Axiom Of Choice

We have the following indirect implication of form equivalence classes:

16 \(\Rightarrow\) 419 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 \(\Rightarrow\) 31	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
31 \(\Rightarrow\) 419	Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.

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.
31:	\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
419:	UT(\(\aleph_0\),cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.)

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