Axiom Of Choice

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 421 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 \(\Rightarrow\) 421	Unions and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002B[2002B], 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 \(\cal V(x) \subseteq \cal B(x)\).
8:	\(C(\aleph_{0},\infty)\):
421:	\(UT(\aleph_0,WO,WO)\): The union of a denumerable set of well orderable sets can be well ordered.

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