On the equivalence of Rudin's lemma and the Boolean prime ideal theorem
type: Item
instance of: scholarly article
title: On the equivalence of Rudin's lemma and the Boolean prime ideal theorem
zbMATH Open document ID: 1490.03028
author: Mengqiao Huang
author: Xiaodong Jia
author: Qingguo Li
published in: Topology and its Applications
publication date: 2022-02-09T00:00:00Z
review text: The authors prove (in Theorem 3.1 of the paper under review) that Mary Ellen Rudin's lemma and its variants listed below are all equivalent, within \(\mathbf{ZF}\) (the Zermelo-Fraenkel set theory minus the axiom of choice (\(\mathbf{AC}\))), to the Boolean prime ideal theorem (\(\mathbf{PIT}\)), i.e. to the principle ``Every ideal in a Boolean algebra is included in a prime ideal''. \begin{itemize} \item[1.] Topological Rudin lemma [\textit{R. Heckmann}, Electron. Notes Theor. Comput. Sci. 298, 215--232 (2013; Zbl 1334.68127), Lemma 3.1]: Let \((X,\tau)\) be a topological space and let \(\mathcal{A}\) be an irreducible subset of the space of all nonempty compact subsets of \(X\) endowed with the upper Vietoris topology. Any closed set \(C\subseteq X\) that meets all members of \(\mathcal{A}\) contains an irreducible closed subset that still meets all members of \(\mathcal{A}\). \item[2.] The \(\sqsubseteq\)-directed version of the topological Rudin lemma [\textit{R. Heckmann}, Electron. Notes Theor. Comput. Sci. 298, 215--232 (2013; Zbl 1334.68127), Corollary 3.2]: Let \((X,\tau)\) be a topological space and let \(\mathcal{A}\) be a \(\sqsubseteq\)-directed family of nonempty compact subsets of \(X\). Any closed set \(C\subseteq X\) that meets all members of \(\mathcal{A}\) contains an irreducible closed subset that still meets all members of \(\mathcal{A}\). \item[3.] The irreducible cutset principle [\textit{M. Erné}, ``Sober spaces, well-filtration and compactness principle'', Preprint, Proposition 3]: Let \((X,\tau)\) be a topological space. For every filtered collection \(\mathcal{A}\) of nonempty compact saturated subsets of \(X\), there is an irreducible (closed) subset \(A\) meeting all members of \(\mathcal{A}\). \item[4.] Order-theoretic Rudin lemma [\textit{R. Heckmann}, Electron. Notes Theor. Comput. Sci. 298, 215--232 (2013; Zbl 1334.68127), Lemma 3.4]: Let \(P\) be a preorder and let \(\mathcal{F}\) be a \(\sqsubseteq\)-directed family of finitary upp…
reviewed by: Eleftherios Tachtsis
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 7471568
zbMATH Keywords: Rudin's lemma
zbMATH Keywords: prime ideal theorem
zbMATH Keywords: domain theory
Wikidata QID: Q113862542
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107970
OpenAlex ID: W4200128497
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DOI: 10.1016/J.TOPOL.2021.107970
description: scientific article; zbMATH DE number 7471568