Uniqueness of completion for metrically generated constructs

Abstract

In this paper, for metrically generated constructs X in the sense of [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] we study completion as a U -reflector R on the subconstruct X 0 of all T 0 -objects, for U some class of embeddings. Roughly speaking we deal with constructs X that are generated by the subclass of their metrizable objects and for various types of completion functors R available in that context, we obtain internal descriptions of the largest class U for which completion is unique. We apply our results to some well known situations. Completion of uniform spaces, of proximity spaces or of non-Archimedian uniform spaces is unique with respect to the class of all epimorphic embeddings, and this class is the largest one. However the largest class of morphisms for which Dieudonné completion of completely regular spaces or of zero dimensional spaces is unique, is strictly smaller than the class of all epimorphic embeddings. The same is true for completion in quantitative theories like uniform approach spaces for which the largest U coincides with the class of all embeddings that are dense with respect to the metric coreflection. Our results on completion for metrically generated constructs explain these differences.

Bicompletion of metrically generated constructs

“…When we apply our completion technique to symmetric structures, which was the more restrictive setting of [7], we recover the list of examples discussed in detail in that paper. We add one more interesting example to that list, namely by describing a completion theory for Lipschitz spaces [13], [14].…”

“…Since the subconstructs of complete objects associated to these morphism classes consist of the classes of respective injective objects [1], the category of uniformly complete objects is a subcategory of the category of metrically complete objects. Analogous to the comparison made in [7] for symmetric metrically generated constructs, a more explicit relationship between uniform and metric bicompletion on M C ξ 0 can be obtained.…”

“…Whereas in [7] two specific types of such completions were constructed for constructs generated by metrics, in this paper we present a far more general completion technique. Our approach here depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms L(R) is known.…”

“…We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R F , for the construct X 0 of T 0 objects, for which the associated firm class of morphisms can be derived from L(R). The methods developed in [7] were only applicable to symmetric structures, whereas here also non symmetric ones are allowed.…”

Constructing the sobrification of an approach space via bicompletion

“…The bicompletion of X is obtained by equipping the set X with the collection of extended quasimetrics {p | p ∈ D}. For more details we refer to [9,10]. In this section we will denote the bicompletion of a qUG 0 -space X by X.…”