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.