# Quotient map that is not closed

Can anyone help me find some example of a closed relation $\sim$ on a Hausdorff space $X$ such that the quotient map $p:X→X/\sim$ is not a closed map?

Here an equivalence relation $\sim$ is closed if the set $\{(x,y):x \sim y \}$ is closed.

• Note that if $X$ is compact, then a closed equivalence relation implies that the quotient map is closed. – Stefan Hamcke Apr 16 '16 at 17:46

Take $X = \mathbb{R} \times \mathbb{R}$ and define $(x_1,y_1) \sim (x_2,y_2)$ if $x_1 = x_2$. Then the quotient map is the projection $\pi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ taking $(x,y) \mapsto x$.
However, it is not closed, since the image of $xy = 1$ is $x \in \mathbb{R}$, $x \neq 0$, which is not closed in $\mathbb{R}$.
• What is $X$ and what is the equivalence relation $\sim$? – Henno Brandsma Apr 16 '16 at 8:28
• @Emily given two points with different first coordinates, you can find (disjoint) open sets around them $U$ and $V$ such that nothing in $U$ is equivalent to anything in $V$. – Forever Mozart Apr 16 '16 at 8:33