Uncountable groups and the geometry of inverse limits of coverings

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with unique path-lifting property ( lifting spaces for short) as replacement covering spaces. particular, consider inverse limit sequence X . It is known that path-connectivity can be expressed means derived functor lim ← 1 , which is, however, notoriously difficult compute when group, π ) uncountable. To circumvent difficulty, express set path-components ˜ in terms functors and applied sequences countable arising from polyhedral approximations A consequence our computation path-connectedness space, implies supplements ˇ where As an application show G ⋅ Ker Z F ˆ = ≠ B n canonical finite rank free groups, group Hawaiian Earring, Baumslag-Solitar intersection kernels homomorphisms