Kolmogorov quotient

Definition

The Kolmogorov quotient of a topological space is defined as its quotient by the equivalence relation of topological indistinguishability, equipped with the quotient topology. Equivalently, it is the image of the initial object with respect to quotient maps to Kolmogorov spaces (which are $T_{0}$ topological spaces).

The Kolmogorov quotient of any topological space is a Kolmogorov space, and every Kolmogorov space is its own Kolmogorov quotient.

Relation between properties and Kolmogorov quotients

Property of topological spaces	Meaning	Property of being topological space whose Kolmogorov quotient satisfies the other property	Meaning
Kolmogorov space	any two points are topologically distinguishable	any topological space	any topological space
T1 space	points are closed	symmetric space	given any two topologically distinguishable points, there is an open subset containing the first but not the second.
Hausdorff space	distinct points can be separated by disjoint open subsets	preregular space	any two topologically distinguishable points can be separated by disjoint open subsets
regular Hausdorff space		regular space