View More View Less
  • 1 Shimane University Department of Mathematics Matsue 690-8504 Japan
Restricted access

Abstract  

We introduce an alternative definition of fibrewise uniformity and discuss consequences deduced from new axioms. By modifying James’ definition of fibrewise uniform structure, which is a slightly strengthened one, we define a new fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities and fibrewise covering uniformities. Moreover, we obtain a characterization of the fibrewise completion of fibrewise generalized uniform space as a fibrewise extension of a fibrewise space. As an application of the fibrewise completion theory, we show that there exists a fibrewise Shanin compactification of a fibrewise space. Finally, we study extendability of fibrewise maps from dense subspaces. That is, for a fibrewise space X, AX dense in X and a fibrewise continuous map f: AY, when can f be extended to the whole space X? Many characterization theorems of extendable fibrewise continuous maps are given.