Resolvability of spaces having small spread or extent

István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

In a recent paper O. Pavlov proved the following two interesting resolvability results:

1. If a space $X$ satisfies $\Delta(X) > \operatorname{ps}(X)$ then $X$ is maximally resolvable.
2. If a $T_3$-space $X$ satisfies $\Delta(X) > \operatorname{pe}(X)$ then $X$ is $\omega$-resolvable.

Here $\operatorname{ps}(X)$ ( $\operatorname{pe}(X)$) denotes the smallest successor cardinal such that $X$ has no discrete (closed discrete) subset of that size and $\Delta(X)$ is the smallest cardinality of a non-empty open set in $X$. In this note we improve (1) by showing that $\Delta(X) >$ $\operatorname{ps}(X)$ can be relaxed to $\Delta(X) \ge$ $\operatorname{ps}(X)$. In particular, if $X$ is a space of countable spread with $\Delta(X) > \omega$ then $X$ is maximally resolvable.

The question if an analogous improvement of (2) is valid remains open, but we present a proof of (2) that is simpler than Pavlov's.

Bibliography

1

O. Pavlov, On resolvability of topological spaces Topology and its Applications 126 (2002) 37-47.

2000 Mathematics Subject Classification: 54A25, 54B05

Key words and phrases: $\kappa$-resolvable space, maximally resolvable space, dispersion character, spread, extent

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