Spaces splittable over the class of Eberlein and descriptive compact spaces (Q683602)

The definition of a splittable topological space \(X\) goes back to \textit{V. V. Tkachuk} [Commentat. Math. Univ. Carol. 33, No. 3, 551--555 (1992; Zbl 0769.54004)]. Based on a characterization of this notion by \textit{A. V. Arkhangel'skij} and \textit{D. B. Shakhmatov} [J. Sov. Math. 50, No. 2, 1497--1512 (1990; Zbl 0701.41026); translation from Tr. Semin. Im. I. G. Petrovskogo 13, 206--227 (1988)], splittability was generalized in the following way. Given a class \(\mathcal{P}\) of topological spaces, a space \(X\) is said to be splittable over \(\mathcal{P}\) if for any \(A \subseteq X\) there exists \(Y \in \mathcal{P}\) and a continuous function \(f: X \rightarrow Y\) such that \(f^{-1}(f(A)) =A\). In the present paper the authors study splittability over several classes of spaces. They show that a scattered pseudocompact space, splittable over the class of Eberlein compact spaces, is Eberlein compact and that a countable compact space, splittable over the class of Eberlein compact spaces, is compact, thus giving (partial) answers to questions in [\textit{D. Jardón} and \textit{V. V. Tkachuk}, Topology Appl. 184, 41--49 (2015; Zbl 1331.54018)]. They also answer another question from this paper, dealing with the class \(\mathcal{P}\) of Corson compact spaces. Other classes of spaces the authors deal with are Rosenthal compact spaces and descriptive compact spaces.