Pseudocompact \(\Delta\)-spaces are often scattered
The following definition was given in [\textit{J. Kąkol} and \textit{A. Leiderman}, Proc. Am. Math. Soc., Ser. B 8, 86--99 (2021; Zbl 1473.54018)]: A space \(X\) is a \(\Delta\)-\emph{space} if for any decreasing sequence \(\mathcal{S}=\{X_n:n\in\omega\}\) of subsets of \(X\) with empty intersection, there exists a sequence \(\{U_n:n\in\omega\}\) of open subsets of \(X\) with empty intersection such that \(X_n\subseteq U_n\) for each \(n\in\omega\). [loc. cit.] contains a study of general \(\Delta\)-spaces, where it was shown that \(X\) is a \(\Delta\)-space if and only if it has the \(\Delta\)-\emph{property}: for any disjoint family \(\mathcal{A}=\{A_n:n\in\omega\}\) of subsets of \(X\) there exists a point-finite family \(\{U_n:n\in\omega\}\) of open sets of \(X\) such that \(A_n\subseteq U_n\) for each \(n\in\omega\). It was also shown in [loc. cit.] that Čech-complete \(\Delta\)-spaces must be scattered. It was established in [\textit{J. Kąkol} and \textit{A. Leiderman}, ibid., Ser. B 8, 267--280 (2021; Zbl 1482.54019)] that each countably compact \(\Delta\)-space is scattered. In the present paper it is shown that in a pseudocompact \(\Delta\)-space all countable subsets must be scattered, implying all pseudocompact \(\Delta\)-spaces of countable tightness are scattered. It is also established that adding a countable set to a pseudocompact \(\Delta\)-space can destroy the \(\Delta\)-property. However, if a countably compact space \(X\) has a \(\Delta\)-subspace \(Y\) such that \(X\setminus Y\) is countable, then \(X\) must be a \(\Delta\)-space. In addition, it is shown that monotonically normal \(\Delta\)-spaces are hereditarily paracompact and a subspace \(X\) of an ordinal space has the \(\Delta\)-property if and only if \(X\) is hereditarily paracompact. A series of ten questions are asked, including the following: If \(X\) is a pseudocompact \(\Delta\)-space, is it true that \(t(X)\leq\omega\)? If \(X\) is a separable pseudocompact \(\Delta\)-space, must \(X\) be scattered? If \(X\) is a Lindelöf scattered subspace of an Eberlein compact space, must \(X\) be a \(\Delta\)-space? If \(X\) is a countably compact \(\Delta\)-space, is it true that every continuous image of \(X\) is a \(\Delta\)-space?
scientific article; zbMATH DE number 7493689
scientific article