Inductively generated formal topologies

Coquand, Thierry; Sambin, Giovanni; Smith, Jan M.; Valentini, Silvio

doi:10.1016/s0168-0072(03)00052-6

Cited by 103 publications

(229 citation statements)

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“…The notion of inductively generated formal topology by Coquand et al [7] gives us a convenient method to define formal topologies.…”

Section: Inductively Generated Formal Topologiesmentioning

confidence: 99%

A point-free characterisation of Bishop locally compact metric spaces

Kawai

2017

JLA

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Abstract:We give a characterisation of Bishop locally compact metric spaces in terms of formal topology. To this end, we introduce the notion of inhabited enumerably locally compact regular formal topology, and show that the category of Bishop locally compact metric spaces is equivalent to the full subcategory of formal topologies consisting of those objects which are isomorphic to some inhabited enumerably locally compact regular formal topology.In the course of obtaining the above equivalence, we show a couple of point-free results which are of independent interest. First, we show that any overt enumerably locally compact regular formal topology admits a one-point compactification, i.e. it can be embedded into a compact overt enumerably completely regular formal topology as the open complement of a formal point. Second, we characterise the class of enumerably completely regular formal topologies as the subtopolgies of the product of countably many copies of the formal unit interval.We work in Aczel's constructive set theory CZF with Regular Extension Axiom and Dependent Choice.2010 Mathematics Subject Classification 03F60 (primary); 06D22, 54E45 (secondary) Keywords: formal topologies, locally compact metric spaces, Bishop constructive mathematics IntroductionIn locale theory (Johnstone [13]), the standard adjunction between the category of topological spaces and that of locales restricts to an equivalence between the category of sober spaces and that of spatial locales. The equivalence allows us to transfer results between general topology and locale theory.Aczel [1] showed that the adjunction is constructively valid by replacing the notion of locale with Sambin's notion of formal topology [19] To overcome this difficulty, Palmgren [17] constructed another embedding, a full and faithful functor M : LCM → FTop, from the category of locally compact metric spaces LCM into that of formal topologies FTop, using the localic completion of generalized metric spaces due to Vickers [21]. Unlike the standard adjunction, the embedding M has important properties that a metric space X is compact if and only if M(X) is compact and that M(X) is locally compact whenever X is locally compact.In our previous work [14, Chapter 4], we characterised the image of the category of compact metric spaces under the embedding M using the notion of compact overt enumerably completely regular formal topology. This means that the category of compact metric spaces is equivalent to the full subcategory of FTop consisting of those formal topologies which are isomorphic to some compact overt enumerably completely regular formal topology.In the present paper, we extend the characterisation to the class of Bishop locally compact metric spaces. We introduce the notion of inhabited enumerably locally compact regular formal topology and show that the class of inhabited enumerably locally compact regular formal topologies characterises the image of Bishop locally compact metric spaces under the embedding M up to isomorphism. Specifically, we show tha...

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“…The notion of inductively generated formal topology by Coquand et al [7] gives us a convenient method to define formal topologies.…”

Section: Inductively Generated Formal Topologiesmentioning

confidence: 99%

A point-free characterisation of Bishop locally compact metric spaces

Kawai

2017

JLA

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“…The notion of formal topology was introduced by Martin-Löf and Sambin to describe locales predicatively; it corresponds impredicatively to that of open locale, and classically to that of locale (see [7,13,14]). Here we give a definition of a formal topology which is more suitable for our purposes, though equivalent to that in [5].…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

“…Now, we are going to show how the abstract characterization of discrete locales in section 5 page 40 of [9] is equivalent to our notion of discrete formal topology. Given that the mentioned characterization of discrete locales makes reference to the diagonal map ∆ P = id P , id P : P → P × P in the category of locales and given that we do not know how to build predicative binary products in the whole category of formal topologies but we only know it in the full sub-category of inductively generated formal topologies in the sense of [5] (see [12]), here we restrict our attention to inductively generated formal topologies. We just recall that if S is a base for P, then S × S is a base for the product formal topology P × P and the corresponding positivity predicate is Pos(p) ≡ ∃ a∈S,b∈S ( (a, b) ≤ p & ( Pos(a) & Pos(b) ) ) for p in P × P. In this context we define the notion of open formal topology map as follows.…”

Section: Atomic O-algebras As Discrete Formal Topologiesmentioning

confidence: 99%

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Constructive version of Boolean algebra

Ciraulo

Maietti

Toto

2012

Logic Journal of IGPL

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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.

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“…A geometric theory gives an inductively generated formal topology [CSSV03] in an obvious way, though with extra complexity due to the standard practice of presenting with a base of opens rather than a subbase. Regarding maps, we cannot apply in any obvious way the topos techniques using generic points.…”

Section: Geometricitymentioning

confidence: 99%

A localic theory of lower and upper integrals

Vickers

2008

Mathematical Logic Qtrly

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An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the nonnegative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. This is a preprint version of the article published as -

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Inductively generated formal topologies

Cited by 103 publications

References 9 publications

A point-free characterisation of Bishop locally compact metric spaces

A point-free characterisation of Bishop locally compact metric spaces

Constructive version of Boolean algebra

A localic theory of lower and upper integrals

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