Fibrewise extensions, shanin compactification and extensions of fibrewise maps

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, A ⊂ X dense in X and a fibrewise continuous map f : A → Y , when can f be extended to the whole space X ? Many characterization theorems of extendable fibrewise continuous maps are given.