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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.
From intuitionistic to pointfree topology: some remarks on the foundation of homotopy theory
 Special issue with papers from the second workshop on formal topology. Annals of Pure and Applied Logic 137
, 2006
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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.
Cauchy nets in the constructive theory of apartness spaces
 Scientiae Math. Japonicae
, 2002
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Computable Separation in Topology, from T0 to T2
"... Abstract: This article continues the study of computable elementary topology started in [Weihrauch and Grubba 2009]. For computable topological spaces we introduce a number of computable versions of the topological separation axioms T0, T1 and T2. The axioms form an implication chain with many equiv ..."
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Abstract: This article continues the study of computable elementary topology started in [Weihrauch and Grubba 2009]. For computable topological spaces we introduce a number of computable versions of the topological separation axioms T0, T1 and T2. The axioms form an implication chain with many equivalences. By counterexamples we show that most of the remaining implications are proper. In particular, it turns out that computable T1 is equivalent to computable T2 and that for spaces without isolated points the hierarchy collapses, that is, the weakest computable T0 axiomWCT0 is equivalent to the strongest computable T2 axiom SCT2. The SCT2spaces are closed under Cartesian product, this is not true for most of the other classes of spaces. Finally we show that the computable version of a basic axiom for an effective topology in intuitionistic topology is equivalent to SCT2. Key Words: computable analysis, computable topology, axioms of separation
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.