Topology

Part 1: set topology, topological spaces, metric spaces, neighbourhood bases, countability axioms, convergence, continuity, topological products, compact spaces, separation axioms, connected spaces, Theorem of Tychonov
Part 2: fundamental group: paths, multiplication of paths, homotopy of paths, fundamental group, fundamental group of a circle, if time permits also: Brouwer Fixed Point Theorem, free groups, Theorem of Seifert and Van Kampen, Introduction to homology

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 create bases and are able to make competent judgements about the fundamental groups and homology of simple topological spaces.

Interested bachelor students are welcome.
MA1001/MA0001 (Analysis 1) , MA1002/MA0002 (Analysis 2), MA1101/MA0004 (Linear Algebra and Discrete Structures 1), MA1102/MA0005 (Linear Algebra and Discrete Structures 2)

online lectures: live streaming and/or video; exercise module: student presentation, group work
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 motivate 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 exercise sheets and solutions 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. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

Online: Blackboard and video

Jänich: Topologie,
Laures, Szymik: Grundkurs Topologie
Bradley, Bryson, Terilla: Topology: A Categorical Approach
Hatcher: Algebraic Topology

The exam consists of one oral exam at the end of the semester, consisting of two parts. In the first part, students are assigned a small problem, for which they have 20 minutes. In the second part (25 minutes), they will be asked to explain their solution, and answer further questions about the content of the lecture.

It tests if the students are able to reproduce and verify definitions and main assertions introduced in the lecture and to apply them to specific examples. In part of the questions, students will be asked for the final result of a calculation or to merely state a particular example, in others they have to present a shorter proof, a complete calculation, or a more involved example.