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  1. On the Existence of Stone-Čech Compactification.Giovanni Curi - 2010 - Journal of Symbolic Logic 75 (4):1137-1146.
  • Fundamental Results for Pointfree Convex Geometry.Yoshihiro Maruyama - 2010 - Annals of Pure and Applied Logic 161 (12):1486-1501.
    Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual (...)
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  • Maximal and Partial Points in Formal Spaces.Erik Palmgren - 2006 - Annals of Pure and Applied Logic 137 (1):291-298.
    The class of points in a set-presented formal topology is a set, if all points are maximal. To prove this constructively a strengthening of the dependent choice principle to infinite well-founded trees is used.
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  • Embedding Locales and Formal Topologies Into Positive Topologies.Francesco Ciraulo & Giovanni Sambin - 2018 - Archive for Mathematical Logic 57 (7-8):755-768.
    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 (...)
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  • A Predicative Completion of a Uniform Space.Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2012 - Annals of Pure and Applied Logic 163 (8):975-980.
  • On the Collection of Points of a Formal Space.Giovanni Curi - 2006 - Annals of Pure and Applied Logic 137 (1):126-146.
    On the collection of points of a formal space.
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  • Locatedness and Overt Sublocales.Bas Spitters - 2010 - Annals of Pure and Applied Logic 162 (1):36-54.
    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 (...)
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  • The Associated Sheaf Functor Theorem in Algebraic Set Theory.Nicola Gambino - 2008 - Annals of Pure and Applied Logic 156 (1):68-77.
    We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.
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  • Quasi-Apartness and Neighbourhood Spaces.Hajime Ishihara, Ray Mines, Peter Schuster & Luminiţa Vîţă - 2006 - Annals of Pure and Applied Logic 141 (1):296-306.
    We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.
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  • Exact Approximations to Stone–Čech Compactification.Giovanni Curi - 2007 - Annals of Pure and Applied Logic 146 (2):103-123.
    Given a locale L and any set-indexed family of continuous mappings , fi:L→Li with compact and completely regular co-domain, a compactification η:L→Lγ of L is constructed enjoying the following extension property: for every a unique continuous mapping exists such that . Considered in ordinary set theory, this compactification also enjoys certain convenient weight limitations.Stone–Čech compactification is obtained as a particular case of this construction in those settings in which the class of [0,1]-valued continuous mappings is a set for all L. (...)
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  • Topological Inductive Definitions.Giovanni Curi - 2012 - Annals of Pure and Applied Logic 163 (11):1471-1483.
    In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...)
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  • The Axiom of Multiple Choice and Models for Constructive Set Theory.Benno van den Berg & Ieke Moerdijk - 2014 - Journal of Mathematical Logic 14 (1):1450005.
  • Formally Continuous Functions on Baire Space.Tatsuji Kawai - 2018 - Mathematical Logic Quarterly 64 (3):192-200.
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  • Relating Bishopʼs Function Spaces to Neighbourhood Spaces.Hajime Ishihara - 2013 - Annals of Pure and Applied Logic 164 (4):482-490.
    We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces.
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  • Two Subcategories of Apartness Spaces.Hajime Ishihara - 2012 - Annals of Pure and Applied Logic 163 (2):132-139.
  • On the T 1 Axiom and Other Separation Properties in Constructive Point-Free and Point-Set Topology.Peter Aczel & Giovanni Curi - 2010 - Annals of Pure and Applied Logic 161 (4):560-569.
    In this note a T1 formal space is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces , strengthening separation properties discussed elsewhere. Finally we relate (...)
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  • Derived Rules for Predicative Set Theory: An Application of Sheaves.Benno van den Berg & Ieke Moerdijk - 2012 - Annals of Pure and Applied Logic 163 (10):1367-1383.
  • Abstract Inductive and Co-Inductive Definitions.Giovanni Curi - 2018 - Journal of Symbolic Logic 83 (2):598-616.
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  • Separation Properties in Neighbourhood and Quasi-Apartness Spaces.Robin Havea, Hajime Ishihara & Luminiţa Vîţă - 2008 - Mathematical Logic Quarterly 54 (1):58-64.
    We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.
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  • 2010 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium '10.Michael Rathjen - 2011 - Bulletin of Symbolic Logic 17 (2):272-329.
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  • On Some Peculiar Aspects of the Constructive Theory of Point-Free Spaces.Giovanni Curi - 2010 - Mathematical Logic Quarterly 56 (4):375-387.
    This paper presents several independence results concerning the topos-valid and the intuitionistic predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined.
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