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Apartness spaces as framework for constructive topology
 Ann. Pure Appl. Logic
, 2003
"... An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonst ..."
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An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.
A Constructive Theory of PointSet Nearness
 in Proceedings of Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization, Seminar in Dagstuhl, Germany, 4–9 June 2000; Springer Lecture Notes in Computer Science
, 2001
"... An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology. ..."
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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.
Research Article Arthritic HandFinger Movement Similarity Measurements: Tolerance Near Set Approach
, 2011
"... License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem considered in this paper is how to measure the degree of resemblance between nonarthritic and arthritic hand movements during rehabilitation exercise. The ..."
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License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem considered in this paper is how to measure the degree of resemblance between nonarthritic and arthritic hand movements during rehabilitation exercise. The solution to this problem stems from recent work on a tolerance space view of digital images and the introduction of image resemblance measures. The motivation for this work is both to quantify and to visualize differences between handfinger movements in an effort to provide clinicians and physicians with indications of the efficacy of the prescribed rehabilitation exercise. The more recent introduction of tolerance near sets has led to a useful approach for measuring the similarity of sets of objects and their application to the problem of classifying image sequences extracted from videos showing fingerhand movement during rehabilitation exercise. The approach to measuring the resemblance between hand movement images introduced in this paper is based on an application of the wellknown Hausdorff distance measure and a tolerance nearness measure. The contribution of this paper is an approach to measuring as well as visualizing the degree of separation between images in arthritic and nonarthritic handfinger motion videos captured during rehabilitation exercise. 1.
Compactness in apartness spaces
"... Abstract. In this note, we establish some results which suggest a possible solution to the problem of finding the right constructive notion of compactness in the context of a not–necessarily–uniform apartness space. ..."
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Abstract. In this note, we establish some results which suggest a possible solution to the problem of finding the right constructive notion of compactness in the context of a not–necessarily–uniform apartness space.
On Complements of Sets and the Efremovič Condition in Pre–apartness Spaces 1
"... Abstract: In this paper we study various properties of complements of sets and the Efremovič separation property in a symmetric pre–apartness space. Key Words: Pre–apartness spaces, Efremovič property Category: F.4.1 The constructive theory of apartness 2 (point–set and set–set) has been developed w ..."
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Abstract: In this paper we study various properties of complements of sets and the Efremovič separation property in a symmetric pre–apartness space. Key Words: Pre–apartness spaces, Efremovič property Category: F.4.1 The constructive theory of apartness 2 (point–set and set–set) has been developed within the framework of Bishop’s constructive mathematics BISH [1, 2, 3, 13] in a series of papers over the past five years [17, 5, 12, 14, 7]. In this paper we derive some basic properties of complements of sets in pre–apartness spaces and discuss a strong separation property. Our starting point is a set X equipped with an inequality relation applicable to points of X, and a symmetric relation ⊲ ⊳ applicable to subsets of X. The inequality satisfies two simple properties x � = y ⇒ y � = x x � = y ⇒¬(x = y). Forapointx of X we write x⊲⊳Sas shorthand for {x} ⊲ ⊳ S. There are three notions of complement applicable to a subset S of X: – the logical complement – the complement – and the apartness complement ¬S = {x ∈ X: x/ ∈ S}, ∼ S = {x ∈ X: ∀s ∈ S (x � = s)},
Apartness, topology, and uniformity: a constructive view
, 2001
"... Abstract. The theory of apartness spaces, and their relation to topological spaces (in the point—set case) and uniform spaces (in the set—set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metris ..."
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Abstract. The theory of apartness spaces, and their relation to topological spaces (in the point—set case) and uniform spaces (in the set—set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by even a uniform structure. 1.
ON STRONG INCLUSIONS AND ASYMMETRIC PROXIMITIES IN FRAMES
, 2010
"... The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification ..."
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The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification of a strong inclusion for biframes then provided a pointfree model also for the nonsymmetric variant. In this paper we show that a strong inclusion can be nonsymmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasiproximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasiuniform frames in a full analogy with the symmetric case.