Splitting squares
A set \(A\subseteq2^\omega\) is a splitting family if for every \(a\in[\omega]^\omega\) there exists \(x\in A\) such that \(\exists^\infty n\in a\) \(x(n)=0\) and \(\exists^\infty n\in a\) \(x(n)=1\), i.e., \(x\) splits \(a\). The set \(A\) is called countably splitting, if for every countable set \(F\subseteq[\omega]^\omega\) we can find \(x\in A\) which splits every element of \(F\). The author proves that for every Borel function \(f:[2^\omega]^n\to2^\omega\) there exists a closed countably splitting family \(A\) such that \(f\restriction[A]^n\) omits a perfect set of values in \(2^\omega\). This result answers in the negative a question of J. Zapletal, and it is somewhat stronger than the question requires.
scientific article; zbMATH DE number 5279133