M6140 Topology

Faculty of Science
Spring 2009
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 M3,01023
  • Timetable of Seminar Groups:
M6140/01: Tue 13:00–13:50 M3,01023
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. Main objectives can be summarized as follows: to master the notions of topological and uniform space; to understand basic properties of topological spaces, in particular separability, connectedness and compactness; to learn about the behaviour of continuous real-valued functions on topological spaces; to familiarize 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. 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
Lecture 2 hours a week, seminar 1 hour a week. Examination written and oral.
Language of instruction
Czech
Further Comments
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 2006, Spring 2007, Spring 2008, 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 2009, recent)
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