Two Lie groups which are isomorphic but not homeomorphic

Question

I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does anyone have an example or a proof?

Answer 1

$\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as groups because both are vector spaces over $\mathbb{Q}$ of the same dimension, but they are not homeomorphic as topological spaces because the former can be disconnected by removing a point and the latter cannot.