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Constructive Order Completeness
, 2004
"... Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an ..."
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Partially ordered sets are investigated from the point of view of Bishop’s constructive mathematics. Unlike the classical case, one cannot prove constructively that every nonempty bounded above set of real numbers has a supremum. However, the order completeness of R is expressed constructively by an equivalent condition for the existence of the supremum, a condition of (upper) order locatedness which is vacuously true in the classical case. A generalization of this condition will provide a definition of upper locatedness for a partially ordered set. It turns out that the supremum of a set S exists if and only if S is upper located and has a weak supremum—that is, the classical least upper bound. A partially ordered set will be called order complete if each nonempty subset that is bounded above and upper located has a supremum. It can be proved that, as in the classical mathematics, R n is order complete. 1
Cauchy nets in the constructive theory of apartness spaces
 Scientiae Math. Japonicae
, 2002
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Constructive Solutions of Continuous Equations
, 2003
"... We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand... ..."
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We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand...
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.