**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Metric spaces: definition, open sets, closed sets, limit points, convergence, completeness, Baire’s theorem, continuity, spaces of continuous functions, completeness, completion of a metric space.
- Topological spaces , open sets, closed sets, bases, sub-bases, continuous functions; examples- metric topology, order topology, subspace topology, product topology.
- Connectedness, locally connected, path connected, locally path connected, connected subsets of the real line.
- Compact spaces, sequential compact spaces, locally compact spaces, Compact subsets of .
- Countability axioms, Hausdorff, regular etc., Urysohn lemma, Urysohn theorem , Tietze extension theorem.
- Tychonoff theorem, one point compactification, Stone-Cech compactification theorem.
- Quotient spaces; group actions on topological spaces.

Additional Topics Nagata Smirnov metrization theorem, paracompact spaces,
introduction to covering spaces, properly discontinuous action.

- Munkres: Topology.
- Simmons: Introduction to Topology and Modern Analysis.