Week 1: Set theory and Logic: Sets and functions, Finite and Infinite sets, and the Axiom of Choice
Week 2: Topological spaces: the basic axioms of topology with examples, Bases and Subbases, Various kinds of topologies on the real line
Week 3: Subspace Topology, Box and Product topology, Metric topology
Week 4: Limit points and closed sets, Continuous functions
Week 5: Connectedness: connected spaces and subspaces, Connectedness of the real line, Intermediate value theorem
Week 6: Connected components, Path-connected, locally connected, and locally path-connected spaces
Week 7: Compact spaces: open cover characterization, Finite intersection property, various notions of compactness
Week 8: Uniform continuity, Heine-Borel theorem, Extreme value theorem, Lebesgue number Lemma
Week 9: Baire spaces and the Baire category theorem
Week 10: Urysohn’s Lemma and Tietze extension theorem on normal spaces
Week 11: Complete metric spaces, Totally bounded metric spaces and compactness, Lebesgue number lemma
Week 12: Function spaces and the Arzela-Ascoli theorem
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