Supra g-closed Sets in Supra Bitopological Spaces

International Mathematical Forum, Vol. 3, 08, no. 4, 75-8 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/imf.08.8 Supra g-closed Sets in Supra Bitopological Spaces R. Gowri Department of Mathematics Government College for Women (Autonomous) Kumbakonam, India A. K. R. Rajayal Departmentof Mathematics Government College for Women (Autonomous) Kumbakonam, India Copyright c 08 R. Gowri and A. K. R. Rajayal. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The intention of this paper is to introduce a new class of sets called supra g-closed sets in supra bitopological spaces. Also, we introduce S τij -T / -spaces in supra bitopological spaces. Mathematics Subject Classification: 54D05, 54D0, 54D08, 54D0 Keywords: Supra bitopology, S τij -g-closed sets, S τij -g-open sets, S τij - T / -space Introduction Kelly [5] was introduced the concept of bitopological spaces. Fukutake [] introduced generalized closed sets in bitopological spaces. The supra topological spaces have been introduced by Mashhour [8] in 983 and also introduced the notion of supra open and supra closed set and characterize those sets using supra closure and supra interior operators respectively. In topological space the arbitrary union condition is enough to have a supra topological space. Here

76 R. Gowri and A. K. R. Rajayal every topological space is a supra topologicl space but the converse is not always true. Ravi, Ramkumar and Kamaraj [9] introduced the concept of supra g-closed sets and obtain some properties of supra g-closed sets. Gowri and Rajayal [3] introduced the concept of supra bitopological spces. In this paper we introduce and investigate S τij -g-closed sets in supra bitopological spaces. Also we introduce S τij -T / -space and detailed study of some of its properties. Preliminaries Definition. [8] (X, S τ ) is said to be a supra topological space if it is satisfying these conditions: () X, S τ () The union of any number of sets in S τ belongs to S τ. Definition. [8] Each element A S τ is called a supra open set in (X, S τ ), and its compliment is called a supra closed set in (X, S τ ). Definition.3 [8] If (X, S τ ) is a supra topological spaces, A X, A, S τa is the class of all intersection of A with each element in S τ, then (A, S τa ) is called a supra subspace of (X, S τ ). Definition.4 [8] The supra closure of the set A is denoted by S τ -cl(a) and is defined as S τ -cl(a) = {B : B is a supra closed and A B}. Definition.5 [8] The supra interior of the set A is denoted by S τ -int(a) and is defined as S τ -int(a) = {B : B is a supra open and B A}. Definition.6 [3] If S τ and S τ are two supra topologies on a non-empty set X, then the triplet (X, S τ, S τ ) is said to be a supra bitopological space. Definition.7 [3] Each element of S τi is called a supra τ i -open sets(briefly S τi )-open sets) in (X, S τ, S τ ). Then the complement of S τi -open sets are called a supra τ i -closed sets(breifly S τi -closed sets), for i =,. Definition.8 [3] If (X, S τ, S τ ) is a supra bitopological space, Y X, Y then (Y, S τ, S τ ) is a supra bitopological subspace of (X, S τ, S τ ) if S τ = {U Y ; U is a S τ open in X} and = {V Y ; V is a S τ open in X}. S τ Definition.9 [3] The S τi -closure of the set A is denoted by S τi -cl(a) and is defined as S τi -cl(a) = {B : B is a S τi closed and A B, for i =, }. Definition.0 [3] The S τi -interior of the set A is denoted by S τi -int(a) and is defined as S τi -int(a)= {B : B is a S τi open and B A, for i =, }.

Supra g-closed sets in supra bitopological spaces 77 Definition. [9] Let (X, µ) be a supra topological space. A subset A of X is supra generalized closed (briefly, supra g-closed) if cl µ (A) U whenever A U and U is supra open. The collection of all supra g-closed sets in X is denoted by GC(X). 3 Supra generalized closed sets in supra bitopological spaces Definition 3. A subset A of a supra bitopological space (X, S τ, S τ ) is said to be supra τ ij -generalized-closed(briefly S τij -g-closed) if S τj -cl(a) U whenever A U and U S τi, where i,j =, and i j. The complement of S τij -g-closed set is said to be S τij -g-open. The family of all S τij -g-closed(resp. S τij -g-open)sets of (X, S τ, S τ ) is denoted by S τij -gc(x)(resp. S τij -go(x)), where i,j =, and i j. Example 3. Let X = {a, b, c}, S τ = {, X, {a, b}, {b, c}}, S τ = {, X, {a, c}, {b, c}}. S τ -gc(x) = {, X, {a}, {b}, {a, c}}. Definition 3.3 A subset A of supra bitopological space (X, S τ, S τ ) is said to be pairwise S τ -g-closed(briefly p-s τ -g-closed) if A is S τ -g-closed and S τ - g-closed. The complement of a pairwise S τ -g-closed set is said to be pairwise S τ -g-open(briefly p-s τ -g-open.) Theorem 3.4 Every S τj -closed set in (X, S τ, S τ ) is S τij -g-closed. Proof: Let A X be a S τj -closed set and A U, where U is S τi -open set. Since A is S τj -closed set, S τj -cl(a) = A and hence S τj -cl(a) U. Thus A is S τij -g-closed. Remark 3.5 The converse of the Theorem 3.4, is not true as can be seen from the following example. Example 3.6 Let X = {,, 3}, S τ = {, X, {, }, {, 3}}, S τ = {, X, {, 3}, {, 3}}. Here {, 3} is S τ -g-closed but it is not S τ -closed. Remark 3.7 The union of two S τij -g-closed sets need not be S τij -g-closed as seen from the following example.

78 R. Gowri and A. K. R. Rajayal Example 3.8 Let X = {a, b, c}, S τ = {, X, {a, b}, {b, c}}, S τ = {, X, {a, c}, {b, c}}. Then {a} and {b} are S τ -g-closed but {a} {b} = {a, b} is not S τ -g-closed. Remark 3.9 The intersection of two S τij -g-closed sets need not be S τij -gclosed as seen from the following example. Example 3.0 Let X = {a, b, c}, S τ = {, X, {a}, {a, b}, {b, c}}, S τ = {, X, {c}, {a, b}, {a, c}}. Then {a, b} and {a, c} are S τ -g-closed but {a, b} {a, c} = {a} is not S τ - g-closed. Theorem 3. Let (X, S τ, S τ ) be a supra bitopological space. If A is S τij - g-closed and B is S τj -closed, then A B is S τij -g-closed. Proof: Let U be a S τi -open such that A B U. Then A U (B) c and so S τj -cl(a) U (B) c. Therefore S τj -cl(a) B U. Since B is S τj -closed. Therefore, S τj -cl(a B) U. Hence, A B is S τij -g-closed. Remark 3. S τ -gc(x) is generally not equal to S τ -gc(x) as can be seen from the following example. Example 3.3 Let X = {a, b, c}, S τ = {, X, {a, b}, {b, c}}, S τ = {, X, {a}, {a, b}, {a, c}, {b, c}}. Then S τ -gc(x) = {, X, {a}, {b}, {c}, {a, c}, {b, c}} and S τ -gc(x) = {, X, {a}, {c}}. Thus S τ -gc(x) S τ -gc(x). Remark 3.4 Let S τ and S τ be two supra topologies on X. If S τ S τ, then S τ -gc(x) S τ -gc(x). Remark 3.5 Intersection of S τij -g-closed set and S τi -open set is neither S τij -g-closed nor S τi -open as can be seen from the following example. Example 3.6 Let X = {a, b, c, d}, S τ = {, X, {a}, {a, d}, {b, c, d}}, S τ = {, X, {b}, {b, c}, {a, c, d}, {b, c}}. Then S τ -gc(x) = {, X, {a}, {b}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, c, d}, {a, b, d}, {b, c, d}, }. We have A = {a, d} is S τ -open and B = {b, c, d} is S τ -g-closed set, their intersection {d} is neither S τ -g-closed set nor S τ -open set.

Supra g-closed sets in supra bitopological spaces 79 Theorem 3.7 Let (Y, S τ, S τ ) be a supra bitopological closed subspace of (X, S τ, S τ ). If A is a S τij -g-closed subset of (Y, S τ, S τ ), then A is S τij -gclosed subset of (X, S τ, S τ ). Proof: Let B be a S τi -open subset of (X, S τ, S τ ) such that A B. Then, A B Y. Since, A is S τij -g-closed and B Y is S τi -open in (Y, S τ, S τ ). Therefore, S τj -cl(a) Y = S τ j -cl(a) B. But Y is a supra bitopological closed subset of (X, S τ, S τ ) and S τj -cl(a) B. Hence, A is S τij -g-closed subset of (X, S τ, S τ ). Theorem 3.8 Let A be a subset of supra bitopological space (X, S τ, S τ ). If A be S τij -g-closed, then S τj -cl(a) A contains no non empty S τi -closed set. Proof: Let A be S τij -g-closed subset of S τj -cl(a) A. Now B S τj -cl(a) A and A B c where A is S τij -g-closed and B c is S τi -open. Thus S τj -cl(a) B c or equivalently B [S τj cl(a)] c. By assumption, B [S τj cl(a)] and so B [S τj cl(a)] [S τj cl(a)] c =. Therefore, B =. Hence, S τj -cl(a) A contains no non empty S τi -closed set. Remark 3.9 The converse of the above Theorem 3.0, is not true as seen from the following example. Example 3.0 Let X = {a, b, c}, S τ = {, X, {a, b}, {b, c}}, S τ = {, X, {a, c}, {b, c}}. Take A = {a, b}. Then S τ -cl(a) A = S τ cl {a, b} {a, b} = {c} which does not contain any non empty S τ -closed set. But {a, b} is not S τ -g-closed. Corollary 3. If A is S τij -g-closed set in (X, S τ, S τ ) then A is S τj -closed if and only if S τj -cl(a) A is S τi -closed. Proof: Assume that A is S τij -g-closed set and S τj -closed. Then S τj -cl(a)= A. That is S τj -cl(a) A = and hence S τj -cl(a) A is S τi -closed. Conversely, suppose S τj -cl(a) A is S τi -closed, then by Theorem 3.0, A is S τij -g-closed, then S τj -cl(a) A contains no non empty S τi -closed set. This implies S τj - cl(a) A =. Hence, A is S τj -closed. Theorem 3. For each x (X, S τ, S τ ), the singleton {x} is S τi -closed or {x} c is S τij -g-closed set. Proof: Suppose that {x} is not S τi -closed, then {x} c is not S τi -open. Then X is the only S τi -open set which contains {x} c and {x} c is S τij -g-closed set. Theorem 3.3 If A be S τij -g-closed set in (X, S τ, S τ ) and A B S τj - cl(a), then B is also S τij -g-closed set.

80 R. Gowri and A. K. R. Rajayal Proof: Assume A is S τij -g-closed set and A B S τj -cl(a). Let B U and U is S τi -open. Given A B. Then A U. Since A is S τij -g-closed set, we have S τj -cl(a) U. Since B S τj -cl(a), S τj -cl(b) S τj -cl(a) U. Hence B is S τij -g-closed set 4 Supra generalized open sets in supra bitopological spaces Definition 4. A subset A of a supra bitopological space (X, S τ, S τ ) is supra τ ij -generalized-open(briefly S τij -g-open) if its complement is S τij -g-closed set. Remark 4. Union of S τij -g-open set and S τi -closed set is neither S τij -gopen nor S τi -closed as can be seen from the following example. Example 4.3 Let X = {a, b, c, d}, S τ = {, X, {a}, {a, d}, {b, c, d}}, S τ = {, X, {b}, {b, c}, {a, c, d}, {b, c}}. Then S τ -go(x) = {, X, {a}, {b}, {c}, {d}, {b, c}, {b, d}, {c, d}, {a, c, d}}. We have A = {b, c} is S τ -closed and B = {a} is S τ -g-open set, their Union {a, b, c} is neither S τ -g-open set nor S τ -closed set. Theorem 4.4 A subset of supra bitopological space in (X, S τ, S τ ) is S τij - g-open set if and only if B S τj -int(a) whenever B is S τi -closed and B A where i,j =, and i j. Proof: Let A be τ ij -g-open set. Let B be a S τi -closed set such that B A. Let A B and B is S τi -closed. Then A c B c and B c is S τi -open, We have A c is τ ij -g-closed. Hence, [S τj cl(a) c ] B c. Therefore B S τj -int(a). Conversely, suppose B S τj -int(a) whenever B A and B is S τi -closed. Let A c U and U is S τi -open. Then U c A and U c is S τi -closed. By hypothesis U c S τj -int(a). Hence, [S τj int(a)] c U. That is [S τj cl(a) c ] U. Consequently, A c is S τij -g-closed set. Hence, A is S τij -g-open. Theorem 4.5 A subset A is S τij -g-closed set then S τj -cl(a) A is S τij -gopen. Proof: Let A be S τij -g-closed set. Let B S τj -cl(a) A where B is S τi - closed set. Since A is S τij -g-closed, we have S τj -cl(a) A does not contain non empty S τi -closed by Theorem 3.0, Consequently, B =. Therefore, S τj - cl(a) A, S τj -int(s τj -cl(a) A), we obtain B S τj -int(s τj -cl(a) A). Hence, S τj -cl(a) A is S τij -g-open.

Supra g-closed sets in supra bitopological spaces 8 Theorem 4.6 Let A and B be subset of supra bitopological space (X, S τ, S τ ) such that S τj -int(a) B A. If A is S τij -g-open, then B is S τij -g-open, where i,j =, and i j. Proof: Let A be S τij -g-open. Let U be a S τi -closed such that U B. Since U B and B A, we have U A. Therefore, U S τj -int(a). Since S τj -int(a) B, we have S τj int(s τj int(a)) S τj int(b). Therefore S τj -int(a) S τj -int(b). Consequently, U S τj -int(b). Hence, B is S τij -g-open 5 Supra T / -space in supra bitopological space Definition 5. A supra bitopological space (X, S τ, S τ ) is said to be supra T / -space(briefly S τij T / -space) if every S τij -g -closed set is S τj -closed. Definition 5. A supra bitopological space (X, S τ, S τ ) is said to be pairwise supra T / -space(briefly p-s τ T / -space) if it is both S τ T / -space and S τ T / -space. Theorem 5.3 Let (X, S τ, S τ ) be a supra bitopological space, then () Every supra T / -space is supra T 0 -space. () Every supra T -space is supra T / -space. Proof: It is obivious. Theorem 5.4 A supra bitopological space (X, S τ, S τ ) is S τij T / -space if and only if {x} is S τj -open or S τi -closed for each x X, where i,j =, and i j Proof: Suppose that {x} is not S τi -closed. Then {x} c is S τij -g-closed by Theorem, 3.4. Since (X, S τ, S τ ) is S τij T / -space, {x} c is S τj -closed. Therefore, {x} is S τj -open. Conversely, let A be a S τij -g-closed set. By assumption, {x} is S τj -open or S τi -closed for any x Sτ j -cl(a). Case(i): Suppose {x} is S τj -open. Since {x} A =, We have x A. Case(ii): Suppose {x} is S τi -closed. If x / A, then {x} S τj -cl(a) A, which is a contradiction to Theorem, 3.0. Hence x A. Thus in both cases, we conclude that A is S τj -closed. Therefore, (X, S τ, S τ ) is an S τij T / -space.

8 R. Gowri and A. K. R. Rajayal 6 Conclusion In this paper, we introduced supra g-closed sets in supra bitopological spaces. Also we investigated the behavior relative to union, intersection and supra subspaces of S τij -g-closed sets as well as S τij -g-open sets. Through S τij -gclosed sets, we introduced a new separation axiom, namely S τij -T / -space. References [] T. Fukutake, On Generalized Closed Sets In Bitopological Spaces, Bull. Fukuka Univ. Ed. Part III, 35 (986), 9-8. [] R. Gowri and S. Vembu, Soft g-closed Sets in Soft Biminimal Spaces, Int. Jr. of Mathematics and its Applications, 5 (07), 365-370. [3] R. Gowri and A. K. R. Rajayal, On supra bitopological spaces, IOSR-JM, 3 (07), 55-58. [4] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh and A. M. Abd El-latif, Supra Generalized Closed Soft Sets with Respect to an soft Ideal in supra soft Topological spaces, Appl. Math. Inf. Sci., 8 (04), no. 4, 73-740. https://doi.org/0.785/amis/080430 [5] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 3 (963), 7-89. https://doi.org/0./plms/s3-3..7 [6] N. Levine, Generalized closed sets topology, Rend. Circ. Mat. Palermo, 9 (970), 89-96. https://doi.org/0.007/bf0843888 [7] H. Maki, K. C. Rao and A. Nagoor Gani, On generalized semi-open and pre-open sets, Pure Appl. Math. Sci., 49 (999), 7-9. [8] A. S. Mashhour, A.A. Allam, F.S. Mahmoud and F.H. Khedr, On Supra topological spaces, Indian Jr. Pure and Appl. Math., 4 (983), no. 4, 50-50. [9] O. Ravi, G. Ramkumar and M. Kamaraj, On supra g-closed sets, International Journal of Advances in Pure and Applied Mathematics, (0), no., 5-66. Received: February 3, 08; Published: February 8, 08