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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nA *complex chart* on $X$ is a [[homeomorphism|05GZ11PWRY]] $\\varphi \\colon U \\to V$ from an open set $U \\subseteq X$ onto an open set $V \\subseteq \\C$.\nTwo charts $\\varphi_1 \\colon U_1 \\to V_1$ and $\\varphi_2 \\colon U_2 \\to V_2$ are *compatible* if the transition map\n$$\\varphi_2 \\circ \\varphi_1^{-1} \\colon \\varphi_1(U_1 \\cap U_2) \\to \\varphi_2(U_1 \\cap U_2)$$\nis [[holomorphic|0DG0Z72CQD]], and likewise $\\varphi_1 \\circ \\varphi_2^{-1}$.\nA *complex atlas* on $X$ is a family of pairwise compatible charts whose domains form an [[open cover|0NVPNC12JY]] of $X$.\nTwo atlases are equivalent if their union is again an atlas.\nA *complex structure* on $X$ is an equivalence class of complex atlases.\nA *Riemann surface* is a topological space $X$ with a complex structure such that:\n\n1. $X$ is [[Hausdorff|0MXB4CKHFW]], and\n2. $X$ is [[connected|0D939CMME7]].\n\n## Remarks\n\nSome authors also require $X$ to be second countable; by Radó's theorem this holds automatically for every Riemann surface.\nSo a Riemann surface is the same as a connected complex manifold of dimension one.\n\nExample: the [[Riemann sphere|0CDMXGPSES]] $\\hat\\C = \\C \\cup \\set{\\infty}$ with the two charts $\\varphi_1(z) = z$ on $\\C$ and $\\varphi_2(z) = 1/z$ on $\\hat\\C \\setminus \\set{0}$, where $\\varphi_2(\\infty) = 0$.\nOn $\\varphi_1(\\C \\setminus \\set{0}) = \\C \\setminus \\set{0}$ the transition map is $z \\mapsto 1/z$, which is holomorphic there; its inverse is the same map.\n\nNon-example: the closed unit disc $\\set{z : \\abs{z} \\le 1}$ is not a Riemann surface for any atlas.\nA boundary point has no open neighbourhood homeomorphic to an open subset of $\\C$, so the domains of charts cannot cover $X$.\n",
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