  In his classical textbook on algebraic topology Edwin Spanier developed the
theory of covering spaces within a more general framework of lifting spaces
(i.e., Hurewicz fibrations with unique path-lifting property). Among other,
Spanier proved that for every space $X$ there exists a universal lifting space,
which however need not be simply connected, unless the base space $X$ is
semi-locally simply connected. The question on what exactly is the fundamental
group of the universal space was left unanswered.
  The main source of lifting spaces are inverse limits of covering spaces over
$X$, or more generally, over some inverse system of spaces converging to $X$.
Every metric space $X$ can be obtained as a limit of an inverse system of
polyhedra, and so inverse limits of covering spaces over the system yield
lifting spaces over $X$. They are related to the geometry (in particular the
fundamental group) of $X$ in a similar way as the covering spaces over
polyhedra are related to the fundamental group of their base. Thus lifting
spaces appear as a natural replacement for the concept of covering spaces over
base spaces with bad local properties.
  In this paper we develop a general theory of lifting spaces and prove that
they are preserved by products, inverse limits and other important
constructions. We show that maps from $X$ to polyhedra give rise to coverings
over $X$ and use that to prove that for a connected, locally path connected and
paracompact $X$, the fundamental group of the above-mentioned Spanier's
universal space is precisely the intersection of all Spanier groups associated
to open covers of $X$, and that the later coincides with the shape kernel of
$X$. Furthermore, we examine in more detail lifting spaces over $X$ that arise
as inverse limits of coverings over some approximations of $X$.
