Axiom Of Choice

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 406 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
407 \(\Rightarrow\) 14	Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2.
14 \(\Rightarrow\) 385	clear
385 \(\Rightarrow\) 406	The axiom of choice and two particular forms of Tychonoff theorem, Alas, O. T. 1969, Portugal. Math.

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

Howard-Rubin Number	Statement
407:	Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\).
14:	BPI: Every Boolean algebra has a prime ideal.
385:	Countable Ultrafilter Theorem: Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.
406:	The product of compact Hausdorf spaces is countably compact. Alas [1994].

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