Constructive metrisability in point-free topology

AbstractThe notion of elementary diameter is introduced to provide, in the context of Locale Theory, a constructive notion of metrisability. Besides foundational aspects, elementary diameters allow to express metrisability in locales more simply with respect to the existing (non-constructive) approach based on diameters. By relying on the presentation of Locale Theory provided by formal topology, the notions to be presented may be conceived as phrased within (Martin-Löf) Type Theory. A type-theoretic version of Urysohn metrisation theorem is thus obtained. As an application, a set (data type) of indexes for the points of locally compact metrisable formal spaces is shown to exist