Definition:Perfectly Normal Space

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

$\left({S, \tau}\right)$ is a perfectly normal space if and only if:

$\left({S, \tau}\right)$ is a perfectly $T_4$ space
$\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

That is:

Every closed set in $T$ is a $G_\delta$ set.
$\forall x, y \in S$, both:
$\exists U \in \tau: x \in U, y \notin U$
$\exists V \in \tau: y \in V, x \notin V$

Also see

  • Results about perfectly normal spaces can be found here.