# 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, and Open Neighbourhoods of a Point

- The Open and Closed Sets of a Topological Space
- The Open and Closed Sets of a Topological Space Examples 1
- The Open and Closed Sets of a Topological Space Examples 2
- The Open Neighbourhoods of Points in a Topological Space
- The Open Neighbourhoods of Points in a Topological Space Examples 1
- The Open Neighbourhoods of Points in a Topological Space Examples 2

###### 2.2. Interior Points of a Set

- The Interior Points of Sets in a Topological Space
- The Interior Points of Sets in a Topological Space Examples 1
- The Interior Points of Sets in a Topological Space Examples 2
- The Interior of Open Sets in a Topological Space
- Basic Theorems Regarding the Interior Points of Sets in a Topological Space

###### 2.3. Accumulation Points of a Set

- Accumulation Points of a Set in a Topological Space
- Accumulation Points, Topological Spaces 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

###### 2.4. The Closure of a Set

- The Closure of a Set in a Topological Space
- The Closure of a Set in a Topological Space Examples 1
- The Closure of a Set in a Topological Space 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

###### 2.5. The Boundary of a Set

- The Boundary 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
- Boundary Criterion for Open/Closed Sets in a Topological Space

###### 2.6. Dense and Nowhere Dense Sets, The Baire Category Theorem

- Dense and Nowhere Dense Sets in a Topological Space
- Dense and Nowhere Dense Sets - Topological Spaces Examples 1 )
- Basic Theorems Regarding Nowhere Dense Sets in a Topological Space

- 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

###### 2.7. Separable Topological Spaces

- Separable Topological Spaces
- Separable Topological Spaces Examples 1 )
- Countable Topological Spaces are Separable Topological Spaces
- The Countable Chain Condition for Topological Spaces

###### 2.8. Hausdorff Topological Spaces

- Hausdorff Topological Spaces
- Hausdorff Topological Spaces Examples 1
- Hausdorff Topological Spaces Examples 2
- Hausdorff Topological Spaces Examples 3

###### 2.9. Other Forms of Separation

## 3. Bases for Topologies

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

- Bases of a Topology
- Bases of a Topology Examples 1
- Bases of a Topology Examples 2
- A Sufficient Condition for a Collection of Sets to be a Base of a Topology
- Generating Topologies from a Collection of Subsets of a Set
- The Lower and Upper Limit Topologies on the Real Numbers

###### 3.2. Subbases of a Topology

###### 3.3. Local Bases of a Point

- Local Bases of a Point in a Topological Space
- Basic Theorems Regarding Local Bases of a Point in a Topological Space

###### 3.4. Comparability of Topologies

###### 3.5. First and Second Countable Topological Spaces

## 4. Sequences and Nets, Continuous Maps, and Homeomorphisms

###### 4.1. Sequences and Nets

- Directed Sets
- Sequences and Nets
- Subsequences and Subnets in a Topological Space
- Convergence of Sequences and Nets in Topological Spaces
- Net Criterion for a Point to be an Accumulation Point of a Set
- Net Criterion for a Point to be in the Closure of a Set
- IFF Criterion for Net Convergence Uniqueness

###### 4.2. Continuous Maps on Topological Spaces

- Generalizing Continuity to Maps on Topological Spaces
- Continuous Maps on Topological Spaces
- Continuity of the Composition of 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
- First Countable Sequential Criterion for Continuity of Maps on Topological Spaces

###### 4.3. The Initial and Final Topologies

###### 4.4. Homeomorphism Between Topological Spaces

- Homeomorphisms on Topological Spaces
- Homeomorphisms on Topological Spaces Examples 1
- Homeomorphisms on Topological Spaces 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

## 5. Topological Subspaces, Sums, Products, and Quotients

###### 5.1. Topological Subspaces

- Topological Subspaces
- Topological Subspaces Examples 1
- Topological Subspaces Examples 2
- Open and Closed Sets in Topological Subspaces
- Basic Theorems Regarding 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

###### 5.2. Topological Sums

###### 5.3. Topological Products

- 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

- Arbitrary Topological Products of Topological Spaces
- A Basis for Arbitrary Topological Products of Topological Spaces
- Box Topological Products of Topological Spaces

###### 5.4. Topological Quotients

- Topological Quotients
- Open and Closed Sets in Topological Quotients
- Topological Quotients in Euclidean Space
- Topological Quotients from Equivalence Relations Defined by Functions

###### 5.5. The Gluing Lemma

## 6. Connectedness and Path Connectedness

###### 6.1. 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

###### 6.2. 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

## 7. Compactness

###### 7.1. 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

###### 7.2. Lindelöf, Countably 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

###### 7.3. Tychonoff's Theorem

## 8. Isotopy and Homotopy

###### 8.1. Isotopy and Ambient Isotopy

###### 8.2. Homotopy

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

###### 8.3 Retracts and Deformation Retracts

## 9. The Fundamental Group of a Topological Space

###### 9.1 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
- The Retract Theorem for Subspaces of a Topological Space
- An Example of a Retract that is NOT a Deformation Retract
- The Fundamental Groups of Connected Graphs
- R2 is NOT Homeomorphic to R3

###### 9.2. 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

###### 9.3. 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

###### 9.4. 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
- The Seifert-van Kampen Theorem Example 4

###### 9.5. The Classification of Connected Compact 1-Manifolds and 2-Manifolds

- Identifying Polygons and Their Quotient Spaces
- The Classification of Connected Compact 1-Manifolds and 2-Manifolds
- The Classification Theorem for Connected Compact 2-Manifolds
- Examples of Classifying Connected Compact 2-Manifolds 1

###### 9.6. Covering Spaces

- Covering Spaces
- Covering Space Examples - R2 \ {(0, 0)}
- Covering Space Examples - The Projective Plane
- Covering Space Examples - The Bouquet of Two Circles
- Covering Maps are Open Maps
- Lifts of Paths
- The Uniqueness of Universal Covers Theorem
- The Fundamental Group of Spaces and Their Covers
- Covering Space Examples - The Torus

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###### References

- 1. An Illustrated Introduction to Topology and Homotopy by Sasho Kalajdzievski.

- 2. General Topology by John L. Kelley.