Implications

We have the following indirect implication of form equivalence classes:

14 \(\Rightarrow\) 423
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
14 \(\Rightarrow\) 153 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
153 \(\Rightarrow\) 10 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
10 \(\Rightarrow\) 423 clear

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.

153:

The closed unit ball of a Hilbert space is compact in the weak topology.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

423:

\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in  \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

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