Box products are often discretely generated

A space X is discretely generated if for any A ⊂ X and x ∈ A ¯ there exists a discrete set D ⊂ A such that x ∈ D ¯ . We prove that if X t is a monotonically normal space for any t ∈ T then the box product ∏ ∐ t ∈ T X t 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.