Axiom Of Choice

We have the following indirect implication of form equivalence classes:

424 \(\Rightarrow\) 19 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
424 \(\Rightarrow\) 94	On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications.
94 \(\Rightarrow\) 34	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
34 \(\Rightarrow\) 19	Sur les fonctions representables analytiquement, Lebesgue, H. 1905, J. Math. Pures Appl.

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

Howard-Rubin Number	Statement
424:	Every Lindel\"{o}f metric space is super second countable. \ac{Gutierres} \cite{2004} and note 159. \iput{super second countable}
94:	\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals has a choice function. Jech [1973b], p 148 prob 10.1.
34:	\(\aleph_{1}\) is regular.
19:	A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1).

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