Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
36 \(\Rightarrow\) 62	On Loeb and weakly Loeb Hausdorff spaces, Tachtsis, E. 2000, Math. Japon.
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
36:	Compact T\(_2\) spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)
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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