Address for Correspondence Department of Science and Humanities, Karpagam College of Engineering, Coimbatore -32, India

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1 Research Paper sb* - CLOSED SETS AND CONTRA sb* - CONTINUOUS MAPS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES A. Poongothai*, R. Parimelazhagan, S. Jafari Address for Correspondence Department of Science and Humanities, Karpagam College of Engineering, Coimbatore -32, India Department of Science and Humanities, Karpagam College of Engineering, Coimbatore -32, India College of Vestsjaelland, South Herrestraede 11, 4200 Slagelse, Denmark ABSTRACT: In this paper, we introduce and study the concept of intuitionistic fuzzy sb * - closed sets and also we study the intuitionistic fuzzy contra sb* - continuous maps and its properties in intuitionistic fuzzy topological spaces. AMS Classification (2000) MSC: 54A40. KEYWORDS: Intuitionistic Fuzzy sb*-closed set, Intuitionistic Fuzzy b - closed set, Intuitionistic Fuzzy sb * -open sets, Intuitionistic Fuzzy contra sb* - continuous maps. 1. INTRODUCTION Zadeh [21] introduced the concept of fuzzy sets. Using fuzzy sets Chang [4] introduced fuzzy topological spaces. Since then various authors have contributed to the development of fuzzy topology. The concept of intuitionistic fuzzy sets was introduced by Atanassov[1,2] as a generalization of fuzzy sets. In the last 30 years various concept of Fuzzy mathematics has been extended for intuitionistic fuzzy sets. Coker [5] introduced the concept of intuitionistic fuzzy topological space. Intuitionistic fuzzy compactness [6], Intuitionistic fuzzy connectedness [20], Intuitionistic fuzzy separation axioms [3], Intuitionistic fuzzy continuity [9], Intuitionistic fuzzy g- closed sets [18] and Intuitionistic fuzzy g-continuity [17] have been generalized for intuitionistic fuzzy topological spaces. Dontchev[7] introduced the concept of contra continuous mapping. Ekici and Kerre[8] introduced the concept of fuzzy contra continuous mapping. Krsteska and Ekici[12] introduced the concept of intuitionistic fuzzy contra continuous mapping and fuzzy contra strongly pre continuous mapping. The authors [13, 14, 15] introduced sb* - closed sets, sb* - continuous maps, sb* irresolute and homeomorphisms and studied some of their basic properties in topological spaces. In the present paper, we introduce the concept of intuitionistic fuzzy sb*-closed set and intuitionistic fuzzy contra sb* - continuous maps and also we obtain some of their characterization and properties. 2. PRELIMINARIES In this section, we recall some basic notions, definitions and properties. Throughout this paper, X and Y denotes the intuitionistic fuzzy topological spaces (X,τ ) and (Y, σ) respectively. For any subset A of a space (X, τ ), the closure of A, the interior of A and the complement of A are denoted by cl(a), int(a) and A c, A (or) X- A respectively. Definition 2.1[1]: Let X be a non empty fixed set. An intuitionistic fuzzy set A in X is an object having the form A = {<x, µ A (x), γ A (x)> : x X} where the functions µ A : X [0,1] and γ A : X [0,1] denote the degree of membership µ A (X) and the degree of non membership γ A (x) of each element x X to the set A respectively, and 0 µ A (x) + γ A (x) 1 for each x X. Definition 2.2[1]: The intuitionistic fuzzy sets 0 ~ = {<x, 0, 1>, x X} and 1 ~ ={<x, 1, 0>, x X} are respectively called empty and whole intuitionistic fuzzy set on X. Two intuitionistic fuzzy sets A = {<x, µ A (x), γ A (x)> : x X} and B = {<x, µ B (x) γ B (x)> : x X} are said to be q-coincident (AqB in short) if and only if there exists an element x X such that µ A (x) > γ B (x) (or) γ A (x) < µ B (x). Definition 2.3[2]: Let X be a non empty set, and let the intuitionistic fuzzy sets be A and B. Let {A j : j J} arbitrary family of intuitionistic fuzzy sets in X. Then (a) A B iff µ A (x) µ B (x) and γ A (x) γ B (x) for all x X; (b) A = {<x, γ A (x), µ A (x)> : x X} ; (c) A j = {<x, µ Aj (x), γ Aj (x) > : x X }; (d) A j = {<x, µ Aj (x), γ Aj (x) > : x X} ; (e) [ ]A = {<x, µ A (x), 1 µ A (x)> : x X} ; (f) < >A= {<x, 1 γ A (x), γ A (x)> : x X} ; (g) A =A, ~ 1 = 0 and 0 ~ 1 ; (h) A B A B, A B A B Definition 2.4[5]: An intuitionistic fuzzy topology τ on a non empty set X is a family τ of intuitionistic fuzzy sets in X satisfying the following axioms: (T 1 ) 0 ~, 1 ~ τ. (T 2 ) G 1 G 2 τ for any G 1, G 2 τ. (T3) G i τ for any arbitrary family {G i : i I} τ. In this case the pair (X, τ ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in τ is known as an intuitionistic fuzzy open set in X. Definition 2.5 [5]: Let A intuitionistic fuzzy set in an intuitionistic fuzzy topological space X. Then int A = {G /G is an intuitionistic fuzzy open set in X and G A} is called intuitionistic fuzzy interior of A; cl A = {G / G is an intuitionistic fuzzy closed set in X and G A} is called an intuitionistic fuzzy closure of A. Lemma 2.6[5]: Let A and B y two intuitionistic fuzzy sets of an intuitionistic fuzzy topological space (X,τ ), then (a) (A q B) A B c. (b) A is an intuitionistic fuzzy closed set in X cl(a) = A. (c) A is intuitionistic open set in X int(a) = A. (d) cl(a c ) = (int(a)) c. (e) int(a c ) = (cl(a)) c. (f) A B int(a) int (B). (g) A B cl(a) cl (B).

2 (h) cl(a B)=cl(A) cl(b). (i) int(a B) = int(a) int(b). Definition 2.7[6]:Let X be a non empty set and c X a fixed element in X. If α (0,1] and β [0,1) are two real numbers such that α + β 1 then (a) C(α, β) = <x, C α, C 1 β > is called an intuitionistic fuzzy point in X, where α denotes the degree of membership if c(α, β) and β denote the degree of non membership of c(α, β). (b) C(β) = <x, 0, 1 C 1 β > is called a vanishing intuitionistic fuzzy point in X, where β denote the degree of non membership of C(β). Definition 2.8[9]: Let A intuitionistic fuzzy set of an intuitionistic fuzzy topological space X. Then A is called (i) an intuitionistic fuzzy α-open set (briefly IFαOS) if A int(cl(int(a))), (ii) an intuitionistic fuzzy semi open set (briefly IFSOS) if A cl(int(a)). Definition 2.9[9]: Let A intuitionistic fuzzy set of an intuitionistic fuzzy topological space X. Then A is called (i) an intuitionistic fuzzy α- closed set (briefly IFαCS) if A c is an intuitionistic fuzzy α-open set, (ii) an intuitionistic fuzzy semi- closed set (briefly IFSCS) if A c is an intuitionistic fuzzy semi open set. Definition 2.10[19]: An intuitionistic fuzzy set A of an intuitionistic fuzzzy topological space X is called intuitionistic fuzzy w-closed if cl(int(a)) O whenever A O and O is intuitionistic fuzzy semi -open. Definition 2.11[11]: An intuitionistic fuzzy set A in an intuitionistic fuzzy topological space X is said to be intuitionistic fuzzy b- closed (briefly IFbCS) if cl(int(a)) int(cl(a)) A. Definition 2.12[16]:An intuitionistic fuzzy set A is an intuitionistic fuzzy weakly generalized closed set (briefly IFwgCS) if cl(int(a)) U whenever A U, U is intuitionistic fuzzy open in X. Definition 2.13[13]: A subset A of a topological space (X, τ ) is called a strongly b -closed set (briefly sb*- closed) if cl(int(a)) U whenever A U and U is b open in X. Definition 2.14[14]: Let X and Y be topological spaces. A map f: X Y is called strongly b* - continuous (briefly sb*- continuous) if the inverse image of every open set in Y is sb* - open in X. Definition 2.15[15]: Let X and Y are topological spaces. A map f: (X,τ ) (Y, σ) is said to be sb* irresolute if the inverse image of every sb* - closed set in Y is sb* - closed set in X. Definition 2.16[15]: A bijection f : (X, τ ) (Y, σ) is called a sb* - homeomorphism if f is both sb* - continuous and sb* - open. Definition 2.17[5]: Let X and Y be two non empty sets and f: X Y be a function. (i). If B = {<y, µ B (y), γ B (y)> : y Y } ; is an intuitionistic fuzzy set in Y, then the pre image of B under f is denoted and defined by f 1 (B) = {<x, f 1 (µ B )(x), f 1 (γ B )(x) > : x X} ;. (ii). If A = {<x, λ A (x), υ A (x)> : x X} ; is an intuitionistic fuzzy set in X, then the image of A under f is denoted and defined by f(a) ={<y, f (λ A )(y), f (υ A )(y) > : y Y }, where f -1 (υ A ) = 1 f (1 υ A ). In (i), (ii), since µ B, γ B, λ A, υ A are fuzzy sets, we explain that f 1 (µ B )(x)a = µ B (f (x)), and f (λ A )(y) = {supλ A (x) if f 1( y) φ, = 0 otherwise}. Definition 2.18[5]: Let A, Ai(i J) intuitionistic fuzzy sets in X and B, Bj (j K) intuitionistic fuzzy sets in Y and f: X Y be a function. Then (i) f 1 ( B j ) = f 1 (B j ); (ii) f 1 ( B j ) = f 1 (B j ); (iii) f 1 ( 1 ~ ) = 1 ~ ; f 1 ( 0 ~ ) = 0 ~ ; (iv) f 1 1 ( B ) = f B ; (v) f( A i ) = f(a i ). Definition 2.19[5,10]: Let X and Y be two intuitionistic fuzzy topological spaces and f : X Y be a function. Then (i) f is intuitionistic fuzzy continuous iff the pre image of each intuitionistic fuzzy open set in Y is an intuitionistic fuzzy open set in X. Definition 2.20[10]: Let f: X Y be a function. The graph g: X X Y of f is defined by g(x) = (x, f(x)), x X. Lemma 2.21[10]: Let g: X X Y be the graph of a function f: X Y. If A is an intuitionistic fuzzy set of X and B is an intuitionistic fuzzy set of Y, then g 1 (A B)(x) = (A f 1 (B)(x)). Definition 2.22[9]: Let f: X Y be mapping. Then f is said to be intuitionistic fuzzy continuous if f 1(B) is an intuitionistic fuzzy open set in X for every intuitionistic fuzzy open set B in Y. Definition 2.23[8]:A mapping f: X Y is called intuitionistic fuzzy contra continuous if f 1 (B) is an intuitionistic fuzzy open set in X, for each intuitionistic fuzzy closed set B in Y. 3. INTUITIONISTIC FUZZY sb*- CLOSED SETS In this section, we introduce a new class of closed sets called intuitionistic fuzzy sb*- closed set in an intuitionistic fuzzy topological space. Also we discuss some of its properties. Definition 3.1: An intuitionistic fuzzy set A in an intuitionistic fuzzy topological space (X, τ) is called an intuitionistic fuzzy sb - closed set(briefly IFsb*- closed) if cl(int(a)) O whenever A O and O is intuitionistic fuzzy b-open. Remark 3.2: Every intuitionistic fuzzy closed set is intuitionistic fuzzy sb* - closed set but the converse is not true. Example 3.3: Let X = {a, b} and τ = 0, 0.5, 0.5>, <b, 0.4, 0.6>}. Then the intuitionistic fuzzy set A = {<a, 0.5, 0.5>, <b, 0.5, 0.5>} is intuitionistic fuzzy sb -closed but it is not intuitionistic fuzzy closed Remark 3.4: Every intuitionistic fuzzy sb -closed set is intuitionistic fuzzy b-closed but the converse is not true. Example 3.5: Let X = {a, b} and τ = 0, 1,U 0.4, 0.6>, <b, 0.3, 0.7>}. Then the intuitionistic fuzzy set A = {<a, 0.4, 0.6>, <b, 0.3, 0.7>} is intuitionistic fuzzy b -closed but it is not intuitionistic fuzzy sb*- Remark 3.6: Every intuitionistic fuzzy w-closed set is intuitionistic fuzzy sb -closed but the converse is not true. Example 3.7: Let X = {a, b} and τ = 0, 0.7, 0.3>, <b, 0.6, 0.4>}. Then the intuitionistic fuzzy

3 set A = {<a, 0.6, 0.4>, <b, 0.7, 0.3>} is intuitionistic fuzzy sb -closed but it is not intuitionistic fuzzy w- Remark 3.8: Every intuitionistic fuzzy α-closed set is intuitionistic fuzzy sb -closed but the converse is not true. Example 3.9: Let X = {a, b} and τ = 0, 0.3, 0.7>, <b, 0.2, 0.7>}. Then the intuitionistic fuzzy set A = {<a, 0.6, 0.3>, <b, 0.6, 0.4>} is intuitionistic fuzzy sb -closed but it is not intuitionistic fuzzy α - Remark 3.10: Every intuitionistic fuzzy sb -closed set is intuitionistic fuzzy wg-closed but the converse is not true. Example 3.11: Let X= {a, b} and τ = 0, intuitionistic fuzzy topology on X where U = {<a, 0.9, 0.1>, <b, 0.7, 0.2>}. Then the intuitionistic fuzzy set A = {<a, 0.4, 0.6>, <b, 0.3, 0.7>} is intuitionistic fuzzy wg-closed but it is not intuitionistic fuzzy sb - Theorem 3.12: Let (X, τ ) intuitionistic fuzzy topological space and A is an intuitionistic fuzzy set of X. Then A is intuitionistic fuzzy sb closed if and only if (AqF ) (cl(int(a))qf)for every intuitionistic fuzzy b closed set F of X. Proof: Necessity: Suppose A intuitionistic fuzzy sb*- closed set.let F intuitionistic fuzzy b closed set of X and (AqF ). Then by lemma 2.6(a), A F c and F c is intuitionistic fuzzy b open in X. Since A is intuitionistic fuzzy sb* - closed, cl(int(a)) F c. Hence by lemma 2.6(a) (cl(int(a))qf). Sufficiency: Let O intuitionistic fuzzy b open set of X such that A O(ie.,) A (O c ) c. Therefore by lemma 2.6(a), (AqO c ) and O c is an intuitionistic fuzzy b closed set in X. Hence by hypothesis (cl(int(a))qo c ).Therefore by lemma 2.6(a), cl(int(a)) (O c ) c.(i.e.,) cl(int(a)) O. Hence A is intuitionistic fuzzy sb -closed in X. Theorem 3.13:Let A intuitionistic fuzzy sb closed set in an intuitionistic fuzzy topological space (X,τ ) and C(α, β) intuitionistic fuzzy point of X such that C(α, β)q cl(int(a)) then cl(int(c(α, β)))qa. Proof: If cl(int(c(α, β)))qa, then by lemma (2.5)(a), cl(int(c(α, β))) A c which implies that A (cl(int(c(α, β)))) c and so cl(int(a)) (cl(int(c(α, β)))) c (C(α, β)) c, since A is intuitionistic fuzzy sb closed in X. Therefore by lemma 2.6(a), (cl(int(a))q(c(α, β))) which is a contradiction. Theorem 3.14: Let A and B be two intuitionistic fuzzy sb -closed sets in an intuitionistic fuzzy topological space, then A B is intuitionistic fuzzy sb - Proof: Let O intuitionistic fuzzy b- open set in X such that A B O. Then A O and B O. So cl(int(a)) O and cl(int(b)) O. Therefore cl(int(a B)) cl(int(a)) cl(int(b)) O. Hence A B is intuitionistic fuzzy sb *- Remark 3.15: The union of any two intuitionistic fuzzy sb* closed sets need not intuitionistic sb* closed set in general as seen from the following example. Example 3.16: Let X = {a, b} and τ = 0, 1,U is an intuitionistic fuzzy topology on X, Where U = {<a, 0.6, 0.4>, <b, 0.8, 0.2>}. The intuitionistic fuzzy set A = {<a, 0.1, 0.9>, <b, 0.8, 0.2> and B = {<a, 0.6, 0.4>, <b, 0.7, 0.3>} are intuitionistic fuzzy sb* closed sets but the union A B is not an intuitionistic fuzzy sb -closed set. Theorem 3.17: Let A intuitionistic fuzzy sb*- closed in an intuitionistic fuzzy topological space (X,τ ) and A B cl(int(a)). Then B is intuitionistic fuzzy sb*-closed in X. Proof: Let A intuitionistic fuzzy sb*-closed set such that A B cl(int(a)). Let U intuitionistic fuzzy b-open set of X such that B U. Since A is intuitionistic fuzzy sb -closed, we have cl(int(a)) U and A B U. Since B cl(int(a)), cl(int(b)) cl(int(a)) U. Therfore cl(int(b)) U. Thus B is intuitionistic fuzzy sb*-closed set in X. 4. INTUITIONISTIC FUZZY CONTRA sb CONTINUOUS MAPS In this section, we introduce a new definition called intuitionistic fuzzy contra sb -continuity in intuitionistic fuzzy topological space. Also we discuss some of its basic properties. Definition 4.1: (i) A mapping f: X Y is called intuitionistic fuzzy contra strongly b* continuous if f 1 (B) is an intuitionistic fuzzy sb* open set in X, for each intuitionistic fuzzy closed set B in Y. (ii) A mapping f: X Y is called intuitionistic fuzzy contra sb*- irresolute if the inverse image of every intuitionistic fuzzy sb* - closed set in Y is intuitionistic fuzzy sb* - open set in X. Remark 4.2: From the above definition, the following implication is true. Intuitionistic fuzzy contra continuity intuitionistic fuzzy contra sb* - continuity. The converse of the above remark is not true as seen from the following example. Example 4.3: Let X = {a, b}, Y = {x, y} and τ = 0, and σ = 0, 1,V are the intuitionistic fuzzy topologies on X and Y, where U = {<a, 0.5, 0.5>, <b, 0.4, 0.6>}and V = {<x, 0.5, 0.5>, <y, 0.5, 0.5>} respectively. Then the mapping f: X Y is defined by f(a) = x and f(b) = y is intuitionistic fuzzy contra sb* continuous but not intuitionistic fuzzy contra continuous. Theorem 4.4: Let f: X Y be a mapping from an intuitionistic fuzzy topological space X into an intuitionistic fuzzy topological space Y. Then the following statements are equivalent: i. f is an intuitionistic fuzzy contra sb* - ii. continuous mapping, f 1 (B) is an intuitionistic fuzzy sb* - closed set in X, for each intuitionistic fuzzy open set B in Y. Proof : (i) (ii): Let f y intuitionistic fuzzy contra sb* - continuous mapping and let B intuitionistic fuzzy open set in Y. Then, B c is an intuitionistic fuzzy closed set in Y. By assumption f 1 (B c )is intuitionistic fuzzy sb* open set in X. (f 1 (B)) c is an intuitionistic fuzzy sb* open set in X, since f 1 (B c )=(f 1 (B)) c. Hence f 1 (B) is intuitionistic fuzzy sb* - closed set in X. (ii) (i): Let B intuitionistic fuzzy closed set in Y. B c is an intuitionistic fuzzy open set in Y. By (ii), f 1 (B c ) = (f 1 (B)) c is an intuitionistic fuzzy sb* - closed set in X. Hence f 1 (B) is an intuitionistic fuzzy sb* -open set in X. Therefore f is intuitionistic

4 fuzzy contra sb* - continuous mapping. Theorem 4.5: Let f: X Y and g: Y Z, where X, Y and Z are intuitionistic fuzzy topological spaces. Then the following statements hold. i. If f: X Y is an intuitionistic fuzzy contra sb* - continuous mapping and g: Y Z is an intuitionistic fuzzy contra continuous mapping, then g f: X Z is an intuitionistic fuzzy sb* - continuous mapping. ii. If f: X Y is an intuitionistic fuzzy sb* irresolute mapping and g: Y Z is an intuitionistic fuzzy contra sb* - continuous mapping, then g f: X Z is an intuitionistic fuzzy contra sb* -continuous mapping. iii. If f : X Y is an intuitionistic fuzzy sb* - closed, surjective mapping and g f: X Z is an intuitionistic fuzzy contra sb* - continuous mapping, then g: Y Z is an intuitionistic fuzzy contra sb*-continuous mapping. iv. If g f: X Z is an intuitionistic fuzzy contra sb*- continuous mapping and g : Y Z is an intuitionistic fuzzy open, injective mapping, then f: X Y is a intuitionistic fuzzy contra sb* - continuous mapping. Proof: i. Let A intuitionistic fuzzy open set in Z. Since g is intuitionistic fuzzy contra continuous, g 1 (A) is intuitionistic fuzzy closed set in Y. Since f is intuitionistic fuzzy contra sb* - continuous, f 1 (g 1 (A)) = (g f ) 1 (A) is intuitionistic fuzzy sb* - open set in X. Hence g f is an intuitionistic fuzzy sb* - continuous mapping. ii. Let A intuitionistic fuzzy open set in Z. Since g is intuitionistic fuzzy contra sb* - continuous, g 1 (A) is intuitionistic fuzzy sb* - closed in X. Since f is intuitionistic fuzzy sb* - irresolute, f 1 (g 1 (A)) = (g f ) 1 (A) is an intuitionistic fuzzy sb* - closed set in X. Hence g f is an intuitionistic fuzzy contra sb* continuous mapping. iii. iv. From the fact that for any surjective mapping g: Y Z, g 1 (A) = f(g f) 1 (A) holds for each intuitionistic fuzzy set A in Z. From the fact that for any open, injective mapping g: Y Z, f 1 (B) = (g f) 1 (g(b)) holds for each intuitionistic fuzzy set B in Y. Theorem 4.6: Let f: X Y be a function and g: X X Y be the graph of the function f, then f is intuitionistic fuzzy contra sb* continuous. Proof: Let B y intuitionistic fuzzy open set in Y. Then f 1 (B) = f 1 ( 1 ~ B) = 1 ~ f 1 (B) = g 1 ( 1 ~ B). Since B is an intuitionistic fuzzy open set in Y, ~ 1 B is an intuitionistic fuzzy open set in X Y. Since g is intuitionistic fuzzy contra sb* - continuous, g 1 ( 1 ~ B) is an intuitionistic fuzzy sb* - closed set in X. Hence f 1 (B) is an intuitionistic fuzzy sb* closed set in X and so f is an intuitionistic fuzzy contra sb* - continuous mapping. Theorem 4.7: If f: X Y is an intuitionistic fuzzy contra sb* - continuous function and g: Y Z is an intuitionistic fuzzy continuous function, then g f: X Z is intuitionistic fuzzy contra sb* - continuous. Proof: Let V intuitionistic fuzzy open set in Z. Then g 1 (V ) is intuitionistic fuzzy open in Y. Since f is intuitionistic fuzzy contra sb* - continuous, f 1 (g 1 (V)) = (g f) 1 (V ) is intuitionistic fuzzy sb* - closed in X. Therefore g f : X Z is intuitionistic fuzzy contra sb* - continuous. Theorem 4.8: Let X and Y be intuitionistic fuzzy topological spaces. Let f: X Y is any mapping. If the graph g: X X Y of f is intuitionistic fuzzy contra sb* irresolute then f is also intuitionistic fuzzy contra sb* irresolute. Proof: Let A intuitionistic fuzzy sb* - open set in Y. By definition, f 1 (A) = 1 ~ f 1 (A) = g 1 ( 1 ~ A). Since g is intuitionistic fuzzy contra sb* - irresolute, g 1 ( 1 ~ A) is intuitionistic fuzzy sb* - closed set in X. 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