Implications

We have the following indirect implication of form equivalence classes:

332 \(\Rightarrow\) 132
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 \(\Rightarrow\) 132 Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic

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.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

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