Implications

We have the following indirect implication of form equivalence classes:

332 \(\Rightarrow\) 288-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
332 \(\Rightarrow\) 343 Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
343 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 288-n clear

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

Howard-Rubin Number Statement
332:  

A product of non-empty compact sober topological spaces is non-empty.

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.

288-n:

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

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