ar X iv : 0 70 7 . 36 08 v 2 [ m at h . G N ] 1 1 O ct 2 00 7 An equivalent condition for a uniform space to be coverable

We prove that an equivalent condition for a uniform space to be cov-erable is that the images of the natural projections in the fundamental inverse system are uniformly open in a certain sense. As corollaries we (1) obtain a concrete way to find covering entourage, (2) correct an error in [3], and (3) show that coverable is equivalent to chain connected and uniformly joinable in the sense of [5]. In [6] and [3] we approached the problem of defining universal covers of locally bad spaces using the following ideas: (1) The appropriate category in which to work in is not topological spaces, but rather uniform spaces and uniformly continuous mappings. (2) The replacement for quotient mappings in this category is bi-uniformly continuous mappings (see below). (3) The appropriate replacement for curves is equivalence classes of chains. We showed in [3] that such a program can be carried out for a large class of uniform spaces called coverable spaces. In particular we constructed, for any uniform space X, a uniform space X, a natural uniformly continuous mapping φ : X → X and a group δ 1 (X) that acts on X. For coverable spaces the mapping φ has many of the properties of a universal covering map, such as lifting and universal properties, and we refer to the space X as the uniform universal cover of X. The group δ 1 (X) (which we called the " deck group " in [3] but which was renamed the " uniform fundamental group " in [7]) is a functorial invariant of uniform structures having properties like the fundamental group in this category. See [3] for many specific theorems and examples, and [4] for some additional applications.