Course detail
Applied Topology
FSI-9APTAcad. year: 2024/2025
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
Mode of study
Not applicable.
Guarantor
Department
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.
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
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)
J.L.Kelly, General Topology, Springer-Verlag, 1975. (EN)
N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989. (EN)
N.M.Martin and S. Pollard,Closure Spacers and Logic, Kluwer Acad. Publ., Dordrecht, 1996. (EN)
R.W. Hall, G.T. Hermann, Y. Kong and R. Kopperman, Digital Topology (Theory and Applications), Springer, 2006 (EN)
S. Vickers, Topology Via Logic, Cambridge University Press, New York, 1989. (EN)
J.L.Kelly, General Topology, Springer-Verlag, 1975. (EN)
N. Bourbali, Elements of Mathematics - General Topology, Chap. 1-4, Springer-Verlag, Berlin, 1989. (EN)
N.M.Martin and S. Pollard,Closure Spacers and Logic, Kluwer Acad. Publ., Dordrecht, 1996. (EN)
R.W. Hall, G.T. Hermann, Y. Kong and R. Kopperman, Digital Topology (Theory and Applications), Springer, 2006 (EN)
S. Vickers, Topology Via Logic, Cambridge University Press, New York, 1989. (EN)
Recommended reading
E. Čech, Topological spaces (Revised by Z. Frolík mand M. Katětov), Academia, Prague, 1966. (EN)
E. Čech, Topologické prostory, Nakladatelství ČSAV, Praha, 1959. (CS)
J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977. (CS)
R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977. (EN)
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, Topologické prostory, Nakladatelství ČSAV, Praha, 1959. (CS)
J. Adámek, V. Koubek a J. Reiterman, Základy obecné topologie, SNTL, Praha, 1977. (CS)
R. Engelking, General Topology,Panstwowe Wydawnictwo Naukowe, Warszawa, 1977. (EN)
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)
Classification of course in study plans
Type of course unit
Lecture
20 hod., 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
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