Section 34: The Urysohn Metrization Theorem

1. A family of continuous functions $\{f_\alpha\}$ separates points from closed sets if for every closed set and a point not in it there is a function in the family such that it is constant on the closed set and takes a different value at the point.
2. The Imbedding Theorem: If $X$ is $T_1$ and $\{f_\alpha\}_{\alpha\in J}$ separates points and closed sets in $X$ then $(f_\alpha(x))_{\alpha\in J}$ is an embedding of $X$ in $\mathbb{R}^J$ . If all functions map $X$ to $[0,1]$ then the embedding is in $[0,1]^J$ .
3. A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^J$ for some $J$ .
4. $\Rightarrow$ The Imbedding Theorem. $\Leftarrow$ A subspace of the product of completely regular spaces is completely regular.
5. The Urysohn Metrization Theorem: Regular + Second-countable $\Rightarrow$ Metrizable.
6. regular + second-countable implies normal + second-countable implies for a countable collection of pairs of basis neighborhoods such that $\overline{B}_m\subseteq B_n$ we can find a continuous function $g_{n,m}$ equal 1 inside $\overline{B}_m$ and 0 outside $B_n$ : this is a countable family of continuous functions that separates points from closed sets.
7. A regular second-countable space is homeomorphic to a subspace of the infinite-dimensional euclidean space $\mathbb{R}^\omega$ .
8. Some facts about metrization that follow:
9. A second-countable space is metrizable iff it is regular.
10. A compact Hausdorff space is metrizable iff it is second-countable.
11. A compact Hausdorff space that is the union of two closed metrizable subspaces is metrizable.
12. The one-point compactification of a locally compact Hausdorff space $X$ is metrizable iff $X$ is second-countable.
13. So, if a locally compact Hausdorff space is second-countable then it is metrizable, but not vice versa: discrete uncountable.
14. A locally metrizable space is a space such that every point has a metrizable neighborhood (in the subspace topology).
15. A compact Hausdorff space is metrizable iff it is locally metrizable.
16. A Lindelöf regular space is metrizable iff it is locally metrizable.
17. $\mathbb{R}_K$ is Lindelöf Hausdorff and locally metrizable, yet not metrizable.