Uniform Gauge Spaces

Abstract

Approach spaces form a local theory. Nevertheless there is also a natural uniform notion of completeness and completion and in this chapter we define the appropriate setting hereto, namely the category of uniform gauge spaces. This setting is linked to uniform spaces. Just as was the case for approach spaces here this category will be a supercategory of both the categories of uniform spaces and uniformly continuous maps and of metric spaces with non-expansive maps. Again, the former is a stable subcategory and the latter is a concretely coreflective subcategory. Finally we also consider the non-symmetric variant, quasi-uniform gauge spaces.