We will follow the book "Topology" by J. Munkres.
Chapter 1: Set Theory and Logic
1. Fundamental concepts
2. Functions
3. Relations
5. Cartesian products
6. Finite sets
7. Countable and Uncountable sets
Chapter 2: Topological Spaces and Continuous Functions
12. Topological Spaces
13. Basis for a Topology (omitting K-topologies)
15. The Product Topology on X x Y
16. The Subspace Topology
17. Closed Sets and Limit Points
18. Continuous Functions
19. The Product Topology (omitting box topology)
20. The Metric Topology
21. The Metric Topology (continued)
22. The Quotient Topology
Chapter 3: Connectedness and Compactness
23. Connected Spaces
24. Connected Subspaces of the Real Line
25. Components and Local Connectedness
26. Compact Spaces
27. Compact Subspaces of the Real Line
28. Limit Point Compactness
29. Local Compactness
Chapter 4: Countability and Separation Axioms
30. The Countability Axioms
31. The Separation Axioms
32. Normal Spaces
33. The Urysohn Lemma
34. The Urysohn Metrization Theorem
35. The Tietze Extension Theorem
Chapter 7: Complete Metric Spaces and Function Spaces
43. Complete Metric Spaces
45. Compactness in Metric Spaces
46. Pointwise and Compact Convergence
Chapter 8: Baire Spaces and Dimension Theory
48. Baire Spaces
49. A Nowhere-Differentiable Function
Chapter 9: The Fundamental Group
51. Homotopy of Paths
52. The Fundamental Group
53. Covering Spaces