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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]].\nThen $D$ is *holomorphically convex* if for every [[compact set|0R6EWFKRCF]] $K \\subseteq D$ the [[holomorphically convex hull|CD6SMMZSW8]] $\\hat K_D$ is compact.\n\n## Remarks\n\nSince $\\hat K_D$ is closed in $D$, the condition says that $\\hat K_D$ has no limit points on the boundary of $D$.\n\nExample: a convex open set $D$.\nFor compact $K \\subseteq D$, the hull $\\hat K_D$ lies in the convex hull $\\operatorname{ch}(K)$, which is compact and contained in $D$.\nA subset of $D$ that is closed in $D$ and contained in the compact set $\\operatorname{ch}(K) \\subseteq D$ is compact.\n\nNon-example: $D = \\set{z \\in \\C^2 : 0 \u003c \\abs{z} \u003c 1}$ and $K = \\set{\\abs{z} = 1/2}$.\nBy [[Hartogs's extension theorem|FSGES670AZ]] every holomorphic $f$ on $D$ extends to the ball, and by the maximum principle $\\abs{f(z)} \\le \\max_K \\abs{f}$ for $0 \u003c \\abs{z} \\le 1/2$.\nSo $\\hat K_D$ contains $\\set{0 \u003c \\abs{z} \\le 1/2}$, which is not compact.\n\nBy the Cartan–Thullen theorem, a domain in $\\C^n$ is holomorphically convex if and only if it is a domain of holomorphy.\n",
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