Noetherian type in topological products
type: Item
instance of: scholarly article
title: Noetherian type in topological products
zbMATH Open document ID: 1302.54008
author: Menachem Kojman
author: David Milovich
author: Santi Spadaro
published in: Israel Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.3966
review text: Inspired by the definition of Noetherian rings one may define any collection, \(\mathcal F\), of sets to be Noetherian if for every~\(F\in\mathcal F\) the family \(\{G\in\mathcal{F}:F\subseteq G\}\) is finite. More generally, the Noetherian type of~\(\mathcal F\) is defined to be the smallest cardinal number~\(\kappa\) such that \(\bigl|\{G\in\mathcal{F}:F\subseteq G\}\bigr|<\kappa\) for all~\(F\in\mathcal F\). The Noetherian type of a topological space~\(X\), written \(\text{Nt}(X)\), then is the minimum Noetherian type is a base for~\(X\). The authors investigate the behaviour of this cardinal function, and some variants, under the formation of products, mostly in the class of compact Hausdorff spaces. They provide two completely regular spaces~\(X\) and~\(Y\) such that the Noetherian type of their product is strictly less than the minimum of the types of~\(X\) and~\(Y\). The question whether this is possible for compact Hausdorff spaces remains open. The bulk of the paper is devoted to the Noetherian type of \(G_\delta\)-modifications (and others) of compact Hausdorff spaces, in particular of Cantor cubes. Using methods from PCF theory they show that the type of the \(G_\delta\)-modification of~\(2^{\aleph_\omega}\) is at most~\(\aleph_4\), and, more generally, the \(G_{\aleph_n}\)-modification has type at most the maximum of~\(\aleph_4\) and~\(\aleph_{n+1}\), and exactly \(\aleph_{n+1}\) if \(n\geq4\).
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 03E04
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 6361332
zbMATH Keywords: Noetherian type
zbMATH Keywords: base for a topological space
zbMATH Keywords: PCF theory
zbMATH Keywords: \(G_\delta\)-modificatioon
zbMATH Keywords: products
reviewed by: K. P. Hart
MaRDI profile type: Publication
OpenAlex ID: W2043481337
arXiv ID: 1012.3966
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DOI: 10.1007/S11856-014-1101-4
description: scientific article; zbMATH DE number 6361332