Polish metric spaces with fixed distance set

TL;DR

This paper investigates the complexity of classifying Polish metric spaces with a fixed set of possible distances, analyzing how properties of the set influence the classification difficulty and the structure of these spaces.

Contribution

It provides a detailed analysis of the complexity and classification of Polish spaces with fixed distances, including examples in various Wadge classes and conditions for belonging to specific classes.

Findings

Determines the complexity of the collection of Polish spaces with fixed distances.

Provides examples of sets in certain Wadge classes.

Characterizes properties of the set of distances that influence space classification.

Abstract

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in $A$, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of $A$, of the relations of isometry and isometric embeddability between these Polish spaces.