We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 335-n \(\Rightarrow\) 333 |
Bases for vector spaces over the two element field and the axiom of choice, Keremedis, K. 1996a, Proc. Amer. Math. Soc. |
| 333 \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 76 | clear |
| 76 \(\Rightarrow\) 425 |
On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 335-n: | Every quotient group of an Abelian group each of whose elements has order \(\le n\) has a set of representatives. |
| 333: | \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of 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 \(x\) and \(|f(x)|\) is odd. |
| 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 \(x\). |
| 425: | For every first countable topological space \((X, \Cal T)\) there is a family \((\Cal D(x))_{x \in X}\) such that \(\forall x \in X\), \(D(x)\) countable local base at \(x\). \ac{Gutierres} \cite{2004} and note 159. |
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