Products of (weakly) discretely generated spaces

A space X is discretely generated at a point x∈X if for any A⊆X with x∈cl(A), there exists a discrete set D⊆A such that x∈cl(D). The space X is discretely generated if it is discretely generated at every point x∈X. We say that X is weakly discretely generated if for any non-closed set A⊆X, there exists a discrete set D⊆A such that cl(D)∖A≠∅. We obtain new results concerning products of spaces belonging to these classes.