Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
109 \(\Rightarrow\) 66 clear
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.
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
109:

Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.

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.

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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