Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
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
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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