EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, 2016, 434-442 ISSN 1307-5543 – www.ejpam.com Some New Regular Generalized Closed Sets in Ideal Topological Spaces Ümit Karabıyık1, Aynur Keskin Kaymakcı 2,∗ 1 Department of Mathematics–Computer Sciences, Faculty of Science, Necmettin Erbakan University, Konya, Turkey 2 Department of Mathematics, Faculty of Science, Selcuk University, Konya, Turkey Abstract. We introduce the notions of saw-Ir g -closed sets and weakly-r gI -closed by using the notion of regular open sets. Further, we study the concept of saw-Ir g -closed sets and their relationships in ideal topological spaces by using these new notions. Furthermore, we introduce and examine some properties of αI -∗-normal space. 2010 Mathematics Subject Classifications: 54A05, 54C05 Key Words and Phrases: Ir g -closed, saw-Ir g -closed, w-r gI -closed, ideal topological spaces 1. Introduction In 1990, Jankovic and Hamlett [3], have initiated the application ideal topological spaces. Khan and Noiri [3] have introduced semi-local functions in ideal topological spaces. Firstly the notion of Ig -closed set is given by Dontchev et al. [1]. In 2007, Navaneethakrishnan and Joseph [9] have introduced some of properties of Ig -closed sets and Ig -open sets by using local function. Recently Karabiyik [5], has defined concept of r gI -closed set which is weaker than Ig -closed set and examined some properties. Also, he has given some characterization of this set. In 2013, Ekici and Ozen [2] introduced weakly-Ir g -closed sets which is a generalized class of τ∗. In this paper, we define saw-Ir g -closed sets and weakly-r gI -closed by have using the notion of regular open sets. We investigated some of their properties. Also, we generalized many concepts which is defined in ideal topological spaces. The relationships of generalized class and various properties are examined. ∗Corresponding author. Email addresses: ukarabiyik@konya.edu.tr (Ü Karabıyık), akeskin@selcuk.edu.tr (A Kaymakcı) http://www.ejpam.com 434 c© 2016 EJPAM All rights reserved. Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 435 2. Preliminaries In this paper, (X ,τ) symbolize topological spaces on which no separation axioms are as- sumed unless clearly stated. For a subset A of X , cl∗(A) and int∗(A) will represent the closure of A and the interior of A in (X ,τ). A subset A of a topological spaces (X ,τ) is said to be reg- ular open [11] (resp. regular closed) if A= int(cl(A)) (resp. A= cl(int(A))). An ideal I on a nonempty set X is a collection of subsets of X which satisfies the following properties [7] (i) A∈ I and B ⊆ A implies B ∈ I and (ii) A∈ I and B ∈ I implies A∪ B ∈ I . A topological spaces (X ,τ) with an ideal I on X is called an ideal topological spaces and is denoted by (X ,τ, I). If Y is a subset of X then IY = {G ∩ Y : G ∈ I} is an ideal on Y and (Y,τ/Y , I/Y ) denote the ideal topological subspaces. Let P(X ) is the set of all subset of X , a set operator ∗ : P(X ) → P(X ), called a local function [6] of A with respect to τ and I , which is defined as: for A ⊂ X , A∗(I ,τ) = {x ∈ X : U ∩ A 6∈ I for every U ∈ τ(X , x)}. We simply write A∗ instead of A∗(I ,τ) in case there is no concision. For every ideal topological spaces (X ,τ, I) there exists a topology τ∗ finer than τ defined as τ∗ = {U ⊆ X : cl∗(X −A) = X −A} generated by the base β(I ,τ) = {U ⊆ J : U ∈ τ and J ∈ I}. A Kuratowski closure operator cl∗(·) for a topology τ∗(I ,τ) called the ∗-topology, finer than τ is defined by cl∗(A) = A∪ A∗ [12]. For a subset A of X , cl∗(A) and int∗(A) will represent the closure of A and the interior of A in (X ,τ∗), respectively. A subset A of an ideal topological spaces (X ,τ, I) is said to be a τ∗-closed [3], if A∗ ⊂ A. A subset A of an ideal topological spaces (X ,τ, I) is said to be a I -open [4], if A ⊂ int(A∗). A subset A of an ideal topological spaces (X ,τ, I) is said to be a I -regular open (resp. I -regular closed) [8], if A = int∗(cl∗(A)) (resp. A = cl∗(int∗(A))). A subset A of an (X ,τ, I) be a ideal topological spaces is said to be a Ir g -closed [10], A∗ ⊂ U whenever A⊆ U and is regular open set in X . A subset A of an (X ,τ, I) be a ideal topological spaces is said to be a r gI -closed [5], if cl∗(A) ⊂ U whenever A⊆ U and is regular open set in X . A subset A of an (X ,τ, I) be a ideal topological spaces is said to be a weakly-Ir g -closed(briefly w-Ir g -closed) [2], if (int(A))∗ ⊂ U whenever A⊆ U and is regular open set in X . 3. Saw-Ir g-Closed Sets In this section, fist of all we introduce the notion called saw-Ir g -closed and give some characterizations of this sets. Definition 1. A subset A of an (X ,τ, I) be a ideal topological spaces is said to be αI -∗-closed if cl∗(int(cl(A))) ⊂ A. The complement of αI -∗-closed set is said to be αI -∗-open. Definition 2. A subset A of an (X ,τ, I) be a ideal topological spaces is said to be: (1) Strongly almost weakly-Ir g -closed(briefly saw-Ir g -closed) if (int(cl(A)))∗ ⊂ U whenever A⊆ U and U is a regular open in X . Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 436 (2) Almost weakly-Ir g -closed(briefly aw-Ir g -closed) if (int(A∗))∗ ⊂ U whenever A ⊆ U and U is a regular open in X . (3) Almost weakly-r gI -closed(briefly aw-r gI -closed) if ((int(cl∗(A))))∗ ⊂ U whenever A ⊆ U and U is a regular open in X . Theorem 1. Let (X ,τ, I) be a ideal topological spaces and A ⊂ X , the following properties are equivalent: (1) A is a saw-Ir g -closed sets (2) cl∗(int(cl(A))) ⊂ U whenever A⊆ U and U is a regular open in X . Proof. (1)⇒ (2) Let A is a saw-Ir g -closed set. Assume that A⊆ U and U is a regular open in X .Then we have (int(cl(A)))∗ ⊂ U . Since int(cl(A)) ⊂ cl(A) ⊂ A ⊂ U . This implies that int(cl(A))∪ (int(cl(A)))∗ = cl∗(int(cl(A))) ⊂ U . (2) ⇒ (1) Let cl∗(int(cl(A))) ⊂ U whenever A ⊆ U and U is a regular open in X . Since (int(cl(A)))∗ ∪ (int(cl(A))) ⊂ U then (int(cl(A)))∗ ⊂ U whenever A ⊆ U and is U is regular open in X . Theorem 2. For a subset A of X the following properties hold: (1) A is open and saw-Ir g -closed then A is Ir g -closed, (2) A is open and w-Ir g -closed then A is Ir g -closed, (3) A is I-open and aw-Ir g -closed then A is Ir g -closed. Proof. (1) Let A be a open and saw-Ir g -closed set in (X ,τ, I). Since A is a open, A ⊂ int(A) ⊂ int(cl(A)). Hence, A∗ ⊂ (int(cl(A)))∗ ⊂ U , A ⊆ U and U is regular open in X . So, we have A is a Ir g -closed. (2) Let A be a open and w-Ir g -closed set in (X ,τ, I). Hence we have A⊂ int(A) and A∗ ⊂ (int(A))∗ ⊂ U . This implies that A is a Ir g -closed. (3) Let A be a I -open and aw-Ir g -closed set in (X ,τ, I). Hence we have A ⊂ int(A∗) and A∗ ⊂ (int(A∗))∗ ⊂ U . This shows that A is a Ir g -closed. Theorem 3. Every αI -∗-closed set is saw-Ir g -closed set. Proof. Let A⊆ U and U is a regular open in X . Since A is a αI -∗-closed, cl∗(int(cl(A))) ⊂ cl∗(int(cl(U))) ⊂ U . Thus, A is a saw-Ir g -closed set in (X ,τ, I). The following example shows that the reverse of Theorem 3 is not true. Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 437 Example 1. Let (X ,τ, I) be a ideal topological space such that X = {a, b, c, d}, τ = {;, X , {b}, {c}, {b, c}, {c, d}, {b, c, d}, {a, c, d}} and I = {;}. Then, A = {a, c} ⊂ X is saw- Ir g -closed set but is not αI -∗-closed set. Remark 1. The intersection of two saw-Ir g -closed set in ideal topological spaces need not be a saw-Ir g -closed set. Example 2. Let (X ,τ, I) be a ideal topological space such that X = {a, b, c}, I = {;, {b}}, and τ= {;, X , {a}, {b}, {a, b}}. Let A= {a, c} and B = {a, b}. From here A and B are saw-Ir g -closed set but A∩ B = {a} is not saw-Ir g -closed set. Theorem 4. Let (X ,τ, I) be a ideal topological spaces A ⊂ X . If A is a saw-Ir g -closed set then (int(cl(A)))∗ − A contains no any nonempty regular closed set. Proof. Let A is a saw-Ir g -closed set in (X ,τ, I). Suppose that U is a closed set. Such that U ⊆ (int(cl(A)))∗ − A. Since X − U is open and A⊂ X − U , then (int(cl(A)))∗ ⊂ X − U . Then, we get U ⊂ X − (int(cl(A)))∗. Hence U ⊂ (int(cl(A)))∗. Thus, U ⊂ (int(cl(A)))∗ ∩ X − (int(cl(A)))∗ = ; and (int(cl(A)))∗ − A contains no any nonempty closed set. Proposition 1. Let (X ,τ, I) be an ideal topological spaces. If A ⊂ B ⊂ cl∗(int(cl(A))) and A is saw-Ir g -closed, then B is saw-Ir g -closed. Proof. Let B ⊂ U and U is a regular open in X . Since A ⊂ U and A is a saw-Ir g -closed set then cl∗(int(cl(A))) ⊂ U . Since, B ⊂ cl∗(int(cl(A))) ⊂ U , we obtain cl∗(int(cl(B))) ⊂ cl∗(int(cl(cl∗(int(cl(A)))))) ⊂ cl∗(int(cl(A))) ⊂ U . Therefore cl∗(int(cl(B))) ⊂ U , B is a saw-Ir g -closed. Corollary 1. Let (X ,τ, I) be an ideal topological spaces and A⊂ X . If A is a saw-Ir g -closed and regular open set, then cl∗(A) is a saw-Ir g -closed. Proof. Let A is saw-Ir g -closed set and regular open set in (X ,τ, I). Then we have A⊂ cl∗(A) ⊂ cl∗(A) = cl∗(int(cl(A))). Hence by Proposition 1 cl∗(A) is a saw-Ir g -closed set in (X ,τ, I). Theorem 5. Let (X ,τ, I) be an ideal topological spaces and A⊂ X . Assume that A is a saw-Ir g - closed set. The following properties are equivalent: (1) A is a αI -∗-closed, (2) (int(cl(A)))∗ − A is a regular closed set. Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 438 Proof. (1) ⇒ (2) Let A be a αI -∗-closed set. Which means that cl∗(int(cl(A))) ⊂ A. This implies (int(cl(A)))∗ ⊂ A and (int(cl(A)))∗−A= ;. Thus, cl∗(int(cl(A)))−A is a regular closed set. (2)⇒ (1) Let cl∗(int(cl(A)))− A be a regular closed set. Since A is a saw-Ir g -closed set in (X ,τ, I), by Theorem 3 (int(cl(A)))∗ − A= ;. Hence, we get cl∗(int(cl(A))) ⊂ A. Thus A is a αI -∗-closed. Corollary 2. Let (X ,τ, I) be an ideal topological spaces and A ⊂ X . If A is a aw-Ir g -closed and τ∗-closed set, then cl∗(int(A∗)) ⊂ U whenever A⊂ U and U is a regular open set in X . Proof. Let A be a aw-Ir g -closed set in X . Suppose that A⊂ U and U is a regular open set in X . We have (int(A∗))∗ ⊂ U . On the other hand since A is a τ∗-closed, we have int(A∗) ⊂ int(A) ⊂ A⊂ U . Hence (int(A∗))∗ ∪ int(A∗) ⊂ U . This implies cl∗(int(A∗)) ⊂ U . Theorem 6. If (X ,τ, I) is any ideal topological spaces where I = {;}, then A is a aw-Ir g -closed if and only if G ⊂ int∗(cl(A∗)) whenever G ⊂ A and G is a regular closed set. Proof. Necessity: let G be a regular closed set in X and G ⊂ A. Then it is well-known that X − G is regular open set and (X − A) ⊂ (X − G). Since X − A is a aw-Ir g -closed, then cl∗(int((X − A)∗)) ⊂ (X − G). From the fact that for I = {;}, A∗ = cl(A). Therefore X − int∗(cl(A∗)) ⊂ (X − G). So, we have G ⊂ int∗(cl(A∗)). Sufficiency: let H be a regular open set in X and (X − A) ⊂ H. Since (X − H) is a regular open set such that (X −H) ⊂ A, then (X −H) ⊂ int∗(cl(A∗)). We have X − int∗(cl(A∗)) = cl∗(int(A∗)) ⊂ H. Thus, (X − A) is a aw-Ir g -closed set. Hence, A is a aw-Ir g -open set in X . 4. Weakly-r gI -Closed Sets In this section, secondly we introduce weakly-r gI -closed sets and investigate their basic properties. Definition 3. A subset A of an (X ,τ, I) be a ideal topological spaces is said to be (i) weakly-r gI -closed(briefly w-r gI -closed) set if (int(cl∗(A))) ⊂ U whenever A⊆ U and U is a regular open in X . (ii) weakly-I r g-closed(briefly w-I r g-closed) set if (int(cl∗(A))) ⊂ U whenever A⊆ U and U is I-regular open in X . Theorem 7. Every w-I r g-closed set is a w-r gI -closed set. Proof. Let A be a w-I r g-closed set. Then (int(cl∗(A))) ⊂ U whenever A ⊆ U and U is a I -regular open in X . Since U is I -regular open, we have U = int∗(cl∗((U))) and int(cl(U)) ⊂ int∗(cl∗((U))). Therefore, U is a regular open in X . This shows that A is a w-r gI -closed set. The following example shows that the reverse of Theorem 7 is not true. Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 439 Example 3. Let (X ,τ, I) be a ideal topological space such that X = {a, b, c}, I = {;, {c}}, and τ = {;, X , {a}, {c}, {a, c}}. Then A = {a} ⊂ X is a w-r gI -closed set but is not w-I r g-closed set, since int(cl∗(A)) = int(cl∗({a})) = {a} where A is contained in the regular open set U. But int∗(cl∗(A)) = int∗(cl∗({a})) 6= {a} which means A is contained there is not I-regular open set U. Remark 2. Let be a (X ,τ, I) be a ideal topological spaces. The following diagram holds for a subset A⊂ X : r gI − closed // �� w− r gI − closed �� w− I r g − closedoo Ir g − closed // w− Ir g − closed aw− r gI − closedoo �� αI − ∗− closed // saw− Ir g − closed // OO aw− Ir g − closed Theorem 8. Let be a (X ,τ, I) be a ideal topological spaces. For every subset A∈ I , A is a r gI -closed sets. Proof. Let A ⊂ U , where U is regular open. Since A∗ = ; for every A ∈ I , we obtain cl∗(A) = A∪ A∗ = A⊂ U . Therefore A is a r gI -closed set. Theorem 9. Let be a (X ,τ, I) be a ideal topological spaces. For every subset A of X , A∗ is a r gI -closed sets. Proof. Let A∗ ⊂ U , where U is a regular open. Since (A∗)∗ ⊂ A∗, we have cl∗(A∗) ⊂ U . Hence A∗ is a r gI -closed set. Theorem 10. Let be a (X ,τ, I) be a ideal topological spaces. If A is a r gI -closed set, then cl∗(A)−A does not contain any nonempty regular closed set. Proof. Let F be a regular closed subset of X , such that F ⊂ cl∗(A)−A where A is r gI -closed set. We get cl∗(A) ⊂ (X − F). This shows that F ⊂ (X − cl∗(A))∩ cl∗(A). Hence F = ;. Theorem 11. (X ,τ, I) be a ideal topological spaces and A ⊂ X . If A is w-r gI -closed set and τ∗-closed, then A is a w-Ir g -closed set. Proof. Let A be a τ∗-closed and w-r gI -closed set in (X ,τ, I). Then, we have A∗ ⊂ A and we have int(A∗) ⊂ int(A). On the other hand, int(cl∗(A)) ⊂ U whenever A ⊂ U and U regular open in X . Hence, int(A) ⊂ int(A) ∪ int(A∗) ⊂ int(A∪ A∗) = int(cl∗(A)). This implies that (int(A))∗ ⊂ int(cl∗(A)) ⊂ U and so A is a w-Ir g -closed set. Theorem 12. In an ideal spaces (X ,τ, I), the union two r gI -closed set in an r gI -closed set. Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 440 Proof. Let A and B r gI -closed set. Suppose A∪B ⊂ U and U is regular open. Then A⊂ U and B ⊂ U by hypothesis, cl∗(A) ⊂ U and cl∗(B) ⊂ U . Therefore cl∗(A∪ B) = cl∗(A)∪ cl∗(B) ⊂ U . This shows that A∪ B is r gI -closed set. Example 4. Let X = {a, b, c},τ = {;, X , {a}, {b}, {a, b}} and I = {;, {b}}. For A = {a, b} and B = {a, c} since X is the only regular open set containing A and B. Therefore A and B are r gI - closed sets. Now, A∩B = {a} is regular open and cl∗(A∩B) = {a, c} 6⊆ {a}. This shows that A∩B is not an r gI -closed set. Theorem 13. (X ,τ, I) be a ideal topological spaces and A⊂ X . If A is w-r gI -closed, B is regular closed and τ∗-closed[3] then, A∩ B is a w-r gI -closed. Proof. Let U be a regular open such that A∩ B ⊂ U . Then we have A⊂ U ∩ (X − B). Since A is a w-r gI -closed and B is a τ∗-closed, then int(cl∗(A)) ⊂ U ∩ (X − B). Also, cl∗(A∩ B) ⊂ cl∗(A)∩ cl∗(B). Therefore cl∗(A∩ B) ⊂ U ∩ (X − B). Since B is τ∗-closed, cl∗(A∩ B) ⊂ U . This shows that A∩ B is a w-r gI -closed. Theorem 14. Let (X ,τ, I) be a ideal topological spaces A ⊂ X . If A is w-r gI -closed set then (int(cl∗(A)))− A contains no any nonempty regular closed set. Proof. This theorem can be proved similar way of Theorem 4. Theorem 15. Let (X ,τ, I) be a ideal topological spaces A subset of X . A is w-r gI -open set if and only if G ⊂ cl(int∗(A)) whenever G ⊂ A and G is regular closed. Proof. Necessity: let G ⊂ A and G be regular closed. Then (X − A) ⊂ (X − G) and (X − G) regular open. Since (X − A) is a w-r gI -closed, (int(cl∗(x − A))) ⊂ (X − G). Hence, we get G ⊂ cl(int∗(A)). Sufficiency: suppose that G ⊂ cl(int∗(A)) whenever G ⊂ A and G is regular closed. Let (X − A) ⊂ U where U is regular open. Then (X − U) ⊂ A. By hypothesis (X − U) ⊂ cl(int∗(A)) and cl(int∗(X − A)) ⊂ U . Therefore A is w-r gI -open. 5. αI -∗-normal Spaces In this section, we talk about the αI -∗-normal space and investigate some properties. Definition 4. An ideal topological spaces (X ,τ, I) is said to be αI -∗-normal if for every pair of disjoint regular closed subsets A, B of X , there exist disjoint αI -∗-open sets U, V of X such that A⊆ U and B ⊆ V . Theorem 16. Let (X ,τ, I) be a ideal topological spaces and A ⊂ X , the following properties are equivalent: (1) X is a αI -∗-normal, Ü Karabıyık, A Kaymakcı / Eur. J. Pure Appl. Math, 9 (2016), 434-442 441 (2) For any disjoint regular closed sets A and B, there exist disjoint saw-Ir g -open sets U and V of X such that A⊆ U and B ⊆ V , (3) For any regular closed set A and any regular open set B containing A, there exists a saw-Ir g - open set U such that A⊆ U ⊆ cl∗(int(cl(A))) ⊆ B. Proof. (1)⇒ (2) The proof is obvious. (2)⇒ (3) Let A be a regular closed and B be a regular open subset of X , such that A⊆ B. Then A and (X − B) are disjoint regular closed set of X . By the hypothesis, there exist a saw-Ir g -open sets U and V of X such that A ⊆ U and (X − B) ⊆ V . Since V is saw-Ir g -open set, (X − B) ⊆ int∗(cl(int(V ))). Hence we have U ∩ int∗(cl(int(V ))) = ;. So, we obtain cl∗(int(cl(U))) ⊆ cl∗(int(cl(X − V ))). This shows that A⊆ U ⊆ cl∗(int(cl(U))) ⊆ B. (3)⇒ (1) Let A and B any disjoint regular closed set of X . Then, A⊆ (X − B) and (X − B) is regular open set. Hence there exist a saw-Ir g -open set G of X such that A⊆ G ⊆ cl∗(int(cl(G))) ⊆ (X − B). Definition 5. A function f : (X ,τ, I)→ (Y,ϕ) is said to be saw-Ir g -continuous if for every closed set F in Y , f −1(F) saw-Ir g -closed in X . Definition 6. A function f : (X ,τ, I)→ (Y,ϕ, J) is called saw-Ir g -irresolute if for every saw-Jr g - closed in Y , f −1(F) saw-Ir g -closed in X . Theorem 17. Let f : X → Y be a saw-Ir g -continuous regular closed and injective function. If Y is normal, then X is αI -∗-normal. Proof. Let A and B any disjoint regular closed set of X . Since f is regular closed injection, f (A) and f (B) are disjoint regular closed sets of Y . By the normality of Y , there exist disjoint open sets U and V such that f (A) ⊆ U and f (B) ⊆ V . Since f is saw-Ir g -continuous, then f −1(U) and f −1(V ) are saw-Ir g -open sets such that A⊆ f −1(U) and B ⊆ f −1(V ). Therefore X is αI -∗-normal by Theorem 16. Theorem 18. Let f : X → Y be a saw-Ir g -irresolute regular closed injection. If Y is αI -∗-normal, then X is αI -∗-normal. Proof. Let A and B any disjoint regular closed set of X . Since f is regular closed injection, f (A) and f (B) are disjoint regular closed sets of Y . Since Y is αI -∗-Normal, by Theorem 16 there exist disjoint saw-Ir g -open U and V such that f (A) ⊆ U and f (B) ⊆ V since f is saw-Ir g - irresolute, then f −1(U) and f −1(V ) are saw-Ir g -open sets such that A⊆ f −1(U) and B ⊆ f −1(V ). Therefore X is αI -∗-normal. Theorem 19. Let f : X → Y be a saw-Ir g -irresolute regular closed injection. If X is αI -∗-normal and Y ⊂ X regular closed, then Y is αI/Y -∗-normal spaces. Proof. Let A and B any disjoint regular closed set of Y . Since Y is regular closed, we have A and B are disjoint regular closed sets of X . Since X is αI -∗-Normal, there exist disjoint saw-Ir g -open U and V such that A⊆ U and B ⊆ V . 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