In the previous question, Some counterexamples are given which shows shat not every perfectly normal space is paracompact. Thanks Henno. I have another question. It may be difficult:
Is every perfectly normal space submetrizable?
A topological space $X$ is called a perfectly normal space if $X$ is a normal space and every closed subset of $X$ is a $G_\delta$-set.
Thanks for your help.
No, Alexandroff's double arrow space (or the split interval), i.e. $[0,1] \times \{0,1\}$ ordered lexicographically, in the order topology) is compact Hausdorff (so normal), hereditarily Lindelöf (which is equivalent to all closed sets being $G_\delta$ for compact spaces) but not metrisable. Hence it is not submetrisable (it would have a $G_\delta$ diagonal otherwise, and a (countably) compact space with a $G_\delta$ diagonal is metrisable).