Axiom Of Choice

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 125 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 \(\Rightarrow\) 125	P-Raüme and Auswahlaxiom, Brunner, N. 1984c, Rend. Circ. Mat. Palermo.

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.
125:	There does not exist an infinite, compact connected \(p\) space. (A \(p\) space is a \(T_2\) space in which the intersection of any well orderable family of open sets is open.)

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