Ideals of nowhere dense sets in some topologies on positive integers
type: Item
instance of: scholarly article
title: Ideals of nowhere dense sets in some topologies on positive integers
zbMATH Open document ID: 1398.54057
author: Marta Kwela
author: Andrzej Nowik
published in: Topology and its Applications
publication date: 2018-10-02T00:00:00Z
review text: The authors study three topologies on \(\mathbb{N}\), the set of natural numbers, defined by taking families of arithmetic progressions as a base. These were defined by \textit{H. Furstenberg} [Am. Math. Mon. 62, No. 5, 353 (1955; Zbl 1229.11009)], \textit{S. W. Golomb} [Am. Math. Mon. 66, 663--665 (1959; Zbl 0202.33001)], and \textit{A. M. Kirch} [Am. Math. Mon. 76, 169--171 (1969; Zbl 0174.25602)]. Furstenberg used all arithmetic progressions \(\{b+an:n\in\mathbb{N}_0\}\) with \(b\leq a\), Golomb demanded \(b<a\) and \(\text{gcd}(a,b)=1\), and Kirch added the demand that \(a\) be square-free to Golomb's conditions. The paper is devoted mainly to the corresponding ideals of nowhere dense sets and starts by establishing some (non-)inclusions. The ideals are tall and \(F_{\sigma\delta}\)-subsets of the Cantor set but not \(F_\sigma\). None of the ideals can be extended to summable ideals.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54D10
zbMATH DE Number: 6946126
zbMATH Keywords: ideal
zbMATH Keywords: nowhere dense set
zbMATH Keywords: connected Hausdorff space
zbMATH Keywords: arithmetic progression
zbMATH Keywords: Furstenberg topology
zbMATH Keywords: Golomb topology
zbMATH Keywords: Kirch topology
MaRDI profile type: Publication
full work available at URL: j.topol.2018.08.015
OpenAlex ID: W2889518477
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DOI: 10.1016/J.TOPOL.2018.08.015
description: scientific article; zbMATH DE number 6946126