Axiom Of Choice

We have the following indirect implication of form equivalence classes:

286 \(\Rightarrow\) 362 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 \(\Rightarrow\) 361	Zermelo's Axiom of Choice, Moore, 1982, page 325
361 \(\Rightarrow\) 362	Zermelo's Axiom of Choice, Moore, 1982, page 325

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:	\(C(\aleph_{0},\infty)\):
361:	In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.
362:	In \(\Bbb R\), every Borel set is analytic. G. Moore [1982], pp 181 and 325.

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