In this proposition:
Where in the proof is the closedness of the normal subgroup $H$ used?
2026-03-31 07:52:17.1774943537
If $H$ and $\frac GH$ are connected so is $G$
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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).