A note on ultracomplete hyperspaces (Q2081683)

The hyperspace of all non-empty compact subsets of a given space \(X\), equipped with the Vietoris topology, is denoted by \(K(X)\). A space \(X\) is \(\omega\)-hyperbounded if the closure of any \(\sigma\)-compact subspace of \(X\) is compact, a notion introduced in [\textit{J. Angoa} et al., Mat. Vesn. 65, No. 3, 306--318 (2013; Zbl 1313.54026)]. A hyperspace \(K(X)\) is ultracomplete whenever in the collection of all compact subspaces of the remainder of some compactification of \( K(X)\) ordered by inclusion there exists a countable cofinal subcollection, i.e. the remainder of a compactification of \( K(X)\) is hemicompact, compare [\textit{R. F. Arens}, Ann. Math. (2) 47, 480--495 (1946; Zbl 0060.39704)]. Tychonoff spaces \(X\) such that their hyperspaces \(K(X)\) are ultracomplete are examined in the paper under review. The author declares (in the abstract) that the most far-reaching results are: \begin{itemize} \item If a space \(X\) is \(\omega\)-hyperbounded and locally compact, then the hyperspace \(K(X^\omega)\) is ultracomplete; \item The hyperspace \(K((X \setminus A)^\omega)\) is ultracomplete and countably compact, whenever \(X\) is a compact space and \(A\) is a countable set containing only \(P\)-points of \(X\). \end{itemize} The paper contains a few historical comments on equivalent notions that are relevant to expressing the results contained therein. For example, ultracomplete spaces were first studied by \textit{V. I. Ponomarev} and \textit{V. V. Tkachuk} [Mosc. Univ. Math. Bull. 42, No. 5, 14--17 (1987; Zbl 0652.54003); translation from Vestn. Mosk. Univ., Ser. I 1987, No. 5, 16--19 (1987)] and named there as strongly complete. In the conclusion, several technical problems were posed. But in my opinion the most interesting is the problem from page 2879, which I would put as follows: For which ultrafilters \(p\in \omega^*\) is the hyperspace \(K((\omega^*\setminus \{p\})^\omega)\) strongly complete?