# Article

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Keywords:
(extended) pseudo-(quasi-)metric space; (quasi-)distance space; preordered space; demi-(quasi-)metric space; cartesian closed topological; CCT hull
Summary:
An existing description of the cartesian closed topological hull of $p\text{\bf MET}^\infty$, 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 $pqs\text{\bf MET}^\infty$, 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 $pq\text{\bf MET}^\infty$, 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 $\text{\bf Prost}$, the category of preordered spaces and relation preserving maps.
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