Axiom Of Choice

We have the following indirect implication of form equivalence classes:

239 \(\Rightarrow\) 425 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 \(\Rightarrow\) 76	clear
76 \(\Rightarrow\) 425	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.

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).
76:	\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).
425:	For every first countable topological space \((X, \Cal T)\) there is a family \((\Cal D(x))_{x \in X}\) such that \(\forall x \in X\), \(D(x)\) countable local base at \(x\). \ac{Gutierres} \cite{2004} and note 159.

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