Open universal sets (Q1868879)

An open set \(U\subseteq X\times Y\) is said to be an open universal set for~\(X\) parametrised by~\(Y\) if for all open sets \(V\subseteq X\) there is \(y\in Y\) such that \(V=\{x:(x,y)\in U\}\). It is shown that the existence of an open universal set puts some restrictions to some of the topological cardinal invariants of the spaces \(X\),~\(Y\). In particular, \(\mathit{w}(X)\leq\text\textit{nw}(Y)\), \(\mathit{hd}(X^n)\leq\text\textit{hL}(Y^n)\), \(\mathit{hL}(X^n)\leq\text\textit{hd}(Y^n)\), and \(\mathit{hc}(X^n)\leq\text\textit{hc}(Y^n)\) for regular Hausdorff spaces \(X\),~\(Y\); if \(X\)~is compact, then also \(\mathit{hL}(X^n)\leq\text\textit{hL}(Y^n)\) and \(\mathit{hd}(X^n)\leq\text\textit{hd}(Y^n)\); if \(X\)~has a~\(G_\delta\)~diagonal, then \(\mathit{hd}(X^\omega)\leq\text\textit{hL}(Y)\), \(\mathit{hL}(X^\omega)\leq\text\textit{hd}(Y)\), and \(\mathit{hc}(X^\omega)\leq\text\textit{hc}(Y)\). The following two statements are shown to be independent of ZFC: (1)~Every compact zero-dimensional space with open universal set parametrised by a~space with the hereditary c.c.c.\ is metrisable. (2)~Every cometrisable space with an open universal set parametrised by a~hereditary c.c.c.\ space is metrisable.