$X$ Hausdorff and $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ is closed implies quotient map is open.

Question

Let $∼$ be an equivalence relation on a topological space X. $\ Y = X/∼ $ equipped with the quotient topology. How to show that if X is Hausdorff and the set $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ is closed then the quotient map is open.

Could someone possibly provide some hints in proving this statement.

Answer 1

This is false, I think: take $X = \mathbb{R}$, $A = \mathbb{Z}$ and let $\sim$ be the relation that identifies $A$ to a point. Then the relation as a subset of $X \times X$ is $(A \times A) \cup \Delta_X$, so this is closed, and $X$ is clearly Hausdorff.

The space $X/\negthinspace\negthinspace\sim$ is not first countable at the point that corresponds to $A$, and if $q$ were open, its image would even be second countable.