Topologie: Grundlagen
Introduction to Topology

Set topology, topological spaces, metric spaces, connectedness, neighborhood bases, countability axioms, separation axioms, convergence, continuity, compactness, Theorem of Tychonov, Theorem of Uryson, homotopy equivalence, fundamental group

At the end of the module, students are able to analyse topological spaces with regard to topological properties like connectedness and compactness. They are able to make competent judgements about the fundamental groups of simple topological spaces. They are able to apply the standard notions of topology in various contexts and to prove some simple results in the field presented in the course.

MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra 1, MA1102 Linear Algebra 2

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercises will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. The practice sessions will focus on presentation of proofs, visualization of concepts, working out sample solutions, and devising problems reflecting the content of the lecture.

Tafelarbeit

K. Jänich, Topologie, Springer.
J. R. Munkres, Topology, Prentice Hall.
B. v. Querenburg, Mengentheoretische Topologie, Springer.
G. Laures, M. Szymik, Grundkurs Topologie, Springer.

Die Prüfungsleistung besteht aus einer Klausur (90 Minuten). In dieser Klausur soll an Beispielen gezeigt werden, dass die Definitionen und Aussagen der Vorlesung ohne Hilfsmittel reproduziert, überprüft und angewandt werden können. Teilweise werden bei Fragestellungen nur Rechenergebnisse oder die Reproduktion einer Aussage verlangt, teilweise soll eine vollständige Argumentationskette oder ein kürzerer Beweis geführt werden.