Resolvability and monotone normality

István Juhász Lajos Soukup and Zoltán Szentmiklóssy

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 $\Delta(X) = \min\{ \vert G\vert : G \ne \emptyset$    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 $\vert X\vert = \Delta(X) = \aleph_{\omega}$ such that no subspace of $X$ is $\omega_2$-resolvable.

Key words and phrases: resolvable spaces, monotonically normal spaces

2000 Mathematics Subject Classification: subjclass: 54A35, 03E35, 54A25

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