Axiom Of Choice

We have the following indirect implication of form equivalence classes:

426 \(\Rightarrow\) 423 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 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 423	clear

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:	\(C(\aleph_{0},\infty)\):
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
423:	\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

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