M6140 Topology

Faculty of Science
Spring 2006
Extent and Intensity
2/1/0. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Guaranteed by
prof. RNDr. Jiří Rosický, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: prof. RNDr. Jiří Rosický, DrSc.
Timetable
Mon 12:00–13:50 N41
  • Timetable of Seminar Groups:
M6140/01: Mon 14:00–14:50 N41, J. Rosický
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 af a metric space and a continuous function. It introduces topological spaces and presents their basic properties connected with 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. Continuous maps: continuous maps, homeomorphisms 3. Subspaces and products: subspaces, products 4. Axioms of separability: Kuratowski spaces, Hausdorff spaces, regular spaces, normal spaces 5. Compact spaces: compactness, basic properties, Tychonoff's theorem 6. Connected spaces: connectedness, components, product of connected spaces, arcwise connected spaces, locally connected spaces, continua, Cantor discontinuum 7. Homotopies: definition, basic properties, simple connected spaces, fundamental group, Brouwer's theorem in dimension 2, fundamental theorem of algebra 8. Real valued functions: completely regular spaces, Urysohn's theorem, Tietze's theorem 9. Locally compact spaces: definition, basic properties, one-point compactification 10. Brouwer's theorem: complexes, triangulation, Sperner's lemma, Brouwer's theorem 11. Uniform spaces: definition, examples, uniform continuity, relations with topological spaces
Literature
  • PULTR, Aleš. Podprostory euklidovských prostorů. Vyd. 1. Praha: SNTL - Státní nakladatelství technické literatury, 1986, 253 s. info
  • CHVALINA, Jan. Obecná topologie. Vyd. 1. Brno: Univerzita J.E. Purkyně, 1984, 193 s. info
  • PULTR, Aleš. Úvod do topologie a geometrie. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1982, 231 s. info
Assessment methods (in Czech)
Výuka: přednáška, Zkouška: ústní
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Spring 2006, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2006/M6140