Implications

We have the following indirect implication of form equivalence classes:

343 \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
343 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 389 clear

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

Howard-Rubin Number Statement
343:

A product of non-empty, compact \(T_2\) topological spaces is non-empty.

62:

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

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.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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