Cauchy Points of Metric Locales

Abstract

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.