Covering by discrete and closed discrete sets

TL;DR

The paper investigates the covering properties of Baire metric spaces with discrete sets, establishing limitations and providing examples of spaces that can or cannot be covered by small collections of discrete sets.

Contribution

It proves that no Baire metric space can be covered by a small number of discrete sets and presents new examples of spaces with such covering properties.

Findings

No Baire metric space can be covered by a small number of discrete sets.

Provides a ZFC example of a regular Baire σ-space covered by a small number of discrete sets.

Offers a consistent example of a normal Baire Moore space with similar covering properties.

Abstract

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