Splittings
The authors investigate possible sizes of splitting families of sets. In general a family \({\mathcal S}\) is said to split the family \({\mathcal A}\) if for every infinite disjoint subfamily \({\mathcal A}'\) there is \(S\in {\mathcal S}\) such that the families \(\{A\in {\mathcal A}': A\subseteq S\}\) and \(\{A\in {\mathcal A}': A\cap S= \emptyset\}\) are both infinite. The minimum size of such an \({\mathcal S}\) is denoted by \({\mathfrak s} ({\mathcal A})\). It is shown that \({\mathfrak s} ([ \omega ]^{<\omega})= \max \{{\mathfrak s},{\mathfrak b}\}\) and hence that \({\mathfrak s}([\omega ]^{<\omega})\) may be \(\omega_1\) while \({\mathfrak c}\) is big. The authors also determine the splitting number of three clopen bases for the Cantor set: \({\mathcal B}_0= \{[ s]: s\in 2^{<\omega}\}\), the canonical base \({\mathcal B}_1\) for the product topology and the family \({\mathcal B}_2\) of all clopen sets. It is easily seen that \({\mathfrak s} ({\mathcal B}_0)\leq {\mathfrak s} ({\mathcal B}_1) \leq {\mathfrak s} ({\mathcal B}_2)\); the authors show that \({\mathfrak s} ({\mathcal B}_1)= {\mathfrak c}\) and that \({\mathfrak s} ({\mathcal B}_0)= {\mathfrak c}\) is not larger than the cofinality of the ideal of measure zero sets; this solves a well-known problem of P. Simon.
cofinality of the ideal of measure zero sets
scientific article; zbMATH DE number 887160
scientific article