Axiom Of Choice

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 173 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
426 \(\Rightarrow\) 76	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
76 \(\Rightarrow\) 173	Paracompactness of metric spaces and the axiom of choice, Howard, P. 2000a, Math. Logic Quart.

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 \(\cal V(x) \subseteq \cal B(x)\).
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 \(x\).
173:	\(MPL\): Metric spaces are para-Lindelöf.

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