# Is the space of left cosets of a topological group completely regular?

Let $$G$$ be a (Hausdorff) topological group and $$H$$ its closed subgroup. Then it's known that $$G/H$$ is a regular topological space. Is it always completely regular?

A reference is good too.

In the case that $$H$$ is compact, we know that the quotient map $$\pi:G\to G/H$$ is perfect and open, therefore $$G/H$$ is completely regular (since it is an open-and-closed map; image of a completely regular space by a map that is both open and closed is completely regular; see Chaber J. "Remarks on open-and-closed mappings").