The hierarchy of Borel universal sets (Q1598685)

A \(\Sigma^0_\alpha\)-subset \(U\) of the product \(X\times Y\) is a \(\Sigma^0_\alpha\)-universal set of \(X\) parametrized by \(Y\) if for every \(\Sigma^0_\alpha\)-subset \(A\) of \(X\) there is a point \(y\) in \(Y\) such that \(A=\{ x : (x,y) \in U \}\). Spaces admitting a \(\Sigma^0_\alpha\)-universal set parametrized by a space of given weight are studied. Some characterizations of such spaces are given. It is pointed out that the case \(\alpha\) finite is different from the case \(\alpha\) infinite. Some examples showing difference from the case \(\alpha=1\) studied in a previous paper by \textit{P. M. Gartside, R. W. Knight} and \textit{J. T. H. Lo} [ibid., 131-145 (2002; Zbl 0990.54003)] are given. Further, some consequences for the case \(X\) compact are presented.