TITLE: If $H$ and $\frac GH$ are connected so is $G$
QUESTION [8 upvotes]: In this proposition:

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

REPLY [8 votes]: 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).