Authorea

\(\lambda\)- 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 ([email protected])

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

(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

cl (A\({}^{C}\)) = [int (A)]\({}^{C}\),
int (A\({}^{C}\)) = [cl (A)]\({}^{C}\),
A is an intuitionistic fuzzy closed set in X \(\Leftrightarrow\) Cl (A) = A
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.

Remark 3.16: The concept of intuitionistic fuzzy \(\lambda\)- closed mappings and intuitionistic fuzzy semi closed mappings are independent as seen from the following examples.

Example 3.17: 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 .4\(>\)},V= {\(<\)u, 0.8,0.2 \(>\), \(<\)v, 0.1, 0 .9 \(>\)}. 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,\(\tau \)) \(?\) (Y,\(\sigma \)) by f(a)=u and f(b)=v then the mapping f(\(<\)a, 0. 5, 0 .5\(>\), \(<\)b, 0.4, 0 .6\(>\))={\(<\)u, 0. 5, 0 .5 \(>\), \(<\)v, 0.4, 0 .6\(>\)} is intuitionistic fuzzy semi closed set and not intuitionistic fuzzy \(\lambda\) - closed set . Hence f is intuitionistic fuzzy semi closed mapping but not intuitionistic fuzzy \(\lambda\) - closed.

Example 3.18: 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, 0.5, 0.2\(>\)}, 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,\(\tau \)) \(?\) (Y,\(\sigma \)) by f(a)=x and f(b)=y then f(U) =f{(\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.2, 0.5 \(>\))} ={(\(<\)u, 0.5, 0.5\(>\), \(<\)v, 0.2, 0.5 \(>\))} is intuitionistic fuzzy \(\lambda\)- closed set but not intuitionistic fuzzy semi closed set.

Hence f is intuitionistic fuzzy \(\lambda\)-closed mappings but not intuitionistic fuzzy semi closed mapping.

Remark 3.19: The concept of intuitionistic fuzzy \(\lambda\)- closed mappings and intuitionistic fuzzy semi pre closed mappings are independent as seen from the following examples.

Example 3.20: 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, 0.3, 0.5 \(>\)}, V= {u, 0 .5, 0.5\(>\), \(<\)v, 0.5, 0 .3 \(>\)}.

Let ={\(\mathop{0}\limits_{\sim } \), \(\mathop{1}\limits_{\sim } \) ,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,\(\tau \)) \(?\) (Y,\(\sigma \)) by f(a)=u and f(b)=v then f(\(<\)a, 0.5, 0.5\(>\), \(<\)b, 0.5, 0.3 \(>\)} =\(<\)u, 0.5, 0.5\(>\), \(<\)v, 0.5, 0.3 \(>\)} is intuitionistic fuzzy \(\lambda\)-closed set, but not intuitionistic fuzzy semi pre closed set. Then f is intuitionistic fuzzy \(\lambda\)-closed mapping, but not intuitionistic fuzzy semi pre closed mapping.

Example 3.21: 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. 4, 0 .6\(>\)} and V = {\(<\)u, 0.2,0.8 \(>\), \(<\)v, 0 .1, 0 .9\(>\)}.

Then f ({\(<\)a, 0. 5, 0 .5 \(>\), \(<\)b, 0. 4, 0 .6\(>\)})={\(<\)u, 0. 5, 0 .5 \(>\), \(<\)v, 0. 4, 0 .6\(>\)} is intuitionistic fuzzy semi pre closed mapping but not intuitionistic fuzzy \(\lambda\)-closed mapping.

Remark 3.22: IF G\(\alpha \)-closed mapping and IF \(\lambda \)-closed mappings are independent to each other.

Example 3.23: Let X = {a,b} and Y={u,v }and intuitionistic fuzzy sets U and V are defined as follows; U= {\(<\) a, 0.4 ,0.6 \(>\), \(<\) b, 0.3,0.7 \(>\) },V={\(<\) a,0.2,0.8 \(>\),\(<\)b, 0.3, 0.7 \(>\) }

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({\(<\)a,0.6,0.4\(>\), \(<\)b,0.7,0.3\(>\)}) = {{\(<\)a,0.6,0.4\(>\), \(<\)b,0.7,0.3\(>\)}) is IF G\(\alpha \)-closed set but not IF-closed set. Hence f is intuitionistic fuzzy \(\lambda\)-mapping but not intuitionistic fuzzy IF G\(\alpha \)-closed mapping.

Example 3.24 : Let X = {a,b} and Y={u,v }and intuitionistic fuzzy sets U and V are defined as follows; U= {\(<\) a, 0.1,0.9\(>\), \(<\) b, 0.3,0.7\(>\)}.V={\(<\) a,0.8,0.2\(>\), \(<\)b, 08, 0.1\(>\)}

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({\(<\)a,0.9,0.1 \(>\), \(<\)b,0.7,0.3 \(>\)}) = {\(<\)a,0.9,0.1 \(>\), \(<\)b,0.7,0.3 \(>\)})\({\rm \ IF\ }{\rm -}{\rm closed\ set}\) in Y but is not \(IF{\rm G}\alpha {\rm closed\ set}\). Hence f is intuitionistic fuzzy \(\lambda\)- mapping but not intuitionistic fuzzy\(\ IF{\rm G}\alpha {\rm closed}\) mapping.

Remark 3.25: IF\({\rm G}\alpha \)-closed mapping \({\rm and\ }\)IF \(\lambda\)-closed mapping are independent to each other as seen from the following example:

Example 3.26: Let X = {a,b} and Y={u,v }and intuitionistic fuzzy sets U and V are defined as follows; U= {\(<\) a, 0.6,0.4 \(>\), \(<\) b, 0.7,0.2 \(>\) }.V={\(<\)u, 0.2, 0.6 \(>\),\(<\)b, 0.2, 0.7 \(>\) }

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,\(\sigma \)) ? (Y,\(\ \sigma \) ) by f(a) =u and f(b) = v

Then f (U)=f({\(<\) a, 0.6,0.4\(>\),\(<\)b, 0.7,0.2 \(>\)}={\(<\)a,0.6,0.4\(>\),\(<\)b,0.7,0.2 \(>\)}in\({\rm \ }\alpha {\rm G}{\rm \ closed\ set\ }\)Y.but not \(\lambda\) -closed set in Y . Hence f is intuitionistic fuzzy IF\(\alpha {\rm G}\)-closed mapping, but not IF\(\lambda\)-closed mapping.

Example 3.27: Let X = {a,b} and Y={u,v }and intuitionistic fuzzy sets U and V are defined as follows; U= {\(<\) a, 0.2,0.8 \(>\), \(<\) b, 0.3,0.7 \(>\) }.V={\(<\)u, 0.6, 0.4 \(>\),\(<\)b, 0.7, 0.3 \(>\) }

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, 0.8,0.2\(>\),\(<\)b, 0.7,0.3 \(>\)}={\(<\)a,0.8,0.2\(>\),\(<\)b,0.7,0.3 \(>\)}is\({\rm \ }\)

\({\rm \ }{\rm -}{\rm closed\ set\ }\)Y, but not \(\alpha {\rm G}\) -closed set in Y . Hence f is intuitionistic fuzzy \(\lambda\)-closed mapping but not intuitionistic fuzzy \(\alpha {\rm G}{\rm \ closed\ mappings}\) .

Remark 3.28: From above examples and remarks we get following diagram of implications.

In this diagram A B means that A implies B.

A B means that B does not implies A.

A B means that A and B are independent to each other.

Theorem 3.29: Let f: X ? Y be a mapping. Then the following are equivalent if Y is an

IF\(\lambda\)- T\({}_{1/2 }\)space.

(i) f is an IF\(\lambda\)- closed map

(ii) \(\lambda\)- cl (f(A)) \(\subseteq \) f(cl (A)) for each IFS A of X.

Proof: (i) \(\Rightarrow \) (ii) Let A be an IFS in X. Then cl(A) is an IFCS in X. (i) implies that f(cl(A)) is an IF\(\lambda\)-CS in Y. Since Y is an IF\(\lambda\)-T\({}_{1/2 }\)space, f(cl(A)) is an IFCS in Y. Therefore \(\lambda\)-cl(f(cl (A))) = f(cl(A)). Now\(\lambda\)- cl (f(A)) \(\subseteq \)\(\lambda\)-cl (f(cl (A))) = f(cl (A)).

Hence \(\lambda\)-cl(f(A))\(\ \subseteq \) f(cl(A)) for each IFS A of X.

(ii)\(\Rightarrow \) (i) Let A be any IFCS in X. Then cl(A) = A. (ii) implies that

\(\lambda\)-cl(f(A)) \(\subseteq \) f(cl(A)) = f(A).But f(A) \(\subseteq \)\(\lambda\)- cl(f(A)). Therefore\(\lambda\)- cl(f(A)) = f(A). This implies f(A) is an IF\(\lambda\)-CS inY. Since every IF\(\lambda\)-CS is an IFCS, f(A) is an IF\(\lambda\)- CS in Y. Hence f is an IF\(\lambda\)- closed map.

Theorem 3.30: Let f : X ? Y be a bijection. Then the following are equivalent if Y is an

IF\(\lambda\)-T\({}_{1/2}\) space

(i) f is an IF\(\lambda\)-closed map

(ii) \(\lambda\)- cl(f(A)) \(\subseteq \) f(cl(A)) for each IFS A of X

(iii) f \({}^{-1}\)(\(\lambda\)-cl(B)) \(\subseteq \) cl(f \({}^{-1}\)(B)) for every IFS B of Y.

Proof: (i) \(\Leftrightarrow \) (ii) is obvious from Theorem 3.15.

(ii) \(\Rightarrow \) (iii) Let B be an IFS in Y. Then f \({}^{-1}\)(B) is an IFS in X. Since f is onto,

\(\lambda\)-cl(B) = \(\lambda\)-cl(f(f\({}^{-1}\) (B))) and (ii) implies \(\lambda\)-cl(f(f \({}^{-1}\)(B))) \(\subseteq \) f(cl(f \({}^{-1}\)(B))). Therefore

\(\lambda\)-cl(B) \(\subseteq \) f(cl(f \({}^{-1}\)(B))). Now f \({}^{-1}\)(\(\lambda\)-cl(B)) \(\subseteq \) f \({}^{-1}\)(f(cl(f \({}^{-1}\)(B))). Since f is one to one,

f \({}^{-1}\)(\(\lambda\)-cl(B)) \(\subseteq \) cl(f \({}^{-1}\)(B)).

(iii) \(\Rightarrow \) (ii) Let A be any IFS of X. Then f(A) is an IFS of Y. Since f is one to one,

(iii) implies that f \({}^{-1}\)(\(\lambda\)-cl(f(A)) \(\subseteq \) cl(f \({}^{-1}\)(A)) = cl(A). Therefore f (f \({}^{-1}\)(\(\lambda\)-cl(f(A)))) \(\subseteq \)f(cl(A)). Since f is onto .\(\lambda\)-cl(f(A)) = f(f \({}^{-1}\)(\(\lambda\)-cl(f(A)))) \(\subseteq \) f(cl(A)).

Theorem 3.31: Let f : X ? Y be an IF\(\lambda\)-closed map. Then for every IFS A of X, f(cl(A)) is an

IF\(\lambda\)-CS in Y.

Proof: Let A be any IFS in X. Then cl(A) is an IFCS in X. By hypothesis, f(cl(A)) is an

IF\(\lambda\)-CS in X.

Theorem 3.32: Let f : X ? Y be an IF\(\lambda\)-closed map where Y is an IF\(\lambda\)-T\({}_{1/2}\) space. Then f is a

IF regular closed map if every IF\(\lambda\)-CS is an IFRCS in Y.

Proof: Let A be an IFRCS in X. since every IFRCS is an IFCS[5], A is an IFCS in X. By hypothesis 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 hence is an IFCS in Y, by hypothesis. This implies that f(A) is an IF regular closed map.

Theorem 3.33 : If every IFS is an IFCS, then an IF\(\lambda\)-closed mapping is an IF \(\lambda\)- continuous mapping.

Proof: Let A be an IFCS in Y. Then f \({}^{-1}\)(A) is an IFS in X. Therefore f\({}^{--1}\) (A) is an IFCS in X. Since every IFCS is an IF\(\lambda\)-CS, f \({}^{-1}\)(A) is an IF\(\lambda\)-CS in X. This implies that f is an IF\(\lambda\)-continuous mapping.

Theorem 3.34 : A mapping f : X ? Y is an IF\(\lambda\)-closed mapping if and only if for every IFS B of Y and for every IFOS U containing f \({}^{-1}\)(B), there is an IF\(\lambda\)-OS A of Y such that B \(\subseteq \)A and

f \({}^{-1}\) (A) \(\subseteq \) U.

Proof: Necessity: Let B be any IFS in Y. Let U be an IFOS in X such that f \({}^{-1}\)(B) \(\subseteq \)U, then U\({}^{c}\)\({}^{ }\)is an IFCS in X. By hypothesis f(U\({}^{c}\)) is an IF\(\lambda\)-CS in Y. Let A = (f(U\({}^{c}\)))\({}^{c}\), then A is\({}^{ }\)an IF\(\lambda\)-OS in Y and B \(\subseteq \)A. Now f \({}^{-1}\)(A) = f \({}^{-1}\)(f(U\({}^{c}\)))\({}^{c}\) = (f \({}^{-1}\)(f(U\({}^{c}\))))\({}^{c}\) \(\subseteq \) U.

Sufficiency: Let A be any IFCS in X, then A\({}^{c}\) is an IFOS in X and f \({}^{-1}\)(f(A\({}^{c}\)))\({}^{c}\)\(\subseteq \) A\({}^{c}\). By hypothesis there exists an IF\(\lambda\)-OS B in Y such that f(A\({}^{c}\)) \(\subseteq \) B and f \({}^{-1}\)(B) \(\subseteq \) A\({}^{c}\). Therefore A \(\subseteq \) (f \({}^{-1}\)(B))\({}^{c}\). Hence B \({}^{c}\)\(\subseteq \) f(A) \(\subseteq \)f(f \({}^{-1}\)(B)) \({}^{c}\) \(\subseteq \) B \({}^{c}\). This implies that f(A) = B\({}^{c}\).Since B\({}^{ c}\) is an IF\(\lambda\)-CS in Y, f(A) is an IF\(\lambda\)-CS in Y. Hence f is an IF\(\lambda\)-closed mapping.

Theorem 3.35: If f: X ? Y is an IF closed map and g: Y? Z is an IF\(\lambda\)-closed map, then g \(o \) f is an IF\(\lambda\)- closed map.

Proof: Let A be an IFCS in X, then f(A) is an IFCS in Y, Since f is an IF closed map. Since g is an IF\(\lambda\)-closed map, g(f(A)) is an IF\(\lambda\)-CS in Z. Therefore g \(o \) f is an IF\(\lambda\)-closed map.

Theorem 3.36: Let f: X ? Y be a bijective map where Y is an IF\(\lambda\)-T\({}_{1/2}\) space. Then the following are equivalent.

(i) f is an IF\(\lambda\)-closed map.

(ii) f (B) is an IF\(\lambda\)-OS in Y for every IFOS B in X.

Proof: (i) \(\Leftrightarrow \) (ii) is obvious.

Definition 3.37: A mapping f: (X,\(\tau \)) \(?\) (Y,\(\sigma \)) is said to be intuitionistic fuzzy \(\lambda\)–open map (IF \(\lambda\)–open map) if f(V) is \(\lambda\)-open set in (Y,\({\rm \ }\sigma {\rm ),for\ every\ open\ set\ in\ (Y,}\sigma {\rm ).}\)

4. CONCLUSION In this paper we have introduced intuitionistic fuzzy \(\lambda\)- open mappings, intuitionistic fuzzy \(\lambda\)-closed mappings and studied some of their properties.

REFERENCES

[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 1986, 87-96.

[2] C. L. Chang, Fuzzy topological spaces, J.Math.Anal.Appl. 24, 1968, 182-190.

[3] D. Coker, An introduction to intuitionistic fuzzy topological space,Fuzzy Sets and Systems, 88, 1997, 81-89.

[4] H. Gurcay, Es. A. Haydar and D. Coker, On fuzzy continuity in intuitionistic fuzzy topological spaces, J.Fuzzy Math.5 \eqref{GrindEQ__2_}, 1997, 365-378.

[5] Joung Kon Jeon, Young Bae Jun, and Jin Han Park, Intuitionistic fuzzy alpha-continuity and intuitionistic fuzzy pre continuity, International Journal of Mathematics and

Mathematical Sciences, 2005, 3091-3101.

[6] R. Santhi and D. Jayanthi, Intuitionistic fuzzy generalized semi-preclosed sets (accepted).

[7] Pushpapalatha A Ph.D. thesis, “Studies on Generalizations of mappings in topological spaces” in Bharathiar University, Coimbatore.

[8] Rajarajeswari P and.Bagyalakshmi G ‘\(\lambda \)-closed sets in intuitionistic fuzzy topological space’ International Journal of Computer Applications. (0975-8887) Volume 34, No. 1, November 2011

[9] Rajarajeswari P and Bagyalakshmi G ‘\(\lambda \)-continuous in intuitionistic fuzzy topological space’ Foundation of Computer Science FCS, New York, USA Volume 1, No. 1, November 2012 International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868

[10] Rajarajeswari P and Bagyalakshmi G ‘Application of \(\lambda \)-continuous in intuitionistic fuzzy topological space’ Foundation of Computer Science FCS, New York, USA, ISSN (0975-887) Volume 94- No. May 2014.

[11] Santhi, R. and Sakthivel, K., Alpha generalized closed mappings in intuitionistic fuzzy topological spaces, Far East Journal of Mathematical Sciences, 43 (2010), 265-275.

[12] Santhi, R. and Sakthivel, K., Intuitionistic fuzzy generalized semi continuous Mappings, Advances in Theoretical and Applied Mathematics, 5(2010), 11-20.

[13] S.S. Thakur & Jyothi Pandy Bajpai, Intuitionistic fuzzy w-closed sets and Intuitionistic fuzzy w-continuity, International Journal of contemporary Advanced Mathematics (IJCM), volume \eqref{GrindEQ__1_}; Issue \eqref{GrindEQ__1_}

[14] Thakur, S. S and Rekha Chaturvedi, Regular generalized closed sets in intuitionistic fuzzy topological spaces, Universitatea Din Bacau,Studii Si Cetari Stiintifice, Seria: Matematica, 2006, 257-272.

[15] M. Thirumalaiswamy and K. M. Arifmohammed, Semi pre Generalized Closed Mappings in Intuitionistic Fuzzy Topological Spaces (submitted).

[16] M. Thirumalaiswamy and K. M. Arifmohammed, Semi pre Generalized Open Sets and Applications of Semipre Generalized Closed Sets in Intuitionistic Fuzzy Topological Spaces (submitted).

[17] Young Bae Jun and Seok- Zun Song, Intuitionistic fuzzy semi-pre open sets and Intuitionistic fuzzy semi-pre continuous mappings, Jour. Of Appl. Math & computing, 2005, 467-474.

[18] L. A. Zadeh, Fuzzy sets, Information and control, 8, 1965.

P.Rajarajeswari ([email protected]) – Department of Mathematics, Chikkanna Government Arts College, Tirupur - 641 602, Tamil Nadu, India

G.Bagyalakshmi ([email protected]) – Department of Mathematics, AJK College of Arts and Science, Coimbatore -641105. Tamil Nadu, India