Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
109 \(\Rightarrow\) 66	clear
66 \(\Rightarrow\) 67	Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
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
109:	Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.
66:	Every vector space over a field has a basis.
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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