\begin{definition}[Definition:Completely Hausdorff Space/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''completely Hausdorff space''' or '''$T_{2 \frac 1 2}$ space''' {{iff}}:
:$\forall x, y \in S, x \ne y : \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x^- \cap N_y^- = \O$
That is:
:$\struct {S, \tau}$ is a '''$T_{2 \frac 1 2}$ space''' {{iff}} every two points in $S$ are separated by closed neighborhoods.
{{NamedforDef|Felix Hausdorff}}
\end{definition}