Axiom Of Choice

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 416 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
149 \(\Rightarrow\) 67	The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic note-26
67 \(\Rightarrow\) 144	Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 416	Constructive order theory, Ern'e, M. 2001, Math. Logic Quart.

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

Howard-Rubin Number	Statement
149:	\(A(F)\): Every \(T_2\) topological space is a continuous, finite to one image of an \(A1\) space.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
144:	Every set is almost well orderable.
416:	Every non-compact topological space \(S\) is the union of a set that is well-ordered by inclusion and consists of open proper subsets of \(S\).

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