Given a function f : X → Y of metric spaces, its asymptotic dimension asdim(f ) is the supremum of asdim(A) such that A ⊂ X and asdim(f (A)) = 0. Our main result is generalizes a result of Bell and Dranishnikov [3] in which f isLipschitz and X is geodesic. We provide analogs of 0.1 for Assouad-Nagata dimension dim AN and asymptotic Assouad-Nagata dimension asdim AN . In case of linearly controlled asymptotic dimension l-asdim we provide counterexamples to three questions of Dranishnikov [14].As an application of analogs of 0.1 we prove Theorem 0.2. If 1 → K → G → H → 1 is an exact sequence of groups and G is finitely generated, then asdim AN (G, d G ) ≤ asdim AN (K, d G |K) + asdim AN (H, d H ) for any word metrics metrics d G on G and d H on H. 0.2 extends a result of Bell and Dranishnikov [3] for asymptotic dimension.

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.

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov and capacity dimension of Buyalo [7]) to Assouad-Nagata dimension. This is done by applying two functors on the Lipschitz category of metric spaces: microscopic and macroscopic. In the second part we identify (among spaces of finite Assouad-Nagata dimension) spaces of Assouad-Nagata dimension at most n as those for which the n-sphere S n is a Lipschitz extensor. Large scale and small scale analogues of that result are given.