Axiom Of Choice

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 114 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
149 \(\Rightarrow\) 67	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26
67 \(\Rightarrow\) 114	Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.

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

Howard-Rubin Number	Statement
149:	\(A(F)\): Every \(T_2\) topological space is a continuous, finite to one image of an \(A1\) space.
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).
114:	Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)

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