Topology of Metric Spaces

Introduce the basic concepts of topology and their application to the study of functional spaces.

This lesson will cover the following topics:

- Metric and topological spaces: definition, limits, and continuity. Open sets, closed sets, neighborhoods. Interior and adherence of a subset, density. Product topology and quotient topology.

- Connectedness: definition, connected components of R. Continuous image of a connected component. Arc-connectedness, convexity in a normed vector space. Connected components

- Compactness: definition. Compact sets in^Rn. Continuous image of a compact set. Bolzano–Weierstrass theorem. Ascoli’s theorem.

- Completeness: Cauchy sequences in a metric space; definition of a complete metric space. Extension of mappings; a metric space is complete. Fixed-point theorem.

- Banach and Hilbert spaces: definition, the finite-dimensional case. Continuous linear maps, topological duality. Examples:^Lp and^C0 spaces. Hilbert spaces, projection onto a closed convex set, duality.

The analysis courses in the first and second years of the bachelor’s program and the first semester of the third year, specifically:

- HAX404X Topology of^{R^n} and Functions of Several Variables

- HAX502X Differential Calculus and Differential Equations

Recommended prerequisites: first semester of the third year of a bachelor's degree program

Hourly volumes:

            CM: 31.5

            TD: 31.5

            Practical Work: -

            Land: -