We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 303 \(\Rightarrow\) 50 |
Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math. |
| 50 \(\Rightarrow\) 14 |
A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar. |
| 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 |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 303: | If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\). |
| 50: | Sikorski's Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141. |
| 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. |
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