Definition:Perfectly T4 Space

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

$T$ is a perfectly $T_4$ space iff:
 * $(1): \quad T$ is a $T_4$ space
 * $(2): \quad$ Every closed set in $T$ is a $G_\delta$ set.

That is:
 * Every closed set in $T$ can be written as a countable intersection of open sets of $T$.