Katia Fokina

The Scott rank of Polish metric spaces

We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q > 0$, there is a relation $R_{<q}(x,y)$ stating that the distance of $x$ and $y $ is less than $q$. We show that compact spaces have Scott rank at most $ω$, and that there are discrete ultrametric spaces of arbitrarily high countable Scott rank.