# 2021-14 Perfectly normal product

Let X, Y be compact spaces. Suppose $$X \times Y$$ is perfectly normal, i.e, for every disjoint closed subsets E, F in $$X \times Y$$, there exists a continuous function $$f: X \times Y \to [0, 1] \subset \mathbb{R}$$ such that $$f^{-1}(0) = E, f^{-1}(1) = F$$. Is it true that at least one of X and Y is metrizable?

(added Sep. 11, 8AM: Assume further that $$X \times Y$$ is Hausdorff.)

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