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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").

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This is an exercise 3.3.F in the book "Topological Groups and Related Structures" by Arhangel'skii and Tkachenko.

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