Implications

We have the following indirect implication of form equivalence classes:

334 \(\Rightarrow\) 147
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
334 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 147 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
334:

\(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 2\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is even.

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).

147:

\(A(D2)\):  Every \(T_2\) topological space \((X,T)\) can be covered by a well ordered family of discrete sets.

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