Covering maps in the uniform category

First, generalized uniform covering maps are classified in terms of subgroups of the uniform fundamental group. Hausdorff, locally uniform joinable, and chain connected covering spaces of a uniformly locally joinable chain connected space are classifieded in terms of closed subgroups of its uniform fundamental group. If the space is also semilocally simply uniform joinable, uniform covering spaces are classified in terms of all subgroups of its uniform fundamental group. Next it is shown that the inverse limit of a strong Mittag-Leer inverse system of Hausdorff uniform covering spaces is a generalized uniform covering space. The question of the converse is investigated. It is necessary for a generalized uniform covering map to have strong approximate uniqueness for it to be the inverse limit of uniform covering maps. It is shown that such a generalized uniform covering map is indeed the inverse limit of uniform covering maps. It is unknown whether all generalized uniform covering maps have strong approximate uniqueness. The analog to the above characterization is investigated for regular generalized uniform covering maps, i.e., generalized uniform covering maps that are induced by group actions. Finally, given a pointed 1-movable continuum represented as the inverse limit of its nerves, it is shown that connected pointed 1-movable generalized covering spaces are the same as the inverse limit of covering spaces of its nerves provided the inverse system of covering spaces satisfy certain properties. I thank Sergey Melikhov for suggesting to look at this issue.