/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 1, 2016, 27-33 ISSN 1307-5543 – www.ejpam.com A Note on αgrw-Closed Sets Ennis Rosas1, N.Selvanayaki2∗, and Gnanambal Ilango3 1 Departamento de Matemáticas, Universidad de Oriente, Cumaná, Venezuela and Facultad de Ciencias Básicas, Universidad del Atlántico, Barranquilla, Colombia. Ãű2 Department of Mathematics, Akshaya College of Engineering and Technology, Coimbatore, Tamil Nadu, India. 3 Department of Mathematics, Government Arts College, Coimbatore, Tamilnadu, India. Abstract. In this paper, some properties of αgrw-closed sets are discussed and also some characteri- zations of αgrw-closed sets are studied in topological spaces. 2010 Mathematics Subject Classifications: 54A05 Key Words and Phrases: αgrw-Closed Sets, rsker(A), s-Normal Space 1. Introduction In 2013, αgrw-closed sets are introduced and studied by Selvanayaki and Gnanambal Ilango [14] and some basic properties of αgrw-closed sets are investigated. The class of αgrw- closed sets properly lies between the class of rw-closed sets and the class of gprw-closed sets. In 2007, Benchalli and Wali [1] have introduced a new type of Kernel known as regular semi kernel.The aim of this paper is to study some properties of αgrw-closed sets and some characterizations of it. Throughout this paper, space (X ,τ) (or simply X ) always means a topological space on which no separation axioms are assumed unless explicitly stated. For a subset A of a space X , cl(A), int(A) and X −A (or Ac)denote the closure of A, the interior of A and the complement of A in X , respectively. 2. Preliminaries Definition 1. A subset A of a topological space (X ,τ) is called (i) regular open [15] if A= int(cl(A)) and regular closed if A= cl(int(A)). ∗Corresponding author. Email addresses: ennisrafael@gmail.com (Ennis Rosas), selvanayaki.nataraj@gmail.com (N. Selvanayaki) and gnanamilango@yahoo.co.in (Gnanambal Ilango) http://www.ejpam.com 27 c© 2016 EJPAM All rights reserved. Ennis Rosas, N. Selvanayaki, Gnanambal Ilango / Eur. J. Pure Appl. Math, 9 (2016), 27-33 28 (ii) semi-open [7] if A⊆ cl(int(A)) and semi-closed if int(cl(A)) ⊆ A. (iii) α-open [13] if A⊆ int(cl(int(A))) and α-closed [12] if cl(int(cl(A))) ⊆ A. Definition 2 ([2]). A subset A of a space (X ,τ) is called regular semi-open if there is a regular open set U such that U ⊆ A⊆ cl(U). The family of all regular semi-open sets of X is denoted by RSO(X ). Definition 3 (Noiri [10]). A subset A of a space (X ,τ) is said to be semi-regular open if it is both semi-open and semi-closed. The family of all semi-regular open sets of X is denoted by SR(X ). On other hand, Maio and Noiri defined a subset A of X to be semi-regular open if A= sint(scl(A)). However, these three notions are equivalent, which is given in the following theorem. Theorem 1 ([10]). For a subset A of a space X , the followings are equivalent: (i) A∈ SR(X )(= RSO(X )), (ii) A= sint(scl(A)), (iii) there exists a regular open set U of X such that U ⊆ A⊆ cl(U). Definition 4. A subset A of a topological space (X ,τ) is called (i) generalized closed (briefly g-closed) [8] if cl(A) ⊆ U whenever A⊆ U and U is open in X . (ii) α-generalized closed (briefly αg-closed)[11] if αcl(A) ⊆ U whenever A⊆ U and U is open in X . (iii) α-generalized regular weakly closed (briefly αgrw-closed)[14] if αcl(A) ⊆ U whenever A⊆ U and U is regular semi-open in X . The set of all αgrw-closed sets in (X ,τ) is denoted by αgrwC(X ). Definition 5 ([9]). A topological space (X ,τ) is said to be s-normal if for each pair of disjoint closed sets A and B, there exists disjoint semi-open sets U, V such that A⊆ U and B ⊆ V . Definition 6 ([1]). The intersection of all regular semi-open subsets of (X ,τ) containing A is called the regular semi-kernel of A and is denoted by rsker(A). Theorem 2 ([3]). If A is open and S is semi-open in a topological space X, then A∩S is semi-open in X . Lemma 1 ([1]). Let A⊆ Y ⊆ X , where X is a topological space and Y is an open subspace of X . If A∈ RSO(X ), then A∈ RSO(Y ). Lemma 2 ([1]). Let Y be regular open in X and U be a subset of Y . Then U is regular semi-open in X if and only if U is regular semi-open in the subspace Y . Lemma 3 ([6]). Let x be a point of (X ,τ). Then {x} is either nowhere dense or pre-open. Lemma 4 ([5]). If A is regular semi-open in (X ,τ), then X − A is also regular semi-open. Lemma 5 ([1]). For any subset A of (X ,τ), A⊆ rsker(A). Ennis Rosas, N. Selvanayaki, Gnanambal Ilango / Eur. J. Pure Appl. Math, 9 (2016), 27-33 29 3. αgrw-Closed Sets Proposition 1. In a space (X ,τ), if RSO(X ) = {;, X }, then every subset of X is an αgrw-closed set. Proof. Let RSO(X ) = {;, X } and A be any subset of X . Suppose A= ;, then A is an αgrw- closed set in X. Suppose A 6= ;, then X is the only regular semi-open set containing A and so αcl(A) ⊆ X . Hence A is αgrw-closed. Remark 1. The converse of the above proposition need not be true as seen from the following example. Example 1. Let X = {a, b, c} with topology τ = {;, {a}, {b, c}, X }. Then every subset of X is αgrw-closed in X but RSO(X ) = {;, {a}, {b, c}, X }. Proposition 2. Every subset of (X ,τ) is αgrw-closed if and only if RSO(X ,τ) ⊆ {F ⊆ X : F c ∈ τα}, where τα is the topology generated by the α-open sets in (X ,τ). Proof. Suppose that every subset of (X ,τ) is αgrw-closed. Let U ∈ RSO(X ,τ). Since U ⊆ U and U is αgrw-closed, we have αcl(U) ⊆ U . Thus U ∈ {F ⊆ X : F c ∈ τα} and hence RSO(X ,τ) ⊆ {F ⊆ X : F c ∈ τα}. Conversely, assume that RSO(X ,τ) ⊆ {F ⊆ X : F c ∈ τα}. Let A be any subset of (X ,τ) such that A⊆ U , where U is regular semi-open. Thus U is α-closed and so αcl(A) ⊆ U . Hence A is αgrw-closed in X. Proposition 3. If A is both open and g-closed in X , then it is αgrw-closed in X. Proof. Let A be open and g-closed in X. Let A ⊆ U and U be regular semi-open in X. Now A⊆ A, we have cl(A) ⊆ A. This implies αcl(A) ⊆ U . Hence A is αgrw-closed in X . Remark 2. If A is both open and αgrw-closed in X , then A need not be g-closed in X . Example 2. Let X = {a, b, c} with topology τ = {;, {a}, {b}, {a, b}, X }.Then A= {a, b} is both open and αgrw-closed but not g-closed. Proposition 4. If A is regular semi-open and αgrw-closed, then A is α-closed. Proof. Suppose A is regular semi-open and αgrw-closed. We have αcl(A) ⊆ A. Since A⊆ αcl(A) always, αcl(A) = A. Hence A is α-closed. Example 3. In Example 2, the set {b, c} is α-closed and αgrw-closed but is not regular semi-open. Corollary 1. Let A be regular semi-open and αgrw-closed in X. Then A∩ F is αgrw-closed in X, where F is α-closed. Ennis Rosas, N. Selvanayaki, Gnanambal Ilango / Eur. J. Pure Appl. Math, 9 (2016), 27-33 30 Proof. Since A is regular semi-open and αgrw-closed then by Proposition 4, we have A is α-closed. Therefore A∩ F is α-closed, since F is α-closed. Hence A∩ F is αgrw-closed. Proposition 5. If A is both open and αg-closed, then A is αgrw-closed. Proof. Let A be an open and αg-closed. Let A⊆ U and U be regular semi-open. Now A⊆ A and by hypothesis αcl(A) ⊆ A. Therefore αcl(A) ⊆ U . Hence A is αgrw-closed. Remark 3. If A is both open and αgrw-closed, then A need not be αg-closed. Example 4. Let X = {a, b, c, d} with topology τ = {;, {a}, {b}, {a, b}, {a, b, c}, X }. Then the subsets {a, b} and {a, b, c} are αgrw-closed and open but not αg-closed. Remark 4. Difference of two αgrw-closed sets is not generally αgrw-closed. Example 5. Let X = {a, b, c, d} with topology τ= {;, {a}, {b}, {a, b}, {a, b, c}, X }. Then the sets A= {a, c, d} and B = {c, d} are αgrw-closed but A− B = {a} is not αgrw-closed. Proposition 6. Let B ⊆ A⊆ X . If A is open in X , then A∈ αgrwC(X ) implies A∈ αgrwC(Y ). Proof. Let A be αgrw-closed in X and let A⊆ G where G is regular semi-open in Y . Then G = U ∩ Y , where U is regular semi-open in X by Lemma 1. This implies A ⊆ U . Since A is αgrw-closed in X , αcl(A) ⊆ U and so αcl(A) ∩ Y ⊆ U ∩ Y . Therefore αclY (A) ⊆ G. Hence A∈ αgrwC(Y ). Proposition 7. Suppose B ⊆ A ⊆ X , B is αgrw-closed relative to A and A is both regular open and αgrw-closed subset of X . Then B is αgrw-closed in X . Proof. Let B ⊆ U and U be regular semi-open in X . Then we have B ⊆ A∩ U . Since A is open and U is semi-open in X by Theorem 2, A∩ U is semi-open in X . Since every regular- open set is regular semi-open and every regular semi-open set is semi-closed, A and U are semi-closed. Therefore A∩ U is semi-closed in X . Thus A∩ U is regular semi-open in X . Also A∩ U ⊆ A ⊆ X and A is open subspace of X by Lemma 1, A∩ U is regular semi-open in A. Since B is αgrw-closed relative to A, αclA(B) ⊆ A∩ U . But αclA(B) = A∩αcl(B). This implies A∩αcl(B) ⊆ A∩ U and we have A∩αcl(B) ⊆ U . Since A is regular open and αgrw-closed by Proposition 4, αcl(A) = A and so αcl(B) ⊆ A. Thus αcl(B) ⊆ U and hence B is αgrw-closed in X. Proposition 8. If a subset A of (X ,τ) is αgrw-closed, then αcl(A) − A contains no non-empty regular closed set. Proof. Suppose that A isαgrw-closed in (X ,τ) and F be a regular closed subset ofαcl(A)−A. Then A⊆ F c . Since every regular open set is regular semi-open and A is αgrw-closed, αcl(A) ⊆ F c . Consequently F ⊆ [αcl(A)]c . Thus F ⊆ αcl(A)∩ [αcl(A)]c = ;. Hence αcl(A)− A contains no non-empty regular closed set. Ennis Rosas, N. Selvanayaki, Gnanambal Ilango / Eur. J. Pure Appl. Math, 9 (2016), 27-33 31 Remark 5. The converse of the above proposition need not be true. In Example 2, let A = {a}. Then αcl(A)−A= {c} does not contain non-empty regular closed set, but A is not an αgrw-closed set. Proposition 9. Let A⊆ Y ⊆ X and Y is regular open in X then A is αgrw-closed in Y whenever A is αgrw-closed in X . Proof. Let A be αgrw-closed in X and Y be regular open subset of X . Let U be any regular semi-open set in Y such that A⊆ U . By Lemma 2, U is regular semi-open in X . Then we have αcl(A) ⊆ U . That is Y ∩ αcl(A) ⊆ Y ∩ U = U . Thus αclY (A) ⊆ U and hence A is αgrw-closed in Y . Proposition 10. A subset A of (X ,τ) is αgrw-closed if and only if αcl(A) ⊆ rsker(A). Proof. Suppose that A is αgrw-closed. Let x ∈ αcl(A). Suppose x /∈ rsker(A), then there is a regular semi-open set U containing A such that x /∈ U . Since U is regular semi-open containing A, we have x /∈ αcl(A), which is a contradiction. Thus αcl(A) ⊆ rsker(A). Conversely, let αcl(A) ⊆ rsker(A). If U is any regular semi-open set containing A, then αcl(A) ⊆ rsker(A) ⊆ U . Therefore A is αgrw-closed. Remark 6 ([4]). In the notion of Lemma 3, we may consider the following decomposition of a given topological space (X ,τ), namely X = X1 ∪ X2, where X1 = {x ∈ X : {x} is nowhere dense} and X2 = {x ∈ X : {x} is preopen}. Proposition 11. For any subset A of (X ,τ), X2 ∩αcl(A) ⊆ rsker(A). Proof. Let x ∈ X2 ∩ αcl(A) and suppose that x /∈ rsker(A). Then there is a regular semi- open set U containing A such that x /∈ U . If F = X −U , then F is regular semi-closed and so F is semi-closed. We have scl({x}) = {x} ∪ int(cl({x})) ⊆ F . Since αcl({x}) ⊆ αcl(A), we have int(cl({x})) ⊆ A∪ int(cl(A)). Again since x ∈ X2, we have x /∈ X1 and so int(cl({x})) 6= ;. Therefore there has to be some point y ∈ A∩ int(cl({x})) and hence y ∈ F∩A, a contradiction. Thus x ∈ rsker(A). Hence X2 ∩αcl(A) ⊆ rsker(A). Proposition 12. For any subset A of (X ,τ), if X1 ∩αcl(A) ⊆ A, then A is αgrw-closed in X . Proof. Suppose that X1 ∩ αcl(A) ⊆ A. Then X1 ∩ αcl(A) ⊆ rsker(A), since A ⊆ rsker(A). Now αcl(A) = X ∩αcl(A) = (X1∪X2)∩αcl(A) = (X1∩αcl(A))∪(X2∩αcl(A)) ⊆ rsker(A), since X1 ∩αcl(A) ⊆ rsker(A) and by Proposition 11. Thus A is αgrw-closed by Proposition 10. Proposition 13. Let X be a regular space in which every regular semi-open subset is open. If A is compact subset of X , then A is αgrw-closed. Proof. Let A⊆ U and U be regular semi-open. By assumption U is open in X . Since A is a compact subset of a regular space X , then there exists a closed set V such that A⊆ V = cl(V ) ⊆ U . Thus cl(V ) ⊆ U and so αcl(A) ⊆ U . Hence A is αgrw-closed. REFERENCES 32 Proposition 14. If (X ,τ) is s-normal and F ∩ A = ;, where F is regular semi-open and A is αgrw-closed , then there exist disjoint semi-open sets S1 and S2 such that A⊆ S1 and F ⊆ S2. Proof. Since F is regular semi-open and F ∩ A= ;. Then A⊆ F c and so αcl(A) ⊆ F c . Thus αcl(A) ∩ F = ;. Since αcl(A) and F are semi-closed and X is s-normal, there exist semi-open sets S1 and S2 such that αcl(A) ⊆ S1 and F ⊆ S2. 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