Implications

We have the following indirect implication of form equivalence classes:

411 \(\Rightarrow\) 374-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 \(\Rightarrow\) 423 clear
423 \(\Rightarrow\) 374-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.

423:

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

374-n:

\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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