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Definition: Let p: E -> B be a covering map; let b_0 be in B. Choose e_0 so that p(e_0) = b_0. Given an element [f] of pi1(B, b_0), let g be the lifting of f to a path in E that begins at e_0. Let h([f]) denote the end point g(1) of g. Then h is a well-defined set map, h: pi1(B, b_0) -> p^(-1)(b_0). We call h the lifting correspondence derived from the covering map p. It depends of course on the choice of the point e_0.
Source: Topology (second edition) by James R. Munkres