The rank problem for metric spaces

Abstract: We construct a complete, second countable metric space with uncountable Scott rank. This answers a well-known question posed by Fokina, Friedman, K{ö}rwien and Nies, and shows that the known upper bound $ω_1$ on the rank, proved by Doucha, is sharp. Further, a Baire category based variant of the rank of metric spaces is developed, and this is used to give new proofs of the known rank bounds based only on standard methods of descriptive set theory.