Axiom Of Choice

We have the following indirect implication of form equivalence classes:

345 \(\Rightarrow\) 154 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 \(\Rightarrow\) 154	Kategoriesatze und multiples Auswahlaxiom, Brunner, N. 1983c, Z. Math. Logik Grundlagen Math.

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.
154:	Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

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