PřF:M6140 Topology - Course Information
M6140 TopologyFaculty of Science
- Extent and Intensity
- 2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
- prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Dr. Michael Lieberman, PhD. (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
- Mon 20. 2. to Mon 22. 5. Tue 14:00–15:50 M6,01011
- Timetable of Seminar Groups:
- 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, B-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.
- 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
- 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
- Teaching methods
- Lectures: theoretical explanation with examples of applications
Exercises: solving theoretical problems focused on practising basic concepts and theorems
- Assessment methods
- Examination oral.
- Language of instruction
- Further Comments
- Study Materials
The course is taught annually.