Abstract:An existing description of the cartesian closed topological hull of $p\text {MET}^\infty $, the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a ``family'' of cartesian closed topological subconstructs of $pqs\text {MET}^\infty $, the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of $pq\text {MET}^\infty $, the category of extended pseudo-quasi-metric spaces and nonexpansive maps (which has recently gained interest in theoretical computer science), and this hull is also shown to be a nice generalization of $\text {Prost}$, the category of preordered spaces and relation preserving maps.
Keywords: (extended) pseudo-(quasi-)metric space, (quasi-)distance space, preordered space, demi-(quasi-)metric space, cartesian closed topological, CCT hull
AMS Subject Classification: 18D15, 18B99, 54C35, 54E99