Neutrosophic Set and Neutrosophic Topological Spaces
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1 IOSR Journal of Mathematics (IOSR-JM) ISS: Volume 3, Issue 4 (Sep-Oct. 2012), PP eutrosophic Set and eutrosophic Topological Spaces 1..Salama, 2 S..lblowi 1 Egypt, Port Said University, Faculty of Sciences Department of Mathematics and Computer Science 2 Department of Mathematics, King bdulaziz University, Saudi rabia bstract: eutrosophy has been introduced by Smarandache [7, 8] as a new branch of philosophy. The purpose of this paper is to construct a new set theory called the neutrosophic set. fter given the fundamental definitions of neutrosophic set operations, we obtain several properties, and discussed the relationship between neutrosophic sets and others. Finally, we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 6] to the case of neutrosophic sets. Possible application to superstrings and space time are touched upon. Keywords: Fuzzy topology; fuzzy set; neutrosophic set; neutrosophic topology I. Introduction The fuzzy set was introduced by Zadeh [9] in 1965, where each element had a degree of membership. The intuitionstic fuzzy set (Ifs for short) on a universe X was introduced by K. tanassov [1, 2, 3] in 1983 as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. fter the introduction of the neutrosophic set concept [7, 8]. In recent years neutrosophic algebraic structures have been investigated. eutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both their classical and fuzzy counterparts, such as a neutrosophic set theory. II. Terminologies We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [7, 8], and tanassov in [1, 2, 3]. Smarandache introduced the neutrosophic components T, I, F which represent the membership, indeterminacy, and non-membership values respectively, where 0, 1 is nonstandard unit interval. 2.1 Definition. [3,4] Let T, I,F be real standard or nonstandard subsets of 0, 1, with Sup_T=t_sup, inf_t=t_inf Sup_I=i_sup, inf_i=i_inf Sup_F=f_sup, inf_f=f_inf n-sup=t_sup+i_sup+f_sup n-inf=t_inf+i_inf+f_inf, T, I, F are called neutrosophic components III. eutrosophic Sets and Its Operations We shall now consider some possible definitions for basic concepts of the neutrosophic set and its operations. 3.1 Definition Let X be a non-empty fixed set. neutrosophic set ( S for short) is an obect having the form x, x, x, x : x X Where x and x ship function (namely x ), the degree of indeterminacy (namely x ship (namely x ) respectively of each element x X to the set. which represent the degree of member ), and the degree of non-member 3.1 Remark neutrosophic x, x, x, x : x X can be identified to an ordered triple,, in 0,1 on. X. 31 P a g e
2 eutrosophic Set nd eutrosophic Topological Spaces 3.2 Remark For the sake of simplicity, we shall use the symbol,, for the S x, x, x, x : x X 3.1 Example Every IFS a non-empty set X is obviously on S having the form x, x,1 x x, x : x X Since our main purpose is to construct the tools for developing neutrosophic set and neutrosophic topology, we must introduce the SS 0 and 1 in X as follows: 0 may be defined as: 01 0 x,0,0,1 : x X 02 0 x,0,1,1 : x X 03 0 x,0,1,0 : x X 04 0 x,0,0,0 : x X 1 may be defined as: 1 1 x,1,0,0 : x X 12 1 x,1,0,1 : x X 13 1 x,1,1,0 : x X 14 1 x,1,1,1 : x X 3.2 Definition Let,, a S on X, then the complement of the set three kinds of complements C 1 C( ) 1 ( x),1 ( x),1 ( x) : x X, C 2 C x,, x, x : x X, C 3 C x,,1 x, x : x X. One can define several relations and operations between SS follows: 3.3 Definition Let X be a non-empty set, and SS and B in the form x, x, x, x C, for short may be defined as, B x, B x, B x, B x, then we may consider two possible definitions for subsets B B may be defined as (1) B x B x, x and x B x x X (2) B x x, x x and x x B B 3.1 Proposition For any neutrosophic set the following are holds (1) 0, 0 0 (2) 1, Definition Let X be a non-empty set, and x, x, x, x, B x, x, x, x (1) B may be defined as: B x, x. x, x. x, x. x I 1 B B B I 2 B x, x B x, x B x, x B x I B x, x x, x x, x x 3 B B B B B B B are SS. Then 32 P a g e
3 (2) B may be defined as: B x, x x, x x, U 1 B B x B x U 2 B x, x B x, x B x, x B x (3) x, x, x,1 x (4) x,1 x, x, x eutrosophic Set nd eutrosophic Topological Spaces We can easily generalize the operations of intersection and union in definition 3.4 to arbitrary family of SS as follow: 3.5 Definition Let : J be an arbitrary family of SS in X, then (1) may be defined as: (i) x, x, x, x J J (ii) x, x, x, x (2) may be defined as: (i),,, J J J (ii),,. J J J 3.6. Definition Let and B are neutrosophic sets then B may be defined as B x,, x x, x B B B 3.2. Proposition For all Btwo, neutrosophic sets then the following are true (1) C B C C B (2) C B C C B 1. EUTROSOPHIC TOPOLOGICL SPCES Here we extend the concepts of fuzzy topological space [4], and intuitionistic fuzzy topological space [5, 7] to the case of neutrosophic sets. 4.1 Definition neutrosophic topology ( T for short) an a non empty set X is a family of neutrosophic subsets in X satisfying the following axioms T 1 O,1, T 2 G1 G2 for anyg1, G2 T 3 Gi G i : i J In this case the pair,, X is called a neutrosophic topological space ( TS for short) and any neutrosophic set in is known as neuterosophic open set ( OS for short) in X. The elements of are called open neutrosophic sets, neutrosophic set F is closed if and only if it C (F) is neutrosophic open. 4.1 Example ny fuzzy topological space X, 0 in the sense of Chang is obviously a TS in the form : 0 wherever we identify a fuzzy set in X whose members ship function is with its counterpart Remark eutrosophic topological spaces are very natural generalizations of fuzzy topological spaces allow more general functions to be members of fuzzy topology. 4.3 Example Let X x and x,0.5,0.5,0.4 : x X B x,0.4,0.6,0.8 : x X 33 P a g e
4 D x,0.5,0.6,0.4 : x X C x,0.4,0.5,0.8 : x X Then the family O,1,, B, C, D n n eutrosophic Set nd eutrosophic Topological Spaces of Ss in X is neutrosophic topology on X. 4.4 Example Let X, 0 be a fuzzy topological space in changes sense such that 0 is not indiscrete suppose now that 0,1 : J then we can construct two TSS on X as follows 0 V a) 0 0,1 V, ( x),0 : J. b) 0,1 V,0, ( x),1 V : J Proposition Let X, be an TS on X, then we can also construct several TSS on X in the following way: a) o, 1 [ ] G : G, b) o, 2 G : G. Proof T are easy. a) 1 T and 3 2 T Let[ ] G : J, G.Since 0, 1 x,, or,, or, G, G G G G G G G G,, we have G [ ] G,, (1 ) or,,(1 ) 0, 1 G b) This similar to (a) G G G G 4.2 Definition Let X, 1, X, 2 be two neutrosophic topological spaces onx. Then 1 is said be contained in 2 (in symbols1 2 ) if G for each G 1. In this case, we also say that 1 is coarser than Proposition Let : J be a family of TSS on X. Then is a neutrosophic topology on X. Furthermore, is the coarsest T on X containing all. Proof. Obvious, s G 4.3 Definition The complement of (C () for short) of OS. is called a neutrosophic closed set ( CS for short) in X. ow, we define neutrosophic closure and interior operations in neutrosophic topological spaces: 4.4 Definition Let X, be TS and x, x, x, x be a S in X. Then the neutrosophic closer and neutrosophic interior of are defined by Cl ( ) K : K is an CSin X and K Int ( ) G : G is an OS in X and G.It can be also shown that It can be also shown that Cl () is CS and Int () is a OS in X a) is in X if and only if Cl (). b) is CSin X if and only if Int( ). 4.2 Proposition For any neutrosophic set in x, we have (a) Cl( C( ) C( Int ( ), (b) Int( C( )) C( Cl( )). Proof. a) Let,, : x X and suppose that the family of neutrosophic subsets contained in are indexed by the family if SS contained in are indexed by the 34 P a g e
5 family,, : i J eutrosophic Set nd eutrosophic Topological Spaces Gi Gi G i. Then we see that Int( ) G, i Gi, and G i C( Int( )) G i, Gi, G i. Since C() and G i and G i for each i J, we C. i.e. Cl( C( )) G i, Gi, G i. Hence Cl( C( ) C( Int ( ), follows hence obtaining () immediately b) This is analogous to (a). 4.3 Proposition Let x, be a TS and (a) Int( ), (b) Cl(),, B be two neutrosophic sets in X. Then the following properties hold: (c) B Int ( ) Int ( B), (d) B Cl( ) Cl( B), (e) Int ( Int ( )) Int ( ) Int ( B), (f) Cl( B) Cl( ) Cl( B), (g) Int( 1 ) 1, (h) Cl( O ) O, Proof (a), (b) and (e) are obvious (c) follows from (a) and Definitions. References [1] K. tanassov, intuitionistic fuzzy sets, in V.Sgurev, ed.,vii ITKRS Session, Sofia(June 1983 central Sci. and Techn. Library, Bulg. cademy of Sciences( 1984)). [2] K. tanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986) [3] K. tanassov, Review and new result on intuitionistic fuzzy sets, preprint IM-MFIS-1-88, Sofia, [4] C.L. Chang, Fuzzy Topological Spaces, J. Math. nal. ppl. 24 (1968) [5] Dogan Coker, n introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems. 88(1997) [6] Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons and Fractals 27(2006) [7] Florentin Smarandache, eutrosophy and eutrosophic Logic, First International Conference on eutrosophy, eutrosophic Logic, Set, Probability, and Statistics University of ew Mexico, Gallup, M 87301, US(2002), smarand@unm.edu [8] F. Smarandache. Unifying Field in Logics: eutrosophic Logic. eutrosophy, eutrosophic Set, eutrosophic Probability. merican Research Press, Rehoboth, M, [9] L.. Zadeh, Fuzzy Sets, Inform and Control 8(1965) P a g e
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