Implications

We have the following indirect implication of form equivalence classes:

407 \(\Rightarrow\) 298
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 \(\Rightarrow\) 298 Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math.
Projective topological spaces, Gleason, A.M. 1958, Illinois J. 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.

298:

Every compact Hausdorff space has a Gleason cover.

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