Resolvability vs. almost resolvability (Q1030207)

It is shown (in ZFC) that for each infinite cardinal \(\kappa\) there is a subspace \(X\) of the Cantor cube \(2^{\kappa}\) such that \(X\) is almost \(2^{\kappa}\)-resolvable (\(X\) contains \(2^{\kappa}\) many dense subsets such that the intersection of any two distinct sets is nowhere dense in \(X\)) but not \(\omega_1\)-resolvable, and \(|X| = \Delta(X) = \kappa\). \(\Delta (X)\) is the dispersion character of \(X\), the minimal cardinality of a non-empty open subset of \(X\). This result improves a consistent result by \textit{W. W. Comfort} and \textit{W. Hu} [Topology Appl. 154, 205--214 (2007; Zbl 1110.54004)].