Implications

We have the following indirect implication of form equivalence classes:

386 \(\Rightarrow\) 18
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
386 \(\Rightarrow\) 10 Products, the Baire category theorem, and the axiom of dependent choice, Herrlich-Keremedis-1999a[1999a], Topology and its Applications.
10 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 18 clear

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

Howard-Rubin Number Statement
386:

Every B compact (pseudo)metric space is Baire.

10:

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

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

18:

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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