Quotient Space of Real Line may not be Kolmogorov

Theorem
Let $\left({\R, \tau}\right)$ be the real numbers with the usual (Euclidean) topology.

Then there exists an equivalence relation $\sim$ on $\R$ such that the quotient space $\left({\R / {\sim}, \tau_\sim}\right)$ is not Kolmogorov.

Proof
By Quotient Space of Real Line may be Indiscrete, there is an equivalence relation $\sim$ on $\R$ such that the quotient space $\left({\R / {\sim}, \tau_\sim}\right)$ has two points and is indiscrete.

It follows directly from the definition of Kolmogorov space that $\left({\R / {\sim}, \tau_\sim}\right)$ is not a Kolmogorov space.