A normal collectionwise Hausdorff, not collectionwise normal space

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.

Topology Proceedings 1 (1976) pp. 295-304: WEAKLY COLLECTIONWISE HAUSDORFF SPACES

Topology Proceedings 2 (1977) pp. 583-592: REMARKS ON lambda-COLLECTIONWISE HAUSDORFF SPACES

A Note on Complete Collectionwise Normality and Paracompactness

1. A question that has aroused considerable interest and which has remained unanswered is the following. Is a normal Moore space metrizable? Both R. H. Bing and F. B. Jones have important results which go a long way toward answering this question. For example, Bing has proved that a collectionwise normal Moore space is metrizable [l ] while Jones has shown that a separable normal Moore space is...

The Constructible Universe

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's ax...