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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[metric spaces|063W5JMY85]].\nAn isometric isomorphism from $X$ to $Y$ is a [[bijective|0SNQEF7Z53]] [[isometry|0878DN3WCS]] $f \\colon X \\to Y$.\nThe spaces $X$ and $Y$ are isometric when there is an isometric isomorphism from $X$ to $Y$.\n\n## Remarks\n\nBecause each isometry is injective, an isometry is an isometric isomorphism if and only if it is surjective.\nThe inverse of an isometric isomorphism is again an isometric isomorphism, and the composition of two of them is one.\nEach isometric isomorphism is a [[homeomorphism|05GZ11PWRY]].\nThe map $x \\mapsto x + 1$ is an isometric isomorphism from $[0, 1]$ to $[1, 2]$ with the usual metric.\n\n## Explanations\n\n### L1 human\n\nIsometric spaces are copies of each other: a bijection matches their points, and it keeps all the distances.\nEach fact about distances in one space is also true in the other space.\n",
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