A Non-Metrizable Collectionwise Hausdorff Tree with No Uncountable Chains and Aronszajn Subtrees

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It is independent of the usual (ZFC) axioms of set theory whether every collec- tionwise 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 ♦∗, which holds in Gödel’s Constructible Universe.