Generalized covering projections and fibrations with unique path lifting

Covering space theory is an important tool of algebraic topology, as it gives rise to a rich interplay between self-homeomorphisms of spaces and the group theory of homotopy classes of loops. However, for spaces that are locally not well behaved, this theory is somewhat limited. To illustrate this, we first revisit a classical example, the Infinite Earring, which is known to not have a universal covering space. For the latter fact, we present a version of a proof from the literature, but with a somewhat more geometric flavor. We then explain why there is a simply connected fibration with unique path lifting over the Infinite Earring and how any fibration with unique path lifting can be turned into a generalized covering projection, via local-path-connectification of its domain, without altering the fundamental group. This leads to a natural question: Can every generalized covering projection be obtained in this way? We answer this question in the negative, using a known example from the literature for which there is a simply connected generalized covering projection and for which the trivial subgroup of the fundamental group is not closed in the quotient topology of the compact-open-topology.