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A semifilter approach to selection principles. (English) Zbl 1121.03060

Summary: In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal \(\mathfrak g\) is a lower bound of the additivity number of the \(\sigma \)-ideal generated by Menger subspaces of the Baire space, and under \(\mathfrak u < \mathfrak g\) every subset \(X\) of the real line with the property \(\text{Split} (\Lambda ,\Lambda )\) is Hurewicz, and thus it is consistent with ZFC that the property \(\text{Split} (\Lambda ,\Lambda )\) is preserved by unions of less than \(\mathfrak b\) subsets of the real line.

MSC:

03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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