EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 3, 2013, 352-362 ISSN 1307-5543 – www.ejpam.com On Decompositions of Continuity and Complete Continuity in Ideal Topological Spaces E. Hatir Konya N.E. University, Education Faculty, Meram, Konya, Turkey Abstract. We define new classes of sets called δβI -open set, δα− I -open set, δβ − I−set, semi∗ − I - open set, sδI − g-closed set in ideal topological spaces. Using these sets, we obtain decompositions of continuity and complete continuity in ideal topological spaces. Also, we investigate some properties of these sets and relationship other generalized sets. 2010 Mathematics Subject Classifications: 54C08, 54C10; 54A05 Key Words and Phrases: ideal topological spaces, δβ-open set, δβI -open set 1. Introduction and Preliminaries Recently, Ekici and Noiri [3] have introduced pre∗− I -open sets to obtain a decomposition of continuity and defined α∗ − I -open sets and showed that the family of α∗ − I -open sets is a topology in ideal topological space. In [14], the authors have studied some new classes of functions in ideal topological spaces. In this paper, we define new classes of sets called δβI - open set, δα− I -open set, δβ− I -set, semi∗− I -open set, sδI− g-closed set in ideal topological spaces. Using these sets, we obtain decompositions of continuity and complete continuity in ideal topological spaces. Also, we investigate some properties of these sets and relationship other generalized sets. Throughout this paper, spaces (X ,τ) and (Y,σ) (or simply X and Y ), always mean topo- logical spaces on which no separation axiom is assumed. For a subset A of a topological space (X ,τ), Cl(A) and Int(A) will denote the closure and interior of A in (X ,τ), respectively. A subset of a space (X ,τ) is said to be regular open (resp. regular closed) [12] if A= Int(Cl(A)) (resp. A= Cl(Int(A))). A is called δ-open [12] if for each x ∈ A, there exists a regular open set G such that x ∈ G ⊂ A. The complement of a del ta-open set is called δ-closed. A point x ∈ X is called a δ-cluster point of A if Int(Cl(U))∩A 6=∅ for each open set U containing x . The set of all δ-cluster points of A is called the δ-closure of A and is denoted by Clδ(A). The δ-interior of A is the union of all regular open sets of X contained in A and it is denoted by Intδ(A). Email address: hatir10@yahoo.com http://www.ejpam.com 352 c© 2013 EJPAM All rights reserved. E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 353 An ideal I on a topological space (X ,τ) is a nonempty collection of subsets of X which satisfies i) A ∈ I and B ⊂ A implies B ∈ I , ii) A ∈ I and B ∈ I implies A∪ B ∈ I . An ideal topological space is a topological space (X ,τ) with an ideal I on X and if P(X ) is the set of all subsets of X , a set operator (·)∗ : P(X ) → P(X ) called a local function [8] of A with respect to τ and I is defined as follows: for A ⊂ X , A∗(I) = � x ∈ X : U ∩ A /∈ I for every U ∈ τ(x) where τ(x) = {U ∈ τ : x ∈ U}. We simply write A∗ instead of A∗(I ,τ). X ∗ is often a proper subset of X . The hypothesis X = X ∗ [6] is equivalent to the hypothesis τ ∩ I = ∅. For every ideal topological space, there exists a topology τ∗(I) or briefly τ∗, finer than τ, generated by β(I ,τ) = {U \ I : U ∈ τ and I ∈ I}, but in general β(I ,τ) is not always a topology [7]. Additionally, Cl∗(A) = A∪ A∗ defines a Kuratowski closure operator for τ∗(I). If I is an ideal on X, then (X ,τ, I) is called an ideal topological space or simply an ideal space. A subset A of an ideal space (X ,τ, I) is said to be R− I -open [14] if A= Int(Cl∗(A)). A point x in an ideal space (X ,τ, I) is called a δI− cluster point of A if Int(Cl∗(U))∩A 6=∅ for each neighborhood U of x . The set of all δI -cluster points of A is called the δI -closure of A and is denoted by δClI(A). A is said to be δI -closed [14] if δClI(A) = A. The complement of δI -closed set is called δI -open set. Lemma 1 ([7]). Let (X ,τ, I) be an ideal topological space and A, B be subsets of X . 1. If A⊂ B, then A∗ ⊂ B∗ 2. If G ∈ τ, then G ∩ A∗ ⊂ (G ∩ A)∗ 3. A∗ = Cl(A∗)⊂ Cl(A). Definition 1. A subset A of an ideal topological space (X ,τ, I) is called a) α-open [10] if A⊂ Int(Cl(Int(A))) b) preopen [9] if A⊂ Int(Cl(A)) c) Pre− I -open [2] if A⊂ Int(Cl∗(A)) d) α− I -open [4] if A⊂ Int(Cl∗(Int(A))) e) δ-preopen [11] if A⊂ Int(Clδ(A)) f) pre∗− I -open [3] if A⊂ Int(δClI(A)) g) α∗− I -open [3] if A⊂ Int(Cl∗(Intδ(A))) h) strongly α− I -open [3] if A⊂ Int(Cl∗(δInt I(A))) i) β∗I -open [3] if A⊂ Cl∗(Int(Clδ(A))) j) t − I -set [5] if Int(Cl∗(A)) = Int(A) k) δβ∗I -open [13] if A⊂ Cl∗(Int(δClI(A))) E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 354 2. δβI -Open Sets Definition 2. A subset A of an ideal space(X ,τ, I) is said to be δβI -open if A⊂ Cl(Int(δClI(A))). Remark 1. The following diagram holds for a subset A of an ideal space (X ,τ, I). open ↓ α− I -open → pre−I -open → pre∗− I -open → δβ∗I -open → δβI -open ↓ ↓ ↓ ↓ ↓ α-open → preopen → δ-preopen → β∗I -open → δβ-open Figure 1: Diagram None of these implications is reversible, as shown in the following example and in [3] Example 1. Let X = {a, b, c, d}, τ = {X ,∅, {a} , {b} , {a, b} , {a, c} , {a, b, c}} and I = P(X ). Then the set {c, d} is δβ−open, but it is not δβI -open. The set {b, d} is δβI -open set, but it is not both β∗I -open and δβ∗I -open. If we take I = {∅}, then the set {c, d} is δβI -open, but it is not pre∗ − I -open. The family of all δβI -open (resp. δβI -closed) sets of X is denoted by δβ IO(X ) (resp. δβ IC(X )). Definition 3. Let (X ,τ, I) be an ideal space. a) The union of all δβI -open sets contained in A is called the δβI -interior of A and is denoted by δβ Int I(A) b) The intersection of all δβI -closed sets containing A is called the δβI -closure of A and is denoted by δβClI(A). Theorem 1. The following properties hold for the δβI -closure of a set A in a space (X ,τ, I). a) A is δβI -closed in X if and only if A= δβClI(A), b) δβClI(A)⊂ δβClI(B) whenever A⊂ B ⊂ X , c) δβClI(A) is δβI -closed, d) δβClI(δβClI(A)) = δβClI(A), e) x ∈ δβClI(A) if A∩ U 6=∅ for every δβI -open set containing x. Proof. Straightforward. We give the following Lemma using in the sequel. Lemma 2. Let A be a subset of a space (X ,τ, I). Then a) δClI(A)∩ U ⊂ δClI(A∩ U), for any δI -open set U in X , E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 355 b) δInt I(A∪ F)⊂ δInt I(A)∪ F, for any δI -closed set F in X . Proposition 1. Let (X ,τ, I) be an ideal space. If A⊂ B ⊂ δClI(A) and B be a δβI -open, then A is δβI -open. Proof. Let A⊂ B ⊂ δClI(A) and B be a δβI -open. Then we have δClI(A) = δClI(B). Thus, A⊂ B ⊂ Cl(Int(δClI(B))) = Cl(Int(δClI(A))) and hence A is δβI -open set. Proposition 2. Let (X ,τ, I) be an ideal space. If A⊂ B ⊂ Cl(A) and A be a δβI -open, then B is δβI -open. Proof. Let A⊂ B ⊂ Cl(A) and A be δβI -open. Then A⊂ Cl(Int(δClI(A))). Since B ⊂ Cl(A), then B ⊂ Cl(Cl(Int(δClI(A)))) = Cl(Int(δClI(A)))⊂ Cl(Int(δClI(B))). Thus B is δβI -open set. Corollary 1. Let (X ,τ, I) be an ideal space. If A is δβI -open, then Cl(A) is δβI -open. Proposition 3. Let (X ,τ, I) be an ideal space and A⊂ X is δβI -closed if and only if C l(Int(δInt I(A)))⊂ A. Proof. Let A∈ δβ IC(X ) ⇐⇒ X − A∈ δβ IO(X ). ⇐⇒ X − A⊂ Cl(Int(δClI(X − A))) =Cl(Int(X −δInt I(A))) = Cl(X − Cl(δInt I(A))) =X − Int(Cl(δInt I(A))) ⇐⇒ Int(Cl(δInt I(A)))⊂ A. Remark 2. The intersection of any two δβI -open sets need not be δβI -open set as shown example below. Example 2. Let X = {a, b, c, d}, τ = {X ,∅, {a} , {b} , {a, b} , {a, c} , {a, b, c}} and I = P(X ). Hence {b, d} , {a, c, d} are δβI -open sets, but the set {d} is not δβI -open. Definition 4. A subset A of an ideal space (X ,τ, I) is said to be δα−I -open if A⊂ Int(Cl(δInt I(A))). The family of all δα− I -open (resp. δα− I -closed) sets of X is denoted by δαIO(X ) (resp. δαIC(X )). It is obvious that every δI -open set is δα− I -open. Proposition 4. Let (X ,τ, I) be an ideal topological space. Then, the family of δα− I -open sets is a topology for X . Proof. It is obvious that ∅ and X are δα− I -open sets. Let A, B be δα− I -open sets. Then A∩ B ⊂Int(Cl(δClI(A)))∩ Int(Cl(δClI(B))) =Int(Cl(δInt I(A))∩ Int(Cl(δInt I(B)))) E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 356 ⊂Int(Cl(δInt I(A)∩ Cl(δInt I(B))) =Int(Cl(Int(δInt I(A))∩ Cl(δInt I(B)))) ⊂Int(Cl(Cl(δInt I(A)∩δInt I(B)))) =Int(Cl(δInt I(A∩ B))) Hence, A∩ B is a δα− I -open set. For the last axiom of topology, let Ai be δα− I -open sets for i ∈ I . Then Ai ⊂ Int(Cl(δInt I(Ai)))⊂ Int(Cl(δInt I(∪i∈IAi))). Thus, ∪i∈IAi ⊂ Int(Cl(δInt I(∪i∈IAi))). This implies that ∪i∈IAi is a δα− I -open set. Proposition 5. Let (X ,τ, I) be an ideal space. If A is δβI -open and B is δα− I -open, then A∩ B is δβI -open. Proof. Let A ∈ δβ IO(X ) and B ∈ δαIO(X ). Then, we have A ⊂ Cl(Int(δClI(A))) and B ⊂ Int(Cl(δInt I(B))), respectively. This implies that A∩ B ⊂Cl(Int(δClI(A)))∩ Int(Cl(δInt I(B))) ⊂Cl(Int((Int(δClI(A))))∩ (Cl(δInt I(B))))) ⊂Cl(Int(Cl(δClI(A)∩δInt I(B))))⊂ Cl(Int(Cl(δClI(A∩ B)))) ⊂Cl(Int(δClI(δClI(A∩ B)))) = Cl(Int(δClI(A∩ B))). Corollary 2. A set A in (X ,τ, I) is a δβI -open if and only if U ∩A∈ δβ IO(X ), for every δI -open set U of (X ,τ, I). Proof. Let A be a δβI -open set. Then we have U ∩ A⊂U ∩ Cl(Int(δClI(A))) =Int(U)∩ Cl(Int(δClI(A)))⊂ Cl(Int(U)∩ Int(δClI(A))) =Cl(Int(U ∩δClI(A)))⊂ Cl(Int(δClI(U ∩ A))) by Lemma 2. Hence U ∩ A∈ δβ IO(X ). Definition 5. A subset A of an ideal space (X ,τ, I) is said to be a) strongly−t − I -set [3] if Int(δClI(A)) = Int(A) b) δβ − t-set [5] if C l(Int(Clδ(A))) = Int(A) c) δβ − t − I -set if C l(Int(δClI(A))) = Int(A) d) δα∗− I -set if Int(Cl(δInt I(A))) = δInt I(A) E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 357 Proposition 6. a) δβ − t-set is a δβ − t − I -set. b) A δβ − t − I -set is a strongly−t − I -set. Proof. Obvious. Proposition 7. Let A and B be subsets of an ideal space (X ,τ, I). If A and B are δβ − t − I -sets, then A∩ B is a δβ − t − I -set. Proof. Let A and B be δβ − t − I -sets. Then Int(A∩ B)⊂Cl(Int(δClI(A∩ B))) ⊂Cl(Int(δClI(A)∩δClI(B))) =Cl(Int(δClI(A))∩ Int(δClI(B))) ⊂Cl(Int(δClI(A)))∩ Cl(Int(δClI(B))) =Int(A)∩ Int(B) = Int(A∩ B) This implies that A∩ B is a δβ − t − I -set. Definition 6. Let (X ,τ, I) be an ideal space. a) A subset A in X is said to be δβ − B − I -set (resp. stronglyB − I -set [3], δβ − B-set [5]) if there is a U ∈ τ and a δβ − t − I -set (resp. strongly−t − I -set, δβ − t-set) V in X such that A= U ∩ V . b) A subset A in X is said to be δ− C-set if there is a δI -open set U in X and a δα∗ − I -set V in X such that A= U ∩ V . Proposition 8. a) A δβ − t − I -set A is a δβ − B− I -set. b) An open set is a δβ − B− I -set. c) A δI -open set is a δ− C-set. Proposition 9. a) A δβ − B-set is a δβ − B− I -set. b) A δβ − B− I -set is a stronglyB− I -set. Remark 3. The converses of the statements in Proposition 6 and Proposition 9 are false as in the following example. Example 3. Let X = {a, b, c, d}, τ= {X ,∅, {a} , {a, c} , {a, b, c} , {c, d} , {c} , {a, c, d}} and I = {∅, {c}}. Hence {c, d} is strongly−t− I -set (resp. strongly B− I -set), but it is not δβ − t− I - set (resp. δβ -B-I-set). {d} is δβ − t − I -set (resp. δβ − B− I -set), but it is not δβ − t-set (resp. δβ − B-set). Lemma 3. Let (X ,τ, I) be an ideal space and A be a subset of X . E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 358 a) If A is open, then δClI(A) = Cl(A), b) If A is closed, then δInt I(A) = Int(A). Proof. a) Since every δI -open set is open, we have Cl(A)⊂ δClI(A) [14]. Conversely, let x /∈ Cl(A). Then there exists an open set U containing x such that U ∩ A=∅. Since A is an open set, Int(Cl(U))∩ A= ∅ and we know that Int(Cl∗(U)) ⊂ Int(Cl(U)), i.e. Int(Cl∗(U))∩ A=∅. This means that x /∈ δClI(A). So, we get the result. b) This follows from (a). Theorem 2. For a subset A of an ideal space (X ,τ, I), the following properties are equivalent; a) A is regular open, b) Int(δClI(A)) = A, c) A is pre∗− I -open and a strongly−t − I -set. Proof. a) =⇒ b). Let A be regular open. Then A is open and by Lemma 3, δClI(A) = Cl(A). Therefore, we have Int(δClI(A)) = Int(Cl(A)) = A. b) =⇒ c). Straightforward. c) =⇒ a). Let A be pre∗− I -open and strongly−t − I -set. Then A⊂ Int(δClI(A)) = Int(A)⊂ A and A is open, A= Int(δClI(A)) = Int(Cl(A)). Theorem 3. Let A be a subset of an ideal space (X ,τ, I). Then the following properties are equivalent; a) A is open, b) A is pre∗− I -open and a strongly B− I -set, c) A is δβI -open and a δβ − B− I -set. Proof. a)⇐⇒ b) It follows from [3, Theorem 33] a) =⇒ c) Diagram 1 and Proposition 8 c) =⇒ a) Let A be a δβI -open and a δβ − B− I -set. Then there exist an open set U and a δβ − t − I -set V in X such that A= U ∩ V . Since V is δβ − t − I -set and A is δβI -open, then A⊂Cl(Int(δClI(A))) = Cl(Int(δClI(U ∩ V ))) ⊂Cl(Int(δClI(U)∩δClI(V ))) = Cl(Int(δClI(U))∩ Int(δClI(V ))) ⊂Cl(Int(δClI(U)))∩ Cl(Int(δClI(V ))) = Cl(Int(δClI(U)))∩ Int(V ). Thus, A=U ∩ V = (U ∩ V )∩ U ⊂ Cl(Int(δClI(U)))∩ Int(V )∩ U E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 359 =U ∩ Int(V ) and U ∩ Int(V )⊂ U ∩ V = A. Hence A= U ∩ Int(V ) and A is an open set. Theorem 4. Let A be a subset of an ideal space (X ,τ, I). Then the following properties are equivalent; a) A is δI -open, b) A is δα− I -open and a δ− C-set. Proof. The proof is similar with Theorem 3 3. Decompositions of Continuity and δI -Continuity Definition 7. a) Let f : (X ,τ, I) → (Y,σ) be a function. If for each V ∈ σ, f −1(V ) is a δβI -open (resp. δβ − B− I -set), then f is said to be δ−β − I -continuous (resp. δβ − B− I - continuous). b) Let f : (X ,τ, I)→ (Y,σ, J) be a function. If for each δI -open set V in Y , f −1(V ) is a δI -open, then f is said to be δ− I -continuous [10]. c) Let f : (X ,τ, I)→ (Y,σ, J) be a function. If for each δI -open set V in Y , f −1(V ) is a δα− I - open (resp. δ− C-set), then f is said to be δα− I -continuous (resp. δ− C-continuous). By Theorem 3, we obtain the following Theorem. Theorem 5. For a function f : (X ,τ, I)→ (Y,σ), the following properties are equivalent; a) f is continuous, b) f is δ− β − I -continuous and δβ − B− I -continuous. Remark 4. δ−β− I -continuity and δβ−B− I -continuity are independent notions of each other. Example 4. Let X = Y = {a, b, c, d}, τ = {X ,∅, {a} , {a, c} , {a, b, c} , {c, d} , {c} , {a, c, d}} and I = {∅, {c}} and σ = {∅, Y, {a, c}}. Define a function f : (X ,τ, I)→ (Y,σ) such that f (x) = x. Then f is δ− β − I -continuous, but it is not δβ − B − I -continuous. If we change the topology on Y as σ1 = {∅, Y, {d}} in the function f : (X ,τ, I)→ (Y,σ1) defined as f (x) = x, then f is δβ − B− I -continuous, but it is not δ− β − I -continuous. Theorem 6. For a function f : (X ,τ, I)→ (Y,σ, J), the following properties are equivalent; a) f is δ− I -continuous, b) f is δα− I -continuous and δ− C-continuous. Remark 5. δα− I -continuity and δ− C-continuity are independent notions of each other. E. Hatir / Eur. J. Pure Appl. Math, 6 (2013), 352-362 360 Example 5. Let X = Y = {a, b, c, d}, τ = {X ,∅, {a} , {a, c} , {a, b, c} , {c, d} , {c} , {a, c, d}} and I = {∅, {c}}. Also let σ = {∅, Y, {b, c}} and J = P(X ). Define a function f : (X ,τ, I)→ (Y,σ, J) such that f (x) = x.Then f is δ− C-continuous, but it is not δα− I -continuous. If we change the topology on Y as σ1 = {∅, Y, {a, c}} and J = P(X ) in the function f : (X ,τ, I)→ (Y,σ1, J) such that f (x) = x, then f is δα− I -continuous, but it is not δ− C-continuous. 4. Decomposition of Complete Continuity Definition 8. A subset A of an ideal space (X ,τ, I) is said to be semi∗− I -open (resp. semi∗− I - closed) set if A⊂ Cl(δInt I(A)) (resp. Int(δClI(A))⊂ A) The intersection of all semi∗− I -closed sets containing A is called semi∗−δ− I -closure of A and denoted by sδClI(A). Theorem 7. Let A be a subset of an ideal space (X ,τ, I). Then sδClI(A) = A∪ Int(δClI(A)). Proof. Since Int(δClI(A∪ Int(δClI(A))))⊂Int(δClI(A)∪δClI(Int(δClI(A)))) =Int(δClI(A))⊂ A∪ Int(δClI(A)), A∪ Int(δClI(A)) is semi∗− I -closed containing A and hence sδClI(A)⊂ A∪ Int(δClI(A)). On the other hand, since sδClI(A) is semi∗− I -closed, Int(δClI(A))⊂ Int(δClI(sδClI(A)))⊂ sδClI(A). Thus A∪ Int(δClI(A))⊂ sδClI(A). Definition 9. A subset A of an ideal space (X ,τ, I) is said to be semi − δI -generalized-closed (briefly, sδI − g-closed) if sδClI(A)⊂ U, whenever A⊂ U and U is pre∗− I -open. Theorem 8. For a subset A of an ideal space (X ,τ, I), the following properties are equivalent; a) A is regular open, b) A is pre∗− I -open and semi−δI -generalized-closed. Proof. (a) =⇒ (b). Let A be a regular open. Since every regular open set is pre∗ − I -open, A is pre∗ − I -open. By sδClI(A) = A∪ Int(δClI(A)) = Int(δClI(A)) = Int(Cl(A)) = A and Theorem 2, A is sδI − g-closed. (b) =⇒ (a). Let A be pre∗ − I -open and a sδI − g-closed set. Then sδClI(A) ⊂ A and hence A is strong semi−I -closed. Therefore, Int(δClI(A)) ⊂ A. Since A is pre∗ − I -open, A⊂ Int(δClI(A)) and Int(δClI(A) = A. Thus, by Theorem 2, A is regular open. To obtain decomposition of complete continuity, we introduce the following new func- tions. REFERENCES 361 Definition 10. A function f : (X ,τ)→ (Y,σ) is said to be completely continuous [1] if for each V ∈ σ, f −1(V ) is regular open in (X ,τ). Definition 11. A function f : (X ,τ, I) → (Y,σ) is said to be pre∗ − I -continuous [3] (resp. contra sδI − g-continuous) if for each V ∈ σ, f −1(V ) is pre∗ − I -open (resp. sδI − g-closed) in (X ,τ, I). By Theorem 8, we obtain the following decomposition of complete continuity. Theorem 9. For a function f : (X ,τ, I)→ (Y,σ), the following properties are equivalent; a) f is completely continuous, b) f is pre∗− I -continuous and contra sδI − g-continuous. Remark 6. By the following example, pre∗ − I -continuity and contra sδI − g-continuity are independent concepts. Example 6. 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