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  "body": "\n## Definition\n\nA *Riemann domain over $\\C^n$* is a pair $(X, \\pi)$ where $X$ is a [[connected|0V032VWTHD]] [[Hausdorff space|0MXB4CKHFW]] and $\\pi \\colon X \\to \\C^n$ is a [[local homeomorphism|MDH71ZZ8XX]].\nLet $W \\subseteq X$ be open and $f \\colon W \\to \\C$ a function.\nThen $f$ is *holomorphic* if $f \\circ (\\pi|_V)^{-1}$ is a [[holomorphic function|507C34W0ZT]] on $\\pi(V)$ for every open set $V \\subseteq W$ on which $\\pi$ is injective.\nThe Riemann domain is *schlicht* if $\\pi$ is injective.\n\n## Remarks\n\nThe charts $\\pi|_V$ make $X$ a complex manifold of dimension $n$ on which $\\pi$ is locally biholomorphic.\nA schlicht Riemann domain is the same as a domain in $\\C^n$, through the identification of $X$ with $\\pi(X)$.\n\nExample: $X = \\C^2$ with $\\pi(w, z_2) = (e^w, z_2)$.\nThe space $X$ is connected and Hausdorff, and $\\pi$ is a local homeomorphism because $w \\mapsto e^w$ is a local homeomorphism of $\\C$ onto $\\C \\setminus \\set{0}$.\nIt is not schlicht, since $\\pi(w + 2\\pi i, z_2) = \\pi(w, z_2)$.\n\nNon-example: $X = \\C^2$ with $\\pi(w, z_2) = (w^2, z_2)$.\nHere $\\pi$ is not injective on any neighbourhood of a point $(0, z_2)$, because $(w, z_2)$ and $(-w, z_2)$ have the same image; so $\\pi$ is not a local homeomorphism there.\n",
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