Quotient of a topological group

Summary

The quotient of a topological group $G$ by a subgroup $H$ is the set $G/H$ of left cosets with the quotient topology. The projection $G \to G/H$ is continuous and open. The quotient is Hausdorff exactly when $H$ is closed, and when $H$ is normal the quotient is again a topological group. Quotients give homogeneous spaces such as spheres, which are quotients of rotation groups.