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