Supported approach spaces

In this paper we work in the category of approach spaces with contractions [14], the objects of which are sets endowed with a numerical distance between sets and points. Approach spaces are to be considered a simultaneous generalization of both quasi-metric and topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the point-to-set distance in a quasi-metric space. The embedding of the category of topological spaces with continuous maps and of quasi-metric spaces with non-expansive maps is extremely nice. Every approach space has both a quasi-metric coreflection as well as a topological coreflection. Different approach spaces though can have the same topological as well as the same quasi-metric coreflection, in other words, in general these coreflections do not determine the approach space. In this paper we investigate approach spaces for which these coreflections, do determine the approach space. We will call such spaces supported. We prove that in the setting of compact approach spaces many examples of supported approach spaces can be found. Thus, compact spaces that are base-regular, which is a weakening of regularity, are always supported. An important feature of supported approach spaces is the behaviour of contractions. On a supported domain contractivity is characterized by the combination of continuity for the topological coreflection and non-expansiveness for the quasi-metric coreflection. This result implies that a supported approach space actually is the infimum of its quasi-metric and its topological coreflection. In the course of our study we also give several more examples of both supported and non-supported approach spaces.