\begin{definition}[Definition:Perfectly Normal Space]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''perfectly normal space''' {{iff}}:
:$\struct {S, \tau}$ is a perfectly $T_4$ space
:$\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
That is:
:Every closed set in $T$ is a $G_\delta$ set.
:$\forall x, y \in S$, ''both'':
::$\exists U \in \tau: x \in U, y \notin U$
::$\exists V \in \tau: y \in V, x \notin V$
\end{definition}