Open Projection and Closed Graph Implies Quotient is Hausdorff

Theorem
Let $\mathcal R\subset X\times X$ be an equivalence relation on a topological space $X$.

Let $X/\mathcal R$ be the quotient space.

Let $p$ denote the quotient mapping.

Let:
 * $\mathcal R$ be closed in $X\times X$
 * $p$ be open

Then $X/\mathcal R$ is Hausdorff.

Also see

 * Hausdorff Space iff Diagonal Set on Product is Closed