Resolvability properties via independent families (Q857050)

For a given cardinal number \(\kappa > 1\), a topological space \(X\) is said to be \(\kappa\)-resolvable if it contains \(\kappa\) many pairwise disjoint dense subspaces. \(\Delta(X)\)-resolvable spaces are called maximally resolvable; here \(\Delta(X)\) is the dispersion character of \(X\), the minimal cardinality of a non-empty open subset of \(X\). \(X\) is extraresolvable if it contains a collection \(\mathcal D\) of dense subsets with \(| \mathcal D| = \Delta(X)^+\), such that the intersection of any two distinct elements of \(\mathcal D\) is nowhere dense in \(X\). This paper deals with the question of Juhász, Soukup and Szentmiklóssy if every extraresolvable Tychonoff space is maximally resolvable. Using some combinatorial principles the authors show that if GCH fails, there are extraresolvable not maximally resolvable Tychonoff spaces. It is also shown that a dense \(\omega\)-resolvable subspace of \(D(\lambda)^I\) is \(\lambda\)-resolvable.