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21
The Dedekind reals in abstract Stone duality
 Mathematical Structures in Computer Science
, 2008
"... Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherentl ..."
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Cited by 9 (5 self)
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Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambdacalculus, of which we provide a selfcontained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using twosided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore’s, because we work constructively, and we also consider backtofront (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.
POINTFREE FORMS OF DOWKER AND MICHAEL INSERTION THEOREMS
"... In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact sp ..."
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In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.
On the Cauchy Completeness of the Constructive Cauchy Reals
 Mathematical Logic Quarterly
"... Intuitionistic set theory without choice axioms does not prove that every Cauchy sequence of rationals has a modulus of convergence, or that the set of Cauchy sequences of rationals is Cauchy complete. Several other related nonprovability results are also shown. ..."
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Cited by 5 (3 self)
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Intuitionistic set theory without choice axioms does not prove that every Cauchy sequence of rationals has a modulus of convergence, or that the set of Cauchy sequences of rationals is Cauchy complete. Several other related nonprovability results are also shown.
Spatiality for formal topologies
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2006
"... We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples. ..."
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Cited by 5 (2 self)
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We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.
Sequentially Continuous Linear Mappings in Constructive Analysis
, 1996
"... this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, I ..."
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Cited by 4 (4 self)
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this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, INT, and RUSS, see [2]. 2 Sequential continuity preserves Cauchyness
THE ASSOCIATED SHEAF FUNCTOR THEOREM IN ALGEBRAIC SET THEORY
"... Abstract. 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 develo ..."
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Abstract. 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. 1.
On the Constructive Dedekind Reals
 Log. Anal
, 2008
"... In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alon ..."
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In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Setlike principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class. 1
THE SEMICONTINUOUS QUASIUNIFORMITY OF A Frame
"... The aim of this note is to show how various facts in classical topology connected with semicontinuous functions and the semicontinuous quasiuniformity have their counterparts in pointfree topology. In particular, we introduce the localic semicontinuous quasiuniformity, which generalizes the semic ..."
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The aim of this note is to show how various facts in classical topology connected with semicontinuous functions and the semicontinuous quasiuniformity have their counterparts in pointfree topology. In particular, we introduce the localic semicontinuous quasiuniformity, which generalizes the semicontinuous quasiuniformity of a topological space (known to be one of the most important examples of transitive compatible quasiuniformities). We show that it can be characterized in terms of the so called spectrum covers, via a construction introduced by the authors in a previous paper. Several consequences are derived.