Definition:Completely Hausdorff Space

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

$\left({S, \tau}\right)$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space :
 * $\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \varnothing$

That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.

That is:
 * $\left({S, \tau}\right)$ is a $T_{2 \frac 1 2}$ space every two points in $S$ are separated by closed neighborhoods.