Fibrewise covering uniformities and completions (Q949826)

By using coverings, the authors introduce the concepts of fibrewise covering uniform space and fibrewise generalized uniform space and fibrewise semi-uniform space (corresponding to the fibrewise (entrourage) uniformity theory by \textit{I. M. James} [Fibrewise topology, Cambridge Tracts in Mathematics 91. Cambridge: Cambridge University Press (2008; Zbl 1147.55001)] and to the generalized (respectively, semi-) uniformity theory by \textit{K. Morita} [in: Topics in general topology, North-Holland Math. Libr. 41, 1--39 (1989; Zbl 0719.54013)]. In their definition, the concept of conjugate pairs of coverings plays an important role. Further, the authors discuss the relationships of fibrewise uniform spaces and fibrewise covering uniform spaces, and of the fibrewise topologies induced from these fibrewise uniform\ (respectively, covering uniform) spaces. The fibrewise completions of fibrewise generalized uniform spaces and fibrewise semi-uniform spaces are also studied.