Let $X,Y$ be perfect topological spaces (e.g. every closed set is $G_\delta$). Is $X\times Y$ perfect?
I know that this isn't true for uncountable products, for example $[0,1]$ is perfect because every metric space is perfect, but $[0,1]^\mathbb{R}$ is not perfect because every countable intersection of open sets containing the zero function has more than one element. Therefore the singleton of the zero function is closed but not $G_\delta$.
However, is it true for finite products? For countable products?
No, even for the square of a compact $T_4$ such space, see here for a proof. Double Arrow space is a concrete and classic counterexample.