Course detail

Applied Topology

FSI-9APTAcad. year: 2020/2021

In the course, the students will be taught fundamentals of the general 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

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977. (CS)
E. Čech, Topological spaces, in: Topological Papers of Eduard Čech, ch. 28, Academia, Prague, 1968, 436 - 472. (EN)
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)
N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989. (EN)
E. Čech, Topological spaces (Revised by Z. Frolík mand M. Katětov), Academia, Prague, 1966. (EN)
J.L.Kelly, General Topology, Springer-Verlag, 1975. (EN)
R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977. (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)

Planned learning activities and teaching methods

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.

Language of instruction

Czech, English

Work placements

Not applicable.

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.

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

The attendance of lectures is not compulsory and, therefore, it will not be checked.

Classification of course in study plans

  • Programme D-APM-K Doctoral, 1. year of study, summer semester, 0 credits, recommended

  • Programme D4P-P Doctoral

    branch D-APM , 1. year of study, summer semester, 0 credits, 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