Int. J. Anal. Appl. (2022), 20:72 On ωθ̃-µ-Open Sets in Generalized Topological Spaces Fatimah Al Mahri∗, Abdo Qahis Department of Mathematics, College of Science and Arts, Najran university, Saudi Arabia ∗Corresponding author: cahis82@gmail.com Abstract. In this paper analogous to [1], we introduce a new class of sets called ωθ̃-µ-open sets in generalized topological spaces which lies strictly between the class of θ̃µ-open sets and the class of ω-µ-open sets. We prove that the collection of ωθ̃-µ-open sets forms a generalized topology. Finally, several characterizations and properties of this class have been given. 1. Introduction One notion that has received much attention lately is the so-called ω-open sets in a topological space (X,τ) was introduced by Hdeib [12], which forms a topology finer than τ. Recently, many topological concepts and several interesting results related to this notion have obtained by many authors such as [3], [10], [9], [2]. A collection µ of subsets of a nonempty set X is a generalized topology (GT) if ∅∈ µ and µ is closed under arbitrary unions, this notion was introduced by Császár in the sense of [5]. We call the pair (X,µ) a generalized topological space (briefly GTS) on X. The elements of µ are called µ-open sets and their complements are called µ-closed sets, see [7], the union of all elements of µ will be denoted by Mµ and a GTS (X,µ) is said to be strong [7] if X ∈ µ. If A is a subset of a GTS (X,µ), then the µ-closure of A, cµ(A), is the intersection of all µ-closed sets containing A and the µ-interior of A, iµ(A), is the union of all µ-open sets contained in A (see [5,7]). It is easy to observe that operators iµ and cµ are idempotent and monotonic A subset A of a GTS (X,µ) is µ-open if and only if A = iµ(A), and and iµ(A) = X \cµ(X \A). Evidently, A is µ-closed if and only if A = cµ(A), cµ(A) is the smallest µ-closed set containing A, iµ(A) is the largest µ-open set contained in A. Over recent years several authors have been working in formulate many topological Received: Nov. 21, 2022. 2010 Mathematics Subject Classification. 54A05, 54C08. Key words and phrases. generalized topology; θ̃µ-open sets; ω-µ-open sets; ωθ̃-µ-open sets; θ̃µ-locally countable; ωθ̃-anti-locally countable. https://doi.org/10.28924/2291-8639-20-2022-72 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-72 2 Int. J. Anal. Appl. (2022), 20:72 concepts to establish new concepts in the structure of GTS, see [4], [8], [6] [11], [17], [15], [13] and others. Then motivated by the notion of ω-open set in a topological space (X,τ), Al Ghour and Wafa Zareer (2016) [1] defined the notions of ω-µ-closed sets and ω-µ-open sets in the structure of GTS as follows : A subset A of GTS (X,µ) is called ω-µ-closed if it contains all its condensation points. The complement of an ω-µ-closed set is called ω-µ-open. The family of all ω-µ-open subsets of X forms a GT on X, denoted by ωµ. Let us now recall some notions defined in [14]. A subset A of GTS (X,τ) is said to be θ̃µ-open if and only if for each x ∈ A, there exists U ∈ µ such that x ∈ U ⊆ cµ(U) ∩Mµ ⊆ A and the collection of all θ̃µ-open subsets of a GTS (X,µ) is denoted by θ̃µ. Then θ̃µ is also a GT included in µ. Analogous to [1] and by using the notion of θ̃µ-open, we introduce the relatively new notions of ω θ̃ -µ-open as a new class of sets . We present several characterizations, properties, and examples related to the new concepts. In section 2, we use the the notion of θ̃µ-open to introduce ωθ̃-µ-open sets in GTS as a new class of sets and we prove that this class lies strictly between the class of θ̃µ-open sets and the class of ω-µ-open sets. Moreover, we give some sufficient conditions for the equivalence between the class of ω θ̃ -µ-open sets and the class of ω-µ-open sets. In section 3, several interesting properties of ω θ̃ -µ-open subsets are discussed via the operations of ω θ̃ -interior and ω θ̃ -closure. Definition 1.1. [16] A GTS (X,µ) is said to be µ-locally indiscrete if every µ-open set in (X,µ) is µ-closed. Definition 1.2. [1] A GTS (X,µ) is called µ-locally countable if Mµ is nonempty and for every point x ∈Mµ, there exists a U ∈ µ such that x ∈ U and U is countable. Definition 1.3. [14] Let (X,µ) be a GTS , A ⊆ X and γ θ̃ : P (X) → P (X) be an operation defined as the following: γ θ̃µ (A) = {x ∈ X : cµ(U) ∩Mµ ∩A 6= ∅ f or all U ∈ µ,x ∈ U}. Theorem 1.1. [1] Let (X,µ) be a GTS. Then Mµ = Mωµ. Theorem 1.2. [1] If (X,µ) is a µ-locally countable GTS, then ωµis the discrete topology on Mµ. 2. ω θ̃ -µ-open sets We begin this section by introducing the following definition. Definition 2.1. Let (X,µ) be a GTS and A ⊆ X. Consider an operation Γω θ̃ : P (X) → P (X) defined as the following: Γω θ̃ (A) = {x ∈ X : U ∩A is uncountable f or all U ∈ θ̃µ and x ∈ U}. A point x ∈ X is called a Int. J. Anal. Appl. (2022), 20:72 3 θ̃µ-condensation point of A if for all U ∈ θ̃µ such that x ∈ U and U ∩A is uncountable. The set of all θ̃µ-condensation points of A is denoted by Γω θ̃ (A). Lemma 2.1. Let (X,µ) be a GTS. The operation Γω θ̃ : P (X) → P (X) has the following properties: (1) if A ⊆ B ⊂ X, then Γω θ̃ (A) ⊆ Γω θ̃ (B) (monotonic property); (2) Γω θ̃ (Γω θ̃ (A)) ⊆ Γω θ̃ (A) for any A ⊆ X (restricting property); (3) if A is any countable subset of X, then Γω θ̃ (A) = ∅. Proof. (1) Let A ⊆ B ⊂ X and x ∈ Γω θ̃ (A). Then U ∩A is uncountable for each U ∈ θ̃µ and x ∈ U. Since A ⊆ B, then U ∩B is uncountable. Thus x ∈ Γω θ̃ (B) and hence Γω θ̃ (A) ⊆ Γω θ̃ (B). (2) Let x ∈ Γω θ̃ (Γω θ̃ (A)). Then U ∩ Γω θ̃ (A) is an uncountable for all U ∈ θ̃µ and x ∈ U. Let y ∈ U ⋂ Γω θ̃ (A). Then y ∈ U and y ∈ Γω θ̃ (A) which implies that U ∩A is an uncountable set. Hence x ∈ Γω θ̃ (A) and therefore Γω θ̃ (Γω θ̃ (A)) ⊆ Γω θ̃ (A). (3) The proof is obvious by Definition 2.1. � Definition 2.2. Let (X,µ) be a GTS and A ⊆ X. Then A is said to be ω θ̃ -µ-closed if Γω θ̃ (A) ⊆ A. The complement of an ω θ̃ -µ-closed set is said to be ω θ̃ -µ-open. The family of all ω θ̃ -µ-open subsets of (X,µ) is denoted by ω θ̃ , where ω θ̃ = {W ⊆ X : Γω θ̃ (X\W ) ⊆ X \W}. The following theorem and lemma give a necessary and sufficient condition for ω θ̃ -µ-open sets. Theorem 2.1. Let (X,µ) be a GTS and W ⊆ X. Then the following statements are equivalent: (1) W is ω θ̃ -µ-open; (2) if for every x ∈ W there exists a U ∈ θ̃µ such that x ∈ U and U \W is a countable set. Proof. (1) ⇒ (2): Suppose W is ω θ̃ -µ-open. Since X\W is ω θ̃ -µ-closed set, then Γω θ̃ (X\W ) ⊆ X\W. This means that for every x ∈ W, x /∈ Γω θ̃ (X \W ) and hence there exists a U ∈ θ̃µ such that x ∈ U and U ∩ (X \W ) = U \W is countable. (2) ⇒ (1): Let x ∈ W. Then by assumption there exists a U ∈ θ̃µ such that x ∈ U and U∩(X\W ) is countable. Which implies that x /∈ Γω θ̃ (X\W ), Γω θ̃ (X\W ) ⊆ X\W and hence X\W is ω θ̃ -µ-closed. Therefore W is ω θ̃ -µ-open set. � Lemma 2.2. A subset W of a GTS (X,µ) is ω θ̃ -µ-open if and only if for every x ∈ W there exists a U ∈ θ̃µ and a countable C ⊆Mµ such that x ∈ U \C ⊆ W. Proof. Necessity. Let W be ω θ̃ -µ-open and x ∈ W. By Theorem 2.1, there exists U ∈ θ̃µ such that x ∈ U and U\W is countable. Let C = U\W. Then C is countable, C ⊆Mµ and x ∈ U∩(X\C) = U ∩ ( X \ (U ∩X \W ) ) = U ∩W ⊆ W and hence x ∈ U \C ⊆ W. Sufficiency. Let x ∈ W. From assumption there exists U ∈ θ̃µ and a countable set C ⊆ Mµ such that x ∈ U \C ⊆ W. Therefore, U \W ⊆ C and U \W is a countable set and this completes the proof. � 4 Int. J. Anal. Appl. (2022), 20:72 Theorem 2.2. Let (X,µ) be a GTS and C ⊆ X. If C is ω θ̃ -µ-closed, then C ⊆ F ∪ B for some ω θ̃ -µ-closed set F and a countable subset B. Proof. Let C be any ω θ̃ -µ-closed set in (X,µ). Then X \C is ω θ̃ -µ-open. By Lemma 2.2, for each x ∈ X \ C, there exist a θ̃µ-open set U containing x and a countable subset B ⊆ Mµ such that x ∈ U\B ⊆ X\C. Thus C ⊆ X\(U\B) = X\(U∩(X\B)) = (X\U)∪B. Let F = X\U. Then F is ω θ̃ -µ-closed such that C ⊆ F ∪B. � Theorem 2.3. Let (X,µ) be a GTS. Then the collection ω θ̃ forms a generalized topology on X. Proof. It is clear that ∅ ∈ ω θ̃ . Let {Wλ : λ ∈ ∆} be a collection of ωθ̃-µ-open subsets of (X,µ) and x ∈ ⋃ λ∈∆ Wλ. There exists an λ0 ∈ ∆ such that x ∈ Wλ0. Since Wλ0 is ωθ̃-open set, then by Lemma 2.2, there exist U ∈ θ̃µ and a countable set C ⊆ Mµ such that x ∈ U \ C ⊆ Wλ0 ⊆ ⋃ λ∈∆ Wλ. By Lemma 2.2, it follows that ⋃ λ∈∆ Wλ is ωθ̃-µ-open. Hence the collection ωθ̃ is generalized topology on X. � The next theorem obtains that the new class of ω θ̃ -µ-open sets lies strictly between the class of θ̃-µ-open sets and the class of ω-µ-open sets. Theorem 2.4. Let (X,µ) be a GTS. Then θ̃µ ⊆ ωθ̃ ⊆ ωµ. Proof. To show that θ̃µ ⊆ ωθ̃, let W ∈ θ̃µ and x ∈ W. Take U = W and C = ∅. Then U ∈ θ̃µ, C ⊆Mµ such that x ∈ U \C ⊆ W. Therefore, by Lemma 2.2, it follows that W ∈ ωθ̃. To show that ω θ̃ ⊆ ωµ, Let W ∈ ωθ̃. By Theorem 2.1, for each x ∈ W there exists a U ∈ θ̃µ such that x ∈ U and U \W is countable. Since θ̃µ ⊆ µ, then U ∈ µ and hence W is ω-µ-open. Therefore W ∈ ωµ. � The following diagram follows immediately from the definitions and Theorem 2.4. θ̃µ −open =⇒ ωθ̃ −µ−openww� ww� µ−open =⇒ ω −µ−open The converse of these implications need not be true in general as shown by the following examples. Example 2.1. Consider X = R, A = {4n : n ∈ N} and µ = {∅, [0, 2], [1, 3] ∪ A, [0, 3] ∪ A}. Then (X,µ) is a generalized topological space and the family of all θ̃µ-open sets is θ̃µ = {∅, [0, 3] ∪ A}. Then [1, 3] ∈ ωµ \ωθ̃, i.e. [1, 3] is ω-µ-open but it is not ωθ̃-µ-open. Also, it is easy to check that Γω θ̃ (R\ [0, 3]) ⊆R\ [0, 3]. Thus [0, 3] ∈ ω θ̃ \ θ̃µ, i.e. [0, 3] is ωθ̃-µ-open but it is not θ̃µ-open Example 2.2. Let X = {a,b,c,d} with GT µ = {∅,{a,b},{a,c},{a,b,c}}. Then {a,c}∈ ω θ̃ \ θ̃µ, i.e. the set {a,c} is ω θ̃ -µ-open but it is not θ̃µ-open. Note that the previous examples show that θ̃µ 6= ωθ̃ 6= ωµ in general. Int. J. Anal. Appl. (2022), 20:72 5 Remark 2.1. The notions of µ-open and ω θ̃ -µ-open sets are independent of each other. For more clarity in Example 2.1, the set [0, 3] is ω θ̃ -µ-open but it is not µ-open and the set [1, 3] ∪A is µ-open but it is not ω θ̃ -µ-open. Theorem 2.5. If a GTS (X,µ) is a µ-locally indiscrete, then µ ⊆ ω θ̃ . Proof. To show that µ ⊆ ω θ̃ , let A ∈ µ and x ∈ A. Take U = A. Since (X,µ) is µ-locally indiscrete, then cµ(U) = U and we have x ∈ U ⊆ cµ(U)∩Mµ ⊆ A. Thus A ∈ θ̃µ and by Theorem 2.4, θ̃µ ⊆ ωθ̃. Therefore A ∈ ω θ̃ . � Lemma 2.3. Let (X,µ) be a GTS. Then Mµ ∈ θ̃µ. Proof. Let A = Mµ and x ∈ A. Then there exists Ux ∈ µ such that x ∈ Ux. Since Ux ⊆ cµ(Ux ) ⋂ Mµ ⊆ A, then A = Mµ ∈ θ̃µ. � For a GT µ on a nonempty set X, let Mω θ̃ = ⋃ {U ⊆ X : U ∈ ω θ̃ }. Thus we have the following theorem. Theorem 2.6. Let (X,µ) be a GTS. Then Mµ = Mω θ̃ Proof. By Lemma 2.3, Mµ ∈ θ̃µ and form Theorem 2.4, θ̃µ ⊆ ωθ̃ and hence Mµ ⊆ Mωθ̃. On the other hand, let x ∈ Mω θ̃ . Since, Mω θ̃ ∈ ω θ̃ , then by Lemma 2.2, there exists a U ∈ θ̃µ and a countable set C ⊆Mµ such that x ∈ U \C ⊆Mω θ̃ . Since U ⊆Mµ and U is µ-open, it follows that x ∈Mµ and hence Mω θ̃ ⊆Mµ. Therefore Mµ = Mω θ̃ . � By Theorem 1.1 and Theorem 2.6, we obtain the following corollary Corollary 2.1. Let (X,µ) be a GTS. Then Mµ = Mω θ̃ = Mωµ We will denote by (τcoc)X, the cocountable topology on a nonempty set X. Theorem 2.7. Let (X,µ) be a GTS. Then (τcoc)U ⊆ ωθ̃ for all U ∈ θ̃µ \{∅}. Proof. Let U ∈ θ̃µ \ {∅}, W ∈ (τcoc)U and x ∈ W. Since W ⊆ U, we have x ∈ U and U \ W = U \ (U ∩V ) for some V ∈ τcoc. Now, U \W = U \ (U ∩V ) = U \V . Thus U \W is countable set and by Theorem 2.1, it follows that W ∈ ω θ̃ . This shows that (τcoc)U ⊆ ωθ̃. � Theorem 2.8. For any GTS (X,µ), the following statements are equivalent. (1) θ̃µ = ωθ̃. (2) (τcoc)U ⊆ θ̃µ for all U ∈ θ̃µ \{∅}. Proof. (1) =⇒ (2): Assume that θ̃µ = ωθ̃ and U ∈ θ̃µ \{∅}. Then by Theorem 2.7, (τcoc)U ⊆ ωθ̃ = θ̃µ. (2) =⇒ (1): Suppose that (τcoc)U ⊆ θ̃µ for all U ∈ θ̃µ \{∅}. It is enough to show that ωθ̃ ⊆ θ̃µ. Let 6 Int. J. Anal. Appl. (2022), 20:72 W ∈ ω θ̃ and x ∈ W. By Lemma 2.2, there exists Ux ∈ θ̃µ and a countable set Cx ⊆ Mµ such that x ∈ Ux \Cx ⊆ W. Thus Ux ∩X \Cx ∈ (τcoc)Ux , where X \Cx ∈ τcoc. From assumption Ux \Cx ∈ (τcoc)Ux ⊆ θ̃µ for all x ∈ W, and so Ux \Cx ∈ θ̃µ. It follows that W = ⋃ {Ux \Cx : x ∈ W}∈ θ̃µ, and hence θ̃µ = ωθ̃. � Proposition 2.1. Let (X,µ) be a GTS. If θ̃µ is a topology on X, then ωθ̃ is a topology. Proof. Suppose that θ̃µ is a topology. By Theorem 2.3, ωθ̃ is generalized topology. It is enough to show that the collection ω θ̃ is closed under finite intersection. Let W, G be ω θ̃ -µ-open sets and x ∈ W ∩ G. Then by Theorem 2.1, there exist U,V ∈ θ̃µ containing x such that U \ W and V \ G are countable sets. Since θ̃µ is a topology, we have x ∈ U ∩ V ∈ θ̃µ. Furthermore, (U∩V )\(W ∩G) = (U∩V )∩ [ X\W ∪X\G ] = [(U∩V )\W )]∪[(U∩V )\G)] ⊂ (U\W )∪(V \G). Therefore, (U ∩V ) \ (W ∩G) is a countable set and hence W ∩G is ω θ̃ -µ-open. � Definition 2.3. Let (X,µ) be a GTS . Then (X,µ) is said to be θ̃µ-locally countable if Mµ is nonempty and for every point x ∈Mµ, there exists a U ∈ θ̃µ such that x ∈ U and U is countable. The following corollary is a direct result from Definition 2.3 and Definition 1.2. Corollary 2.2. Let (X,µ) be a GTS. If (X,µ) is θ̃µ-locally countable, then (X,µ) is µ-locally countable. Theorem 2.9. If (X,µ) is a θ̃µ-locally countable GTS, then ωθ̃ is the discrete topology on Mµ. Proof. It is enough to show that every singleton subset of Mµ is ωθ̃-µ-open. Since (X,µ) is θ̃µ-locally countable, then for each x ∈ Mµ, there exists a U ∈ θ̃µ such that x ∈ U and U is countable. By Theorem 2.7, we have (τcoc)U ⊆ ωθ̃. Therefore U \ (U \{x}) = {x}∈ ωθ̃. � The following corollary is a direct result of Theorem 2.9. Corollary 2.3. Let (X,µ) be a strong GTS. If (X,µ) is a θ̃µ-locally countable, then ωθ̃ is the discrete topology on X. Proposition 2.2. If (X,µ) is a θ̃µ-locally countable GTS, then ωθ̃ = ωµ. Proof. Since (X,µ) is θ̃µ-locally countable, then by Theorem 2.9, ωθ̃ is the the discrete topology on Mµ. From Corollary 2.2 and Theorem 1.2, we get ωθ̃ = ωµ. � Corollary 2.4. Let (X,µ) be a GTS. If Mµ is a countable nonempty set, then ωθ̃ is the discrete topology on Mµ. Proof. Since Mµ is countable nonempty set, then for x ∈ Mµ, there exists U ∈ θ̃µ such that U is countable set. Thus (X,µ) is θ̃µ-locally countable. From Theorem 2.9, we get ωθ̃ is the discrete topology on Mµ. � Int. J. Anal. Appl. (2022), 20:72 7 3. Further properties of ω θ̃ -µ-open sets Definition 3.1. Let (X,µ) be a GTS and A ⊆ X. A point x ∈ X is called an ω θ̃ -closure point of A if and only if U ∩A 6= ∅ for all U ∈ ω θ̃ and x ∈ U. Consider the following operations are defined as follows: (1) γω θ̃ (A) = {x ∈ X : U ∩A 6= ∅, f or all U ∈ ω θ̃ and x ∈ U}; (2) cω θ̃ (A) = ∩{F : A ⊆ F,F is ω θ̃ -µ-closed in X}. Lemma 3.1. Let (X,µ) be a GTS. Then cω θ̃ (A) = γω θ̃ (A) for any A ⊆ X. Proof. It is enough to show that γω θ̃ (A) is the smallest ω θ̃ -µ-closed set containing A. Clearly A ⊆ γω θ̃ (A). Further γω θ̃ (A) is ω θ̃ -µ-closed, that is X\γω θ̃ (A) is ω θ̃ -µ-open because for each x ∈ X\γω θ̃ (A) there is Ux ∈ ωθ̃ such that x ∈ Ux and Ux ∩A = ∅. Now, for any y ∈ Ux implies y ∈ X \γωθ̃ (A) so that X \γω θ̃ (A) = ⋃ x∈X\γω θ̃ (A) Ux ∈ ωθ̃. Finally if A ⊆ F and F is any ω θ̃ -µ-closed, then X \ F is ω θ̃ -µ-open and (X \ F ) ∩ A = ∅ so that X \F ⊆ X \γω θ̃ (A) and hence γω θ̃ (A) ⊆ F. Therefore γω θ̃ (A) is the smallest ω θ̃ -µ-closed set containing A, and by Definition 3.1(2), γ ωθ̃ (A) = c ωθ̃ (A). � The proof of the following theorem is straightforward and thus omitted. Theorem 3.1. For subsets A,B of GTS (X,µ), the following properties hold: (1) if A ⊆ B ⊂ X, then cω θ̃ (A) ⊆ cω θ̃ (B); (2) A ⊆ cω θ̃ (A) for A ⊆ X; (3) cω θ̃ (cω θ̃ (A)) = cω θ̃ (A) for A ⊆ X; (4) A is ω θ̃ -µ-closed if and only if cω θ̃ (A) = A. Definition 3.2. Let (X,µ) be a GTS and A ⊆ X. Then we define the following notions: (1) c θ̃µ (A) = ∩{F : A ⊆ F,F is θ̃µ-closed in X}; (2) cωµ(A) = ∩{F : A ⊆ F,F is ω-µ-closed in X}. The proof of the following corollary is straightforward and thus omitted. Corollary 3.1. For a subset A of a GTS (X,µ), the following properties hold: (1) A is θ̃µ-closed if and only if cθ̃µ(A) = A; (2) A is ω-µ-closed if and only if cωµ(A) = A. Lemma 3.2. Let (X,µ) be a GTS. Then γ θ̃µ (A) ⊆ c θ̃µ (A) for any A ⊆ X. Proof. Let x /∈ c θ̃µ (A). Then x ∈ X \c θ̃µ (A) so that there is U ∈ θ̃µ satisfying x ∈ U and U ∩A = ∅. Since U ∈ θ̃µ, then there is V ∈ µ such that x ∈ V ⊆ cµ(V ) ∩Mµ ⊆ U and cµ(V ) ∩Mµ ∩A = ∅, consequently x /∈ γ θ̃ (A). Thus we have γ θ̃ (A) ⊆ c θ̃ (A). � 8 Int. J. Anal. Appl. (2022), 20:72 Theorem 3.2. Let (X,µ) be a GTS and A ⊆ X. Then the following properties hold: (1) cωµ(A) ⊆ cωθ̃ (A) ⊆ cθ̃µ(A); (2) If A is θ̃µ-closed, then A is ωθ̃-µ-closed; (3) If A is ω θ̃ -µ-closed, then A is ω-µ-closed. Proof. (1) To show that cωµ(A) ⊆ cωθ̃ (A), let x /∈ cωθ̃ (A) and so there is a U ∈ ωθ̃ containing x such that U∩A = ∅. From Theorem 2.4, we have ω θ̃ ⊆ ωµ, U ∈ ωµ, and hence x /∈ cωµ(A). To show that cω θ̃ (A) ⊆ c θ̃µ (A), let x /∈ c θ̃µ (A) and so there is a U ∈ θ̃µ containing x such that U ∩A = ∅. From Theorem 2.4, we have θ̃µ ⊆ ωθ̃, U ∈ ωθ̃, and hence x /∈ cωθ̃ (A). (2) Suppose that A is θ̃µ-closed. Then by Corollary 3.1(1), cθ̃µ(A) = A. Thus by (1), cωθ̃ (A) = A and hence A is ω θ̃ -µ-closed. (2) Suppose that A is ω θ̃ -µ-closed. Then by Theorem 3.1(4), cω θ̃ (A) = A . Thus by (1), cωµ(A) = A and hence A is ω-µ-closed. � Proposition 3.1. Let (X,µ) be a θ̃µ-locally countable GTS and A ⊆ X. Then cωµ(A) = cωθ̃ (A) Proof. By Theorem 3.2(1), cωµ(A) ⊆ cωθ̃ (A). Let x ∈ cωθ̃ (A). Then U ∩A 6= ∅ for all U ∈ ωθ̃ and x ∈ U. Since (X,µ) is a θ̃µ-locally countable, then by Theorem 2.9, ωθ̃ is the discrete topology on Mµ and hence ωµ = ωθ̃. Which implies that x ∈ cωµ(A) and cωθ̃ (A) ⊆ cωµ(A). Hence cωµ(A) = cω θ̃ (A). � Theorem 3.3. Let (X,µ) be a µ-locally indiscrete GTS and let A ⊆ X. Then the following properties hold. (1) cµ(A) = cθ̃µ(A); (2) cω θ̃ (A) ⊆ cµ(A); (3) If A is µ-closed in (X,µ), then A is θ̃µ-closed in (X,µ). (4) If A is µ-closed in (X,µ), then A is ω θ̃ -µ-closed in (X,µ). Proof. (1) Clearly cµ(A) ⊆ cθ̃µ(A). To show that cθ̃µ(A) ⊆ cµ(A), let x /∈ cµ(A). Then there exists U ∈ µ such that x ∈ U and U ∩A = ∅. Since (X,µ) is a µ-locally indiscrete, cµ(U) = U. It follows that U ⊆ cµ(U) ∩Mµ ⊆ U and hence U ∈ θ̃µ. Thus x /∈ cθ̃µ(A). (2) Since (X,µ) is µ-locally indiscrete. then by Theorem 2.5, µ ⊆ ω θ̃ and hence cω θ̃ (A) ⊆ cµ(A). (3) Suppose that A is µ-closed in (X,µ), then cµ(A) = A. Thus by (1), A = cθ̃µ(A) and hence A is θ̃µ-closed in (X,µ). (4) Suppose that A is µ-closed in (X,µ), then cµ(A) = A. Thus by (2), A = cω θ̃ (A) and hence A is ω θ̃ -µ-closed in (X,µ). � Definition 3.3. A GTS (X,µ) is said to be ω θ̃ -anti-locally countable if the intersection of any two ω θ̃ -µ-open sets is either empty or uncountable. The following lemma is used to prove the theorem which is stated below. Int. J. Anal. Appl. (2022), 20:72 9 Lemma 3.3. Let (X,µ) be ω θ̃ -anti-locally countable and A ⊆ X. If A ∈ ω θ̃ , then c θ̃µ (A) = cω θ̃ (A). Proof. Suppose that ∅ 6= A ⊆ X and A ∈ ω θ̃ . By Theorem 3.2(1), cω θ̃ (A) ⊆ c θ̃µ (A). To Show that c θ̃µ (A) ⊆ cω θ̃ (A), let x ∈ c θ̃µ (A) and W ∈ ω θ̃ such that x ∈ W. Then by Lemma 2.2, there exists U ∈ θ̃µ and a countable set C ⊆ Mµ such that x ∈ U \C ⊆ W. Since x ∈ U ∩ cθ̃µ(A), U ∩A 6= ∅. Choose y ∈ U ∩ A. Since A ∈ ω θ̃ , there exists V ∈ θ̃µ and a countable set D ⊆ Mµ such that y ∈ V \D ⊆ A. Since y ∈ U ∩V and (X,µ) is ω θ̃ -anti-locally countable, then U ∩V is uncountable. Thus, (U \C) ∩ (V \D) 6= ∅ and hence A∩W 6= ∅. Therefore, x ∈ cω θ̃ (A). � A subset A of GTS (X,µ) is said to be θ̃µ-clopen(resp. ωθ̃-µ-clopen) if it is both θ̃µ-open and θ̃µ-closed (resp. ωθ̃-µ-open and ωθ̃-µ-closed). In the following, by using Lemma 3.3, we prove the main result in this section. Theorem 3.4. Let (X,µ) be ω θ̃ -anti-locally countable and A ⊆ X. Then, A is θ̃µ-clopen if and only if A is ω θ̃ -µ-clopen. Proof. ⇒) Suppose that A is θ̃µ-clopen, then A and X \A are θ̃µ-open. Since θ̃µ ⊆ ωθ̃, then A and X \A are ω θ̃ -µ-open, and hence A is ω θ̃ -µ-clopen. ⇐) Suppose that A is ω θ̃ -µ-clopen. Since A and X \A are ω θ̃ -µ-open, the by Lemma 3.3, c θ̃µ (A) = cω θ̃ (A) and c θ̃µ (x \A) = cω θ̃ (X \A). Since A is ω θ̃ -µ-clopen., then c θ̃µ (A) = cω θ̃ (A) = A and cω θ̃ (X \A) = X \A. Therefore, c θ̃µ (A) = A and c θ̃µ (X \A) = X \A and hence A and X \A are θ̃µ-closed sets. This means that A is θ̃µ-clopen. � Definition 3.4. Let (X,µ) be a GTS and A ⊆ X. Then, we define the following notions: (1) iω θ̃ (A) = ∪{U ⊆ X : U ⊆ A, U is ω θ̃ -µ-open}; (2) i θ̃ (A) = ∪{U ⊆ X : U ⊆ A, U is θ̃µ-open}; (3) iωµ(A) = ∪{U ⊆ X : U ⊆ A, U is ω-µ-open}. Theorem 3.5. For subsets A,B of GTS (X,µ), the following properties hold: (1) if A ⊆ B ⊂ X, then iω θ̃ (A) ⊆ iω θ̃ (B); (2) for A ⊆ X, then iω θ̃ (A) ⊆ A; (3) iω θ̃ (iω θ̃ (A)) = iω θ̃ (A) for A ⊆ X; (4) A is ω θ̃ -µ-open if and only if iω θ̃ (A) = A. Proof. The proof is obvious � 10 Int. J. Anal. Appl. (2022), 20:72 Corollary 3.2. Let (X,µ) be a GTS and A ⊆ X. Then i θ̃µ (A) ⊆ iω θ̃ (A) ⊆ iωµ(A). Proof. To show that i θ̃µ (A) ⊆ iω θ̃ (A), let x ∈ i θ̃µ (A). Then there is U ∈ θ̃µ such that x ∈ U ⊆ A. By Theorem 2.4, U is ω θ̃ -µ-open. Thus x ∈ iω θ̃ (A). To show that iω θ̃ (A) ⊆ iωµ(A), let x ∈ iωθ̃ (A). Then there is U ∈ ω θ̃ such that x ∈ U ⊆ A. Then by Theorem 2.4, U is ω-µ-open and hence x ∈ iωµ(A) � Theorem 3.6. Let (X,µ) be a GTS and A ⊆ X. Then the following properties hold: (1) cω θ̃ (X \A) = X \ iω θ̃ (A); (2) iω θ̃ (X \A) = X \cω θ̃ (A). Proof. (1) Let x ∈ cω θ̃ (X \A) and U ∈ ω θ̃ with x ∈ U. Since x ∈ cω θ̃ (X \A), U ∩ (X \A) 6= ∅. This implies that x /∈ iω θ̃ (A) and hence x ∈ X \ iω θ̃ (A). Conversely, for x ∈ X \ iω θ̃ (A), x /∈ iω θ̃ (A), and then U ∩ (X \A) 6= ∅ for all U ∈ ω θ̃ and x ∈ U which implies x ∈ cω θ̃ (X \A). (2) Let x ∈ X \ cω θ̃ (A) if and only if x /∈ cω θ̃ (A) if and only if there is U ∈ ω θ̃ with x ∈ U such that U ∩A = ∅ if and only if x ∈ iω θ̃ (X \A). � 4. Conclusion In this paper, we introduced the notion of ω θ̃ -µ-open sets in the sense of generalized topology given in [5]. We have proved that the collection of ω θ̃ -µ-open sets forms a generalized topology on X that lies between the class of θ̃µ-open sets and the class of ω-µ-open sets. The relationships of ωθ̃-µ-open and other well-known generalized open sets are given. 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