Week 1: Definition and examples of topological spaces, Examples of topological spaces, Basis for topology, Subspace Topology, Product Topology.
Week 2: Continuous maps, Continuity of addition and multiplication maps, ring of continuous functions, Continuous maps to a product, Projection from a point.
Week 3: Closed subsets, Closure, Joining continuous maps, Metric spaces, Connectedness.
Week 4: Connected components, Path connectedness.
Week 5: Connectedness of GL(n,R)^+, Connectedness of GL(n,C), SL(n,C), SL(n,R), Hausdorff topological spaces, Compactness.
Week 6: SO(n) is connected, Compact metric spaces, Lebesgue Number Lemma, Locally compact spaces.
Week 7:One point compactification, One point compactification (continued), Uniqueness of one point compatification, Quotient topology, Quotient topology on G/H.
Week 8: Grassmannian, Normal topological spaces, Urysohn’s Lemma, Tietze Extension Theorem, Regular and Second Countable spaces, Urysohn’s Metrization Theorem.
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