Abstract: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 $Split (\Lambda ,\Lambda )$ is Hurewicz, and thus it is consistent with ZFC that the property $Split (\Lambda ,\Lambda )$ is preserved by unions of less than $\frak b$ subsets of the real line.
Keywords: Menger property, Hurewicz property, property $Split(\Lambda ,\Lambda )$, semifilter, multifunction, small cardinals, additivity number
AMS Subject Classification: 03A, 03E17, 03E35, 54D20