Solution of Shapirovskii's question (Q2502957)

Boris Shapirovskii's the in mid-1970s posed the following question: Suppose that \(X\) and~\(Y\) are compact Hausdorff spaces such that \(\pi_\chi(x,X)>w(Y)\) for all \(x\in X\) and let \(f:X\to Y\) be a~continuous surjection. Do there exist two disjoint closed sets in~\(X\) mapping onto~\(Y\)? If \(X\) is an extremally disconnected or dyadic space, then the answer is positive and this result was already known to Shapirovskii. The author proves that there are always two disjoint open sets in~\(X\), each mapping densely into~\(Y\). From work of \textit{B.~Balcar} and \textit{P.~Simon} [Topology Appl.~41 No.~1--2, 133--145 (1991; Zbl 0752.54013)] he derives a~zero-dimensional counterexample in which \(Y\) is the space of ultrafilters of a~subalgebra of the Boolean algebra of clopen sets in~\(X\) and the surjective continuous map is given by Stone duality. Next the author proves that the answer for the modified Shapirovskii question in which we assume \(\pi_\chi(x,X)>w(X)^+\) for all \(x\in X\) is positive when \(X\) is zero-dimensional.