Czech

2

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

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

Not applicable.

The course is taught through lectures explaining the basic principles and theory of the discipline.

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.

1. Closure operators, open and closed sets
2. Continuous mappings and homeomorphisms
3. Closure operators in geometry, algebra and logic
4. Topological spaces, neighbourhoods, báses and subbases
5. Separation axioms
6. Convergence
7. Metric and metrizable spaces
8. Compact spaces
9. Connected spaces
10.Applications of topological spaces in computer science (digital topology)

Not applicable.

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 attendance at seminars is required and will be checked regularly by the teacher supervising a seminar. If a student misses a seminar due to excused absence, he or she will receive problems to work on at home and catch up with the lessons missed.

Not applicable.

Not applicable.

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)

20 hours, optionally

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