A tale of two \((s)\)-ities (Q1978978)

Let us first recall two notions introduced by \textit{Szpilrajn-Marczewski} in 1935. A set \(M\) in a complete separable metric space is said to have property \((s)\) if every perfect set has a perfect subset which is either a subset of \(M\) or is disjoint from \(M\). The set \(M\) has property \((s_0)\) if every perfect set has a perfect subset which is disjoint from \(M\). It is known that the family of sets having property \((s)\) is a \(\sigma \)-algebra, and the family of sets having property \((s_0)\) is a \(\sigma \)-ideal. The author answers in the negative the following question posed by \textit{J. Brown}: Let \(X\) and \(Y\) be two uncountable complete separable metric spaces. Does every \((s)\)-set in \(X\times Y\) belong to the \(\sigma \)-algebra generated by all sets of the form \(A\times B\) where \(A\subset X\) and \(B\subset Y\) are \((s)\)-sets? The construction uses the fact that the Boolean algebra \((s)/(s_0)\) is complete. The proof is motivated by John Morgan's theory of category bases.