Implications

We have the following indirect implication of form equivalence classes:

303 \(\Rightarrow\) 386
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
303 \(\Rightarrow\) 50 Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math.
50 \(\Rightarrow\) 14 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
14 \(\Rightarrow\) 385 clear
385 \(\Rightarrow\) 386 Products, the Baire category theorem, and the axiom of dependent choice, Herrlich-Keremedis-1999a[1999a], Topology and its Applications.

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

Howard-Rubin Number Statement
303:

If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\).

50:

Sikorski's  Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.

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.

386:

Every B compact (pseudo)metric space is Baire.

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