Axiom Of Choice

We have the following indirect implication of form equivalence classes:

16 \(\Rightarrow\) 32 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
16 \(\Rightarrow\) 352	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
352 \(\Rightarrow\) 32	Disasters in metric topology without choice, Keremedis, K. 2002, Comment. Math. Univ. Carolinae

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

Howard-Rubin Number	Statement
16:	\(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function.
352:	A countable product of second countable spaces is second countable.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.

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