For Hausdorff spaces, H-closed = D-pseudocompact for every ultrafilter D

We prove that, for an arbitrary topological space X, the following conditions are equivalent: (a) Every open cover of X has a finite subfamily with dense union; (b) X is D-pseudocompact, for every ultrafilter D. Locally, our result asserts that if X is weakly initially λ-compact, and 2μ ≤ λ, then X is D-pseudocompact, for every ultrafilter D over any set of cardinality ≤ μ. As a consequence, if 2μ ≤ λ, then the product of any family of weakly initially λ-compact spaces is weakly initially μ-compact.