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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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We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.
Quotient spaces and coequalisers in formal topology
 J. UCS
"... Abstract: We give a construction of coequalisers in formal topology, a predicative version of locale theory. This allows for construction of quotient spaces and identification spaces in constructive topology. ..."
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Abstract: We give a construction of coequalisers in formal topology, a predicative version of locale theory. This allows for construction of quotient spaces and identification spaces in constructive topology.
PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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Cited by 2 (1 self)
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
A discrete duality between apartness algebras and apartness frames
, 2008
"... 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 apa ..."
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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 and Vîta [3], and we prove a discrete duality for them.
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.
Right Type Departmental Bulletin Paper
, 2004
"... Within Bishop’s constructive mathematics, we briefly introduce apartness spaces and discuss various continuity properties of mappings between these spaces. The theory of proximity spaces originated apparently in 1908 at the mathematical congress in Rome, when Riesz [24] presented some ideas in his ‘ ..."
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Within Bishop’s constructive mathematics, we briefly introduce apartness spaces and discuss various continuity properties of mappings between these spaces. The theory of proximity spaces originated apparently in 1908 at the mathematical congress in Rome, when Riesz [24] presented some ideas in his ‘theory of enchainment’ which have become the basic concepts of the theory. In the early 1950’s, the subject was rediscovered by Efrernovic [13], [14] when he axiomatically characterized the proximity relation ‘A is near $B $ ’ for subsets of any set $X $. Recently there has been quite an intense investigation of topological structures in image processing, mostly in connection with the analysis of connectivity and the operation of thinning (see e.g. [3, 12, 19, 21], etc.). An interesting attempt to introduce richer structures than those of topology, and replacing thus local ’ continuity properties by a global notion of nearness, has been done in [20] where the authors contemplated the so called semiproximity spaces as a theoretical tool in the image processing studies. See also $[18, 25] $. This paper outlines a few aspects of the theory of apartness spaces. We work entirely