Resolvability and monotone normality

Abstract

A space $X$ is said to be $\kappa$-resolvable (resp. almost $\kappa$-resolvable) if it contains $\kappa$ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets). $X$ is maximally resolvable iff it is $\Delta(X)$-resolvable, where $\Delta(X) = \min\{|G| : G \ne \emptyset {open}\}.$ We show that every crowded monotonically normal (in short: MN) space is $\omega$-resolvable and almost $\mu$-resolvable, where $\mu = \min\{2^{\omega}, \omega_2 \}$. On the other hand, if $\kappa$ is a measurable cardinal then there is a MN space $X$ with $\Delta(X) = \kappa$ such that no subspace of $X$ is $\omega_1$-resolvable. Any MN space of cardinality $< \aleph_\omega$ is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space $X$ with $|X| = \Delta(X) = \aleph_{\omega}$ such that no subspace of $X$ is $\omega_2$-resolvable.