Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces.Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connected, then every Peano covering map is equivalent to the projection e X/H → X, where H is a subgroup of the fundamental group of X and e X equipped with the topology used in [2], [15] and introduced in [23, p.82]. The projection e X/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on e X for which one has a characterization of e X/H → X having the unique path lifting property if H is a normal subgroup of π 1 (X). Namely, H must be closed in π 1 (X). Such groups include π(U , x 0 ) (U being an open cover of X) and the kernel of the natural homomorphism π 1 (X, x 0 ) →π 1 (X, x 0 ).

In Rips Complexes and Covers in the Uniform Category [3] we define, following James [9], covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. In this paper we investigate when these covering maps are induced by group actions. Also, as an application of our results we present an exposition of Prajs' [15] homogeneous curve that is path-connected but not locally connected.

In "Rips complexes and covers in the uniform category" (Brodskiy et al., preprint [4]) the authors define, following James (1990) [5], covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.

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It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version is equivalent to shape injectivity for Peano spaces.