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In this proposition:

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

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    $\begingroup$ Why did we assume $e \in A$? Where do we use this? $\endgroup$
    – roi_saumon
    Apr 30, 2020 at 23:11

1 Answer 1

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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).

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