Preservation of properties of a map by forcing
Let be a continuous map such as an open map, a closed map or a quotient map. We study under what circumstances remains an open, closed or quotient map in forcing extensions.
A non-metrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees
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 , which holds in Gödel’s Constructible Universe.
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