Axiom Of Choice

We have the following indirect implication of form equivalence classes:

388 \(\Rightarrow\) 131 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
388 \(\Rightarrow\) 106
106 \(\Rightarrow\) 126	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
126 \(\Rightarrow\) 131	clear

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

Howard-Rubin Number	Statement
388:	Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.
106:	Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
131:	\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

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