Let $X$ be a topological space, let $\sim$ be an equivalence relation on $X$, and let $\pi: X \to X/{\sim}$ be the quotient map. The quotient space $X/{\sim}$ is Hausdorff if and only if the set
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\begin{align*}
R = \{(x, y) \in X \times X : x \sim y\}
\end{align*}
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is closed in $X \times X$ (with the product topology).