Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
286 \(\Rightarrow\) 40	S´eminaire d’Analyse 1992, Morillon, 1991b,
40 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 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
286:	Extended Krein-Milman Theorem: Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point. H. Rubin/J. Rubin [1985], p. 177-178.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
418:	DUM(\(\aleph_0\)): The countable disjoint union of metrizable spaces is metrizable.

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