Definition of a continuous group homomorphism

Question

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.

Answer 1

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.