Axiom Of Choice

We have the following indirect implication of form equivalence classes:

292 \(\Rightarrow\) 119 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
292 \(\Rightarrow\) 90	The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. Math.
90 \(\Rightarrow\) 118	Horrors of topology without AC: A non-normal orderable space, van Douwen, E.K. 1985, Proc. Amer. Math. Soc.
118 \(\Rightarrow\) 119	Horrors of topology without AC: A non-normal orderable space, van Douwen, E.K. 1985, Proc. Amer. Math. Soc.

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

Howard-Rubin Number	Statement
292:	\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
118:	Every linearly orderable topological space is normal. Birkhoff [1967], p 241.
119:	van Douwen's choice principle: \(C(\aleph_{0}\),uniformly orderable with order type of the integers): Suppose \(\{ A_{i}: i\in\omega\}\) is a set and there is a function \(f\) such that for each \(i\in\omega,\ f(i)\) is an ordering of \(A_{i}\) of type \(\omega^{*}+\omega\) (the usual ordering of the integers), then \(\{A_{i}: i\in\omega\}\) has a choice function.

Comment:

Back