Axiom Of Choice

We have the following indirect implication of form equivalence classes:

410

$\Rightarrow$ 10 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
410 $\Rightarrow$ 411	clear
411 $\Rightarrow$ 412	clear
412 $\Rightarrow$ 10	The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.

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

Howard-Rubin Number	Statement
410:	RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.
411:	RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.
412:	RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.
10:	$C(\aleph_{0},< \aleph_{0})$ : Every denumerable family of non-empty finite sets has a choice function.

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