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We study a natural generalization of covering projections defined in terms of unique lifting properties. A map p : E -+ X has the continuous path-covering property if all paths in X lift uniquely and continuously (rel. basepoint) with respect to the compactopen topology. We show that maps with this property are closely related to fibrations with totally path-disconnected fibers and to the natural quotient topology on the homotopy groups. In particular, the class of maps with the continuous path-covering property lies properly between Hurewicz fibrations and Serre fibrations with totally path-disconnected fibers. We extend the usual classification of covering projections to a classification of maps with the continuous path-covering property in terms of topological pi 1: for any pathconnected Hausdorff space X, maps E -+ X with the continuous path-covering property are classified up to weak equivalence by subgroups H <= pi 1(X, x0) with totally pathdisconnected coset space pi 1(X, x0)/H. Here, weak equivalence refers to an equivalence relation generated by formally inverting bijective weak homotopy equivalences.

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Fundamenta Mathematicae




Polish Academy of Sciences, Institute of Mathematics