Axiom Of Choice

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 362 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 \(\Rightarrow\) 361	Zermelo's Axiom of Choice, Moore, 1982, page 325
361 \(\Rightarrow\) 362	Zermelo's Axiom of Choice, Moore, 1982, page 325

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:	\(C(\aleph_{0},\infty)\):
361:	In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.
362:	In \(\Bbb R\), every Borel set is analytic. G. Moore [1982], pp 181 and 325.

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