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Definition: Let f be a loop in R^2, and let a be a point not in the image of f. Set g(s)=[f(s_ - a] / ||f(s) - a||; then g is a loop in S^1. Let p: R -> S^1 be the standard covering map, and let g* be a lifting of g to S^1. Because g is a loop, the difference g*(1) - g*(0) is an integer. This integer is called the winding number of f with respect to a.
Source: Topology (second edition) by James R. Munkres