Proposition 5.24.3.label Let $X$ be a $T_{1}$ space, then the following are equivalent:
- (1)
$X$ is completely regular.
- (2)
There exists $\cf \subset C(X; [0, 1])$ that separates points and closed sets.
Proof. (2) $\Rightarrow$ (1): Let $E \subset X$ closed and $x \in X \setminus E$, then there exists $f \in \cf$ such that $x \not\in \ol{f(E)}$. By Urysohn’s lemma, there exists $\phi \in C([0, 1]; [0, 1])$ such that $\phi(f(x)) = 1$ and $\phi(f(E)) = 0$.$\square$
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