On a \(\sigma \)-ideal of compact sets
A~partially ordered set \((D,{\leq})\) is Tukey reducible to a~partially ordered set \((E,{\leq})\), written \((D,{\leq})\leq_T(E,{\leq})\), if there is a~mapping \(f:D\to E\) such that the images of unbounded subsets of~\(D\) under~\(f\) are unbounded in~\(E\). We write \((D,{\leq})<_T(E,{\leq})\) if \((D,{\leq})\leq_T(E,{\leq})\) and \((E,{\leq})\nleq_T(D,{\leq})\). Recall that \(\mathcal{I}_{1/n}=\{H\subseteq\omega:\sum_{h\in H}1/h<\omega\}\) and \(\mathcal{M}\cap\mathcal{K}(2^\omega)\) is the \(\sigma\)-ideal of compact sets consisting of compact meager subsets of~\(2^\omega\). It is known that \(\mathcal{M}\cap\mathcal{K}(2^\omega)\leq_T\mathcal{I}_{1/n}\). In the paper under the review the author presents a~construction of a~\(G_\delta\) \(\sigma\)-ideal~\(\mathfrak{I}\) of compact sets for which he proves that \(\mathcal{M}\cap\mathcal{K}(2^\omega)<_T\mathfrak{I}\nleq_T\mathcal{I}_{1/n}\). These results give negative answers to a~question of \textit{A.~Louveau} and \textit{B. Veličković} [``Analytic ideals and cofinal types'', Ann. Pure Appl. Logic 99, No.~1--3, 171--195 (1999; Zbl 0934.03061)] and a~question of \textit{S.~Solecki} and \textit{S.~Todorčević} [``Cofinal types of topological directed orders'', Ann. Inst. Fourier 54, No.~6, 1877--1911 (2004; Zbl 1071.03034)].
\(G_\delta\) \(\sigma\)-ideal of compact sets
scientific article; zbMATH DE number 5710138