Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
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
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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