I have been working with Topological Groups and have noticed that the subgroups of real interest in the topic are the closed subgroups. I figured this is because closed subgroups inherit most of the nice properties of the groups we work on and the quotients as well. I just wanted to know if there was any other reasons that I may be overlooking or unaware of?
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2As you said, it is nice. For example, the Closed Subgroup Theorem. – Dietrich Burde Sep 14 '22 at 19:52
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2It's because we want the quotient to be Hausdorff: https://math.stackexchange.com/questions/2787022/quotient-of-a-topological-group-by-a-closed-subgroup-is-hausdorff – Qiaochu Yuan Sep 14 '22 at 20:21
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1Also worth pointing out is the fact that open subgroups are closed – Alessandro Codenotti Sep 15 '22 at 13:18