Implications

We have the following indirect implication of form equivalence classes:

411 \(\Rightarrow\) 288-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
411 \(\Rightarrow\) 412 clear
412 \(\Rightarrow\) 10 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
10 \(\Rightarrow\) 288-n clear

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

Howard-Rubin Number Statement
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.

288-n:

If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

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