EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 4093-4111 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some New Types of Fuzzy Closed Sets, Separation Axioms, and Compactness via Double Fuzzy Topologies Fahad Alsharari1, Osama M. Taha2, Islam M. Taha2,3,∗ 1 Department of Mathematics, College of Science, Jouf University, Sakaka, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt 3 Department of Basic Sciences, Higher Institute of Engineering and Technology, Egypt Abstract. In this article, we first defined a stronger form of (r, s)-generalized fuzzy semi-closed sets ⟨briefly, (r, s)-gfsc sets⟩ called (r, s)− g⊛fsc sets and investigated some of its features. More- over, we showed that (r, s)−fsc set⇒ (r, s)− g⊛fsc set⇒ (r, s)− gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ∗) and (V, η, η∗), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r, s)-GFS-regular and (r, s)-GFS-normal spaces with the help of (r, s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r, s)-gfso sets were defined and the relationships between them were introduced. 2020 Mathematics Subject Classifications: 03E72, 54A05, 54A40, 54C08, 54D15 Key Words and Phrases: Intuitionistic fuzzy set, double fuzzy topology, (r, s) − gfsc set, (r, s)− g⊛fsc set, continuity, (r, s)-GFS-regular space, (r, s)-GFS-normal space, compactness 1. Introduction and preliminaries The theory of fuzzy set was first presented by Zadeh [46]. Since then it has been improved and applied in most all the branches of technology and science, where theory of sets and mathematical logic play an important role. Also, many applications of these theory contributed to solving several practical problems in mathematics, social science, engineering, economics, etc. In recent years, many authors have contributed to fuzzy sets theory in the different directions in mathematics such as geometry, topology, algebra, operation research, see [31, 48]. The notion of fuzzy sets was used to introduce fuzzy topological spaces in [15]. The study in [15] was particularly important in the development of the field of fuzzy topology, see [3, 14, 16, 19, 26, 27]. The authors of [4–10, 21, 28, 36, 39] ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5573 Email addresses: f.alsharari@ju.edu.sa (F. Alsharari), osama.taha2015@yahoo.com (O. M. Taha), imtaha2010@yahoo.com (I. M. Taha) https://www.ejpam.com 4093 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4094 studied topological structures inspired by the hybridizations of soft sets [33] with fuzzy sets [46] and rough sets [24]. The concept of an intuitionistic fuzzy set was initiated by Atanassov [11, 12], which is a generalization of a fuzzy set. Coker [17, 18] introduced the concept of an intuitionistic fuzzy topological space based on the sense of Chang [15]. Later, Samanta and Mondal [34, 35] gave the definition of an intuitionistic fuzzy topological space based on the sense of Šostak [45]. The name (intuitionistic) was replaced with the name (double) by Garcia and Rodabaugh [25]. The concept of (r, s) − gfc sets was introduced and investigated by Abbas [1]. Thereafter, the concept of (r, s) − sgfc sets was introduced by Zahran et al. [47] on double fuzzy topological space based on the sense of Šostak. Also, Taha [42] defined the concept of (r, s)−gfsc sets and some characterizations were given. So far, lots of spectacular and creative studies about the theories of an intuitionistic fuzzy set have been considered by some scholars, see e. g. [2, 13, 20, 22, 23]. The organization of this article is as follows: • In Section 2, as a stronger form of (r, s)− gfsc sets [42], the notion of (r, s)− g⊛fsc sets is introduced and some properties are investigated. Moreover, we introduce new types of fuzzy mappings between double fuzzy topological spaces and relationships are obtained. • In Section 3, we define new types of fuzzy separation axioms with the help of (r, s)− gfsc sets and establish some of their properties. • In Section 4, some new types of compactness in double fuzzy topological spaces are defined and the relationships between them are specified. • In the end, we give some conclusions and make a plan for future works in Section 5. Throughout this article, nonempty sets will be denoted by V , U , etc. The family of all fuzzy sets on U is denoted by IU , and for µ ∈ IU , µc(u) = 1− µ(u), for all u ∈ U (where I = [0, 1], I1 = [0, 1), and I◦ = (0, 1]). Also, for t ∈ I, t(u) = t, for all u ∈ U. A fuzzy point ut on U is a fuzzy set, defined as follows: ut(k) = t if k = u, and ut(k) = 0 for all k ∈ U − {u}. ut is said to belong to a fuzzy set µ, denoted by ut ∈ µ, if t ≤ µ(u). The family of all fuzzy points on U is denoted by Pt(U). A fuzzy set µ is a quasi-coincident with λ, denoted by µqλ, if there is u ∈ U , such that µ(u) + λ(u) > 1, if µ is not quasi-coincident with λ, we denote µqλ. The following results and notions will be used in the next sections: F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4095 Lemma 1. [27] Let U be a nonempty set and ν, µ ∈ IU . Then, (i) νqµ iff there is ut ∈ ν such that utqµ, (ii) ν ∧ µ ̸= 0 if νqµ, (iii) νqµ iff ν ≤ µc, (iv) µ ≤ ν iff ut ∈ µ implies ut ∈ ν iff utqµ implies utqν iff utqν implies utqµ, (v) utq ∨ δ∈∆ νδ iff there is δ0 ∈ ∆ such that utqνδ0 . Definition 1. [35, 47] A double fuzzy topology on U is a pair (η, η∗) of the mappings η, η∗ : IU → I, which satisfy the following conditions. (i) η(ν) + η∗(ν) ≤ 1, for each ν ∈ IU . (ii) η(ν1 ∧ ν2) ≥ η(ν1) ∧ η(ν2) and η∗(ν1 ∧ ν2) ≤ η∗(ν1) ∨ η∗(ν2), for each ν1, ν2 ∈ IU . (iii) η( ∨ δ∈∆ νδ) ≥ ∧ δ∈∆ η(νδ) and η∗( ∨ δ∈∆ νδ) ≤ ∨ δ∈∆ η∗(νδ), for each {νδ}δ∈∆ ⊂ IU . The triplet (U, η, η∗) is said to be a double fuzzy topological space ⟨briefly, dfts⟩ in the sense of Šostak. η∗(ν) and η(ν) may be interpreted as gradation of nonopenness and openness for ν ∈ IU , respectively. In a dfts (U, η, η∗), the interior of ν ∈ IU , the closure of ν ∈ IU , the semi-closure of ν ∈ IU and the semi-interior of ν ∈ IU will be denoted by Iη,η∗(ν, r, s), Cη,η∗(ν, r, s), SCη,η∗(ν, r, s) and SIη,η∗(ν, r, s), respectively [20, 29, 34]. Definition 2. [29, 30] Let (U, η, η∗) be a dfts, ν ∈ IU , r ∈ I◦, and s ∈ I1, then we have (i) ν is called an (r, s)-fsc ⟨resp., (r, s)-fpc and (r, s)-frc⟩ set if ν ≥ Iη,η∗(Cη,η∗ (ν, r, s), r, s) ⟨resp., ν ≥ Cη,η∗(Iη,η∗ (ν, r, s), r, s) and ν = Cη,η∗(Iη,η∗ (ν, r, s), r, s)⟩. (ii) ν is called an (r, s)-fso ⟨resp., (r, s)-fpo and (r, s)-fro⟩ set if ν ≤ Cη,η∗(Iη,η∗ (ν, r, s), r, s) ⟨resp., ν ≤ Iη,η∗(Cη,η∗ (ν, r, s), r, s) and ν = Iη,η∗(Cη,η∗ (ν, r, s), r, s)⟩. Definition 3. [1, 42, 47] Let (U, η, η∗) be a dfts, µ, ν ∈ IU , r ∈ I◦, and s ∈ I1, then we have (i) µ is called an (r, s)-generalized fuzzy closed ⟨briefly, (r, s)-gfc⟩ set if Cη,η∗(µ, r, s) ≤ ν whenever µ ≤ ν and η(ν) ≥ r, η∗(ν) ≤ s. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4096 (ii) µ is called an (r, s)-semi generalized fuzzy closed ⟨briefly, (r, s)-sgfc⟩ set if SCη,η∗(µ, r, s) ≤ ν whenever µ ≤ ν and ν is (r, s)-fso set. (iii) µ is called an (r, s)-generalized fuzzy semi-closed ⟨briefly, (r, s)-gfsc⟩ set if SCη,η∗(µ, r, s) ≤ ν whenever µ ≤ ν and η(ν) ≥ r, η∗(ν) ≤ s. Definition 4. [34, 47] Let h : (U, τ, τ∗)→ (V, η, η∗) be a mapping, then h is said to be (i) DF-continuous if τ(h−1(λ)) ≥ η(λ) and τ∗(h−1(λ)) ≤ η∗(λ) for each λ ∈ IV . (ii) DF-open if η(h(ν)) ≥ τ(ν) and η∗(h(ν)) ≤ τ∗(ν) for each ν ∈ IU . (iii) DF-closed if η(hc(ν)) ≥ τ(νc) and η∗(hc(ν)) ≤ τ∗(µc) for each ν ∈ IU . Definition 5. [1, 29, 42] Let h : (U, τ, τ∗)→ (V, η, η∗) be a mapping, r ∈ I◦, and s ∈ I1, then h is said to be (i) DFS-continuous ⟨resp., DFGS-continuous and DFG-continuous⟩ if h−1(µ) is (r, s)- fso ⟨resp., (r, s)-gfso and (r, s)-gfo⟩ set for each µ ∈ IV with η(µ) ≥ r , η∗(µ) ≤ s. (ii) DFGS-irresolute ⟨resp., DF-irresolute⟩ if h−1(µ) is (r, s)-gfso ⟨resp., (r, s)-fso⟩ set for each µ ∈ IV is (r, s)-gfso ⟨resp., (r, s)-fso⟩ set. (iii) DFS-open ⟨resp., DFGS-open and DFG-open⟩ if h(ν) is (r, s)-fso ⟨resp., (r, s)- gfso and (r, s)-gfo⟩ set for each ν ∈ IU with τ(ν) ≥ r , τ∗(ν) ≤ s. (iv) DFS-closed ⟨resp., DFGS-closed and DFG-closed⟩ if h(ν) is (r, s)-fsc ⟨resp., (r, s)-gfsc and (r, s)-gfc⟩ set for each ν ∈ IU with τ(νc) ≥ r , τ∗(νc) ≤ s. The basic results and notions that we need in the next sections are found in [1, 32, 42– 44, 47]. 2. A stronger novel form of (r, s)− gfsc sets Here, we introduce and study a stronger form of (r, s)− gfsc sets called (r, s)− g⊛fsc sets. Also, we show that (r, s)− fsc set [29] ⇒ (r, s)− g⊛fsc set ⇒ (r, s)− gfsc set [42], but the converse may not be true. After that, we introduce new types of fuzzy mappings between double fuzzy topological spaces and relationships are obtained. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4097 Definition 6. Let (V, η, η∗) be a dfts, ν, ρ ∈ IV , r ∈ I◦, and s ∈ I1, then we have: (i) ρ is called an (r, s)-strongly generalized fuzzy semi-closed ⟨briefly, (r, s)− g⊖fsc⟩ if SCη,η∗(ρ, r, s) ≤ ν whenever ρ ≤ ν and ν is (r, s)− gfo set, (ii) ρ is called an (r, s)-strongly∗ generalized fuzzy semi-closed ⟨briefly, (r, s)− g⊛fsc⟩ if SCη,η∗(ρ, r, s) ≤ ν whenever ρ ≤ ν and ν is (r, s)− gfso set. Remark 1. (i) A fuzzy set ρ ∈ IV is (r, s)− g⊖fso if ρc is (r, s)− g⊖fsc set. (ii) A fuzzy set ρ ∈ IV is (r, s)− g⊛fso if ρc is (r, s)− g⊛fsc set. Remark 2. From the previous definition, we can summarize the relationships among different types of fuzzy closed subsets as in the next diagram. (r, s)− fsc → (r, s)− g⊛fsc ↓ ↓ (r, s)− sgfc (r, s)− g⊖fsc ↓ ↓ (r, s)− gfsc Remark 3. The converses of the above implications may not be true, as shown by Ex- amples 1, 2, 3 and 4. Example 1. Let V = {v1, v2, v3, v4} and ρ, ν ∈ IV defined as follows: ρ = { v1 1.0 , v2 1.0 , v3 1.0 , v4 0.0} and ν = { v1 0.0 , v2 0.0 , v3 1.0 , v4 1.0}. Also, (η, η ∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ = ν, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ = ν, 1, otherwise. Thus, ρ is (12 , 1 2)− g⊛fsc set, but it is not (12 , 1 2)− fsc set. Example 2. Let V = {v1, v2, v3, v4} and ρ, λ1, λ2, λ3 ∈ IV defined as follows: ρ = { v1 1.0 , v2 0.0 , v3 1.0 , v4 0.0}, λ1 = { v1 0.0 , v2 1.0 , v3 1.0 , v4 1.0}, λ2 = { v1 0.0 , v2 1.0 , v3 1.0 , v4 0.0} and λ3 = { v1 0.0 , v2 0.0 , v3 1.0 , v4 0.0}. Also, (η, η∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 4 , if µ ∈ {λ1, λ2, λ3}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 4 , if µ ∈ {λ1, λ2, λ3}, 1, otherwise. Thus, ρ is (14 , 1 4)− g⊖fsc set, but it is not (14 , 1 4)− g⊛fsc set. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4098 Example 3. Let V = {v1, v2, v3} and ν,µ1, µ2 ∈ IV defined as follows: ν = { v1 1.0 , v2 0.0 , v3 0.0}, µ1 = { v1 0.0 , v2 0.0 , v3 1.0} and µ2 = { v1 1.0 , v2 1.0 , v3 0.0}. Also, (η, η ∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 1, otherwise. Thus, ν is (12 , 1 2)− sgfc set, but it is not (12 , 1 2)− g⊛fsc set. Example 4. Let V = {v1, v2, v3} and ν, µ1, µ2 ∈ IV defined as follows: ν = { v1 1.0 , v2 0.0 , v3 1.0}, µ1 = { v1 1.0 , v2 0.0 , v3 0.0} and µ2 = { v1 1.0 , v2 1.0 , v3 0.0}. Also, (η, η ∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 3 , if µ ∈ {µ1, µ2}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 3 , if µ ∈ {µ1, µ2}, 1, otherwise. Thus, ν is (13 , 1 3)− gfsc set, but it is not (13 , 1 3)− g⊛fsc set. Remark 4. In general, (r, s)−gfc sets [1] and (r, s)−g⊛fsc sets are independent concepts, as shown by Example 5. Example 5. Let V = {v1, v2, v3, v4} and ρ, ν, µ1, µ2 ∈ IV defined as follows: ρ = { v1 0.0 , v2 1.0 , v3 0.0 , v4 0.0}, ν = { v1 1.0 , v2 1.0 , v3 0.0 , v4 1.0}, µ1 = { v1 1.0 , v2 0.0 , v3 0.0 , v4 0.0} and µ2 = { v1 1.0 , v2 1.0 , v3 0.0 , v4 0.0}. Also, (η, η∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 1, otherwise. Thus, ρ is (12 , 1 2)− g⊛fsc set, but it is not (12 , 1 2)− gfc set. Also, ν is (12 , 1 2)− gfc set, but it is not (12 , 1 2)− g⊛fsc set. Remark 5. In general, any intersection of (r, s) − g⊛fso sets is not (r, s) − g⊛fso, and any union of (r, s)− g⊛fsc sets is not (r, s)− g⊛fsc, as shown by Example 6. Example 6. Let V = {v1, v2, v3, v4} and ν, ρ, µ1, µ2, µ3 ∈ IV defined as follows: ν = { v1 1.0 , v2 0.0 , v3 1.0 , v4 1.0}, ρ = { v1 0.0 , v2 1.0 , v3 1.0 , v4 1.0}, µ1 = { v1 1.0 , v2 0.0 , v3 0.0 , v4 0.0}, µ2 = { v1 0.0 , v2 1.0 , v3 0.0 , v4 0.0} and µ3 = { v1 1.0 , v2 1.0 , v3 0.0 , v4 0.0}. Also, (η, η ∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 3 , if µ ∈ {µ1, µ2, µ3}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 3 , if µ ∈ {µ1, µ2, µ3}, 1, otherwise. Thus, µ1 and µ2 are (13 , 1 3)− g⊛fsc sets, but µ1 ∨µ2 is not (13 , 1 3)− g⊛fsc. Also, ρ and ν are (13 , 1 3)− g⊛fso sets, but ρ ∧ ν is not (13 , 1 3)− g⊛fso. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4099 Theorem 1. Let (V, η, η∗) be a dfts, µ, λ ∈ IV , r ∈ I◦, and s ∈ I1, then λ is (r, s)−g⊛fsc set iff every µ is (r, s) − gfso set and λ ≤ µ, there is ρ is (r, s) − fsc set, such that λ ≤ ρ ≤ µ. Proof. (⇒) Let λ be an (r, s) − g⊛fsc, λ ≤ µ and µ be an (r, s) − gfso set, then SCη,η∗(λ, r, s) ≤ µ. Put ρ = SCη,η∗(λ, r, s), there is ρ is (r, s)−fsc set such that λ ≤ ρ ≤ µ. (⇐) Assume that λ ≤ µ and µ is (r, s) − gfso set, then by hypothesis, there is ρ is (r, s)− fsc set such that λ ≤ ρ ≤ µ, therefore, SCη,η∗(λ, r, s) ≤ µ. So, λ is (r, s)− g⊛fsc set. Proposition 1. Let (V, η, η∗) be a dfts, µ, λ ∈ IV , r ∈ I◦, and s ∈ I1, then the following properties holds. (i) If λ is (r, s)− g⊛fsc and λ ≤ µ ≤ SCη,η∗(λ, r, s), then µ is (r, s)− g⊛fsc set. (ii) If λ is (r, s)− g⊛fso and SIη,η∗(λ, r, s) ≤ µ ≤ λ, then µ is (r, s)− g⊛fso set. (iii) If one of the following two cases holds: (a) λ is (r, s)− g⊛fsc and (r, s)− gfso. (b) λ is (r, s)− g⊛fsc and η(λ) ≥ r, η∗(λ) ≤ s. Then, λ is (r, s)− fsc set. Proof. (i) Let ν be an (r, s)− gfso set and µ ≤ ν, then λ ≤ ν. Since λ is (r, s)− g⊛fsc set, hence SCη,η∗(λ, r, s) ≤ ν, but µ ≤ SCη,η∗(λ, r, s). Then, SCη,η∗(µ, r, s) ≤ ν. So, µ is (r, s)− g⊛fsc set. (ii) and (iii) are easily proved by a similar way. Theorem 2. Let (V, η, η∗) be a dfts, ν ∈ IV , s ∈ I1, and r ∈ I◦, then the following statements are equivalent. (i) ν is (r, s)− fro set. (ii) ν is (r, s)− g⊛fsc set and η(ν) ≥ r, η∗(ν) ≤ s. Proof. (i) ⇒ (ii) Let µ ∈ IV be an (r, s)− gfso set and ν ≤ µ. Since ν is (r, s)− fro set, then ν ∨ Iη,η∗(Cη,η∗(ν, r, s), r, s) = ν ≤ µ. So, SCη,η∗(ν, r, s) ≤ µ, and hence ν is (r, s)− g⊛fsc set. (ii) ⇒ (i) Since ν is (r, s) − g⊛fsc set and η(ν) ≥ r, η∗(ν) ≤ s, then by Proposition 1(iii), ν is (r, s)− fsc set. But, ν is (r, s)− fpo set. Therefore, ν is (r, s)− fro set. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4100 Theorem 3. Let (V, η, η∗) be a dfts, ρ, µ, ν ∈ IV , s ∈ I1, and r ∈ I◦, then the following statements are equivalent. (i) ν is (r, s)− g⊛fso set. (ii) For any µ is (r, s)− gfsc set and µ ≤ ν, then µ ≤ SIη,η∗(ν, r, s). (iii) For any µ is (r, s) − gfsc set and µ ≤ ν, there is ρ is (r, s) − fso set such that µ ≤ ρ ≤ ν. Proof. (i) ⇒ (ii) Let µ be an (r, s) − gfsc set and µ ≤ ν. Then, νc ≤ µc, which is (r, s) − gfso set. Hence, SCη,η∗(ν c, r, s) ≤ µc implies µ ≤ (SCη,η∗(ν c, r, s))c. Then, µ ≤ SIη,η∗(ν, r, s). (ii) ⇒ (iii) Let µ be an (r, s) − gfsc set and µ ≤ ν. Then, by hypothesis µ ≤ SIη,η∗(ν, r, s). Put SIη,η∗(ν, r, s) = ρ. Hence, µ ≤ ρ ≤ ν. (iii) ⇒ (i) Let µ be an (r, s)− gfso set and νc ≤ µ. Then, µc ≤ ν and by hypothesis, there is ρ is (r, s) − fso set such that µc ≤ ρ ≤ ν, that is, νc ≤ ρc ≤ µ. Therefore, by Theorem 1, νc is (r, s)− g⊛fsc set. Hence, ν is (r, s)− g⊛fso set. Definition 7. Let h : (U, τ, τ∗)→ (V, η, η∗) be a mapping, then h is said to be (i) Strongly∗ double fuzzy generalized semi-continuous ⟨briefly, S∗DFGS-continuous⟩ if h−1(ν) is (r, s)− g⊛fso set for each ν ∈ IV and η(ν) ≥ r , η∗(ν) ≤ s . (ii) S∗DFGS-irresolute if h−1(ν) is (r, s)− g⊛fso set for each ν ∈ IV is (r, s)− g⊛fso set. (iii) S∗DFGS-open if h(ρ) is (r, s)−g⊛fso set for each ρ ∈ IU and τ(ρ) ≥ r , τ∗(ρ) ≤ s . (iv) S∗DFGS-closed if h(ρ) is (r, s)− g⊛fsc set for ρ ∈ IU and τ(ρc) ≥ r , τ∗(ρc) ≤ s . Remark 6. From the previous definitions, we can summarize the relationships among different types of DF-continuity as in the next diagram. DF − continuity ↙ ↘ DFG − continuity DFS − continuity ↓ ↓ DFGS − continuity ←− S∗DFGS − continuity F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4101 Remark 7. The converses of the above implications may not be true, as shown by Ex- amples 7 and 8. Example 7. Let V = {v1, v2, v3, v4} and ρ, ν ∈ IV defined as follows: ρ = { v1 0.0 , v2 0.0 , v3 1.0 , v4 1.0} and ν = { v1 0.0 , v2 0.0 , v3 0.0 , v4 1.0}. Define η, η∗, τ, τ∗ : IV −→ I as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ = ρ, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ = ρ, 1, otherwise, τ(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ = ν, 0, otherwise, τ∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ = ν, 1, otherwise. Thus, the identity mapping idv : (V, η, η∗) → (V, τ, τ∗) is S∗DFGS-continuous, but it is not DFS-continuous. Example 8. Let V = {v1, v2, v3} and µ1, µ2, µ3 ∈ IV defined as follows: µ1 = { v1 0.0 , v2 0.0 , v3 1.0}, µ2 = { v1 1.0 , v2 1.0 , v3 0.0} and µ3 = { v1 0.0 , v2 1.0 , v3 1.0}. Define η, η∗, τ, τ∗ : IV −→ I as follows: η(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ ∈ {µ1, µ2}, 1, otherwise, τ(µ) =  1, if µ ∈ {0, 1}, 1 2 , if µ = µ3, 0, otherwise, τ∗(µ) =  0, if µ ∈ {0, 1}, 1 2 , if µ = µ3, 1, otherwise. Thus, the identity mapping idv : (V, η, η∗)→ (V, τ, τ∗) is DFGS-continuous, but it is not S∗DFGS-continuous. Lemma 2. Every S∗DFGS-irresolute mapping is S∗DFGS-continuous. Remark 8. The converse of Lemma 2 may not be true, as shown by Example 9. Example 9. Let V = {v1, v2}. Define η, η∗, τ, τ∗ : IV −→ I as follows: η(ρ) =  1, if ρ ∈ {0, 1}, 1 2 , if ρ ∈ {0.1, 0.3}, 0, otherwise, η∗(ρ) =  0, if ρ ∈ {0, 1}, 1 2 , if ρ ∈ {0.1, 0.3}, 1, otherwise, F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4102 τ(ρ) =  1, if ρ ∈ {0, 1}, 1 2 , if ρ = 0.1, 0, otherwise, τ∗(ρ) =  0, if ρ ∈ {0, 1}, 1 2 , if ρ = 0.1, 1, otherwise. Thus, the identity mapping idv : (V, η, η∗) → (V, τ, τ∗) is S∗DFGS-continuous, but it is not S∗DFGS-irresolute. 3. Some novel higher separation axioms Here, we are going to give the definitions of two types of higher fuzzy separation axioms with the help of (r, s)− gfsc sets [42] called (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩ spaces and establish some of their properties. Definition 8. A dfts (U, η, η∗) is said to be (i) (r, s)-GFS-regular iff utqµ for each µ ∈ IU is (r, s)− gfsc set implies that, there is νδ ∈ IU with η(νδ) ≥ r, η∗(νδ) ≤ s for δ ∈ {1, 2}, such that ut ∈ ν1, µ ≤ ν2 and ν1qν2. (ii) (r, s)-GFS-normal iff µ1qµ2 for each (r, s)−gfsc sets µδ ∈ IU for δ ∈ {1, 2} implies that, there is νδ ∈ IU with η(νδ) ≥ r and η∗(νδ) ≤ s, such that µδ ≤ νδ and ν1qν2. Theorem 4. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then the following statements are equivalent. (i) (U, η, η∗) is (r, s)-GFS-regular space. (ii) If ut ∈ λ for each λ ∈ IU is (r, s) − gfso, there is µ ∈ IU with η(µ) ≥ r and η∗(µ) ≤ s, such that ut ∈ µ ≤ Cη,η∗(µ, r, s) ≤ λ. (iii) If utqλ for each λ ∈ IU is (r, s)−gfsc, there is µδ ∈ IU with η(µδ) ≥ r, η∗(µδ) ≤ s for δ ∈ {1, 2}, such that ut ∈ µ1, λ ≤ µ2 and Cη,η∗(µ1, r, s)qCη,η∗(µ2, r, s). Proof. (i)⇒ (ii) Let ut ∈ λ for each λ ∈ IU is an (r, s)−gfso, then utqλ c for (r, s)−gfsc set λc. Since (U, η, η∗) is (r, s)-GFS-regular, there is µ, ν ∈ IU with η(µ) ≥ r, η∗(µ) ≤ s and η(ν) ≥ r, η∗(ν) ≤ s such that ut ∈ µ, λc ≤ ν and µqν. It implies ut ∈ µ ≤ νc ≤ λ. Since η(ν) ≥ r and η∗(ν) ≤ s, ut ∈ µ ≤ Cη,η∗(µ, r, s) ≤ λ. (ii)⇒ (iii) Let utqλ for each λ ∈ IU is an (r, s)−gfsc, then ut ∈ λc for (r, s)−gfso set λc. By (ii), there is µ ∈ IU with η(µ) ≥ r, η∗(µ) ≤ s such that ut ∈ µ ≤ Cη,η∗(µ, r, s) ≤ λc. Since η(µ) ≥ r and η∗(µ) ≤ s, then µ is (r, s) − gfso and ut ∈ µ. Again, by (ii), there is µ1 ∈ IU with η(µ1) ≥ r, η∗(µ1) ≤ s such that ut ∈ µ1 ≤ Cη,η∗(µ1, r, s) ≤ µ ≤ Cη,η∗(µ, r, s) ≤ λc. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4103 It implies λ ≤ (Cη,η∗(µ, r, s)) c = Iη,η∗(µ c, r, s) ≤ µc. Put µ2 = Iη,η∗(µ c, r, s), then η(µ2) ≥ r, η∗(µ2) ≤ s. So, Cη,η∗(µ2, r, s) ≤ µc ≤ (Cη,η∗(µ1, r, s)) c, that is, Cη,η∗(µ1, r, s)qCη,η∗(µ2, r, s). (iii) ⇒ (i) It is trivial. In a similar way, we can prove Theorem 5. Theorem 5. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then the following statements are equivalent. (i) (U, η, η∗) is (r, s)-GFS-normal space. (ii) If ν ≤ λ for each ν ∈ IU is (r, s) − gfsc and λ ∈ IU is (r, s) − gfso set, there is µ ∈ IU with η(µ) ≥ r and η∗(µ) ≤ s, such that ν ≤ µ ≤ Cη,η∗(µ, r, s) ≤ λ. (iii) If λ1qλ2 for each (r, s) − gfsc sets λδ ∈ IU for δ ∈ {1, 2}, there is µδ ∈ IU with η(µδ) ≥ r and η∗(µδ) ≤ s, such that λδ ≤ µδ and Cη,η∗(µ1, r, s)qCη,η∗(µ2, r, s). Theorem 6. If h : (U, τ, τ∗) → (V, η, η∗) is DF-irresolute, DF-open and bijective map, and (U, τ, τ∗) is (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩ space, then (V, η, η∗) is (r, s)- GFS-regular ⟨resp., (r, s)-GFS-normal⟩ space. Proof. Let vtqµ for each µ ∈ IV is (r, s) − gfsc. Since h is DF-irresolute, DF-open and bijective map, then by Theorem 4.11 [42], h is DFGS-irresolute. Hence, h−1(µ) is (r, s)− gfsc set. Put vt = h(ut). Then, utqh −1(µ). Since (U, τ, τ∗) is (r, s)-GFS-regular, there is µδ ∈ IU with τ(µδ) ≥ r, τ∗(µδ) ≤ s and δ ∈ {1, 2} such that ut ∈ µ1, h −1(µ) ≤ µ2 and µ1qµ2. Since h is DF-open and bijective map, we have vt ∈ h(µ1), µ = h(h−1(µ)) ≤ h(µ2), h(µ1)qh(µ2). Hence, (V, η, η∗) is (r, s)-GFS-regular space. The other case follows similar lines. Theorem 7. If h : (U, τ, τ∗) → (V, η, η∗) is DF-continuous, DFGS-irresolute closed and injective map, and (V, η, η∗) is (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩, then (U, τ, τ∗) is (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩. Proof. Let utqλ for each λ ∈ IU is (r, s) − gfsc. Since h is DFGS-irresolute closed, h(λ) is (r, s) − gfsc. Since h is injective, utqλ implies h(ut)qh(λ). Since (V, η, η∗) is (r, s)-GFS-regular, there is µδ ∈ IU with η(µδ) ≥ r, η∗(µδ) ≤ s and δ ∈ {1, 2} such that h(ut) ∈ µ1, h(λ) ≤ µ2 and µ1qµ2. Since h is DF-continuous, ut ∈ h−1(µ1), λ ≤ h−1(µ2) with η(h−1(µδ)) ≥ r, η∗(h−1(µδ)) ≤ s and δ ∈ {1, 2} and h−1(µ1)qh −1(µ2). Hence, (U, τ, τ∗) is (r, s)-GFS-regular. The other case follows similar lines. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4104 Theorem 8. If h : (U, τ, τ∗) → (V, η, η∗) is DFGS-irresolute, DF-open, DF-closed and surjective map, and (U, τ, τ∗) is (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩, then (V, η, η∗) is (r, s)-GFS-regular ⟨resp., (r, s)-GFS-normal⟩. Proof. Let vt ∈ µ for each µ ∈ IV is (r, s) − gfso. Since h is DFGS-irresolute and surjective then, there is u ∈ h−1({v}) such that ut ∈ h−1(µ) with (r, s)− gfso set h−1(µ). Since (U, τ, τ∗) is (r, s)-GFS-regular, by Theorem 4, there is ν ∈ IU with τ(ν) ≥ r, τ∗(ν) ≤ s such that ut ∈ ν ≤ Cτ,τ∗(ν, r, s) ≤ h−1(µ). It implies vt ∈ h(ν) ≤ h(Cτ,τ∗(ν, r, s)) ≤ µ. Since h isDF-open andDF-closed, then η(h(ν)) ≥ r, η∗(h(ν)) ≤ s and η(hc(Cτ,τ∗(ν, r, s))) ≥ r. Hence, vt ∈ h(ν) ≤ Cη,η∗(h(ν), r, s) ≤ Cη,η∗(h(Cτ,τ∗(ν, r, s)), r, s) ≤ µ. Thus, (V, η, η∗) is (r, s)-GFS-regular. The other case follows similar lines. 4. Novel types of compactness Here, several types of compactness in double fuzzy topological spaces were introduced and the relationships between them were studied. Definition 9. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)-fuzzy compact iff for each family {λj ∈ IU | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ λj . Definition 10. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)-fuzzy GS-compact iff for each family {λj ∈ IU | λj is (r, s) − gfso}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ λj . Lemma 3. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy GS- compact, then µ is (r, s)-fuzzy compact. Proof. Follows from Definitions 9 and 10. Theorem 9. Let h : (U, τ, τ∗) → (V, η, η∗) be a DFGS-continuous mapping, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy GS-compact, then h(µ) is (r, s)-fuzzy compact. Proof. Let {λj ∈ IV | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋 with h(µ) ≤ ∨ j∈𭟋 λj , then {h−1(λj) ∈ IU | h−1(λj) is (r, s) − gfso} (by h is DFGS-continuous), such that µ ≤∨ j∈𭟋 h−1(λj). Since µ is (r, s)-fuzzy GS-compact, there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ h−1(λj). Thus, h(µ) ≤ ∨ j∈𭟋◦ λj . Hence, the proof is completed. Definition 11. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)-fuzzy almost compact iff for each family {λj ∈ IU | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Cη,η∗(λj , r, s). F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4105 Definition 12. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)-fuzzy almost GS-compact iff for each family {λj ∈ IU | λj is (r, s)− gfso}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Cη,η∗(λj , r, s). Lemma 4. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy almost GS-compact, then µ is (r, s)-fuzzy almost compact. Proof. Follows from Definitions 11 and 12. Lemma 5. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy compact (resp., GS-compact), then µ is (r, s)-fuzzy almost compact (resp., almost GS-compact). Proof. Follows from Definitions 9, 10, 11 and 12. Remark 9. The converse of Lemma 5 may not be true, as shown by Example 10. Example 10. Let V = I, k ∈ N − {1}, and ρ, λk ∈ IV defined as follows: ρ(v) = { 1, if v = 0, 1 2 , otherwise, λk(v) =  0.8, if v = 0, kv, if 0 < v ≤ 1 k , 1 , if 1 k < v ≤ 1. Also, (η, η∗) defined on V as follows: η(µ) =  1, if µ ∈ {0, 1}, 2 3 , if µ ≤ ρ, k k+1 , if µ ≤ λk, 0, otherwise, η∗(µ) =  0, if µ ∈ {0, 1}, 1 3 , if µ ≤ ρ, 1 k+1 , if µ ≤ λk, 1, otherwise. Thus, V is (12 , 1 2)-fuzzy almost compact, but it is not (12 , 1 2)-fuzzy compact. Theorem 10. Let h : (U, τ, τ∗) → (V, η, η∗) be a DF-continuous mapping, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy almost GS-compact, then h(µ) is (r, s)-fuzzy almost compact. Proof. Let {λj ∈ IV | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋 with h(µ) ≤ ∨ j∈𭟋 λj , then {h−1(λj) ∈ IU | h−1(λj) is (r, s) − gfso} (by h is DFGS-continuous), such that µ ≤∨ j∈𭟋 h−1(λj). Since µ is (r, s)-fuzzy almost GS-compact, there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Cτ,τ∗(h −1(λj), r, s). Since h is DF-continuous mapping, it follows µ ≤ ∨ j∈𭟋◦ Cτ,τ∗(h −1(λj), r, s) F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4106 ≤ ∨ j∈𭟋◦ h−1(Cη,η∗(λj , r, s)) = h−1( ∨ j∈𭟋◦ Cη,η∗(λj , r, s)). Thus, h(µ) ≤ ∨ j∈𭟋◦ Cη,η∗(λj , r, s). Hence, the proof is completed. Definition 13. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)- fuzzy nearly compact iff for each family {λj ∈ IU | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Iη,η∗(Cη,η∗(λj , r, s), r, s). Definition 14. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1, then µ ∈ IU is called an (r, s)- fuzzy nearly GS-compact iff for each family {λj ∈ IU | λj is (r, s) − gfso}j∈𭟋, such that µ ≤ ∨ j∈𭟋 λj , there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Iη,η∗(Cη,η∗(λj , r, s), r, s). Lemma 6. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy nearly GS-compact, then µ is (r, s)-fuzzy nearly compact. Proof. Follows from Definitions 13 and 14. Lemma 7. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy compact (resp., GS-compact), then µ is (r, s)-fuzzy nearly compact (resp., nearly GS-compact). Proof. Follows from Definitions 9, 10, 13 and 14. Remark 10. The converse of Lemma 7 may not be true, as shown by Example 11. Example 11. Let V = I, 0 < k < 1, and ν, ρ, λk ∈ IV defined as follows: ν(v) = { 1 2 , if 0 ≤ v < 1, 1, if v = 1, ρ(v) = { 1, if v = 0, 1 2 , if 0 < v ≤ 1, λk(v) = { v k , if 0 ≤ v ≤ k, 1−v 1−k , if k < v ≤ 1. Also, (η, η∗) defined on V as follows: η(µ) =  1, if µ ∈ {ν, ρ, 0, 1}, max({1− k, k}), if µ = λk, 0, otherwise, η∗(µ) =  0, if µ ∈ {ν, ρ, 0, 1}, min({k, 1− k}), if µ = λk, 1, otherwise. Thus, V is (12 , 1 2)-fuzzy nearly compact, but it is not (12 , 1 2)-fuzzy compact. F. Alsharari, O. M. Taha, I. M. Taha / Eur. J. Pure Appl. Math, 17 (4) (2024), 4093-4111 4107 Theorem 11. Let h : (U, τ, τ∗)→ (V, η, η∗) be a DF-continuous and DF-open mapping, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy nearly GS-compact, h(µ) is (r, s)-fuzzy nearly compact. Proof. Let {λj ∈ IV | η(λj) ≥ r and η∗(λj) ≤ s}j∈𭟋 with h(µ) ≤ ∨ j∈𭟋 λj , then {h−1(λj) ∈ IU | h−1(λj) is (r, s) − gfso} (by h is DFGS-continuous), such that µ ≤∨ j∈𭟋 h−1(λj). Since µ is (r, s)-fuzzy nearly GS-compact, there is a finite subset 𭟋◦ of 𭟋, such that µ ≤ ∨ j∈𭟋◦ Iτ,τ∗(Cτ,τ∗(h −1(λj), r, s), r, s). Since h is DF-continuous and DF-open, it follows h(µ) ≤ ∨ j∈𭟋◦ h(Iτ,τ∗(Cτ,τ∗(h −1(λj), r, s), r, s)) ≤ ∨ j∈𭟋◦ Iη,η∗(h(Cτ,τ∗(h −1(λj), r, s)), r, s) ≤ ∨ j∈𭟋◦ Iη,η∗(h(h −1(Cη,η∗(λj , r, s))), r, s) ≤ ∨ j∈𭟋◦ Iη,η∗(Cη,η∗(λj , r, s), r, s). Hence, the proof is completed. Lemma 8. Let (U, η, η∗) be a dfts, r ∈ I◦, and s ∈ I1. If µ ∈ IU is (r, s)-fuzzy soft nearly GS-compact (resp., nearly compact), then µ is (r, s)-fuzzy soft almost GS-compact (resp., almost compact). Proof. Follows from Definitions 11, 12, 13 and 14. Remark 11. We can summarize the relationships among different types of fuzzy com- pactness as in the next diagram. GS-compactness → compactness ↓ ↓ nearly GS-compactness → nearly compactness ↓ ↓ almost GS-compactness → almost compactness REFERENCES 4108 5. Conclusion and future work In this article, we have introduced a novel class of generalizations of fuzzy closed subsets called “(r, s)−g⊛fsc sets” via double fuzzy topologies and some characterizations have been discussed. Moreover, we have defined novel types of fuzzy mappings and the relationship between these mappings have been introduced with the help of some problems. Also, we have shown that (r, s)− fsc ⇒ (r, s)− g⊛fsc ⇓ ⇓ (r, s)− sgfc (r, s)− g⊖fsc ⇓ ⇓ (r, s)− gfsc but in general, the converses of the above implications may not be true. Thereafter, “(r, s)-GFS-regular” and “(r, s)-GFS-normal” spaces have been defined as two new no- tions of higher fuzzy separation axioms and some characterizations of these separation axioms have been obtained. In the end, several novel types of fuzzy compactness in the frame of double fuzzy topologies have been introduced and some properties have been given. Also, the relationship between them have been explored. In the upcoming papers, we shall discuss the concepts given here in the frames of a fuzzy idealization [38, 40] and fuzzy soft r-minimal structures [37, 41]. Moreover, we will study the main properties of classical compactness in the frame of double fuzzy topologies. Acknowledgements We would like to thank the reviewers and editors whose constructive comments and suggestions helped to improve this paper. References [1] S. E. Abbas. (r, s)-generalized intuitionistic fuzzy closed sets. J. Egyptian Math. Soc., 14:331–351, 2006. [2] S. E. Abbas and B. Krsteska. 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