Commun.Fac.Sci.Univ.Ank.Series A1 Volume 62, Number 2, Pages 67—74 (2013) ISSN 1303—5991 GENERALIZED NEIGHBOURHOOD SYSTEMS OF FUZZY POINTS SEVDA SAGIRO¼ GLU,¼ ERDAL GÜNER AND EDA KOÇYIGIT¼ Abstract. We define the generalized fuzzy neighbourhood systems on the set of fuzzy points in a nonempty set X and investigate their prop- erties by using a new interior operator. With the help of these concepts we introduce generalized fuzzy continuity, which include many of the variations of fuzzy continuity already in the literature, as special cases. 1. Introduction A neighbourhood system assigns each object a (possibly empty, finite or infinite) family of nonempty subsets. Such subsets, called neighbourhoods, represent the semantics of near. Formally, neighbourhoods play the most fundemantel role in mathematical analysis. Informally, it is a common and intuitive notion. It is in databases [10,20], in rough sets [27], in logic [5], in texts of genetic algorithms [14], and many others. This paper introduces generalized neighbourhood systems on the set of fuzzy points of a nonempty set. The fundemantal idea of fuzzy sets was first introduced by Zadeh [35]. Chang [9] is known as the initiator of the notion of fuzzy topology. In 1976, the fuzzy topology was redifined in somewhat different way by Lowen [15]. Then many attempts have been made to extend various branches of mathematics to the fuzzy settings. We focus our work to extend the notions of the generalized neighbourd system to the fuzzy settings. To generalize the notions of topolgy, the initial attempts can be seen in [18] and [16], respectively, i.e., supratopologies and minimal structures. Recently, Császár [11] introduced the notions of generalized topologies (brieflyGT) and generalized neighbourhood systems (briefly GNS). In [1], fuzzy supratopology and, recently, in [26], generalized fuzzy topology were defined as generalizations of fuzzy topology introduced by Chang. In addition, as a generalization of fuzzy topology introduced by Lowen, fuzzy minimal structure was defined in [3]. The neighbourhood and q-neighbourhood of a fuzzy point in a fuzzy topological space Received by the editors Octob. 21, 2012; Accepted: Dec. 08, 2013. 2000 Mathematics Subject Classification. 54A05, 54A40, 54C05. Key words and phrases. Generalized fuzzy topology, generalized fuzzy neighbourhood system, fuzzy ( , 0)-continuity. c 2013 Ankara University 67 68 SEVDA SAGIRO¼ GLU,ERDALGÜNERANDEDAKOÇYI¼ GIT¼ in the sense of Chang was introduced by Pu and Liu [19]. An earlier study on neighbourhood of fuzzy points can be found in [13]. In this paper, we define the generalized fuzzy neighbourhood systems on the set of fuzzy points in a nonempty set X and investigate their properties by using a new interior operator which corresponds to the notion of the interior operator in general form and gives us the way to show that every generalized fuzzy topology can be generated by a generalized fuzzy neighbourhood system. In addition, we introduce generalized fuzzy continuity with the help of generalized fuzzy neighbourhood sys- tems. These notions lead us to give a general form to various concepts discussed in the literature. 2. Preliminaries Let X be an arbitrary nonempty set. A fuzzy set A in X is a function on X into the interval I = [0, 1] of the real line. The class of all fuzzy sets in X will be denoted by IX and symbols A, B, ... is used for fuzzy sets in X. The complement of a fuzzy set A in X is 1X A. The fuzzy sets in X taking on respectively the constant values 0 and 1 are denoted by 0X and 1X , respectively. A fuzzy set A is nonempty X if A = 0X . For two fuzzy sets A, B I , we write A B if A (x) B (x) for 6 2X ≤ ≤ each x X. For a family Aj j J I , the union C = jAj and the intersection 2 f g 2 [ D = jAj, are defined by C (x) = sup Aj (x) and D (x) = inf Aj (x) for each \ J f g J f g x X. For a fuzzy set A in X, the set x X : A (x) > 0 is called the support of A.2A fuzzy singleton or a fuzzy point withf 2 support x andg value (0 < 1) is ≤ denoted by x. The fuzzy point x is said to be contained in a fuzzy set A, denoted by x A, iff A (x) , whereas the notion xqA means that x is quasi-concident 2 ≤ with A, i.e., xqA implies + A (x) > 1. X Y 1 Let f be a function from X to Y, A I and B I . Then f (B) and f (A) 1 2 2 are defined as; f (B)(x) = B (f (x)) for x X and 2 1 sup A (x) , f (y) = x f 1(y) 6 ; f (A)(y) = 2 ( 0 , otherwise for y Y, respectively. Throughout2 this paper, by a fuzzy topological space (shortly fts) we mean a fts X (X, o) , as initiated by Chang [9], i.e., o I satisfy (a) 0X , 1X o, (b) If Aj o 2 2 for each j J = , then j J Aj o and (c) If A, B o, then A B o. The elements of2o are6 ; called fuzzy[ 2 open2 sets and their complements2 are\ called2 fuzzy closed sets. We shall denote the fuzzy interior and fuzzy closure of a fuzzy set X A I with ioA and coA, respectively, i.e. ioA = U : U A, A o and 2 [ f ≤ 2 g coA = F : F A, 1X A o . A fuzzy set V is called a neighbourhood of \ f ≥ 2 g fuzzy point x iff there exists U o such that x U V and V is called a 2 2 ≤ q-neighbourhood of x iff there exists U o such that xqU V. The fuzzy set 2 ≤ GENERALIZED NEIGHBOURHOOD SYSTEMS OF FUZZY POINTS 69 theoretical and fuzzy topological concepts used in this paper are standard and can be found in Zadeh [35], Chang [9], Pu and Liu [19]. The family of all fuzzy semiopen [4] (resp. fuzzy preopen [32], fuzzy - [32], fuzzy -open [7], fuzzy semi-preopen [34], fuzzy regular open [4]) sets of (X, o) shall be denoted by F So (resp. F P o, F o, F o, F SP o, F Ro). Fuzzy minimal structures are defined and investigated in [3]. A subfamily m X I is said to be a fuzzy minimal structure on X iff 1X m for each I and the elements of m are called fuzzy m-open sets. A fuzzy2 supratopology2 [1] is a X subfamily g of I , satisfying 0X , 1X g and arbitrary union of members of g 2 belongs to g. In addition, if g satisfies these conditions except 1X g, then g is said to be a generalized fuzzy topology (briefly GFT) in [26]. 2 In the sequal, the set of all fuzzy points in X is denoted by . P 3. Generalized Fuzzy Neighbourhood Systems Let us define IX : 2 satisfy V (x) for V (x) P! ≤ 2 Then we shall say that V (x) is a generalized fuzzy neighbourhood (briefly 2 GFN) of the fuzzy point x and is a generalized fuzzy neighbourhood system (briefly GFNS) on the set of fuzzy points in X. We denote by ( ) the collection of all GFNS’son . P P X For an arbitrary fuzzy set A I , write x ,A iff there exists V (x) satisfying V A. 2 2 P 2 ≤ X Definition 3.1. Let ( ) and A I . Then define the fuzzy set { A as: 2 P 2 sup , I satisfying x ,A x ,A 9 2 2 P ({ A)(x) = 2P ( 0 , otherwise for all x in X. { A is called the interior of A on . Lemma 3.2. Let ( ) . Then 2 P (a) { 0X = 0X , X (b) { A A, for A I , ≤ 2 X (c) A B implies { A { B, for all A, B I . ≤ ≤ 2 Proof. (a) Since ,0 = , we have ({ 0X )(x) = 0 for all x in X. Thus { 0X = 0X . P X ; (b) Clearly { 0X 0X . Let A = 0X and an arbitrary x in X. If ({ A)(x) = 0, ≤ 6 then { A A. If ({ A)(x) = sup := t > 0, then there exists I satisfying ≤ x ,A 2 2P x ,A. In this case, let xt ,A for j J = , then there exists Vj xt 2 P j 2 P 2 6 ; 2 j satisfying tj Vj (x) A (x) for each j J. Thus t = supj J tj A (x) . Therefore ≤ ≤ 2 2 ≤ { A A. ≤ (c) A B implies ,A ,B. Therefore { A { B. ≤ P P ≤ 70 SEVDA SAGIRO¼ GLU,ERDALGÜNERANDEDAKOÇYI¼ GIT¼ X Proposition 1. Let an arbitrary (Aj)j J I . Then { Aj { Aj . 2 j J ≤ j J 2 2 ! S S Proof. Clearly Aj j J Aj for each j J. Then { Aj { Aj for each ≤ [ 2 2 ≤ j J 2 ! S j J by Lemma 3.2(c) . Hence { Aj { Aj . 2 j J ≤ j J 2 2 ! S S X Lemma 3.3. Let ( ) and g = G I : G = { G . Then g is a GFT on X. 2 P 2 Proof. Clearly, 0X g by Lemma 3.2(a) . Let G = j J Gj and Gj g for j 2 [ 2 2 2 J = . Then { G G is clear by Lemma 3.2(b) . On the other hand we have 6 ; ≤ G { Gj since Gj g for each j J. In addition, { Gj { G is clear by ≤ j J 2 2 j J ≤ 2 2 Proposition1S Therefore G { G. Hence G g. S ≤ 2 So it is clear that every GFNS generates a GFT.