2007/06/26 by N. Brodskiy, Brodskiy, N., J. Dydak +5 · 1 citation
Mathematics · #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG) #math.GN #math.MG
paper · pdf · doi:10.48550/arxiv.0706.3937
26 pages. The paper was split and the second part is available at arXiv:0802.4304
arxiv created 2008/03/03 · arxiv updated 2009/12/01
James \citeJam introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \citeBP3 introduced a theory of covers for uniform spaces generalizing their results for topological groups \citeBP1-\citeBP2. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section \refSECTION-Comparison, \citeBP3 generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions. In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc \citeKraMin.