TITLE: Example of topological spaces $A \subset X$ with $X \setminus A$ not homeomorphic to $(X/A) \setminus (A/A)$
QUESTION [5 upvotes]: This is a homework question. I am asked to show that if $A \subset X$ is closed, then $X \setminus A$ is homeomorphic to $(X/A) \setminus (A/A)$. I have done this, and I now have to show by example that this is false if we do not require that $A$ is closed. Could someone point me in the right direction in finding such an example?

REPLY [5 votes]: HINT: Take $X=\Bbb R$ and $A=\Bbb Q$. (On further thought: it may be a little easier to take $A=\Bbb R\setminus\Bbb Q$.)