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 \(\Rightarrow\) 29 | Zermelo's Axiom of Choice, Moore, 1982, page 324 |
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: | \(C(\aleph_{0},\infty)\): |
| 29: | If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup^{}_{x\in S} A_{x}| = |\bigcup^{}_{x\in S} B_{x}|\). Moore, G. [1982], p 324. |
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