T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space

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Theorem

Let $T = \struct {S, \tau}$ be a $T_3$ topological space.

Let $\BB$ be a $\sigma$-locally finite basis.

Then:

$T$ is a perfectly $T_4$ space

Proof

From T3 Space with Sigma-Locally Finite Basis is T4 Space:

$T$ is a $T_4$ space

It remains to show that every closed set in $T$ is a $G_\delta$ set

Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis where $\BB_n$ is a locally finite set of subsets for each $n \in \N$.

Let $F$ be closed in $T$.

By definition of closed set:

$X \setminus F$ is open in $T$.

Let $G = X \setminus F$.

Lemma 1

$G$ is an $F_\sigma$ set

$\Box$

From Complement of F-Sigma Set is G-Delta Set:

$F$ is a $G_\delta$ set

Since $F$ was arbitrary, we have:

every closed set in $T$ is a $G_\delta$ set.

It follows that $T$ is a perfectly $T_4$ space by definition.

$\blacksquare$