Definition:Perfectly T4 Space

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

$T$ is a perfectly $T_4$ space if and only if:

$(1): \quad T$ 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 $T$.

Also see

  • Results about perfectly $T_4$ spaces can be found here.