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Saturated Sets in Fuzzy Topological Spaces Computational and Applied Mathematics Journal 2015; 1(4): 180-185 Published online July 10, 2015 (http://www.aascit.org/journal/camj) Saturated Sets in Fuzzy Topological Spaces K. A. Dib, G. A. Kamel Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt Email address [email protected] (K. A. Dib), [email protected] (G. A. Kamel) Citation K. A. Dib, G. A. Kamel. Saturated Sets in Fuzzy Topological Spaces. Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 180-185. Abstract Keywords The properties of the lattice , in the - fuzzy spaces, not only control the properties of Limit Points, the operations on the family of -fuzzy subsets but also affect some essential properties in Closed Sets, - fuzzy topological spaces. For example, the concepts of limit point and closed subset are Saturated Sets, important concepts in topology in the theory and in the applications. In ordinary topology Saturated Fuzzy Topology, every closed subset contains all its limit points. But in the fuzzy case the situation is Fuzzy Topology different: only the saturated fuzzy subset contains all its limit points. It is obtained the formulation of the saturated subsets and it is shown that the family of saturated subsets in fuzzy topological space defines a (saturated) fuzzy topology. In this article, we trace the variation in some properties of some fuzzy topological spaces due to the lattice variation. Received: May 30, 2015 Most solutions of practical problems in analysis are limit points of some sequences. The Revised: June 16, 2015 results of this article for the saturated set affects the simplicity of finding solutions for Accepted: June 17, 2015 many practical problems in fuzzy case. 1. Introduction The concept of fuzzy subsets has been introduced by Zadeh (1965) [16]. Zadeh defined the fuzzy subset A of X as a function from X into the unit interval of real numbers [0,1]. Goguen (1967) generalized the definition of the fuzzy subset, introducing the L -fuzzy subset of X as a function A X → L, where L is a complete and completely distributive complemented lattice [9]. The minimal and maximal elements of L are denoted by 0and 1respectively (or simply 0 and 1). Since then, the theory of fuzzy mathematics has been developed in many directions and many applications were discovered in a wide variety of fields. Chang (1968) introduced the concept of fuzzy topology (C- fuzzy topology) [2]. Wong (1973) noted that new concepts of convergence and clustering are needed in order to develop the theory further in this context [15]. Since then, an entensive work on fuzzy topological space has been carried out by many researchers; they have used different lattices for the membership values of elements. In [4], it was given a trail to correct the deviation in the definitions of convergence and clustering in fuzzy topological spaces by studying the P -fuzzy topological spaces, where P L,,, is a sublattice lattice from the lattice of power set . In this work, we will demonstrate how the properties of the lattice L not only control the properties of the operations of the family of fuzzy subsets but also control some properties of the fuzzy topological structure on this family. In particular, we will trace the variation in the properties of the limit points and closed sets in fuzzy topological spaces resulting from the lattice variation. It is studied the (saturated) fuzzy subsets which contains all its limit points and obtained its formulation in the fuzzy topological spaces and P L. Computational and Applied Mathematics Journal 2015; 1(4): 180-185 181 2. Preliminaries operations on the sets, consequently, the operations on P L satisfy the same mentioned properties of the 2.1. In this Paper, We Shall Consider the operations on . Following Two Lattices P (4) The difference operation on the family of P - fuzzy subsets ( -fuzzy subsets) is defined as The first lattice is the closed unit interval [0,1] of real P L follows: numbers with the usual order and it is denoted by . The A#B ) "A ) #B ) ; ) $ X. elements of this lattice satisfy the inequality 0 ; for 2.2. Fuzzy Point every nonzero numbers r, s. The complementary operation is ! For any lattice , the fuzzy point is defined as follows: defined by → ; r " 1 # r, for every r $ . This is an important lattice, which is widely used in the literature. Definition (2.2.1) [15]. Let be a lattice and X be a nonempty set. A fuzzy subset of is called a fuzzy The operations on the family of fuzzy subsets are H X point, if defined as follows: Let A , B are -fuzzy subsets of X, then: for all and H) " 0; ) )< H)< " r ; 0 = & 1 (i) and is denoted by . A & B if A ) &B ) ; for all ) $ X. H " H)<, r It follows that iff and . (ii) H)<, r " Hy<, s )< " y< r " s A B) "A) B); for all ) $ X. Therefore, the fuzzy point is different from the H)<, r (iii) fuzzy point for every . A.B) "A) .B); for all ) $ X. H)<, s r (iv) / A ) " 1 # A); for all ) $ X. 2.2.1. Properties of the Family of Fuzzy Points Relative the Given (v) ϕ ?@ " 3A@, B; B $ C6 ) " 0 and X) " 1, for all ) Lattices The second lattice is denoted by which is a I- For the first lattice the family has the P L " [0,1] DE subfamily of the power set and it is defined by following properties: (i) H), s &H), r ; if s & r, 3 6 P 2 P ; P " 4 5 0 $ 4 . (ii) H), s H), r " H), s r , H), s .H), r " , The complementary operation is H), s . r P → P (iii) If and ! is the complement of , then defined by ! 3 6 ; for each The 0 = = 1 r r 4 " # 4 0 4 $ P . ! , ! . algebraic structure P L,,,! forms a complete and H), r .H), r H), 1 H), r H), r 0 completely distributive complemented lattice with as Therefore, for every nonzero r $ [0,1], there exists a 306 fuzzy element contained in by taking the smallest element and as the greatest element. In the H)<; s H)<, r F lattice , ! ; for all , where ! is ; for example The family has no disjoint P 7 7 " 0 7 $ P 7 s = s " G. DE the complement of 8. The importance of this lattice is that it fuzzy points and it is linearly ordered. generates a family of fuzzy subsets ; the properties of II- For the second lattice the family has P L"P DE its algebraic operations are very close to the properties of the the following properties: algebraic operations on the subsets in the ordinary case. This (i) H), 7 2 H), H ; if 7 2 H, lattice was used in the articles [4]. (ii) H), 7 H), H " H), 7 H and H), 7 The operations on the -fuzzy subsets of are P X H), H " H ),7H, defined as follows: Let A , B are P -fuzzy subsets of X, (iii) There are disjoint fuzzy points, for example: then: ! ; , H), 7 H), 7 " 0 7 306 (iv) This family is not linearly ordered. a A 5 B if A) 5 B); for all ) $ X. The fuzzy point H), 7 is contained in the fuzzy subset , if and it is not contained in , if b A B) "A) B); for all ) $ X. A 7 2 A), A 7 I JK and 7 JK. But the fuzzy point H), 7 ; 7 I A) c A B) "A) B); for all ) $ X. contains the fuzzy point H), A). In this lattice one can easily show that: ϕ ; if (d) / H)<, 7 H)<,H " 7 H " A ) " L # A) 306; for all ) $ X. 306; 7, H $ P . In particular, there are the following cases: (e) / = ϕ, and / A A A A " X. Remarks 2.1.1. 1 1 ϕ H L)< , M0 , OP H L)< , 306 L ,1OP " ; and (1) It’s worth to noting that P 30,16 " 3306, 30,166 , N N which is generated by the lattice , is L " 30,16 1 2 ϕ isomorphic to the lattice . H L)< , M0 , OP H L)< , 306 M ,1OP " L N N (2) On the first lattice " [0,1] one can define different complementary operations, as shown in Example (3.1). while (3) The supremum and infimum operations on the family G W of fuzzy subsets ; for arbitrary lattice , are . P L H R)< , S0 , UV H R)< , 306 S , 1UV 306 coincides with ordinary union and ordinary intersection T T 182 K. A. Dib and G. A. Kamel: Saturated Sets in Fuzzy Topological Spaces The fuzzy points must be regarded fuzzy subsets ` and is a C-fuzzy topology on the H)<, 7 , H)<,H W ZG "W as different points for even if they have the same family of fuzzy subsets `. According to the definition of the 7 H, G part . complementary operations on the lattices and the family )< W G If the lattice , the family of fuzzy points of closed sets of the topology is L " P ZW forms a lattice, which is DX " 3H), 30, r6; r $ 6 isomorphic to by the correspondence . ϕ W G T , r Y H),30,r6 aW " 3 , R , AF ; r $ bR_V,RTV,R_Vc6 We say that the fuzzy point H), r is a fuzzy element of the fuzzy subset if . while the family of closed sets of the topology is J J) " r ZG 2.2.2. Limit Point of Fuzzy Subsets ϕ W W W . aG " 3 , R , AF ; r $ bR V,R V,R Vc6 Chang (1968) introduced the definition of topology on d T G fuzzy subsets of the base set X, using the lattice " [0,1] Obviously, the closure of the fuzzy subset varies as the the unit interval of real numbers [2].
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