\begin{definition}[Definition:Completely Hausdorff Space/Definition 1]
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}}:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$ 
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.
{{NamedforDef|Felix Hausdorff}}
\end{definition}