Axiom Of Choice

We have the following indirect implication of form equivalence classes:

175 \(\Rightarrow\) 13 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
175 \(\Rightarrow\) 13	Horrors of topology without AC: A non-normal orderable space, van Douwen, E.K. 1985, Proc. Amer. Math. Soc. note-49

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

Howard-Rubin Number	Statement
175:	Transitivity Condition: For all sets \(x\), there is a set \(u\) and a function \(f\) such that \(u\) is transitive and \(f\) is a one to one function from \(x\) onto \(u\).
13:	Every Dedekind finite subset of \({\Bbb R}\) is finite.

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