The obstruction to the existence of various notions of ``generalized covering map" is almost always the unique path lifting property. The topology of a generalized covering space defined in terms of unique lifting properties with respect to locally path-connected spaces (specifically, in the sense of Fischer-Zastrow) must agree with the standard ``whisker topology" used in classical covering space theory.
For a path-connected metric space \(X\), we characterize the unique path lifting property for the whisker topology (equivalently, the existence of a generalized covering) relative to an arbitrary subgroup \(H\) of the fundamental group. Our characterization is given in terms of a closure-like property of \(H\) that involves test maps from a certain ``dyadic" one-dimensional Peano continuum. The simple nature of this closure-like property sheds light on the difference between the unique path lifting property, homotopy path-Hausdorff property, and a number of other properties of recent interest.
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