Topology WM-MA-TP

Metric space.
Metric space, metric, ball, open and closed sets, topology, interior and closure of a set, dense and co-dense sets, bounded sets, limits of sequences. Base for a topological space, equivalent metrics.
Operations on metric spaces.
Subspaces, Cartesian products.
Continuous mappings.
Definition of a continuous mapping, equivalent conditions. Homeomorphisms. Notion of a topological property.
Totally bounded spaces and complete spaces.
Definition of a totally bonded metric space and of a complete metric space, Cantor theorem and Baire theorem. Banach fixed-point theorem.
Compact spaces.
Definition of a compact space, convergence in compact metric spaces, subspaces and Cartesian products of compact spaces, images of compact spaces under continuous mappings.
Connected spaces.
Definition of a connected space. Cartesian products of connected spaces. Images of connected spaces under continuous mappings.
Cardinal functions.
Separable, second-countable and Lindelof metric spaces.