Let $X$ be a uniform space with uniformity $\mathcal{U}$ . $X$ is said to be separated or Hausdorff if it satisfies the following separation axiom: $$\bigcap \mathcal{U}=\Delta,$$ where $\Delta$ is the diagonal relation on $X$ and $\bigcap \mathcal{U}$ is the intersection of all elements (entourages) in $\mathcal{U}$ . Since $\Delta\subseteq \bigcap \mathcal{U}$ , the separation axiom says that the only elements that belong to every entourage of $\mathcal{U}$ are precisely the diagonal elements $(x,x)$ . Equivalently, if $x\ne y$ , then there is an entourage $U$ such that $(x,y)\notin U$ .

The reason for calling $X$ separated has to do with the following assertion:

$X$ is separated iff $X$ is a Hausdorff space under the topology $T_{\mathcal{U}}$ induced by $\mathcal{U}$ .

Recall that $T_{\mathcal{U}}=\lbrace A\subseteq X\mid \mbox{for each }x\in A\mbox{, there is }U\in \mathcal{U}\mbox{, such that }U[x]\subseteq A\rbrace$ , where $U[x]$ is some uniform neighborhood of $x$ where, under $T_{\mathcal{U}}$ , $U[x]$ is also a neighborhood of $x$ . To say that $X$ is Hausdorff under $T_{\mathcal{U}}$ is the same as saying every pair of distinct points in $X$ have disjoint uniform neighborhoods.