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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.

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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 $p(y)$.

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

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