Quotient of a topological group

Remarks

The projection $\pi$ is continuous and an open map: for $U$ open in $G$, the set $\pi^{-1}(\pi(U)) = UH$ is a union of the open sets $Uh$. If $H$ is normal, then $G/H$ is a topological group, and $\pi$ is a continuous homomorphism. The space $G/H$ is Hausdorff if and only if $H$ is closed in $G$. Example: $\R/\Z$ is a compact Hausdorff topological group, isomorphic to the circle group through $x + \Z \mapsto e^{2\pi i x}$. Non-example of the Hausdorff property: $\R/\Q$ carries the indiscrete topology, because $\Q$ is dense, so every nonempty open set of $\R$ meets every coset of $\Q$.