Section 31: The Separation Axioms

1. Regular space: a $T_1$ -space such that a closed subset and a point not in it can be separated by two open sets.
2. A space is regular iff it is $T_1$ and any neighborhood of a point contains the closure of a neighborhood of the point.
3. Normal space: a $T_1$ -space such that any two closed disjoint subsets can be separated by two open neighborhoods.
4. A space is normal iff it is $T_1$ and any neighborhood of a closed set contains the closure of a neighborhood of the set.
5. Refining of topology and separation axioms:
6. Neither regularity nor normality of the space is refining-preserved property.
7. Subspaces and separation axioms:
8. A subspace of a regular space is regular.
9. A subspace of a normal space need NOT be normal.
10. A closed subspace of a normal space is normal.
11. Products and separation axioms:
12. The product of regular spaces is regular.
13. The product of even two normal spaces need NOT be normal.
14. If the product of any number of spaces is Hausdorff (regular, normal), so is each space in the product.
15. Continuous functions and separation axioms:
16. A closed continuous image of a normal space is normal.
17. The image of a *-space under a perfect map (see §26) is a *-space, where * stands for either one:
18. Hausdorff
19. regular
20. locally compact
21. second-countable
22. Topological groups and separation axioms: let $G$ be a topological group, $X$ be a space, then
23. An action $\alpha$ of $G$ on $X$ is a map $G\cdot X\to X$ such that 0) it is continuous, i) $e\cdot x=x$ (do nothing), ii) $g\cdot(g'\cdot x)=(g\cdot g')\cdot x$ .
24. The orbit space of the action $\alpha$ is the quotient space denoted by $X|G$ given by $x~g\cdot x$ , $g\in G$ .
25. If $G$ is compact then the quotient map is a perfect map.
26. If $G$ is compact and $X$ is Hausdorff (regular, normal, locally compact, second-countable), so is $X|G$ .