Fuzzy soft boundary. Azadeh Zahedi Khameneh, Adem Kılıçman, Abdul Razak Salleh

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1 Annals of Fuzzy Mathematics and Informatics Volume 8, No. 5, (November 2014), pp ISSN: (print version) ISSN: (electronic version) c Kyung Moon Sa Co. Fuzzy soft boundary Azadeh Zahedi Khameneh, Adem Kılıçman, Abdul Razak Salleh Received 4 December 2013; Revised 17 April 2014; Accepted 6 May 2014 Abstract. In this paper, we continue study on fuzzy soft topology. Fuzzy soft interior, fuzzy soft closure, and fuzzy soft continuity are considered deeply. The concept of fuzzy soft boundary is introduced and some of its basic properties are also studied AMS Classification: 54A05, 54A40, 54B10, 54C05 Keywords: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft boundary, Fuzzy soft continuous, Fuzzy soft product topology. Corresponding Author: Azadeh Zahedi Khameneh (azadeh503@gmail.com) 1. Introduction We live in the world of uncertainties where the most of the problems which we deal are vague rather than precise. Even the language used in our daily life situations is usually full of imprecise phrases. During past decades, different mathematical theories such as probability theory, fuzzy set theory [25], and rough set theory [16] were introduced to deal with various types of uncertainties. A wide range of problems can be solved by these methods although they cannot be applied effectively to model imprecise information related to some parameters. Due to lack of parameterization tools in all previous theories, Molodtsove[14] introduced the concept of soft set in This concept can be seen as an approximation description of some objects based on some parameters by a given set-valued map. Furthermore, Molodtsove[14] pointed out that theory of soft set is more general than the former theories dealing with uncertainty. He also mentioned several directions for soft set s application such as smoothness of functions, game theory, operations research and so on. Since then, works on soft set theory have progressed rapidly. In 2000, Maji et al.[10] published the first research paper on soft set theory and expanded the theory of soft set fundamentally. They extended some basic concepts from classical set theory to the soft set theory, and then in [11] discussed practical application of soft sets in decision-making problems. In [2], Aktaş and Çaǧman applied the concept of soft set to introduce the new notion soft group. In [6], Irfan

2 Ali et al. developed the soft set theory and introduced some new operations for it. Shabir and Naz[22], defined the concept of soft topology and studied some basic topological properties of soft spaces. For more details about soft set theory and soft topology see [7, 4, 21, 5]. But dealing with imprecise and vague information in real life problems encouraged researchers to consider soft set theory in a fuzzy environment. In [12], Maji et al. combined the concepts of fuzzy set and soft set and introduced a new hybrid concept called fuzzy soft set. Later in [18], they applied fuzzy soft set to solve a decision making problem. Kharal and Ahmad[1] also studied some properties of fuzzy soft set and extended some operations in classical set theory to fuzzy soft set theory. Then in [8], they introduced the concept of fuzzy soft map and studied the concept of image and pre-image of a fuzzy soft set. Majumdar and Samanta[13] generalised the concept of fuzzy soft set by attaching the degree of possibility of the relationship between each object and parameter. Topological studies of fuzzy soft sets was begun by Tanya and Kandemir[24]. They applied classical definition of topology to construct a topology over a fuzzy soft set and called this new topological space fuzzy soft topology. Furthermore, they studied some fundamental topological structures such as interior, closure, and base for the fuzzy soft topology. Later Simsekler and Yuksel[23] studied fuzzy soft topological space in the sense of Tanay and Kandemir[24]. But they defined the concept of fuzzy soft topology over a fuzzy soft set with a fixed parameter set and considered some topological concepts such as base, subbase, neighborhood, and Q-neighborhood for fuzzy soft topological spaces. Roy and Samanta[19] remarked a new definition of fuzzy soft topology. They proposed the notion of fuzzy soft topology over an ordinary set by applying fuzzy soft subsets of it where parameter set is supposed fixed everywhere. Then in [20], they continued study on fuzzy soft topology and defined the concept of fuzzy soft point and different neighborhood structures of a fuzzy soft point. The concept of soft quasi-coincidence for fuzzy soft sets was considered by Atmaca and Zorlutuna[3]. They also studied the fundamental topological notions such as interior and closure for a fuzzy soft sets by applying this new concept. Recently, Zahedi et al.[26] introduced the concept of product fuzzy soft topology and studied some of its properties. They also considered the Hausdorff property of finite product of fuzzy soft Hausdorff spaces. Osmanoglu and Tokat [15] introduced fuzzy soft compactness and some basic definitions and theorems by using basic properties of fuzzy soft topology. Li and Cui [9] discussed topologies on intuitionistic fuzzy soft sets. Although different topological structures of a fuzzy soft space have been studied, the concept of boundary of a fuzzy soft set has received less attention. In ordinary topology the boundary of a set is considered as an exact set. But in real-world experiences, where we mostly deal with qualitative explanation of phenomena, we cannot determine the boundary of a set exactly. In fact, the boundary of a set is mostly a vague area instead of accurate line. So we need to consider the concept of boundary in a fuzzy soft space where we cannot judge exactly which points belong to the boundary zone. To obtain this aim, the present paper is organized as the following. Firstly in section 2, we recall some definitions and properties of fuzzy soft set theory which are used along this work. Then in section 3 the concepts of fuzzy soft interior, fuzzy soft closure, and fuzzy soft continuity are studied in the fuzzy 688

3 soft topological spaces. Finally in section 4, the concept of fuzzy soft boundary is introduced and some of its properties are studied. 2. Preliminaries Let X and E are used to show the set of objects and parameters, respectively. Let A E, and I X, where I = [0, 1], denotes the set of all fuzzy subsets of X. Definition 2.1 ([14]). A pair (F, A) is called a soft set over X if F is a mapping given by F : A 2 X such that for all a A, F (a) X. Definition 2.2 ([12]). A pair (f, A) is called a fuzzy soft set over X if f is a mapping given by f : A I X. So a A, f(a) is a fuzzy subset of X with membership function f(a) := f a : X [0, 1]. We denote the fuzzy soft set (f, A) by f A and abbreviate the terminology fuzzy soft set by F.S-set. Moreover, we call the F.S-set f A as a crisp F.S-set if a A, the value set of f(a) is a subset of {0, 1}. The soft set (F, A) can be seen as the F.Sset (χ F, A) where χ F refers to the characteristic function of set F (a). In addition, during this paper we use the notation F.S(X, E) to show the set of all F.S-sets over X with regard to the parameter set E. Definition 2.3 ([12, 24]). (Rules of fuzzy soft set) For two F.S-sets f A and g B over the common universe X where A, B E we have, i. f A is a F.S-subset of g B (or g B is a F.S-superset of f A ) shown by f A g B (or g B f A ) if: (a) A B, (b) for all a in A, f a (x) g a (x), x X. ii. f A = g B if f A g B and g B f A. iii. The complement of F.S-set f A is denoted by fa c and given by the mapping f c : A I X such that fa c = 1 f a, a A. iv. f A = Φ A (null F.S-set with respect to A) if a A, f a (x) = 0 for all x X. v. f A = X A (absolute F.S-set with respect to A) if a A, f a (x) = 1 for all x X. If A = E, the null and absolute fuzzy soft set is denoted by Φ and X, respectively. vi. The union of f A and g B, denoted by f A g B, is a F.S-set over X with membership function (f g)(c) where f c (x) if c A B (f g) c (x) = g c (x) if c B A max{f c (x), g c (x)} if c A B for all x X and c A B. vii. The intersection of f A and g B, denoted by f A g B, is a F.S-set over X with membership function (f g)(c) where (f g) c (x) = min{f c (x), g c (x)} for all x X and e A B. viii. f A AND g B is a F.S-set over X and is defined by h A B where A B is the cartesian product of parameter sets A and B, and h is the map h : A B I X such that for all x X, a A, and b B, we have h (a,b) (x) = min{f a (x), g b (x)}. 689

4 ix. f A OR g B is a F.S-set over X defined by k A B where k is the map k : A B I X such that for all x X, a A, and b B, we have k (a,b) (x) = max{f a (x), g b (x)}. Regarding to vii, the intersection of two F.S-sets f A and g B is defined if the parameter sets of them meet each other. To avoid of such a difficulty, in [17] the concept of extended intersection of two F.S-sets has been considered. Proposition 2.4 ([1]). If f A, g B, and h C are some F.S-sets over X where A, B, and C are subsets of the parameter set E, then (1) [f A g B ] h C = [f A h C ] [g B h C ], (2) [f A g B ] h C = [f A h C ] [g B h C ]. Proposition 2.5 ([26]). If f E and g E are two F.S-sets over X, then (1) [f E g E ] c = f c E g c E, (2) [f E g E ] c = f c E g c E. Definition 2.6 ([8]). Let X 1 and X 2 be universal sets and E 1 and E 2 be corresponding parameter sets. Suppose that f A is a F.S-set over X 1 and g B is a F.S-set over X 2 where A E 1 and B E 2. If Ψ U : X 1 X 2 and Ψ P : E 1 E 2 are ordinary functions where U refers to universal set and P refers to parameter set, then i. The map Ψ : F.S(X 1, E 1 ) F.S(X 2, E 2 ) is called a F.S-map from X 1 to X 2 and for any y X 2 and e P (E 1 ) E 2, the image of f A under Ψ is the F.S-set Ψ(f A ) over X 2 defined as below: [ ] [Ψ(f)](e )(y) = sup x Ψ U 1 (y) sup f(e) e Ψ 1 P (e ) A if Ψ 1 P (e ) A and Ψ 1 U (y), otherwise [Ψ(f)](e )(y) = 0. ii. Let Ψ : F.S(X 1, E 1 ) F.S(X 2, E 2 ) be a F.S-map from X 1 to X 2. Then the inverse image of F.S-set g B under Ψ, denoting by Ψ 1 (g B ), is a F.S-set over X 1. For all x X 1 and e E 1 it is defined as below: { [Ψ 1 gψp (g)](e)(x) = (e)(ψ U (x)) if Ψ P (e) B 0 otherwise. (x) We also show the F.S-map Ψ by the ordered pair (Ψ P, Ψ U ) of ordinary maps Ψ U and Ψ P. Proposition 2.7 ([8]). Let Ψ : F.S(X 1, E 1 ) F.S(X 2, E 2 ) be a F.S-map as introduced in Definition 2.6. Let {f ia } i I and {g ib } i I are two families of F.S-sets in X 1 and X 2, respectively. Then (1) Ψ(Φ E1 ) = Φ E2 and Ψ 1 (Φ E2 ) = Φ E1. (2) Ψ( X 1 ) X 2 and Ψ 1 ( X 2 ) = X 1. (3) Ψ[ i f ia] = i Ψ(f ia) and Ψ 1 [ i g ib] = i Ψ 1 (g ib ). (4) Ψ[ i f ia] i Ψ(f ia) and Ψ 1 [ i g ib] = i Ψ 1 (g ib ). (5) For each i I, [Ψ(f ia )] c c Ψ(f ia ) and [Ψ 1 (g ib )] c = Ψ 1 c (g ib ). 690

5 (6) For any i, j I, if f ia f j A, then Ψ(f ia ) Ψ(f j A ) and if g ib g j B, then Ψ 1 (g ib ) Ψ 1 (g j B ). Definition 2.8 ([26]). Let f E1 F.S(X 1, E 1 ) and g E2 F.S(X 2, E 2 ) be two F.Ssets over X 1 and X 2, respectively. The cartesian product of f E1 and g E2, denoted by f E1 g E2, is a F.S-set over X 1 X 2 defined as below: f g : E 1 E 2 I X 1 I X 2 (e, e ) f(e) g(e ) such that f(e) g(e ) is the fuzzy product of fuzzy sets f(e) and g(e ) where f(e) g(e ) : X 1 X 2 [0, 1] (x, y) min{f e (x), g e (y)} Proposition 2.9 ([26]). Let f E1, f 1E1, and f 2E1 F.S(X 1, E 1 ); and g E2, g 1E2, and F.S(X 2, E 2 ). Then g 2E2 (1) X 1 X 2 = X 1 X 2. (2) f E1 Φ E2 = Φ E1 g E2 = Φ E1 E 2 = Φ E1 Φ E2. (3) e E 1, e E 2, (f X 2 ) (e,e )(x, y) = f e (x), ( X 1 g) (e,e ) = g e (y), where x X 1 and y X 2. (4) [f 1E1 f 2E1 ] [g 1E2 g 2E2 ] = [f 1E1 g 1E2 ] [f 2E1 g 2E2 ]. Specially we have f E1 g E2 = [f E1 X 2 ] [ X 1 g E2 ]. 3. Fuzzy soft topology The concept of fuzzy soft topology firstly introduced by Tanay and Kandemir[24]. They defined the concept of fuzzy soft topology as a topology over the given fuzzy soft set f A. So a fuzzy soft topology in the sense of Tanay and Kandemir[24] is the collection τ of F.S-subsets of f A closed under arbitrary supremum and finite infimum. It also contains Φ A and X A. But Roy and Samanta[19] redefined the concept of fuzzy soft topology. The most significant reason for such a change is to make sure that the DeMorgan Laws hold in the new definition of fuzzy soft topology. Here we recall the definition of fuzzy soft topology as introduced in [19]. Definition 3.1 ([19]). A fuzzy soft topology over X denoted by τ, is a collection of fuzzy soft subsets of X such that: i. X and Φ τ. ii. The union of any number of F.S-sets in τ belongs to τ. iii. The intersection of any two F.S-sets in τ belongs to τ. The triplet (X, E, τ) is called a fuzzy soft topological space, F.S-topological space in brief, and each element of τ is called a fuzzy soft open set, F.S-open set in brief, in X. The complement of a F.S-open set is called a F.S-closed set. We show the collection of all F.S-closed sets in X by τ c. Theorem 3.2. Let (X, E, τ) be a F.S-topological space. Then (1) Φ and X are F.S-closed sets. (2) The intersection of any number of F.S-closed sets is a F.S-closed set. (3) The union of any two F.S-closed sets is a F.S-closed set. 691

6 Proof. Follows from Proposition 2.5. Figure 1. F.S-open set U E in τ F.S Definition 3.3. Let τ 1 and τ 2 be two F.S-topologies over X. We say that τ 1 is coarser than τ 2, or τ 2 is finer than τ 1, if τ 1 τ 2. Example 3.4 ([26]). Let X and E be the universal and parameter set, respectively. Take τ = {Φ, X} (say trivial F.S-topology over X), then τ is clearly the coarsest one. Take τ = F.S(X, E) (say discrete F.S-topology over X), then τ is the finest one. Example 3.5 ([26]). Let R be the set of all real numbers with the usual topology τ and E = [0, 1) R. If U = (a, b) R is an open interval in R, then we define the crisp F.S-set U E over R related to the open interval U R by the mapping U : E = [0, 1) I R α U(α) := U α : R {0, 1} such that for all x R, { 1 x (a, b) U α (x) = 0 x / (a, b). The family {U E } of such crisp F.S-sets forms a F.S-topology over R denoted by τf.s. In fact, the F.S-open set U E in τf.s is a parameterization extension of open interval U = (a, b) R (see Fig. 1). Example 3.6. Consider the real line R as a topological space with usual topology τ and let E = [0, 1). The collection τ F.S = {f : E I R : α E, (f(α)) 1 (0, 1] τ} is a F.S-topology over R. Moreover τf.s τ F S where τf.s is the F.S-topology over R introduced earlier in Example

7 Example 3.7. Let (X, γ) be a fuzzy topological space and µ γ be a fuzzy open set in X. The characteristic function of α-cut set of µ is denoted by χ µα and is defined by the mapping χ µα : X {0, 1} such that for all x X { 1 µ(x) > α χ µα (x) = 0 µ(x) α. Now let E = [0, 1). We define the crisp F.S-set (χ µ, E) as below: χ µ : E = [0, 1) I X α χ µ (α) := χ µα : X {0, 1}. Then the collection γ = {χ µ : [0, 1) I X : µ γ} is a F.S-topology over X. Theorem 3.8. Let (X, E, τ) be a F.S-topological space. For any e E, the collection τ e = {(f(e)) 1 (0, 1] : f τ, e E} forms a topology over X with respect to e. Proof. Follows from Definition 3.1. Theorem 3.9. If {τ α : α I} be a family of F.S-topologies over X, then α I τ α is also a F.S-topology over X. Proof. It is straightforward. It is clear that α I τ α is the coarsest F.S-topology over X. Remark The α I τ α is not a F.S-topology over X in general case. This is shown in the following example. Example Let X = {x 1, x 2 } and E = {e 1, e 2 }. Let τ = {Φ, X, f 1E, f 2E, f 3E } and δ = {Φ, X, g 1E, g 2E, g 3E, g 4E } be two F.S-topologies over X where and f 1E = {(e 1, { 1 x x 2 }), (e 2, { 0 x x 2 })} f 2E = {(e 1, { 1 x x 2 }), (e 2, { 1 x x 2 })} f 3E = {(e 1, { 1 x x 2 }), (e 2, { 0 x x 2 })} g 1E = {(e 1, { 1 x x 2 }), (e 2, { 0.5 x x 2 })} g 2E = {(e 1, { 0 x x 2 }), (e 2, { 1 x x 2 })} g 3E = {(e 1, { 0 x x 2 }), (e 2, { 0.5 x x 2 })} g 4E = {(e 1, { 1 x x 2 }), (e 2, { 1 x x 2 })}. Then τ δ = {Φ, X, f 1E, f 2E, f 3E, g 1E, g 2E, g 3E, g 4E } is not a F.S-topology over X since f 1E g 1E / τ δ. 693

8 Definition 3.12 ([26]). Let (X, E, τ) be a F.S-topological space. The collection B of F.S-open subsets of X is called a F.S-base for τ if every F.S-open sets in τ can be written as a union of members of B. Definition 3.13 ([26]). Let (X 1, E 1, τ 1 ) and (X 2, E 2, τ 2 ) be two F.S-topological spaces. The F.S-topology τ, generated by B = {f E1 g E2 : f E1 τ 1, g E2 τ 2 }, is called F.S-product topology over X 1 X 2 and denoted by (X, E, τ ) where X = X 1 X 2 and E = E 1 E 2. Lemma Let f E1 F.S(X 1, E 1 ) and g E2 F.S(X 2, E 2 ). Then we have [f E1 g E2 ] c = [f c E 1 X 2 ] [ X 1 g c E 2 ] Proof. Take (x, y) X 1 X 2 and (e, e ) E 1 E 2. By Proposition 2.9 (3), we have (f g) c (e,e ) (x, y) = 1 (f g) (e,e )(x, y) = 1 min{f e (x), g e (y)} = max{1 f e (x), 1 g e (y)} = max{f c e (x), g c e (y)} = max{(f c X 2 ) (e,e ) (x, y), ( X 1 g c ) (e,e )(x, y)} = [(f c X 2 ) ( X 1 g c )] (e,e )(x, y) This means that [f E1 g E2 ] c = [f c E 1 X 2 ] [ X 1 g c E 2 ]. Theorem Let (X, E, τ) be a F.S-topological space. If f E and g E are two F.S-closed sets in τ, then f E g E is a F.S-closed set in τ. Proof. Follows from Lemma F.S-Closure and F.S-Interior. Definition Let (X, E, τ) be a F.S-topological space. i. Fuzzy soft closure, F.S-closure in brief, of f E is denoted by Clf E and is defined as the intersection of all F.S-closed super sets of f E. So Clf E = g E fe g E where g E s are F.S-closed sets in (X, E, τ). ii. Fuzzy soft interior, F.S interior in brief, of f E is denoted by Intf E and is defined as the union of all F.S-open subsets of f E. So Intf E = h E fe h E where h E s are F.S-open sets in (X, E, τ). Theorem 3.17 ([23]). Let (X, E, τ) be a F.S-topological space and f E and g E be two F.S-sets over X. Then (1) ClΦ = Φ and Cl X = X. (2) f E Clf E and Clf E is the smallest F.S-closed set containing the F.S-set f E. (3) f E is a F.S-closed set if and only if f E = Clf E. (4) Cl(Clf E ) = Clf E. (5) if f E g E, then Clf E Clg E. (6) Cl(f E g E )=Clf E Clg E. 694

9 (7) Cl(f E g E ) Clf E Clg E. Proof. See [23]. Theorem 3.18 ([23]). Let (X, E, τ) be a F.S-topological space and f E and g E are two F.S-sets over X. Then (1) IntΦ = Φ and Int X = X. (2) Intf E f E and Intf E is the biggest F.S-open set contained in the F.S-set f E. (3) f E is a F.S-open set if and only if f E = Intf E. (4) Int(Intf E ) = Intf E. (5) if f E g E, then Intf E Intg E. (6) Int(f E g E ) Intf E Intg E. (7) Int(f E g E ) = Intf E Intg E. Proof. See [23]. Corollary For any F.S-set f E in the F.S-topological space (X, E, τ), Intf E f E Clf E. Proof. Follows from Theorems 3.17 and Theorem Let (X, E, τ) be a F.S-topological space and {f ie } i I be a family of F.S-sets over X. Then (1) (2) i Clf ie Cl[ i f ie]. i Intf ie Int[ i f ie]. Proof. Follows from Theorems 3.17 and Proposition Let (X, E, τ) be a F.S-topological space and f E be a F.S-set over X. Then (1) Clf c E = (Intf E) c. (2) Intf c E = (Clf E) c. Proof. (1) (Intf E ) c = [ g E ] c : g E τ, g E f E = g c E : g E τ, g c E f c E (2) It is similar to 1. = Clf c E Theorem If (X, E, τ) is a F.S-topological space and f E and g E are two F.Ssets over X. Then (1) Cl(f E g E ) Clf E Clg E. (2) Int(f E g E ) Intf E Intg E. Proof. (1) Since f E Clf E and g E Clg E, then it is easily to see that f E g E Clf E Clg E. 695

10 Cl(f E g E ) Clf E Clg E follows from Theorems 3.15 and (2) By Proposition 3.21, Lemma 3.14, and Theorem 3.17 we have [Int(f E g E )] c = Cl[(f E g E ) c ] = Cl[(f c E X) ( X g c E)] = Cl(f c E X) Cl( X g c E) [Clf c E Cl X] [Cl X Clg c E] = [Clf c E X] [ X Clg c E] = [(Intf E ) c X] [ X (Intg E ) c ] = (Intf E Intg E ) c So [Int(f E g E )] c (Intf E Intg E ) c. This implies that Int(f E g E ) Intf E Intg E F.S-Continuity. Definition 3.23 ([26]). Let Ψ : (X 1, E 1, τ 1 ) (X 2, E 2, τ 2 ) be a F.S-map as introduced in Definition 2.6. Ψ is called i. A F.S-continuous map if and only if g E2 τ 2, Ψ 1 (g E2 ) τ 1. ii. A F.S-open map if and only if f E1 τ 1, Ψ(f E1 ) τ 2. Theorem The F.S-map Ψ : (X 1, E 1, τ 1 ) (X 2, E 2, τ 2 ) is F.S-continuous if and only if g E2 τ c 2, Ψ 1 (g E2 ) τ c 1. Proof. Follows from Proposition 2.7 (5). Theorem If Ψ : (X 1, E 1, τ 1 ) (X 2, E 2, τ 2 ) is a F.S-continuous map, then g E2 F.S(X 2, E 2 ), Cl[Ψ 1 (g E2 )] Ψ 1 (Clg E2 ). Proof. Follows from Proposition 2.7 and Theorems 3.24 and Definition Let Ψ 1 : F.S(X 1, E 1 ) F.S(X 2, E 2 ) and Ψ 2 : F.S(X 2, E 2 ) F.S(X 3, E 3 ) be two F.S-maps. The mapping Ψ = Ψ 2 oψ 1 : F.S(X 1, E 1 ) F.S(X 3, E 3 ) is called the F.S-composition map such that for any f E1 F.S(X 1, E 1 ) it is defined as [ ] [(Ψ 2 oψ 1 )(f)](α)(z) = sup x (Ψ 2U oψ 1U ) 1 (z) sup f(β) β (Ψ 2P oψ 1P ) 1 (α) where z X 3, α E 3 ; and Ψ 1P : E 1 E 2, Ψ 2P : E 2 E 3, Ψ 1U : X 1 X 2, and Ψ 2U : X 2 X 3 are ordinary maps. Moreover For x X 1 and β E 1 we have [(Ψ 2 oψ 1 ) 1 (g)](β)(x) = g[(ψ 2P oψ 1P )(β)][(ψ 2U oψ 1U )(x)] where g E3 F.S(X 3, E 3 ). Lemma If Ψ 1 : F.S(X 1, E 1 ) F.S(X 2, E 2 ) and Ψ 2 : F.S(X 2, E 2 ) F.S(X 3, E 3 ) are two F.S-maps, then (Ψ 2 oψ 1 ) 1 = Ψ 1 1 oψ (x)

11 Proof. Take f E3 F.S(X 3, E 3 ). Then for x X 1 and e E 1, we have [(Ψ 2 oψ 1 ) 1 (f)](e)(x) = f[(ψ 2P oψ 1P )(e)][(ψ 2U oψ 1U )(x)] = f Ψ2P (Ψ 1P (e))[ψ 2U (Ψ 1U (x))] = [Ψ 1 2 (f)](ψ 1P (e))(ψ 1U (x))] = Ψ 1 1 [Ψ 1 2 (f)](e)(x) = [(Ψ 1 1 oψ 1 2 )(f)](e)(x) Thus (Ψ 2 oψ 1 ) 1 = Ψ 1 1 oψ 1 2. Theorem If Ψ 1 : (X 1, E 1, τ 1 ) (X 2, E 2, τ 2 ) and Ψ 2 : (X 2, E 2, τ 2 ) (X 3, E 3, τ 3 ) are two F.S-continuous maps, then Ψ 2 oψ 1 : (X 1, E 1, τ 1 ) (X 3, E 3, τ 3 ) is a F.S-continuous map. Proof. Follows from Lemma Lemma Let for i=1,2, Ψ i : F.S(X, E) F.S(Y i, V i ), Ψ ip : E V i, and Ψ iu : X Y i be some F.S-maps and ordinary maps, respectively as introduced in Definition 2.6. If Ψ 1 Ψ 2 : F.S(X, E) F.S(Y 1, V 1 ) F.S(Y 2, V 2 ) is a F.S-map defined by (Ψ 1 Ψ 2 )(f E ) = Ψ 1 (f E ) Ψ 2 (f E ) where f E F.S(X, E), then for g 1V1 F.S(Y 1, V 1 ) and g 2V2 F.S(Y 2, V 2 ), (Ψ 1 Ψ 2 ) 1 (g 1V1 g 2V2 ) = Ψ 1 1 (g 1V 1 ) Ψ 1 2 (g 2V 2 ) Proof. Consider the two following ordinary maps and Ψ 1P Ψ 2P : E V 1 V 2 e (Ψ 1P (e), Ψ 2P (e)) Ψ 1U Ψ 2U : X Y 1 Y 2 x (Ψ 1U (x), Ψ 2U (x)). Take g 1V1 F.S(Y 1, V 1 ) and g 2V2 F.S(Y 2, V 2 ). For given e E and x X, we have [(Ψ 1 Ψ 2 ) 1 (g 1 g 2 )](e)(x) = (g 1 g 2 )((Ψ 1P Ψ 2P )(e))((ψ 1U Ψ 2U )(x)) = (g 1 g 2 )(Ψ 1P (e), Ψ 2P (e))(ψ 1U (x), Ψ 2U (x)) = min{g 1 (Ψ 1P (e))(ψ 1U (x)), g 2 (Ψ 2P (e))(ψ 2U (x))} = min{ψ 1 1 (g 1)(e)(x), Ψ 1 2 (g 2)(e)(x)} = [Ψ 1 1 (g 1) Ψ 1 2 (g 2)](e)(x) Thus (Ψ 1 Ψ 2 ) 1 (g 1V1 g 2V2 ) = Ψ 1 1 (g 1V 1 ) Ψ 1 2 (g 2V 2 ). Theorem Let Ψ i : (X, E, τ) (Y i, V i, τ i ) be two F.S-maps (for i = 1, 2). The F.S-map Ψ 1 Ψ 2 : (X, E, τ) (Y 1 Y 2, V 1 V 2, τ ) is continuous if and only if Ψ 1 and Ψ 2 are F.S-continuous. 697

12 Proof. : Let Ψ 1 and Ψ 2 be two F.S-continuous maps. Let g 1V1 g 2V2 be a F.S-open set in F.S-topological space (Y 1 Y 2, V 1 V 2, τ ) where τ is the F.S-product topology over Y 1 Y 2 introduced in [26]. By applying Lemma 3.29 we have (Ψ 1 Ψ 2 ) 1 (g 1V1 g 2V2 ) = Ψ 1 1 (g 1V 1 ) Ψ 1 2 (g 2V 2 ) τ. This means that Ψ is a F.S-continuous. : Take g 1V1 τ 1. Then by Lemma 3.29, for e V and x Y we have (Ψ 1 Ψ 2 ) 1 (g 1 Ỹ2)(e)(x) = [Ψ 1 1 (g 1) Ψ 1 2 (Ỹ2)](e)(x) = [Ψ 1 1 (g 1) X](e)(x) = Ψ 1 1 (g 1)(e)(x) This means that Ψ 1 1 (g 1V 1 ) = (Ψ 1 Ψ 2 ) 1 (g 1V1 Ỹ2) τ. Similarly if g 2V2 τ 2, then Ψ 1 2 (g 2V 2 ) = (Ψ 1 Ψ 2 ) 1 (Ỹ1 g 2V2 ) τ. This completes the proof. Lemma Let for i=1,2, Ψ i : F.S(X i, E i ) F.S(Y i, V i ) be two F.S-maps and Ψ ip : E i V i and Ψ iu : X i Y i be some ordinary maps as introduced in Definition 2.6. If Ψ 1 Ψ 2 : F.S(X 1, E 1 ) F.S(X 2, E 2 ) F.S(Y 1, V 1 ) F.S(Y 2, V 2 ) is a F.S-map defined by (Ψ 1 Ψ 2 )(f 1E1 f 2E2 ) = Ψ 1 (f 1E1 ) Ψ 2 (f 2E2 ) where f 1E1 F.S(X 1, E 1 ) and f 2E2 F.S(X 2, E 2 ), then (Ψ 1 Ψ 2 ) 1 (g 1V1 g 2V2 ) = Ψ 1 1 (g 1V 1 ) Ψ 1 2 (g 2V 2 ) Proof. It is similar to Lemma Theorem Let Ψ i : F.S(X i, E i, τ i ) F.S(Y i, V i, γ i ) where i = 1, 2 be two F.S-maps. The F.S-map Ψ 1 Ψ 2 : (X 1 X 2, E 1 E 2, τ ) (Y 1 Y 2, V 1 V 2, γ ) is continuous if and only if Ψ 1 and Ψ 2 are F.S-continuous. Proof. Follows from Lemma Fuzzy soft Boubdary Definition 4.1. Let (X, E, τ) be a F.S-topological space and f E be a F.S-set over X. We define the boundary of f E, denoted by Bdf E, as below Bdf E = Clf E Clf c E Proposition 4.2. If (X, E, τ) is a F.S-topological space, then (1) Bdf E is a F.S-closed set. (2) Bdf E Clf E. (3) BdΦ = Φ and Bd X = Φ. Proof. Follows from Definition 4.1. Proposition 4.3. If (X, E, τ) is a F.S-topological space, then (1) Bdf E = Bdf c E. (2) [Bdf E ] c = Intf c E Intf E. (3) Bdf E f E Clf E. 698

13 (4) Bdf E f E if f E be a F.S-closed set, and so BdClf E Clf E for all F.S-set f E. (5) Bdf E f E c if f E be a F.S-open set, and so BdIntf E Clf E c for all F.S-set f E. (6) BdBdf E Bdf E. (7) BdIntf E Bdf E. (8) BdClf E Bdf E. Proof. (1) It follows from Definition 4.1. (2) It follows from Definition 4.1, Proposition 2.5, and Proposition (3) It follows from Definition 4.1. (4) If f E be a F.S-closed set, then Theorem 3.17 (3), implies that Clf E = f E. So we have Bdf E = Clf E Clf c E = f E Clf c E which implies that Bdf E f E. Moreover if f E F.S(X, E), then we have BdClf E = ClClf E Cl(Clf E ) c = Clf E Cl(Clf E ) c So BdClf E Clf E. (5) It is similar to 4. (6) It is clear since Bdf E is a F.S-closed set. (7) BdIntf E = Cl(Intf E ) Cl(Intf E ) c = ClIntf E ClClf E c = ClIntf E Clf E c Clf E Clf E c = Bdf E (8) Thus BdIntf E Bdf E. BdClf E = Cl(Clf E ) Cl(Clf E ) c = ClClf E ClIntfE c = Clf E ClIntfE c Clf E Clf E c = Bdf E So BdClf E Bdf E. Example 4.4. Let X = R, E = [0, 1), and τf.s be the F.S-topology over R as introduced in Example 3.5. We define the crisp F.S-set V E over R related to the half-open interval [a, b) R by the mapping V : E = [0, 1) I R as below { 1 x [a, b) V α (x) = 0 x / [a, b) where α E and x R. By applying Definition 3.16, the closure and the interior of the crisp F.S-set V E, denoted by ClV E and IntV E respectively, are defined by the mappings ClV : E = [0, 1) I R and IntV : E = [0, 1) I R such that for any x R and α E we have { 1 x [a, b] (ClV ) α (x) = 0 x / [a, b], and { 1 x (a, b) (IntV ) α (x) = 0 x / (a, b). 699

14 Moreover by Definition 4.1, the boundary of crisp F.S-set V E is defined by the mapping BdV : E = [0, 1) I R as below { 1 x {a, b} (BdV ) α (x) = 0 x / {a, b} where x R and α E. Remark 4.5. In general topology, it is well-known that BdA = if and only if A is an open and a closed set, both in X. But in F.S-topology, it may not hold in general. This is shown by the following example. Example 4.6. Take the set of real numbers R with usual topology τ. Consider the F.S-topological space (R, [0, 1), τ F.S ) as introduced earlier in Example 3.6. Let a, b, c R such that a < b < c and let f E be a F.S-set over R such that for any α E, the mapping f α : R [0, 1] is defined as below 0 x a x a a < x c f α (x) = c a x b c b c < x < b 0 x b where x R. It is clear that the complement of f E, denoted by fe c, given by the mapping f c : E = [0, 1) I R such that x R and α E is defined as below 1 x a fα(x) c = c x c a a < x c c x c b c < x < b 1 x b. Regarding to Example 3.6, f E is a F.S-open and a F.S-closed set both, in F.Stopological space (R, [0, 1), τ F.S ) since α E, f 1 α (0, 1] = (a, b) τ and fα c 1 (0, 1] = (, c) (c, + ) τ. So f E = Clf E and fe c = Clf E c. This implies that Bdf E = f E f E c. Thus x R and α E, 0 x a, x b (Bdf) α (x) = min{f α (x), fα(x)} c = f α (x) a < x a+c 2, c+b 2 x < b fα(x) c a+c 2 < x < c+b 2 means that Bdf E Φ. Theorem 4.7. If (X, E, τ) is a F.S-topological space, then (1) Bd[f E g E ] Bdf E Bdg E. (2) Bd[f E g E ] Bdf E Bdg E. Proof. (1) Bd[f E g E ] = Cl[f E g E ] Cl[f E g E ] c = Cl[f E g E ] Cl[f c E g c E] [Clf E Clg E ] [Clf c E Clg c E] = [Clf E (Clf c E Clg c E)] [Clg E (Clf c E Clg c E)] = [Bdf E Clg c E] [Bdg E Clf c E] Bdf E Bdg E 700

15 (2) Bd[f E g E ] = Cl[f E g E ] Cl[f E g E ] c = Cl[f E g E ] Cl[f c E g c E] [Clf E Clg E ] [Clf c E Clg c E] = [(Clf E Clg E ) Clf c E] [(Clf E Clg E ) Clg c E] = [Bdf E Clg E ] [Bdg E Clf E ] Bdf E Bdg E Theorem 4.8. Let (X, E, τ) be a F.S-topological space and f E be a F.S-set over X. If (Bdf) e (x) = 0, then f e (x) = 0 or f e (x) = 1. Proof. Let x X such that for some e E we have (Bdf) e (x) = 0, means min{(clf) e (x), (Clf c ) e (x)} = 0. Then (Clf) e (x) = 0 or (Clf c ) e (x) = 0 which implies that f e (x) = 0 or fe c (x) = 0. Thus f e (x) = 0 or f e (x) = 1. Corollary 4.9. In the F.S-topological space (X, E, τ), the F.S-set f E F.S-set if Bdf E = Φ. is a crisp Proof. Let Bdf E = Φ. Theorem 4.8 implies that f e (x) = 0 or f e (x) = 1, x X and for all e E. This means that the set value of mapping f(e) is a subset of {0, 1} i.e., f E is a crisp F.S-set. Theorem If Ψ : (X 1, E 1, τ 1 ) (X 2, E 2, τ 2 ) is a F.S-continuous map, then g E2 F.S(X 2, E 2 ), Bd[Ψ 1 (g E2 )] Ψ 1 [Bdg E2 ] Proof. If g E2 have is a F.S-set over X 2, then by Proposition 2.7 and Theorem 3.25 we Bd[Ψ 1 (g E2 )] = Cl[Ψ 1 (g E2 )] Cl[Ψ 1 (g E2 )] c = Cl[Ψ 1 (g E2 )] Cl[Ψ 1 (g c E 2 )] Ψ 1 [Clg E2 ] Ψ 1 [Clg c E 2 ] = Ψ 1 [Clg E2 Clg c E 2 ] = Ψ 1 [Bdg E2 ] Thus Bd[Ψ 1 (g E2 )] Ψ 1 [Bdg E2 ]. Theorem If (X, E, τ) is a F.S-topological space, then (1) Bd[f E g E ] Clf E Clg E. (2) Bd[f E g E ] [Bdf E Clg E ] [Clf E Bdg E ]. Proof. (1) By applying Proposition 4.2 we have Bd[f E g E ] Cl[f E g E ]. So Bd[f E g E ] Clf E Clg E follows from Theorem 3.22 (1). 701

16 (2) By Lemma 3.14, Theorem 3.17, Theorem 3.22 (1), Proposition 2.4 (2), and Proposition 2.9 (4) we have Bd[f E g E ] = Cl[f E g E ] Cl[f E g E ] c = Cl[f E g E ] Cl[(fE c X) ( X g E)] c [Clf E Clg E ] [Cl(f E c X) Cl( X g E)] c [Clf E Clg E ] [(ClfE c X) ( X Clg E)] c = [(Clf E Clg E ) (Clf c E X)] [(Clf E Clg E ) ( X Clg c E)] = [(Clf E Clf c E) (Clg E X)] [(Clf E X) (Clg E Clg c E)] = [Bdf E Clg E ] [Clf E Bdg E ] Hence Bd[f E g E ] [Bdf E Clg E ] [Clf E Bdg E ]. 5. Conclusions The aim of this work is to introduce and to study the concept of fuzzy soft boundary. We define fuzzy soft boundary as an parameterization extension of the concept of boundary in the classical sense and then consider some properties of it. The fuzzy soft topology is also considered and closure, interior, and continuity are studied in a fuzzy soft topological space. Acknowledgements. This work was partially supported by the Institute for Mathematical Research, Universiti Putra Malaysia under the Grant number References [1] B. Ahmad and A. Kharal, On fuzzy soft sets, Adv. Fuzzy Syst. 2009, Art. ID , 6 pp. [2] H. Aktaş and N. Çaǧman, Soft sets and soft groups, Inform. Sci. 177 (2007) [3] S. Atmaca and I. Zorlutuna, On fuzzy soft topological spaces, Ann. Fuzzy Math. Inform. 5(2) (2013) [4] N. Çaǧman, S. Karataş and S. Enginoǧlu, Soft topology, Comput. Math. Appl. 62 (2011) [5] H. Hazra, P. Majumdar and S. K. Samanta, Soft topology, Fuzzy Inf. Eng. 4(1) (2012) [6] M. Irfan Ali, F. Feng, X. Liu and W.K. Min, On some new operations in soft set theory, Comput. Math. Appl. 57(9) (2009) [7] M. Irfan Ali, M. Shabir and Munzza Naz, Algebraic structures of soft sets associated with new operations, Comput. Math. Appl. 61(9) (2011) [8] A. Kharal and B. Ahmad, Mappings on fuzzy soft classes, Adv. Fuzzy Syst. 2009, Art. ID , 6 pp. [9] Z. Li and R. Cui, On the topological structure of intuitionistic fuzzy soft sets, Ann. Fuzzy Math. Inform. 5(1) (2013) [10] P. K. Maji, R. Biswas and A. R. Roy, soft set theory, Comput. Math. Appl. 45 (2003) [11] P. K. Maji, R. Biswas and A. R. Roy, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) [12] P. K. Maji, R. Biswas and A. R. Roy, Fuzzy soft set, J. Fuzzy Math. 9(3) (2001) [13] P. Majumdar and S. K. Samanta, Generalized fuzzy soft sets, Comput. Math. Appl. 59(4) (2010) [14] D. Molodtsov, Soft set theory first-results, Comput. Math. Appl. 37 (1999) [15] I. Osmanoglu and D. Tokat, Compact fuzzy soft spaces, Ann. Fuzzy Math. Inform. 7(1) (2014)

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