Axiom Of Choice

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 146 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
213 \(\Rightarrow\) 85	clear
85 \(\Rightarrow\) 62	clear
62 \(\Rightarrow\) 146	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26

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

Howard-Rubin Number	Statement
213:	\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(\|y\| = \aleph_{1})\) then \(X\) has a choice function.
85:	\(C(\infty,\aleph_{0})\): Every family of denumerable sets has a choice function. Jech [1973b] p 115 prob 7.13.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
146:	\(A(F,A1)\): For every \(T_2\) topological space \((X,T)\), if \(X\) is a continuous finite to one image of an A1 space then \((X,T)\) is an A1 space. (\((X,T)\) is A1 means if \(U \subseteq T\) covers \(X\) then \(\exists f : X\rightarrow U\) such that \((\forall x\in X) (x\in f(x)).)\)

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