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The regularlocallycompact coreflection of stably locally compact locale
 Journal of Pure and Applied Algebra
, 2001
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally comp ..."
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The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,
Multi Lingual Sequent Calculus and Coherent Spaces
 Fundamenta Informaticae
, 1997
"... We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic ge ..."
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We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalized in this fashion.
Stably Compact Spaces and Closed Relations
, 2001
"... Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and ..."
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Cited by 15 (3 self)
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Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and study the resulting categorical properties. Apart from extending ordinary continuous maps, these morphisms have a number of pleasing properties, the most prominent, perhaps, being that they correspond to preframe homomorphisms on the localic side. We exploit this Stonetype duality to establish that the category of stably compact spaces and closed relations has bilimits.
Computably based locally compact spaces
, 2003
"... ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalen ..."
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Cited by 9 (6 self)
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ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth’s effectively given domains and Jung’s Strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the waybelow relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott’s domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.
Uniform Approximation of Topological Spaces
 TOPOLOGY AND ITS APPLICATIONS 83
, 1996
"... We sharpen the notion of a quasiuniform space to spaces which carry with them functional means of approximating points, opens and compacts. Assuming nothing but sobriety, the requirement of uniform approximation induces that such spaces are compact ordered (in the sense of Nachbin). We study un ..."
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We sharpen the notion of a quasiuniform space to spaces which carry with them functional means of approximating points, opens and compacts. Assuming nothing but sobriety, the requirement of uniform approximation induces that such spaces are compact ordered (in the sense of Nachbin). We study uniformly approximated spaces with the means of topology, uniform topology, order theory and locale theory. In each case it turns out that one can give a succinct and meaningful characterization. This leads us to believe that uniform approximation is indeed a concept of central importance.
The patch frame of the Lawson dual of a stably continuous frame.Unpublished research note
 School of Computer Science, St Andrews University
, 2000
"... Continuous maps of compact regular locales form a coreflective subcategory of the category of perfect maps of stably compact locales. The coreflection of a stably compact locale is given by the frame of Scott continuous nuclei and is referred to as its patch. We show that the patch of a stably compa ..."
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Continuous maps of compact regular locales form a coreflective subcategory of the category of perfect maps of stably compact locales. The coreflection of a stably compact locale is given by the frame of Scott continuous nuclei and is referred to as its patch. We show that the patch of a stably compact locale is isomorphic to the patch of its Lawson dual.
Priestley Duality for Strong Proximity Lattices
"... In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zerodimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone’s topology ..."
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In 1937 Marshall Stone extended his celebrated representation theorem for Boolean algebras to distributive lattices. In modern terminology, the representing topological spaces are zerodimensional stably compact, but typically not Hausdorff. In 1970, Hilary Priestley realised that Stone’s topology could be enriched to yield orderdisconnected compact ordered spaces. In the present paper, we generalise Priestley duality to a representation theorem for strong proximity lattices. For these a “Stonetype ” duality was given in 1995 in joint work between Philipp Sünderhauf and the second author, which established a close link between these algebraic structures and the class of all stably compact spaces. The feature which distinguishes the present work from this duality is that the proximity relation of strong proximity lattices is “preserved ” in the dual, where it manifests itself as a form of “apartness. ” This suggests a link with constructive mathematics which in this paper we can only hint at. Apartness seems particularly attractive in view of potential applications of the theory in areas of semantics where continuous phenomena play a role; there, it is the distinctness between different states which is observable, not equality. The idea of separating states is also taken up in our discussion of possible morphisms for which the representation theorem extends to an equivalence of categories.
SEMANTIC SPACES IN PRIESTLEY FORM
, 2006
"... To my family. ii Table of Contents Table of Contents iii Abstract vi ..."
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To my family. ii Table of Contents Table of Contents iii Abstract vi
Bitopological duality for distributive lattices and Heyting algebras
 Mathematical Structures in Computer Science
, 2010
"... It is widely considered that the beginning of duality theory was Stone’s groundbreaking work in the mid 30ies on the dual equivalence of the category Bool of Boolean algebras and Boolean algebra homomorphism and the category Stone of compact Hausdorff zerodimensional spaces, which became known as S ..."
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It is widely considered that the beginning of duality theory was Stone’s groundbreaking work in the mid 30ies on the dual equivalence of the category Bool of Boolean algebras and Boolean algebra homomorphism and the category Stone of compact Hausdorff zerodimensional spaces, which became known as Stone spaces, and continuous functions. In 1937 Stone [7] extended this to the dual equivalence of the category DLat of bounded distributive lattices and bounded lattice homomorphisms and the category Spec of what later became known as spectral spaces and spectral maps. Spectral spaces provide a generalization of Stone spaces. Unlike Stone spaces, spectral spaces are not Hausdorff (not even T1), and as a result, are more difficult to work with. In 1970 Priestley