 Examples and Results

6.6.3 Applications of

π_{1}S^{1}6.7 The Seifert van Kampen Theorem

6.7.1 Examples

Exercises