#
PřF:M6140 Topology - Course Information

## M6140 Topology

**Faculty of Science**

spring 2012 - acreditation

The information about the term spring 2012 - acreditation is not made public

**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. Mgr. Michal Kunc, Ph.D. (lecturer)
**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 **Prerequisites**-
**M3100**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, M-MA)
- Mathematics (programme PřF, N-MA)

**Course objectives**- The course presents one of the basic disciplines of modern mathematics. It naturally generalizes the well-known concepts of a metric space and a continuous function. After passing the course, students should: master the notions of topological and uniform space; understand basic properties of topological spaces, in particular separation axioms, connectedness and compactness; be able to reason about the behaviour of continuous real-valued functions on topological spaces; be familiar with a proof of Brouwer's fix-point theorem and with homotopy theory, including the use of fundamental groups to prove 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**- PULTR, Aleš.
*Podprostory euklidovských prostorů*. 1. vyd. Praha: Státní nakladatelství technické literatury, 1986. 253 s. info - CHVALINA, Jan.
*Obecná topologie*. 1. vyd. Brno: Rektorát UJEP, 1984. 193 s. info - PULTR, Aleš.
*Úvod do topologie a geometrie. 1*. 1. vyd. Praha: Státní pedagogické nakladatelství, 1982. 231 s. info

- PULTR, Aleš.
**Teaching methods**- Lectures: theoretical explanation with examples of applications

Exercises: solving theoretical problems focused on practising basic concepts and theorems **Assessment methods**- Examination written and oral.
**Language of instruction**- Czech
**Further Comments**- The course is taught annually.

The course is taught: every week.

- Enrolment Statistics (spring 2012 - acreditation, recent)
- Permalink: https://is.muni.cz/course/sci/spring2012-acreditation/M6140