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If the identity element e of a topological group G has a compact neighbourhood, then every element of G has a compact neighbourhood also -- so G is locally compact:
The topology of a topological group is determined by the collection of neighborhoods of the identity element of the group

I see that the cited proof does not require the Hausdorffness of G.

Can you please explain me why then a locally compact group is usually defined as a group whose underlying topological space is locally compact and also Hausdorff?
https://en.wikipedia.org/wiki/Locally_compact_group

Why necessarily Hausdorff? Is this a mere convention, or will the definition become inconsistent if the Hausdorffness is dropped?

Michael_1812
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    Mere convention. – Tyrone Jan 19 '21 at 16:21
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    I presume, though, that many nice results about topological groups that require a point-set-topological proof will fail without requiring that the space is Hausdorff. For a topological group to be Hausdorff, it suffices that it is $T_1$ (edit: maybe even $T_0$), and on a space that even fails to be $T_1$ all your typical results will break down. – Jeroen van der Meer Jan 19 '21 at 16:24
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    Come to think of it, I don't really know any natural nontrivial examples of non-Hausdorff topological groups. – Jeroen van der Meer Jan 19 '21 at 16:24
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    Thank you, guys. I think both the comment by Tyrone and the last comment by @JeroenvanderMeer answer my question. – Michael_1812 Jan 19 '21 at 19:12
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    I believe that if $G$ is not Hausdorff, then the closure of ${1}$ is a normal subgroup and the quotient $G/\overline{{1}}$ is Hausdorff. Moreover anything topological one might say about $G$ should translate into something meaningful about $G/\overline{{1}}$ so the former may safely be ignored. – Ruy Jan 19 '21 at 19:30
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    With regards local compactness: there are several definitions which people use. One asks that each point have a compact neighbourhood. The more stringent definition asks that each point have a neighbourhood base of compact neighbourhoods. These two definitions agree in Hausdorff spaces and also in regular spaces. Every topological group is regular and hence these two definitions agree in this context. They are not equivalent elsewhere. – Tyrone Jan 19 '21 at 19:37

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