= Uncountable discrete one-point compactification counterexample
The <one-point compactification> $D\cup\{\infty\}$ of an uncountable <discrete space> is a <hereditarily normal space> but not a <perfectly normal space>. A subspace missing $\infty$ is discrete; one containing it is closed, hence compact Hausdorff and normal. However, every neighborhood of $\infty$ misses only finitely many points of $D$. A countable intersection misses at most countably many points, so the closed singleton $\{\infty\}$ is not a <G-delta set>.
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