Topologie

The scope of the course is to provide a general introduction to topological spaces and their properties. Moreover, we will in the end of the course introduce the fundamental group which gives a tool of determining when two topological spaces are the same (homeomorphic).

We will cover the following topics:

• Topological spaces and their basis.

• The order, product, subspace, quotient and metric topology.

• Closed sets and limit points.

• Continuous functions and homeomorphisms.

• Compact spaces.

• Connected spaces.

• Separation axioms: Hausdorff, normal, regular.

• Topological invariants.

• The Tychonoff theorem.

• The Arzela-Ascoli theorem.

• The Stone-Weierstrass theorem.

• Homotopy of paths.

• The fundamental group, functionality.