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A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees. (English) Zbl 1150.54005
Summary: 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.
##### MSC:
 54A35 Consistency and independence results in general topology 54E35 Metric spaces, metrizability 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
##### Keywords:
tree; collectionwise Hausdorff; metrizable; Aronszajn tree
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