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I suspect this isn't a terribly difficult question, but I don't know the answer and I'd guess someone has already looked into it.

Is it possible for a Lie group on a non-Hausdorff manifold to exist? To be a bit more precise (since the above is nonstandard terminology), do 2nd countable locally Euclidean topological groups exist? If they do dropping the requirement of 2nd countability, that would also be interesting, but such spaces can actually be rather messy so a 2nd countable example is preferred if it exists.

The question is little more than a curiosity, but I found that I couldn't answer it with any standard examples, nor could I come up with an obvious proof.

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up vote 13 down vote accepted

A topological group is Hausdorff if and only if it is $T_1$. Any locally Euclidean space is $T_1$, so the answer is no.

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'doh! I knew this particular theorem, but I wasn't even thinking about applying it. –  Logan Maingi Sep 18 '11 at 6:01
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