#
PřF:M6140 Topology - Course Information

## M6140 Topology

**Faculty of Science**

Autumn 2020

**Extent and Intensity**- 2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).

Taught partially online. **Teacher(s)**- prof. RNDr. Jiří Rosický, DrSc. (lecturer)

Mgr. Jan Jurka (seminar tutor) **Guaranteed by**- prof. RNDr. Jiří Rosický, DrSc.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Timetable**- Wed 12:00–13:50 M4,01024
- Timetable of Seminar Groups:

*J. Jurka* **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**- On-line lectures: theoretical explanation with examples of applications

Exercises: solving theoretical problems focused on practising basic concepts and theorems **Assessment methods**- Course ends by an oral exam. Presence at the course recommended. Homeworks are given, handed in exercises.
**Language of instruction**- Czech
**Further Comments**- Study Materials

The course is taught annually.

- Enrolment Statistics (recent)

- Permalink: https://is.muni.cz/course/sci/autumn2020/M6140