38 research outputs found

    Overlap Algebras as Almost Discrete Locales

    Full text link
    Boolean locales are almost discrete. In fact, spatial Boolean locales are the same thing as discrete spaces. This does not make sense intuitionistically, since (non-trivial) discrete locales fail to be Boolean. We show that Sambin's "overlap algebras" have good enough features to be called "almost discrete locales". Keywords. Strongly dense sublocales, almost discrete spaces, overlap algebras, constructive topology

    σ\sigma-locales in Formal Topology

    Full text link
    A σ\sigma-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that σ\sigma-frames, actually σ\sigma-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every σ\sigma-frame LL is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over LL. We then give a constructive characterization of the smallest (strongly) dense σ\sigma-sublocale of a given σ\sigma-locale, thus providing a ``σ\sigma-version'' of a Boolean locale. Our development depends on the axiom of countable choice.Comment: Paper presented at the conference Continuity, Computability, Constructivity - From Logic to Algorithms (CCC 2017), Nancy, France, June 26-30 201

    Positivity relations on a locale

    Get PDF
    This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)

    Embedding locales and formal topologies into positive topologies

    Get PDF
    A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales

    Overlap Algebras: a Constructive Look at Complete Boolean Algebras

    Get PDF
    The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean algebra, has powersets and other natural structures as instances. In this paper we study the category of overlap algebras as an extension of the category of sets and relations, and we establish some basic facts about mono-epi-isomorphisms and (co)limits; here a morphism is a symmetrizable function (with classical logic this is just a function which preserves joins). Then we specialize to the case of morphisms which preserve also finite meets: classically, this is the usual category of complete Boolean algebras. Finally, we connect overlap algebras with locales, and their morphisms with open maps between locales, thus obtaining constructive versions of some results about Boolean locales.Comment: Postproceedings of CCC2018: Continuity, Computability, Constructivity. Faro, Portugal, 24-28 Sep 201

    Constructive version of Boolean algebra

    Full text link
    The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds classically to that of map preserving arbitrary joins; we provide a description of atomic set-based overlap algebras in the language of formal topology, thus giving a predicative characterization of discrete locales; we show that the power-collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap algebra and overlap morphism in various ways to provide constructive versions of the category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page

    A constructive Galois connection between closure and interior

    Full text link
    We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.Comment: This is a revised version. Content is reorganized so to separate clearly what requires an impredicative proof from what can be proven also predicatively. Moreover, some results are given in a more general form and some counterexamples are adde

    Reducibility, a constructive dual of spatiality

    Get PDF
    An intuitionistic analysis of the relationship between pointfree and pointwise topology brings new notions to light that are hidden from a classical viewpoint. In this paper, we study one of these, namely the notion of reducibility for a pointfree topology, which is classically equivalent to spatiality. We study its basic properties and we relate it to spatiality and to other concepts in constructive topology. We also analyse some notable examples. For instance, reducibility for the pointfree Cantor space amounts to a strong version of Weak K\uf6nig\u2019s Lemma

    Overlap Algebras as Almost Discrete Locales

    Get PDF
    Boolean locales are "almost discrete", in the sense that a spatial Boolean locale is just a discrete locale (that is, it corresponds to the frame of open subsets of a discrete space, namely the powerset of a set). This basic fact, however, cannot be proven constructively, that is, over intuitionistic logic, as it requires the full law of excluded middle (LEM). In fact, discrete locales are never Boolean constructively, except for the trivial locale. So, what is an almost discrete locale constructively? Our claim is that Sambin's overlap algebras have good enough features to deserve to be called that. Namely, they include the class of discrete locales, they arise as smallest strongly dense sublocales (of overt locales), and hence they coincide with the Boolean locales if LEM holds
    corecore