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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]].\nA point $x \\in X$ is *normal* if the [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is an [[integrally closed domain|H20P600HQS]].\nThe space $X$ is *normal* if every point of $X$ is normal.\n\n## Remarks\n\nA normal space is [[reduced|831N2Q4GCB]], since an integral domain has no nonzero nilpotents.\n\nExample: the cusp $A = \\set{z_1^2 = z_2^3} \\subseteq \\C^2$ is not normal at $0$.\nIts local ring is isomorphic to $\\C\\{t^2, t^3\\}$ by $z_1 \\mapsto t^3$, $z_2 \\mapsto t^2$.\nThe element $t = z_1 / z_2$ of the field of fractions satisfies $t^2 - z_2 = 0$, so it is integral, but it does not lie in $\\C\\{t^2, t^3\\}$.\n\nNon-example of another kind: the cross $\\set{z_1 z_2 = 0}$ is not normal at $0$, because its local ring has the zero divisors $z_1$ and $z_2$, so it is not a domain.\n\nExample: $\\C^n$ is normal, since the ring $\\C\\{z_1, \\dots, z_n\\}$ of convergent power series is factorial, and factorial domains are integrally closed.\n\nFor a reduced space $X$, there is an equivalent form.\nThe space $X$ is normal if and only if the following holds for every open $U \\subseteq X$.\nEvery function on $U$ that is holomorphic on the regular points and locally bounded is holomorphic on $U$.\n",
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