Course: Topology 1

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Course title Topology 1
Course code KAG/T1
Organizational form of instruction Lecture + Exercise
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 3
Language of instruction Czech
Status of course Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Vanžurová Alena, doc. RNDr. CSc.
Course content
1. Structures on sets. 2. Topological structures, open sets, interior, exterior, boundary points, bounded sets, base, subbase, Hausdorff space, first and second countability, continuous mappings, examples of topological structures, subspaces. 3. Structures on Euclidean spaces, topology of Euclidean spaces, examples of open sets, epsilon-delta definition of a continuous function, examples of continuous and non-continuous mappings. 4. Comparison of topologies, final and initial topologies, product topology, quotient topology, examples: factorization of the squere. 5. Metric topology, open ball, properties of the metric topology, bopunded sets. 6. Compact topological spaces, continuous mappings of compact spaces, extrema of continuous functions, examples: criterion of compactness in Euclidean spaces, spheres. 7. Connected spaces, examples of connected and non-connected spaces. 8. Applications: Topological groups, topological vector spaces, manifolds.

Learning activities and teaching methods
Lecture, Work with Text (with Book, Textbook)
Learning outcomes
Generalization of properties of metric spaces to topological spaces, metrizability, continuous mappings.
1. Knowledge Define basic topological concepts
Prerequisites
Principles of the set theory and metric spaces.

Assessment methods and criteria
Mark, Oral exam

Recommended literature
  • Engelking R. (1977). General Topology. Warszawa.
  • Kolomogorov, Fomin. (1975). Úvod do teorie funkcí a funkcionální analýzy. SNTL Praha.
  • Krupka D., Krupková O. (1990). Topologie a geometrie. SPN Praha.
  • Štěrbová, M. (1989). Úvod do obecné topologie. UP Olomouc.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Teaching Training in Descriptive Geometry for Secondary Schools (2015) Pedagogy, teacher training and social care 1 Winter
Faculty of Science Mathematics (1) Mathematics courses 3 Winter
Faculty of Science Mathematics and Applications (1) Mathematics courses 3 Winter
Faculty of Science Discrete Mathematics (2015) Mathematics courses 2 Winter
Faculty of Science Applied Mathematics (2014) Mathematics courses 1 Winter