Fibrewise extensions, Shanin compactification and extensions of fibrewise maps (Q1046833)

In a previous paper, the authors studied fibrewise covering uniformities and completions (corresponding to fibrewise uniform structures defined by James). In this paper, in order to extend the theory of generalized uniformities (g-uniformities) to fibrewise g-uniformities, the authors introduce an alternative definition of fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities (a concept stronger than fibrewise uniform structures) and fibrewise covering uniformities. Generalizing fibrewise covering uniform spaces, they define fibrewise generalized uniform spaces and obtain a characterization of the fibrewise completion of fibrewise generalized uniform spaces as a fibrewise extension of fibrewise spaces. As an application of the fibrewise completion theory, in this paper, the fibrewise theory of Shanin compactifications is studied. The authors show that there exists a fibrewise Shanin compactification of a fibrewise space. Finally, they define the notion of fibrewise semi-uniformities and study the extendibility of fibrewise maps from dense subspaces. Many characterization theorems for extendible fibrewise continuous maps are given.