Axiom Of Choice

We have the following indirect implication of form equivalence classes:

286 $\Rightarrow$ 78 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 $\Rightarrow$ 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 $\Rightarrow$ 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
43:	$DC(\omega)$ (DC), Principle of Dependent Choices: If $S$ is a relation on a non-empty set $A$ and $(\forall x\in A) (\exists y\in A)(x S y)$ then there is a sequence $a(0), a(1), a(2), \ldots$ of elements of $A$ such that $(\forall n\in\omega)(a(n)\mathrel S a(n+1))$. See Tarski [1948], p 96, Levy [1964], p. 136.
78:	Urysohn's Lemma: If $A$ and $B$ are disjoint closed sets in a normal space $S$, then there is a continuous $f:S\rightarrow [0,1]$ which is 1 everywhere in $A$ and 0 everywhere in $B$. Urysohn [1925], pp 290-292.

Comment:

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Implication	Reference
286 \(\Rightarrow\) 40	S´eminaire d’Analyse 1992, Morillon, 1991b,
40 \(\Rightarrow\) 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 78	The Axiom of Choice, Jech, [1973b] The Axiom of Choice, Jech, [1973b]

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.
43:	\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136.
78:	Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.