Cofinal types of topological directed orders.
type: Item
instance of: scholarly article
title: Cofinal types of topological directed orders.
zbMATH Open document ID: 1071.03034
author: Stevo Todorčević
author: Sławomir Solecki
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116163
review text: 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.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 2162445
zbMATH Keywords: Tukey order
zbMATH Keywords: analytic ideals
zbMATH Keywords: \(\sigma\)-ideals of compact sets
zbMATH Keywords: Tukey reducibility
zbMATH Keywords: measure and category
MaRDI profile type: Publication
OpenAlex ID: W2122005522
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DOI: 10.5802/AIF.2070
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162445