Computable sets : located and overt locales

Abstract

What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in locale theory in a constructive, or topos theoretic, context. We show that the two notions are intimately connected. We propose a definition of located closed sublocale motivated by locatedness of subsets of metric spaces. A closed sublocale of a compact regular locale is located iff it is overt. Moreover, a closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. For Baire space metric locatedness corresponds to having a decidable positivity predicate. Finally, we show that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all are results are valid in any predicative topos