We have the following indirect implication of form equivalence classes:

218 \(\Rightarrow\) 425
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
218 \(\Rightarrow\) 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
218:

\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then  there  is  a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\).

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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