Axiom Of Choice

We have the following indirect implication of form equivalence classes:

388 \(\Rightarrow\) 155 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
388 \(\Rightarrow\) 106
106 \(\Rightarrow\) 78	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
78 \(\Rightarrow\) 155	Geordnete Lauchli Kontinuen, Brunner, N. 1983a, Fund. Math.

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.
78:	Urysohn's Lemma: If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.
155:	\(LC\): There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. Continuum \(\equiv\) compact, connected, Hausdorff space; and strongly connected \(\equiv\) every continuous real valued function is constant.)

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