Implications

We have the following indirect implication of form equivalence classes:

286 \(\Rightarrow\) 330
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 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 330 clear

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.

165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

330:

\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See Form 67.)

Comment:

Back