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
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. 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
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.
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 2009, Spring 2010, Spring 2011, Spring 2012, 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.