General Topology (M.S. and Ed.D.)

Preparatory Course: Math 5303

1.

The basic concepts: topological spaces, bases, subbases, open sets, closed sets, limit points, closure, interior, boundary, convergence, dense subsets, continuous functions, homeomorphisms, topological imbeddings.

2.

Various kinds of topologies: the discrete topology, the indiscrete topology, the metric topology, the order topology, the subspace topology, the product topology, the box topology, open maps, closed maps, quotient maps, and the quotient topology.

3.

Connectivity: connected and path-connected spaces, locally connected and locally path-connected spaces, components and path-components, connected subsets of the real line.

4.

Compactness: compact spaces, limit point, sequential, and countable compactness, local compactness, Tychonoff's Theorem.

5.

Countability and separation axioms: the first and second axioms of countability, separable spaces, Lindelöf spaces, Hausdorff spaces, regular spaces, normal spaces, the Urysohn Lemma, the Tietze Extension Theorem, the Urysohn Metrization Theorem.

REFERENCE: James R. Munkres, Topology: A First Course, Chapters 1-4, 5.1