Axiom Of Choice

We have the following indirect implication of form equivalence classes:

218 \(\Rightarrow\) 106 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
218 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 106	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.

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

Howard-Rubin Number	Statement
218:	\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(\|f(x)\|\) is relatively prime to \(n\).
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

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