Localic completion of generalized metric spaces II: Powerlocales

The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. Applications: (1) A localic completion is always open, and is compact iff its generalized metric space is totally bounded. (2) The Heine-Borel Theorem is proved in a strong form with continuous maps to the powerlocdales of R, (x, y) 7→ the closeed interval [x, y]. (3) Every localic completion is a triquotient surjective image of a locale of Cauchy sequences. The work is constructive in the topos-valid sense.