In the quotient $G/H$, why we must suppose that $H$ is closed?

Question

We have the following known statement:

Theorem: If $G$ is a topological group and $H\subseteq G$ is a closed invariant subgroup of $G$, then $G/H$ (of course with the quotient topology) is a topological group.

I'm reading the proof of this over and over again and I don't see why we must suppose that $H$ is closed. Why it's neccesary? I don't want an example of a quotient $G/H$ that isn't a topological group and $H$ is not closed. What I want to know is where (in the proof) we use that $H$ is closed in order to prove that $G/H$ is a topological group.

Any suggestion?

Thanks.

Answer 1

You need that $H$ is closed for $G/H$ to be a Hausdorff group (if $G$ is Hausdorff). Perhaps the author you're reading assumes this as part of the definition of a topological group.