Implications

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 270
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
384 \(\Rightarrow\) 14 "Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
14 \(\Rightarrow\) 270 Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic

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

Howard-Rubin Number Statement
384:

Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.

14:

BPI: Every Boolean algebra has a prime ideal.

270:

\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

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