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

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

2011-07-07

arxiv.org/abs/1107.1435v2

Mathematics

General Topology

v. 2: added some results, some remarks, various minor improvements. 7 pages. v. 1: 4 pages

Scientific paper

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.

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