Section 32: Normal Spaces

1. Hausdorff + Compact $\Rightarrow$ Normal
2. Hausdorff + Locally Compact $\Rightarrow$ Regular (Completely Regular: exercise 7 of §33).
3. Compact implies a closed subset is compact; in a Hausdorff space two compacts can be separated.
4. Regular + Lindelöf $\Rightarrow$ Normal
5. Cover each of two closed sets with a countable collection of open sets (Lindelöf) such that their closures do not intersect the other set (regular); subtract from nth open set the union of closures of all open sets from 1 to n covering the other set.
6. Metric $\Rightarrow$ Normal
7. Cover each point in two disjoint closed sets with a ball such that if your double its radius it still does not intersect the other set. That’s the beauty of a metric!
8. Ordered $\Rightarrow$ Normal (in the order topology)
9. The product of two ordered (even well-ordered) spaces need NOT be normal: $S_\Omega\times\overline{S}_\Omega$ is not normal.
10. Well-ordered: (a,b]=(a,b+1) are open and form a basis, cover each closed set with such intervals that do not intersect the other set.
11. General case (ordered): covered, for example, in Steen, Seebach, Counterexample 39, 1-6.
12. A completely normal space is a space such that every its subspace is normal.
13. A pair of subsets is separated (wtf? what is not called separated or separation?) if they are disjoint and neither one contains a limit point of the other.
14. A space is completely normal iff every pair of separated subsets can be separated by neighborhoods.
15. Subspaces and complete normality:
16. A subspace of a completely normal space is completely normal.
17. Products and complete normality:
18. The product of even two completely normal spaces needs NOT to be normal.
19. Completely normal spaces:
20. a regular second-countable space
21. a metric space
22. an ordered space