tree; collectionwise Hausdorff; metrizable; Aronszajn tree
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.
 Devlin K.J., Shelah S.: Souslin properties and tree topologies
. Proc. London Math. Soc. (3) 39 (1979), 2 237-252. MR 0548979
| Zbl 0432.54029
 Iwasa A.: Metrizability of trees. doctoral dissertation, Department of Mathematics, University of South Carolina, 2001.
 Kunen K.: Set Theory: An Introduction to Independence Proofs
. North-Holland, Amsterdam, 1980. MR 0597342
| Zbl 0534.03026
 Nyikos P.J.: Metrizability, monotone normality, and other strong properties in trees
. Topology Appl. 98 (1999), 269-290. MR 1720006
| Zbl 0969.54026