
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.
Consistency and independence results in general topology, Maximally resolvable space, Other combinatorial set theory, independent family, Partitions of sets, extraresolvable space, Extraresolvable space, Independent family, maximally resolvable space, Cardinality properties (cardinal functions and inequalities, discrete subsets), Consistency and independence results, Geometry and Topology, Souslin number, ω-resolvable space, \(\kappa\)-resolvable space
Consistency and independence results in general topology, Maximally resolvable space, Other combinatorial set theory, independent family, Partitions of sets, extraresolvable space, Extraresolvable space, Independent family, maximally resolvable space, Cardinality properties (cardinal functions and inequalities, discrete subsets), Consistency and independence results, Geometry and Topology, Souslin number, ω-resolvable space, \(\kappa\)-resolvable space
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