3
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ Note that if $X$ is compact, then a closed equivalence relation implies that the quotient map is closed. $\endgroup$ – Stefan Hamcke Apr 16 '16 at 17:46
8
$\begingroup$

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 $\mathbb{R}$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ What is $X$ and what is the equivalence relation $\sim$? $\endgroup$ – Henno Brandsma Apr 16 '16 at 8:28
  • $\begingroup$ But how can we check that this relation is closed? $\endgroup$ – Emily Apr 16 '16 at 8:31
  • $\begingroup$ @HennoBrandsma I have given an equivalence. I hope it is clear now. $\endgroup$ – Seven Apr 16 '16 at 8:33
  • 1
    $\begingroup$ @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 $V$. $\endgroup$ – Forever Mozart Apr 16 '16 at 8:33
  • 1
    $\begingroup$ I saw I didn't answer this part in math.stackexchange.com/a/1744671/4280, so I added the general facts there as well. $\endgroup$ – Henno Brandsma Apr 16 '16 at 8:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.