Additivity of metrizability and related properties

A topological property P is called n-additive in nth power (or weakly n-additive) if a topological space X has P as soon as Xn = ∪{Yi: i ϵ n} where all Yi have P. If P is n-additive in nth power for all natural n ⩾ 1, we say that P is weakly finitely additive. in question we deal with in this paper is whether metrizability is weakly finitely additive. It was proved by Tkachuk (1994) that it is so in the class of regular spaces with Souslin property. Metrizability was also proved by Tkachuk (1994) to be weakly finitely additive in the class of Hausdorff compact spaces. We generalize this last result, showing that metrizability is weakly finitely additive in the class of regular pseudocompact spaces. o prove that if Xn is a regular Lindelöf space then it is metrizable if represented as a union of its n metrizable subspaces. w that there is an example of a Tychonoff nonmetrizable space X such that Xn is a union of two metrizable subspaces for all n ⩾ 1. The method of constructing this example can be used to solve several problems stated by Tkachuk (1994).