Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
218 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 381	Disjoint unions of topological spaces and choice, Howard, P. 1998b, Math. Logic Quart.
381 \(\Rightarrow\) 418	Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart.
418 \(\Rightarrow\) 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
218:	\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(\|f(x)\|\) is relatively prime to \(n\).
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
381:	DUM: The disjoint union of metrizable spaces is metrizable.
418:	DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.
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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