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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[topological groups|92700QAB5Y]].\nAn *isomorphism of topological groups* from $G$ to $H$ is a map $f \\colon G \\to H$ that is both a [[group isomorphism|08MG7H8JW5]] and a [[homeomorphism|0QT00PB5G2]].\nEquivalently, $f$ is a bijective [[continuous group homomorphism|9J5PQMG3HJ]] whose inverse $f^{-1} \\colon H \\to G$ is continuous.\nThe groups $G$ and $H$ are *isomorphic* if such a map exists.\n\n## Remarks\n\nExample: $\\log \\colon (\\,(0, \\infty), \\cdot\\,) \\to (\\R, +)$.\nIt is bijective, $\\log(xy) = \\log x + \\log y$, and both $\\log$ and its inverse $\\exp$ are continuous.\nNon-example: the identity map from $(\\R, +)$ with the discrete topology to $(\\R, +)$ with the usual topology.\nIt is a continuous bijective homomorphism, but its inverse is not continuous, since $\\set{0}$ is open in the discrete topology and not in the usual one.\n",
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