COURSE AIMS AND OBJECTIVES: Introduce students to the structures of metric and topological spaces.
COURSE DESCRIPTION AND SYLLABUS (by weeks):
1. Basic and more complex examples from mathematical analysis and motivation for the concept of metric space;
2. Metric spaces. Examples, open and closed sets, equivalent metrics, continuous mappings;
3. Topological spaces. Topological structures, basis, subbasis, subspaces, product of spaces, quotion space, homeomorphism;
4. Hausdorff's spaces. Examples, properties, continuous mapping on compact space, compactness in Rn , uniform continuous mappings and compactness;
5. Connected spaces.
6. Complete metric spaces. Banach's theorem, Cantor's theorem, Baire's theorem, completeness of metric space;
7. Arzela-Ascolli's theorem.