Axiom Of Choice

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 382 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 \(\Rightarrow\) 381	Disjoint unions of topological spaces and choice, Howard, P. 1998b, Math. Logic Quart.
381 \(\Rightarrow\) 382	clear

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).
381:	DUM: The disjoint union of metrizable spaces is metrizable.
382:	DUMN: The disjoint union of metrizable spaces is normal.

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