Implications

We have the following indirect implication of form equivalence classes:

66 \(\Rightarrow\) 381
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
66 \(\Rightarrow\) 67 Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
67 \(\Rightarrow\) 381 Disjoint unions of topological spaces and choice, Howard, P. 1998b, Math. Logic Quart.

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

Howard-Rubin Number Statement
66:

Every vector space over a field has a basis.

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.

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