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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