Applied Topology (FSI-9APT)

Academic year 2022/2023
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
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.
Learning outcomes and competences:
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
Prerequisites:
All knowledge of the courses oriented on algebra and analysis that are taught in the bachelor's and master's study of Mathematical Engineering.
Course contents:
In the course, the students will be taught fundamentals of the general topology with respec to applications in geometry, analysis, algebra and computer science.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and methods of applied topology including examples.
Assesment methods and criteria linked to learning outcomes:
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.
Controlled participation in lessons:
The attendance of lectures is not compulsory and, therefore, it will not be checked.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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
Literature - fundamental:
2. E. Čech, Topological spaces, in: Topological Papers of Eduard Čech, ch. 28, Academia, Prague, 1968, 436 - 472.
4. N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989.
5. J.L.Kelly, General Topology, Springer-Verlag, 1975.
6. N.M.Martin and S. Pollard,Closure Spacers and Logic, Kluwer Acad. Publ., Dordrecht, 1996.
7. S. Vickers, Topology Via Logic, Cambridge University Press, New York, 1989.
8. R.W. Hall, G.T. Hermann, Y. Kong and R. Kopperman, Digital Topology (Theory and Applications), Springer, 2006
Literature - recommended:
1. J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977.
2. E. Čech, Topologické prostory, Nakladatelství ČSAV, Praha, 1959.
3. 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
4. E. Čech, Topological spaces (Revised by Z. Frolík mand M. Katětov), Academia, Prague, 1966.
5. R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 S
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 S