Cofinal types of topological directed orders.
A directed set \(D\) is Tukey reducible to a directed set \(E\), written \(D\leq_TE\), if there is \(f:D\to E\) such that the preimage of each bounded subset of \(E\) is bounded in \(D\). The authors investigate the structure of the Tukey order among partial orders belonging to two major classes: the class of analytic \(P\)-ideals of subsets of \(\mathbb N\) with inclusion and the class of \(\sigma\)-ideals of compact sets with inclusion. In fact, they define a natural family of the so-called basic orders containing both of these classes and establish results in this greater generality. \((D,{\leq})\) is said to be basic if \(D\) is a separable metric space, each pair of elements of \(D\) has a least upper bound, the operation of the least upper bound is continuous, each bounded sequence has a convergent subsequence, and each converging sequence has a bounded subsequence. The authors prove that if the topology on a basic order is analytic, then it is Polish. This is used to prove that analytic \(P\)-ideals are the only ideals on \(\mathbb N\) which can be made into analytic basic orders by strengthening topology on them. Further they prove that if there is a Tukey map between basic orders, then there is also a definable one. They define a general type of property of basic orders which are invariant under Tukey reducibility. Using these general invariants they establish several results on Tukey non-reducibility between concrete partial orders. One of these results concerns the relation between measure and category. The authors define and study a functor \(I\to D(I)\) between ideals on \(\mathbb N\) and systems of monotone compact sets. They prove that for analytic \(P\)-ideals the functor \(D\) respects the Tukey order, \(D(I)\leq_TI\), and \(D(I)\leq_TD(I_{1/n})\), where \(I_{1/n}=\{x\subseteq\mathbb N: \sum_{n\in x}1/(n+1)<\infty\}\). Finally, they prove that each analytic ideal has a cofinal \(G_\delta\) subset.
scientific article; zbMATH DE number 2162445