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Localization in bundles of uniform spaces. (English) Zbl 0741.54007

In this paper, uniformities on spaces are defined by families of pseudometrics, and a morphism \(f: X\to Y\) of uniform spaces is required to satisfy the — highly restrictive — condition that the pseudometrics on \(X\) and \(Y\) are indexed by the same set \(I\) (if not, there are no morphisms between \(X\) and \(Y\)) and \(f\) is contractive with respect to each member of \(I\). Working in this category, the authors show how to construct directed colimits of uniform spaces; and they set up an adjunction between presheaves of uniform spaces on a base space \(T\) and uniform bundles over \(T\). No examples or applications are given, other than the example of a trivial bundle \(T\times Y\to T\).

MSC:

54B40 Presheaves and sheaves in general topology
54E15 Uniform structures and generalizations
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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