# PřF:M6140 Topology - Course Information

## M6140 Topology

**Faculty of Science**

Autumn 2024

**Extent and Intensity**- 2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
**Teacher(s)**- doc. Lukáš Vokřínek, PhD. (lecturer)
**Guaranteed by**- doc. Lukáš Vokřínek, PhD.

Department of Mathematics and Statistics – Departments – Faculty of Science

Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science **Prerequisites**-
**M3100**Mathematical Analysis III ||**M3100F**Mathematical Analysis III

Mathematical analysis: metric spaces, continuous functions **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Mathematics (programme PřF, B-MA)

**Course objectives**- The course presents one of the basic disciplines of modern mathematics. Introduces topological spaces which naturally generalize the well-known concepts of a metric space and a continuous function. Presents the separation axioms, the concepts of connectedness and compactness. Explains the concept of homotopy and introduces the fundamental group including its use. Finally, it presents uniform spaces and uniformly continuous functions.
**Learning outcomes**- Understanding the concept of continuity formalized by means of topological and uniform spaces;

grasping the concepts of separation, connectedness and compactness;

ability to see the topological background of the theory of continuous real-valued functions and metric spaces;

familiarity with the concept of homotopy, including the fundamental group and its use for proving Brouwer's fix-point theorem and the fundamental theorem of algebra. **Syllabus**- 1. Topological spaces: definition, examples
- 2. Continuity: continuous maps, homeomorphisms
- 3. Basic topological constructions: subspaces, quotient spaces, products, sums
- 4. Separation axioms: T0-spaces, T1-spaces, Hausdorff spaces, regular spaces, normal spaces
- 5. Real-valued functions: completely regular spaces, Urysohn's lemma, Tietze's theorem
- 6. Compact spaces: compactness, basic properties, Tychonoff's theorem
- 7. Compactification: locally compact spaces, one-point compactification, Čech-Stone compactification
- 8. Connectedness: connected spaces, components, product of connected spaces, arcwise connected spaces, locally connected spaces, continua, 0-dimensional spaces
- 9. Uniform spaces: definition, basic properties, uniformly continuous maps, compact uniform spaces, metrizability, uniformizability
- 10. Homotopy: definition, basic properties, simply connected spaces, fundamental group, Brouwer's theorem in dimension 2, fundamental theorem of algebra
- 11. Brouwer's theorem: complexes, triangulation, Sperner's lemma, Brouwer's theorem

**Literature**- L. Vokřínek, Topologie

*required literature***Teaching methods**- Lectures: theoretical explanation based on given literature suplemented with examples and applications. It will be in presence or, in the case of need, on-line.

Exercises: solving theoretical problems focused on practising basic concepts and theorems **Assessment methods**- Course ends by an oral exam. Exams will be in presence or, in the case of need, online using Zoom. Presence at the course recommended. Homeworks are given, handed in exercises.
**Language of instruction**- Czech
**Further Comments**- The course is taught annually.

The course is taught: every week.

- Enrolment Statistics (recent)

- Permalink: https://is.muni.cz/course/sci/autumn2024/M6140