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Cauchy nets in the constructive theory of apartness spaces
 Scientiae Mathematicae Japonicae
, 2002
"... Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation o ..."
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Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation of Bishop’s Lemma on locatedness. 1 Introduction Axioms for a constructive theory of apartness between sets were introduced in [12], where the particular example of a uniform space was discussed in detail. In the present paper we discuss Cauchy and convergent sequences in the framework of that theory. By constructive mathematics we mean mathematics developed with intuitionistic logic
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