Axiom Of Choice

We have the following indirect implication of form equivalence classes:

43 \(\Rightarrow\) 78 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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
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.

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