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

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.

💡 Research Summary

The paper investigates the relationship between three notions in general topology: weak initial λ‑compactness, D‑pseudocompactness (for ultrafilters D), and H‑closedness in Hausdorff spaces. A space X is called weakly initially λ‑compact if every open cover of cardinality ≤ λ has a finite subfamily whose union is dense in X. D‑pseudocompactness means that for any I‑indexed family of non‑empty open sets (Oi)i∈I, there exists a point x∈X such that for every neighbourhood U of x the set {i∈I | U∩Oi≠∅} belongs to the ultrafilter D; such an x is a D‑limit point of the family.

The central result (Theorem 1) shows that if X is weakly initially λ‑compact and 2^μ ≤ λ, then X is D‑pseudocompact for every ultrafilter D on a set of size ≤ μ. The proof proceeds by contradiction: assuming X fails to be D‑pseudocompact, one constructs for each point x an open neighbourhood Ux and a set Zx∈D such that Zx={i∈I | Ux∩Oi=∅}. The family {VZ | Z∈D}, where VZ is the union of all Ux with Zx=Z, forms an open cover of X. Weak initial λ‑compactness yields a finite subfamily VZ1,…,VZn whose union is dense. The intersection Z=Z1∩…∩Zn still belongs to D, so picking i∈Z gives a contradiction because Oi is disjoint from each VZj. Hence X must be D‑pseudocompact.

Proposition 2 provides the converse direction under the presence of a regular ultrafilter: if X is D‑pseudocompact for some regular ultrafilter D over μ, then X is weakly initially μ‑compact. This uses known results (e.g., Lipparini’s Corollary 15).

Combining these, Corollary 3 states that if 2^μ ≤ λ, then the product of any family of weakly initially λ‑compact spaces is weakly initially μ‑compact. The argument uses the fact that D‑pseudocompactness is productive (Ginsburg–Saks) and Proposition 2 to translate back to weak initial compactness.

Theorem 5 establishes a suite of equivalent conditions for an arbitrary space X: (1) X is H(i) (every open filter on X has a non‑empty adherence), (2) X is weakly initially λ‑compact for every infinite cardinal λ, (3) X is D‑pseudocompact for every ultrafilter D, and (4) for each infinite λ there exists a regular ultrafilter D over λ such that X is D‑pseudocompact. When X is Hausdorff (or Hausdorff and regular), these are further equivalent to (5) X being H‑closed (i.e., closed in every Hausdorff space containing it) and (6) X being compact. Thus, in Hausdorff spaces, H‑closedness, compactness, universal D‑pseudocompactness, and universal weak initial compactness coincide.

The paper also generalizes the framework to an arbitrary family 𝔽 of subsets of X. Theorem 8 shows that if X is 𝔽‑