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

TL;DR

The paper establishes an equivalence between open cover conditions and ultrafilter-based pseudocompactness in topological spaces, with implications for product spaces and compactness properties.

Contribution

It proves a new equivalence involving $D$-pseudocompactness and open covers, extending understanding of compactness in Hausdorff and other spaces.

Findings

Open cover with dense union iff $D$-pseudocompact for all ultrafilters

Weakly initially $ heta$-compact spaces have $D$-pseudocompactness under certain conditions

Product of weakly initially $ heta$-compact spaces is weakly initially $ heta$-compact

Abstract

We prove that, for an arbitrary topological space $X$, the following two conditions are equivalent: (a) Every open cover of $X$ has a finite subset with dense 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 $D$-\brfrt pseudocompact, for every ultrafilter $D$ over any set of cardinality $ \leq \mu$. As a consequence, if $2^ \mu \leq \lambda $, then the product of any family of weakly initially $\lambda$-compact spaces is weakly initially $\mu$-compact.