Course detail

# Applied Topology

In the course, the students will be taught fundamentals of the general topology with respec to applications in geometry, analysis, algebra and computer science.

Language of instruction

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Entry knowledge

All knowledge of the courses oriented on algebra and analysis that are taught in the bachelor's and master's study of Mathematical Engineering.

Rules for evaluation and completion of the course

Students are to pass an exam consisting of the written and oral parts. During the exam, their knowledge of the concepts introduced and of the basic propertief of these concepts will be assessed. Also their ability to use theoretic results for solving concrete problems will be evaluated.
The attendance of lectures is not compulsory and, therefore, it will not be checked.

Aims

The aim of the course is to make the students acquitant with basics of topology and with topological methods frequently used in other mathematical disciplines and in computer science.
The students will acquire knowledge of basic topological concepts and their properties and will understand the important role topology playes in mathematical analysis. They will also learn to solve simple topological problems and apply the results obtained into other mathematical disciplines and computer science

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

E. Čech, Topological spaces, in: Topological Papers of Eduard Čech, ch. 28, Academia, Prague, 1968, 436 - 472. (EN)
N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989. (EN)
J.L.Kelly, General Topology, Springer-Verlag, 1975. (EN)
N.M.Martin and S. Pollard,Closure Spacers and Logic, Kluwer Acad. Publ., Dordrecht, 1996. (EN)
S. Vickers, Topology Via Logic, Cambridge University Press, New York, 1989. (EN)
R.W. Hall, G.T. Hermann, Y. Kong and R. Kopperman, Digital Topology (Theory and Applications), Springer, 2006 (EN)

J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977. (CS)
E. Čech, Topologické prostory, Nakladatelství ČSAV, Praha, 1959. (CS)
T. Y. Kong and A. Rosenfeld, Digital topology: introduction and survey, Computer Vision, Graphics, and Image Processing 48(3), 1989, 357 - 393. Publisher Academic Press Professional, Inc. San Diego, CA, USA (EN)
E. Čech, Topological spaces (Revised by Z. Frolík mand M. Katětov), Academia, Prague, 1966. (EN)
R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977. (EN)

Classification of course in study plans

• Programme D-APM-K Doctoral, 1. year of study, summer semester, recommended
• Programme D-APM-P Doctoral, 1. year of study, summer semester, recommended

#### Type of course unit

Lecture

20 hours, optionally

Teacher / Lecturer

Syllabus

1. Basic concepts of set theory
2. Axiomatic system of closure operators
3. Čech closure operators
4. Continuous mappings
5. Kuratowski closure operators and topologies
6. Basic properties of topological spaces
7. Compactness and connectedness
8. Metric spaces
9. Closure operators in algebra and logic
10. Introduction to digital topology