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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[topological groups|92700QAB5Y]].\nA *continuous group homomorphism* from $G$ to $H$ is a map $f \\colon G \\to H$ such that:\n\n1. $f$ is a [[group homomorphism|0JYTG5K9Y4]], that is, $f(xy) = f(x) f(y)$ for all $x, y \\in G$;\n2. $f$ is [[continuous|0VGFHTM75Y]].\n\n## Remarks\n\nExample: $f \\colon (\\R, +) \\to (\\C \\setminus \\set{0}, \\cdot)$, $f(t) = e^{2\\pi i t}$.\nIt satisfies $f(s + t) = f(s) f(t)$, and it is continuous as a composition of continuous functions.\nNon-example: the identity map from $(\\R, +)$ with the usual topology to $(\\R, +)$ with the discrete topology.\nIt is a homomorphism, but the preimage of the open set $\\set{0}$ is $\\set{0}$, which is not open in the usual topology.\nA homomorphism $f$ is continuous if and only if it is continuous at the identity $e$ of $G$.\nIndeed, near a point $a$ one writes $f(x) = f(a) f(a^{-1} x)$ and uses that translations are homeomorphisms.\n",
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