Covering by discrete and closed discrete sets. (English) Zbl 1161.54001
For a space \(X\), dis\((X)\) denotes the smallest infinite cardinal \(\kappa\) such that \(X\) can be covered by \(\kappa\) many discrete subspaces, and \(\Delta(X)\) is the smallest cardinality of a non-empty open set in \(X\). I. Juhász and Z. Szentmiklóssy [Topology Appl. 155, 2102–2104 (2008; Zbl 1155.54007)] asked if dis\((X) \geq \Delta(X)\) for any compactum \(X\). The author investigates this inequality for Baire spaces and shows that it is true for two classes of such spaces; one of them contains Baire metric spaces, and the other paracompact Baire \(\sigma\)-spaces. However, there are Baire spaces \(X\) such that dis\((X) < \Delta(X)\). Several related questions are posed.
Reviewer: Ljubiša D. Kočinac (Niš)
MSC:
| 54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
| 54E18 | \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc. |
| 54E30 | Moore spaces |
| 54E52 | Baire category, Baire spaces |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
Citations:
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