Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
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
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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