Quotient Space of Real Line may not be Kolmogorov

Theorem

Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.

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

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

$\blacksquare$