Axiom Of Choice

We have the following indirect implication of form equivalence classes:

334 \(\Rightarrow\) 232 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
334 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 232	Paracompactness of metric spaces and the axiom of choice, Howard, P. 2000a, Math. Logic Quart.

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

Howard-Rubin Number	Statement
334:	\(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of sets such that for all \(x\in X\), \(\|x\|\ge 2\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(\|f(x)\|\) is even.
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).
232:	Every metric space \((X,d)\) has a \(\sigma\)-point finite base.

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