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A space $X$ is called perfectly $\kappa$-normal if the closure of any open set (that is, every canonical closed set) is a zero-set.

How can i prove this proposition directly?

$Proposition$: A Cartesian product of metric spaces is perfectly $\kappa$-normal.

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    @BrianM.Scott: Ščepin introduced the notions of $\kappa$-metrizability and proved that $\kappa$-metrizability is productive [ On $\kappa$-metrizable space, MAth. USSR-Izv. 14 (1980), 407-440 ]. Since every metrizable space is $\kappa$-metrizable and every $\kappa$-metrizable space is perfectly $\kappa$-normal then A cartesian product of metric spaces is perfectly $\kappa$-normal.2012-11-26

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