# Topology Topics

## 1. The Definition of a Topological Space and Examples of Topologies

###### 1.1. Introduction to Topological Spaces

- The Relative Complement and Complement of a Set
- De Morgan's Laws for the Intersections and Unions of Sets
- Topological Spaces
- The Discrete and Indiscrete Topologies
- Topologies on a Finite 3-Element Set
- The Topology of Open Intervals on the Set of Real Numbers
- The Topology of Closed Intervals on the Set of Real Numbers
- The Initial and Final Segment Topologies
- The Cofinite Topology
- The Countable Complement Topology
- Nested Topologies
- The K-Topology
- The Union and Intersection of Two Topologies

## 2. Open and Closed Sets, Open Neighbourhoods, Interior Points, Accumulation Points, Closure, Boundary Points, and Hausdorff Spaces

###### 2.1. Open Sets, Closed Sets, Interior Points, and Accumulation Points

- The Open and Closed Sets of a Topological Space ( Examples 1 | Examples 2 )
- The Open Neighbourhoods of Points in a Topological Space ( Examples 1 | Examples 2 )
- The Interior Points of Sets in a Topological Space ( Examples 1 | Examples 2 )
- The Interior of Open Sets in a Topological Space
- Basic Theorems Regarding the Interior Points of Sets in a Topological Space
- Accumulation Points of a Set in a Topological Space ( Examples 1 )
- Basic Theorems Regarding the Accumulation Points of Sets in a Topological Space
- Criterion for a Set of a Topological Space to be Closed
- Open Sets, Closed Sets, Interior and Accumulation Points Review

###### 2.2. The Closure of a Set, The Boundary of a Set, and Hausdorff Topological Spaces

- The Closure of a Set in a Topological Space ( Examples 1 | Examples 2 )
- The Closure of a Set Equals the Union of the Set and Its Accumulation Points
- The Closure of Closed Sets in a Topological Space
- Basic Theorems Regarding the Closure of Sets in a Topological Space
- A Comparison of the Interior and Closure of a Set in a Topological Space
- The Boundary of a Set in a Topological Space ( Examples 1 )
- The Boundary of Any Set is Closed in a Topological Space
- The Boundary of Clopen Sets in a Topological Space
- Basic Theorems Regarding the Boundary of a Set in a Topological Space
- Hausdorff Topological Spaces ( Examples 1 | Examples 2 | Examples 3 )
- Normal Topological Spaces
- Closure, Boundary, and Hausdorff Topological Spaces Review

## 3. Bases and Generated Topologies

###### 3.1. Bases of a Topology

- Bases of a Topology ( Examples 1 | Examples 2 )
- A Sufficient Condition for a Collection of Sets to be a Basis of a Topology
- Generating Topologies from a Collection of Subsets of a Set
- The Lower and Upper Limit Topologies on the Real Numbers
- Comparable Topologies on a Set
- Local Bases of a Point in a Topological Space
- Basic Theorems Regarding Local Bases of a Point in a Topological Space
- First Countable Topological Spaces
- Second Countable Topological Spaces
- All Second Countable Topological Spaces are First Countable
- Subbases of a Topology ( Examples 1 )
- Basic Theorems Regarding Subbases of a Topology
- Bases of Topologies, Local Bases of Points, and Subbases of Topologies Review

###### 3.2. Dense and Nowhere Dense Sets, Separable Topological Spaces

- Dense and Nowhere Dense Sets in a Topological Space ( Examples 1 )
- Basic Theorems Regarding Nowhere Dense Sets in a Topological Space
- Separable Topological Spaces ( Examples 1 )
- Countable Topological Spaces are Separable Topological Spaces
- Second Countable Topological Spaces are Separable Topological Spaces
- Sets of the First and Second Categories in a Topological Space
- Basic Theorems Regarding Sets of the First Category in a Topological Space
- The Baire Category Theorem

## 4. Continuous Maps on Topological Spaces

###### 4.1. Continuous Maps on Topological Spaces

- Generalizing Continuity to Maps on Topological Spaces
- Continuous Maps on Topological Spaces
- The Open Neighbourhood Definition of Continuous Maps on Topological Spaces
- Equivalent Statements Regarding Continuous Maps on Topological Spaces
- The Closed Set Definition of Continuous Maps on Topological Spaces
- Summary of Equivalent Statements Regarding Continuous Maps on Topological Spaces
- Continuity of the Composition of Continuous Maps on Topological Spaces
- First Countable Sequential Criterion for Continuity of Maps on Topological Spaces
- Continuous Maps on Topological Spaces Review

## 5. Homeomorphisms Between Topological Spaces

###### 5.1. Homeomorphisms Between Topological Spaces

- Open and Closed Maps on Topological Spaces
- Homeomorphisms on Topological Spaces ( Examples 1 | Examples 2 )
- The Interior of a Set under Homeomorphisms on Topological Spaces
- The Closure of a Set under Homeomorphisms on Topological Spaces
- The Set of Accumulation Points under Homeomorphisms on Topological Spaces
- The Boundary of a Set under Homeomorphisms on Topological Spaces
- First Countability under Homeomorphisms on Topological Spaces
- Second Countability under Homeomorphisms on Topological Spaces
- The Hausdorff Property under Homeomorphisms on Topological Spaces
- Separability under Homeomorphisms on Topological Spaces
- Homeomorphisms on Topological Spaces Review

## 6. Topological Subspaces and Topological Quotients

###### 6.1. Topological Subspaces

- Initial Topologies
- Final Topologies
- Topological Subspaces ( Examples 1 | Examples 2 )
- Open and Closed Sets in Topological Subspaces
- Restricted Metric Subspaces as Topological Subspaces
- Hereditary Properties of Topological Spaces
- Heredity of First Countability on Topological Subspaces
- Heredity of Second Countability on Topological Subspaces
- Heredity of the Hausdorff Property on Topological Subspaces
- Nonheredity of Separability on Topological Subspaces
- Weakly Hereditary Properties of Topological Subspaces
- Topological Subspaces Review

###### 6.2. Topological Quotient Spaces

- Topological Quotients
- Open and Closed Sets in Topological Quotients
- Topological Quotients in Euclidean Space
- Topological Quotients from Equivalence Relations Defined by Functions
- Topological Sums of Topological Spaces
- The Gluing Lemma
- Gluing Topological Spaces to Themselves
- Gluing Disjoint Topological Spaces
- Construction of the Möbius Strip
- Topological Quotients Review

## 7. Topological Products

###### 7.1. Topological Product Spaces

- Finite Topological Products of Topological Spaces
- Projection Mappings of Finite Topological Products
- The Open and Closed Sets of Finite Topological Products
- The Interior of Sets in Finite Topological Products
- The Closure of Sets in Finite Topological Products
- The Set of Accumulation Points in Finite Topological Products
- Dense Sets in Finite Topological Products
- First Countability of Finite Topological Products
- Second Countability of Finite Topological Products
- Separability of Finite Topological Products
- The Hausdorff Property on Finite Topological Products
- Metrizability of Finite Topological Products
- Finite Topological Products Review

## 8. Connectedness and Path Connectedness

###### 8.1. Connected and Disconnected Topological Spaces

- Connected and Disconnected Topological Spaces
- Disconnected Topological Spaces Homeomorphic to the Topological Sum of their Separation
- Continuous Two-Valued Function Criterion for Disconnected Topological Space
- Clopen Set Criterion for Disconnected Topological Spaces ( Examples 1 )
- Connected and Disconnected Sets in Topological Spaces
- The Connectedness of the Closure of a Set
- Common Point Criterion for Connectedness of Unions of Topological Subspaces
- Common Connector Criterion for Connectedness of Unions of Topological Subspaces
- Preservation of Connectivity under Continuous Maps
- Connected/Disconnected Criterion for Non-Homeomorphic Topological Spaces
- Non-Homeomorphic Topological Spaces Classified by Connectivity
- Connectedness of Finite Topological Products
- Connectedness of Box Topological Products
- Connected and Disconnected Topological Spaces Review

###### 8.2. Path Connected Topological Spaces

- Path Connected Topological Spaces
- Path Connectivity of Connected Topological Spaces
- Path Connectivity of Countable Unions of Connected Sets
- Path Connectivity of the Range of a Path Connected Set under a Continuous Function
- Path Connectedness of Arbitrary Topological Products
- Path Connectedness of Open and Connected Sets in Euclidean Space
- Locally Connected and Locally Path Connected Topological Spaces
- Path Connected Topological Spaces Review

## 9. Compactness

###### 9.1. Compactness

- Covers of Sets in a Topological Space
- Compactness of Sets in a Topological Space
- Compactness of Finite Sets in a Topological Space
- Finite Intersection Property Criterion for Compactness in a Topological Space
- Preservation of Compactness under Continuous Maps
- Closed Sets in Compact Topological Spaces
- Compact Sets in Hausdorff Topological Spaces
- Homeomorphisms Between Compact and Hausdorff Spaces
- Quotients from Equivalence Relations defined by Functions from Compact to Hausdorff Spaces
- Compactness Review

###### 9.2. Lindelöf, Countable Compact, and Bolzano-Weierstrass Spaces

- Lindelöf and Countably Compact Topological Spaces
- The Lindelöf Lemma
- Bolzano Weierstrass Topological Spaces
- Hausdorff Spaces Are BW Spaces If and Only If They're Countably Compact
- Compact Spaces as BW Spaces
- The Lebesgue Number Lemma
- Metric Spaces Are Compact Spaces If and Only If They're BW Spaces
- Metric Spaces Are Compact Spaces If and Only If They're Countably Compact
- Local Compactness in a Topological Space* One-Point Compactification of a Topological Space
- The Hausdorff Property on One-Point Compactifications of a Topological Space
- Lindelöf, Countably Compact, and BW Spaces Review

###### 9.3. Tychonoff's Theorem

## 10. Separation Axioms

###### 10.1. The Separation Axioms

- T0 (Kolmogorov) Topological Spaces
- T1 (Fréchet) Topological Spaces
- T2 (Hausdorff) Topological Spaces
- T3 (Regular Hausdorff) Topological Spaces
- T4 (Normal Hausdorff) Topological Spaces
- Heredity of the T0, T1, T2, and T3 Properties on Topological Subspaces
- Arbitrary Products of T0, T1, T2, and T3 Topological Spaces
- A T2 Space That Is NOT a T3 Space
- A T3 Space That Is NOT a T4 Space
- Compact T2 Spaces are T3 Spaces
- Compact T2 Spaces are T4 Spaces

## 11 Isotopy and Homotopy

###### 11.1. Isotopy and Ambient Isotopy

###### 11.2. Homotopy

- Homotopically Equivalent Topological Spaces
- Homeomorphic Topological Spaces are Homotopically Equivalent

###### 11.3 Retracts and Deformation Retracts

## 12. The Fundamental Group of a Topological Space

###### 12.1 The Fundamental Group of a Topological Space

Products of Paths Relative to {0, 1} in a Topological Space

Associativity of Products of Paths in a Topological Space

Constant Paths in a Topological Space

Inverse Paths in a Topological Spaces

The Fundamental Group of a Topological Space at a Point

Fundamental Groups under Homeomorphisms on Topological Spaces

Simply Connected Topological Spaces

The Fundamental Groups of a Path Connected Topological Space are Isomorphic

The Fundamental Groups of Discrete and Indiscrete Topological Spaces

The Induced Mapping from the Fundamental Groups of Two Topological Spaces

The Fundamental Group of a Topological Product

The Fundamental Group of the Cylinder and the Torus

An Example of a Retract that is NOT a Deformation Retract

###### 12.2. Brouwer's Fixed Point Theorem

Brouwer's Fixed Point Theorem

Solutions to Systems of Equations by Brouwer's Fixed Point Theorem

The Jordan Curve Theorem

###### 12.3. Group Presentations and Tietze Transformations

Words Over a Set

Group Presentations

A Group Presentation for Zn

A Group Presentation for S3

Determining Whether Two Groups are Isomorphic by their Group Presentations

Tietze Transformations

Tietze Transformations Example 1

Tietze Transformations Example 2

The Free Product of Groups

###### 12.4. The Seifert-van Kampen Theorem

The Seifert-van Kampen Theorem

The Seifert-van Kampen Theorem Example 1

The Seifert-van Kampen Theorem Example 2

The Seifert-van Kampen Theorem Example 3