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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