We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 426 \(\Rightarrow\) 8 |
On first and second countable spaces and the axiom of choice, Gutierres, G 2004, Topology and its Applications. |
| 8 \(\Rightarrow\) 9 | Was sind und was sollen die Zollen?, Dedekind, [1888] |
| 9 \(\Rightarrow\) 10 | Zermelo's Axiom of Choice, Moore, 1982, 322 |
| 10 \(\Rightarrow\) 423 | clear |
| 423 \(\Rightarrow\) 374-n | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 426: | If \((X,\cal T) \) is a first countable topological space and \((\cal B(x))_{x\in X}\) is a family such that for all \(x \in X\), \(\cal B(x)\) is a local base at \(x\), then there is a family \(( \cal V(x))_{x\in X}\) such that for every \(x \in X\), \(\cal V(x)\) is a countable local base at \(x\) and \(\cal V(x) \subseteq \cal B(x)\). |
| 8: | \(C(\aleph_{0},\infty)\): |
| 9: | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
| 10: | \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. |
| 423: | \(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function. |
| 374-n: | \(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable. |
Comment: