If $H$ and $\frac GH$ are connected so is $G$

Question

In this proposition:

Where in the proof is the closedness of the normal subgroup $H$ used?

Answer 1

It's not used or required. But it's common to only talk about $G/H$ when $H$ is a closed subgroup, because otherwise $G/H$ will not be Hausdorff. In particular, the book in which that proof appears defines topological groups to be Hausdorff (see page 84).