Theorem 5.24.6 (Uryson Metrisation Theorem).label Let $X$ be a second countable regular space, then $X$ is metrisable.
Proof. By Proposition 5.9.3, $X$ is normal. Let $\cb \subset 2^{X}$ be a countable base for $X$, and let
\[\mathcal{S}= \bracsn{(E, F) \in \mathcal{B}^2 | \ol{E} \subset F}\]
By Urysohn’s Lemma, for each $(E, F) \in \mathcal{S}$, there exists $f_{EF}\in C(X; [0, 1])$ such that $f|_{E} = 1$ and $f|_{F^c}= 0$. For any $x \in X$ and $U \in \cn^{o}_{X}(x)$, there exists $E, F \in \mathcal{B}$ such that $x \in E \subset \ol{E}\subset F \subset U$. Thus $f_{EF}(x) = 1$ and $f_{EF}|_{U^c}= 0$. Therefore
\[\cf = \bracsn{f_{EF}|(E, F) \in \mathcal{S}}\subset C(X; [0, 1])\]
is a countable family of continuous functions that separate points and closed sets. By Definition 5.24.4, $X$ embeds into $[0, 1]^{\cf}$, which is metrisable by Proposition 8.1.3.$\square$
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