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  "body": "\n## Definition\n\nLet $\\Omega_1 \\subset \\C^n$ and $\\Omega_2 \\subset \\C^m$ be domains, that is, nonempty connected open sets.\nA [[holomorphic map|3MQTWKBBBT]] $F \\colon \\Omega_1 \\to \\Omega_2$ is *proper* if $F^{-1}(K)$ is [[compact|0TVJZYB6SH]] for every compact set $K \\subset \\Omega_2$.\nThe same condition defines proper holomorphic maps between complex manifolds.\n\n## Remarks\n\nEquivalent form: $F$ is proper if and only if it has the following property.\nFor every sequence $(z_k)$ in $\\Omega_1$ that eventually leaves every compact subset of $\\Omega_1$, the sequence $(F(z_k))$ eventually leaves every compact subset of $\\Omega_2$.\nFor bounded domains this says that $\\operatorname{dist}(F(z), \\partial\\Omega_2) \\to 0$ as $\\operatorname{dist}(z, \\partial\\Omega_1) \\to 0$.\n\nExample: let $\\mathbb{D}$ be the unit disc and $F(z) = z^k$ with $k \\ge 1$.\nA compact $K \\subset \\mathbb{D}$ lies in a closed disc $\\set{\\abs{w} \\le r}$ with $r \u003c 1$, and $F^{-1}(K)$ is a closed subset of $\\set{\\abs{z} \\le r^{1/k}}$, hence compact.\nThe map $(z_1, z_2) \\mapsto (z_1, z_2^2)$ of the unit polydisc in $\\C^2$ to itself is proper for the same reason.\n\nNon-example: the inclusion $F \\colon \\set{\\abs{z} \u003c 1/2} \\to \\mathbb{D}$ is holomorphic but not proper.\nFor the compact set $K = \\set{\\abs{w} \\le 1/2}$, the preimage $F^{-1}(K) = \\set{\\abs{z} \u003c 1/2}$ is not compact.\n\nA proper holomorphic map between domains of the same dimension $n$ is surjective, its fibres are finite, and it is a finite branched covering.\nA [[biholomorphism|FX92PP3HZR]] is a proper holomorphic map.\nThe proper holomorphic self-maps of $\\mathbb{D}$ are exactly the finite [[Blaschke products|9N6R5ADJXT]].\n",
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