We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 379
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
426 \(\Rightarrow\) 8 On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
8 \(\Rightarrow\) 379 clear

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

Howard-Rubin Number Statement
426:

If \((X,\cal T) \) is a first countable topological space and \((\cal B(x))_{x\in X}\) is a family such that for all \(x \in X\), \(\cal B(x)\) is a local base at \(x\), then there is a family \(( \cal V(x))_{x\in X}\) such that for every \(x \in X\), \(\cal V(x)\) is a countable local base at \(x\) and \(\cal V(x) \subseteq \cal B(x)\).

8:

\(C(\aleph_{0},\infty)\):

379:

\(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\).

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