In Engelking's book, there is exercise (p. 49, ex. 1.5.K), where is written that $T_1$ space $X$ is perfectly normal if and only if for every open set $W$ from $X$ where exist sequence $W_1,W_2,\ldots$ of open subsets of $X$ such that $W$ is the union of $W_i$ and $\text{cl}(W_i)$ is a subset of $W$ for any $i=1,2,\ldots$
$\Rightarrow$ implication is not hard to prove. If right proposition holds, it's also obvious that $X$ is perfect space (every open set is an $F_\sigma$ set). But I can't prove that $X$ is normal also. Thank, for any help.
Apply lemma 1.5.15 (p. 43) to get the normality. The fact that the $W_i$ cover $W$ clearly means they cover a closed subset $F$ too. It's Theorem 1 in this note too, for people who are so unfortunate not to have a copy of Engelking.