Implications

We have the following indirect implication of form equivalence classes:

292 \(\Rightarrow\) 118
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.

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.

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