A survey of Kolmogorov quotients

TL;DR

This survey explores the properties and relationships of Kolmogorov quotients across various topological spaces, highlighting cases where they are homeomorphic or relate to metric spaces, with applications to groups and uniform spaces.

Contribution

It provides a comprehensive overview of the relationship between topological spaces and their Kolmogorov quotients, including new insights into pseudometric spaces and topological groups.

Findings

Kolmogorov quotient often homeomorphic to original space

In pseudometric spaces, the quotient is a metric space

Application to topological groups and uniform spaces

Abstract

Every topological space has a Kolmogorov quotient that is obtained by identifying topologically indistinguishable points, that is, points that are contained in exactly the same open sets. In this survey, we look at the relationship between topological spaces and their Kolmogorov quotients. In most natural examples of spaces, the Kolmogorov quotient is homeomorphic to the original space. A non-trivial relationship occurs, for example, in the case of pseudometric spaces, where the Kolmogorov quotient is a metric space. We also look at the topological indistinguishability relation in the context of topological groups and uniform spaces.