Implications

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 173
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 \(\Rightarrow\) 76 clear
76 \(\Rightarrow\) 173 Paracompactness of metric spaces and the axiom of choice, Howard, P. 2000a, 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).

76:

\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

173:

\(MPL\): Metric spaces are para-Lindelöf.

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