**MTH ****304: Metric Spaces and Topology**** ****(4)**

**Pre-requisites**: MTH 303 Real Analysis I

Definition, open sets, closed sets, limit points, convergence, completeness, Baire’s theorem, continuity, spaces of continuous functions

Compactness, sequential compactness, compact metric spaces, compact-open topology, Ascoli’s theorem

Completeness, space filling curve, nowhere differentiable functions

*Topology*

Definition and examples of topology, base, subbase, weaker and stronger topology

Order topology, subspace topology, product and box topology

Continuity, homeomorphisms, quotient topology

Compact spaces, examples, Tychonoff’s theorem and locally compact spaces, limit point compactness, local compactness

Connected spaces, components, path components, totally disconnected spaces, locally connected spaces, examples

Countability axioms, separation axioms, completely regular and normal spaces, Urysohn’s lemma, Tietze extension theorem, Urysohn embedding theorem, Stone-Cech compacitification

*Suggested Books*:

- G. F. Simmons,
*Introduction to Topology and Modern Analysis*, Tata McGraw Hill, 2008 - J. R. Munkres,
*Topology (2nd Edn)*, Dorling Kindersley, 2006

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