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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
18. Compactification
Notes
Complete lecture notes
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Part 1 (pages 122-124)
Part 2 (pages 125-126)
Part 3 (pages 127-129)
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Page 122: Definition of a compactification.
Page 123: Some examples.
Page 124: Existence of compactifications.
Page 125: Ordering of compactifications.
Page 126: Maximality of the Cech-Stone compactification.
Page 127: One-point compactifications and locally Hausdorff spaces.
Page 128: Existence of one-point compactifications.
Page 129: Closing remarks.
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17. Tychonoff theorem
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19. Quotient spaces