Axiom Of Choice

We have the following indirect implication of form equivalence classes:

424 \(\Rightarrow\) 194 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 \(\Rightarrow\) 194	clear

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.
194:	\(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979].

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