Implications

We have the following indirect implication of form equivalence classes:

333 \(\Rightarrow\) 418
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
333 \(\Rightarrow\) 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.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

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.

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