Abstract: 
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If a paracompact Hausdorff space $X$ admits a (classical)
universal covering space, then the natural homomorphism
$\varphi:\pi_1(X)\rightarrow \check{\pi}_1(X)$ from the
fundamental group to its first shape homotopy group is an
isomorphism.
 We present a partial converse to this result:
a path-connected topological space $X$ admits a {\it generalized}
universal covering space if $\varphi:\pi_1(X)\rightarrow
\check{\pi}_1(X)$ is injective.


 This generalized notion of
universal covering $p:\widetilde{X}\rightarrow X$ enjoys most of the
usual properties, with the possible exception of evenly covered
neighborhoods: the space $\widetilde{X}$ is path-connected, locally
path-connected and simply-connected and the continuous surjection
$p:\widetilde{X}\rightarrow X$
 is universally characterized by
 the usual general lifting properties.
 (If $X$ is first countable, then $p:\widetilde{X}\rightarrow X$ is
 already characterized by the unique lifting of paths and their homotopies.)
 In particular, the group of covering transformations
$G=\mathop{\rm Aut}(\widetilde{X}\stackrel{p}{\rightarrow}X)$ is isomorphic
to $\pi_1(X)$ and it acts freely and transitively on every fiber.
If $X$ is locally path-connected, then the quotient $\widetilde{X}/G$
is homeomorphic to $X$. If $X$ is Hausdorff or metrizable, then
so is $\widetilde{X}$, and in the latter case $G$ can be made to act
by isometry. If $X$ is path-connected, locally path-connected and
semilocally simply-connected, then $p:\widetilde{X}\rightarrow X$
agrees with the classical universal covering.

A necessary condition for the standard construction to yield a generalized universal covering is that $X$ be homotopically Hausdorff, which is also
sufficient if $\pi_1(X)$ is countable. Spaces $X$ for which
$\varphi:\pi_1(X)\rightarrow \check{\pi}_1(X)$ is known to be
injective include all subsets of closed surfaces,
 all 1-dimensional separable metric spaces (which we prove to be
 covered by topological $\mathbb{R}$-trees), as well as so-called trees of manifolds
 which arise, for example,
 as boundaries of certain
Coxeter groups.

We also obtain generalized regular coverings, relative to some
special normal subgroups of $\pi_1(X)$, and provide the
appropriate relative version of being homotopically Hausdorff,
along with its corresponding properties.

1. Department of Mathematical Sciences
   Ball State University
   Muncie, IN 47306, U.S.A.
   
2. Institute of Mathematics
   University of Gdańsk
   Wita Stwosza 57
   80-952 Gdańsk, Poland