Question: Is it true that if $X$ is a compact Hausdorff space that is perfectly normal, then $X \times X$ is also perfectly normal?
Recall that a topological space $X$ is called perfectly normal if and only if $X$ is normal and every closed set $A \subseteq X$ is a $G_\delta$-set ($A$ is a $G_\delta$-set if and only if it is the intersection of countably many open sets of $X$).
Let $X$ be a compact Hausdorff space. In particular, this means that $X$ is compact. Consequently, $X \times X$ is compact and Hausdorff. Thus, to show that $X \times X$ is perfectly normal, it is enough to show that each closed set in $X \times X$ is a $G_\delta$-set. Thus, my question is really the following:
Question*: Is it true that If $X$ is a compact Hausdorff and perfectly normal space, then every closed set in $X \times X$ is a $G_\delta$-set?
If this turns out to be false, that's too bad. However, if this turns out to be true, then I have $2$ follow-up questions, where we get rid of some assumptions regarding compactness or Hausdorffness.
Bonus question 1: If $X$ is a topological space such that every closed set is a $G_\delta$-set, then every closed set in $X \times X$ is a $G_\delta$-set.
Bonus question 2: If $X$ is a Hausdorff topological space such that every closed set is a $G_\delta$-set, then every closed set in $X \times X$ is a $G_\delta$-set.
As always, any help with any of the above questions will be greatly appreciated.
Best Answer
The answer is "no". This follows from Corollary 2 of
Katětov, Miroslav. "Complete normality of Cartesian products." Fundamenta Mathematicae 35.1 (1948): 271-274.
See http://matwbn.icm.edu.pl/ksiazki/fm/fm35/fm35125.pdf.
Assume it were true that for any perfectly normal compact Hausdorff space $X$ also $X \times X$ is perfectly normal. Then also $X \times X \times X \times X$ would be perfectly normal, hence also $Y = X \times X \times X$ because it embeds into $X \times X \times X \times X$. Since perfectly normal spaces are heritarily normal (which is denoted as "completely normal" in the above paper), Corollary 2 implies that $X$ is metrizable.
In other words, if $\mathcal{P}$ is a class of perfectly normal compact Hausdorff spaces such that
$$(\ast) \phantom{x} X \in \mathcal{P} \Rightarrow X \times X \in \mathcal{P}$$
then $\mathcal{P}$ is contained in the class $\mathcal{CM}$ of compact metrizable spaces. In fact, $\mathcal{CM}$ is the biggest class having property $(\ast)$.