Axiom Of Choice

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 332 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
345 \(\Rightarrow\) 14	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
14 \(\Rightarrow\) 123	Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr.
123 \(\Rightarrow\) 331	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
331 \(\Rightarrow\) 332	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
345:	Rasiowa-Sikorski Axiom: If \((B,\land,\lor)\) is a Boolean algebra, \(a\) is a non-zero element of \(B\), and \(\{X_n: n\in\omega\}\) is a denumerable set of subsets of \(B\) then there is a maximal filter \(F\) of \(B\) such that \(a\in F\) and for each \(n\in\omega\), if \(X_n\subseteq F\) and \(\bigwedge X_n\) exists then \(\bigwedge X_n \in F\).
14:	BPI: Every Boolean algebra has a prime ideal.
123:	\(SPI^\): Uniform weak ultrafilter principle:* For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).
331:	If \((X_i)_{i\in I}\) is a family of compact non-empty topological spaces then there is a family \((F_i)_{i\in I}\) such that \(\forall i\in I\), \(F_i\) is an irreducible closed subset of \(X_i\).
332:	A product of non-empty compact sober topological spaces is non-empty.

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