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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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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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Continuing the study of apartness in lattices, begun in [8], this paper deals with axioms for a product a-frame and with their consequences. This leads to a reasonable notion of proximity in an a-frame, abstracted from its counterpart in the theory of set-set apartness. |
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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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This paper is dedicated to Prof. Dr. Günter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community.At first sight it appears highly unlikely that Urysohn's Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant constructive version (...) |
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Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic. |
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In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof-technique is extracted and then applied in several different situations. |
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Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them. |
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The paper deals with proximal convergence and Leader's theorem, in the constructive theory of uniform apartness spaces. |
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A natural extension theorem for strongly continuous mappings, the morphisms in the category of apartness spaces, is proved constructively. |
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The anti-Specker property, a constructive version of sequential compactness, is used to prove constructively that a pointwise continuous, order-dense preference relation on a compact metric space is uniformly sequentially continuous. It is then shown that Ishihara's principle BD-ℕ implies that a uniformly sequentially continuous, order-dense preference relation on a separable metric space is uniformly continuous. Converses of these two theorems are also proved. |
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The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) |
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