EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 4, 2020, 758-765 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On β-local Functions in ideal topological spaces P. L. Powar1,∗, T. Noiri2, Shikha Bhadauria3 1 Department of Mathematics and Computer Science, R. D. University, Jabalpur, India 2 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan 3 Department of Mathematics and Computer Science, R. D. University, Jabalpur, India Abstract. In this paper, by using β-open sets in [1] we introduce and investigate the concepts of the β-local function, Is∗g-β-closed sets and Ig-β-closed sets in an ideal topological space. In addition to the properties, an operation cl∗β is defined and the properties are obtained similarly with the local function in [8]. 2020 Mathematics Subject Classifications: 54C10, 54A05, 54D15, 54D30 Key Words and Phrases: β-open set, β-local function, operation cl∗β , Is∗g-β-closed set, Ig-β- closed set. 1. Introduction Kuratowski [11] has introduced the concept of an ideal topological space in 1930. Further, Jankovic and Hamlet [8] have studied ideal topological spaces and obtained their significant properties. They introduced the concept of I-open sets and studied topologies via ideals quite extensively. Abd-El-Monsef et al.[2] further explored the ideas of I-open sets. The concept of Ig-closed sets has been given by Dontchev et al. [6] in 1999 and the idea of Is∗g-closed sets was first introduced by Khan and Hamza [9]. The concepts of the s-local function was first introduced by Abd. El-Monsef et al. [3] and further investigated by Khan and Noiri [10]. Recently, Al-Omari and Noiri [5] have introduced and investigated the notion of local function Γ∗ in an ideal topological space and showed that Γ∗ is equivalent to the δ-local function due to Hatir et al. [7]. In this paper, by using β-open sets in [1] we introduce and investigate the concepts of the β-local function, Is∗g-β-closed sets and Ig-β-closed sets in an ideal topological space. And also, an operation cl∗β is defined and the properties are obtained similarly with the local function in [8]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3856 Email addresses: pvjrdvv@rediffmail.com (P. L. Powar), t.noiri@nifty.com (T. Noiri), shikhabhadauriamaths@gmail.com (Shikha Bhadauria) https://www.ejpam.com 758 c© 2020 EJPAM All rights reserved. P. L. Powar, T. Noiri, Shikha Bhadauria / Eur. J. Pure Appl. Math, 13 (4) (2020), 758-765 759 2. Preliminaries Throughout this paper (X, τ) and (X, τ, I) denote a topological space and an ideal topological space, respectively. The collection of closed sets in X is denoted by τF . For any subset A of X the closure and the interior of A are denoted by cl(A) and Int(A), respectively. We now recall certain definitions, which would be required for our study. Definition 1. [8] An ideal I on a topological spaces (X, τ) is a nonempty collection of subsets of X, which satisfies the following conditions: • A ∈ I and B ∈ I implies A ∪B ∈ I, • A ∈ I and B ⊂ A implies B ∈ I. Then the triplet (X, τ, I) is called an ideal topological space. Definition 2. [8] Let (X, τ, I) be an ideal topological space. For a set A ⊂ X, A∗(X, τ) ={x ∈ X : A ∩ U /∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}, is called the local- function of A with respect to I and τ . A∗(X, τ) is simply denoted by A∗. Definition 3. [5] Let (X, τ, I) be an ideal topological space. For a set A ⊂ X, Γ∗(A)(I, τ) ={x ∈ X : A ∩U /∈ I for every regular open set U containing x} is called the local function Γ∗ of A with respect to I and τ . Definition 4. [3],[10] Let (X, τ, I) be an ideal topological space and A be a subset of X. Then (A)∗ s (I, τ) ={x ∈ X : A ∩ U /∈ I for every U ∈ SO(X,x)} is called the semi-local function of A with respect to I and τ , where SO(X,x) = {U ∈ SO(X)|x ∈ U}. When there is no ambiguity we write A∗s for (A)∗ s (I, τ). Definition 5. Let (X, τ) be a topological space. A subset A of X is said to be: (i) β-open [1] if A ⊂ cl(Int(cl(A))), (ii) semi-open [12] if A ⊂ cl(Int(A)), (iii) regular-open [13] if A = Int(cl(A)). The family of all β-open (resp. semi-open, regular open) sets in X is denoted by βO(X) (resp. SO(X), RO(X)). Definition 6. [1] Let (X, τ) be a topological space. A subset A of X is said to be β-closed if its complement is β-open. Definition 7. [4] Let (X, τ) be a topological space and A be a subset of X. The β-closure of A is defined by the intersection of all β-closed sets containing the set A and it is denoted by βcl(A). P. L. Powar, T. Noiri, Shikha Bhadauria / Eur. J. Pure Appl. Math, 13 (4) (2020), 758-765 760 3. β-local functions In order to define the generalized version of the local function [8], we now introduce the concept of the β-local function. Definition 8. Let (X, τ, I) be an ideal topological space. For a set A ⊂ X, A∗ β(I, βO(X)) ={x ∈ X : A ∩ U /∈ I for every U ∈ βO(x)}, where βO(x) = {U ∈ βO(X) : x ∈ U}, is called the β-local function of A with respect to I and βO(X). A∗ β(I, βO(X)) is simply denoted by A∗ β. Example 1. Let X = {a, b, c, d} be a nonempty set with the topology τ = {φ,X, {a}, {a, b, c}}. Then the collection of closed sets is τF = {X,φ, {b, c, d}, {d}}. Applying Definition 5, we compute the collection βO(X)= {φ,X, {a}, {a, b}, {a, d}, {a, b, c}, {c, d, a}, {d, a, b}, {a, c}}. Next, we consider I = {φ, {b}}. If A = {c, d} ⊂ X then it may be easily verified that A∗ β = {c, d} and A∗ = {b, c, d}. We need the following lemma for our analysis. Lemma 1. [4] Let A be a subset of a topological space (X, τ). Then x ∈ βcl(A) if and only if A ∩ U 6= φ for every U in βO(x). Theorem 1. Let (X, τ, I) be an ideal topological space and A,B be subsets of X. Then the following properties hold: (1). If A ⊂ B then A∗ β ⊂ B∗ β, (2). (A ∪B)∗β = A∗ β ∪B∗ β, (3). (A ∩B)∗β ⊂ A∗ β ∩B∗ β, (4). (A∗ β)∗β ⊂ A∗ β, (5). A∗ β = βcl(A∗ β) ⊂ βcl(A). Proof. (1). If x /∈ B∗ β, then there exists U ∈ βO(x) such that U ∩B ∈ I. Since A ⊂ B, U ∩A ∈ I and hence x /∈ A∗ β. This shows that A∗ β ⊂ B∗ β. (2). Let x ∈ (A∪B)∗β, then using Definition 8, we have U∩(A∪B) /∈ I for every U ∈ βO(x) and (U ∩A) ∪ (U ∩B) /∈ I. Now, since I is an ideal, three cases can be possible: case I (U ∩A) /∈ I and (U ∩B) /∈ I, case II (U ∩A) ∈ I and (U ∩B) /∈ I, case III (U ∩A) /∈ I and (U ∩B) in I. It may be seen easily that for all three cases x ∈ A∗ β ∪B∗ β. Therefore, we have (A∪B)∗β ⊂ A∗ β ∪B∗ β. By (1), A∗ β ⊂ (A∪B)∗β and B∗ β ⊂ (A∪B)∗β. Hence, A∗ β ∪B∗ β ⊂ (A∪B)∗β and we obtain A∗ β ∪B∗ β = (A ∪B)∗β. (3). Since, A∩B ⊂ A and A∩B ⊂ B, by (1), (A∩B)∗β ⊂ (A)∗β and (A∩B)∗β ⊂ (B)∗β and hence, P. L. Powar, T. Noiri, Shikha Bhadauria / Eur. J. Pure Appl. Math, 13 (4) (2020), 758-765 761 (A ∩B)∗β ⊂ (A)∗β ∩ (B)∗β. (4). Let x ∈ (A∗ β)∗β, then by Definition 8, we have, A∗ β ∩ U /∈ I for every U ∈ βO(x) and A∗ β ∩ U 6= φ. Now, let y ∈ A∗ β ∩ U , then y ∈ U and U ∈ βO(y). Since y ∈ A∗ β, A ∩ U /∈ I. Hence, x ∈ A∗ β. Therefore, (A∗ β)∗β ⊂ A∗ β. (5). We know that A∗ β ⊂ βcl(A∗ β). We show that βcl(A∗ β) ⊂ A∗ β. Let x ∈ βcl(A∗ β). Then by Lemma 1, we have, A∗ β ∩U 6= φ for every U ∈ βO(x). Now, let y ∈ A∗ β ∩ U , then we have y ∈ A∗ β and y ∈ U ∈ βO(X). Therefore, we have, A ∩ U /∈ I and hence, x ∈ A∗ β. Therefore, βcl(A∗ β) ⊂ A∗ β and hence, A∗ β = βcl(A∗ β). This implies A∗ β is β-closed. Next, we show that A∗ β ⊂ βcl(A). If x /∈ βcl(A), then there exists U ∈ βO(x) such that U ∩A = φ ∈ I and x /∈ A∗ β. Therefore, A∗ β ⊂ βcl(A). Remark 1. Let (X, τ, I) be an ideal topological space and A ⊂ X. Then the following holds: (1). If I = {φ}, then A∗ β = βcl(A), (2). If K ∈ I, then K∗ β = φ and hence, {φ}∗β = φ, (3). It is not necessary that 3(a). A ⊂ A∗ β or 3(b). A∗ β ⊂ A, (4). A∗ β(I, βO(X)) = (A)∗ s (I, τ) if SO(X) = βO(X), (5). A∗ β(I, βO(X)) = A∗(I, τ) if βO(X) = τ(X). In order to verify 3(a). and 3(b). of Remark 1, we explore the following example: Example 2. LetX = {a, b, c, d} be a nonempty set with the topology τ = {φ,X, {a}, {a, b, c}} and the collection of closed sets is τF = {φ,X, {b, c, d}, {d}}. Next, by applying Defi- nition 5, we compute the collection βO(X)= {φ,X, {a}, {a, b}, {a, d}, {a, c}, {a, b, c}, {c, d, a}, {d, a, b}}. Considering I = {φ, {b}} and A,B ⊂ X where, A = {b, c, d} and B = {a, b, c} then by applying Definition 8, A∗ β = {c, d} and B∗ β = X. In view of above assertions 3(a) and 3(b) have been verified. Lemma 2. Let (X, τ, I) be an ideal topological space. Then the following properties hold: (1). RO(X) ⊂ τ ⊂ SO(X) ⊂ βO(X), (2). A∗ β ⊂ A∗s ⊂ A∗ ⊂ Γ∗(A) for every subset A of X. Proof. (1). The proof is obvious by the Definition 5. (2). First, we show that A∗ β ⊂ A∗s . Let x ∈ A∗ β . Then, A ∩ U /∈ I for every U ∈ βO(x). Since SO(X) ⊂ βO(X), A ∩ U /∈ I for every U ∈ SO(x) and x ∈ A∗s . Hence, we have A∗ β ⊂ A∗s . Similarly, by using the fact that RO(X) ⊂ τ ⊂ SO(X), we may establish A∗s ⊂ A∗ and A∗ ⊂ Γ∗(A). Definition 9. Let (X, τ, I) be an ideal topological space. We define cl∗β(A) = A ∪ A∗ β for every subset A of X. P. L. Powar, T. Noiri, Shikha Bhadauria / Eur. J. Pure Appl. Math, 13 (4) (2020), 758-765 762 Theorem 2. Let (X, τ, I) be an ideal topological space. Then the following properties hold: (1). A ⊂ cl∗β(A), (2). cl∗β(φ) = φ and cl∗β(X) = X, (3). A ⊂ B implies cl∗β(A) ⊂ cl∗β(B), (4). cl∗β(A) ∪ cl∗β(B) = cl∗β(A ∪B), (5). (cl∗β(A))∗β ⊂ cl∗β(A) = cl∗β(cl∗β(A)). Proof. (1). This follows directly by the Definition 9. (2). We have cl∗β(φ) = {φ} ∪ {φ}∗β = φ. Similarly, it may be verified that cl∗β(X) = X. (3). Given, A ⊂ B. By Definition 9, cl∗β(A) = A ∪ A∗ β and cl∗β(B) = B ∪ B∗ β. Next, by Theorem 1 (1), we have A∗ β ⊂ B∗ β. Therefore, we obtain A ∪ A∗ β ⊂ B ∪ B∗ β and hence cl∗β(A) ⊂ cl∗β(B). (4). By Theorem 1 (2), cl∗β(A ∪B) = (A ∪B) ∪ (A∗ β ∪B∗ β) = cl∗β(A) ∪ cl∗β(B). Hence, cl∗β(A ∪B) = cl∗β(A) ∪ cl∗β(B). (5). First, we show that (cl∗β(A))∗β ⊂ cl∗β(A). Let if possible x /∈ cl∗β(A). This implies x /∈ A∗ β and there exists U ∈ βO(x) such that U ∩A ∈ I and we conclude that U ∩A∗ β = φ and φ ∈ I. For if A∗ β∩U 6= φ then there exists y ∈ A∗ β ∩ U and U ∈ βO(y). Then y ∈ A∗ β implies U ∩ A /∈ I, which is a contradiction as U ∩A ∈ I. Hence, U ∩A∗ β = φ. Now, we obtain (A ∪A∗ β) ∩ U = (A ∩ U) ∪ (A∗ β ∩ U) ∈ I. This implies that (cl∗β(A)) ∩ U ∈ I. By Definition 8, we obtain, x /∈ (cl∗β(A))∗β. Hence, we obtain (cl∗β(A))∗β ⊂ cl∗β(A). Next, we show that cl∗β(cl∗β(A)) = cl∗β(A). Now, we have cl∗β(cl∗β(A)) = cl∗β(A)∪ (cl∗β(A))∗β. Since (cl∗β(A))∗β ⊂ cl∗β(A), we obtain cl∗β(cl∗β(A)) ⊂ cl∗β(A). It is obvious that cl∗β(A) ⊂ cl∗β(cl∗β(A)). Therefore, cl∗β(A) = cl∗β(cl∗β(A)). Theorem 3. Let (X, τ, I) be an ideal topological space. Let τ∗β = {U ⊂ X : cl∗β(X \U) = X \ U}. Then τ∗β is a topology for X such that τ∗ ⊂ τ∗β and βO(X) ⊂ τ∗β . Proof. By Theorem 2, we obtain that cl∗β(A) = A ∪ A∗ β is a Kuratowski Closure Operator. Therefore, τ∗β is the topology for X generated by cl∗β. First, we show that τ∗ ⊂ τ∗β . By Lemma 2(2), for every subset A of X, cl∗β(A) = A∪A∗ β ⊂ A ∪ A∗ = cl∗(A). Let A be a τ∗-closed set, then cl∗(A) = A and cl∗β(A) ⊂ A. Hence cl∗β(A) = A and A is τ∗β -closed. Secondly, we show that βO(X) ⊂ τ∗β . Suppose that A is β-closed. If x /∈ A, then by Lemma 1, there exists U in βO(x) such that U ∩ A = φ ∈ I. Hence x /∈ A∗ β. This shows A∗ β ⊂ A. Therefore, cl∗β(A) = A∪A∗ β = A and A is τ∗β -closed. We obtain that βO(X) ⊂ τ∗β . Definition 10. Let (X, τ, I) be an ideal topological space. A subset A of X is said to be Ig-β-closed if A∗ β ⊂ U whenever A ⊂ U and U in βO(X). Theorem 4. For a subset A of an ideal topological space (X, τ, I), the following properties are equivalent: P. L. Powar, T. Noiri, Shikha Bhadauria / Eur. J. Pure Appl. Math, 13 (4) (2020), 758-765 763 (1). A is Ig-β-closed; (2). cl∗β(A) ⊂ U whenever A ⊂ U and U is β-open; (3). For every x ∈ cl∗β(A), βcl({x}) ∩A 6= φ; (4). cl∗β(A)−A contains no nonempty β-closed set; (5). A∗ β −A contains no nonempty β-closed set. Proof. (1). ⇒ (2). By hypothesis, A is Ig-β-closed. Therefore, A∗ β ⊂ U whenever A ⊂ U and U in βO(X). This implies A∗ β ∪ A ⊂ U and hence, cl∗β(A) ⊂ U whenever A ⊂ U and U ∈ βO(X). (2). ⇒ (3). Suppose x ∈ cl∗β(A). If βcl({x}) ∩ A = φ, then A ⊂ (X − βcl({x})), where (X − βcl({x})) is β-open and by hypothesis, cl∗β(A) ⊂ (X − βcl({x})). Therefore, cl∗β(A) ∩ βcl({x}) = φ, which is a contradiction, since x ∈ cl∗β(A). Hence, for every x ∈ cl∗β(A), βcl({x}) ∩A 6= φ (3). ⇒ (4). Let if possible, F ⊂ cl∗β(A) − A, where F is a nonempty β-closed set and x ∈ F . This implies F ⊂ X − A and F ∩ A = φ. Therefore, βcl({x}) ∩ A = φ, which is a contradiction to our hypothesis as βcl({x}) ∩ A 6= φ. Hence, cl∗β(A) − A contains no nonempty β-closed set. (4). ⇒ (5). The proof is obvious, since A∗ β ⊂ cl∗β(A). (5). ⇒ (1). Let A ⊂ U and U is any β-open set of X. By Theorem 1(5), A∗ β is β-closed and A∗ β ∩ (X −U) ⊂ A∗ β −A, where, A∗ β ∩ (X −U) is β-closed. By (5), A∗ β ∩ (X −U) = φ. Therefore, A∗ β ⊂ U and hence A is Ig-β-closed. 4. Is∗g-β-closed sets In this section, the notion of Is∗g-β-closed sets is defined with an illustrative example. Moreover, some properties of these closed sets has been also explored. Definition 11. Let (X, τ, I) be an ideal topological space. A subset A of X is said to be Is∗g-β-closed (resp. Is∗g-closed [9]) if A∗ β ⊂ U (resp. A∗ ⊂ U) whenever A ⊂ U and U is semi-open. The complement of an Is∗g-β-closed set is said to be Is∗g-β-open. The family of Is∗g-β-closed (resp. Is∗g-closed) sets is denoted by Is∗g βC(X) (resp. Is∗gC(X)). Theorem 5. Let (X, τ, I) be an ideal topological space and A a subset of X. If A is Is∗g-closed, then it is Is∗g-β-closed. But the converse is not always true. Proof. Suppose that A is Is∗g-closed. For every U ∈ SO(X) containing A, we have A∗ ⊂ U and by Lemma 2(2), A∗ β ⊂ A∗ ⊂ U . This shows that A is Is∗g-β-closed. Example 3. Let X = {a, b, c, d} be a nonempty set with the topology τ = {φ,X, {b, c}, {a, b, c}, {b}, {a, b}} and the collection of closed sets is τF = {X,φ, {a, c, d}, {c, d}, {a, d}, {d}}. Applying the Definition 5, we compute the collection βO(X)= {φ,X, {a, b}, {b}, {b, c}, {b, d}, {a, b, c}, {b, c, d}, {d, a, b}}. Considering I = {φ, {a}} and applying Definition 11, we com- pute the collection of Is∗gβC(X)= {φ,X, {a, c, d}, {c, d}, {a, d}, {a, c}, {d}, {a}, {c}} and Is∗gC(X) = { φ, X, {a, c, d}, {c, d}, {a, d}, {d}, {a}}. It can be verified that the subsets {a, c} and {c} of X are Is∗gβ-closed but not Is∗g-closed. REFERENCES 764 Theorem 6. Let (X, τ, I) be an ideal topological space and A,B be subsets of X. (1). If A and B are Is∗g-β-closed, then A ∪B is Is∗g-β-closed. (2). If A is closed in X, then A is Is∗g-β-closed. (3). If U is open in X and A is Is∗g-β-open, then U ∩A is Is∗g-β-open. Proof. (1). Let A ∪B ⊂ U and U ∈ SO(X). Then, we know that A ⊂ U and B ⊂ U. Since A and B both are Is∗g-β-closed, we have A∗ β ⊂ U and B∗ β ⊂ U. Hence, A∗ β ∪B∗ β ⊂ U . Now by Theorem 1(2), (A∪B)∗β = A∗ β ∪B∗ β ⊂ U. Hence, we obtain A∪B is Is∗g-β-closed. (2). Let A ⊂ U and U ∈ SO(X). By Lemma 2, A∗ β ⊂ A∗ ⊂ Cl(A) = A ⊂ U . This shows that A is Is∗g-β-closed. (3). The proof is a direct consequence of (1) and (2). 5. Conclusion The concept of the β-local function, the operation cl∗β and Ig-β-closed sets have been introduced with illustrative examples. Moreover, certain properties have been also studied and explored. It may be concluded that the concept of the topology τ∗β is more generalized version of τ∗ and β-open sets, which may be further useful to enrich the class of continuous functions. References [1] M. E. Abd El-Monsef, S. N. EL-Deeb and R.A. Mahmoud, β-open and β-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77− 90. [2] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef, On I-open sets and I-continuous functions, Kyungpook Math. J., 32(1)(1992), 21− 30. [3] M. E. Abd El-Monsef, E. F. Lashien and A. A. Nasef, Some topological operators via ideals, Kyungpook J. Math., 32(2)(1992), 273− 284. [4] M. E. Abd-E-Monsef, R. A. Mohmoud and E. R. Lashin, β-closure and β-interior, J. Fac. Ed. Ain Shans Univ., 10(1986), 235− 245. [5] A. Al-Omari and T. Noiri, Local function Γ∗ in ideal topological spaces, Sci. Stud. Res. Ser. Math. Inform., 26(1)(2016), 5− 16. [6] J. Dontchev, M. Ganster and T. Noiri, Unified operation approach of generalized closed sets via topological ideals, Math. Japon., 49(1999), 395− 402. [7] E. Hatir, A. Al-Omari and S. Jafari, δ-local functions and its properties in ideal topological spaces, Fasciculi Math., 53(2014), 53− 64. [8] D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Amer. Math. Monthly, 97(4)(1990), 295− 310. REFERENCES 765 [9] M. Khan and M. Hamza, Is∗g-closed sets in ideal topological spaces, Glob. J. Pure Appli. Math., 7(1)(2011), 89− 99. [10] M. Khan and T. Noiri, Semi-local functions in ideal topological spaces, J. Adv. Res. Pure Math., 2(1)(2010), 36− 42. [11] K. Kuratowski, Topology I, Warszawa (1933). [12] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70(1963), 36− 41. [13] P. L. Powar and K. Rajak, Some new concepts of continuity in generalized topological space, Int. J. Com. Appl., 38(5)(2012), 12− 17.