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  "body": "\n## Statement\n\nLet $U$ be a nonempty open subset of $\\mathbb{C}$ with $U \\neq \\mathbb{C}$.\nSuppose that $U$ is simply connected: $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.\nLet $\\mathbb{D} = \\{ z \\in \\mathbb{C} : |z| \u003c 1 \\}$.\nThen there is a biholomorphic map $f$ from $U$ onto $\\mathbb{D}$, that is, a bijective [[holomorphic|0DG0Z72CQD]] map with a holomorphic inverse.\nMoreover, for each $z_0 \\in U$ there is exactly one such $f$ with $f(z_0) = 0$ and $f'(z_0)$ a positive real number.\n\n## Remarks\n\nA biholomorphic map is [[conformal|0VHNKP1N3K]], so $U$ and $\\mathbb{D}$ are conformally equivalent.\nThe case $U = \\mathbb{C}$ is excluded: a holomorphic map $\\mathbb{C} \\to \\mathbb{D}$ is bounded, so it is constant by [[Liouville's theorem|0EHKVRF4J6]].\nThe uniqueness follows from the [[Schwarz lemma|0MB48XDBY3]].\nRiemann stated the theorem in his thesis of 1851, and Osgood gave the first rigorous proof in 1900.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that every simply connected open set in the plane, except the plane itself, looks like the unit disc to complex analysis.\nThe set can have a very irregular boundary, and the map still exists.\nFor the upper half-plane, the [[Cayley transform|0X45ZM731G]] is such a map.\n",
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