Quotient Space of Real Line may not be Kolmogorov

Theorem
Let $(\R, \tau)$ be the real numbers with the usual topology.

Then there is an equivalence relation $\sim$ on $\R$ such that the quotient space $(\R / {\sim}, \tau_\sim)$ 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 $(\R / {\sim}, \tau_\sim)$ has two points and is indiscrete.

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