\begin{definition}[Definition:Completely Normal Space]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''completely normal space''' {{iff}}:
:$\struct {S, \tau}$ is a $T_5$ space
:$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
That is:
:$\forall A, B \subseteq S, A^- \cap B = A \cap B^- = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$ 
:$\forall x, y \in S$, ''both'':
::$\exists U \in \tau: x \in U, y \notin U$
::$\exists V \in \tau: y \in V, x \notin V$
{{SeparationAxiomNameVariants||T5|$T_5$||completely normal}}
\end{definition}