Quantale-Valued Uniformizations of Quantale-Valued Generalizations of Approach Groups

Abstract

We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.