DUALITY BEYOND SOBER SPACES - TOPOLOGICAL-SPACES AND OBSERVATION FRAMES

We introduce observation frames as an extension of ordinary frames. Th e aim is to give an abstract representation of a mapping from observab le predicates to all predicates of a specific system, A full subcatego ry of the category of observation frames is shown to be dual to the ca tegory of J(o) topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predi cate logic of observation frames with both infinite conjunctions and d isjunctions, just like there is a geometric logic for (ordinary) frame s with infinite disjunctions but only finite conjunctions. This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topolog ical spaces and of observation frames in order to obtain dualities for various subcategories of J(o) spaces. These involve nonsober spaces.