Proposition 5.24.5.label Let $X$ be a $T_{1}$ space, then the following are equivalent:
- (1)
$X$ is completely regular.
- (2)
There exists a uniformity $\fU$ on $X$ that induces the topology on $X$.
Proof. (1) $\Rightarrow$ (2): By Definition 6.1.17.
(2) $\Rightarrow$ (1): By Definition 5.24.4, $X$ embeds into $[0, 1]^{C(X; [0, 1])}$, which is a uniform space. The subspace uniformity on $X$ then induces the topology on $X$.$\square$
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