On a \(\sigma \)-ideal of compact sets
type: Item
instance of: scholarly article
title: On a \(\sigma \)-ideal of compact sets
zbMATH Open document ID: 1196.03060
author: Tamás Mátrai
published in: Topology and its Applications
publication date: 2010-05-19T00:00:00Z
review text: 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)].
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 5710138
zbMATH Keywords: \(G_\delta\) \(\sigma\)-ideal of compact sets
zbMATH Keywords: Tukey reducibility
MaRDI profile type: Publication
full work available at URL: j.topol.2009.06.014
OpenAlex ID: W1986441031
cites work: Q4289973
cites work: Topology and descriptive set theory
cites work: Hereditary properties of the class of closed sets of uniqueness for trigonometric series
cites work: Q4732461
cites work: Descriptive set theory and harmonic analysis
cites work: Q4692760
cites work: The Structure of σ-Ideals of Compact Sets
cites work: Analytic ideals and cofinal types
cites work: Descriptive set Theory of Families of Small Sets
cites work: Kenilworth
cites work: A \(G_\delta \) ideal of compact sets strictly above the nowhere dense ideal in the Tukey order
cites work: \(G_{\delta }\) ideals of compact sets
cites work: Cofinal types of topological directed orders.
DOI: 10.1016/J.TOPOL.2009.06.014
description: scientific article; zbMATH DE number 5710138