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Lecturer(s)
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Peška Patrik, RNDr. Ph.D.
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Mikeš Josef, prof. RNDr. DrSc.
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Course content
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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.
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Learning activities and teaching methods
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Lecture, Work with Text (with Book, Textbook), Activating (Simulations, Games, Dramatization)
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Learning outcomes
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The aim of the course is to introduce the fundamental concepts and methods of Topology, focusing on properties of topological and metric spaces and their role in analysis and geometry. Students will: - understand the concept of a topological space and topological structures, - work with open and closed sets, closure and interior, - understand continuity in both topological and metric settings, - study constructions of topologies (product, quotient, initial and final topologies), - understand compactness and connectedness and their consequences, - learn metric topology and its properties, - gain insight into applications in topological groups, vector spaces, and manifolds.
After completing the course, the student: Knowledge: - understands the basic concepts of Topology and metric topology, - knows properties of open and closed sets, compactness, and connectedness, - understands constructions of topological spaces (product, quotient, subspaces), - understands the role of topology in other areas of mathematics. Skills: - works with topological structures and compares topologies, - determines closure, interior, and boundary of sets, - verifies continuity of mappings in different settings, - applies criteria of compactness and connectedness, - works with metric spaces and their properties. Competences: - is capable of abstract mathematical thinking, - translates intuitive geometric ideas into formal language, - understands the importance of topological methods in analysis and geometry, - is prepared for further study in advanced areas (e.g. differential geometry).
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Prerequisites
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Knowledge of Mathematical Analysis (in particular sequences, limits, and continuity) and Linear Algebra is assumed. Basic familiarity with Analytic Geometry is an advantage.
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Assessment methods and criteria
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Mark, Oral exam, Student performance
To successfully complete the course, the student must: - actively participate in lectures and/or tutorials, - complete assigned tasks during the semester, - demonstrate understanding of the subject, - be able to solve both practical and theoretical problems, - pass the final exam (or obtain course credit, if applicable).
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Recommended literature
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Engelking R. (1977). General Topology. Warszawa.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. Olomouc.
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Kelley J. L. (2017). General Topology. Dover Books on Mathematics.
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Kolomogorov, Fomin. (1975). Úvod do teorie funkcí a funkcionální analýzy. Praha.
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Krupka D., Krupková O. (1990). Topologie a geometrie. Praha.
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Matoušek, M. (2005). Úvod do topologie. Praha.
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Pultr, A. (1982). Úvod do topologie a geometire I.. Praha.
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Štěrbová, M. (1989). Úvod do obecné topologie. Olomouc.
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Weintraub S. H. (2014). Fundamentals of Algebraic Topology.
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