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1. Some set theory
2. Metric spaces
3. Open sets
4. Basis, subbasis, subspace
5. Closed sets
6. Continuous functions
7. Connectedness
8. Path connectedness
9. Separation axioms
10. Urysohn lemma
11. Tietze extension theorem
12. Urysohn metrization theorem
13. Metrization of manifolds
14. Compact spaces
15. Heine-Borel theorem
16. Compact metric spaces
17. Tychonoff theorem
18. Compactification
19. Quotient spaces
7. Connectedness
Notes
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Videos
Part 1 (page 60)
Part 2 (page 61)
Part 3 (pages 62-63)
Part 4 (page 64)
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Page 60: Motivation. Definition of connected space.
Page 61: Connectedness of intervals.
Page 62: Connectedness and homeomorphisms. Intermediate value theorem. Topological invariants.
Page 63: Further properties of connected spaces.
Page 64: Connected components.
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6. Continuous functions
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8. Path connectedness