Quotient Space of Hausdorff Space is not necessarily Hausdorff

Theorem
Let $T = \left({S, \tau}\right)$ be a Hausdorff space.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $T_\mathcal R := \left({S / \mathcal R, \tau_\mathcal R}\right)$ be the quotient space of $S$ by $\mathcal R$.

Then $T_\mathcal R$ is not necessarily also a Hausdorff space.

Proof
Consider the Euclidean space $\left({\R, \tau}\right)$, where $\R$ is the real number line and $\tau$ the topology induced by the Euclidean metric.

By Real Number Space satisfies all Separation Axioms, $\left({\R, \tau}\right)$ is a Hausdorff space.

By Quotient Space of Real Line may not be Kolmogorov, there is a relation $\mathcal R$ on $\R$ such that the quotient space $\left({\R / \mathcal R, \tau_\mathcal R}\right)$ is not a Kolmogorov space.

Thus the theorem holds by Sequence of Implications of Separation Axioms.