General topology

All lectures: All lectures at the Academy of Intelligence on the discipline: general topology. These fascinating lectures provide us with the opportunity to explore the deep principles and phenomena underlying our knowledge and our desire to understand the world.
1. Introduction to General Topology, Branches of Topology
2. 1. Algebra of Sets. Mappings of Sets. Equivalence Relation. Ordering, Linear Ordering, Well-Ordering
3. 2. Topological Spaces. Base of a Topology. Subbase of a Topology. Topology of a Linear Order. Metric Spaces. Normed Spaces. Topology of a Metric Space. Metrizable Spaces.
4. Closure, interior, and boundary of a subset. Neighborhood of a subset. Point of contact. Limit, interior, and boundary points. Continuous mappings of topological spaces. Homeomorphism.
5. 4. Weak (initial) topology with respect to a family of maps. Cartesian product of a family of sets. Tychonoff product topology. Products of maps. The Cantor perfect set.
6. 5. Final topology with respect to a family of mappings. Sum of spaces and sums of mappings. Quotient space. Quotient mapping.
7. 6. Countability axioms (character, weight of a space). Separability (density of a space). Separation axioms. Normal spaces. Urysohn's lemma
8. 7. The Brouwer–Tietze–Urysohn extension theorem for functions. An example of a regular non-normal space. Peano curve. Tychonoff spaces. Metrizable spaces.
9. 8. Urysohn's Metrization Theorem. Topological Properties Defined by Covers. Compact, Finally Compact, and Paracompact Spaces.
10. 9. The shrinking lemma. Partition of unity. Compact spaces. Alexander's lemma. The First Tychonoff Theorem. Compactifications.
11. Lecture 10. Stone-Čech Compactification. Local Compactness. Alexandroff One-Point Compactification. Metrizable Compact Spaces.
12. 11. Complete metric spaces. Baire category theorem. Totally bounded metric spaces. Criterion for compactness of a metrizable space in terms of metrics
13. 12. The uniform metric on a product. Spaces of mappings. Topologies of uniform and pointwise convergence.
14. 13. The Stone-Weierstrass theorem. Compact-open topology on C(X, Y ). Maps X × Y → Z and X → C(Y, Z). Connectedness. Connected components
15. 14. Hereditary Disconnectedness. Zero-Dimensionality. Path-Connectedness. Homotopy.
16. 15. Homotopy equivalence. Contractible spaces. Fundamental group. The role of the base point for the fundamental group
17. 16. Induced homomorphism. The fundamental group is a homotopy invariant. Coverings. The fundamental group of the circle. Retraction of a closed disk onto the boundary circle
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