Isomorphism of almost locally compact Polish metric structures

TL;DR

This paper proves that isomorphism of almost locally compact Polish metric structures can be classified by countable structures, extending previous results and resolving a question about the classification of isometry in locally compact Polish spaces.

Contribution

It generalizes a known classification result to a broader class of structures and corrects a gap in prior work on the isometry classification problem.

Findings

Isomorphism on Borel classes is classifiable by countable structures.

Extends classification results to almost locally compact Polish metric structures.

Resolves an open question about isometry classification of locally compact Polish spaces.

Abstract

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always classifiable by countable structures. This allows to remove a gap in the proof presented in \cite{Ma} of a positive answer to a question of Gao and Kechris, who asked whether isometry of locally compact Polish metric spaces is classifiable by countable structures.