Axiom Of Choice

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 419 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
113 \(\Rightarrow\) 8	Tychonoff's theorem implies AC, Kelley, J.L. 1950, Fund. Math. Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
8 \(\Rightarrow\) 27	clear
27 \(\Rightarrow\) 31	clear
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
113:	Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.
8:	\(C(\aleph_{0},\infty)\):
27:	\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.
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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