We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
384 \(\Rightarrow\) 14 |
"Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math. |
14 \(\Rightarrow\) 99 |
Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr. |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
384: | Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter. |
14: | BPI: Every Boolean algebra has a prime ideal. |
99: | Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\). Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S. |
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