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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[complex analytic space|HXSZA44J5J]] whose underlying topological space is [[Hausdorff|0S1FJ8VEX9]] and [[second countable|03PPNCM9GG]].\nLet $\\mathcal{O}(X)$ be the algebra of global holomorphic functions on $X$, and let $f(x) \\in \\C$ denote the value of $f \\in \\mathcal{O}(X)$ at $x \\in X$.\nFor a [[compact set|0R6EWFKRCF]] $K \\subseteq X$, its [[holomorphically convex hull|CD6SMMZSW8]] is $\\widehat{K} = \\set{x \\in X : \\abs{f(x)} \\le \\sup_K \\abs{f} \\text{ for all } f \\in \\mathcal{O}(X)}$.\nThe space $X$ is a *Stein space* 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\nA Stein space whose underlying space is a complex manifold is a Stein manifold.\nExample: every closed complex subspace $X$ of $\\C^N$ is Stein.\nThe coordinate functions restricted to $X$ separate points, and $\\widehat{K}$ is a closed subset of $X$ contained in a closed ball of $\\C^N$, hence compact.\nNon-example: the complex projective line $\\mathbb{P}^1$ is a complex space whose global holomorphic functions are constant, so clause 2 fails.\nAn equivalent form: $X$ is Stein if and only if $H^q(X, \\mathcal{F}) = 0$ for every coherent analytic sheaf $\\mathcal{F}$ on $X$ and every $q \\ge 1$.\nThe direction from Stein to vanishing is Cartan's theorem B.\n",
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