EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 30-41 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Compactness in Fuzzy Bitopological Spaces Ahlam Ahmed Alharbi1,2, Adem Kilicman2,∗ 1 Department of Mathematics, Faculty of Science, Taibah University, Madinah 42353, Kingdom of Saudi Arabia 2 Department of Mathematics and Statistics, Faculty of Science University Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia Abstract. The main objective of this research is to study some types of generalized closed sets in fuzzy bitopology including (i, j)−gα−cld, (i, j)−gs−cld, (i, j)−gp−cld, and (i, j)−gβ−cld. We then present basic theorems for determining their relationships and explain their properties, such as closure and interior. In addition, there are many interesting counterexamples. The last part of the research focuses on compactness as an application of the types of fuzzy generalized closed sets in fuzzy bitopological spaces and their types and explores the relationships between these concepts, their important theories, and some relevant counterexamples. This approach provides a better characterization of fuzzy compactness and allows for more precise characterization in fuzzy bitopology. The results of this study are new to the domain of fuzzy bitopology. 2020 Mathematics Subject Classifications: 54A40, 57S40, 03B52, 03E72, 47S40 Key Words and Phrases: Fuzzy bitopological spaces (fbts), fuzzy generalized closed sets ((i, j)− g − cld), fuzzy generalized closure operator ((i, j) − g − cl), fuzzy generalized interior operator ((i, j) − g − int), fuzzy generalized continuous ((i, j) − g − conts), fuzzy generalized irresolute ((i, j)− g − irres), and fuzzy generalized compact ((i, j)− g − compact) 1. Introduction In this project, we prioritized our study on fuzzy bitopology, which was derived from a fuzzy topology first introduced in 1965 by Zadeh [23]. Following this, many researchers have applied fundamental ideas on fuzzy settings from a general topology and improved the concept of fuzzy topology. Chang (1968) introduced fuzzy concepts into fuzzy topology [9]. Kandil (1989) introduced fuzzy bitopological spaces [11]. In addition, generalized fuzzy closed sets were established in a fuzzy topology by Balasubramanian and Sundaram in 1997 [7]. Some scholars have presented many important papers on the development types of fuzzy sets; for example, Singal and Prakash presented a study of a fuzzy pre-open set [20]. Balasubramanian developed a theory of fuzzy β open set [6]. Ahmad and Athar ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5027 Email addresses: aasehli@taibahu.edu.sa (A. A. Alharbi), akilic@upm.edu.my (A. Kilicman) https://www.ejpam.com 30 © 2024 EJPAM All rights reserved. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 31 found important results on fuzzy semi open sets [2]. In addition, Hakeem and Latha introduced new results for fuzzy α open set [15]. Furthermore, extensive research has been conducted on the concept of generalized closed sets in fuzzy space [14, 24]. Subsequently, many studies have introduced the use of generalized closed sets in fuzzy topologies, such as El-Shafei [10]. Some studies have applied these to the concept of functions that contribute to enriching this research area too [13, 18, 19]. On the other hand, earlier research on compactness informed our study of this topic [1, 21]. A recent study discussed the properties of compactness, but in another field, as G-metric spaces [12] and fuzzy soft space as [22]. In addition, Jamal et al. studied several properties of compact space using regular open sets [16]. The study explores the concept of generalized closed sets in fuzzy bitopological spaces, a flexible framework for studying topological properties and partial membership. It provides a smooth transition between open and closed sets, offering a more flexible definition of closure than traditional closed sets. Also, delves into their interrelationships and highlights important theories and counterexamples. Moreover, because the generalized closed sets have many applications in a range of topological concepts, such as neighborhoods, which are discussed and explained in detail in reference [4], they were applied to connectedness as in [3], also to functions as in [5], but we aim to apply them to another topological topic, that is compactness. It provides better characterizations of fuzzy openness and fuzzy compactness and allows for more precise characterizations, which are important properties in fuzzy bitopology. Finally, the research is organized as follows. The first section (Introduction) looks at the subject’s background and related studies. In Section 2 (preliminaries), we briefly dis- cuss several important concepts pertinent to our investigation. The concept of generalized closed sets is presented in Section 3 (Types of Fuzzy Generalized Closed Groups in Fuzzy Bitopology Space), important theorems and distinctive properties are discussed, and some interesting counterexamples are introduced. Then, we provide crucial definitions of fuzzy generalized compactness in Section 4 (Types of Fuzzy Generalized Compactness in Fuzzy Bitopological Spaces). In Section 5 (Conclusion), we summarize our results. 2. Preliminaries In the following part, we go over important antecedent notions that are essential to the development of this paper. Definition 1. [17] Suppose the set X is not empty and the I sign represents the unit period [0, 1], then the following defined as: (1) an operator with X domain and I range is known as a fuzzy set E, where E(x) ∈ (0, 1] when x ∈ E, and E(x) = 0 in case x ̸∈ E. (2) a set D is including E indicated via E ⊆ D if E(x) ≤ D(x), whenever x ∈ X (3) E and D combination indicated by E ∨D if (E ∨D)(x) = max{E(x), D(x)} ∀ x ∈ X. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 32 (4) the intersection of E, D indicated by E∧D if (E∧D)(x) = min{E(x), D(x)} ∀ x ∈ X. (5) the completeness of E denoted via Ec such that (E(x))c = 1− E(x), ∀ x ∈ X. The following definitions explain the meaning of fuzzy topology and fuzzy bitopological spaces. Definition 2. [17] A fuzzy topology of X is a class of fuzzy groups δ ∈ I that holds the coming three conditions: 1. 0 and 1 contained in δ, where 0(x) = 0, 1(x) = 1, whenever x ∈ X. 2. For any E,D ∈ δ, E ∧D ∈ δ. 3. For any (Ei∈I) ∈ δ, ∨i∈IEi ∈ δ. The term ”fuzzy topological space,” or ”fts,” refers to the pair (X, δ). The components of δ are named fuzzy open sets. If F c ∈ δ, a fuzzy set F is mean as fuzzy closed. The collection including all fuzzy closed sets in fuzzy topology δ denote by Fδ. Definition 3. [11] A fuzzy bitopological spaces, or fbts for short, (X, δ1, δ2) since X is not empty, δ1, and δ2 are fuzzy topological spaces on X. Over this dissertation X perform fuzzy bitopology (X, δ1, δ2), and Y to (Y, σ1, σ2), where i ̸= j, and i, j ∈ {1, 2}. In the section that follows, the definitions of fuzzy set interiors and closings are covered. Definition 4. [17] Closing and internal of any fuzzy set M of (X, δ) are indicated also defined as follows: cl(M) = ∧ {F :M ≤ F, F c ∈ δ} int(M) = ∨ {O : O ≤M,O ∈ δ}, respectively. The closing, internal, and complements ofM of X are indicated by δi−cl(M), δi−int(M), and M c i , respectively, with regard to fuzzy topology δi. Additionally, we designate the class of all fuzzy δj-closed by the mathematical symbol Fδj . One of the work’s core tenets is the definition of the fuzzy generalized closed set, which as following: Definition 5. [7] Any fuzzy set N of X is termed fuzzy generalized closed when closure N is subset of U , wherever N is subset of U and U is fuzzy open. i.e., N is fuzzy generalized closed if cl(N) ≤ U , wherever N ≤ U , U is fuzzy open. One of the fundamental ideas in this research is continuous and irresolute mapping, in addition to compactness, they are defined as follows: Definition 6. [17] Let (X, δ) and (Y, σ) be an fts and f a function from X to Y . Then f is fuzzy δ−continuous if and only if f−1(V ) ∈ δ, ∀V ∈ σ. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 33 Definition 7. [8] A mapping f : (X, δ) −→ (Y, σ) is said to be fuzzy δ − α − irresolute if f−1(V ) is fuzzy α−open set in X for each fuzzy α−open set V in Y . Definition 8. [9] (1) Any fuzzy topology (X, τ) is named fuzzy compact when every fuzzy open covering X has a limited subcover. (2) Any fuzzy set B of (X, τ) is named a fuzzy compact subset of X when every fuzzy open covering B has a limited subcover. An important property in the study of compactness is the finite intersection property, which was define as: Definition 9. [9] A class {Ai} of fuzzy groups of X is entitled having finite intersection characteristic (in sum,F.I.P ) when all finite subclass {Ai1 , Ai2 , ..., Ain} has a non empty intersection Ai1 ∩Ai2 ∩ ... ∩Ain ̸= ϕ 3. Types of Fuzzy Generalized Closed Classes in Fuzzy Bitopology Space In the following section, we discuss some types of fuzzy generalized closed groups, theorems, and relationships, and examine their closure and interiors in an fbts. Definition 10. Any fuzzy set H of fbts (X, τ1, τ2), where i, j ∈ {0, 1}, i ̸= j is called: (1) fuzzy (i, j)−generalized α−closed (in sum, (i, j) − gα − cld) if τj − αcl(H) ≤ W , wherever H ≤W , W ∈ τi. (2) fuzzy (i, j)−generalized semi−closed (in sum, (i, j) − gs − cld ) if τj − scl(H) ≤ W , wherever H ≤W , W ∈ τi. (3) fuzzy (i, j)−generalized pre−closed (in sum, (i, j) − gp − cld ) if τj − pcl(H) ≤ W , wherever H ≤W , W ∈ τi (4) fuzzy (i, j)−generalized β−closed (in sum, (i, j) − gβ − cld ) if τj − βcl(H) ≤ W , wherever H ≤W , W ∈ τi. (5) the complement of the above sets are called fuzzy (i, j) − gα − open, (i, j) − gs − open, (i, j)− gp− open, and (i, j)− gβ − open. Remark 1. (1) We denote the class for every fuzzy (i, j)− gα−open, (i, j)− gs−open, (i, j) − gp−open and (i, j) − gβ−open (resp, fuzzy (i, j) − gα−cld, (i, j) − gs−cld, (i, j)− gp−cld, and (i, j)− gβ−cld) sets in (X, τi, τj) by Ofgφ (i,j) and Ffgφ (i,j) resp. Also, we gave the names (i, j)−gφ−cld and (i, j)−gφ−open to all fuzzy types of generalized closed and open groups, respectively. (2) In all sections of this research i, j ∈ {0, 1}, i ̸= j Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 34 From the above Definition10 we conclude the following: Proposition 1. Any fuzzy subset E of (X, τ1, τ2) considered fuzzy (i, j) − gφ−open ⇔ F ≤ τj − φ− int(E), wherever F ∈ Fτi, and F ≤ E, where i, j ∈ {0, 1}, i ̸= j. Proof. Assume E is fuzzy (i, j) − gφ−open. Then Ec is (i, j) − gφ−cld, thus the condition relation is hold for Ec. Therefore, by using the complent we find τj − φ− cl(Ec) = (τj − φ− int(E))c ≤ F c which implies F ≤ τj − φ− int(E). Conversely, by using Definition 10 and taking the complement for both sides in condition we find Ec is fuzzy (i, j)− gφ−cld. For that E is fuzzy (i, j)− gφ−open. In the section that follows, we define the terms ”closure” and ”interior” of fuzzy gen- eralized closed sets in fbts field, as well as the key theories, connections between these notions, and their complement. Definition 11. For all fbts (X, τ1, τ2), E ∈ IX , (i, j) − gφ − closure and (i, j) − gφ − interior in regard to E are indicated and defined as shown: (i) (i, j)− gφ− cl(E) = ∧ {F : E ≤ F, F is (i, j)− gφ− cld } (ii) (i, j)− gφ− int(E) = ∨ {O : O ≤ E,O is (i, j)− gφ− open }. Theorem 1. If E is a fuzzy subset of (X, τ1, τ2). Then the coming conditions are met: (1) ( (i, j)− gφ− int(E) )c = (i, j)− gφ− cl(Ec) (2) ( (i, j)− gφ− cl(E) )c = (i, j)− gφ− int(Ec). Proof. It is clear from the complement low and De Morgan theorem. Theorem 2. If (X, τ1, τ2) is fbts. Then the next statements are satisfied: (1) Every fuzzy (i, j)− g − cld is fuzzy (i, j)− gα− cld. (2) Every fuzzy (i, j)− gα− cld is fuzzy (i, j)− gp− cld and fuzzy (i, j)− gs− cld. (3) Every fuzzy (i, j)− gp− cld or fuzzy (i, j)− gs− cld is fuzzy (i, j)− gβ − cld. Proof. It is clear from Definition 10 and the relations between types of fuzzy sets where Remark 2. The following diagram explaining the relations between all types generalized closed sets in (X, τi, τj), i, j ∈ {0, 1}, i ̸= j: Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 35 Figure 1: Explain the relations between (i, j)− gφ−cld sets. The examples that follow demonstrate that the above diagram’s opposite is not typi- cally true. Example 1. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.7, 0.5}, H(a, b) = {0.7, 0.6}, R(a, b) = {0.2, 0.4}, S(a, b) = {0.6, 0.6}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2)−g−cld never fuzzy τ2 − g−cld, since S ≤ H ∈ τ2 and cl2(S) ̸≤ H . The coming example show that fuzzy (1, 2)− gα− cld ⇏ fuzzy (1, 2)− g − cld. Example 2. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.5, 0.4}, H(a, b) = {0.7, 0.5}, R(a, b) = {0.4, 0.3}, S(a, b) = {0.3, 0.4}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2) − gα−cld never fuzzy (1, 2)− g−cld, since S ≤ E ∈ τ1, and cl2(S) = Hc ̸≤ E. In the following example we explain that fuzzy (1, 2)−gs−cld⇏ fuzzy (1, 2)−gα−cld. Example 3. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.7, 0.5}, H(a, b) = {0.5, 0.4}, R(a, b) = {0.4, 0.3}, S(a, b) = {0.5, 0.5}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2) − gs−cld never fuzzy (1, 2)− gα−cld, since S ≤ E ∈ τ1, and α− cl2(S) = Hc ̸≤ E. The next example shows that fuzzy (1, 2)− gp− cld ⇏ fuzzy (1, 2)− gα− cld. Example 4. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.7, 0.5}, H(a, b) = {0.6, 0.8}, R(a, b) = {0.4, 0.3}, S(a, b) = {0.2, 0.4}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2) − gp−cld never fuzzy (1, 2)− gα−cld, since S ≤ E ∈ τ1, and α− cl2(S) = Rc ̸≤ E. The example below indicates that fuzzy (1, 2)− gβ − cld ⇏ fuzzy (1, 2)− gs− cld. Example 5. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.5, 0.7}, H(a, b) = {0.6, 0.5}, R(a, b) = {0.4, 0.3}, S(a, b) = {0.5, 0.5}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2) − gβ−cld never fuzzy (1, 2) − gs−cld, since S ≤ E ∈ τ1, and s − cl2(S) = F (a, b) = {0.6, 0.5} ̸≤ E. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 36 Likewise, the following example shows that fuzzy (1, 2)−gβ−cld⇏ fuzzy (1, 2)−gp− cld. Example 6. Suppose E,H,R, and S fuzzy subsets of X = {a, b} as follows: E(a, b) = {0.5, 0.7}, H(a, b) = {0.4, 0.6}, R(a, b) = {0.3, 0.4}, S(a, b) = {0.5, 0.5}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then we can see that S is fuzzy (1, 2) − gβ−cld never fuzzy (1, 2) − gp−cld, since S ≤ E ∈ τ1, and p − cl2(S) = F (a, b) = {0.6, 0.5} ̸≤ E. Theorem 3. Assume (X, τ1, τ2) is fbts and E is fuzzy τi−open (resp, τi − cld). Then, the statements below are equal: (1) E is fuzzy (i, j)− gφ− cld (resp, fuzzy (i, j)− gφ− open). (2) E is fuzzy τj − φ− cld (resp, fuzzy τj − φ− open). Proof. Suppose E ∈ τi, and fuzzy (i, j)− gφ−cld. Then τj −φ− cl(E) ≤ E, and hence E is fuzzy τj − φ−cld. Conversely, it is obvious in Theorem 2, also from Figure (1). Theorem 4. Let E ∈ τi and be fuzzy (i, j)−gα−cld. Then E∧F is fuzzy (i, j)−gφ−cld, wherever F ∈ Fτj . Proof. As E ∈ τi, and fuzzy (i, j)− gα−cld, then by Theorem 3 E is fuzzy τj −α−cld. After that, E ∧ F is fuzzy τj − α−cld, which implies that it is fuzzy (i, j)− gα−cld. Therefore by Figure (1) we conclude E ∧ F is fuzzy (i, j)− gφ−cld. Corollary 1. Suppose A ∈ Fi, and fuzzy (i, j)− gα− open. Thereafter A ∨ F is fuzzy (i, j)− gφ−open, whenever F ∈ τj . Theorem 5. Finite union of fuzzy (i, j)− gφ−cld of (X, τ1, τ2) is fuzzy (i, j)− gφ−cld. Proof. Assume E, andD are fuzzy (i, j)−gφ−cld in fbts (X, τ1, τ2). Then E∨D is fuzzy (i, j)−gφ−cld. It follows from the fact τj−φ−cl(E∨D) = τj−φ−cl(E)∨τj−φ−cl(D). Corollary 2. If E, and D are fuzzy (i, j)− gφ−open. Thereafter E ∧D is fuzzy (i, j)− gφ−open. Remark 3. The finite intersection of fuzzy (i, j)− gφ−cld in fbts (X, τ1, τ2) is not fuzzy (i, j)− gφ−cld in general. We show that by the following example for the specific type that is fuzzy (i, j)− gα−cld. Suppose E,H,R,D1, and D2 are fuzzy subsets of X = {a, b} as below: E(a, b) = {0.6, 0.6}, H(a, b) = {0.7, 0.8}, R(a, b) = {0.6, 0.7}, D1(a, b) = {0.5, 0.7}, D2(a, b) = {0.8, 0.5}. Assume τ1 = {0, 1, E}, and τ2 = {0, 1, H,R}. Then D1 and D2 are fuzzy (1, 2)− gα− cld, but D1 ∧D2 is not fuzzy (1, 2)− gα− cld. Corollary 3. (1) The finite itersection of fuzzy (i, j)−gφ−open in fbts (X, τ1, τ2) is fuzzy (i, j)− gφ−open. (2) The finite union of fuzzy (i, j)−gφ−open in fbts (X, τ1, τ2) is not fuzzy (i, j)−gφ−open in general. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 37 4. Types of Fuzzy Generalized Compactness in Fuzzy Bitopological Spaces This section introduces the idea of generalized compactness in fuzzy bitoplogy and characterize it in terms of important theorems and some properties. Definition 12. The space X of fbts (X, δ1, δ2) is named fuzzy (i, j)− gφ−compact when all fuzzy (i, j)− gφ−open cover for X has a finite subcover. In addition, A fuzzy subset A of fbts (X, δ1, δ2) is called fuzzy (i, j)− gφ−compact subset of X when all fuzzy (i, j)− gφ−open cover for A has a finite subcover. Example 7. Suppose A(a, b) = {0.5, 0.5} is fuzzy subset of X = {a, b} , and the fuzzy topologies δ1 = {0, 1}, δ2 = {0, 1, A}. Then X is fuzzy (1, 2) − gφ−compact space. Fur- thermore, A is fuzzy (1, 2)− gφ−compact subset of X. Corollary 4. In any fbts (X, δ1, δ2) if δi is a fuzzy indiscrete topology, then (X, δ1, δ2) is fuzzy (i, j)− gφ−compact, and any subset of it is fuzzy (i, j)− gφ−compact. Theorem 6. All fuzzy (i, j)− gφ−cld subset of fuzzy (i, j)− gφ−compact space is (i, j)− gφ−compact. Proof. Assume E is fuzzy (i, j)−gφ−cld, and {Gi : i ∈ I} is fuzzy (i, j)−gφ−open cover for E. Then, Ec is fuzzy (i, j)− gφ−open, and hence {Gi, E c : i ∈ I} is (i, j)− gφ−open cover for X. Then ∃ finite subcover to X, which is {Gij , E c : j = 1, 2, ..., n}, and hence ∃ finite subcover of E, which is {Gij : i ∈ I, j = 1, 2, ..., n}. Therefore, E is fuzzy (i, j)− gφ−compact. Corollary 5. All fuzzy δj−cld subset of fuzzy (i, j)− gφ−compact space is fuzzy (i, j)− gφ−compact too. Theorem 7. If (X, δ1, δ2) is fuzzy (i, j)− gφ−compact space, thus it is fuzzy δj−compact space. Proof. Suppose {Gj : j ∈ I} is an open cover of (X, δj). Then from Figure(1) and Theorm 2, {Gi : i ∈ I} is consider fuzzy (i, j)− gφ−open cover to X, after that {Gi} has finite subcover. Therefore, X is fuzzy δj−compact space. Theorem 8. If (X, δ1, δ2) is fuzzy δi−cld and δj−compact space. After that, it is fuzzy (i, j)− gφ−compact. Proof. Assume {Gi : i ∈ I} is fuzzy (i, j) − gφ−open cover for X. As X is δi−cld, then by Theorem 3 {Gi : i ∈ I} is fuzzy δj−open cover to X, but X is δj−compact, after that ∃ finite subcover. Therefore X is fuzzy (i, j)− gφ−compact space. Theorem 9. In fbts (X, δ1, δ2). The next explanations are true: (1) ∀ fuzzy (i, j)−gβ−compact is fuzzy (i, j)−gp−compact and fuzzy (i, j)−gs−compact. Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 38 (2) ∀ fuzzy (i, j)−gp−compact or fuzzy (i, j)−gs−compact is fuzzy (i, j)−gα−compact. (3) ∀ fuzzy (i, j)− gα− compact is fuzzy δj−compact. Proof. Obviously from Definition 12 and the relations between types of (i, j)−gφ−cld sets in Theorem 2 and Figure (1). The diagram below explains the relationships between all types of fuzzy (i, j)−gφ−compact: Figure 2: Explain the relations between (i, j)− gφ−compact. Remark 4. In general, the opposite of the aforementioned graph is not true, and this is clear from Definition 12 and the relations between (i, j) − gφ−cld sets in Theory 2 and Examples 2 through 1. In addition, we can see that the concepts of the fuzzy (i, j) − gp−compact space and (i, j)− gs−compact space are independent. Theorem 10. If E, D are fuzzy (i, j)− gφ−compact subsets of (X, δ1, δ2). Then E ∧D is fuzzy (i, j)− gφ−compact. Proof. Suppose {Gi : i ∈ I} is fuzzy (i, j)−gφ−open cover of E∧D. Since E∧D ≤ E, and E ∧ D ≤ D, then {Gi : i ∈ I} ≤ {Ui : i ∈ I, such that E ≤ ∪i=1Ui} ∧ {Vi : i ∈ I, such that D ≤ ∪i=1Vi}. Then by Corollary 2, and as E,D are fuzzy (i, j) − gφ−compact, then {Gi} has a finite subcover {Gij : j = 1, 2, ..., n}. Therefore E ∧ D is fuzzy (i, j)− gφ−compact. Definition 13. A mapping f : (X, δ1, δ2) → (Y, σ1, σ2) is named fuzzy (i, j)−generalized φ− continuous (shortly, (i, j) − gφ − conts) when the opposite image of each fuzzy open of (Y, σj) is fuzzy (i, j)− gφ− open of X. By using the complement, we find: Theorem 11. Suppose f : (X, δ1, δ2) → (Y, σ1, σ2). Then f is fuzzy (i, j)− gφ− conts ⇔ ∀ fuzzy closed set V at (Y, σj), f −1(V ) is fuzzy (i, j)− gφ− cld set at X. Theorem 12. The portrait (i, j)−gφ−conts of fuzzy (i, j)−gφ−compact is fuzzy δj−compact. Proof. Suppose f : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j) − gφ − conts, surjective mapping, and (X, δ1, δ2) is fuzzy (i, j)− gφ−compact space. Assume that {Bj : j ∈ I} is Ahlam Ahmed Alharbi, Adem Kilicman / Eur. J. Pure Appl. Math, 17 (1) (2024), 30-41 39 δj−open cover for Y , thus {f−1(Bj) : j ∈ I} is fuzzy (i, j) − gφ−open cover for X, then it has finite subcover for X, and since f is surjective mapping, so ∃ {B1, B2, ..., Bn} finite subcover for Y . Therefore Y is fuzzy δj−compact. Corollary 6. The δj−continuous image of (i, j)− gφ−compact is δj−compact. Definition 14. A mapping f : (X, δ1, δ2) → (Y, σ1, σ2) is named fuzzy (i, j)−generalizedφ− irresolute mapping (shortly, (i, j)− gφ− irres) when the opposite image of all fuzzy (i, j)− gφ− open set of X is fuzzy (i, j)− gφ− open of Y . By using the complement we find: Theorem 13. Suppose f : (X, δ1, δ2) → (Y, σ1, σ2). Then f is fuzzy (i, j)− gφ− irres ⇔ for all fuzzy (i, j)− gφ−cld V at Y, f−1(V ) is fuzzy (i, j)− gφ− cld at X. Theorem 14. If f : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)− gφ− irres mapping, and E is fuzzy (i, j)− gφ−compact set of X. Thus f(E) is fuzzy (i, j)− gφ−compact of Y . Proof. Suppose f : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)− gφ− irres, onto mapping, also {Vi : i ∈ I} is fuzzy (i, j)− gφ−open cover of f(E). As f is onto, then f(E) ≤ f(∪n j=1f −1(Vij )) ≤ ∪n j=1Vij . So, f(E) is fuzzy (i, j)− gφ−compact at Y . Corollary 7. When f : (X, δ1, δ2) → (Y, σ1, σ2) is fuzzy (i, j)−gφ− irres, onto mapping, and X is fuzzy (i, j)− gφ−compact. After that, Y is fuzzy (i, j)− gφ−compact. Theorem 15. If (X, δ1, δ2) is fbts. So X is fuzzy (i, j) − gφ−compact ⇔ ∀ {Fi} of fuzzy (i, j) − gφ−cld sets of X satisfying F.I.P (Definition 9) has itself a non empty intersection. Proof. Suppose (X, δ1, δ2) is fuzzy (i, j) − gφ−compact, {Fi : i ∈ I} is fuzzy (i, j) − gφ−cld sets of X satisfying F.I.P, and ∩{Fi : i ∈ I} = ϕ. Then X = ∪{Fi c : i ∈ I}. let U = (Fi c) is fuzzy (i, j)− gφ−open cover for X. As X is fuzzy (i, j)− gφ−compact, then U most contain finite subcover of X and X = (∩n j=1Fij ) c, which implies ∩n j=1Fij = ϕ. This runs counter to the hypothesis that Fi has F.I.P. Conversely, assume X is not compact and ∩ {Fi : i ∈ I} ≠ ϕ, where {Fi} is collection of fuzzy (i, j)− gφ−cld subsets at X has F.I.P. Then ∃ U = {Gi : i ∈ I} is (i, j)− gφ−open cover for X, that is lacking finite subcover of X, then {X −Gi1 , X −Gi2 , ..., X −Gin} is a class of (i, j) − gφ−cld sets has F.I.P, and hence ∩ {X − Gi} = X − ∪ Gi ̸= ϕ, then X ̸= ∪i=1Gi. The reality that U is a fuzzy (i, j)− gφ−open cover for X is in conflict with this. REFERENCES 40 5. Conclusion and Future Studies In this research, we explore the relationships between different types of generalized closed sets in a new domain, which is a fuzzy bitopological space. In addition, we explored the interconnections between these sets by some counterexamples. After that, we scrutinize the fundamental theorems and distinctive characteristics associated with these concepts. Also, we applied them to fuzzy compactness and studied their theorems, properties, and relationships. Through this in-depth analysis, we contribute to a better comprehension of these key ideas in the context of fuzzy bitopological spaces. This work also opens up new horizons for the future study of these sets in other fields of fuzzy sets, such as regular sets, study them in more than two topologies, or in another domain, such as fuzzy soft spaces. References [1] A. Abu Safiya, A. Fora and M. Warner. Compactness and weakly induced bifuzzy topological spaces. Fuzzy Sets and Systems, pp. 89–96, 1994. [2] B. Ahmad and Athar Kharal. Fuzzy sets, fuzzy s-open and s-closed mappings. Ad- vances in Fuzzy Systems, Article ID 303042, 5 pages, 2009. [3] A. Alharbi and A. Kilicman. Generalized Connectedness in Fuzzy Bitopological Spaces.Applied Mathematics and Information Sciences, pp. 1019–1023, 2023. [4] A. Alharbi and A. Kilicman. Note on Generalized Neighborhoods Structures in Fuzzy Bitopological Spaces. European Journal of Pure and Applied Mathematics, pp. 1980– 1990, 2023. [5] A. Alharbi and A. Kilicman. Generalized Different Types of Mappings in Fuzzy Bitopological Spaces. European Journal of Pure and Applied Mathematics, pp. 2613- 2631, 2023. [6] G. Balasubramanian. Fuzzy β−open sets and fuzzy β−separation axioms. Kyber- netika, vol. 35, pp. 215–223, 1999. [7] G. Balasubramanian and P. Sundaram. On some generalizations of fuzzy continuous functions. Fuzzy Sets and Systems, pp. 93–100, 1997. [8] Y. Beceren and T. Noiri. Some functions defined by semi−open and β−open sets. Chaos, Solitons and Fractals, pp. 1225–1231, 2008. [9] C. Chang. Fuzzy topological spaces. J. Math. Anal Appl, pp. 182–190, 1968. [10] M. El-Shafei. Some applications of generalized closed sets in fuzzy topological spaces. Kyngpook Math, pp. 13–19, 2005. [11] A. Kandil. Biproximities and fuzzy bitopological spaces. Simon Stevin, pp. 45-66, 1989. REFERENCES 41 [12] A. D. Kusumaningati, and M. Jakfar, Properties of Compact Set in G-metric Space. International Journal of Research in Engineering, Science and Management, 6(9), pp. 1–8, 2023. [13] A. Mashhour, A. Allam and A. Zahran. Fuzzy g−continuous and fuzzy g−open map- ping. Bulletin. Assiut University, pp. 93–106, 1985. [14] N. Nakajima. Generalized fuzzy sets. Fuzzy Sets and Systems, pp. 307–314, 1989. [15] Hakeem. Othman and S. Latha. New results of fuzzy alpha-open sets fuzzy alpha- continuous mappings. Int. J. Contemp. Math. Sciences, pp. 1415–1422, 2009. [16] J. Oudetallah, R. Alharbi, and I. M. Batiha. On r−Compactness in Topological and Bitopological Spaces. Axioms, 12(2), pp. 210, 2023. [17] N. Palaniappan. Fuzzy topology. Alpha Science International Ltd, pp. 1–177, 2002. [18] S. Parimala, B. Vijayalakshmi, and V. Chandrasekar. On fuzzy almost generalized b-continuous mappings in Šostak’s sense. In AIP Conference Proceedings, vol. 2177, No.1, pp. 020106, 2019. [19] Sadanand. Potil, A. Madabhavi, S. Sadugol and G. Madagi. On fuzzy gµ−closed map, fuzzy gµ−continuous maps and fuzzy gµ−irresolute mapgings in fuzzy topological spaces. Conference on Mathematics, Statistics and its Application, pp. 214–227, 2010. [20] M. K. Singal and Niti. Prakash. Fuzzy preopen sets and fuzzy preseparation axioms. Fuzzy Sets and Systems, pp. 273–281, 1991. [21] R. Srivastava and M. Srivastava. On compactness in bifuzzy topological spaces. Fuzzy Sets and Systems, pp. 285–292, (2001). [22] I. Taha. Compactness on fuzzy soft r-minimal spaces. International Journal of Fuzzy Logic and Intelligent Systems, 21(3), 251-258, 2021. [23] L. Zadeh. Fuzzy sets, Information and control, pp. 338–353, 1965. [24] A. Zahran and A. Almograbi. Generalized ψρ−operations on fuzzy topological. Jour- nal of Abstract and Applied Analysis, 12 pages, 2011.