EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 4624 ISSN 1307-5543 – ejpam.com Published by New York Business Global Double Fuzzy δ-Continuous Functions in Double Fuzzy Topological Spaces Wadei AL-Omeri Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan Abstract. In this paper, we introduce (r, s)-δ-fuzzy closed sets in double fuzzy topological spaces and investigate some of their properties. Moreover, we introduce the concept of double fuzzy δ-continuous functions. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the new concepts are introduced and established with some interesting counterexamples. 2020 Mathematics Subject Classifications: 54A40, 45D05, 03E72 Key Words and Phrases: Fuzzy sets, double fuzzy topological spaces, (r, s)-fuzzy δ-closed sets, (r, s)-fuzzy δ-continuous functions 1. Preliminaries The concept of fuzzy sets was introduced by Zadeh in his classical paper [1]. In 1968, Chang [2] used fuzzy sets to introduce the notion of fuzzy topological spaces. Çoker [3, 4] defined the intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. Later on, Demirci and Çoker [5] defined intuitionistic fuzzy topological spaces which is a generaliza- tion of fuzzy topological spaces and intuitionistic fuzzy topological spaces. Mondal and Samanta [6] succeeded to make the topology itself intuitionistic. The resulting structure is given the new name ”intuitionistic gradation of openness”. The name ”intuitionistic” did not continue due to some doubts that were thrown about the suitability of this term. These doubts were quickly ended in 2005 by Gutiérrez García and Rodabaugh [7]. They proved that this term is unsuitable in mathematics and applications. Therefore, they replaced the word ”intuitionistic” by ”double” and renamed its related topologies. The notion of intu- itionistic gradation of openness is given the new name ”double fuzzy topological spaces” see [8–10]. The fuzzy type of the notion of topology can be studied in the fuzzy mathematics see [11–13], which has many applications in different branches of mathematics and physics theory. For example, fuzzy topological spaces can be applied in the modeling of spatial objects such as rivers, roads, trees, and buildings. Since double fuzzy topology forms an DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.4624 Email addresses: wadeimoon1@hotmail.com (W. F. AL-Omeri) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 2 of 15 extension of fuzzy topology and general topology, we think that our results can be applied in modern physics and GIS Problems. In this paper, we define and study (r, s)-δ-fuzzy closed sets in double fuzzy topological spaces and investigate some of their properties. Moreover, we introduce the concepts of double fuzzy δ-continuous functions. Several interesting properties and characterizations are introduced and discussed. Throughout this paper, let X be a nonempty set and I be the closed unit interval [0, 1], I0 = (0, 1] and I1 = [0, 1). The family of all fuzzy subsets on X is denoted by IX . By 0 and 1, we denote the smallest and the greatest fuzzy subsets on X. For a fuzzy subset ρ ∈ IX , 1 − ρ denotes it’s complement. Given a function f : X → Y , f(ρ) and f−1(ρ) define the direct image and the inverse image of f , by f(ρ)(y) = ∨ f(x)=y ρ(x) and f−1(ν)(x) = ν(f(x)), for each ρ ∈ IX , ν ∈ IY and x ∈ X, respectively. For fuzzy subsets ρ and ϕ in X, we write ρqϕ to mean that ρ is quasi coincident (q-coincident) with ϕ that is, there exists at least one point x ∈ X such that ρ(x) + ϕ(x) > 1. Negation of such a statement is denoted as ρqϕ. Notions and notations not described in this paper are standard and usual. Definition 1.1. [8, 14, 15] The pair of functions T , T ∗ : IX → I is called a double fuzzy topology on X if it satisfies the following conditions: (i) T (ρ) + T ∗(ρ) ≤ 1, (ii) T ∗(ρ1 ∧ ρ2) ≥ T ∗(ρ1) ∧ T ∗(ρ2) and T (ρ1 ∧ ρ2) ≤ T (ρ1) ∨ T (ρ2), (iii) T ( ∨ i∈I ρi) ≥ ∧ i∈I T (ρi) and T ∗( ∨ i∈I ρi) ≤ ∨ i∈I T ∗(ρi) for each ρI ∈ IX ,i ∈ I. The triplet (X, T , T ∗) is called a double fuzzy topological space (DFTS, for short). T (ρ) and T ∗(ρ) may be interpreted as a gradation of openness and gradation of non-openness for ρ. A function f : (X, T1, T ∗ 1 ) → (Y, T2, T ∗ 2 ) is said to be double fuzzy continuous if T1(f−1(ν)) ≥ T2(ν) and T ∗ 1 (f −1(ν)) ≤ T ∗ 2 (ν) for each ν ∈ IY . Theorem 1.2. [8, 14] Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1, and ρ ∈ IX , we define an operator CT ,T ∗ : IX × I1 × I0 −→ IX as follows: CT ,T ∗(ρ, r, s) = ∧ {ϕ ∈ IX |ρ ≤ ϕ, T (1− ϕ) ≥ r, T ∗(1− ϕ) ≤ s}. For each ρ, ϕ ∈ IX , r, r1 ∈ I0 and s, s1 ∈ I1, the operator CT ,T ∗ satisfies the following statements: (i) CT ,T ∗(0, r, s) = 0. (ii) ρ ≤ CT ,T ∗(ρ, r, s). (iii) CT ,T ∗(ρ, r, s) ∧ CT ,T ∗(ϕ, r, s) = CT ,T ∗(ρ ∧ ϕ, r, s). (iv) CT ,T ∗(ρ, r, s) ≤ CT ,T ∗(ρ, r1, s1), if r ≤ r1 and s ≥ s1. W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 3 of 15 (v) CT ,T ∗(CT ,T ∗(ρ, r, s), r, s) = CT ,T ∗(ρ, r, s). Theorem 1.3. (see [12-14]) Let (X, T , T ∗) be an DFTS. Then for each r ∈ I0, s ∈ I1, and ρ ∈ IX , we define an operator IT ,T ∗ : IX × I1 × I0 −→ IX as follows: IT ,T ∗(ρ, r, s) = ∨ {ϕ ∈ IX : ϕ ≤ ρ, T (ϕ) ≥ r, T ∗(ϕ) ≤ s}. For each ρ, ϕ ∈ IX , r, r1 ∈ I0 and s, s1 ∈ I1, the operator IT ,T ∗ satisfies the following statements: (i) IT ,T ∗(1− ρ, r, s) = 1− CT ,T ∗(ρ, r, s). (ii) IT ,T ∗(1, r, s) = 1. (iii) IT ,T ∗(ρ, r, s) ≤ ρ. (iv) IT ,T ∗(ρ, r, s) ∨ IT ,T ∗(ϕ, r, s) = IT ,T ∗(ρ ∨ ϕ, r, s). (v) IT ,T ∗(ρ ∨ ϕ, r, s) ≥ IT ,T ∗(ρ ∨ ϕ, r1, s1) if r ≤ r1 and s ≥ s1. (vi) IT ,T ∗(IT ,T ∗(ρ, r, s), r, s) = IT ,T ∗(ρ, r, s). (vii) IT ,T ∗(CT ,T ∗(ρ, r, s), r, s) = ρ, then CT ,T ∗(IT ,T ∗(1− ρ, r, s), r, s) = 1− ρ. Definition 1.4. [16] Let (X, T , T ∗) be an DFTS. ρ ∈ IX , xi ∈ FP (X), r ∈ I0, s ∈ I1, a fuzzy set ρ is called (r, s)-Q-neighborhood of xi, if T (ρ) ≥ r, T ∗(ρ) ≤ s, and xiqρ. Definition 1.5. [17] Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1 and ρ ∈ IX is called (r, s)-fuzzy regular open ((r, s)-FRO, for short) if ρ = IT ,T ∗(CT ,T ∗(ρ, r, s), r, s). A fuzzy set ρ is called a (r, s)-fuzzy regular closed ((r, s)-FRC, for short) iff 1 − ρ is a (r, s)-FRO set. 2. (r, s)-fuzzy δ-closed sets Definition 2.1. Let (X, T , T ∗) be a Double Fuzzy Topological Spaces. Then for each r ∈ I0, s ∈ I1, and ρ ∈ IX , x(α,β) ∈ FP (X), where FP (X) is the family of all fuzzy points in X. A double fuzzy point x(α,β) is said to be a double fuzzy δ-cluster point of a fuzzy set ρ in an DFTS X iff every (r, s)-fuzzy regular open set containing a double fuzzy point x(α,β) (having same support as x(α,β)) has non-null intersection with ρ, or if for every (r, s)-fuzzy regular open Q-neighborhood ϕ of x(α,β) is q-coincident with ρ, by other words IT ,T ∗(CT ,T ∗(ϕ, r, s), r, s)qρ. Definition 2.2. Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1, and ρ ∈ IX , where ρ be a fuzzy subset of an DFTS X. Let ϕ be a fuzzy subset of X satisfying the following conditions: (a) Every double fuzzy point x(α,β) in ϕ is a double fuzzy δ-cluster point of ρ, (b) If ν is a double fuzzy set, such that ϕ ≤ ν, then there is a double fuzzy point. x(α,β) in ν which is not a double fuzzy δ-cluster point of ρ. W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 4 of 15 Any double fuzzy subset of X having same support as ϕ is defined to be (r, s)-fuzzy δ-closure of ρ. Definition 2.3. The set of all double fuzzy δ-cluster points of ρ is called the (r, s)-fuzzy δ-closure of ρ and denoted by δCT ,T ∗ . Thus a property related with the notation δCT ,T ∗ , will always imply that the property holds for all (r, s)-fuzzy δ-closure of ρ. A double fuzzy set ρ is said to be a (r, s)-fuzzy δ-closed set if ρ = δCT ,T ∗(ρ, r, s). The complement of a (r, s)-fuzzy δ-closed set is said to be a (r, s)-fuzzy δ-open set. Definition 2.4. Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1, and ρ ∈ IX , the δCT ,T ∗ and IT ,T ∗ operators are define as follows: (i) δCT ,T ∗(ρ, r, s) = ∧ {ϕ ∈ IX |ρ ≤ ϕ, ϕ is (r, s)− FRC}. (ii) δIT ,T ∗(ρ, r, s) = ∨ {ϕ ∈ IX |ρ ≥ ϕ, ϕ is (r, s)− FRO}. From the above definition, we have the following relations: (i) δCT ,T ∗(1− ρ, r, s) = 1− δIT ,T ∗(ρ, r, s), (ii) 1− δCT ,T ∗(ρ, r, s)) = δIT ,T ∗(1− ρ, r, s). Note: (r, s)-fuzzy δ-closure of a fuzzy set in an DFTS is not unique. However, it is unique up to it’s support, and any (r, s)-FRO set of an DFTS X is a (r, s)-fuzzy open set of X. Therefore, the set of all (r, s)-FRO sets is subset of the set of all (r, s)-fuzzy open sets of an DFTS. So, if a double fuzzy point x(α,β) of a double fuzzy set ρ in an DFTS X is a double fuzzy δ-cluster point of ρ, then x(α,β) is also a double fuzzy cluster point of ρ. Proposition 2.5. Let (X, T , T ∗) be an DFTS. Then, for each r, r1 ∈ I0, s, s1 ∈ I1, and ρ ∈ IX , the operator δIT ,T ∗ satisfies the following statements: (i) δIT ,T ∗(0, r, s) = 0, δIT ,T ∗(1, r, s) = 1. (ii) δIT ,T ∗(ρ, r, s) ≤ ρ. (iii) δIT ,T ∗(ρ, r, s) ≤ δIT ,T ∗(ρ, r1, s1), if r ≥ r1 and s ≤ s1. (iv) δIT ,T ∗(ρ ∧ ϕ, r, s) = δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s). (v) δIT ,T ∗(δIT ,T ∗((ρ, r, s), r, s)) = δIT ,T ∗(ρ, r, s). Proof. (i), (ii), (v) are obvious by using the definition 2.4. (iii) Let r ≥ r1 and s ≤ s1. Then for each (r, s)-FRO and (r1, s1)-FRO sets, we have: δI(ρ, r, s) = ∨ {ϕ ∈ IX |ρ ≥ ϕ, ϕ is (r, s)− FRO} = ∨ {ϕ ∈ IX |ρ ≥ ϕ, ϕ is (r1, s1)− FRO} W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 5 of 15 = δI(ρ, r1, s1). (iv) Since ρ ∧ ϕ ≤ ρ and ρ ∧ ϕ ≤ ϕ, then we have δI(ρ ∧ ϕ, r, s) ≤ δI(ρ, r, s), and δI(ρ ∧ ϕ, r, s) ≤ δI(ϕ, r, s). Thus, δI(ρ ∧ ϕ, r, s) ≤ δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s). Conversely, it is clear that δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s) ≤ ρ ∧ ϕ. Also, T (δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s)) ≥ T (δIT ,T ∗(ρ, r, s)) ∧ T (δIT ,T ∗(ϕ, r, s)) ≥ r ∧ r = r, and T ∗(δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s)) ≤ T ∗(δIT ,T ∗(ρ, r, s)) ∧ T ∗(δIT ,T ∗(ϕ, r, s)) ≤ s ∧ s = s. By the definition of δIT ,T ∗ , we get δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s) ≤ δI(ρ ∧ ϕ, r, s). Hence, δIT ,T ∗(ρ ∧ ϕ, r, s) = δIT ,T ∗(ρ, r, s) ∧ δIT ,T ∗(ϕ, r, s). Proposition 2.6. Let (X, T , T ∗) be an DFTS. Then, for each r, r1 ∈ I0, s, s1 ∈ I1, and ρ ∈ IX , the operator δCT ,T ∗ satisfies the following statements: (i) δCT ,T ∗(0, r, s) = 0, δCT ,T ∗(1, r, s) = 1. (ii) δCT ,T ∗(ρ, r, s) ≥ ρ. (iii) δCT ,T ∗(ρ, r, s) ≤ δCT ,T ∗(ρ, r1, s1), if r ≤ r1 and s ≥ s1. (iv) δCT ,T ∗(ρ ∨ ϕ, r, s) = δCT ,T ∗(ρ, r, s) ∨ δCT ,T ∗(ϕ, r, s). (v) δCT ,T ∗(δCT ,T ∗((ρ, r, s), r, s)) = δCT ,T ∗(ρ, r, s). Proof. Similar to proposition 2.5. W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 6 of 15 Lemma 2.7. (i) For any double fuzzy set ρ in an double fuzzy topology (T , T ∗) on X, IT ,T ∗(CT ,T ∗(ρ, r, s)) is a (r, s)-FRO set. (ii) double fuzzy set ρ in an double fuzzy topology (T , T ∗) on X, such that xiqρ, IT ,T ∗(CT ,T ∗(ρ, r, s)) is a (r, s)-FRO Q-neighborhood of xi. Proof. (i) It’s enough to show that IT ,T ∗(CT ,T ∗(ρ, r, s)) = IT ,T ∗(CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s))). Since IT ,T ∗(CT ,T ∗(ρ, r, s)) ≤ CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s))), and we have IT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s)) ≤ IT ,T ∗(CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s))). Thus IT ,T ∗(CT ,T ∗(ρ, r, s)) ≤ IT ,T ∗(CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s))). Conversely, since IT ,T ∗(CT ,T ∗(ρ, r, s)) ≤ CT ,T ∗(ρ, r, s), and we have CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s)) ≤ CT ,T ∗(CT ,T ∗(ρ, r, s)) = CT ,T ∗((ρ, r, s), r, s). Thus IT ,T ∗(CT ,T ∗(IT ,T ∗(CT ,T ∗(ρ, r, s))) ≤ IT ,T ∗(CT ,T ∗(ρ, r, s)). Hence IT ,T ∗(CT ,T ∗(ρ, r, s)) is a (r, s)-FRO set. (ii) Clearly, IT ,T ∗(ρ, r, s) ≤ IT ,T ∗(CT ,T ∗(ρ, r, s)). Since ρ is a (r, s)-FRO set, we have ρ = IT ,T ∗(ρ, r, s) ≤ IT ,T ∗(CT ,T ∗(ρ, r, s)). By (1), IT ,T ∗(CT ,T ∗(ρ, r, s)) is a (r, s)-FRO set. Therefore IT ,T ∗(CT ,T ∗(ρ, r, s) is a (r, s)-FRO Q-neighborhood of xi. A (r, s)-FRC set is not always (r, s)-fuzzy δ-closed set. For example, Example 2.8. Let X = {a, b} and the fuzzy set µ1 define as follows: µ1(a) = 0.5, µ1(b) = 0.6, W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 7 of 15 The double fuzzy space (T , T ∗) is define on X as follows:: T (ρ) =  1, if ρ ∈ {0, 1}, 1 2 , if ρ = µ1, 0, otherwise. T ∗(ρ) =  0, if ρ ∈ {0, 1}, 1 2 , if ρ = µ1, 1, otherwise. Since µ1 = CT ,T ∗(IT ,T ∗(µ1, r, s), r, s). Then, µ1 is (12 , 1 2)-FRC set, but is not (12), 1 2)-fuzzy δ-closed set. Theorem 2.9. For any DFS ρ in an DFTS (X, T , T ∗), we have: (i) CT ,T ∗(ρ, r, s) ≤ δCT ,T ∗(ρ, r, s). (ii) δIT ,T ∗(ρ, r, s) ≤ IT ,T ∗(ρ, r, s). Proof. It is Obvious. Theorem 2.10. Let ρ, ϕ ∈ IX . The finite union of (r, s)-fuzzy δ-closed sets is also (r, s)-fuzzy δ-closed, where r ∈ I0 and s ∈ I1. That is, if ρ = δCT ,T ∗(ρ, r, s) and ϕ = δCT ,T ∗(ϕ, r, s), then ρ ∨ ϕ = δCT ,T ∗(ρ ∨ ϕ, r, s). Proof. Clearly (ρ ∨ ϕ) ≤ δCT ,T ∗(ρ ∨ ϕ, r, s). We will show that δCT ,T ∗((ρ ∨ ϕ), r, s) ≤ ρ ∨ ϕ. Let xi for each i ∈ I be a double fuzzy point. Suppose that xi ∈ δCT ,T ∗((ρ∨ϕ), r, s). Then for any (r, s)-fuzzy regular Q-neighborhood γ of xα ∈ γq(ρ ∨ ϕ). Thus γqρ or γqϕ. Hence xα ∈ δCT ,T ∗(ρ, r, s) ∨ δCT ,T ∗(ϕ, r, s). That is, xα ∈ (ρ ∨ ϕ). Corollary 2.11. If ρ is a (r, s)-fuzzy δ-closed set in an DFTS (X, T , T ∗), then ρ is double fuzzy closed. The converse don’t holds, the following example shows it. Example 2.12. Let X = {a, b} and the fuzzy set µ1 is fuzzy set define by µ1(a) = 0.5 and µ1(b) = 0.3, Take the (T , T ∗) on X as in Example 2.8. Since µ1 = δCT ,T ∗(µ1, r, s). Then, µ1 is (12), 1 2)-fuzzy δ-closed set, but is not (12 , 1 2)-FRC set. Theorem 2.13. If ρ is a (r, s)-fuzzy δ-open set in an DFTS (X, T , T ∗), then the double fuzzy closure and (r, s)-fuzzy δ-closure are the same, i.e. CT ,T ∗(ρ, r, s) = δCT ,T ∗(ρ, r, s). Proof. By Theorem 2.9, it is sufficient to show that δCT ,T ∗(ρ, r, s) ≤ CT ,T ∗(ρ, r, s). Take any xi ∈ δCT ,T ∗(ρ, r, s). Suppose that xi /∈ CT ,T ∗(ρ, r, s). Then there exists a (r, s)- fuzzy open Q-neighborhood ϕ of xi such that ϕq̃ρ. Since ϕq̃ρ, we have ϕ ≤ 1− ρ. Since (1− ρ) is a (r, s)-fuzzy δ-closed set, CT ,T ∗(ϕ, r, s) ≤ CT ,T ∗(1− ρ, r, s) = 1− ρ. Therefore, IT ,T ∗(CT ,T ∗(ϕ, r, s)) ≤ IT ,T ∗(1− ρ, r, s) ≤ 1− ρ, i.e. IT ,T ∗(CT ,T ∗(ϕ, r, s))q̃ρ. By Lemma 2.7, IT ,T ∗(CT ,T ∗(ρ, r, s)) is a (r, s)-FRO set Q-neighborhood of xi such that IT ,T ∗(CT ,T ∗(ϕ, r, s))q̃ρ. Hence xi /∈ δCT ,T ∗(ρ, r, s). W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 8 of 15 Theorem 2.14. For any (r, s)-fso set ρ, CT ,T ∗(ρ, r, s) = δCT ,T ∗(ρ, r, s). Proof. It’s enough to show that δCT ,T ∗(ρ, r, s) ≤ CT ,T ∗(ρ, r, s). Take any xi ∈ δCT ,T ∗(ρ, r, s) and suppose that xi /∈ CT ,T ∗(ρ, r, s). Now, xi /∈ CT ,T ∗(ρ, r, s), then there exists a (r, s)-fuzzy open Q-neighborhood ν of xi such that ϕq̃ρ. By definition of (r, s)-fso set, there exists a T (ϕ) ≥ r and T ∗(ϕ) ≤ s such that ϕ ≤ ρ ≤ CT ,T ∗(ϕ, r, s) whenever, r ∈ I0, s ∈ I1. Thus ν ≤ 1− ρ ≤ 1− ϕ. Hence, CT ,T ∗(ν, r, s) ≤ CT ,T ∗(1− ρ, r, s) ≤ CT ,T ∗(1− ϕ, r, s) = 1− ϕ. Also, IT ,T ∗(CT ,T ∗(ν, r, s)) ≤ IT ,T ∗(CT ,T ∗(1− ρ, r, s)) ≤ IT ,T ∗(CT ,T ∗(1− ϕ, r, s)) = IT ,T ∗(1− ϕ, r, s) ≤ 1− ϕ, i.e. IT ,T ∗(CT ,T ∗(ν, r, s)) ≤ 1− ϕ. Therefore, ϕ ≤ (1− IT ,T ∗(CT ,T ∗(ν, r, s))). Hence, ρ ≤ CT ,T ∗(ϕ, r, s) ≤ CT ,T ∗(1− IT ,T ∗(CT ,T ∗(ν, r, s))) = (1− IT ,T ∗(CT ,T ∗(ν, r, s)). Because of (1 − (1 − IT ,T ∗(CT ,T ∗(ν, r, s)))) ≥ r and (1 − (1 − IT ,T ∗(CT ,T ∗(ν, r, s))) ≤ s. Thus IT ,T ∗(CT ,T ∗(ν, r, s))q̃ρ. By Lemma 2.7, IT ,T ∗(CT ,T ∗(ν, r, s)) is a (r, s)-fuzzy regular open Q-neighborhood ν of xi such that IT ,T ∗(CT ,T ∗(ν, r, s))q̃ρ. Hence, xi /∈ δCT ,T ∗(ρ, r, s). Proposition 2.15. Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1, and ρ ∈ IX . Let δIT ,T ∗ : IX × I0 × I1 −→ IX be a function satisfy the conditions (1-5) in proposition 2.5 such that T , T ∗ : IX → I be functions defined by T (ρ) = ∨ {r ∈ I|δIT ,T ∗(ρ, r, s) = ρ} and T ∗(ρ) = ∧ {s ∈ I|δIT ,T ∗(ρ, r, s) = ρ}. Then ζ = (T , T ∗) is double fuzzy topology in X. Proof. (a) Since (r, s) ∈ I0 × I1 and we have r+ s ≤ 1. Hence s ≤ 1− r, Thus, 1− T (ρ) = 1− ∨ {r ∈ I|δIT ,T ∗(ρ, r, s) = ρ} = ∧ {1− r ∈ I|δIT ,T ∗(ρ, r, s) = ρ} ≥ ∧ {s ∈ I|δIT ,T ∗(ρ, r, s) = ρ} = T ∗ . (b) Suppose that T (ρ1 ∧ ρ2) ≤ T (ρ1) ∧ T (ρ2). Then there is a ξ ∈ I such that T (ρ1 ∧ ρ2) ≤ ξ ≤ T (ρ1) ∧ T (ρ2). W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 9 of 15 Since, ξ ≤ T (ρi) = ∨ {r ∈ I|δIT ,T ∗(ρξ, r, s) = ρi}, for each i = 1, 2 there exist (r1, s1), (r2, s2) ∈ I0 × I1 such that ξ ≤ ri ≤ T (ρi) and δIT ,T ∗(ρi, ri, si) = ρi. Now, for each i = 1, 2, let r = r1 ∧ r2 and s = s1 ∨ s2. Since (r1, s1), (r2, s2) ∈ I0 × I1, we have: 1− r = 1− (r1 ∧ r2) = (1− r1) ∨ (1− r2) ≥ s1 ∨ s2 = s Hence, (r, s) ∈ I0 × I1. Since r ≤ ri and s ≥ si for each i = 1, 2 we have, δIT ,T ∗(ρξ, r, s) ≥ δIT ,T ∗(ρi, ri, si) = ρξ. Hence, δIT ,T ∗(ρi, r, s) = ρi for each i = 1, 2, we get δIT ,T ∗(ρ1 ∧ ρ2, r, s) = δIT ,T ∗(ρ1, r, s) ∧ δIT ,T ∗(ρ2, r, s) = ρ1 ∧ ρ2. Thus, ξ ≥ T (ρ1 ∧ ρ2) = ∨ {r∗ ∈ I|δIT ,T ∗(ρ1 ∧ ρ2, r, s) = ρ1 ∧ ρ2} ≥ r = r1 ∧ r2 ≥ ξ. Which is a contradiction. Hence, T (ρ1 ∧ ρ2) ≥ T (ρ1) ∧ T (ρ2). Next, suppose that T ∗(ρ1 ∧ ρ2) ≥ T ∗(ρ1) ∨ T ∗(ρ2). Then, there is a ξ ∈ I such that T ∗(ρ1 ∧ ρ2) ≥ ξ ≥ T ∗(ρ1) ∨ T ∗(ρ2). Since ξ ≥ T ∗(ρi) = ∧ {s ∈ I|δIT ,T ∗(ρi, r, s) = ρξ} for each i = 1, 2 there are (r1, s1), (r2, s2) ∈ I0 × I1 such that ξ ≥ sξ ≥ T (ρi) and δIT ,T ∗(ρi, ri, si) = ρi. Now, let r = r1 ∧ r2 and s = s1 ∨ s2. Then (r, s) ∈ I0 × I1 and δIT ,T ∗(ρi, r, si) = ρi for each i = 1, 2. δIT ,T ∗(ρ1 ∧ ρ2, r, s) = δIT ,T ∗(ρ1, r, s) ∧ δIT ,T ∗(ρ2, r, s) = ρ1 ∧ ρ2. Thus, ξ ≥ T ∗(ρ1 ∧ ρ2) = ∧ {r∗ ∈ I|δIT ,T ∗(ρ1 ∧ ρ2, r, s) = ρ1 ∧ ρ2} ≤ s = s1 ∧ s2 ≤ ξ. Which is a contradiction. Hence, T ∗(ρ1 ∧ ρ2) ≤ T ∗(ρ1) ∨ T ∗(ρ2). (c) First suppose that, T ( ∨ ρi) ≤ ∧ T (ρi). Then, there exist ξ such that T ( ∨ ρi) ≤ ξ ≤ ∧ T (ρi). Since ξ ≤ ∨ {r ∈ I|δIT ,T ∗(ρi, r, s) = ρi} for each i there exist (ri, si) ∈ I0×I1 W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 10 of 15 such that ξ ≤ ri ≤ T (ρi) and δIT ,T ∗(ρi, ri, si) = ρi. Now, let r = ∧ ri and s = ∨ si. Since (ri, si) ∈ I0 × I1 for each i we have, 1− r = 1− ∧ (ri) = ∨ (1− ri) ≥ ∨ si = s. Hence, (r, s) ∈ I0 × I1. And since r ≤ ∧ ri and s ≥ ∨ si for each i, δIT ,T ∗(ρi, r, s) ≥ δIT ,T ∗(ρi, ri, si) = ρi. Hence, δIT ,T ∗( ∨ ρi, r, s) ≥ δIT ,T ∗(ρi, r, s) ≥ ρi for each i. So, δIT ,T ∗( ∨ ρi, r, s) ≥ ∨ ρi, And hence, δIT ,T ∗( ∨ ρi, r, s) = ∨ ρi. Thus, ξ ≥ T ( ∨ ρi) = ∨ {r∗ ∈ I|δIT ,T ∗( ∨ ρi, r ∗, s∗) = ∨ ρi} ≥ r = ∧ ri ≥ ξ. Which is a contradiction. Hence, T ( ∨ ρi) ≥ ∧ T (ρi). Similar for s. Therefor, ζ = (T , T ∗) is double fuzzy topology in X. Proposition 2.16. Let (X, T , T ∗) be an DFTS. Then, for each r ∈ I0, s ∈ I1, and ρ ∈ IX . Let δCT ,T ∗ : IX × I0 × I1 −→ IX be a function satisfy the conditions (1-5) in proposition 2.6 such that T , T ∗ : IX → I be a function defined by: T (ρ) = ∨ {r ∈ I|δCT ,T ∗(ρ, r, s) = ρ} and T ∗(ρ) = ∧ {s ∈ I|δCT ,T ∗(ρ, r, s) = ρ}. Then ζ = (T , T ∗) is a double fuzzy family of closed sets in X. Proof. Similar to proposition 2.15 3. (r, s)-Fuzzy δ-continuous functions Definition 3.1. A function f FROm an DFTS (X, T1, T ∗ 1 ) into an DFTS (Y, T2, T ∗ 2 ) is said to be double fuzzy δ-continuous at a double fuzzy point xα,β in X, if f−1(υ) is a (r, s)-fuzzy δ-open set, for each r ∈ I0, s ∈ I1, and υ is a (r, s)-FRO set in IY such that T2(υ) ≥ r and T ∗ 2 (υ) ≤ s. i.e for each double fuzzy point xαinX and for any (r, s)-FRO set Q-neighborhood γ of f(xα) in IY , there exists a (r, s)-FRO set Q-neighborhood β of xα such that f(β) ≤ γ . Theorem 3.2. Let f be a function FROm an DFTS (X, T1, T ∗ 1 ) into an DFTS (Y, T2, T ∗ 2 ). Then the following two conditions are equivalent W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 11 of 15 (i) f is double fuzzy δ-continuous, (ii) for each double fuzzy point xi in X and each (r, s)-FRO set ρ containing f(xi), there exists a (r, s)-FRO set ϕ containing xi where r ∈ I0, s ∈ I1 such that f(ϕ) ≤ ρ. Proof. (1) ⇒ (2) Let xi be a fuzzy point in (X, T1, T ∗ 1 ) and ρ be a (r, s)-FRO set containing f(xi), where r ∈ I0, s ∈ I1. Since every (r, s)-FRO set is a fuzzy open set, it follows FROm (1) that there exists a double fuzzy open nbd ϕ of xi such that, f(IT1,T ∗ 1 (CT1,T ∗ 1 (ϕ, r, s))) ≤ IT2,T ∗ 2 (CT2,T ∗ 2 (ρ, r, s)) = ρ. Taking IT1,T ∗ 1 (CT1,T ∗ 1 (ϕ1, r, s))) = ϕ (being a (r, s)-FRO set containing xi, we obtain (2). (2) ⇒ (1) Let xi be a double point of X and ρ be a double fuzzy open nbd contain- ing f(xi). Then IT2,T ∗ 2 (CT2,T ∗ 2 (ρ, r, s)) is a (r, s)-FRO set containing f(xi) and ρ ≤ IT2,T ∗ 2 (CT2,T ∗ 2 (ρ, r, s)). By (2), there is a (r, s)-FRO set ϕ containing xi such that f(ϕ) ≤ IT2,T ∗ 2 (CT2,T ∗ 2 (ρ, r, s)) (i.e., since a (r, s)-FRO set is a double fuzzy open set) and IT1,T ∗ 1 (CT1,T ∗ 1 (ϕ, r, s)) = ϕ there is a double fuzzy open set ϕ containing xi such that ,f(IT1,T ∗ 1 (CT1,T ∗ 1 (ϕ, r, s))) ≤ IT2,T ∗ 2 (CT2,T ∗ 2 (ρ, r, s)). Theorem 3.3. For a function f : (X, T1, T ∗ 1 ) −→ (Y, T2, T ∗ 2 ). The following statements are equivalent: (i) f is double fuzzy δ-continuous, (ii) f(δCT1,T ∗ 1 (ρ, r, s)) ≤ δCT2,T ∗ 2 (f(ρ), r, s), for each r ∈ I0, s ∈ I1, and ρ ∈ IX . (iii) δCT2,T ∗ 2 (f(ϕ)−1, r, s) ≤ f−1(δCT1,T ∗ 1 (ϕ, r, s)), for each r ∈ I0, s ∈ I1, and ϕ ∈ IY . (iv) for every (r, s)-fuzzy δ-closed set ϕ in IY , r ∈ I0, s ∈ I1, f−1(ϕ) is (r, s)-fuzzy δ-closed set in IX . (v) for every (r, s)-fuzzy δ-open set ϕ in IY , r ∈ I0, s ∈ I1, f−1(ϕ) is (r, s)-fuzzy δ-open set in IX . Proof. (1) ⇒ (2) Let ρ ∈ IX , xα ∈ δCT ,T ∗(ρ, r, s) and γ be a (r, s)-regular open Q-neighborhood of f(xα). Then there exists a (r, s)-regular open Q-neighborhood β of xα such that f(γ) ≤ β. Since xα ∈ δCT ,T ∗(ρ, r, s), we have βqρ. Then f(β)qf(ρ). Thus γqf(ρ) and hence f(xα) ∈ δCT ,T ∗(f(ρ), r, s). So f(δCT ,T ∗(ρ, r, s)) ≤ δCT ,T ∗(f(ρ), r, s). (2) ⇒ (3) Let ϕ ∈ IY . By using (2), f(δCT1,T ∗ 1 (f−1(ϕ), r, s)) ≤ δCT1,T ∗ 1 (f(f−1(ϕ)), r, s) ≤ δCT2,T ∗ 2 (ϕ, r, s). Hence, δCT1,T ∗ 1 (f−1(ϕ), r, s) ≤ f−1(δCT2,T ∗ 2 (ϕ, r, s)). W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 12 of 15 (3) ⇒ (4) We have ϕ = δCT2,T ∗ 2 (ϕ, r, s). Now by (3), δCT1,T ∗ 1 (f−1(ϕ), r, s) ≤ f−1(δCT2,T ∗ 2 (ϕ, r, s)) = f−1(ϕ). Therefore, δCT1,T ∗ 1 (f−1(ϕ), r, s) = f−1(ϕ). Hence f−1(ϕ) is a (r, s)-fuzzy δ-closed. (4) ⇒ (5) Let ϕ ∈ IY , and ϕ be a (r, s)-fuzzy δ-open. Then, 1 − ϕ is a (r, s)-fuzzy δ-closed in IY . By (4), f−1(1 − ϕ) is a (r, s)-fuzzy δ-closed in IX . Since f−1(1 − ϕ) = 1− f−1(ϕ), f−1(ϕ) is (r, s)-fuzzy δ-open in IX . (5) ⇒ (1) The proof is clear. Theorem 3.4. For a function f : (X, τ1, τ ∗ 1 ) → (Y, τ2, τ ∗ 2 ). The following statements are equivalent: (i) f is double fuzzy δ-continuous function, (ii) f−1(µ) is an (r0, s1)-fuzzy δ-closed set in IX for each µ ∈ IY , r0 ∈ I0, s1 ∈ I1, (iii) f−1(µ) is an (r0, s1)-fuzzy δ-open set in IX for each µ ∈ IY ,r0 ∈ I0, s1 ∈ I1. Proof. (1) ⇒ (2) Let µ be an (r0, s1)-fuzzy δ-closed set in IX for each µ ∈ IY , r0 ∈ I0, s1 ∈ I1. Then µ = δCT1,T ∗ 1 (µ, r0, s1). But, by Theorem 3.3 we have δCT1,T ∗ 1 (f−1(µ), r0, s1) ≤ f−1(δCT1,T ∗ 1 (µ, r0, s1)) = f−1(µ). Hence f−1(µ)= δCT1,T ∗ 1 (µ, r0, s1). So we get, f−1(µ) is an (r0, s1)-fuzzy δ-closed set in IX . (2) ⇒ (3) Trivial by taking the complement of (r0, s1)-fuzzy δ-closed set to be an (r0, s1)-fuzzy δ-open set. (3) ⇒ (1) Let x(α,β) be a fuzzy point in IX and let γ be an (r0, s1)-fuzzy regular-open q- neighborhood of f(x(α,β)) for each γ ∈ I, r0 ∈ I0, s1 ∈ I1. But, γ is an (r0, s1)-fuzzy δ-open set in IY and by hypothesis, f−1(γ) is an (r0, s1)-fuzzy δ-open set in IX . Since x(α,β) q f−1(γ) so we get, f−1(γ) is (r0, s1)- fuzzy δ-neighborhood of x(α,β). Therefore, there exists an (r0, s1)- fuzzy regular-neighborhood of µ of x(α,β)such that µ≤ f−1(γ, r0, s1). Hence,f(µ) ≤ γ. Theorem 3.5. Let f : (X, τx, τx∗) → (Y, τy, τy∗) be a bijection function. Then the follow- ing statements are equivalent: (i) f is double fuzzy δ-continuous function, (ii) δIτx, τx∗(λ, r0, s1)=f(δIτx, τx∗(λ, r0, s1)) for each (r0, s1)-fuzzy set λ in IX ,r0 ∈ I0, s1 ∈ I1. Proof. (1) ⇒ (2) Let λ be an (r0, s1)-fuzzy set in IX ,r0 ∈ I0, s1 ∈ I1. Then f(λ) (r0, s1)-fuzzy set λ in IY . But, f is one- to-one, then f−1(δIτx, τx∗(f(λ), r0, s1) ≤ δIτy, τy∗(f − 1(f(λ)), r0, s1) = δIτx, τx∗(λ, r0, s1). Since f is onto, so W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 13 of 15 δIτx, τx∗(f(λ), r0, s1) =f(f−1(Iτy, τy∗(f(λ), r0, s1))) ≤ f(Iτx, τx∗(λ, r0, s1). (2) ⇒ (1) Let λ be an (r0, s1)-fuzzy set in IY , r0 ∈ I0, s1 ∈ I1. Then f−1(λ) is an (r0, s1)-fuzzy set λ in IY . But by hypothesis, δIτy, τy∗ (λ, r0, s1) = δIτx, τx∗(f(f − 1(λ, r0, s1) ≤ f(δIτy, τy∗(f −1(λ), r0, s1)) ) for each (r0, s1)-fuzzy set λ in IX , r0 ∈ I0, s1 ∈ I1. But f is one -to-one function, then f−1(δIτy, τy∗(λ, r0, s1) ≤ f−1(f(δIτx, τx∗(f −1(λ), r0, s1))) = δIτy, τy∗(f −1(λ), r0, s1) Hence by Theorem 3.3, f is double fuzzy δ-continuous function. Remark 3.6. The concepts of a (r, s)-fuzzy δ-continuous and a (r, s)-fuzzy-continuous are independent to each other for each r ∈ I0, s ∈ I1. Example 3.7. Let X = [0, 1] and f : (X, T1, T ∗ 1 ) → (X, T2, T ∗ 2 ) be the identity function (1)- define ρ1 and β1 as follows: ρ1(0) = 0.3, ρ1(1) = 0.7, β1(0) = 0.8, β1(1) = 0.2, And the two spaces (T1, T ∗ 1 ) and (T2, T ∗ 2 ) are define as follows: T1(ρ) =  1, if ρ ∈ {0, 1}, 1 3 , if ρ = ρ1, 0, otherwise. T ∗ 1 (ρ) =  0, if ρ ∈ {0, 1}, 2 3 , if ρ = ρ1, 1, otherwise. and T2(ρ) =  1, if ρ ∈ {0, 1}, 1 3 , if ρ = ρ1, 8 10 , if ρ = β1, 0, otherwise. T ∗ 2 (ρ) =  0, if ρ ∈ {0, 1} 2 3 , if ρ = ρ1, 2 10 , if ρ = β1, 1, otherwise. Then, f is double fuzzy δ-continuous function but not double fuzzy continuous function. (2)- define ρ1 and β1 as follows: ρ1(0) = 0.3, ρ1(1) = 0.7, β1(0) = 0.5, β1(1) = 0.5, And the two spaces (T1, T ∗ 1 ) and (T2, T ∗ 2 ) are defined as follows: T1(ρ) =  1, if ρ ∈ {0, 1}, 1 3 , if ρ = ρ1, 1 2 , if ρ = β1, 0, otherwise. T ∗ 1 (ρ) =  0, if ρ ∈ {0, 1}, 2 3 , if ρ = ρ1, 1 2 , if ρ = β1, 1, otherwise. W. F. AL-Omeri / Eur. J. Pure Appl. Math, 18 (2) (2025), 4624 14 of 15 and T2(ρ) =  1, if ρ ∈ {0, 1}, 1 3 , if ρ = ρ1, 0, otherwise. T ∗ 2 (ρ) =  0, if ρ ∈ {0, 1} 2 3 , if ρ = ρ1, 1, otherwise. Then, f is double fuzzy continuous function but not double fuzzy δ-continuous function. Theorem 3.8. Let (X, T1, T ∗ 1 ), (Y, T2, T ∗ 2 ) and (Z, T3, T ∗ 3 ) be an DFTSs. For the functions f : (X, T1, T ∗ 1 ) → (Y, T2, T ∗ 2 ) and g : (X, T1, T ∗ 1 ) → (Z, T3, T ∗ 3 ). If f and g are (r, s)-fuzzy δ-continuous functions, then g ◦ f (r, s)-fuzzy δ-continuous. Proof. Straightforward. Corollary 3.9. Let f : (X, T1, T ∗ 1 ) → (Y, T2, T ∗ 2 ) be double fuzzy regular δ-continuous function, then δCT1,T ∗ 1 (f(ϕ)−1, r, s) ≤ f−1(δCT2,T ∗ 2 (ϕ, r, s)), for each T2(ϕ) ≥ r, T ∗ 2 (ϕ) ≤ s, and r ∈ I0, s ∈ I1. 4. conclusion The main purpose of this paper is to introduce a new concept in double fuzzy set theory, namely a (r, s)-fuzzy δ-closed sets and a double fuzzy δ-continuous functions. 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