Fubini properties for filter-related \(\sigma\)-ideals.
It is shown that if \(\mathcal G, \mathcal F\subset [\omega]^{\omega}\) are two Borel filters with respect to the standard topology inherited from \(2^{\omega}\) and \(\mathcal F\) is a non-countably generated \(p\)-filter, then there is a set \(B\subset [\omega]^{\omega}\times [\omega]^{\omega}\), Borel in the standard topology, with ``vertical'' sections meager in the Ellentuck topology \(\tau_{\mathcal F}\) (generated by open sets \(\{X\in [\omega]^{\omega}:s\subset X\subset s\cup A\}\), \(s\in [\omega]^{<\omega}\), \(A\in\mathcal F\)) and complements of ``horizontal'' sections meager in \(\tau_{\mathcal G}\). Some finer results are given which give the same kind of example for \(\mathcal G\) with the Baire property and \(\mathcal F\) an arbitrary ultrafilter in \(\omega\). The above mentioned result would answer a question posed by Recław if the assumption on \(\mathcal F\) to be a \(p\)-filter could be omitted.
scientific article; zbMATH DE number 2037256