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    \(\lambda\)- CLOSED MAPS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES

    P. Rajarajeswari, G. Bagyalakshmi

    Abstract. The purpose of this paper is to introduce and study the concepts of intuitionistic fuzzy contra weakly generalized continuous map- pings in intuitionistic fuzzy topological space. Some of their properties are explored.

    2010 AMS Classi?cation: 54A40, 03E72

    Keywords: Intuitionistic fuzzy topology, intuitionistic fuzzy weakly generalized closed set, intuitionistic fuzzy weakly generalized open set and intuitionistic fuzzy contra weakly generalized continuous mappings.

    Corresponding Author: G. Bagyalakshmi (g_bagyalakshmi@yahoo.com)

    ABSTRACT: In this paper we introduce the concept intuitionistic fuzzy \(\lambda\)-open maps and intuitionistic fuzzy \(\lambda\)-closed maps in intuitionistic fuzzy topological space and study some of their properties.

    Keywords: Intuitionistic fuzzy topology, intuitionistic fuzzy \(\lambda\)-closed maps, intuitionistic fuzzy \(\lambda\)-open maps.

    AMS subject classification (2000): 54A40, 03F55

    1. INTRODUCTION

    After the introduction of fuzzy sets by L.A Zadeh [20] in 1965, there have been a number of generalizations of this fundamental concept. The concepts of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy set in 1983, Coker [3] introduced the notion of intuitionistic fuzzy topology in 1997. This approach provides a wide field for investigation in the area of fuzzy topology and its application. The aim of this paper is to introduce intuitionistic fuzzy \(\lambda\)-open maps, intuitionistic fuzzy \(\lambda\)-closed maps and studied some of their properties.

    2. PRELIMINARIES

    Definition 2.1 ([1]) Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS) A in X is an object having the form A = {\(<\)x, \(\mu \)\({}_{A }\)(x), \(\upsilon \)\({}_{A}\) (x) \(>\) : x? X}, where the function \(\mu \)\({}_{A}\) : X ? [0,1] and \(\upsilon \)\({}_{A }\):X? [0,1] denotes the degree of membership \(\mu \)\({}_{A}\)(x) and the degree of non membership \(\upsilon \)\({}_{A}\) (x) of each element x? X to the set A respectively and 0= \(\mu \)\({}_{A}\) (x)+ \(\upsilon \)\({}_{A}\) (x) = 1 for each x? X.

    Definition 2.2 ([1]) Let A and B be intuitionistic fuzzy sets of the form

    A = {\(<\)x, \(\mu \)\({}_{A }\)(x), \(\upsilon \)\({}_{A}\)(x) \(>\): x ? X}, and form B= {\(<\)x, \(\mu \)\({}_{B}\)(x), \(\upsilon \)\({}_{B}\) (x) \(>\): x? X}.Then

    (a) A \(\subseteq\) B if and only if \(\mu \)\({}_{A}\)(x) = \(\mu \)\({}_{B }\)(x) and \(\nu \)\({}_{A}\)(x) = \(\nu \)\({}_{B}\)(x) for all x \(\in\) X

    (b) A = B if and only if A \(\subseteq\) B and B \(\subseteq\) A

    (c) A\({}^{c}\) = {\(\langle\) x, \(\upsilon \)\({}_{A}\)(x), \(\mu \)\({}_{A}\)(x) \(\rangle\) / x \(\in\) X}

    (d) A \(\cap\) B = {\(\langle\) x, \(\mu \)\({}_{A}\)(x) \(\wedge\) \(\mu \)\({}_{B}\)(x), \(\upsilon \)\({}_{A}\) (x) \(\vee\) \(\upsilon \)\({}_{B}\) (x) \(\rangle\) / x \(\in\) X}

    (e) A \(\cup\) B = {\(\langle\) x, \(\mu \)\({}_{A}\)(x) \(\vee\) \(\mu \)\({}_{B}\)(x), \(\upsilon \)\({}_{A}\) (x) \(\wedge\) \(\upsilon \)\({}_{B}\) (x) \(\rangle\) / x \(\in\) X}.

    The intuitionistic fuzzy sets \(\mathop{0}\limits_{\sim } \) = { \(<\)x, 0,1\(>\) x \(\in\) X } and \(\mathop{1}\limits_{\sim } \)={ \(<\)x, 1, 0 \(>\) x \(\in\) X } are respectively the empty set and whole set of X.

    Definition 2.39 ([5]) An intuitionistic fuzzy topology (IFT) on X is a family of IFS which satisfying the following axioms.

    (i) \(\mathop{0}\limits_{\sim } \), \(\mathop{1}\limits_{\sim } \) \(\in\)\(\tau \)

    (ii) G\({}_{1}\) \(\cap\) G\({}_{2}\) \(\in\) \(\tau \) for any G\({}_{1}\), G\({}_{2}\) \(\in \)\(\tau \)

    (iii) \(\cup\) G\({}_{i}\) \(\in\) \(\tau \) for any family {G\({}_{i }\)/ i \(\in\) I} \(\subseteq \) \(\tau \)

    In this case the pair (X, \(\tau \)) is called an intuitionistic fuzzy topological space (IFTS) and each intuitionistic fuzzy set in \(\tau \) is known as an intuitionistic fuzzy open set (IFOS) in X.

    The complement A of an IFOS in an IFTS (X\({}_{,}\) \(\tau \)) is called an intuitionistic fuzzy closed set (IFCS) in (X\({}_{,}\) \(\tau \)).

    Definition 2.4([3]) Let (X\({}_{,}\) \(\tau \)) be an intuitionistic fuzzy topology and

    A = {\(<\)x, \(\mu \)\({}_{A }\)(x), \(\upsilon \)\({}_{B}\) (x) \(>\): x ? X}, be an intuitionistic fuzzy set in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by

    int (A) = \(\cup\) {G/ G is an intuitionistic fuzzy open set in X and G \(\subseteq\) A}

    cl (A) =\(\cap\) { K/ K in an intuitionistic fuzzy closed set in X and A\(\subseteq\) K }

    Definition 2.5 ([11]) Let f be a mapping from an IFTS (X, \(\tau \)) into an IFTS (Y,\(\sigma \)). Then f is said to be an

    (i) intuitionistic fuzzy open mapping (IF open mapping) if f(A) is an IFOS in Y for every IFOS A in X.

    (ii) intuitionistic fuzzy closed mapping (IF closed mapping) if f(A) is an IFCS in Y for every IFCS A in X.

    Remark 2.6 ([3]) For any intuitionistic fuzzy set A in (X\({}_{,}\) \(\tau \)), we have

    1. cl (A\({}^{C}\)) = [int (A)]\({}^{C}\),

    2. int (A\({}^{C}\)) = [cl (A)]\({}^{C}\),

    3. A is an intuitionistic fuzzy closed set in X \(\Leftrightarrow\) Cl (A) = A

    4. A is an intuitionistic fuzzy open set in X \(\Leftrightarrow\) int (A) = A

    Definition 2.7 ([6]) An intuitionistic fuzzy set A = {\(<\)x, \(\mu \)\({}_{A }\)(x), \(\upsilon \)\({}_{B}\) (x) \(>\): x ? X} in an intuitionistic fuzzy topology space (X, \(\tau \)) is said to be

    (i) intuitionistic fuzzy semi closed if int (cl (A)) \(\subseteq\) A

    (ii) intuitionistic fuzzy pre closed if cl (int (A))\(\subseteq\) A

    Definition 2.8 ([5]) Let X and Y are nonempty sets and f: X\(?\)Y is a function

    (a) If B = {\(<\) y, \(\mu \)\({}_{B }\)(y), \(\upsilon \)\({}_{B}\) (y) \(>\) : y\(\in\)Y} is an intuitionistic fuzzy set in Y, then the pre image of B under f denoted by f\({}^{-1}\)(B), is defined by

    f\({}^{-1 }\)(B) =={\(<\)x,f \({}^{-1}\)(\(\mu \) \({}_{B}\)(x)),f \({}^{-1}\)(\(\upsilon \)\({}_{B}\)(x)) \(>\) : x \(\in\) X}

    (b) If A= {\(<\){x, \(\mu \) \({}_{A}\) (x), \(\upsilon \)\({}_{B}\) (x),)\(>\) / x\(\in\) X} is an intuitionistic fuzzy set in X, the image of A under f denoted by f(A) is the intuitionistic fuzzy set in Y defined by

    f(A) = {\(<\)y, f (\(\mu \)\({}_{A }\)(y)), f (\(\upsilon \)\({}_{A}\)(y)) \(>\) : y \(\in\) Y} where f (\(\upsilon \)\({}_{A}\)) = 1-f(1-(\(\upsilon \)\({}_{A}\)))

    Definition 2.9 ([9]) An intuitionistic fuzzy set A of an intuitionistic topology space (X, ) is called an

    (i) intuitionistic fuzzy \(\lambda\)-closed set (IF \(\lambda\)-CS) if A \(\supseteq \) cl(U) whenever A\(\supseteq \)U and U is intuitionistic fuzzy open set in X.

    (ii) intuitionistic fuzzy \(\lambda\)-open set (IF \(\lambda\)-OS) if the complement \(A^{c} \) is an intuitionistic fuzzy \(\lambda\)-closed set A.

    Definition 2.10: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,\(\tau\)) called

    (i) intuitionistic fuzzy generalized closed set [15] (intuitionistic fuzzy g – closed) if cl(A) \(\subseteq\) U whenever A \(\subseteq\) U and U is intuitionistic fuzzy semi open

    (ii) intuitionistic fuzzy g – open set [14], if the complement of an intuitionistic fuzzy g – closed set is called intuitionistic fuzzy g - open set.

    (iii) intuitionistic fuzzy semi open (resp. intuitionistic fuzzy semi closed) [6] if there exists an intuitionistic fuzzy open (resp. intuitionistic fuzzy closed) such that U\(\subseteq\)A \(\subseteq\) Cl(U) (resp. int(U) \(\subseteq\) A \(\subseteq\) U).

    Definition 2.11: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,\(\tau \)) is called

    (i) an intuitionistic fuzzy w-closed [14] if cl(A) \(\subseteq\) O whenever A\(\subseteq\)O and O is intuitionistic fuzzy semi open. (X,\(\tau \))

    (ii) an intuitionistic fuzzy generalized \(\alpha \)-closed set [8] (IFG\(\alpha \)CS if\(\ \alpha \)cl(A) \(\subseteq\) O whenever A\(\subseteq\)O and O is IF\(\alpha \)OS in(X,\(\tau \))

    (iii) an intuitionistic fuzzy \(\alpha \)-generalized closed set [12] (IF\(\alpha \)GCS if\(\ \alpha \)cl(A) \(\subseteq\) O whenever A\(\subseteq\)O and O is IFOS in(X,\(\tau \))

    (iv) an intuitionistic fuzzy regular closed set [4 ] (IFRCS in short) if A = cl(int(A)),

    (v) an intuitionistic fuzzy regular open set [4 ](IFROS in short) if A = int(cl(A)),

    Definition 2.12 ([10]) A mapping f: (X,\(\tau \)) ? (Y, \(\sigma \)) is said to be intuitionistic fuzzy \(\lambda\) –continuous if the inverse image of every intuitionistic fuzzy closed set of Y is intuitionistic fuzzy \(\lambda\) -closed in X.

    Definition 2.13 ([11]) A topological space (X, \(\tau \)) is called intuitionistic fuzzy \(\lambda\) - T\({}_{1/2}\)\({}_{ }\)space

    (IF \(\lambda\) - T\({}_{1/2}\) space in short) if every intuitionistic fuzzy \(\lambda\)-closed set is intuitionistic fuzzy closed in X.

    Definition 2.14: A mapping f: (X,\(\tau \)) ? (Y, \(\sigma \)) is said to be

    (i) an intuitionistic fuzzy w-closed [15 ] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy w-closed set in Y

    (ii) an intuitionistic fuzzy regular closed [17 ] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy regular closed set in Y.

    (iii) an intuitionistic fuzzy generalized \(\alpha \)-closed [8] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy generalized \(\alpha \)-closed set in Y

    (iv) an intuitionistic fuzzy \(\alpha \)-generalized closed [12] if image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy \(\alpha \)-generalized closed set in Y

    Definition 2.15 ([10]) Let A be an IFS in an IFTS (X, \(\tau \)). Then the Intuitionistic fuzzy \(\lambda\)-interior and intuitionistic fuzzy \(\lambda\)-closure of A are defined as follows.

    \(\lambda\)-int(A) = \(\cup \){G G is an IF\(\lambda\)-OS in X and G \(\subseteq \) A}

    \(\lambda\)-cl(A) = n{K K is an IF\(\lambda\)-CS in X and A\(\subseteq \) K }

    3. INTUITIONISTIC FUZZY \(\lambda\)-CLOSED MAPPINGS

    Definition 3.1: A mapping f: (X, ) ? (Y, \(\sigma \)) is said to be intuitionistic fuzzy \(\lambda\)-closed map (IF \(\lambda\)– closed map) if f(V) is \(\lambda\)-closed in (Y, \(\sigma \)) for every closed set V in (X, \(\tau \)).

    Theorem 3.2: Every IF closed map is an IF \(\lambda\)-closed map but not conversely.

    Proof: Let f: X ? Y be an IF closed map. Let A be an IFCS in X. Then f(A) is an IFCS in Y. Since every IFCS is an IF \(\lambda\)–CS, f(A) is an IF\(\lambda\)–CS in Y[8].Hence f is an IF\(\lambda\)-closed map.

    Remark 3.3: The converse above theorem need not be true as seen from the following example.

    Example 3.4: Let X = { a, b} and Y = {u, v} intuitionistic fuzzy sets U and V are defined as follows; U={\(<\)a,0.5,0.5.\(>\),\(<\)b,0.2,0.8\(>\)} V={\(<\)u,0.5,0.5\(>\), \(<\)v,0.3,0.6\(>\)}

    Let \(\tau \) ={0,1,U} and \(\sigma \) ={0,1,V} be Intuitionistic fuzzy topologies on X and Y respectively. Define a map f: (X, \(\tau \)) ? (Y,\(\ \sigma \) ) by f(a) =u and f(b) = v.

    Then f (U)=f({\(<\)a,0.5,0.5\(>\), \(<\)b,0.8,0.2\(>\)}) = {\(<\)u,0.5,0.5\(>\), \(<\)v,0.8,0.2 \(>\)} is \(\lambda\) -closed set in Y But {\(<\)u,0.5,0.5\(>\), \(<\)v,0.8,0.2\(>\)} is not closed set in Y. Hence f is intuitionistic fuzz \(\lambda\) - closed mapping in Y but not intuitionistic fuzzy closed in Y

    Theorem 3.6: Let f : X ? Y be an IF\(\lambda\)-closed map where Y is an IF\(\lambda\)-T\({}_{1/2}\) space, then f is an IF closed map if every IF\(\lambda\)-CS is an IFCS in Y.

    Proof: Let f be an IF\(\lambda\)-closed map. Then for every IFCS A in X, f(A) is an IF\(\lambda\)-CS in Y. Since Y is an IF\(\lambda\)-T\({}_{1/2}\) space, f(A) is an IF\(\lambda\)-CS in Y and by hypothesis f(A) is an IFCS in Y. Hence f is an IF closed map.

    Theorem 3.7: Every IF pre closed map is IF \(\lambda\) -closed map.

    Proof: Let f: X ? Y be an IF pre closed map. Let A be an IFCS in X. By assumption f(A) is an IF pre closed set in Y. Since every IF pre closed set is an IF \(\lambda\)–CS [6] f(A) is an IF\(\lambda\)–CS in Y. Hence f is an IF\(\lambda\)-closed map.

    Remark 3.8: The converse above theorem need not be true as seen from the following example:

    Example 3.9: Let X = {a,b} and Y={u,v} and intuitionistic fuzzy sets U and V are defined as follows; U= {\(<\) a, 0.5,0.5\(>\), \(<\) b, 0.2,0.8 \(>\)} and V={\(<\)u,0.5,0.5\(>\), \(<\)v, 0.3, 0.7\(>\)}

    Let ={0,1,U} and \(\sigma \)={0,1,V} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, ) ? (Y,\(\ \sigma \) ) by f(a) =u and f(b) = v.

    Then f (U)=f({\(<\)a,0.5,0.5\(>\), \(<\)b,0.8,0.2\(>\)}) = {\(<\)u,0.5,0.5\(>\), \(<\)v,0.8,0.2 \(>\)} is IF \(\lambda\)-closed set in Y but not IF preclosed set in Y. Hence f is intuitionistic fuzzy \(\lambda\) -closed map but not intuitionistic fuzzy pre closed map.

    Remark 3.10: IF \(\lambda\) -closed map and IF w-closed map are independent to each other as seen from the following example.

    Example 3.11: Let X = {a,b} and Y={u,v} and intuitionistic fuzzy sets U and V are defined as follows U = {\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.5,0.4\(>\)} and V = {\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.5, 0.2\(>\)}.

    Let ={0,1,U} and \(\sigma \) ={0,1,V} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, ) ? (Y,\(\ \sigma \) ) by f(a) =u and f(b) = v.

    Then f(U) =f(\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.4,0.5\(>\)} = (\(<\)u, 0.5, 0.5\(>\), \(<\)v, 0.4,0.5\(>\)} is intuitionistic fuzzy IF \(\lambda\)-closed set but not IF w-closed set Hence f is intuitionistic fuzzy \(\lambda\)-closed mapping but not intuitionistic fuzzy w-closed mappings.

    Example 3.12: Let X = {a,b} and Y={u,v} and intuitionistic fuzzy sets U and V are defined as follows. U={\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.5,0.5 \(>\)} and V = {\(<\)u, 0.5, 0.5\(>\), \(<\)v, 0.4, 0.6 \(>\)}.

    Let ={0,1,U} and \(\sigma \) ={0,1,V} be intuitionistic fuzzy topologies on X and Y respectively. Define a map f:(X, ) ? (Y,\(\ \sigma \) ) by f(a) =u and f(b) = v .

    f(U)=f{( \(<\) a,0.5,0.5\(>\),\(<\)b, 0.5, 0.5\(>\))}= {\(<\)u,0.5,0.5\(>\),\(<\)v, 0.5, 0.5\(>\) } is IF w-closed set not IF \(\lambda\)-closed set. Hence f is intuitionistic fuzzy w- closed mapping but not intuitionistic fuzzy \(\lambda\) - closed mappings.

    Remark 3.13: Intuitionistic fuzzy g-closed mappings and Intuitionistic fuzzy \(\lambda\) - closed mappings are independent as seen from the following examples.

    Example 3.14: Let X = {a, b}, Y={u, v} and intuitionistic fuzzy sets U and V are defined as follows. U={\(<\)a,0.5,0.5\(>\),\(<\)b,0.6,0.3}, V={\(<\)a,0.5,0.5\(>\),\(<\)b,0.2,0.6}.

    Let \(\tau \) = {\(\mathop{0}\limits_{\sim } \),\(\mathop{1}\limits_{\sim } \), U } and \(\sigma \) ={\(\mathop{0}\limits_{\sim } \) \(\mathop{1}\limits_{\sim } \), V} be intuitionistic fuzzy topologies on X and Y respectively.Define map: f: (X,\(\tau \)) ? (Y,\(\sigma \)) by f(a)=u and f(b)=v then f{(\(<\)a,0.5,0.5\(>\),\(<\)b,0.3,0.6\(>\)}= {\(<\)a,0.5,0.5\(>\),\(<\)b,0.3,0.6 } is intuitionistic fuzzy g-closed set but not intuitionistic fuzzy –\(\lambda\) closed set. Hence f is intuitionistic fuzzy g-closed mapping and not intuitionistic fuzzy \(\lambda\)- mapping.

    Example 3.15 : Let X X= {a,b } and Y={u, v} and intuitionistic fuzzy sets U and V are defined as follows U={\(<\)a,0.5,0.5\(>\),\(<\)b,0.5, 0.2 \(>\)} and V={\(<\)a,0.5,0.5\(>\), \(<\)b,0.5,0.4\(>\)}. Let \(\tau \)= { \(\mathop{0}\limits_{\sim } \) \(\mathop{1}\limits_{\sim } \) U} and \(\sigma \) = { \(\mathop{0}\limits_{\sim } \) \(\mathop{1}\limits_{\sim } \) ,V} be intuitionistic fuzzy topologies on X and Y respectively. Define the map f : X (X,\(\tau \)) \(?\) (Y,\(\sigma \)) by f(a)=u and f(b)=v. Then f{(\(<\)a,0.5,0.5\(>\), \(<\)b,0.2, 0.5 \(>\)} = {\(<\)a,0.5,0.5\(>\), \(<\)b,0.2, 0.5 \(>\)} is intuitionistic fuzzy \(\lambda\)-closed set but not intuitionistic fuzzy g closed set. Hence f is intuitionistic fuzzy \(\lambda\)- mappings and not intuitionistic fuzzy g-closed mappings.