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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[Riemann surfaces|5K0N8QMV6Q]].\nA map $f \\colon X \\to Y$ is a *(holomorphic) branched covering* if:\n\n1. $f$ is a [[holomorphic map|T19P7J53A1]] that is not constant, and\n2. $f$ is proper, that is, $f^{-1}(K)$ is [[compact|00R0X1WGPX]] for every compact $K \\subseteq Y$.\n\nA point $x \\in X$ is a *ramification point* of $f$ if $f$ is not injective on any neighbourhood of $x$; its image is a *branch point*.\n\n## Remarks\n\nEvery branched covering $f$ is surjective, and its ramification points form a closed discrete set $R \\subseteq X$.\nThere is an integer $d \\ge 1$, the degree, such that $f$ restricts to a $d$-sheeted [[covering map|3XZRRE165N]] from $X \\setminus f^{-1}(f(R))$ onto $Y \\setminus f(R)$.\nEvery fibre has exactly $d$ points when counted with multiplicity, where the multiplicity at $x$ is the $k$ such that $f$ looks like $z \\mapsto z^k$ near $x$ in suitable charts.\nIf $X$ is compact, every non-constant holomorphic map $X \\to Y$ is a branched covering.\nIn topology, a branched covering of surfaces is defined in the same way with continuous maps that look locally like $z \\mapsto z^k$.\n\nExample: $f \\colon \\C \\to \\C$, $f(z) = z^2$.\nIt is holomorphic and non-constant, and $f^{-1}(K)$ is closed and bounded for compact $K$, since $\\abs{z}^2 = \\abs{f(z)}$.\nIts degree is $2$, its only ramification point is $0$, and the only [[branch point|XP5JC1VXKZ]] is $0$.\n\nNon-example: $\\exp \\colon \\C \\to \\C$ is holomorphic and non-constant but not proper.\nThe preimage of the compact set $\\set{1}$ is $2\\pi i \\Z$, which is not compact; clause 2 fails.\n",
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