Implications

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 222
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 \(\Rightarrow\) 63 clear
63 \(\Rightarrow\) 70 clear
70 \(\Rightarrow\) 222 The strength of the Hahn-Banach theorem, Pincus, D. 1972c, Lecture Notes in Mathematics

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.

63:

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

70:

There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24.

222:

There is a non-principal measure on \(\cal P(\omega)\).

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