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International Journal of Conceptions on Computing and Information Technology Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808 g* λ* - closed sets in topological spaces N Murugavalli A Pushpalatha Dept. of Mathematics, Sri Eshwar College of Engineering, Coimbatore, India. [email protected] Dept. of Mathematics, Government Arts College, Udumalpet, Tamilnadu, India. [email protected] Abstract— In this paper, we introduce the new notion g**-closed sets and g**-open sets in Topological spaces. Also we introduce a new space called and Tg** -space. We study some properties of g**- closed sets and g**-open sets in Topological spaces. Keywords- g**-closed sets ,g**-open sets and Tg* *- space I. INTRODUCTION N. Levine [12] introduced generalized closed sets in general topology as a generalization of closed sets. This concept was found to be useful and many results in general topology were improved. Many researchers like Balachandran, Sundaram and Maki [4], Bhattacharyya and Lahiri [5], Arockiarani [1], Dunham [9], Gnanambal [10], Malghan [13], Palaniappan and Rao [17], Park [18] Arya and Gupta [2] and Devi [7] have worked on generalized closed sets, their generalizations and related concepts in general topology.P.Sundaram and A.Pushpalatha introduced λ-closed sets in topological space[19]. N.Murugavalli and A.Pushpalatha introduced g*-closed sets in topological space[15]. Now, we introduce the concept of g**- closed sets and g -open sets in Topological space and study some of their properties. * * II. PRELIMINARIES Definition 1.1[19]: A subset A of a topological space X is called a -closed set in X if A cl(G) whenever A G and G is open in X. Definition1.2[15]: A subset A of a topological space X is called a g*-closed set in X if A cl(G) whenever A G and G is g-open in X. Definition 1.3: A subset S of X is called (a) semi-open [11] if there exists an open set G such that G S cl(G) and semiclosed [6] if there exists an closed set F such that int(F) S F. Equivalently, a subset S of X is called semi-open if S cl(int(S)) and semi-closed if Sint(cl(S)) [4]. Definition 1.4: A subset S of X is called (a) generalized closed(g-closed)set if cl(S) G whenever S G and G is open in X and generalized open if Sc is gclosed set in X [12]. Generalized closure of S, denoted by c*(S), is defined as the intersection of all g-closed sets containing S in X [9]. (b) semi generalized closed (sg-closed) if scl(S)G whenever SG and G is semi open in X and semi generalized open if Sc is sg-closed set in X [5]. (c) generalized semi closed (gs-closed) if scl(S)G whenever SG and G is open in X and generalized semi open if Sc is gs-closed set in X[3]. (d) generalized semi-preclosed set (gsp-closed) if spcl(S) G whenever S G and G is open in X [8]. III. G* Λ* - CLOSED SETS IN TOPOLOGICAL SPACES In this section, we introduce the concept of g**-closed sets in topological spaces and study some of its properties. Definition 2.1: A subset A of a topological space X is called a g**-closed set in X if A cl(G) whenever A G and G is g**-open in X. Theorem 2.2: If a subset A of a topological space X is closed in X, then it is g**-closed but not conversely. Proof: Assume that A be a closed set in X. Let G be an g*-open set in X such that AG. Then cl(A) cl(G). But cl(A) = A. Therefore A cl(G) and hence A is g**-closed in X. The following example shows that the converse of the above theorem need not be true. Example 2.3: Let X = {a, b, c} be a topological space with the topology = {, X, {a}, {a, c}}. Then the set {a,b} is g**-closed but not closed. Theorem 2.4: If A and B are g**-closed sets in a topological space X, then A B is g**-closed in X. (b) -closed if cl(int(cl(S))) S and an -open if Sint(cl(int(S))) [16]. (c) pre closed if cl (int(S)) S and pre-open if Sint(cl(S)) [14]. 101 | 1 2 4 Proof: Let A B G, where G is g**-open. A G, B G A cl(G), B cl(G), since A, B are g**-closed. International Journal of Conceptions on Computing and Information Technology Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808 A B cl(G) A B is g**-closed. Theorem 2.5: If A is open and g**-closed, then A is closed. Proof: Assume that A is open and g**-closed in X. Then A cl(A), since A is open and g**-closed. But A cl(A) always. Therefore A = cl(A). Thus A is closed set in X. Remark 2.17: The concepts of -closed sets and g**closed sets are independent as seen from the following examples.For, in Example 2.3 the set {c} are -closed, but not g**-closed. In example 2.10, The sets {a,c} & {b,c} are g**-closed but not -closed. From the definitions and from the above results, we obtain the following diagram: p-closed sg-closed gsp-closed Remark 2.6: If A and B are g**-closed in X, then their union is also g**-closed in X as seen from the following example. Example 2.7: Let X = {a, b, c} be a topological space with topology = { , X, {a, b}}. The sets {a,b} and {b} are g**closed, their union {a, b} is also g**-closed in X. g**-closed s-closed -closed * * Remark 2.8: The concepts of g -closed set and -closed set are independent as seen from the following examples. Example 2.9: Let X = {a, b, c} be a topological space with the topology = {, X, {a}}. The sets {a}, {a,c} &{a,b}are g**-closed but not -closed. Example 2.10: Let X = {a, b, c} be a topological space with the topology = {, X, {a, b}}. The sets {a}& {b} are closed but not g**-closed. Remark 2.11: The concepts of semi-closed sets and closed sets are independent as seen from the following examples. Example 2.12: Let X = {a, b, c} be a topological space with the topology = {, X, {a}, {a, c}}. The set {c} is semiclosed but not g**-closed. The set {a,b} is g**-closed but not semi-closed. Remark 2.13: The concepts of sg-closed set and g**closed are independent. For, in Example 2.9, the sets {a},{a,c} and {a,b} are g**-closed, but not sg-closed. In Example 2.12, the set {b} is sg-closed but not g**-closed set in X. Remark 2.14: The concepts of semi pre-closed sets and g**-closed sets are independent as seen from the following examples. For, in Example 2.9 the set {a},{a,c} and {a,b} are g**-closed, but not sp-closed.In example 2.10, The sets {a} & {b} are sp-closed but not g**-closed. sp-closed Tg** - space : A topological space X is called a Tg** - space if every g**-closed set in X is g*-closed set in X. IV. Remark 2.16: The concepts of b-closed sets and g**closed sets are independent as seen from the following examples. For, in Example 2.9 the set {a},{a,c} and {a,b} are g**-closed, but not b-closed.In example 2.10, The sets {a} & {b} are b-closed but not g**-closed. G* Λ* - CLOSED SETS IN TOPOLOGICAL SPACES In this section, we introduce the concept of g**-open sets, which is weaker than that of open set in topological spaces and study some of its properties. Definition 3.1: A subset A of a topological space X is called g**-open if Ac is g**-closed in X. Theorem 3.2: A subset A of a topological space X is g**open if and only if A int(F) whenever A F and F is gclosed in X. Proof: Assume that A is g**-open and F is a g-closed set in X such that A F. Ac Fc Ac cl(Fc), since Ac is g**-closed . Ac (int(F))c * * Remark 2.15: The concepts of gsp-closed sets and g closed sets are independent as seen from the following examples. For, in Example 2.9 the set {a} is g**-closed, but not gsp-closed.In example 2.10, The sets {a} & {b} are gspclosed but not g**-closed. -closed A int(F) Conversely assume that A int(F) whenever A F and F is g-closed in X. We have to prove that Ac is g**-closed in X. Let Ac G, where G is g-open in X. Then A Gc, which implies A int(Gc) by our hypothesis. 102 | 1 2 4 A (cl(G))c, since (cl(G))c = int(Gc) Ac cl(G) Ac is g**-closed. Thus A is g**-open. International Journal of Conceptions on Computing and Information Technology Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808 Theorem 3.3: If a subset A of a topological space X is open in X, then it is g**-open but not conversely. [4] Proof: Let A be a open set in X. Let A F, where F is closed. Since every open set is g-open set, then int(A) int(F). But int(A) = A. Therefore A int(F). Thus A is g**open in X. [5] The following example shows that the converse of the above theorem need not be true. [7] Example 3.4: Let X = {a, b, c} be a topological space with the topology = {, X, {b},{b,c}}. The set {c} is g**-open in X but not open in X. Theorem 3.5: If A and B are g**-open sets in X, then A B is also a g**-open set in X. [6] [8] [9] [10] [11] * * Proof: Assume that A and B are g -open sets in X. Then Ac and Bc are g**-closed sets in X and so Ac Bc = (A B)c is g**-closed. Thus A B is g**-open set in X. [12] [13] * * Remark 3.6: If A and B are g -open sets then their intersection, A B is also g**-open as seen from the following example. Example 3.7: In Example 2.10 the sets {a} and {b} are g -open, their intersection {a,b} is also g**-open in X. [14] [15] * * REFERENCES [1] [2] [3] [16] I.Arockiarani, Studies on generalizations of generalized closed sets and maps in topological spaces, Ph.D Thesis, BharathiarUniversity,Coimbatore, (1997). S.P.Arya and R.Gupta, On strongly continuous mappings, Kyungpook Math., J. 14(1974), 131-143. Arya, S. 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