Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
406 \(\Rightarrow\) 10	The axiom of choice and two particular forms of Tychonoff theorem, Alas, O. T. 1969, Portugal. Math.
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
406:	The product of compact Hausdorf spaces is countably compact. Alas [1994].
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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