Operation Approaches on α-γ-open Sets in Topological Spaces

Transcript

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 10, Operation Approaches on α-γ-open Sets in Topological Spaces N. Kalaivani Department of Mathematics VelTech HighTec Dr.Rangarajan Dr.Sakunthala Engineering College Chennai, India G. SaiSundaraKrishnan Department of Applied Mathematics and Computational Sciences PSG College of Technology Coimbatore, India Abstract In this paper the notion of α-γ-open sets in a topological space together with its corresponding interior and closure operators are introduced. Further some of their basic properties are studied. Mathematics Subject Classification: 54A05, 54A10 Keywords: α-γ-open set,τ α γ -int(a), τ α γ -cl(a) 1 Introduction O. Najastad[10] introduced α-open sets in a topological space and studied some of their properties. The concept of semiopen sets, preopen sets and semi-preopen sets were introduced respectively by Levine[8], Mashhour[9] and Andrijevic[1]. Andrijevic[2] introduced a new class of topology generated by preopen sets and the corresponding closure and interior operators. Kasahara defined the concept of an operation on topological spaces and introduced α - closed graphs of an operation. Ogata[11] called the operation α as γ operation and introduced the notion of τ γ which is the collection of all γ- open sets in a topological space (X,τ). In this paper in section 3 we introduce the notion of τ α γ which is the collection of all α-γ-open sets in a topological space (X,τ). Further we introduce

2 492 N. Kalaivani and G. SaiSundaraKrishnan the concept of τ α γ interior and τ α γ closure operators and study some of their properties. 2 Preliminaries In this section we recall some of the basic Definitions and Theorems. Definition 2.1 Let( X,τ) be a topological space and A be a subset of X. Then A is said to be (i)[10] α- open set if A int(cl(int(a))) (ii)[7] semi-open set if A cl(int(a)) (iii)[9] pre-open set if A int(cl(a)) (iv)[9] semi-preopen set if A cl(int(cl( A))) Definition 2.2 Let(X,τ) be a topological space,an operation γ on the topology τ is a mapping from τ on to the power set P(X) of X such that V V γ for each V τ,where V γ denotes the value of γ at V. Definition 2.3 Let (X,τ) be a topological space and A be a subset of X and γ be an operation on τ. Then A is said to be: (i) [11] a γ- open set if for each x A there exists an open set U such that x U and U γ A. τ γ denotes the set of all γ-open sets in(x,τ). (ii) [14] γ- semi-open if and only if A τ γ cl(τ γ int(a)). (iii)[12] γ- preopen if and only if A τ γ int(τ γ cl(a)). (iv) [12] γ- semi preopen if and only if A τ γ cl(τ γ int(τ γ cl(a))) Definition 2.4 (i)[14] Let (X,τ ) be a topological space and γ be an operation on τ.then τ γ - interior of A is defined as the union of all γ- open sets contained in A and it is denoted τ γ int(a). That is τ γ int(a) = {U : Uis aγ open set andu A} (ii)[11] Let (X,τ ) be a topological space and γ be an operation on τ. Then τ γ - closure of A is defined as the intersection of all γ- closed sets containing A and it is denoted by τ γ cl(a). That is τ γ cl(a) = {F : F is aγ closed set anda F } Theorem 2.5 Let (X,τ ) be a topological space. Then (i)[12] A subset A is γ- preclosed if and only if τ γ cl(τ γ int(a)) A (ii)[12] A subset A is γ- semi preclosed if and only if τ γ int(τ γ cl(τ γ int(a))) A 3 α-γ-open set Definition 3.1 Let (X,τ) be a topological space and γ be an operation on τ. Then a subset A of X is said to be a α-γ- open set if and only if A τ γ int(τ γ cl(τ γ int(a))) Example 3.2 Let X = {a, b, c, d}, τ = {φ, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c},

3 Operation approaches 493 {a, b, d}}. We define an operation γ : τ P (X) as follows: for every A τ, { int(cl(a)) ifa {a} A γ = cl(a) if A = {a} Then τ γ = {φ, X, {a}, {c}, {a, c}, {a, b, d}} and τ α γ = {φ, X, {a}, {c}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}} Theorem 3.3 Let (X,τ) be a topological space and γ be an operation on τ. Then every γ- open set in (X,τ) isaα-γ-open set. However, the converse need not be true. Proof: Proof is straight forward from the Definition 3.1 In Example 3.2 {a, b}, {a, d}, {a, b, c}, {a, c, d} are α-γ- open sets but not γ- open sets. Theorem 3.4 Let (X,τ) be a topological space and γ be an operation on τ and {A α : α J} be the family of α-γ- open sets in (X,τ ). Then α J A α is also a α-γ-open set. Proof : Given {A α : α J} be the family of α-γ- open sets in (X,τ ). Then for each A α,a α τ γ int(τ γ cl(τ γ int(a α ))). This implies that A α [τ γ int (τ γ cl (τ γ int (A α )))] and hence A α [τ γ int(τ γ cl(τ γ int( A α )))]. Therefore we have α J A α is also a α-γ- open set. Remark 3.5(i) Let (X,τ) be a topological space and γ be an operation on τ. If A, B are any two α-γ- open sets in (X,τ), then the following example shows that A B need not be a α-γ-open set. Let X = {a, b, c},τ = {φ, X, {a}, {b}, {a, b}, {a, c}},define an operation γ on τ such that { cl(a) if b / A A γ = A if b A Then τ α γ = {φ, X, {b}, {a, b}, {a, c}}. A={a, b} and B ={a, c} are α-γ-open sets but A B = {a} is not a α-γ-open set. (ii) The following example shows that the concepts of α-open set and α-γ-open set are independent. Let X = {a, b, c}, τ = {φ, X, {a}, {b}, {a, b}, {a, c}}, the α-open sets are {φ, X, {a}, {b}, {a, b}, {a, c}}. We define an operation γ on τ such that γ(b) =Cl(B). Then τ γ = {φ, X, {b}, {a, c}} and τ α γ = {φ, X, {b}, {a, c}}. Here {a}, {a, b} are α-open sets but not α-γ-open sets. Similarly in example 3.2 {a, d}, {a, c, d} are α-γ-open sets but not α-open sets. Theorem 3.6 If (X,τ) is a γ-regular space, then the concept of α-γ-open set and α-open set coincide. Proof : Proof follows from the Proposition 2.4 [9] and the Theorem 3.6 [9]. Definition 3.7 Let (X,τ ) be a topological space and γ be an operation on τ

4 494 N. Kalaivani and G. SaiSundaraKrishnan and A be a subset of X. A is said to be α-γ- closed if and only if X Ais α-γ- open,which is equivalently A is α-γ- closed if and only if A τ γ cl(τ γ int(τ γ cl(a))). Theorem 3.8 Let (X, τ) be a topological space and γ be an operation on τ. (i) Every α-γ- open set is γ-semi-open. (ii) Every α-γ- open set is γ-preopen. (iii) Every α-γ- open set is γ-semi preopen. Proof:(i) Let A be a α-γ- open set in (X,τ). Then it follows that A τ γ int(τ γ cl(τ γ int(a))) and hence A τ γ cl(τ γ int(a)). Therefore A is γ-semi-open. (ii) Let A be a α-γ- open set in (X,τ ). Since τ γ int(a) A, implies that τ γ cl(τ γ int(a)) τ γ cl(a) and hence τ γ int(τ γ cl(τ γ int(a))) τ γ int(τ γ cl(a)).this implies that A τ γ int(τ γ cl(a)).therefore A is γ-preopen. (iii)proof is obvious using the(i),(ii)results, Definition 3.11[10] and Remark 3.2[10] Remark 3.9 Let X = {a, b, c}, τ = {φ, X, {a}, {c}, {a, b}, {a, c}},define an operation γ on τ such that { A ifa = {a} A γ = A {c} if A {a} Then τ γ = {φ, X, {a}, {c}, {a, c}}, τ γ SO(X) ={φ, X, {a}, {c}, {a, b}, {a, c}, {b, c}} and τ α γ = {φ, X, {a}, {c}, {a, c}}.here {a, b} and {b, c} are γ-semi-open sets but they are not α-γ-open sets. Remark 3.10 Let X = {a, b, c}, τ = {φ, X, {a}, {b}, {a, b}, {a, c}},define an operation γ on τ such that { A ifb A A γ = cl(a) ifb / A Then τ γ = {φ, X, {b}, {a, b}, {a, c}}, τ γ PO(X) ={φ, X, {a}, {b}, {a, b}, {a, c}, {b, c}}, τ γ SPO(X) ={φ, X, {a}, {b}, {a, b}, {a, c}, {b, c}}and τ α γ = {φ, X, {b}, {a, b}, {a, c}}. Here {a} and {b, c} are γ-preopen sets, γ- semi preopen sets but they are not α-γ-open sets. Theorem 3.11 Let A be a subset of a topological space (X,τ ). If B is a γ-semi-open set of X such that B A τ γ int(τ γ cl(b)), then A is a α-γ-open set of X. Proof: Given B A and B is a γ-semi-open set, implies that τ γ int(b) τ γ int(a) and B τ γ cl(τ γ int(b)). This implies that B τ γ cl(τ γ int(a)) and hence τ γ cl(b) τ γ cl(τ γ int(a)). Therefore τ γ int(τ γ cl(b)) τ γ int(τ γ cl(τ γ int(a)))). Hence by assumption A is a α-γ-open set of X. Theorem 3.12 A subset A is α-γ- open if and only if it is γ-semi-open and

5 Operation approaches 495 γ-preopen. Proof: By Theorem 3.8(i) and (ii) it follows that if A is α-γ- open then A is γ-semi- open and γ-preopen. Conversely if A is γ-semi- open and γ-preopen, then A τ γ cl(τ γ int(a)) and A τ γ int(τ γ cl(a)). This implies that A τ γ int(τ γ cl(τ γ int(a))). Therefore A is α-γ-open. Remark 3.13 The following statements are equivalent for subsets of a topological space (X,τ): (i) Every γ- preopen set is γ-semi-open. (ii) A subset A of X is α-γ- open if and only if it is γ-preopen. Proof: (i) (ii) If A is α-γ-open then by the Theorem 3.8 (ii) A is γ-preopen. Conversely if A is γ- preopen, then by (i) and Theorem 3.12, A is α-γ-open. (ii) (i) Proof follows from the Theorem Similarly we can prove the following Remark. Remark 3.14 The following statements are equivalent for subsets of a topological space (X,τ ): (i) Every γ-semi-open set is γ-preopen. (ii) A subset A of X is α-γ-open if and only if it is γ-semi-open. Theorem 3.15 Let A be a subset of a topological space (X,τ).Then A is γ- clopen if and only if it is α-γ-open and γ-preclosed. Proof: If A is γ-clopen,then by Theorem 3.3 and Theorem 2.12 [12] A is α-γopen and γ-preclosed. Conversely if A is α-γ-open and γ-preclosed then A τ γ int(τ γ cl(τ γ int(a))) and τ γ cl(τ γ int(a)) A implies that A τ γ int(a). This implies that A is γ-open. Since A τ γ int(a), τ γ cl(a) τ γ cl(τ γ int(a)) A. Hence τ γ cl(a) A. Therefore A is γ-clopen. Definition 3.16 (i) Let( X,τ) be a topological space and γ be an operation on τ and A be a subset of X. Then τ α γ - interior of A is the union of all α-γ-open sets contained in A and it is denoted by τ α γ int(a). That is τ α γ -int (A) = {U : Uisaα γ open set and U A} (ii)let (X,τ) be a topological space, S be a subset of X and x be a point of X. Then x is called an α-γ- interior point of S if there exists V τ α γ such that x V S. The set of all α-γ- interior points of S is called α-γ-interior of S and is also denoted by α-γ-int(s). Remark 3.17 Let ( X,τ) be a topological space and γ be an operation on τ. Let A,B be subsets of X. Then the following holds good: (i) τ α γ int(a) is the largest α-γ- open subset of X contained in A. (ii) A is α-γ-open if and only if τ α γ int(a) =A (iii) τ α γ int (τ α γ int(a)) = τ α γ int(a) (iv) If A B then τ α γ int(a) τ α γ int(b) (v) τ α γ int(a) τ α γ int(b) τ α γ int(a B) Proof: (i) follows from the Definition (ii) follows from the Definition 3.16 and Theorem 3.4.

6 496 N. Kalaivani and G. SaiSundaraKrishnan (iii) follows from (ii). (iv) follows from the Definition (v) follows from the Theorem 3.4 and (i). Definition 3.18 Let (X,τ) be a topological space and γ be an operation on τ. Let A be a subset of X. Then τ α γ -closure of A is the intersection of α-γ-closed sets containing A and it is denoted by τ α γ cl(a). That is τ α γ cl(a) = {F : Fisaα γ closed set and A F } Remark 3.19 (i)if A is a subset of (X,τ). Then τ α γ cl(a) isaα-γ- closed set containing A. (ii) A is α-γ- closed if and only if τ α γ cl(a) =A. Proof: (i)follows from the Definition (ii) follows from the Definition 3.18 and Definition 3.7. Theorem 3.20 Let A and B be subsets of (X,τ ). Then the following statements hold: (i) τ α γ cl(τ α γ cl(a)) = τ α γ cl(a) (ii)if A B, then τ α γ cl(a) τ α γ cl(b) (iii)τ α γ cl(a) τ α γ cl(b) τ α γ cl(a B) (iv)τ α γ cl(a B) τ α γ cl(a) τ α γ cl(b) Proof: (i) Proof follows from the Definition (ii) Given A B implies that A τ α γ cl(b) and by (i) τ α γ cl(a) τ α γ cl(b). (iii) A A B and B A B implies that τ α γ cl(a) τ α γ cl(a B) and τ α γ cl(b) τ α γ cl(a B). This implies that τ α γ cl(a) τ α γ cl(b) τ α γ cl(a B) (iv)a τ α γ cl(a), B τ α γ cl(b) and A B τ α γ cl(a) τ α γ cl(b). This implies that τ α γ cl(a B) τ α γ cl(τ α γ cl(a)) τ α γ cl(τ α γ cl(b)). Hence τ α γ cl(a B) τ α γ cl(a) τ α γ cl(b). Theorem 3.21 Let (X,τ) be a topological space and γ be an operation on τ. Then for a point x X,x τ α γ cl(a) if and only if V A φ for any V τ α γ such that x V. Proof: Let F 0 be the set of all y X such that V A φ for every V τ α γ and y V. To prove this theorem it is enough to prove that F 0 = τ α γ cl(a). Let x τ α γ cl(a). Let us assume that x / F 0 then there exists a α-γopen set U of x such that U A = φ. This implies that A X U and hence τ α γ cl(a) X U. Therefore, x / τ α γ cl(a) which is a contradiction and hence τ α γ cl(a) F 0. Conversely, let F be a set such that A F and (X F) τ α γ. Let x / F then we have x (X F) and (X F) A = φ. This implies x / F 0. Therefore F 0 F. Hence F 0 τ α γ cl(a). Hence the proof. Theorem 3.22 Let (X,τ ) is a topological space and A X. Then the following statements hold: (i) τ α γ int(x A) =X τ α γ cl(a)

7 Operation approaches 497 (ii) τ α γ cl(x A) =X τ α γ int(a) Proof: Proof of (i) and (ii) is obvious. Definition 3.23 A subset B x of a topological space (X,τ ) is said to be the α-γ-neighbourhood of a point x X if there exists an α-γ- open set U such that x U B x. Theorem 3.24 A subset of a topological space (X,τ )isα-γ-open if and only if it is α-γ- neighbourhood of each of its points. Proof: The proof follows from the Definition 3.16 and Definition Remark 3.25 Let(X,τ ) be a topological space and γ be an operation on τ and A be a subset of X. Then from the Theorem 3.3 and the Definition 3.18 we have A τ α γ cl(a) τ γ cl(a). References [1] D. Andrijevic, semi-preopen sets, Math. Vesnik., (1986), [2] D. Andrijevic, On the Topology Generated by Pre-open Sets,Math. Vesnik, 39,(1987), [3] K. Balachandran, P. Sundaram and H. Maki, Generalized locally closed sets and GLC-Continuous functions,indian J.Pure.Appl.Math.27 (1996), [4] M. Ganster and I.L. Reilly, Locally closed sets and LC-Continuous functions,internat.j.math.sci. 12 (1989), [5] D.S. Jankovic, On functions with α-closed graphs,glasnik Math., 18 (1983),141. [6] S. Kasahara, Operation- compact spaces,math. Japonica, 24 (1979), 97. [7] N. Levine, Semi- open sets and semi- continuity in topological spaces,amer.math. Monthly,70(1963), 36. [8] N. Levine, Generalized closed sets in topology,rend. Circ. Math. Palerno, 2(1970), 89. [9] A.S. Mashhour, M.E. Abd EI-Monsef and S.N.EI-Deep, On Precontinuous and Weak Continuous Mappings,Proc., Math., Phys., Soc., Egypt,53(1982), [10] O. Njastad, On some Classes of nearly open sets,pacific J. Math 15,(1965),

8 498 N. Kalaivani and G. SaiSundaraKrishnan [11] H. Ogata, Operations on Topological spaces and associated topology,math.japonica36(1) (1991), [12] G. Sai Sundara Krishnan and K.Balachandran, On a class of γ- preopen sets in a Topological space,east Asian Math. J. 22(2),(2006), [13] G. Sai Sundara Krishnan, M.Ganster and K.Balachandran, Operation approaches on semi-open sets and applications,kochi.j. Math. 2,(2007), [14] G. Sai Sundara Krishnan and K. Balachandran, On γ-semi-open sets in Topological spaces,bull.cal.math.soc., 98,(6),(2006), Received: September, 2012