Axiom Of Choice

We have the following indirect implication of form equivalence classes:

113 \(\Rightarrow\) 418 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 \(\Rightarrow\) 418	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:	\(C(\aleph_{0},\infty)\):
418:	DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.

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