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Course, academic year 2023/2024
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General Topology 1 - NMMA335
Title: Obecná topologie 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Class: M Bc. OM
M Bc. OM > Zaměření MA
M Bc. OM > Povinně volitelné
Classification: Mathematics > Topology and Category
Incompatibility : NMAT039
Interchangeability : NMAT039
Is incompatible with: NMMA345
Is interchangeable with: NMMA345, NMAT039
Annotation -
An elementary course in general topology for bachelor's program in General Mathematics. Recommended for specialization Mathematical Analysis.
Last update: G_M (16.05.2012)
Course completion requirements -

You need to have "zapocet" to take the exam. The exam is oral and its content is captured in the sylabus.

"Zapocet" is given for active participation on the seminar.

Last update: Vejnar Benjamin, doc. Mgr., Ph.D. (29.10.2019)
Literature -

R. Engelking, General Topology, PWN Warszawa 1977

J. L. Kelley, General Topology, D. Van Nostrand, New York 1957

J. Dugundji, Topology, Boston 1966 (1978)

J. I. Nagata, Modern General Topology, North-Holland 1985 (1968, 1975)

E. Čech, Topological Spaces, Academia, Praha 1966

Last update: Vejnar Benjamin, doc. Mgr., Ph.D. (29.10.2019)
Syllabus -

1. A concept of a topological space

Open and closed sets; interior and closure; neighborhood systems; base of topology, base of neighborhoods of a point; countable weight and countable character, separability; convergence of sequences and nets (* of filters) and Hausdorff spaces (* T_0-, T_1-spaces); continuous mappings; examples of metrizanle and non-metrizable spaces.

2. Operations on topological spaces

Subspace, sum and quotient, product; projective (weal, initial) generation, inductive (strong, final) generation; preserving properties; countable product of metrizable (completely metrizable, compact metrizable) spaces, Hulbert cube.

3. Completely regular spaces - embedding into a power of reals

Embedding lemma (diagonal product); complete regularity and its preserving by products and subspaces; embedding into Tichonov cube (power of reals); embedding of separable metrizable space into Hilbert cube (* metrizablity of T_3-spaces with countable base).

4. Normal spaces - extension of real-valued functions

Normal space and example of metrizable spaces; * counterexamples to preserving by subspaces and products; Urysohn lemma; Urysohn extension theorem; complete regularity of T_4-spaces.

5. Compact and Lindelof spaces

Definition by means of covers; characterization by means of nets (* filters, ultrafilters); preserving by continuous images, by closed subspaces; countable and sequential compactness; example of metrizable spaces; extremes and boundedness of real-valued functions; normality of Lindelof spaces; *product of Lindelof spaces that is not Lindelof.

6. Function spaces on compact sets

Space C(K); algebras and lattices of continuous functions; Stone-Weierstrass theore; consequences.

7. Tichonov theorem and Cech-Stone compactification, extension of meppings

Proof of compactness of products; compactness of Tichonov cube; compactifications; Cech-Stone compactification; extension of continuous mappings; * ultrafilters and beta-hull of N.

8. Cech-completeness and Baire theorem

Topological completeness of metrizable spaces; completion of metrizable space; Cech-completeness; examples of locally compact and completely metrizable spaces; Baire theorem; *uniform space and its completion.

9. Topological groups

Topological group; uniformities on topological groups; complete regularity.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (08.12.2017)
Entry requirements -

Knowledge of the theory of metric spaces.

Last update: Vejnar Benjamin, doc. Mgr., Ph.D. (29.10.2019)
 
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