Axiom Of Choice

We have the following indirect implication of form equivalence classes:

331 \(\Rightarrow\) 102 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
331 \(\Rightarrow\) 332	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
332 \(\Rightarrow\) 343	Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal
343 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 102	The Axiom of Choice, Jech, 1973b, page 162 problem 11.12

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

Howard-Rubin Number	Statement
331:	If \((X_i)_{i\in I}\) is a family of compact non-empty topological spaces then there is a family \((F_i)_{i\in I}\) such that \(\forall i\in I\), \(F_i\) is an irreducible closed subset of \(X_i\).
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.
102:	For all Dedekind finite cardinals \(p\) and \(q\), if \(p^{2} = q^{2}\) then \(p = q\). Jech [1973b], p 162 prob 11.12.

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