# The Scott rank of Polish metric spaces

## TL;DR

This paper investigates the Scott rank in Polish metric spaces, showing compact spaces have rank at most omega, while some ultrametric spaces can have arbitrarily high countable Scott rank.

## Contribution

It extends the notion of Scott rank to Polish metric spaces and characterizes the rank bounds for compact and ultrametric spaces.

## Key findings

- Compact spaces have Scott rank at most ω.
- Existence of discrete ultrametric spaces with arbitrarily high countable Scott rank.

## Abstract

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 $\omega$, and that there are discrete ultrametric spaces of arbitrarily high countable Scott rank.