Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
302 \(\Rightarrow\) 392 clear
392 \(\Rightarrow\) 393 clear
393 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 119 clear

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

Howard-Rubin Number Statement
302:

Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.

392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

393:

\(C(LO,WO)\): Every linearly ordered set of non-empty well orderable sets has a choice function.

165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

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.

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