  In this paper we develop a new approach to the study of uncountable
fundamental groups by using Hurewicz fibrations with the unique path-lifting
property (lifting spaces for short) as a replacement for covering spaces. In
particular, we consider the inverse limit of a sequence of covering spaces of
$X$. It is known that the path-connectivity of the inverse limit can be
expressed by means of the derived inverse limit functor $\varprojlim^1$, which
is, however, notoriously difficult to compute when the $\pi_1(X)$ is
uncountable.To circumvent this difficulty, we express the set of
path-components of the inverse limit, $\widehat X$, in terms of the functors
$\varprojlim$ and $\varprojlim^1$ applied to sequences of countable groups
arising from polyhedral approximations of $X$.
  A consequence of our computation is that path-connectedness of lifting space
implies that $\pi_1(\tilde X)$ supplements $\pi_1(X)$ in $\check\pi_1(X)$ where
$\check\pi_1(X)$ is the inverse limit of fundamental groups of polyhedral
approximations of $X$. As an application we show that $\mathcal G\cdot
\ker_{\mathbb Z}(\widehat F)= \widehat F\ne\mathcal G\cdot
\ker_{B(1,n)}(\widehat F)$, where $\widehat F$ is the canonical inverse limit
of finite rank free groups, $\mathcal G$ is the fundamental group of the
Hawaiian Earring, and $\ker_A(\widehat F)$ is the intersection of kernels of
homomorphisms from $\widehat{F}$ to $A$.
