EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5782 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Introduction to Mixed H ( θ(µ, ν) ) -Open Sets Generated by Hereditary Classes in Generalized Topological Spaces Fahad Alsharari1,∗, Abdo Qahis2 1 Department of Mathematics, College of Science, Jouf University, Sakaka 72311, Saudi Arabia 2 Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia Abstract. In [1], Kim and Min introduced the operation γ∗ and H(θ)-open sets within the con- text of generalized topological spaces, utilizing a hereditary class H. In this study, we extend these concepts by employing two generalized topologies, µ and ν, along with a hereditary class H. Specif- ically, we introduce and investigate the mixed operation γ∗(µ, ν) (denoted briefly as γ∗(µ, ν)) and the mixed H(θ(µ, ν))-open sets (denoted as H(θ(µ, ν))-open sets). We explore the interrelation- ships between γ∗(µ, ν), γ∗, and the µ-closure, as well as the connections between H(θ(µ, ν))-open sets, θ(µ, ν)-open sets, and µ-open sets. Additionally, we define the concepts of Hr(µ, ν)-regular open sets and H(µ, ν)-regular open sets. Finally, we examine properties and characterizations of H(θ(µ, ν))-open sets in terms of Hr(µ, ν)-regular open sets and H(µ, ν)-regular open sets. 2020 Mathematics Subject Classifications: 54A05, 54C08s Key Words and Phrases: Hereditary Classes H, mixed operation γ∗(µ, ν), mixed H ( θ(µ, ν) ) - open sets, Hr(µ, ν)-regular open sets, H(µ, ν)-regular 1. Introduction Á. Császár formulated the idea of generalized topology and generalized open sets in [2], along with the notion of θ-open sets and their properties. For further details, one can refer to [3–5]. In [6], he also introduced the concept of hereditary classes in generalized topological spaces. Specifically, a subset H ⊆ P(X) (where P(X) denotes the power set of a non-empty set X) is termed a hereditary class on X if it satisfies the condition that for any A ⊆ B and B ∈ H, it follows that A ∈ H. Building on these foundations of generalized topology and hereditary classes, authors in [1] introduced the concepts of H(θ)-open sets and the operator γ∗. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5782 Email addresses: f.alsharari@ju.edu.sa (F. Alsharari), cahis82@gmail.com (A. Qahis) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 2 of 10 In this study, we extend these ideas by examining two generalized topologies, denoted as µ and ν, within the framework of a hereditary class H. We introduce and investigate the concepts of the mixed operator γ∗(µ, ν) (denoted briefly as γ∗(µ, ν)) and mixed H(θ(µ, ν))- open sets (referred to as H(θ(µ, ν))-open sets). Our exploration includes a detailed study of their properties and the relationships between γ∗(µ, ν), the operator γ∗, and the µ- closure. In addition, we examine the interconnections between sets that are H(θ(µ, ν))- open, θ(µ, ν)-open, and µ-open. We also present various properties and characterizations of these concepts in terms of Hr(µ, ν)-regular open sets and H(µ, ν)-regular sets. 2. Preliminaries Let X ̸= ∅ and let P(X) be its power set. A family µ ⊆ P(X) is called a generalized topology (GT) on X if: ∅ ∈ µ,⋃ i∈I Ui ∈ µ for any collection {Ui}i∈I ⊆ µ. This concept was first introduced by Á. Császár in [2]. A pair (X,µ) is then referred to as a generalized topological space (GTS) on X. The elements of µ are called µ-open sets, while their complements are called µ-closed sets. The union of all elements of µ is denoted by Mµ, as stated in [7]. A GTS (X,µ) is is said to be strong [8] if X ∈ µ. For a subset A of a GTS (X,µ), the µ-closure of A, denoted cµ(A), is defined as the intersection of all µ-closed sets that contain A. The µ-interior of A, denoted iµ(A), is the union of all µ-open sets that are contained within A (see [2, 7]). Now, considering a hereditary class H, an operator ()∗ : P(X) → P(X) was introduced in [3]. Specifically, c∗ : P(X) → P(X) is defined using ()∗ by c∗(A) = A ∪ A∗, where A∗ = {x ∈ X | A ∩ M /∈ H, ∀M ∈ µ, x ∈ M}. Here, x /∈ A∗ if and only if there exists M ∈ µ such that x ∈ M and M ∩A ∈ H. Recalling definitions and notations from [3], let µ be a GT on X and P(X) be the power set of X. A collection θ ⊆ P(X) is defined as follows: A ∈ θ if for each x ∈ A, there exists M ∈ µ containing x such that M ⊆ cµ(M) ⊆ A. The family θ is a GT on X included in µ, and the elements of θ are θ(µ)-open sets, with complements called θ(µ)-closed sets. For A ⊆ X, the operation γθ : P(X) → P(X) is defined in [3], by γθ(A) = {x ∈ X | cµ(M) ∩A ̸= ∅, ∀M ∈ µ, x ∈ M}. Kim and Min in [5], extended the study to θ-open using a hereditary class H: a collection H(θ) ⊆ P(X) is defined such that A ∈ H(θ) if for each x ∈ A, there exists M ∈ µ containing x with M ⊆ c∗µ(M) ⊆ A. The family H(θ) is a GT on X included in µ, with elements termed H(θ)-open sets and their complements H(θ)-closed sets. Additionally, the operation γ∗ : P(X) → P(X) is defined in [5] as γ∗(A) = {x ∈ X | c∗µ(M) ∩A ̸= ∅,∀M ∈ µ, x ∈ M}. In [4], Á. Császár and Makai Jr. introduced θ(ν1, ν2)-open sets as a means of combining two generalized topologies (GTs), ν1 and ν2, on a set X. A subset A ⊆ X belongs to F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 3 of 10 θ(ν1, ν2) if, for every x ∈ A, there exists M ∈ ν1 such that x ∈ M ⊆ cν2(M) ⊆ A. Moreover, the family θ(ν1, ν2) itself forms a GT contained within ν1 on X. The sets in θ(ν1, ν2) are called θ(ν1, ν2)-open sets, while their complements are referred to as θ(ν1, ν2)- closed sets. Subsequently, in [9], Abdo Qahis and Awn Alqahtani introduced a modification of this concept, defining the class of θ̃(ν1, ν2)-open sets. A subset A ⊆ X is said to be mixed θ̃(ν1, ν2)-open (or simply θ̃(ν1, ν2)-open) if, for every x ∈ A, there exists M ∈ ν1 such that x ∈ M and M ⊆ cν2(M) ∩Mν1 ⊆ A. In conclusion of this section, we review the following important facts due to their significance to the content of our paper. Theorem 1. [6] Let µ be a GTS on X and H a hereditary class on X. Then A∗ ⊆ c∗µ(A) ⊆ cµ(A) for any A ⊆ X. In [10], the authors introduced the operator i∗ : P(X) → P(X), defined by i∗(A) = X \ c∗(X \A) for A ⊆ X. Theorem 2. [10] Let µ be a GT on X and H a hereditary class. Then for A ⊆ X, (i) c∗µ(A) = X \ i∗µ(X \A). (ii) iµ(A) ⊆ i∗µ(A) ⊆ A. Lemma 1. [11] Let µ and ν be two GTs on a nonempty set X and A ⊆ X. Then the following statements hold: (i) x ∈ iθ(µ,ν)(A) if and only if there exists a µ-open set M containing x such that M ⊆ cν(M) ⊆ A. (ii) If A is ν-open in X, then γθ(µ,ν)(A) = cµ(A). Definition 1. [1] Let µ be GT on a nonempty set X, and H a hereditary class on X. Then (X,µ) is H-regular if and only if for every x ∈ X and every µ-open set U containing x, there exists a µ-open set V containing x such that x ∈ V ⊆ c∗(V ) ⊆ U . Theorem 3. [11] Let µ and ν be measures on a nonempty set X. Then X is (µ, ν)-regular if and only if for every x ∈ X and every µ-open set U containing x, there exists a µ-open set V containing x such that x ∈ V ⊆ cν(V ) ⊆ U . 3. Properties of the Mixed Operator γ∗(µ, ν) In [4], Császár and Makai Jr introduced an operation γθ(µ,ν) : P(X) → P(X), utilizing two generalized topologies µ and ν on X. According to their definition, x ∈ γθ(µ,ν)(A) if and only if cν(M) ∩ A ̸= ∅ for every µ-open set M containing x. If x /∈ Mµ, then by definition x ∈ γθ(µ,ν)(A). Additionally, x /∈ γθ(µ,ν)(A) if and only if there exists M ∈ µ with x ∈ M such that cν(M) ∩A = ∅. F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 4 of 10 Definition 2. Let µ and ν be two GT’s on a nonempty set X, and H a hereditary class on X. An operation γ∗(µ, ν) : P(X) → P(X) is defined as follows: for every A ⊆ X, γ∗(µ, ν)(A) = {x ∈ X : c∗ν(M) ∩A ̸= ∅, ∀M ∈ µ and x ∈ M}. If x /∈ Mµ, then by definition x ∈ γ∗(µ, ν)(A). According to Definition 2, x /∈ γ∗(µ, ν)(A) if and only if there exists M ∈ µ and x ∈ M such that c∗ν(M) ∩A = ∅. The following is an immediate consequence that can be obviously obtained. Corollary 1. Let µ and ν be two GT’s on a nonempty set X such that µ = ν, and let H be a hereditary class on X. For any A ⊆ X, the following statements hold: (i) γ∗(µ, ν)(A) = γ∗(A). (ii) If H = {∅}, then γ∗(µ, ν)(A) = γ∗(A) = γθ(A). Theorem 4. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. Then for any A ⊆ X, we have γ∗(µ, ν)(A) ⊆ γθ(µ,ν)(A). Proof. Let x ∈ γ∗(µ, ν)(A). For each µ-open set M containing x, we have c∗ν(M)∩A ̸= ∅. Since c∗ν(M) ⊆ cν(M), it follows that cν(M)∩A ̸= ∅. Therefore, x ∈ γθ(µ,ν)(A), and so γ∗(µ, ν)(A) ⊆ γθ(µ,ν)(A). The following example demonstrates that, in general, γ∗(µ, ν)(A) ̸= γθ(µ, ν)(A). Example 1. Let X = {a, b, c, d}. Consider two generalized topologies: µ = {∅, {b, d}} and ν = {∅, {a, b}, {b, c}, {a, b, c}}, and a hereditary class H = {∅, {b}} on X. For a set A = {a, c}, we have cν({b, d}) = X, Mµ = {b, d}, and cν({b, d}) ∩ A ̸= ∅. Thus, γθ(µ,ν)(A) = X. Since Mµ = {b, d}, it is clear that a, c ∈ γ∗(µ, ν)(A). Noting that {b, d}∗(H, ν) = {d}, we find c∗ν({b, d}) ∩ A = ∅, hence b, d /∈ γ∗(µ, ν)(A). Therefore, γ∗(µ, ν)(A) = {a, c} and γ∗(µ, ν)(A) ⊂ γθ(µ,ν)(A). Corollary 2. Let µ and ν be two GT’s on a nonempty set X and let H be a hereditary class on X. If H = {∅}, then γ∗(µ, ν)(A) = γθ(µ,ν)(A) for any A ⊆ X. Theorem 5. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. For any subsets A and B of X, the following properties hold: (i) γ∗(µ, ν)(∅) = ∅. (ii) If A ⊆ B, then γ∗(µ, ν)(A) ⊆ γ∗(µ, ν)(B). (iii) A ⊆ cµ(A) ⊆ γ∗(µ, ν)(A). Proof. (1) and (2) are obvious. (3) For x ∈ cµ(A) and any µ-open set M containing x, we have M ∩ A ̸= ∅. Conse- quently, c∗ν(M) ∩A ̸= ∅. Therefore, x ∈ γ∗(µ, ν)(A), implying that cµ(A) ⊆ γ∗(µ, ν)(A). The following example shows that, in general, cµ(A) ̸= γ∗(µ, ν)(A). F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 5 of 10 Example 2. Let X = {a, b, c, d}. Consider two generalized topologies: µ = {∅, {a}}, ν = { ∅, {a, b}, {b, c}, {a, b, c} } on X, and a hereditary class H = {∅, {b}}. For a set A = {b, c, d}, since A is µ-closed, cµ(A) = A. Given Mµ = {a}, it follows by the definition of the operator γ∗(µ, ν) that X −Mµ = {b, c, d} ⊆ γ∗(µ, ν)(A). Next, we show that a ∈ γ∗(µ, ν)(A). Since M = {a} ∈ µ and {a}∗(H, ν) = {a, d}, we have c∗ν(M) ∩ A ̸= ∅. Thus, a ∈ γ∗(µ, ν)(A), implying cµ(A) ⊂ γ∗(µ, ν)(A) = X. Thus cµ(A) ̸= γ∗(µ, ν)(A). The following Corollary follows immediately from Theorem 5(iii), and Theorem 1. Corollary 3. Let µ and ν be two GT’s on a nonempty set X, and H a hereditary class on X. For A ⊆ X, A∗ ⊆ c∗µ(A) ⊆ γ∗(µ, ν)(A). Theorem 6. Let µ and ν be two GT’s on a nonempty set X, H a hereditary class on X, and A ⊆ X. Then γ∗(µ, ν)(A) is µ-closed. Proof. Let x ∈ X − γ∗(µ, ν)(A). This means there exists Mx ∈ µ such that c∗ν(Mx) ∩ A = ∅. Since Mx ⊆ c∗ν(Mx), it follows that Mx ∩ A = ∅. Therefore, every y ∈ Mx implies y ∈ X − γ∗(µ, ν)(A), implying X − γ∗(µ, ν)(A) = ⋃ x∈X−γ∗(µ,ν)(A)Mx. Thus, X − γ∗(µ, ν)(A) is µ-open, hence γ∗(µ, ν)(A) is µ-closed. Theorem 7. Let µ and ν be two GT’s on a nonempty set X, and H a hereditary class on X. If A is ν-open in X, then γ∗(µ, ν)(A) = cµ(A). Proof. From (iii) of Theorem 5, we have cµ(A) ⊆ γ∗(µ, ν)(A). For the converse inclusion, suppose x ∈ γ∗(µ, ν)(A). For each M ∈ µ such that x ∈ M and c∗ν(M) ∩ A ̸= ∅. Since c∗ν(M) ⊆ cν(M), it follows that cν(M) ∩ A ̸= ∅. Thus, there exists y ∈ cν(M) ∩ A. Since A is ν-open and contains y, we have M ∩ A ̸= ∅, implying x ∈ cµ(A). Therefore, γ∗(µ, ν)(A) ⊆ cµ(A). Combining this with the earlier inclusion, we conclude γ∗(µ, ν)(A) = cµ(A). The following Corollary follows from Lemma 1(ii) and Theorem 7. Corollary 4. Let µ and ν be two GT’s on a nonempty set X, H a hereditary class on X, and A ⊆ X. If A ∈ ν, then γ∗(µ, ν)(A) = cµ(A) = γθ(µ,ν)(A). 4. H ( θ(µ, ν) ) -Open Sets Definition 3. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. We define the collection H ( θ(µ, ν) ) ⊆ P(X) such that A ∈ H ( θ(µ, ν) ) if and only if for each x ∈ A, there exists M ∈ µ such that x ∈ M ⊆ c∗ν(M) ⊆ A. The elements ofH ( θ(µ, ν) ) are called mixedH ( θ(µ, ν) ) -open (briefly, H ( θ(µ, ν) ) -open) , and their complements are called mixed H ( θ(µ, ν) ) -closed (briefly, H ( θ(µ, ν) ) -closed). F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 6 of 10 Remark 1. Consider µ and ν to be two GT’s on a nonempty set X, and let H be a hereditary class on X. If µ = ν, then H ( θ(µ, ν) ) = H(θ). Theorem 8. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. Then θ(µ, ν) ⊆ H ( θ(µ, ν) ) ⊆ µ. Proof. To show that θ(µ, ν) ⊆ H ( θ(µ, ν) ) , let A ∈ θ(µ, ν) and x ∈ A. Then there exists M ∈ µ such that x ∈ M ⊆ cν(M) ⊆ A. Since c∗ν(M) ⊆ cν(M), we have x ∈ M ⊆ c∗ν(M) ⊆ A. Therefore, A is an H ( θ(µ, ν) ) -open set. Next, to show that H ( θ(µ, ν) ) ⊆ µ, suppose A ∈ H ( θ(µ, ν) ) and let x ∈ A. Then there exists a µ-open set Mx such that x ∈ Mx ⊆ c∗ν(Mx) ⊆ A. Therefore, A = ⋃ x∈AMx ∈ µ. Remark 2. Based on Theorem 8, we can illustrate the following diagram. The following example demonstrates that the above implications are not reversible in general. Example 3. Let X = {a, b, c, d}. Consider two generalized topologies µ = { ∅, {a, b}, {b, c}, {a, b, c} } and ν = { ∅, {b, d} } and a hereditary class H = { ∅, {b} } . Note that: (i) For a set A = {a, b, c}, we have Ma = Mb = {a, b} ∈ µ and Mc = {b, c} ∈ µ. Then M∗ a (H, ν) = M∗ b (H, ν) = M⋆ c (H, ν) = {a, c} and c∗v(Ma) = c∗v(Mb) = c∗v(Mc) = {a, b, c} ⊆ A; (ii) Since cν({a, b}) = cν({b, c}) = cν({a, b, c}) = X ⊈ A, then θ(µ, ν) = {∅}. (iii) From (1) and (2), we show that the set A is H ( θ(µ, ν) ) -open but it is not θ(µ, ν)-open. Also, it is easy to check that B = {a, b} is µ-open but it is not H ( θ(µ, ν) ) -open. Theorem 9. Let µ and ν be two GTs on a nonempty set X, and H be a hereditary class on X. Then the family H ( θ(µ, ν) ) is a GT contained in µ on X. Proof. Firstly, ∅ ∈ H ( θ(µ, ν) ) is obvious. Now, let {Aα ⊆ X : Aα ∈ H ( θ(µ, ν) ) } for α ∈ Λ. Consider x ∈ ∪αAα. Then there exists some α0 ∈ Λ such that for some µ-open set M containing x, we have M ⊆ c∗ν(M) ⊆ Aα0 . This implies there exists x ∈ M ∈ µ such that M ⊆ c∗ν(M) ⊆ ∪αAα and so ∪αAα ∈ H ( θ(µ, ν) ) . Theorem 10. Let µ and ν be two GT’s on a nonempty set X, H a hereditary class on X, and A ⊆ X. Then, A is H(θ(µ, ν))-closed if and only if γ∗(µ, ν)(A) = A. F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 7 of 10 Proof. Let A be H ( θ(µ, ν) ) -closed in X. Since X−A ∈ H ( θ(µ, ν) ) , for each x ∈ X−A, there exists M ∈ µ such that x ∈ M ⊆ c∗ν(M) ⊆ X − A. Thus, c∗ν(M) ∩ A = ∅, implying x /∈ γ∗(µ, ν)(A). Therefore, γ∗(µ, ν)(A) ⊆ A, implying that γ∗(µ, ν)(A) = A. For the reverse inclusion, suppose γ∗(µ, ν)(A) = A and let x ∈ X−A = X−γ∗(µ, ν)(A). Then there exists M ∈ µ such that x ∈ M and c∗ν(M)∩A = ∅. Hence, x ∈ M ⊆ c∗ν(M) ⊆ X −A, showing that X −A is H ( θ(µ, ν) ) -open. Therefore, A is H ( θ(µ, ν) ) -closed. From Theorem 10 and Theorem 7, the following Corollary is directly obtained. Corollary 5. Let µ and ν be two GT’s on a nonempty set X, H a hereditary class on X, and A ⊆ X be H ( θ(µ, ν) ) -closed. If A ∈ ν, then A is µ-closed. Definition 4. Let µ and ν be two GT’s on a nonempty set X, and H a hereditary class on X. The H ( θ(µ, ν) ) -closure of A ⊆ X, denoted by cHθ(µ,ν)(A), is the intersection of all H ( θ(µ, ν) ) -closed sets containing A. The H ( θ(µ, ν) ) -interior of A, denoted by iH(θ(µ,ν))(A), is the union of all H ( θ(µ, ν) ) -open sets contained in A. Theorem 11. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. Then γ∗(µ, ν)(A) ⊆ cHθ(µ,ν)(A). Proof. Let x /∈ cHθ(µ,ν)(A). Then there exists an H ( θ(µ, ν) ) -open set W containing x such that W ∩ A = ∅. Since W ∈ H ( θ(µ, ν) ) , there exists M ∈ µ containing x such that x ∈ M ⊆ c∗ν(M) ⊆ W ⊆ X − A. This implies c∗ν(M) ∩ A = ∅ and hence x /∈ γ∗(µ, ν)(A). Therefore, γ∗(µ, ν)(A) ⊆ cHθ(µ,ν)(A). Definition 5. Let µ and ν be two GTs on a nonempty set X, and let H be a hereditary class on X. A subset A ⊆ X is called H(µ, ν)-regular open (briefly, Hr(µ, ν)-open) if A = iµ(c ∗ ν(A)). Similarly, A is called H(µ, ν)-regular closed (briefly, Hr(µ, ν)-closed) if cµ(i ∗ ν(A)) = A. Theorem 12. Let µ and ν be two GTs on a nonempty set X, H a hereditary class on X, and A ⊆ X. If A ∈ H ( θ(µ, ν) ) and x ∈ A, then there exists a Hr(µ, ν)-open set U such that x ∈ U ⊆ c∗ν(U) ⊆ A. Proof. Since A ∈ H ( θ(µ, ν) ) and x ∈ A, there exists a µ-open set M such that x ∈ M ⊆ c∗ν(M) ⊆ A. Define U = iµ ( c∗ν(M) ) . Then U is Hr(µ, ν)-open, M ⊆ U , and c∗ν(U) = c∗ν ( iµ ( c∗ν(M) )) ⊆ c∗ν(M). This implies x ∈ M ⊆ U ⊆ c∗ν(U) ⊆ c∗ν(M) ⊆ A. Thus, we have x ∈ U ⊆ c∗ν(U) ⊆ A for some Hr(µ, ν)-open set U . Since everyH(µ, ν)-regular open set is µ-open inX, the following Corollary is evidently obtained. Corollary 6. Let µ and ν be two GT’s on a nonempty set X, H a hereditary class on X, and A ⊆ X. Then, A ∈ H ( θ(µ, ν) ) and x ∈ A if and only if there exists a Hr(µ, ν)-open set U such that x ∈ U ⊆ c∗ν(U) ⊆ A. F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 8 of 10 Definition 6. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. A set X is said to be H(µ, ν)-regular (or simply H(µ, ν)-regular) if for every x ∈ X and every µ-closed set F with x /∈ F , there exist sets U ∈ µ, V ∈ ν∗ such that: x ∈ U, F ⊆ V, and U ∩ V = ∅. Theorem 13. Let µ and ν be GT’s on a nonempty set X, and H a hereditary class on X. Then X is H(µ, ν)-regular if and only if for every x ∈ X and every µ-open set U containing x, there exists a µ-open set V containing x such that x ∈ V ⊆ c∗ν(V ) ⊆ U . Proof. Assume X is H(µ, ν)-regular. For x ∈ X and a µ-open set U containing x, there exist disjoint sets V ∈ µ andW ∈ ν∗ such that x ∈ V , (X−U) ⊆ W . Since V ⊆ X−W and X−W is ν∗-closed, we have c∗ν(V ) ⊆ X−W . This implies c∗ν(V )∩(X−U) ⊆ c∗ν(V )∩W = ∅, hence x ∈ V ⊆ c∗ν(V ) ⊆ U . Conversely, suppose F is a µ-closed set and x /∈ F for x ∈ X. Since X −F is a µ-open set containing x, by hypothesis, there exists a µ-open set V containing x such that x ∈ V , V ⊆ c∗ν(V ) ⊆ X − F , c∗ν(V ) ∩ F = ∅, and F ⊆ X − c∗ν(V ). As X − c∗ν(V ) ∈ ν∗ and V ∩ ( X − c∗ν(V ) ) = ∅, it follows that X is H(µ, ν)-regular. Remark 3. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X such that µ = ν. If X is H(µ, ν)-regular, then X is also H-regular. Proposition 1. Let µ and ν be two GT’s on a nonempty set X, and H a hereditary class on X. If X is (µ, ν)-regular, then X is H(µ, ν)-regular. Proof. Let X be (µ, ν)-regular. Consider x ∈ X and an µ-closed set F such that x /∈ F . Then X − F is a µ-open set containing x. By Theorem 3, there exists a µ-open set V containing x such that x ∈ V ⊆ cν(V ) ⊆ X − F. Since c∗ν(V ) ⊆ cν(V ), it follows that x ∈ V ⊆ c∗ν(V ) ⊆ X − F. By Theorem 13, X is H(µ, ν)-regular. Theorem 14. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. If X is H(µ, ν)-regular, then the following hold: (i) For any A ⊆ X, γ∗(µ, ν)(A) = cµ(A). (ii) Every µ-open set is H(θ(µ, ν))-open. F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 9 of 10 Proof. (1) By Theorem 5(iii), we have cµ(A) ⊆ γ∗(µ, ν)(A). To show the reverse inclu- sion, let x ∈ γ∗(µ, ν)(A) and let U be any µ-open set containing x. FromH(µ, ν)-regularity, there exists a µ-open set V such that x ∈ V ⊆ c∗ν(V ) ⊆ U . Since x ∈ γ∗(µ, ν)(A), it follows that c∗ν(V ) ∩A ̸= ∅. Thus, U ∩A ̸= ∅, implying x ∈ cµ(A). (2) Let M be a µ-open set. From (1), we have γ∗(µ, ν)(X−M) = cµ(X−M) = X−M . By Theorem 10, X −M is H(θ(µ, ν))-closed, which means M is H(θ(µ, ν))-open. The next result follows from Theorem 8 and Theorem 14(ii). Corollary 7. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. If X is H(µ, ν)-regular, then µ = H(θ(µ, ν)). Definition 7. Let µ and ν be two GT’s on a nonempty set X, and let H be a hereditary class on X. We define the following notions: ℓH(θ(µ,ν))(A) = {x ∈ X : c∗ν(M) ⊆ A for some µ-open set M containing x}. ℓθ(µ,ν))(A) = {x ∈ X : cν(M) ⊆ A for some µ-open set M containing x}. ℓH(θ)(A) = {x ∈ X : c∗µ(M) ⊆ A for some µ-open set M containing x}. Proposition 2. For any two GT’s ν1 and ν2 on a nonempty set X, we have ℓθ(ν1,ν2)(A) ⊆ ℓH(θ(µ,ν))(A) for any A ⊆ X. Proof. Let x ∈ ℓθ(µ,ν)(A). Then there exists a µ-open set M containing x such that cν(M) ⊆ A. Since c∗ν(M) ⊆ cν(M), it follows that c∗ν(M) ⊆ A. Therefore, x ∈ ℓH(θ(µ,ν))(A). Remark 4. Let µ and ν be two GTs on a nonempty set X, and let A ⊆ X. If µ = ν, then ℓH(θ(µ,ν))(A) = ℓH(θ)(A). Theorem 15. Let ν1 and ν2 be two GT’s on a nonempty set X and A ⊆ X. Then the following properties hold: (i) iH(θ(µ,ν)(A) = X − cH(θ(µ,ν)(X −A) and cH(θ(µ,ν)(A) = X − iH(θ(µ,ν)(X −A). (ii) ℓH(θ(µ,ν)(A) = X − γ∗(µ, ν)(X −A) and γ∗(µ, ν)(A) = X − ℓH(θ(µ,ν)(X −A). Proof. The proof is obvious. The following Corollary comes directly from Definition 4 and Definition 7. Corollary 8. Let µ and ν be two GTs on a nonempty set X and A ⊆ X. Then iH(θ(µ,ν))(A) if and only if there exists a µ-open set M containing x such that M ⊆ c∗ν(M) ⊆ A. F. Alsharari, A. Qahis / Eur. J. Pure Appl. Math, 18 (2) (2025), 5782 10 of 10 5. Conclusion This study aimed to introduce and examine the operation γ∗(µ, ν) and H ( θ(µ, ν) ) - open sets within generalized topological spaces. Several significant results regarding these concepts were established. We thoroughly investigated the relationships among γ∗(µ, ν), γ∗, and the µ-closure, as well as those among H ( θ(µ, ν) ) -open sets, θ(µ, ν)-open sets, and µ-open sets. Finally, we have derived various properties and characterizations of H ( θ(µ, ν) ) -open sets in terms of the concept of H(µ, ν)-regularity. 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