Covering by discrete and closed discrete sets

Details Covering by discrete and closed discrete sets Covering by discrete and closed discrete sets

2008-09-10

arxiv.org/abs/0809.1872v1

Topology Appl. 156 (2009), no. 4, 721--727

Mathematics

General Topology

12 pages, to appear on Topology and its Applications

Scientific paper

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.

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