Box products are often discretely generated

A space X is discretely generated if for any A⊂X and there exists a discrete set D⊂A such that . We prove that if Xt is a monotonically normal space for any t∈T then the box product ∏∐t∈TXt is discretely generated. In particular, every finite product of monotonically normal spaces is discretely generated. We establish the same conclusion for any finite product of Hausdorff spaces with a nested local base at every point. We also show that any dyadic discretely generated compact space is metrizable; besides, under CH, every discretely generated compact space has a dense set of points of countable π-character.