Covering by discrete and closed discrete sets

Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\mathrm{\Delta }(X)$, where $\mathrm{\Delta }(X)$ 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 $\sigma$-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.