MTH307: (Topology)

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• 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 ${\mathbb{R}}^n$.
• 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.

Reference Texts:

1. Munkres: Topology.
2. Simmons: Introduction to Topology and Modern Analysis.