Can we classify complete metric spaces up to isometry?

TL;DR

This paper surveys and introduces new results on classifying complete metric spaces up to isometry, covering both separable and non-separable cases, with simplified proofs and new research directions.

Contribution

It provides a unified presentation of existing results on separable spaces and introduces novel findings and methods for non-separable spaces in metric space classification.

Findings

Existing theorems on separable spaces summarized

New results and techniques for non-separable spaces introduced

Simplified proofs of key theorems provided

Abstract

We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research.