Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 14 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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.