Some covering properties for \(\Psi\)-spaces. (English) Zbl 1199.54140

Bonanzinga, Maddalena; Matveev, Mikhail

Mat. Vesn. 61, No. 1, 3-11 (2009).

The star-selection properties introduced by the reviewer [Publ. Math. 55, No. 3–4, 421–431 (1999; Zbl 0932.54022)], and similar to the classical Menger, Hurewicz and Rothberger covering properties, are studied for \(\Psi\)-spaces, \(\Psi(\mathcal A)\) generated by an almost disjoint family \(\mathcal A\) of infinite subsets of \(\omega\). The authors also introduce partition-selection properties and investigate these properties for \(\Psi\)-spaces. Some results: (1) \(\Psi(\mathcal A)\) is strongly star-Menger (resp., strongly star-Hurewicz) if and only if \(| \mathcal A| <\mathfrak d\) (resp. \(| \mathcal A| <\mathfrak b\)); (2) if \(| \mathcal A| <\aleph_\omega\), then \(\Psi(\mathcal A)\) is star-Menger if and only if it is strongly star-Menger; (3) under \(\mathfrak b<\mathfrak d\), there is a \(\Psi\)-space that is partition-Menger, but not partition-Hurewicz.

Reviewer: Ljubiša Kočinac (Niš)

MSC:

54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)
54A35	Consistency and independence results in general topology

Keywords:

star-selection principles; partition-selection principles; almost disjoint family; \(\Psi\)-space

Citations:

Zbl 0932.54022