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arXiv 0809.18722008-09-10math.GN

离散集和闭离散集的覆盖

Covering by discrete and closed discrete sets

Santi Spadaro

备注：: 12 pages, to appear on Topology and its Applications
发表信息：: Topology Appl. 156 (2009), no. 4, 721--727

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摘要：

假设基数$κ$是相对于空间$X$的\emph{small}，如果$κ<Δ(X)$，其中$Δ(X)$是$X$中非空开集的最小基数。证明了Baire度量空间不能被少量离散集覆盖，并给出了一些推广。我们展示了一个ZFC的例子，经常Baire $σ$ -空间和一个一致的例子，一个正常的Baire摩尔空间，可以覆盖的一个小数目的离散集。最后，我们对线性序空间做一些评论。

英文摘要：

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.

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