Resolvability of spaces having small spread or extent

Abstract

In a recent paper O. Pavlov proved the following two interesting resolvability results: (1) If a space $X$ satisfies $Δ(X) > \ps(X)$ then $X$ is maximally resolvable. (2) If a $T_3$-space $X$ satisfies $Δ(X) > \pe(X)$ then $X$ is $ω$-resolvable. Here $\ps(X)$ ($\pe(X)$) denotes the smallest successor cardinal such that $X$ has no discrete (closed discrete) subset of that size and $Δ(X)$ is the smallest cardinality of a non-empty open set in $X$. In this note we improve (1) by showing that $Δ(X) >$ $\ps(X)$ can be relaxed to $Δ(X) \ge$ $\ps(X)$. In particular, if $X$ is a space of countable spread with $Δ(X) > ω$ 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.