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

2.1. 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

• T0 (Kolmogorov) Topological Spaces
• T1 (Fréchet) Topological Spaces
• T2 (Hausdorff) Topological Spaces
• T3 (Regular Hausdorff) Topological Spaces
• T4 (Normal Hausdorff) 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

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

• Subbases of a Topology
• Subbases of a Topology Examples 1

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

• Comparable Topologies on a Set
• Criterion for Coarser/Finer Topologies

3.5. First and Second Countable Topological Spaces

• First Countable Topological Spaces

• Second Countable Topological Spaces
• Second Countable Criterion for the Existence of an Accumulation Point of Uncountable Sets
• All Second Countable Topological Spaces are First Countable
• Second Countable Topological Spaces are Separable Topological Spaces

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

• Initial Topologies
• Final Topologies

4.4. Homeomorphism Between Topological Spaces

• Open and Closed Maps on 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

5.1. 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

• Topological Sums of Topological Spaces

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

• The Gluing Lemma
• Gluing Topological Spaces to Themselves
• Gluing Disjoint Topological Spaces
• Construction of the Möbius Strip
• Topological Quotients Review

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

7.1. 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

• Alexander's Subbasis Theorem
• Tychonoff's Theorem for Arbitrary Products of Compact Sets

8.1. Isotopy and Ambient Isotopy

• Isotopic Embeddings on Topological Spaces
• Isotopic and Non-Isotopic Embeddings on the Bounded Cone

• Ambient Isotopic Embeddings on Topological Spaces
• Basic Theorems Regarding Ambient Isotopies

8.2. Homotopy

• Homotopic Mappings Relative to a Subset of a Topological Space
• Basic Theorems Regarding Homotopies

• Homotopically Equivalent Topological Spaces
• Homeomorphic Topological Spaces are Homotopically Equivalent

8.3 Retracts and Deformation Retracts

• Retract Subspaces of a Topological Space
• Deformation Retract Subspaces of a Topological Space
• Basic Theorems Regarding Deformation Retracts

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

• Covering Transformations
• The Covering Transformation Theorem for Finding Fundamental Groups
• The Covering Transformation Theorem - Fund. Group of the Circle
• The Covering Transformation Theorem - Fund. Group of the Projective Plane
• The Covering Transformation Theorem - Fund. Group of the Torus

• Regular Covering Spaces
• Constructing Base Spaces

• List of Fundamental Groups of Common Spaces