Covering by discrete and closed discrete sets

Abstract

Say that a cardinal number $κ$ is \emph{small} relative to the space $X$ if $κ<Δ(X)$, where $Δ(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $σ$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.