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  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of complex dimension $n$ that is [[second countable|03PPNCM9GG]], and let $\\mathcal{O}(X)$ be the algebra of [[holomorphic functions|507C34W0ZT]] on $X$.\nFor a [[compact set|0R6EWFKRCF]] $K \\subseteq X$, its [[holomorphically convex hull|CD6SMMZSW8]] is\n$$\\widehat{K} = \\set{x \\in X : \\abs{f(x)} \\le \\sup_{K} \\abs{f} \\text{ for all } f \\in \\mathcal{O}(X)}.$$\nThe manifold $X$ is a *Stein manifold* if:\n\n1. $X$ is holomorphically convex: $\\widehat{K}$ is compact for every compact $K \\subseteq X$.\n2. $X$ is holomorphically separable: for all $x \\ne y$ in $X$ there is $f \\in \\mathcal{O}(X)$ with $f(x) \\ne f(y)$.\n\n## Remarks\n\nSome authors add a third condition: at every point $x$ there are $f_1, \\dots, f_n \\in \\mathcal{O}(X)$ that form local coordinates near $x$.\nThis condition follows from the first two.\nExample: $X = \\C^n$ is Stein.\nThe coordinate functions separate points.\nThe hull $\\widehat{K}$ is closed, and $\\abs{z_j} \\le \\max_K \\abs{z_j}$ on $\\widehat{K}$ for each $j$.\nSo $\\widehat{K}$ lies in a compact polydisc, and it is compact.\nNon-example: a compact connected complex manifold $X$ of positive dimension is not Stein.\nEvery $f \\in \\mathcal{O}(X)$ is constant, by the maximum principle, so clause 2 fails.\nA domain in $\\C^n$ is a Stein manifold if and only if it is a domain of holomorphy.\n",
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