We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 407 \(\Rightarrow\) 14 |
Effective equivalents of the Rasiowa-Sikorski lemma, Bacsich, P. D. 1972b, J. London Math. Soc. Ser. 2. |
| 14 \(\Rightarrow\) 123 |
Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr. |
| 123 \(\Rightarrow\) 331 |
Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal |
| 331 \(\Rightarrow\) 332 |
Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal |
| 332 \(\Rightarrow\) 343 |
Topologie, Analyse Nonstandard et Axiome du Choix, Morillon, M. 1988, Universit\'e Blaise-Pascal |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 407: | Let \(B\) be a Boolean algebra, \(b\) a non-zero element of \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such that for each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there exists an ultrafilter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\), if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\). |
| 14: | BPI: Every Boolean algebra has a prime ideal. |
| 123: | \(SPI^*\): Uniform weak ultrafilter principle: For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\). |
| 331: | If \((X_i)_{i\in I}\) is a family of compact non-empty topological spaces then there is a family \((F_i)_{i\in I}\) such that \(\forall i\in I\), \(F_i\) is an irreducible closed subset of \(X_i\). |
| 332: | A product of non-empty compact sober topological spaces is non-empty. |
| 343: | A product of non-empty, compact \(T_2\) topological spaces is non-empty. |
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