## M6140 Topology

Faculty of Science
Autumn 2020
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Taught partially online.
Teacher(s)
prof. RNDr. Jiří Rosický, DrSc. (lecturer)
Mgr. Jan Jurka (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
Timetable
Wed 12:00–13:50 M4,01024
• Timetable of Seminar Groups:
M6140/01: Wed 18:00–18:50 M3,01023, J. Jurka
Prerequisites
M3100 Mathematical Analysis III || M3100F 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. Introduces topological spaces which naturally generalize the well-known concepts of a metric space and a continuous function. Presents the separation axioms, the concepts of connectedness and compactness. Explains the concept of homotopy and introduces the fundamental group including its use. Finally, it presents uniform spaces and uniformly continuous functions.
Learning outcomes
Understanding the concept of continuity formalized by means of topological and uniform spaces;
grasping the concepts of separation, connectedness and compactness;
ability to see the topological background of the theory of continuous real-valued functions and metric spaces;
familiarity with the concept of homotopy, including the fundamental group and its use for proving Brouwer's fix-point theorem and 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
required literature
• L. Vokřínek, Topologie
recommended literature
• PULTR, Aleš. Podprostory euklidovských prostorů. 1. vyd. Praha: Státní nakladatelství technické literatury, 1986. 253 s. info
• PULTR, Aleš. Úvod do topologie a geometrie. 1. 1. vyd. Praha: Státní pedagogické nakladatelství, 1982. 231 s. info
Teaching methods
On-line lectures: theoretical explanation with examples of applications
Exercises: solving theoretical problems focused on practising basic concepts and theorems
Assessment methods
Course ends by an oral exam. Presence at the course recommended. Homeworks are given, handed in exercises.
Language of instruction
Czech