More On λ κ closed sets in generalized topological spaces

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1 Journal of Algorithms and Computation journal homepage: More On λ κ closed sets in generalized topological spaces R. Jamunarani, 1, P. Jeyanthi 2 and M. Velrajan 3 1,2 Research Center, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur , Tamil Nadu, India. 3 Research Center, Department of Mathematics, Aditanar College of Arts and Science,, Tiruchendur , Tamil Nadu, India ABSTRACT In this paper, we introduce λ κ closed sets and study its properties in generalized topological spaces. ARTICLE INFO Article history: Received 2, February 2014 Received in revised form 10, July 2014 Accepted 19, December 2014 Available online 30, December 2014 Keyword: Generalized topology, µ open set, µ closed set, quasi-topology, strong space, Λ κ set, λ κ open set,λ κ closed set. AMS subject Classification: Primary 54 A Introduction The theory of generalized topology was introduced by Császár in [1]. The properties of generalized topology, basic operators, generalized neighborhood systems and some con- jamunarani1977@gmail.com jeyajeyanthi@rediffmail.com Journal of Algorithms and Computation 45 (2014) PP

2 36 R. Jamunarani / Journal of Algorithms and Computation 45 (2014) PP structions for generalized topologies have been studied by the same author in [1, 2, 3, 4, 5, 6]. It is well known that generalized topology in the sense of Császár [1] is a generalization of topology on a nonempty set. On the other hand, many important collections of sets related with topology on a set form a generalized topology. In this paper we define several subsets in a generalized topological spaces and study their properties. A nonempty family µ of subsets of a set X is said to be a generalized topology [2] if µ and arbitrary union of elements of µ is again in µ. The pair (X,µ) is called a generalized topological space and elements of µ are called µ open sets. A X is µ closed if X A is µ open. By a space (X,µ), we always mean a generalized topological space. If X µ, (X,µ)iscalledastrong [3]space. Clearly, (X,µ)isstrongifandonlyif isµ closedifand only if c µ ( ) =. In a space (X,µ), if µ is closed under finite intersection, (X,µ) is called a quasi-topological space [5]. Clearly, every strong, quasi-topological space is a topological space. ForA X, c µ (A)isthesmallest µ closedset containingaandi µ (A)isthelargest µ open set contained in A. Moreover, X c µ (A) = i µ (X A), for every subset A of X. A subset A of a space (X,µ) is said to be α open [4]( resp. σ open [4], π open [4], b-open [7], β open[4]) ifa i µ c µ i µ (A) (resp. A c µ i µ (A), A i µ c µ (A), A i µ c µ (A) c µ i µ (A), A c µ i µ c µ (A) ). A subset A of a space (X,µ) is said to be α closed (resp. σ closed, π closed, b-closed, β closed) if X A is α open (resp. σ open, π open, b-open, β open). Let (X,µ) be a space and ζ = {µ,α,σ,π,b,β}. For κ ζ, we consider the space (X,κ), throughout the paper. For A M κ = {B X B µ}, the subset Λ κ (A) is defined by Λ κ (A) = {G A G,G κ}. The proof of the following lemma is clear. Lemma 1.1. Let A,B and B α,α be subsets of M κ in a space (X,κ). Then the following properties are hold. (a) B Λ κ (B). (b) If A B then Λ κ (A) Λ κ (B). (c) Λ κ (Λ κ (B)) = Λ κ (B). (d) If A k, then A = Λ κ (A). (e) Λ κ ( {B α α }) = {Λ κ (B α ) α }. (f) Λ κ ( {B α α }) {Λ κ (B α ) α }. 2 More on λ κ closed sets In a space (X,κ), a subset B of M κ is called a Λ κ set if B = Λ κ (B). We state the following theorem without proof. Theorem 2.1. For subsets A and A α, α of M κ in a space (X,κ), the following hold. (a) Λ κ (A) is a Λ κ set. (b) If A κ, then A is a Λ κ set.

3 37 R. Jamunarani / Journal of Algorithms and Computation 45 (2014) PP (c) If A α is a Λ κ set for each α, then {A α α } is a Λ κ set. (d) If A α is a Λ κ set for each α, then {A α α } is a Λ κ set. A subset A of M κ in a space (X,κ) is said to be a λ κ closed set if A = T C, where T is a Λ κ set and C is a κ closed set. The complement of a λ κ closed set is called a λ κ open set. We denote the collection of all λ κ open (resp., λ κ closed) set of X by λ κ O(X) (resp., λ κ C(X)). The following theorem gives the characterization of λ κ closed sets. Lemma 2.2. Let A M κ be a subset in a space (X,κ). Then the following are equivalent. (a) A is a λ κ closed set. (b)a = T c κ (A), where T is a Λ κ set. (c) A = Λ κ (A) c κ (A). Let (X,κ) be a space. A point x M κ is called a λ κ cluster point of A if for every λ κ open set U of M κ containing x we have A U. The set of all λ κ cluster points of A is called the λ κ closure of A and is denoted by c λκ (A). Lemma 2.3 gives some properties of c λκ, the easy proof of which is omitted. Lemma 2.3. Let (X,κ) be a space and A,B M κ. Then the following properties hold. (a) A c λκ (A). (b) c λκ (A) = {F A F and F is λ κ closed}. (c) If A B, then c λκ (A) c λκ (B). (d) A is a λ κ closed set if and only if A = c λκ (A). (e) c λκ (A) is a λ κ closed set. Let (X,κ) be a space and A M κ. A point x M κ is said to be a κ limit point of A if for each κ open set U containing x, U {A {x}}. The set of all κ limit points of A is called a κ derived set of A and is denoted by D κ (A). Let (X,κ) be a space and A M κ. A point x M κ is said to be a λ κ limit point of A if for each λ κ open set U containing x, U {A {x}}. The set of all λ κ limit points of A is called a λ κ derived set of A and is denoted by D λκ (A). Theorem 2.4 gives some properties of λ κ derived sets and Theorem 2.5 gives the characterization of λ κ derived sets. Theorem 2.4. Let (X,κ) be a space and A,B M κ. Then the following hold. (a) D λκ (A) D κ (A). (b) If A B, then D λκ (A) D λκ (B). (c) D λκ (A) D λκ (B) D λκ (A B) and D λκ (A B) D λκ (A) D λκ (B). (d)d λκ D λκ (A) A D λκ (A). (e) D λκ (A D λκ (A)) A D λκ (A). Proof. (a) Since every κ open set is a λ κ open set, it follows. (b) Let x D λκ (A). Let U be any λ κ open set containing x. Then U {A {x}} and so V {B {x}}, since A B. Therefore, x D λκ (B). (c) Since A B A,B we have D λκ (A B) D λκ (A) D λκ (B). Since A,B A B,

4 38 R. Jamunarani / Journal of Algorithms and Computation 45 (2014) PP we have D λκ (A) D λκ (B) D λκ (A B). (d) Let x D λκ D λκ (A) A and U be a λ κ open set containing x. Then U (D λκ (A) {x}).lety U (D λκ (A) {x}).since y D λκ (A)andx y U, U (A {y}). Let z U (A {y}). Then z U (A {y}) implies that z U and z A {y} and so z y. Since x A, z U (A {x}) and so U (A {x}). Therefore, x D λκ (A). (e) Let x D λκ (A D λκ (A)). If x A, the result is clear. Suppose x A. Since x D λκ (A D λκ (A)) A, then for λ κ open set U containing x, U ((A D λκ (A)) {x}). Thus U (A {x}) or U (D λκ (A) {x}). Now it follows from (d) that U (A {x}). Hence, x D λκ (A). Therefore, in all the cases D λκ (A D λκ (A)) A D λκ (A). Theorem 2.5. Let (X,κ) be space and A X. Then c λκ (A) = A D λκ (A). Proof. Since D λκ (A) c λκ (A), A D λκ (A) c λκ (A). On the other hand, let x c λκ (A). If x A, the proof is complete. If x / A, then each λ κ open set U containing x intersects A at a point distinct from x. Therefore, x D λκ (A). Thus, c λκ (A) A D λκ (A) and so c λκ (A) = A D λκ (A) which completes the proof. Let (X,κ) be a space and A X. Then i λκ (A) is the union of all λ κ open set contained in A. Theorem 2.6 gives some properties of i λκ. Theorem 2.6. Let (X,κ) be a space and A,B X. Then the following hold. (a) A is a λ κ open set if and only if A = i λκ (A). (b) i λκ (i λκ (A)) = i λκ (A). (c) i λκ (A) = A D λκ (X A). (d) X i λκ (A) = c λκ (X A). (e) X c λκ (A) = i λκ (X A). (f) A B then i λκ (A) i λκ (B). (g) i λκ (A) i λκ (B) i λκ (A B) and i λκ (A) i λκ (B) i λκ (A B). Proof. (c) If x A D λκ (X A), then x / D λκ (X A) and so, there exists a λ κ open set U containing x such that U (X A) =. Then x U A and hence x i λκ (A). That is, A D λκ (X A) i λκ (A).Ontheother hand, if x i λκ (A),then x / D λκ (X A), since i λκ (A) is a λ κ open set and i λκ (A) (X A) =. Hence, i λκ (A) = A D λκ (X A). (d) X i λκ (A) = X (A D λκ (X A)) = (X A) D λκ (X A) = c λκ (X A). Let (X,κ) be a space and A X. Then b κ (A) = A i κ (A) is said to be κ border of A. Let (X,κ) be a space and A X. Then b λκ (A) = A i λκ (A) is said to be λ κ border of A. Theorem 2.7 gives some properties of b λκ. Theorem 2.7. Let (X,κ) be a space and A X. Then the following hold. (a) b λκ (A) b κ (A). (b) A = i λκ (A) b λκ (A). (c) i λκ (A) b λκ (A) =.

5 39 R. Jamunarani / Journal of Algorithms and Computation 45 (2014) PP (d) A is a λ κ open set if and only if b λκ (A) =. (e) b λκ (i λκ (A)) =. (f) i λκ (b λκ (A)) =. (g) b λκ (b λκ (A)) = b λκ (A). (h) b λκ (A) = A c λκ (X A). (i) b λκ (A) = D λκ (X A). Proof. (f) If x i λκ (b λκ (A)), then x b λκ (A). On the other hand, since b λκ (A) A, x i λκ (b λκ (A)) i λκ (A). Hence x i λκ (A) b λκ (A) which contradicts (c). Thus, i λκ (b λκ (A)) =. (h) b λκ (A) = A i λκ (A) = A (X c λκ (X A)) = A c λκ (X A). (i) b λκ (A) = A i λκ (A) = A (A D λκ (X A)) = D λκ (X A). Let (X,κ) beaspace anda X. Then F κ (A) = c κ (A) i κ (A) is said to bethe κ frontier of A. Let (X,κ) be a space and A X. Then F λκ (A) = c λκ(a) i λκ (A) is said to be the λ κ frontier of A. Theorem 2.8 gives some properties of F λκ. Theorem 2.18 Let (X,κ) be a space and A X. Then the following hold. (a) F λκ (A) F κ (A). (b) c λκ (A) = i λκ (A) F λκ (A). (c) i λκ (A) F λκ (A) =. (d) b λκ (A) F λκ (A). (e) F λκ (A) = b λκ (A) D λκ (A). (f) A is a λ κ open set if and only if F λκ (A) = D λκ (A). (g) F λκ (A) = c λκ (A) c λκ (X A).(h) F λκ (A) = F λκ (X A). (i) F λκ (A) is a λ κ closed set. (j) F λκ (F λκ (A)) F λκ (A). (k) F λκ (i λκ (A)) F λκ (A). (l) F λκ (c λκ (A)) F λκ (A). (m) i λκ (A) = A F λκ (A). Proof. (b) i λκ (A) F λκ (A) = i λκ (A) (c λκ (A) i λκ (A)) = c λκ (A). (c) i λκ (A) F λκ (A) = i λκ (A) (c λκ (A) i λκ (A)) =. (e) Since i λκ (A) F λκ (A) = i λκ (A) b λκ (A) D λκ (A), F λκ (A) = b λκ (A) D λκ (A). (g) F λκ (A) = c λκ (A) i λκ (A) = c λκ (A) c λκ (X A). (i)c λκ (F λκ (A)) = c λκ (c λκ (A) c λκ (X A)) c λκ (c λκ (A)) c λκ (c λκ (X A)) = F λκ (A). Hence F λκ (A) is a λ κ closed set. (j) F λκ (F λκ (A)) = c λκ (F λκ (A) c λκ (X F λκ (A)) c λκ (F λκ (A)) = F λκ (A). (l) F λκ (c λκ (A)) = c λκ ((c λκ (A)) i λκ (c λκ (A)) = c λκ (A) i λκ (c λκ (A)) c λκ (A) i λκ (A) = F λκ (A). (m) A F λκ (A) = A (c λκ (A) i λκ (A)) = i λκ (A).

6 40 R. Jamunarani / Journal of Algorithms and Computation 45 (2014) PP Let (X,κ) be a space and A X. Then E κ (A) = i κ (X A) is said to be κ exterior of A. Let (X,κ) be a space and A X. Then E λκ (A) = i λκ (X A) is said to be λ κ exterior of A. Theorem 2.9 gives some properties of E λκ. Theorem 2.9. Let (X,κ) be a space and A X. Then the following hold. (a) E κ (A) E λκ (A) where E κ (A) denotes the exterior of A. (b) E λκ (A) is λ κ open. (c) E λκ (A) = i λκ (X A) = X c λκ (A). (d) E λκ (E λκ (A)) = i λκ (c λκ (A)). (e) If A B,then E λκ (A) E λκ (B). (f) E λκ (A B) E λκ (A) E λκ (B). (g)e λκ (A B) E λκ (A) E λκ (B). (h) E λκ (X) =. (i) E λκ ( ) = X. (j) E λκ (A) = E λκ (X E λκ (A)). (k) i λκ (A) E λκ (E λκ (A)). (l) X = i λκ (A) E λκ (A) F λκ (A). Proof. (d)e λκ (E λκ (A)) = E λκ (X c λκ (A)) = i λκ (X (X x λκ (A)) = i λ (c λ (A)). (j)e λκ (X E λκ (A)) = E λκ (X i λκ (X A)) = i λκ (X (X i λκ (X A))) = i λκ (i λκ (X A)) = i λκ (X A) = E λ (A). (k) i λκ (A) i λκ (c λκ (A)) = i λκ (X i λκ (X A)) = i λκ (X E λκ (A)) = E λκ (E λκ (A)).

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