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We prove that, for an arbitrary topological space $X$, the following twoconditions are equivalent: (a) Every open cover of $X$ has a finite subset withdense union (b) $X$ is $D$-pseudocompact, for every ultrafilter $D$.  Locally, our result asserts that if $X$ is weakly initially$\lambda$-compact, and $2^ \mu \leq \lambda $, then $X$ is $D$-\brfrtpseudocompact, for every ultrafilter $D$ over any set of cardinality $ \leq\mu$. As a consequence, if $2^ \mu \leq \lambda $, then the product of anyfamily of weakly initially $\lambda$-compact spaces is weakly initially$\mu$-compact.

For Hausdorff spaces, $H$-closed = $D$-pseudocompact for all ultrafilters $D$