Axiom Of Choice

We have the following indirect implication of form equivalence classes:

114 \(\Rightarrow\) 389 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
114 \(\Rightarrow\) 90	Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 139
139 \(\Rightarrow\) 389

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
114:	Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
91:	\(PW\): The power set of a well ordered set can be well ordered.
79:	\({\Bbb R}\) can be well ordered. Hilbert [1900], p 263.
139:	Using the discrete topology on 2, \(2^{\cal P(\omega)}\) is compact.
389:	\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function. \ac{Keremedis} \cite{1999b}.

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