On Complements of Sets and the Efremovi Condition in Pre–apartness Spaces 1.pdf

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On Complements of Sets and the Efremovi Condition in Pre–apartness Spaces 1.pdf

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On Complements of Sets and the Efremovi Condition in Pre–apartness Spaces 1

On Complements of Sets and the Efremovic? Condition in Pre–apartness Spaces1 Luminit?a Simona V??t?a? (Department of Mathematics Statistics University of Canterbury, Christchurch, New Zealand l.vita@math.canterbury.ac.nz) Abstract: In this paper we study various properties of complements of sets and the Efremovic? separation property in a symmetric pre–apartness space. Key Words: Pre–apartness spaces, Efremovic? property Category: F.4.1 The constructive theory of apartness2 (point–set and set–set) has been de- veloped 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). For a point x of X we write x S as shorthand for {x} S. There are three notions of complement applicable to a subset S of X : – the logical complement ?S = {x ∈ X : x /∈ S} , – the complement ～ S = {x ∈ X : ?s ∈ S (x = s)} , – and the apartness complement ?S = {x ∈ X : x S} . The pair (X, ) is called a symmetric pre–apartness space if the following axioms are satisfied. 1 C. S. Calude, H. Ishihara (eds.). Constructivity, Computability, and Logic. A Collection of Papers in Honour of the 60th Birthday of Douglas Bridges. 2 The motivation for this theory lay in the classical theory of nearness and proximity; see [8, 9, 11]. Journal of Universal Computer Science, vol. 11, no. 12 (2005), 2159-2164 submitted: 29/9/05, accepted: 1/11/05, appeared: 28/12/05 ? J.UCS B1 X ?. B2 S T ? S ?～ T . B3 R (S ∪ T ) ? R S ∧R T. B4 ?S ? ～T ? ?S ? ?T . Throughout this paper, unless otherwise specified, X will stand for a symmetric pre–apartnes