Axiom Of Choice

We have the following indirect implication of form equivalence classes:

123 \(\Rightarrow\) 154 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
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
343 \(\Rightarrow\) 154	Products of compact spaces and the axiom of choice II , Hall, E. 2002, Math. Logic Quart.

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 \(X_i\).
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.
154:	Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

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