Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not sensible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined. Contents 1 The notion of formal topology 3 1.1 Concrete topological spaces . . . . . . . . . . . . . . . . . . . . . 3 1.2 Formal topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Three problems and their solution 7 2.1 Formal topologies wi...

Introduction Stone duality between Boolean algebra and compact totally disconnected spaces provides an algebraic and point-free presentation of a large class of topological spaces. A generalisation of this presentation, also due to Stone, is described in [Johnstone, Stone]: the class of spaces is larger, we get all compact Hausdorff spaces, and the representation is still algebraic, using divisible archimedian rings. These rings are now torsion free, contrary to the case of Boolean rings. Actually, these rings appear implicitely in analysing problems of measure on Boolean algebras [Tarski]. We give here a variation of the treatment of [Johnstone, Stone] 1 which can be seen also as a constructive "real" version of Gelfand duality in the style of [BM]. Our main result is a localic proof of the fact that the uniform norm of the Gelfand transform of an element is equal to its norm. 1 Preordered Ring We start from a ring A with a p

Abstract. We present a constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Gelfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.

The goal of this paper is to analyse two remarkable notes by Stone [StoI, StoII]. Both describe a compact space in term of some algebra of functions over this space. This description

Abstract. 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 constructive locale theory. We show that the two notions are intimately connected. Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact ’ is translated as compact and overt. We propose a definition of located predicate on subspaces in formal topology. We call a sublocale located if it can be presented by a formal topology with a located predicate. We prove that a closed sublocale of a compact regular locale has a located predicate iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. Finally, we show by elementary methods 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 our results are valid in any predicative topos. 1.

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. 1.