On the Borel hierarchies of countable products of Polish spaces (Q1174598)

Let \(X\) be a set of cardinality \(c\). Let \({\mathcal T}_1\) be a Polish topology on \(X\), and let \({\mathcal T}_2\) be the discrete topology on \(X\). Let \(X_i\) be the countably infinite product of \((X,{\mathcal T}_i)\) with the product topology. Let \({\mathcal B}\) denote the family of Borel sets in \(X_1\). Define \(\Pi_0=\Sigma_0=\{A\subseteq X_2\): \(A\) is clopen\}, let \(\Sigma_\mu\) consist of countable unions of sets from \(\bigcup_{\nu<\mu}\Pi_\nu\), and let \(\Pi_\mu\) consist of complements of sets in \(\Sigma_\mu\). Define a hierarchy \(\Sigma^*_ \mu\), \(\Pi^*_ \mu\) in \(X_2\) in the same way but starting from \(\Pi^*_0=\Sigma^*_0=\Pi_0\cap{\mathcal B}\). From a set-theoretic assumption which is inconsistent with ZFC but which holds in the Lévy-Solovay model [\textit{R. M. Solovay}, Ann. Math. (2) 92, 1--56 (1970; Zbl 0207.00905)] the author proves a conjecture of \textit{A. Maitra} which implies that for \(0\leq\mu<\omega_1\), \(\Sigma^*_\mu=\Sigma_\mu\cap{\mathcal B}\). In a postscript the author remarks that \textit{V. V. Srivasta} (unpublished) has proved this in ZFC.