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I was reading about Lie groups, and whenever they are defining a Lie group homomorphism they state it is a group homomorphism between Lie groups that is continuous. I tried looking for a definition of continuity of a group homomorphism but I couldn't find any concrete definition. I might be wrong, but what I understood was that a group homomorphism between Lie groups is continuous when the codomain is connected. What condition should a group homomorphism of Lie groups have to accomplish so that I know it is continuous? Is there any generalized condition a group homomorphism between any two groups (not necessarily Lie) needs to satisfy so that we can say it is continuous in its domain? Thanks in advance.

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    $\begingroup$ A Lie group comes with an associated topology: saying that the morphism is continuous is simply saying that the application is continuous with respect to said topology $\endgroup$ Commented Jun 8, 2017 at 19:55
  • $\begingroup$ Thanks. I'll review more about the continuity of maps between differentiable manifolds that preserve a group structure. $\endgroup$
    – Cami77
    Commented Jun 8, 2017 at 20:49

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Since Lie groups are also manifolds, as the comment says they have a topology.

So if $G$ and $H$ are Lie groups, a map $f: G \rightarrow H$ is a continuous group homomorphism if it satisfies

$$f(xy)=f(x)f(y) \ \forall x, y\in G \textrm { (homomorphism) }$$ and $$\forall U \subset H \textrm{ open }, f^{-1}(U)\subset G \textrm{ is open } \textrm{ (continuity)}.$$

I think Lie groups are by definition smooth manifolds, and then a Lie group homomorphism would be a continuous group homomorphism that is in fact smooth and not just continuous.

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  • $\begingroup$ I agree, saying it's smooth seems more accurate for these homomorphisms. I only have an initial differential geometry background and reading about topology it's still a little tricky, but this helped. Thanks! $\endgroup$
    – Cami77
    Commented Jun 8, 2017 at 23:40

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