Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
270 \(\Rightarrow\) 62	Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic
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
270:	\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.
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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