Axiom Of Choice

We have the following indirect implication of form equivalence classes:

239 \(\Rightarrow\) 147 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
239 \(\Rightarrow\) 427	clear
427 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 147	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26

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

Howard-Rubin Number	Statement
239:	AL20(\(\mathbb Q\)): Every vector \(V\) space over \(\mathbb Q\) has the property that every linearly independent subset of \(V\) can be extended to a basis. Rubin, H./Rubin, J. [1985], p.119, AL20.
427:	\(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis. \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
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).
147:	\(A(D2)\): Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

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