Notes on Interval Valued Fuzzy RW-Closed, Interval Valued Fuzzy RW-Open Sets in Interval Valued Fuzzy Topological Spaces

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 3, Number 1 (2013), pp Research India Publications Notes on Interval Valued Fuzzy RW-Closed, Interval Valued Fuzzy RW-Open Sets in Interval Valued Fuzzy Topological Spaces R. Indira, K. Arjunan and N. Palaniappan 1Department of Mathematics, T.K. Government ARTS College, Vrithachalam, Tamilnadu, India. ganesan.natesh@gmail.com 2.Department of Mathematics, H.H. The Rajahs College, Pudukkottai , Tamilnadu, India. arjunan_1975@yahoo.co.in & arjunan.karmegam@gmail.com 3Professor of Mathematics, Alagappa University, Karaikudi , Tamilnadu, India. palaniappan.nallappan@gmail.com ABSTRACT In this paper, we study some of the properties of interval valued fuzzy rw-closed and interval valued fuzzy rw-open sets in interval valued fuzzy topological spaces and prove some results on these.note interval valued is denoted as (i, v) AMS SUBJECT CLASSIFICATION: 03F55, 08A72, 20N25. KEY WORDS: (i, v)-fuzzy subset, (i, v)-fuzzy topological spaces, (i, v)-fuzzy rw-closed, (i, v)-fuzzy rw-open. INTRODUCTION The concept of a fuzzy subset was introduced and studied by L.A.Zadeh [20] in the year The subsequent research activities in this area and related areas have found applications in many branches of science and engineering. The following papers have motivated us to work on this paper. C.L.Chang [4] introduced and studied fuzzy topological spaces in 1968 as a generalization of topological spaces. Many researchers like R.H.Warren [19], K.K.Azad [1], G.Balasubramanian and P.Sundaram [2, 3], S.R.Malghan and S.S.Benchalli [12,

2 24 R. Indira, K. Arjunan and N. Palaniappan 13] and many others have contributed to the development of fuzzy topological spaces. Tapas kumar mondal and S.K.Samanta [17] have introduced the topology of interval valued fuzzy sets. We introduce the concept of interval valued fuzzy rw-closed and interval valued fuzzy rw-open sets in interval valued fuzzy topological spaces and established some results. 1.PRELIMINARIES: 1.1 Definition: An interval number a on [0, 1] is a closed subinterval of [0, 1], that is, a = [a, a ] such that 0 a a 1 where a and a are the lower and upper end points of a respectively. Note that 0 = [0, 0] and 1 = [1, 1]. 1.2 Definition: For any interval numbers a = [a, a ] and b = [b, b ] on [0, 1], we define a b if and only if a b and a b, a = b if and only if a = b and a = b, a b if and only if a b and a b, (iv) ka = [ka, ka ], whenever 0 k 1, (v) (vi) a b = [min{ a, b }, min {a, b }], a b = [max{ a, b }, max {a, b }], (vii) (a ) c = 1 a = [1 a, 1 a ]. 1.3 Definition: [17] Let X be any nonempty set. A mapping A : X D[0, 1] is called an interval valued fuzzy subset (briefly, (i, v)-fuzzy subset ) of X, where D[0, 1] denotes the family of all closed subintervals of [0, 1] and A (x) = [A (x), A (x)] for all x X, where A and A are fuzzy subsets of X such that A (x) A (x) for all x X. Thus A (x) is an interval ( a closed subset of [0, 1] ) and not a number from the interval [0, 1] as in the case of fuzzy subset. 1.1 Remark: Let D X be the set of all interval valued fuzzy subset of X. 1.4 Definition: Let A ={ x, A (x) / x X }, B = { x, B (x) / x X } D X. We define the following relations and operations: A B if and only if A (x) B (x), for all x in X. A = B if and only if A (x) = B (x), for all x in X. (Ā) c = 1 A = { x, 1 A (x) } / x X }. (iv) A B = { x, min{ A (x), B (x) } / x X }.

3 Notes on Interval Valued Fuzzy RW-Closed 25 (v) A B = { x, max { A (x), B (x)} / x X }. 1.1 Theorem: [17] Let A, B, C D X. The following results hold good. 0 A 1, A B = B A, A B = B A, A ( B C )= ( A B ) ( A C), (iv) A ( B C ) = ( A B ) ( A C ), (v) A 1 = A, A 1 = 1, (vi) 1 ( A B ) = ( 1 A ) ( 1 B ), (vii) 1 ( A B ) = ( 1 A ) ( 1 B ) (viii) A B = A ( 1 B ) (ix) A 0 = 0, A 0 = A. 1.5 Definition: [17] Let X be a set and be a family of (i, v)-fuzzy subsets of X. The family is called an (i, v)-fuzzy topology on X if and only if satisfies the following axioms 0, 1, If { A i ; i I}, then i I A, i If A 1, A 2, A 3,.. A n, then i n i 1 A. i The pair ( X, ) is called an (i, v)-fuzzy topological space. The members of are called (i, v)-fuzzy open sets in X. An (i, v)-fuzzy set A in X is said to be (i, v)-fuzzy closed set in X if and only if ( A) c is an (i, v)-fuzzy open set in X. 1.6 Definition: Let (X, ) be an (i, v)-fuzzy topological space and A be an (i, v)-fuzzy set in X. Then { B :( B) c and B A } is called (i, v)- fuzzy closure of A and is denoted by cl( A). 1.2 Theorem: Let A and B be two (i, v)-fuzzy sets in (i, v)-fuzzy topological space (X, ). Then the following results are true, (iv) cl( A) is an (i, v)-fuzzy closed set in X cl( A) is the least (i, v)-fuzzy closed set containing A A is an (i, v)-fuzzy closed if and only if A = cl( A) cl( 0 ) = 0, 0 is the empty (i, v)-fuzzy set

4 26 R. Indira, K. Arjunan and N. Palaniappan (v) (vi) cl( cl( A) ) = cl( A) cl( A B ) = cl( A) cl( B) (vii) cl( A) cl( B) cl( A B ) 1.7 Definition: Let (X, ) be an (i, v)-fuzzy topological space and A be an (i, v)-fuzzy set in X. Then { B : B and B A } is called (i, v)-fuzzy interior of A and is denoted by int( A). 1.3 Theorem: Let (X, ) be an (i, v)-fuzzy topological space, A and B be two (i, v)-fuzzy sets in X. The following results hold good, (iv) (v) (vi) int ( A) is an (i, v)-fuzzy open set in X int( A) is the largest (i, v)-fuzzy open set in X which is less than or equal to A. A is an (i, v)-fuzzy open set if and only if A =int ( A) A B implies int ( A) int( B) int (int ( A) ) = A int( A B ) = int( A) int( B) (vii) int( A) int( B) int( A B ) (viii) int( 1 A ) = 1 cl( A) (ix) cl( 1 A)= 1 int( A). 1.8 Definition: Let (X, ) be an (i, v)-fuzzy topological space and A be (i, v)-fuzzy set in X. Then A is said to be (iv) (v) (vi) (i, v)-fuzzy semiopen if and only if there exists an (i, v)-fuzzy open set V in X such that V A cl( V). (i, v)-fuzzy semiclosed if and only if there exists an (i, v)-fuzzy closed set V in X such that int( V) A V. (i, v)-fuzzy regular open set of X if int( cl( A) ) = A. (i, v)-fuzzy regular closed set of X if cl( int( A) ) = A. (i, v)-fuzzy regular semiopen set of X if there exists an (i, v)-fuzzy regular open set V in X such that V A cl( V). We denote the class of (i, v)-fuzzy regular semiopen sets in (i, v)-fuzzy topological space X by IVFRSO(X). (i, v)-fuzzy generalized closed (ivfg-closed) if cl( A) V whenever A V and V is (i, v)-fuzzy open set and A is (i, v)-fuzzy generalized open if 1 A is (i, v)-fuzzy generalized closed.

5 Notes on Interval Valued Fuzzy RW-Closed Theorem: The following are equivalent: A is an (i, v)-fuzzy semiclosed set, A C is an (i, v)-fuzzy semiopen set, (iv) int( cl( A) ) A cl( int( A C ) ) A 1.5 Theorem: Any union of (i, v)-fuzzy semiopen sets is an (i, v)-fuzzy semiopen set and any intersection of (i, v)-fuzzy semiclosed sets is an (i, v)- fuzzy semiclosed. 1.2 Remark: Every (i, v)-fuzzy open set is an (i, v)-fuzzy semiopen but not conversely. (iv) Every (i, v)-fuzzy closed set is an (i, v)-fuzzy semi-closed set but not conversely. The closure of an (i, v)-fuzzy open set is (i, v)-fuzzy semiopen set. The interior of an (i, v)-fuzzy closed set is (i, v)-fuzzy semi-closed set. 1.6 Theorem: An (i, v)-fuzzy set A of an (i, v)-fuzzy topological space X is an (i, v)-fuzzy regular open if and only if A C is (i, v)-fuzzy regular closed set. 1.3 Remark: Every (i, v)-fuzzy regular open set is an (i, v)-fuzzy open set but not conversely. Every (i, v)-fuzzy regular closed set is an (i, v)-fuzzy closed set but not conversely. 1.7 Theorem: The closure of an (i, v)-fuzzy open set is an (i, v)-fuzzy regular closed. The interior of an (i, v)-fuzzy closed set is an (i, v)-fuzzy regular open set. 1.8 Theorem: Every (i, v)-fuzzy regular semiopen set is an (i, v)-fuzzy semiopen set but not conversely. Every (i, v)-fuzzy regular closed set is an (i, v)-fuzzy regular semiopen set but not conversely. Every (i, v)-fuzzy regular open set is an (i, v)-fuzzy regular semiopen set but not conversely. 1.9 Theorem: Let (X, ) be an (i, v)-fuzzy topological space and A be (i, v)- fuzzy set in X. Then the following conditions are equivalent:

6 28 R. Indira, K. Arjunan and N. Palaniappan A is (i, v)-fuzzy regular semiopen A is both (i, v)-fuzzy semiopen and (i, v)-fuzzy semi-closed. A c is (i, v)-fuzzy regular semiopen in X. 1.9 Definition: An (i, v)-fuzzy set A of an (i, v)-fuzzy topological space ( X, ) is called: (i, v)-fuzzy g-closed if cl( A) V whenever A V and V is (i, v)- fuzzy open set in X. (i, v)-fuzzy g-open if its complement A c is (i, v)-fuzzy g-closed set in X. (i, v)-fuzzy rg-closed if cl( A) V whenever A V and V is (i, v)- fuzzy regular open set in X. (iv) (i, v)-fuzzy rg-open if its complement A c is (i, v)-fuzzy rg-closed set in X. (v) (i, v)-fuzzy w-closed if cl( A) V whenever A V and V is (i, v)- fuzzy semi open set in X. (vi) (i, v)-fuzzy w-open if its complement A c is (i, v)-fuzzy w-closed set in X. (vii) (i, v)-fuzzy gpr-closed if pcl( A) V whenever A V and V is (i, v)-fuzzy regular open set in X. (viii) (i, v)-fuzzy gpr -open if its complement A c is (i, v)-fuzzy gpr-closed set in X Definition: Let (X, ) be an (i, v)-fuzzy topological space. An (i, v)- fuzzy set A of X is called (i, v)-fuzzy regular w-clsoed(briefly, (i, v)-fuzzy rw-closed ) if cl( A) U whenever A U and U is (i, v)-fuzzy regular semiopen in (i, v)-fuzzy topological space X. NOTE: We denote the family of all (i, v)-fuzzy regular w-closed sets in (i, v)-fuzzy topological space X by IVFRWC(X) Definition: An (i, v)-fuzzy set A of an (i, v)-fuzzy topological space X is called an (i, v)-fuzzy regular w-open (briefly, (i, v)-fuzzy rw-open) set if its complement A C is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. NOTE: We denote the family of all (i, v)-fuzzy rw-open sets in (i, v)-fuzzy topological space X by IVFRWO(X).

7 Notes on Interval Valued Fuzzy RW-Closed SOME PROPERTIES: 2.1 Theorem: Every (i, v)-fuzzy closed set is an (i, v)-fuzzy rw-closed set in an (i, v)-fuzzy topological space X. Proof: Let A be an (i, v)-fuzzy closed set in an (i, v)-fuzzy topological space X. Let B be an (i, v)-fuzzy regular semiopen set in X such that A B. Since A is (i, v)-fuzzy closed, cl( A) = A. Therefore cl( A) = A B. Hence A is (i, v)-fuzzy rw-closed in (i, v)-fuzzy topological space X. Remark: The converse of the above Theorem need not be true in general. 2.1 Example: Let X = { 1, 2, 3 }. Define an (i, v)-fuzzy set A = { < 1, [0.6, 0.6] >, < 2, [0, 0] >, < 3, [0, 0] > }. Let ={ 1, 0, A}. Then (X, ) is an (i, v)-fuzzy topological space. Define an (i, v)-fuzzy set B = { <1, [0, 0]>, <2, [0.6, 0.6]>, < 3, [0, 0] > }. Then B is an (i, v)-fuzzy rw-closed set but it is not an (i, v)-fuzzy closed set in (i, v)-fuzzy topological space X. Remark: (i, v)-fuzzy generalized closed sets and (i, v)-fuzzy rw-closed sets are independent. 2.2 Example: Let X = { 1, 2, 3, 4 }. Define (i, v)-fuzzy sets A, B, C in X by A = { < 1, [1, 1]>, <2, [0, 0]>, <3, [0, 0]>, <4, [0, 0]> }, B = { <1, [0, 0]>, <2, [1, 1]>, <3, [0, 0]>, <4, [0, 0]> }, C = { <1, [1, 1]>, <2, [1, 1]>, <3, [0, 0]>, <4, [0, 0]> }. Consider ={ 0, 1, A, B, C }. Then (X, ) is an (i, v)-fuzzy topological space. In this (i, v)-fuzzy topological space X, the (i, v)-fuzzy set D : X D[0, 1] define by D = { <1, [0, 0]>, <2, [0, 0]>, <3, [1, 1]>, <4, [0, 0]> }. Then D is an (i, v)-fuzzy generalized closed set in (i, v)-fuzzy topological space X. In this (i, v)-fuzzy topological space, the (i, v)-fuzzy set E : X D[0, 1] define by E = { <1, [1, 1]>, <2, [0, 0] >, <3, [1, 1]>, <4, [0, 0]> }. Then E is an (i, v)-fuzzy regular semiopen set containing D, but E does not contain cl( D) which is C C. Therefore E is not an (i, v)-fuzzy rw-clsoed set in (i, v)-fuzzy topological space X. 2.3 Example: Let X = I = [0, 1]. Define an (i, v)-fuzzy set D in X by D(x) = [0.5, 0.5] if x = 2/3 [0, 0] otherwise. Let = { 0, 1, D }. Then (X, ) is an (i, v)-fuzzy topological space. Let A(x) = [0.3, 0.3] if x = 2/3 [0, 0] otherwise. Then A is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. Now cl( A) = D C and D is an (i, v)-fuzzy open set containing A but D does not contain cl( A)

8 30 R. Indira, K. Arjunan and N. Palaniappan which is D C. Therefore A is not an (i, v)-fuzzy generalized closed. Remark: (i, v)-fuzzy rw-closed sets and (i, v)-fuzzy semi-closed sets are independent. 2.4 Example: consider the (i, v)-fuzzy topological space (X, ) defined in Example 2.1. Then the (i, v)-fuzzy set A = { < 1, [ 1, 1 ] >, < 2, [0, 0] >, < 3, [0, 0] > } is an (i, v)-fuzzy rw-closed but it is not an (i, v)-fuzzy semiclosed set in (i, v)-fuzzy topological space X. 2.5 Example: Consider the (i, v)-fuzzy topological space (X, ) defined in Example 2.2. In this (i, v)-fuzzy topological space X, the (i, v)-fuzzy set μ :X D[0, 1] is define by μ = {< 1, [1, 1]>, <2, [0, 0]>, <3, [1, 1]>, <4, [0, 0]>}. Then μ is an (i, v)-fuzzy semi-closed in (i, v)-fuzzy topological space X. μ is also (i, v)-fuzzy regular semiopen set containing μ which is does not contain cl( μ) = B c ={<1, [1, 1]>, <2, [0, 0]>, <3, [1, 1]>, <4, [1, 1]>}. Therefore μ is not an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. 2.2 Theorem: Every (i, v)-fuzzy w-closed set is (i, v)-fuzzy rw-closed. Proof: The proof follows from the Definition 1.10 and the fact that every (i, v)-fuzzy regular semi open set is (i, v)-fuzzy semi open. Remark: The converse of Theorem 2.2 need not be true as from the following example. 2.6 Example: Let X = { 1, 2 } and = { 0, 1, A } be an (i, v)-fuzzy topology on X, where A = { < 1, [0.7, 0.7] >, < 2, [0.6, 0.6] > }. Then the (i, v)-fuzzy set B = { < 1, [0.7, 0.7] >, < 2, [0.8, 0.8] > } is (i, v)-fuzzy rw-closed but it is not (i, v)-fuzzy w-closed. 2.3 Theorem: Every (i, v)-fuzzy rw-closed set is (i, v)-fuzzy rg-closed. Proof: The proof follows from the Definition 1.10 and the fact that every (i, v)-fuzzy regular open set is (i, v)-fuzzy regular semi open. Remark: The converse of Theorem 2.3 need not be true as from the following example. 2.7 Example: Let X = {1, 2, 3, 4 } and (i, v)-fuzzy sets A, B, C, D defined as follows A = {<1, [0.9, 0.9]>, < 2, [0, 0] >, < 3, [0, 0] >, < 4, [0, 0] > }, B = { <1, [0, 0]>, < 2, [0.8, 0.8] >, < 3, [0, 0]>, < 4, [0, 0]

9 Notes on Interval Valued Fuzzy RW-Closed 31 >}, C = { < 1, [0.9, 0.9] >, < 2, [0.8, 0.8] >, <3, [0, 0]>, < 4, [0, 0]> }, D = { <1, [0.9, 0.9]>, <2, [0.8, 0.8]>, <3, [0.7, 0.7]>, <4, [0, 0]>}, = { 1, 0, A, B, C, D } be an (i, v)-fuzzy topology on X. Then the (i, v)- fuzzy set E = { < 1, [0, 0] >, < 2, [0, 0] >, < 3, [0.7, 0.7] >, < 4, [0, 0] > } is (i, v)-fuzzy rg-closed but it is not (i, v)-fuzzy rw-closed. 2.4 Theorem: Every (i, v)-fuzzy rw-closed set is (i, v)-fuzzy gpr-closed. Proof: Let A is an (i, v)-fuzzy rw closed set in (i, v)-fuzzy topological space ( X, ). Let A O, where O is (i, v)-fuzzy regular open in X. Since every (i, v)-fuzzy regular open set is (i, v)-fuzzy regular semi open and A is (i, v)- fuzzy rw-closed set, we have cl( A) O. Since every (i, v)-fuzzy closed set is (i, v)-fuzzy pre closed, pcl( A) cl( A). Hence pcl( A) O which implies that A is (i, v)-fuzzy gpr-closed. Remark: The converse of Theorem 2.4 need not be true as from the following example. 2.8 Example: Let X = { 1, 2, 3, 4, 5 } and (i, v)-fuzzy sets A, B, C defined as follows A = { < 1, [0.9, 0.9] >, < 2, [0.8, 0.8] >, < 3, [0, 0] >, < 4, [0, 0] >, < 5, [0, 0] > }, B = { < 1, [0, 0] >, < 2, [0, 0] >, < 3, [0.8, 0.8] >, < 4, [0.7, 0.7]>, < 5, [0, 0] > }, C = {< 1, [0.9, 0.9] >, < 2, [0.8, 0.8] >, < 3, [0.8, 0.8] >, < 4, [0.7, 0.7] >, < 5, [0, 0] > }. Let = { 1, 0, A, B, C } be an (i, v)-fuzzy topology on X. Then the (i, v)- fuzzy set D = { < 1, [0.9, 0.9] >, < 2, [0, 0] >, < 3, [0, 0] >, < 4, [0, 0] >, < 5, [0, 0] > } is (i, v)-fuzzy gpr-closed but it is not (i, v)-fuzzy rwclosed. 2.5 Theorem: If A is an (i, v)-fuzzy regular open and (i, v)-fuzzy rg-closed in (i, v)-fuzzy topological space (X, ), then A is (i, v)-fuzzy rw-closed in X. Proof: Let A is an (i, v)-fuzzy regular open and (i, v)-fuzzy rg-closed in X. We prove that A is an (i, v)-fuzzy rw-closed in X. Let U be any (i, v)-fuzzy regular semi open set in X such that A U. Since A is (i, v)-fuzzy regular open and (i, v)-fuzzy rg-closed, we have cl( A) A. Then cl( A) A U. Hence A is (i, v)-fuzzy rw-closed in X. 2.6 Theorem: If A and B are (i, v)-fuzzy rw-closed sets in (i, v)-fuzzy topological space X, then union of A and B is (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. Proof: Let C be an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X such that ( A B) C. Now A C and B C.

10 32 R. Indira, K. Arjunan and N. Palaniappan Since A and B are (i, v)-fuzzy rw-closed sets in (i, v)-fuzzy topological space X, cl( A) C and cl( B) C. Therefore(cl( A) cl( B)) C. But (cl( A) cl( B)) = cl( A B ). Thus cl( A B) C. Hence A B is an (i, v)- fuzzy rw-closed set in (i, v)-fuzzy topological space X. 2.7 Theorem: If A and B are (i, v)-fuzzy rw-closed sets in (i, v)-fuzzy topological space X, then the intersection of A and B need not be an (i, v)- fuzzy rw-closed set in (i, v)-fuzzy topological space X. Proof: Consider the (i, v)-fuzzy topological space (X, ) defined in Example 2.2. In this (i, v)-fuzzy topological space X, the (i, v)-fuzzy sets G 1, G 2 : X D[0, 1] are defined by G 1 = { < 1, [0, 0] >, < 2, [0, 0] >, < 3, [1, 1] >, < 4, [1, 1] > } and G 2 = { < 1, [1, 1] >, < 2, [1, 1] >, < 3, [1, 1] >, < 4, [0, 0] > }. Then G 1 and G 2 are the (i, v)-fuzzy rw-closed sets in (i, v)-fuzzy topological space X. Let D = G 1 G 2. Then D = {< 1, [0, 0]>, < 2, [0, 0]>, <3, [1, 1]>, <4, [0, 0]>}. Then D = G 1 G 2 is not an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. 2.8 Theorem: If an (i, v)-fuzzy subset A of (i, v)-fuzzy topological space X is both (i, v)-fuzzy regular open and (i, v)-fuzzy rw-closed, then A is an (i, v)-fuzzy regular closed set in (i, v)-fuzzy topological space X. Proof: Suppose an (i, v)-fuzzy subset A of (i, v)-fuzzy topological space X is both (i, v)-fuzzy regular open and (i, v)-fuzzy rw-closed. As every (i, v)-fuzzy regular open set is an (i, v)-fuzzy regular semiopen set and A A, we have cl( A) A. Also A cl( A). Therefore cl( A) = A. That is A (i, v)-fuzzy closed. Since A is (i, v)-fuzzy regular open, int ( A ) = A. Now cl( int ( A ) ) = cl( A ) = A. Therefore A is an (i, v)-fuzzy regular closed set in (i, v)- fuzzy topological space X. 2.9 Theorem: If an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is both (i, v)-fuzzy regular semiopen and (i, v)-fuzzy rw-closed, then A is an (i, v)-fuzzy closed set in (i, v)-fuzzy topological space X. Proof: Suppose an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is both (i, v)-fuzzy regular semiopen and (i, v)-fuzzy rw-closed. Now A A, we have cl( A) A. Also A cl( A). Therefore cl( A) = A and hence A is an (i, v)-fuzzy closed set in (i, v)-fuzzy topological space X. 2.1 Corollary: If A is an (i, v)-fuzzy regular semi open and (i, v)-fuzzy rwclosed in (i, v)-fuzzy topological space (X, ). Suppose that F is (i, v)-fuzzy closed in X then A F is (i, v)-fuzzy rw-closed in X.

11 Notes on Interval Valued Fuzzy RW-Closed 33 Proof: Suppose A is both (i, v)-fuzzy regular semi open and (i, v)-fuzzy rwclosed set in X and F is (i, v)-fuzzy closed in X. By Theorem 2.9, A is (i, v)-fuzzy closed in X. So A F is (i, v)-fuzzy closed in X. Hence A F is (i, v)-fuzzy rw-closed in X Theorem: If an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is both (i, v)-fuzzy open and (i, v)-fuzzy generalized closed, then A is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. Proof: Suppose an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is both (i, v)-fuzzy open and (i, v)-fuzzy generalized closed. Now A A, by hypothesis we have cl( A) A. Also A cl( A). Therefore cl( A) = A. That is A is an (i, v)-fuzzy closed set and hence A is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X, as every (i, v)-fuzzy closed set is an (i, v)-fuzzy rw-closed set Theorem: If an (i, v)-fuzzy subset C is both (i, v)-fuzzy regular open and (i, v)-fuzzy rw-closed set in an (i, v)-fuzzy topological space X, then C need not be an (i, v)-fuzzy generalized closed set in (i, v)-fuzzy topological space X. Proof: Consider the example, let X ={1, 2, 3} and the (i, v)-fuzzy sets A, B, C: X D[0, 1] be defined as A = { <1, [1, 1]>, <2, [0, 0]>, <3, [0, 0]> }, B = { <1, [0, 0]>, <2, [1, 1]>, <3, [0, 0]>} and C = { < 1, [1, 1] >, < 2, [1, 1] >, < 3, [0, 0] > }. Consider = { 0, 1, A, B, C }. Then (X, ) is an (i, v)-fuzzy topological space. In this (i, v)-fuzzy topological space X, C is both (i, v)-fuzzy open and (i, v)-fuzzy rw-closed set in (i, v)- fuzzy topological space X but it is not (i, v)-fuzzy generalized closed Theorem: Let A be an (i, v)-fuzzy rw-closed set of an (i, v)-fuzzy topological space X and suppose A B cl( A). Then B is also an (i, v)- fuzzy rw-closed set in (i, v)-fuzzy topological space X. Proof: Let A B cl( A) and A be an (i, v)-fuzzy rw-closed set of (i, v)- fuzzy topological space X. Let E be any (i, v)-fuzzy regular semiopen set such that B E. Then A E and A is (i, v)-fuzzy rw-closed, we have cl( A) E. But cl( B) cl( A) and thus cl( B) E. Hence B is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X Theorem: In an (i, v)-fuzzy topological space X if IVFRSO(X) = { 0, 1 }, where IVFRSO(X) is the family of all (i, v)-fuzzy regular semiopen sets then every (i, v)-fuzzy subset of X is (i, v)-fuzzy rw-closed.

12 34 R. Indira, K. Arjunan and N. Palaniappan Proof: Let X be an (i, v)-fuzzy topological space and IVFRSO(X) = { 0, 1, }. Let A be any (i, v)-fuzzy subset of X. Suppose A = 0. Then 0 is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. Suppose A 0. Then 1 is the only (i, v)-fuzzy regular semiopen set containing A and so cl( A) 1. Hence A is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. Remark: The converse of the above Theorem 2.13 need not be true in general. 2.9 Example: Let X = { 1, 2, 3 } and the (i, v)-fuzzy sets A, B : X D[0, 1] be defined as A = { <1, [1, 1]>, <2, [0, 0]>, <3, [0, 0]> } and B = { <1, [0, 0]>, <2, [1, 1] >, < 3, [1, 1] > }. Consider = { 0, 1, A, B }. Then (X, ) is an (i, v)-fuzzy topological space. In this (i, v)-fuzzy topological space X, every (i, v)-fuzzy subset of X is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X, but IVFRSO = { 0, 1, A, B } Theorem: If A is an (i, v)-fuzzy rw-closed set of (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A) ) = 0, then cl( A) A does not contain any non-zero (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Proof: Suppose A is an (i, v)-fuzzy rw-closed set of (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A)) = 0. We prove the result by contradiction. Let B be an (i, v)-fuzzy regular semiopen set such that cl( A) A B and B 0. Now B cl( A) A, i.e B 1 A which implies A 1 B. Since B is an (i, v)-fuzzy regular semiopen set, by Theorem 1.9, 1 B is also (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Since A is an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X, by definition cl( A) 1 B. So B 1 cl( A). Therefore B cl( A) ( 1 cl( A) ) = 0, by hypothesis. This shows that B = 0 which is a contradiction. Hence cl( A) A does not contain any non-zero (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. 2.2 Corollary: If A is an (i, v)-fuzzy rw-closed set of (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A) ) = 0, then cl( A) A does not contain any non-zero (i, v)-fuzzy regular open set in (i, v)-fuzzy topological space X. Proof: Follows form the Theorem 2.14 and the fact that every (i, v)-fuzzy regular open set is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X.

13 Notes on Interval Valued Fuzzy RW-Closed Corollary: If A is an (i, v)-fuzzy rw-closed set of (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A) ) = 0, then cl( A) A does not contain any non-zero (i, v)-fuzzy regular closed set in (i, v)-fuzzy topological space X. Proof: Follows form the Theorem 2.14 and the fact that every (i, v)-fuzzy regular closed set is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X Theorem: Let A be an (i, v)-fuzzy rw-closed set of (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A) ) = 0 Then A is an (i, v)- fuzzy closed set if and only if cl( A) A is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Proof: Suppose A is an (i, v)-fuzzy closed set in (i, v)-fuzzy topological space X. Then cl( A) = A and so cl( A) A = 0, which is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Conversely, suppose cl( A) A is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Since A is (i, v)-fuzzy rw-closed, by Theorem 2.14 cl( A) A does not contain any-non zero (i, v)-fuzzy regular open set in (i, v)-fuzzy topological space X. Then cl( A) A = 0. That is cl( A) = A and hence A is an (i, v)-fuzzy closed set in (i, v)-fuzzy topological space X Theorem: If an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is (i, v)-fuzzy open, then it is (i, v)-fuzzy rw-open but not conversely. Proof: Let A be an (i, v)-fuzzy open set of (i, v)-fuzzy topological space X. Then A C is (i, v)-fuzzy closed. Now by Theorem 2.1, A C is (i, v)-fuzzy rwclosed. Therefore A is an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Remark: The converse of the above Theorem need not be true in general Example: Let X = { 1, 2, 3 }. Define an (i, v)-fuzzy subset A in X by A = { < 1, [1, 1] >, < 2, [1, 1] >, < 3, [0, 0] > }. Let = { 0, 1, A }. Then (X, ) is an (i, v)-fuzzy topological space. Define an (i, v)-fuzzy set B in X by B = { < 1, [0, 0] >, < 2, [1, 1] >, < 3, [0, 0] > }. Then B is an (i, v)-fuzzy rw-open set but it is not (i, v)-fuzzy open set in (i, v)-fuzzy topological space X. Remark: By Remark 1.3, it has been proved that every (i, v)-fuzzy regular open set is an (i, v)-fuzzy open set but not conversely. By Theorem 2.16, every (i, v)-fuzzy open set is an (i, v)-fuzzy rw-open set but not conversely and hence every (i, v)-fuzzy regular open set is an (i, v)-fuzzy rw-open set but not conversely.

14 36 R. Indira, K. Arjunan and N. Palaniappan 2.17 Theorem: An (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is (i, v)-fuzzy rw-open if and only if D int( A), whenever D A and D is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Proof: Suppose that D int( A), whenever D A and D is an (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. To prove that A is (i, v)-fuzzy rw-open in (i, v)-fuzzy topological space X. Let A C B and B is any (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. Then B C A. By Theorem 1.9, B C is also (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X. By hypothesis, B C int( A) which implies ( int( A) ) C B. That is cl( A C ) B, since cl( A C ) = ( int( A) ) C. Thus A C is an (i, v)-fuzzy rw-closed and hence A is (i, v)-fuzzy rw-open in (i, v)-fuzzy topological space X. Conversely, suppose that A is (i, v)-fuzzy rw-open. Let B A and B is any (i, v)-fuzzy regular semiopen in (i, v)-fuzzy topological space X. Then A C B C. By Theorem 1.9, B C is also (i, v)-fuzzy regular semiopen. Since A C is (i, v)-fuzzy rw-closed, we have cl( A C ) B C and so B int( A), since cl( A C ) = ( int( A) ) C Theorem: If A and B are (i, v)-fuzzy rw-open sets in an (i, v)-fuzzy topological space X, then A B is also an (i, v)-fuzzy rw-open set in (i, v)- fuzzy topological space X. Proof: Let A and B be two (i, v)-fuzzy rw-open sets in an (i, v)-fuzzy topological space X. Then A C and B C are (i, v)-fuzzy rw-closed sets in (i, v)- fuzzy topological space X. By Theorem 2.6, A C B C is also an (i, v)-fuzzy rw-closed set in (i, v)-fuzzy topological space X. That is ( A C B C ) = ( A B) C is an (i, v)-fuzzy rw-closed set in X. Therefore A B is also an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X Theorem: The union of any two (i, v)-fuzzy rw-open sets in an (i, v)- fuzzy topological space X is generally not an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Proof: Consider the (i, v)-fuzzy topological space (X, ) defined as in Example In this (i, v)-fuzzy topological space X, the (i, v)-fuzzy sets D 1, D 2 : X D[0, 1] are defined by D 1 = { < 1, [1, 1] >, < 2, [0, 0] >, < 3, [0, 0] > } and D 2 = { < 1, [0, 0] >, < 2, [0, 0] >, < 3, [1, 1] > }. Then D 1 and D 2 are the (i, v)-fuzzy rw-open sets in (i, v)-fuzzy topological space X. Let E = D 1 D 2. Then E = { <1, [1, 1]>, <2, [0, 0]>, <3, [1, 1]> }. Then E = D 1 D 2 is not an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X.

15 Notes on Interval Valued Fuzzy RW-Closed Theorem: If int( A) B A and A is an (i, v)-fuzzy rw-open set in an (i, v)-fuzzy topological space X, then B is also an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Proof: Suppose int( A) B A and A is an (i, v)-fuzzy rw-open set in an (i, v)-fuzzy topological space X. To prove that B is an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Let F be any (i, v)-fuzzy regular semiopen set in (i, v)-fuzzy topological space X such that F B. Now F B A. That is F A. Since A is (i, v)-fuzzy rw-open set of (i, v)-fuzzy topological space X, F int( A), by Theorem By hypothesis int( A) B. Then int( int( A) ) int( B). That is int( A) int( B). Then F int( B). Again by Theorem 2.17, B is an (i, v)-fuzzy rw-open set in (i, v)- fuzzy topological space X Theorem: If an (i, v)-fuzzy subset A of an (i, v)-fuzzy topological space X is (i, v)-fuzzy rw-closed and cl( A) ( 1 cl( A) ) = 0, then cl( A) A is an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Proof: Let A be an (i, v)-fuzzy rw-closed set in an (i, v)-fuzzy topological space X and cl( A) ( 1 cl( A) ) = 0. Let B be any (i, v)-fuzzy regular semiopen set of (i, v)-fuzzy topological space X such that B ( cl( A) A ). Then by Theorem 2.14, cl( A) A does not contain any non-zero (i, v)- fuzzy regular semiopen set and so B = 0. Therefore B int( cl( A) A ). By Theorem 2.17, cl( A) A is (i, v)-fuzzy rw-open Theorem: Let A and B be two (i, v)-fuzzy subsets of an (i, v)-fuzzy topological space X. If B is an (i, v)-fuzzy rw-open set and A int( B), then A B is an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Proof: Let B be an (i, v)-fuzzy rw-open set of an (i, v)-fuzzy topological space X and A int( B). That is int( B) ( A B). Also int( B) ( A B) B and B is an (i, v)-fuzzy rw-open set. By Theorem 2.20, A B is also an (i, v)-fuzzy rw-open set in (i, v)-fuzzy topological space X. Remark: Every (i, v)-fuzzy w-open set is (i, v)-fuzzy rw-open but its converse may not be true Example: Let X = { 1, 2 } and = { 1, 0, A } be an (i, v)-fuzzy topology on X, where A={<1, [0.7, 0.7]>, <2, [0.6, 0.6]>}. Then the (i, v)- fuzzy set B ={<1, [0.2, 0.2]>, <2, [0.1, 0.1]> } is (i, v)-fuzzy rw-open in (X, ) but it is not (i, v)-fuzzy w-open in (X, ).

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