I would like to check whether the quotient space is compact or not. I know compact space's quotient space is also compact because projection map is continuous. But, what about the quotient space which is not compact?
I would like to formally prove the following cases ($\mathbb{R}$'s topology is usual Euclid topology here).
$\mathbb{R}/\mathbb{Q}$ is compact
Thank you in advance.
There is a general result asserting that if $G$ is topological group and $H$ is a dense subgroup, then $G/H$ has the trivial topology, i.e. the only open sets are the empty set and $G/H$ itself. Thus, $\mathbb{R}/\mathbb{Q}$ has a trivial topology. In particular, it is compact.
For a proof of the above general result, see here.