Definition:Urysohn Space

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

$\left({X, \vartheta}\right)$ is an Urysohn space or $T_{2 \frac 1 2}$ space iff:
 * $\forall x, y \in X: x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U^- \cap V^- = \varnothing$

That is, for any two points $x, y \in X$ there exist open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively whose closures are disjoint.

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

Variants of Name
Earlier (pre-1970) treatment of this subject tends to refer to this as a completely Hausdorff space, and what we define as a completely Hausdorff space as an Urysohn space.