General Topology [SMATB216]

Learning outcomes

General topology is roughly speaking "the geometry of analysis", allowing to adapt the notions of continuity and convergence to the needs of a mathematical problem. This course is an introduction to the basic notions of topology, which are essential to be able to attend a course of functional analysis.

Content

The course has seven chapters: Fundamental definitions and properties (open and closed sets, neighborhoods, convergent sequences). Bases, sub-bases and local bases (fundamental neighborhoods)(fundaments of a topology). Continuous Functions (global, local, and sequential continuity) and equivalent topological spaces. Countability (working with sequences) and separation (unique limits). Compactness (open cover and extraction of a finite subcover) (Alexander's theorem and Heine-Borel theorem, properties, compactness in R^{n}) and sequential compactness (extraction of a convergent subsequence). Product spaces (product topology, Tychonoff's product compactness theorem). Metric spaces (metric topology, compactness in metric spaces (compactness equivalent to sequential compactness), complete metric spaces (compact = closed and totally bounded).