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Int Jr. of Mathematics Sciences & Applications Vol.3, No.1, January-June 2013 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com 1 2* Boundary set on Bigeneralized Topological Space * M. MURUGALINGAM, ** R. GLORY DEVA GNANAM * Department of Mathematics, Thiruvalluvar college, Papanasam. Email: [email protected] ** Department of Mathematics, C.S.I Jayaraj Annapackiam college, Nallur. Email: [email protected] Abstract In this paper we define 1 2* boundary set in a bigeneralized topological space we study the properties of the set. Key words: 12* boundary point, I.Introduction: In 1963, J.C. Kely [1] defined the study of Bitopological spaces. The concept of generalized topology was introduced by A. Csaszar [2] in 2002. In 2010, C. Boonpok [3] introduced the concept of bigeneralized topological spaces. In 2013, sompong and muangchan [4] introduced the concepts of boundary set on Bigeneralized topological space. In this paper, we introduce the concepts of 12* boundary point, and study their properties in bigeneralized topological spaces. II. Preliminaries Let X be a non-empty set and be a collection of subsets of X. Then is called a generalized topology on X if and arbitrary union of elements of in . A space with a generalized topology is called a generalized topological space. In a generalized topological space (X, ) the following are true i) Cl (A) = X – int (X – A) ii) Int (A) = X - Cl (X – A) iii) Cl (X – A) = X – int (A) and int (X – A) = X – Cl (A) iv) If X – A then Cl (A) = A and if A then int (A) = A. v) If A B then Cl (A) Cl (B) and int (A) int (B) vi) A Cl (A) and int (A) A Let X be a non – empty set and 1, 2 be generalized topologies on X. The triple (X, 1,2 ) is called a bigeneralized topological space (briefly BGTS). III. 1 2* Boundary set 141 M. MURUGALINGAM, R. GLORY DEVA GNANAM Definition: 3.1 A subset A of a bigeneralized topological space (X, 1,2 ) is called 1 2* closed if A = 1 Cl (A) 2 Cl (A). The complement of 1 2* closed set is called 1 2* open. The 1 2* closure of a set A is defined in the usual way. Result: 3.2 Let (X, 1, 2) be a bigeneralized topological space and let A be a subset of X. Then 1 2* Cl (A) = 1 Cl A 2 Cl A. In this section, we introduce the concept 1 2* boundary set and study their properties in bigeneralized topological space. Definition: 3.3 Let (X, 1, 2) be a BGTS. Let A be a subset of X and x X. Then x is called 1 2* boundary point of A if x (1 2* Cl A 1 2* Cl (X – A)). We denote the set of all 1 2* – boundary point of A by 1 2* Bd (A). From definition we have 1 2* Bd (A) = (1 Cl A 2 Cl A) (1 Cl (X – A) 2 Cl (X – A)) Example: 3.2 Let X = {a, b, c, d} 1 = {, X, {a, b}, {b, c}} and 2 = {, X, {a, c}, {a, b}} Now, 1 closed sets are {, X, {c}, {a}} and 2 closed sets are {, X, {b}, {c}} 1 2 Bd ({a, b}) = {c} and 1 2 Bd ({b, c}) = {a} Theorem: 3.3 Let (X, 1, 2) be a bigeneralized topological space. Let A be a subset of X. Then 12* Bd (A) = 1 2* Bd (X – A) Proof: 1 2* Bd (A) = (1 Cl A 2 Cl A) (1 Cl (X – A) 2 Cl (X – A)) = (1 Cl (X – (X – A)) 2 Cl (X – (X – A))) (1 Cl (X – A) 2 Cl (X – A)) = 1 2* Bd (X – A) Theorem: 3.4 Let (X, 1, 2) be a bigeneralized topological space and A, B be a subset of X. Then the following are true. i) 1 2* Bd (A) = (1 Cl (A) 2 Cl (A)) \ (1 int (A) 2 int (A)) ii) 1 2* Bd (A) (1 int (A) 2 int (A)) = iii) 1 2* Bd (A) (1 int (X – A) 2 int (X – A)) = iv) 1 Cl A 2 Cl A = 1 2* Bd (A) (1 int (A) 2 int (A)) v) X = (1 int (X – A) 2 int (X – A)) 1 2* Bd (A) (1 int (A) 2 int (A)) is a pair wise disjoint Union. Proof: i) 1 2* Bd (A) = (1 Cl (A) 2 Cl (A)) (1 Cl (X – A) 2 Cl (X – A)) = (1 Cl (A) 2 Cl (A)) (X – (1 int (A) 2 int (A))) = (1 Cl (A) 2 Cl (A)) \ (1 int (A) 2 int (A)) 142 1 2* Boundary set on Bigeneralized… From (i), it’s obvious that 1 2* Bd (A) (1 int (A) 2 int (A)) = iii) 1 2* Bd (A) (1 int (X – A) 2 int (X – A)) = 1 2* Bd (X – A) (1 int (X – A) 2 int (X – A)) = iv) 1 Cl A 2 Cl A = [1 Cl (A) 2 Cl (A) \ 1 int (A) 2 int (A)] (1 int (A) 2 int (A)) = 1 2* Bd (A) (1 int (A) 2 int (A)) v) (1 int (X – A) 2 int (X – A)) 1 2* Bd (A) (1 int (A) 2 int (A)) = [X – ((1 Cl (A) 2 Cl (A))] (1 Cl (A) 2 Cl (A)) = X. By ii) and (iii) 1 2* Bd (A) (1 int (A) 2 int (A)) = and 1 2* Bd (A) (1 int (X – A) 2 int (X – A)) = Now, (1 int (A) 2 int (A)) (1 int (X – A) 2 int (X – A)) A (X – A) = Therefore, X = (1 int (A) 2 int (A)) 1 2* Bd (A) (1 int (X – A) 2 int (X – A)) is a pair wise disjoint union. ii) Theorem: 3.5 Let (X, 1, 2) be a BGTS and A be a subset of X. Then i) A is 1 2* closed if and only if 1 2* Bd (A) A. ii) A is 1 2* open if and only if 1 2* Bd (A) X – A. Proof: i) Assume that A is 1 2* closed, Thus 1 Cl A 2 Cl A = A Let x 1 2* Bd (A). Then x (1 Cl A 2 Cl A) (1 Cl (X – A) 2 Cl (X – A)) x A (1 Cl (X – A) 2 Cl (X – A)) xA Therefore, 1 2* Bd (A) A. Conversely let 1 2* Bd (A) A Then 1 2* Bd (A) (X – A) = Now, (1 Cl A 2 Cl A) (X – A) = (1 Cl A 2 Cl A) [(1 Cl (X – A) 2 Cl (X – A)) (X – A)] = 1 2* Bd (A) (X – A) = Therefore, 1 Cl A 2 Cl A A Also A 1 Cl A 2 Cl A That implies 1 Cl A 2 Cl A = A Hence A is 1 2* closed. ii) Assume that A is 1 2* open, Thus 1 int (A) 2 int (A) = A Now 1 2* Bd (A) A = [(1 Cl (A) 2 Cl (A)) – (1 int (A) 2 int (A))] A = (1 Cl (A) 2 Cl (A)) – A) A = Hence 1 2* Bd (A) X – A 143 M. MURUGALINGAM, R. GLORY DEVA GNANAM Conversely, Assume that 1 2* Bd (A) X – A Thus 1 2* Bd (A) A = . This implies, [(1 Cl (A) 2 Cl (A)) \ (1 int (A) 2 int (A))] A = Since A (1 Cl (A) 2 Cl (A)), A – (1 int (A) 2 int (A)) = But 1 int (A) 2 int (A) A. Hence A = 1 int (A) 2 int (A). A is 1 2* open. Theorem: 3.6 Let (X, 1, 2) be a bigeneralized topological space and A be a subset of X. Then 1 2* Bd (A) = if and only if A is 1 2* closed and 1 2* open. Proof: Let 1 2* Bd (A) = Then by theorem 3.5 A is 1 2* closed and 1 2* open. Conversely, let 1 2* closed and 1 2* open. Then 1 Cl A 2 Cl A = A and 1 Cl (X – A) 2 Cl (X – A) = X – A 1 2* Bd (A) = (1 Cl A 2 Cl A) (1 Cl (X – A) 2 Cl (X – A)) = A (X – A) = Definition: 3.7 Let (X, 1, 2) be a bigeneralized topological space. A subset A of X is said 1 2* dense if 1 2* Cl A = X. Theorem: 3.8 Let (X, 1, 2) be a bigeneralized topological space. If A is 1 2* closed and X – A is 2* Bd (A) = A Proof: Since A is 1 2* closed 1 2* Bd (A) A. Since X – A is 1 2* dense, 1 Cl (X – A) 2 Cl (X – A) = X Clearly A 1 Cl A 2 Cl A. Hence A (1 Cl A 2 Cl A) X This implies, A (1 Cl A 2 Cl A) (1 Cl (X – A) 2 Cl (X – A)) This implies, A 1 2* Bd (A). Hence A = 1 2* Bd (A) 1 2* dense then 1 References 1. J.C. Kelly, Bitopoligicaal spaces, Pro. London math soc., 313 (1963), 71-89. 2. A.Csazar, Generalized topology generalized continuity, Acta math. Hungar 96 (2002), 351 – 357. 3. C. Boonpok, Weakly open Functions on Bigneralized Topoligical spaces, Int. J. Math. Analysis, 4(18) (2010), 891-897 4. Supunnee sompong and sa-at muangchan, Boundary set on Bigeneralized Topoligical spaces, Int. Journal of Math. Analysis, vol- 7, 2013, No.2, 85 -89 144