A new class of generalized delta semiclosed sets using grill delta space

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1 A new class of generalized delta semiclosed sets using grill delta space K.Priya 1 and V.Thiripurasundari 2 1 M.Phil Scholar, PG and Research Department of Mathematics, Sri S.R.N.M.College, Sattur , Tamil Nadu, India. 2 Assistant Professor, PG and Research Department of Mathematics, Sri S.R.N.M.College, Sattur , Tamil Nadu, India. Abstract In this paper we apply the notion of generalized semiclosed sets and obtain a new class of generalized delta semiclosed sets using grill delta space. Also we investigate the properties of generalized delta semiclosed sets. Keywords: Grill, topology, operator Φ, G-gs Mathematical Subject Classification: 54A10 1 Introduction The concept of grill topological spaces depended on the two operators are Φ and Ψ. This concept was first introduced by Choquet [2] in Also, for the investigation of many topological notions similar compactifications, proximity spaces and extension problems of different kinds, Rodyna A. Hosny[6] introduce the concept -set using grill and obtain grill delta space. In this paper, we explore the concept of generalized semiclosed sets to define a new class of generalized delta semiclosed sets via grill delta space. 2 Preliminaries Throughout this paper (X, τ ) (or simply X) represent topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of X, cl(a), int(a) and A c denote the closure of A, the interior of A and the complement of A respectively. Definition 2.1. [1] A collection G of non empty subsets of a space X is called grill on X if (i) A G and A B X B G (ii) A, B X and A B G A G or B G. Definition 2.2. [4]The -interior of a subset A of X is the union of all regular open set of X contained in A and is denoted by int (A). The subset A is called -open if A = int (A), i.e. a set is -open if it is the union of regular open sets. The complement of a -open is called Alternatively, a set A (X, τ ) is called -closed if A = cl (A), where cl (A) = {x X : int(cl(u)) A φ, U τ and x U} Definition 2.3. [6]Let (X, τ, G) be a grill delta space. We define a mapping Φ : P(X) P(X) denoted by Φ G (A, τ ) (for A P(X)) or Φ G (A) or simply Φ (A), called the operator associated with grill G and the topology τ, and is defined by Φ G = {x X : A U x G, U x O(X, τ )} Definition 2.4. [6]Let (X, τ,g) be a grill delta space. We define a map Ψ G : P(X) P(X) by Ψ G (A) = X \ (Φ (X \ A)) or Ψ (A) = A Φ (A) for all A P(X). 461

2 Definition 2.5. [6]Corresponding to a grill delta space (X, τ, G) there exists a unique topology (say) on X given by = {U X : Ψ (X \U) = (X \U)} where for any A X, Ψ (A) = A Φ (A) = cl (A). Theorem 2.6. [6]Let (X, τ, G) be a grill delta space. Then (i) A B ) Φ (A) Φ (B). (ii) Φ(A B) = Φ (A) Φ (B), for any A X. (iii) Φ (Φ (A)) Φ (A) = cl (Φ (A)) cl (A), for any A, B X. (iv) A X and A G Φ (A) = φ. Definition 2.7. [5] A subset A of X is called generalized -semiclosed (briefly gs closed) if scl(a) U whenever A U and U is -open in X. The family of all gs-closed subsets of the space X is denoted by GSC(X). 3 Generalized delta semiclosed sets with respect to a grill delta space Definition 3.1. Let (X, τ, G) be a grill delta space. Then a subset A of X is said to be gs-closed with respect to grill G (G-gs-closed, for short) if Φ (A) U and U is -open in X. Definition 3.2. A subset A of X is said to be G-gs-open if X \ A is G-gsclosed. Proposition 3.3. For any (X, τ, G) grill delta space, (i) Every closed set in X is G-gs (ii) For any set A in X, Φ (A) is G-gs (iii) Every -closed set is G-gs (iv) Any non member of G is G-gs (v) Every gs-closed is G-gs (i)let A be a closed set. Then cl(a) = A. Let U be a -open set in X such that A U. Then, Φ (A) = cl (A) cl(a) = A U Φ (A) U A is G-gs (ii)let A be a subset in X. Then Φ (Φ (A)) Φ (A) U Φ (A) is G-gsclosed. (iii)let A be -closed set then cl (A) = A A Φ = A Φ (A) A. Therefore, Φ (A) U whenever A U and U is -open in X. This implies A is G-gs (iv)let A G then Φ (A) = φ A is G-gs (v)let A be a gs-closed set and U be a -open in X, such that A U, then scl(a) U, consider Φ (A) cl (A) scl(a) U A is G-gs Thus every gs-closed set is G-gs Remark 3.4. The converse of above proposition 3.3(v) need not be true as shown in the following example. Example 3.5. Let X = {a, b, c}, τ = {φ, X, {a}, {b}, {a, b}}, τ = {φ, X, {a}, {b}, {a, b}} and G = {{a}, {b}, {a, b}, X}, then (X, τ ) is a space and G is a grill on X. Let A = {a, b} then Φ (A) = {a, b} U, where U is -open in X. Therefore, A is G-gs But scl(a) = X does not subset of {a, b}. Therefore, A is not gs Definition 3.6. Let X be a grill delta space and (φ )A X. Then [A] = {B X : A B φ } is a grill on X, called the principal grill generated by A. Proposition 3.7. In case of [X] principal grill generated by X, it is known that τ = τ [X] so that any [X]-gs-closed set becomes simply a gs-closed set and vice-versa. 462

3 Theorem 3.8. Let (X, τ, G) be a grill delta space. If a subset A of X is G-gs-closed then cl (A) U whenever A U and U is -open. Let A be a G-gs-closed set and U be a -open in X such that A U then Φ (A) U A Φ (A) U cl (A) U. Thus cl (A) U whenever A U and U is -open. Theorem 3.9. Let (X, τ, G) be a grill delta space. If a subset A of X is G-gs-closed then for all x cl (A), cl ({x}) A φ. Let x cl (A). If cl ({x}) A = φ A X \ cl ({x}) then by Theorem 3.8, cl (A) X \ cl ({x}) which is a contradiction to our assumption that x cl (A). Therefore, cl ({x}) A φ. Theorem Let (X, τ, G) be a grill delta space. If a subset A of X is G-gs-closed then cl (A) \ A contains no non empty closed set of (X, τ ). Moreover Φ (A) \ A contains no non-empty closed set of (X, τ ). Let F be a closed set contained in cl (A) \ A and let x F, since F A = φ we get cl ({x}) A = φ which is a contradiction to fact that cl ({x}) A φ. cl (A) \ A contains no non-empty closed set of (X, τ ). Since Φ (A) \ A = cl (A) \ A, Φ (A) \ A contains no non-empty closed set of (X, τ ). Corollary Let (X, τ ) be a T 1 -space and G be a grill on X. Then every G-gs-closed set is Let A be a G-gs-closed set and x Φ (A). Then x cl (A). By Theorem 3.8, cl ({x}) A φ, {x} A φ, x A. Therefore, Φ (A) A. Thus A is Corollary Let (X, τ ) be a T 1 -space and G be a grill on X. Then A( X) is G-gs-closed set if and only if A is Proposition Let G be a grill on a space (X, τ ) and A be a G-gs-closed set. Then the following are equivalent: (i) A is (ii) cl (A) \ A is closed in (X, τ ). (iii) Φ (A) \ A is closed in (X, τ ) (i) (ii)let A be -closed then cl (A) \ A = φ so cl (A) \ A is a closed set. (ii) (iii)since cl (A) \ A = Φ (A) \ A, Φ (A) \ A is closed in (X, τ ). (iii) (i)let Φ (A) \ A be closed in (X, τ ). Since A is G-gs-closed by theorem 3.10, Φ (A) \ A = φ. So A is Lemma Let (X, τ ) be a space and G be a grill on X. If A( X) is -dense in itself, then Φ (A) = cl (Φ (A)) = cl (A) = cl (A). Let A be -dense in itself A Φ (A) cl (A) cl (Φ (A)) = Φ (A) cl (A) cl (A) = Φ (A) = cl (Φ (A)). Now by definition cl (A) = A Φ (A) = A cl (A) = cl (A). Therefore, Φ (A) = cl (Φ (A)) = cl (A) = cl (A). Theorem Let G be a grill on a space (X, τ ). If A( X) is -dense in itself and G-gs-closed, then A is gs Follows from Lemma Corollary For a grill G on a space (X, τ ). Let A( X) be -dense in itself. Then A is G-gs-closed if and only if A is gs Follows from proposition 3.3(v) and theorem Theorem For any grill on a space (X, τ ) the following are equivalent: (i) Every subset of X is G-gs (ii) Every -open subset of (X, τ ) is 463

4 (i) (ii)let A be -open in (X, τ ). Then by (i), A is G-gs-closed so that Φ (A) A A is (ii) (i)let A X and U be -open in (X, τ ) such that A U. Since U is -open by (ii), Φ (U) U. Now A U Φ (A) Φ (U) U A is G-gs Theorem For any subset A of a space (X, τ ) and a grill G on X. If A is G-gs-closed then A (X \ Φ (A)) is G-gs Let A (X \ Φ (A)) U, where U is -open in X. Then X \ U X \ (A (X \ Φ (A))) = Φ (A) \ (A). Since A is G-gs-closed, by theorem 3.8, we have X \ U = φ, that is X = U. Since X is the only -open set containing A (X \ Φ (A)), A (X \ Φ (A)) is G-gs Proposition For any subset A of a space (X, τ ) and a grill G on X, the following are equivalent: (i) A (X \ Φ (A)) is G-gs (ii) Φ (A) \ A is G-gs-open. Follows from the fact that X \ (Φ (A) \ A) = A (X \ Φ (A)). Theorem Let (X, τ ) be a space, G be a grill on X and A, B be subsets of X such that A B cl(a). If A is G-gs-closed, then B is G-gs Let B U, where U is -open in X. Since A is G-gs-closed, Φ (A) U cl (A) U. Now, A B cl (A) cl (A) cl (B) cl (A). Thus cl (B) U and hence B is G-gs Corollary closure of every G-gs-closed set is G-gs Theorem Let G be a grill on a space (X, τ ) and A, B be subsets of X such that A B Φ (A). If A is G-gsclosed, Then A and B are gs A B Φ (A) A B cl (A) and hence by theorem 3.20, B is G-gs Again, A B Φ (A) Φ (A) Φ (B) Φ (Φ (A)) Φ (A) Φ (A) = Φ (B). Thus A and B are -dense in itself and hence by Theorem 3.15, A and B are gs Theorem Let G be a grill on a space (X, τ ). Then a subset A of X is G-gs-closed if and only if F int (A) whenever F A and F is closed. Let A be G-gs-open and F A, where F is closed in (X, τ ). Then X \ A X \ F Φ (X \ A) X \ F cl (X \ A) X \ F F int (A). Conversely, X \A U where U is open in (X, τ ) X \U int (A) cl (X) U. Thus (X \A) is G-gs-closed and hence A is G-gs-open. Conclusion In this paper, we introduce a new class of generalized delta semiclosed sets using grill delta space and study some of their properties. References [1] A. A. Nasefa, A. A. Azzam, Some topological operators Via grills, Journal of Linear and Topological Algebra, Vol. 05, No. 03, 2016, [2] G. Choquet, Sur les notions de filter et grills, Comptes Rendus Acad. Sci. Paris, 224(1947), [3] I. Arockiarani and V. Vinodhini, A new class of generalized semiclosed sets using grills, Scientia Magna, Vol. 8 (2012), No. 2,

5 [4] N. V. Velicko, H-closed topological spaces, Amer.Math.Soc.Transl., 78(1968), [5] S. S. Benchalli and Umadevi I. Neeli, Generalized Delta semiclosed sets in Topological spaces, Int. Jl. Appl. Mahs, Vol. 24. No. 1(2011), [6] Rodyna A. Hosny, -Sets with Grill, International Mathematical Forum, Vol. 7, 2012, no. 43,