Implications

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 331
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
123 \(\Rightarrow\) 331 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
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 \(X_i\).

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