Definition:Completely Hausdorff Space/Definition 2

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 :
 * $\forall x, y \in S, x \ne y : \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$

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

Also see

 * Equivalence of Definitions of Completely Hausdorff Space