Axiom Of Choice

We have the following indirect implication of form equivalence classes:

16 \(\Rightarrow\) 374-n 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 \(\Rightarrow\) 32	L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
32 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 423	clear
423 \(\Rightarrow\) 374-n	clear

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.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
423:	\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.
374-n:	\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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