General Prerequisites: Part A Topology, including the notions of a topological space, a continuous function and a basis for a topology. A basic knowledge of Set Theory, including cardinal arithmetic, ordinals and the Axiom of Choice, will also be useful.
Course Overview: The aim of the course is to present a range of major theory and theorems, both important and elegant in themselves and with important applications within topology and to mathematics as a whole. Central to the course is the general theory of compactness, compactifications and Tychonoff's theorem, one of the most important in all mathematics (with applications across mathematics and in mathematical logic) and computer science.
Course Synopsis: Separation axioms. Subbases. Lindelöf and countably compact spaces. Seperable spaces. Filters and ultrafilters. Convergence in terms of filters. Tychonoff's theorem. Compactifications, in particular the Alexandroff One-Point Compactification and the Stone-Čech Compactification. Proper maps. Completeness, connectedness and local connectedness. Components and quasi-components. Urysohn's metrization theorem. Paracompactness. Stone's Theorem; that metric spaces are paracompact. Totally disconnected compact spaces and Stone duality.
