Implications

We have the following indirect implication of form equivalence classes:

334 \(\Rightarrow\) 382
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
334 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 381 Disjoint unions of topological spaces and choice, Howard, P. 1998b, Math. Logic Quart.
381 \(\Rightarrow\) 382 clear

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

Howard-Rubin Number Statement
334:

\(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 2\), 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 even.

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.

382:

DUMN:  The disjoint union of metrizable spaces is normal.

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