Week 1: Topological spaces: Prerequisites, Open sets and Topology.
Week 2: Topological spaces: Topologies on R, Comparison of topologies, Closed sets, Bases and subbsaes for a topology.
Week 3: Neighborhoods, Interior and Closure of sets, Limit points, The boundary of a set, Dense sets.
Week 4: Creating new topological spaces: The subspace topology, The product topology, The quotient topology.
Week 5: Alternative methods of defining a topology in terms of Kuratowski closure/interior operator, First and Second countable spaces, Separable spaces.
Week 6: Continuous functions and homeomorphisms, Non- homeomorphic spaces.
Week 7: Connectedness: A first approach to connectedness, Distinguishing topological spaces via connectedness, Connected subspaces of the real line.
Week 8: Connectedness: Components, Path connectedness, Local connectedness.
Week 9: Compactness: Open covering and compact spaces, Basic properties of compactness. Compactness and finite intersection property.
Week 10: Compactness: B-W compactness, Limit point compactness, Local compactness, One-point compactification
Week 11: The separation axioms T0, T1, T2, T3, T4; their characterizations and basic properties. Urysohn’s lemma, Tietze extension theorem.
Week 12: The Tychnoff Theorem, The Stone-Cech Compactification
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