On Kolmogorov quotients

Abstract

<p>Every topological space has a Kolmogorov quotient that is obtained by identifying points if they 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. The author's interest in the subject was sparked by study of abstract model theory, specically the paper [1] by X. Caicedo, where Kolmogorov quotients are used in a topological proof of Lindström's theorem. We omit the proofs in this extended abstract; a full version [2] with detailed proofs is in preparation.<br /></p>