Title




Supported approach spaces


Author






Abstract




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 quasimetric and topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the pointtoset distance in a quasimetric space. The embedding of the category of topological spaces with continuous maps and of quasimetric spaces with nonexpansive maps is extremely nice. Every approach space has both a quasimetric coreflection as well as a topological coreflection. Different approach spaces though can have the same topological as well as the same quasimetric 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 baseregular, 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 nonexpansiveness for the quasimetric coreflection. This result implies that a supported approach space actually is the infimum of its quasimetric and its topological coreflection. In the course of our study we also give several more examples of both supported and nonsupported approach spaces. 


Language




English


Source (journal)




Quaestiones mathematicae.  Pretoria


Publication




Pretoria
:
2023


ISSN




03799468
16073606


DOI




10.2989/16073606.2023.2247724


Volume/pages




46
:s:[1]
(2023)
, p. 161190


ISI




001098712000005


Full text (Publisher's DOI)






Full text (open access)






Full text (publisher's version  intranet only)





