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 ${\underleftarrow{\lim }}^1$, which is, however, notoriously difficult to compute when the fundamental group, ${\pi }_1(X)$, is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit $\tilde{X}$ of a sequence of covering spaces in terms of the functors $\underleftarrow{\lim }$ and ${\underleftarrow{\lim }}^1$ applied to sequences of countable groups arising from polyhedral approximations of X.

A consequence of our computation is that path-connectedness of a lifting space, $\tilde{X}$, 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 $\mathcal{G}\cdot {\mathrm{Ker}}_{\mathbb{Z}}(\hat{F})=\hat{F}\ne \mathcal{G}\cdot {\mathrm{Ker}}_{B(1,n)}(\hat{F})$, where $\hat{F}$ is the canonical inverse limit of finite rank free groups, $\mathcal{G}$ is the fundamental group of the Hawaiian Earring, $B(1,n)$ is the Baumslag-Solitar group, and ${\mathrm{Ker}}_A(\hat{F})$ is the intersection of kernels of homomorphisms from $\hat{F}$ to A.

在本文中，我们通过使用具有独特路径提升特性（简称提升空间）的Hurewicz 纤维作为覆盖空间的替代，开发了一种研究不可数基本群的新方法。特别地，我们考虑X的覆盖空间序列的逆极限。已知逆极限的路径连通性可以通过导出的逆极限函子来表示${\underleftarrow{\mathrm{林}}}^1$，然而，这是出了名的难以计算，当基本群， ${\pi }_1(X)$，不可数。为了规避这个困难，我们表达了逆极限的路径分量集$\stackrel{～}{X}$ 用函子表示的覆盖空间序列 $\underleftarrow{\mathrm{林}}$ 和 ${\underleftarrow{\mathrm{林}}}^1$应用于由X 的多面体逼近产生的可数群序列。

我们计算的结果是提升空间的路径连通性， $\stackrel{～}{X}$， 暗示 ${\pi }_1(\stackrel{～}{X})$ 补充剂 ${\pi }_1(X)$ 在 ${\check{\pi }}_1(X)$ 在哪里 ${\check{\pi }}_1(X)$是X的多面体近似的基本群的逆极限。作为一个应用程序，我们表明$\mathcal{G}\cdot {\mathrm{克尔}}_{\mathbb{Z}}(\widehat{F})=\widehat{F}\ne \mathcal{G}\cdot {\mathrm{克尔}}_{乙(1,n)}(\widehat{F})$， 在哪里 $\widehat{F}$ 是有限秩自由群的规范逆极限， $\mathcal{G}$ 是夏威夷耳环的基本组， $乙(1,n)$ 是 Baumslag-Solitar 群，并且 ${\mathrm{克尔}}_{\mathrm{一种}}(\widehat{F})$ 是来自同态的核的交集 $\widehat{F}$到A。