The Cartesian Closed Topological Hull of the Category of Approach Uniform Spaces
Mark Nauwelaerts · Rocky Mountain Journal of Mathematics · 2001
The category AUnif of approach uniform spaces and uniform contractions properly combines uniform spaces and extended pseudo-metric spaces but (like Unif) lacks convenience, such as cartesian closedness.This paper therefore considers its cartesian closed topological hull, which is first described as a subcategory of SAULim, the category of semi-approach uniform limit spaces and uniform contractions.This hull is then also given a description inside the topological universe hull of AUnif and is shown to be a reasonable generalization of the corresponding hull of Unif.Furthermore, some referencing notes are provided with respect to similar results that can be obtained when starting from qAUnif (where symmetry assumptions are omitted). Introduction.It is often desirable and useful for a (concrete) category to have extra properties in addition to just being nicely topological, such as being cartesian closed topological (CCT).However, many categories are not cartesian closed, which has inspired a theory of CCT extensions of such (failing) categories, where the least such CCT extension of a given concrete category, the CCT hull of a category, is especially interesting.For instance, in [2], Adámek and Reiterman constructed the CCT hull of Unif, the category of uniform spaces (and uniformly continuous maps), and in [3], they described the CCT hull of the category (p)MET (∞) of (extended pseudo-)metric spaces (and nonexpansive maps).Later the author added to these results by describing the CCT hull of the category qUnif of quasi-uniform spaces (and uniformly continuous maps) [16] (and thereby also adding to an alternative characterization of the CCT hull of Unif by Alderton and Schwarz [4]) and by describing the CCT hull of the category pqMET ∞ of extended