Axiom Of Choice

We have the following indirect implication of form equivalence classes:

192 \(\Rightarrow\) 375 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
192 \(\Rightarrow\) 43	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
43 \(\Rightarrow\) 375

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
192:	\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).
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.
375:	Tietze-Urysohn Extension Theorem: If \((X,T)\) is a normal topological space, \(A\) is closed in \(X\), and \(f: A\to [0,1]\) is continuous, then there exists a continuous function \(g: X\to [0,1]\) which extends \(f\).

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