Definition:Perfectly Normal Space

Definition
Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a perfectly normal space :
 * $\struct {S, \tau}$ 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:
 * $\exists U \in \tau: x \in U, y \notin U$
 * $\exists V \in \tau: y \in V, x \notin V$