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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
11. Tietze extension theorem
Notes
Complete lecture notes
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Videos
Part 1 (page 82)
Part 2 (page 83)
Part 3 (page 84)
Part 4 (page 85)
Part 5 (page 86)
Part 6 (page 87)
Page links
Page 82: Introduction.
Page 83: Sequences of functions.
Page 84: Limits of uniformly convergent sequences.
Page 85: Approximated extension of a function.
Page 86: Proof of the Tietze extension theorem.
Page 87: Tietze extension theorem for unbounded functions.
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10. Urysohn lemma
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12. Urysohn metrization theorem