Geodesic
A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 515-523.

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It is independent of the usual (ZFC) axioms of set theory whether every collectionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add ``or has an Aronszajn subtree,'' the statement remains ZFC-independent. This is done by constructing a tree as in the title, using the set-theoretic hypothesis $\diamondsuit^*$, which holds in Gödel's Constructible Universe.
Classification : 03E05, 54A35, 54E35, 54F05
Keywords: tree; collectionwise Hausdorff; metrizable; Aronszajn tree 
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Iwasa, Akira; Nyikos, Peter J. A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 515-523. http://geodesic.mathdoc.fr/item/CMUC_2006__47_3_a13/