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.


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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