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Definition: A map pi from a Hausdorff space, X, to the complex plane C is an n-sheeted branched covering if X contains no isolated points, the fibers of pi are finite, and if there is a finite set delta of points of C called branch points, such that the map (X-pi^-1(delta)) -> (C-delta) is an n-sheeted covering space.
Source: Algebra, second edition by Michael Artin