Topology I

This is an introduction to set-theoretic and algebraic topology. The discussion of set-theoretic topology will be kept rather short covering the following topics:

topological spaces and continuous maps; (categorical) constructions of topological spaces;

(path-)connectedness; separation axioms: Hausdorff, regular, normal; countability axioms; Urysohn’s Lemma, Tietze’s Extension Theorem; Urysohn’s Metrization Theorem; (possibly: the Alexandroff and the Stone–compactification.)

The bulk of the course focuses on algebraic topology; specifically:

homotopy the fundamental group 𝜋1(𝑋,∗), the Seifert-van Kampen Theorem, the theory of covering spaces; first steps in (singular) (co)homology, possibly including the Eilenberg–Steenrod axioms, Mayer–Vietoris, etc.

Semester: SoSe 2024