# If $A$ is closed in $X\times X$ and $p$ is an open map, then $X/\sim$ is Hausdorff

Let $$X$$ be a Hausdorff space and $$\sim$$ an equivalence relation on $$X$$.
Let $$A=\{(x, y)\in X\times X\mid x\sim y\}\subseteq X\times X$$. Prove:

Let $$p:X\to X/\sim$$ denote the canonical projection. If $$A$$ is closed in $$X\times X$$ and $$p$$ is an open map, then $$X/\sim$$ is Hausdorff.

Take $$p(x), p(y) \in X / \sim$$ distinct, so that $$(x, y) \in (X \times X) - A$$. Because $$A$$ is closed, we can find an open neighborhood of $$(x, y)$$ of the form $$U \times V$$, disjoint from $$A$$. Then $$p(U)$$ and $$p(V)$$ are disjoint open neighborhoods of $$p(x)$$ and $$p(y)$$.

• How Can i by that conclude that X/~ is Hausdorff? – Jasmin Dec 10 '19 at 23:57
• By showing that any two points admit disjoint open neighborhoods – Bart Michels Dec 11 '19 at 11:48