Implications

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 330
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 \(\Rightarrow\) 328 clear
328 \(\Rightarrow\) 330 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).

328:

\(MC(WO,\infty)\): For  every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)

330:

\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).  (See Form 67.)

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