By using coverings we introduce the concepts of fibrewise covering uniform space and its generalizations (fibrewise generalized
uniform space and fibrewise semi-uniform space), and study the fibrewise completions of fibrewise generalized uniform spaces
and fibrewise semi-uniform spaces.
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.