On the equivalence of Rudin's lemma and the Boolean prime ideal theorem
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 upper sets of \(P\). Any lower set \(L\) that meets all members of \(\mathcal{F}\) has a directed lower subset \(A\) that still meets all members of \(\mathcal{F}\). \item[5.] Rudin's lemma [\textit{A. Jung}, Cartesian closed categories of domains. Darmstadt: Technische Hochschule Darmstadt (1988; Zbl 0663.18004), Theorem 4.11]: If \(\{F_{i}:i\in I\}\) is a \(\sqsubseteq\)-directed family of nonempty finite sets in a poset \(P\), then there is a directed set \(A\subseteq\bigcup_{i\in I}F_{i}\) that meets all \(F_{i}\). \end{itemize} It follows that Rudin's lemma is not provable in \(\mathbf{ZF}\) and is strictly weaker than \(\mathbf{AC}\) in \(\mathbf{ZF}\); the standard reference-model of \(\mathbf{ZF}\) in which \(\mathbf{PIT}\) is true but \(\mathbf{AC}\) is false is the basic Cohen model -- see model \(\mathcal{M}1\) of \textit{P. Howard} and \textit{J. E. Rubin} [Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)].
scientific article; zbMATH DE number 7471568