Definition:Completely Hausdorff Space

Definition
Let $T = \struct {S, \tau}$ be a topological space.

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

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

Also known as
Some sources give this as $T_{\frac 5 2}$ space, which of course evaluates to the same as a $T_{2 \frac 1 2}$ space.