Abstract. If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism ϕ : π 1 (X) →π 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 generalized universal covering space if ϕ :This generalized notion of universal covering p : X → X enjoys most of the usual properties, with the possible exception of evenly covered neighborhoods: the space X is path-connected, locally path-connected and simply-connected and the continuous surjection p : X → X is universally characterized by the usual general lifting properties. (If X is first countable, then p : X → X is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations G = Aut( X p → X) is isomorphic to π 1 (X) and it acts freely and transitively on every fiber. If X is locally path-connected, then the quotient X/G is homeomorphic to X. If X is Hausdorff or metrizable, then so is 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 : X → 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 π 1 (X) is countable. Spaces X for which ϕ : π 1 (X) →π 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 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 π 1 (X), and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties.General Assumption. Throughout this article, we consider a pathconnected topological space X with base point x 0 ∈ X. Recall that a continuous map p : X → X is called a covering of X, and X is called a covering space of X, if for every x ∈ X there is an open subset U of X with x ∈ U and such that U is evenly covered by p, that is, p −1 (U ) is the disjoint union of open subsets of X each of which is mapped homeomorphically onto U by p.In the classical theory, one assumes that X is, in addition, locally pathconnected and wishes to classify all path-connected covering spaces of X and to find among them a universal covering space, that is, a covering p : X → X with the property that for every covering q : X → X by a path-connected space X there is a covering q : X → X such that q•q = p. If X is locally pathconnected, we have the following well-known result, which can be found, for example, in [22] and [25]:

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces. While the condition "semilocally simply connected" plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition "homotopically Hausdorff" instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.Comment: 17 pages, 8 figure

Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\pi_1$-shape injective.Comment: 12 pages, 5 figure