Compact covering mappings on a Borel space (Q2721642)

A surjective mapping \(f:X\to Y\) is a compact covering mapping if every compact set in \(Y\) is the image of a compact set in \(X\); \(f\) is inductively perfect if there is \(X'\subseteq X\) such that \(f(X')=f(X)\) and for each compact set \(K\subseteq Y\), \(X'\cap f^{-1}(K)\) is compact. Every inductively perfect mapping is a compact covering. The inverse is true if we assume that \(X\) is a \(G_\delta\) or \(Y\) is an \(F_\sigma\) subset of the Cantor space \(2^\omega\) but if we demand \(X\) and \(Y\) be arbitrary Borel sets the statement is undecidable in ZFC. In [\textit{G. Debs} and \textit{J. Saint Raymond}, Fundam. Math. 167, No. 3, 213-249 (2001; Zbl 0968.03052)] the authors proved that if \(\aleph_1^L<\aleph_1\) then every compact covering mapping from a \(\Delta^1_1\) set onto a \(\Pi^0_3\) set is inductively perfect. In the paper under review the authors prove that if \(\aleph_1^L<\aleph_1\) (\(\aleph_2^L<\aleph_1\)) then every compact covering mapping from a \(\Delta^1_1\) space onto a \(\Sigma^0_3\) space (\(\Sigma^0_4\) space) is inductively perfect.