| Course title | Topology |
|---|---|
| Course code | KAG/TOP |
| Organizational form of instruction | Lecture + Lesson |
| Level of course | Master |
| Year of study | not specified |
| Semester | Winter |
| Number of ECTS credits | 5 |
| Language of instruction | Czech |
| Status of course | Compulsory |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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1. Structures on sets. 2. Topological structures, open sets, interior, exterior, closure, closed sets, base, subbase, Hausdorff spaces, first and second countable spaces, continuous mappings, examples of topological structures, subspaces. 3. Structures on Euclidean space, the topology on Euclidean space, examples of open sets, the definition of continuous mappings, examples of continuous and non-continuous mappings. 4. Comparison of topologies, final and initial topology, product topology, factor topology, examples. 5. Metric topology, open sphere, properties of metric topology, bounded sets. 6. Compact topological spaces, continuous mappings of compact topological spaces, extrema of continuous functions, examples: criteria of compactness in Euclidean spaces, spheres. 7. Connected spaces, examples - counterexamples. 8. Application: topological group, topological vector spaces, manifolds.
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| Learning activities and teaching methods |
| Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook) |
| Learning outcomes |
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Introduction to properties of topological spaces which are generalizations of metric spaces and play an important role in mathematical analysis.
1. Knowledge The student understands the basics of set topology and can illustrate the knowledge with simple examples. The student is capable to construct new topological spaces from known topological spaces. |
| Prerequisites |
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Preliminaries in the theory of sets and metric spaces.
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| Assessment methods and criteria |
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Mark, Oral exam
A student has to understand the basics of the subject and is capable to solve some practical tasks during the oral exam. |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Science | Study plan (Version): Mathematics (2023) | Category: Mathematics courses | 2 | Recommended year of study:2, Recommended semester: Winter |