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An existing description of the cartesian closed topological hull of $p{\mathbf{\text{MET}}}^{\infty }$, 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 $pqs{\mathbf{\text{MET}}}^{\infty }$, 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...