Ohio completeness and products (Q2469554)

In the paper under review basic properties (closed subspace and products) of Ohio complete spaces are studied. A space \(X\) is \textsl{Ohio complete} [introduced by \textit{A. V. Arhangel'skiĭ}, Topology Appl. 150, 79--90 (2005; Zbl 1075.54012)], if every compactification \(\gamma X\) of \(X\) has the following property: there exists a \(G_\delta\) subset \(B\subseteq \gamma X\) such that \(X\subseteq B\) and for every \(y\in B\setminus X\) there exists a \(G_\delta\) subset \(C\subseteq\gamma X\) containing \(y\) and disjoint from \(X\). All Čech-complete spaces are clearly Ohio complete. It is very interesting that, as the authors prove, ``every compactification'' in the definition of Ohio completeness cannot be replaced by ``some compactification''. It is also proved that \(C^*\)-embedded closed subspaces of Ohio complete spaces are Ohio complete. As a consequence, any closed subspace of a normal Ohio complete space is again Ohio complete. However the main result of the paper is the following: if an uncountable product of spaces is a closed subspace of some Ohio complete space, then all but countably many of its factors are compact. As a consequence, the authors obtain the following nice corollary: if \(X\) is an Ohio complete space, then \(\mathbb{R}^\kappa\) cannot be embedded in \(X\) as a closed subspace for any uncountable cardinal \(\kappa\).