Metrizability, monotone normality, and other strong properties in trees

Abstract

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.