Course detail

Applied Topology

FSI-9APTAcad. year: 2016/2017Summer semesterNot applicable.. year of study2  credits

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

Learning outcomes of the course unit

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.

Recommended optional programme components

Not applicable.

Recommended or required reading

E. Čech, Topological spaces (Revised by Z. Frolík mand M. Katětov), Academia, Prague, 1966.
J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977.
E. Čech, Topological spaces, in: Topological Papers of Eduard Čech, ch. 28, Academia, Prague, 1968, 436 - 472.
E. Čech, Topologické prostory, Nakladatelství ČSAV, Praha, 1959.
R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977.
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
N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989.
J.L.Kelly, General Topology, Springer-Verlag, 1975.
N.M.Martin and S. Pollard,Closure Spacers and Logic, Kluwer Acad. Publ., Dordrecht, 1996.
S. Vickers, Topology Via Logic, Cambridge University Press, New York, 1989.
R.W. Hall, G.T. Hermann, Y. Kong and R. Kopperman, Digital Topology (Theory and Applications), Springer, 2006

Planned learning activities and teaching methods

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

Assesment methods and criteria linked to learning outcomes

The exam will consist of a written part and an oral one.

Language of instruction

Czech, English

Work placements

Not applicable.

Course curriculum

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)


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.

Specification of controlled education, way of implementation and compensation for absences

Since it is not obligatory for studentsn to be present at lectures, the presence will not be checked.

Type of course unit



20 hours, optionally

Teacher / Lecturer


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