## M6140 Topology

Faculty of Science
Spring 2008
Extent and Intensity
2/1/0. 3 credit(s) (fasci plus compl plus > 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
Timetable
Tue 11:00–12:50 UM
• Timetable of Seminar Groups:
M6140/01: Tue 13:00–13:50 UM, M. Kunc
Prerequisites
M3100 Mathematical Analysis III
Mathematical analysis: continuous functions, metric spaces
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
Course objectives
The course presents one of basic disciplines of modern mathematics. It naturally follows the well-known concepts of a metric space and a continuous function. It introduces topological spaces and presents their basic properties, in particular separability, connectedness and compactness. There are considered real valued continuous functions on topological spaces as well. Finally, there is proved Brouwer's fix-point theorem and there is shown how the fundamental group provides a simple proof of 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
Assessment methods (in Czech)
Výuka: přednáška (účast nepovinná), Zkouška: písemná a ústní
Language of instruction
Czech