Definition 5.1.17 (Separated). Let $(X, \fU)$ be a uniform space, then the following are equivalent:
$X$ is T0.
$X$ is T1.
$X$ is Hausdorff.
$X$ is regular.
$\Delta = \bigcap_{U \in \fU}U$.
If the above holds, then $X$ is separated.
Proof. $(1) \Rightarrow (5)$: Let $x, y \in X$ with $x \ne y$. Assume without loss of generality that there exists $U(x) \in \cn(x)$ such that $y \not\in U$. In which case, $(x, y) \not\in U$ and $\Delta \supset \bigcap_{U \in \fU}U$.
$(5) \Rightarrow (2)$: By Proposition 5.1.14, $\ol \Delta \subset \bigcap_{U \in \fU}\ol U = \Delta$, so $\ol \Delta$ is closed. By (6) of Definition 4.8.1, $X$ is Hausdorff.
$(1) \Rightarrow (4)$: $X$ is T1 and satisfies (2) of Definition 4.9.1 by Proposition 5.1.16, so $X$ is regular.
$(4) \Rightarrow (3) \Rightarrow (2) \Rightarrow (1)$: (T3) $\Rightarrow$ (T2) $\Rightarrow$ (T1) $\Rightarrow$ (T0).$\square$