Topology Topics

1. The Definition of a Topological Space and Examples of 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
2.2. Interior Points of a Set
2.3. Accumulation Points of a Set
2.4. The Closure of a Set
2.5. The Boundary of a Set
2.6. Dense and Nowhere Dense Sets, The Baire Category Theorem
2.7. Separable Topological Spaces
2.8. Hausdorff Topological Spaces
2.9. Other Forms of Separation

3. Bases for Topologies

4. Sequences and Nets, Continuous Maps, and Homeomorphisms

4.1. Sequences and Nets
4.2. Continuous Maps on Topological Spaces
4.3. The Initial and Final Topologies
4.4. Homeomorphism Between Topological Spaces

5. Topological Subspaces, Sums, Products, and Quotients

5.1. Topological Subspaces
5.2. Topological Sums
5.3. Topological Products
5.4. Topological Quotients
5.5. The Gluing Lemma

6. Connectedness and Path Connectedness

7. Compactness

8. Isotopy and Homotopy

9. The Fundamental Group of a Topological Space

9.1 The Fundamental Group of a Topological Space
9.2. Brouwer's Fixed Point Theorem
9.3. Group Presentations and Tietze Transformations
9.4. The Seifert-van Kampen Theorem
9.5. The Classification of Connected Compact 1-Manifolds and 2-Manifolds
9.6. Covering Spaces
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