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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?

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up vote 7 down vote accepted

$\mathbb{R}$ 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.

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The proof of the first fact you quoted requires the axiom of choice yes? –  user38268 Sep 4 '12 at 0:22
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Yes, if you take away the axiom of choice, there is a model in which the opposite conclusion to the OP's original question holds (it's consistent with ZF that all group homomorphisms between Lie groups are continuous). –  user29743 Sep 4 '12 at 0:47
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