Projection homomorphism of an abelian group is closed

Question

Let $G$ be a an abelian topological group and let $\Gamma$ be a discrete subgroup. Is the projection homomorphism $\pi:G\to G/\Gamma$ closed?

Answer 1

No. For instance, taking $G=\mathbb{R}$ and $\Gamma=\mathbb{Z}$, the set $\{n+1/2^n:n\in\mathbb{Z}_+\}$ is closed in $\mathbb{R}$ but is image in $\mathbb{R}/\mathbb{Z}$ is not.