EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1980-1990 ISSN 1307-5543 – ejpam.com Published by New York Business Global Note on Generalized Neighborhoods Structures 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, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia Abstract. This article’s main aim is to study the concepts of the generalized neighborhood and generalized quasi-neighborhood in fuzzy bitopological spaces. It also introduces fundamental the- orems for determining the relationships between them. Additionally, some significant examples were examined to demonstrate the significance of the interconnections, some theorems were also introduced to study some main properties of neighborhood structures. Finally, we also studied the concepts of closure, interior, and each of their critical theories and properties by generalized neighborhood systems in fuzzy bitopological spaces. 2020 Mathematics Subject Classifications: 03B52, 03E72, 54A40, 54E55, 57N40, 94D05 Key Words and Phrases: Fuzzy bitopological spaces (fbts), fuzzy generalized closed sets (g − closed), fuzzy closure operator (cl), fuzzy interior operator (int), fuzzy generalized neigborhood (Ngφ), and fuzzy generalized quasi neigborhood (NgφQ) 1. Introduction In this project, we have prioritized our study on fuzzy bitopology, which derived from fuzzy topology that was first introduced in 1965 by the scientist Zadeh [8]. Following this, many researchers applied fundamental ideas on fuzzy settings from general topology and improved the concept of fuzzy topology. Such as Chang, in 1968 introduced some fuzzy concepts in fuzzy topology [4]. In addition, in 1989 Kandil introduced fuzzy bitopological spaces [1]. Also, generalized fuzzy closed groups were established in fuzzy topology in 1997 by Balasubramanian and Sundaram [6]. After that, many scientists applied the notion of a generalized closed set in fuzzy space and in 2005 El-Shafei introduced some applications of it [9]. Also, in 2009 Xuzhu Wang et al presented a book that contains all the basic operations in fuzzy science [12]. As Zahran and El-Maghrabi studied in 2011 some operations on it in fuzzy space [13]. Then in 2017 Benchalli et al studied delta ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4808 Email addresses: aasehli@taibahu.edu.sa (Ahlam Ahmed Alharbi), akilic@upm.edu.my (Adem Kilicman) https://www.ejpam.com 1980 © 2023 EJPAM All rights reserved. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1981 generalized beta closed in topological spaces [3]. After one year, Kandil et al defined the concept of locally pairwise closed sets and studied some of their properties [7]. In 2019 Ramaboopathi and Dharmalingam introduced a new class of generalized closed sets in bitopological spaces [11], as in the same year Andal and Thiripurasundari introduced a new concept of fuzzy generalized pi closed in fuzzy bitopological spaces [2]. Finally, in 2021 Das et al introduced the idea of γ generalized fuzzy quasi neighborhood of a fuzzy point [5]. 2. Preliminaries In the following part, we go over important antecedent notions that are essential to the development of this paper. Definition 1. [10] 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. (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 as (E(x))c = 1− E(x), ∀ x ∈ X. The following definitions explain the meaning of fuzzy topology and fuzzy bitopological spaces. Definition 2. [10] A fuzzy topology of X is a class of fuzzy groups δ ∈ I which 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, then F is mean as fuzzy closed. The collection including whole fuzzy closed groups in fuzzy topology δ denote by Fδ. Definition 3. [1] 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 which follows, the definitions of fuzzy set interiors and closings are covered. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1982 Definition 4. [10] 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 of M 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. [6] Any fuzzy group E of X is termed fuzzy generalised closed when closure E is subset of W , wherever E is subset of W , W is fuzzy open. i.e., E is fuzzy generalised closed in case of cl(E) ≤ W , wherever E ≤ W , W is fuzzy open. Definition 6. [10] (1) A fuzzy point xλ is claimed that quasi-coincident with E, shown by xλ q E if λ > Ec(x), or λ+E(x) > 1 and xλ is claimed does not quasi-coincident with E if λ+E(x) ≤ 1 and we write E q xλ. (2) E is claimed quasi-coincident with C indicated as E q C if there exists x ∈ X so that E(x) > Cc(x) or E(x) +C(x) > 1, and E is claimed does not quasi-coincident with C if there exists x ∈ X so that E(x) + C(x) ≤ 1 and we write E q C. If E q C (resp, E q C) is true, then E and C are quasi-coincident (resp, not quasi- coincident)with each other at x. 3. Generalized Neighborhoods Structures at Fuzzy Bitopological Spaces This section introduces the idea of generalized neighborhoods concepts by using (∈) relationship and quasi coincident concept (q) in fuzzy bitopological spaces and characterize it in terms of important theorems and some properties. Definition 7. A fuzzy subgroup E of fbts (X, δ1, δ2) is known as: (1) Fuzzy (i, j)−generalized φ−closed (in sum, (i, j)− gφ− closed) if δj −φ− cl(E) ≤ U where E ≤ U , U ∈ δi, and φ including the types (alpha (α), semi (s), pre (p), and beta (β)). (2) The supplement of the fuzzy (i, j) − gφ − closed set is referred to (i, j) − gφ − open set in X. Remark 1. (1) The universal set of all fuzzy (i, j) − gφ−open, and (i, j) − gφ−closed sets of fbts (X, δ1, δ2) is represented by Ofgφ (i,j), F fgφ (i,j), and so forth. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1983 (2) Also, the family of all gφ−open,and gφ−closed subsets of X pertaining to the fuzzy topology δi is indicated Ofgφ i , and Ffgφ i , i = 1, 2. Proposition 1. A fuzzy group E in fbts (X, δ1, δ2) is fuzzy (i, j) − gφ-open ⇐⇒ F ≤ δj − φ− int(E) wherever F c ∈ δi, F ≤ E. Proof. Assume E is fuzzy (i, j) − gφ−open, F c ∈ δi, when F ≤ E. Then Ec ≤ F c. As Ec is fuzzy (i, j) − gφ−closed, thus δj − φ − cl(Ec) = (δj − φ − int(E))c ≤ F c that indicates F ≤ δj − φ− int(E). Conversely, assume E is fuzzy set of X, F ∈ Fi so F ≤ δj − φ − int(E), F ≤ E. After adding the supplement to both sides, we find (δj − φ − int(E))c ≤ F c so Ec ≤ F c and F c is fuzzy open in δi, thus E c is fuzzy (i, j)− gφ−closed (Defention7). Hence E is fuzzy (i, j)− gφ−open. Definition 8. A fuzzy group E in fbts (X, δ1, δ2) is known as: (1) Fuzzy (i, j)−generalizedφ−neighborhood (shortly, (i, j)−gφ−nbd) of fuzzy singleton set xr if ∃ fuzzy (i, j)− gφ− open set C so xr ∈ C ≤ E. The family of all fuzzy (i, j)− gφ−nbds of fuzzy singleton set xr, will be denoted by Ngφ (i,j)(xr). (2) Fuzzy (i, j)− generalizedφ−Q− neighborhood (shortly, (i, j)− gφQ− nbd) of fuzzy singleton xr if ∃ fuzzy (i, j)− gφ− open set C so xr q C ≤ E. The family of all fuzzy (i, j)− gφQ− nbds of fuzzy singleton xr, will be denoted by NgφQ (i,j) (xr). Remark 2. In general, every fuzzy δ−Q−neighborhood of a fuzzy point does not include the point itself. The coming example show that: Example 1. Assume x0.7 is fuzzy point of X = {a, b, c} and E is fuzzy set of X defined as E(a) = 0.4, E(b) = 0.5, E(c) = 0.3. Let δ = {0, 1, E} on X. Then E ∈ NQ δ (x0.7) but 0.7 ≰ E(x), and hence x0.7 ̸∈ E but x1−0.7=0.3 ∈ E. Corollary 1. In fbts (X, δ1, δ2) every fuzzy (i, j) − gφQ − nbd of fuzzy point xr of X is equivlant to (i, j)− gφ− nbd of fuzzy point x1−r. Theorem 1. (1) Every fuzzy δj − nbd of fuzzy point xr is fuzzy (i, j)− g − nbd of xr. (2) Every fuzzy (i, j)− g − nbd of fuzzy point xr is fuzzy (i, j)− gα− nbd of xr. (3) Every fuzzy (i, j)− gα−nbd of xr is fuzzy (i, j)− gs−nbd and (i, j)− gp−nbd of xr. (4) Every fuzzy (i, j)− gs− nb or (i, j)− gp− nbd of xr is fuzzy (i, j)− gβ − nbd of xr. Proof. (1) Suppose that E ∈ Nj(xr), thus ∃C ∈ δj , so xr ∈ C ≤ E. As every δj − open set is (i, j)−g−open set, then ∃C is fuzzy (i, j)−g−open set, so xr ∈ C ≤ E, and hence E ∈ Ng (i,j)(xr). A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1984 (2) Suppose that E ∈ Ng (i,j)(xr), thus ∃C is fuzzy (i, j) − g − open set, so xr ∈ C ≤ E, and since every fuzzy (i, j)− g − open is fuzzy (i, j)− gα− open, then ∃C is fuzzy (i, j)− gα− open set, so xr ∈ C ≤ E, and hence E ∈ Ngα (i,j)(xr). (3) Suppose that E ∈ Ngα (i,j)(xr), thus ∃C is fuzzy (i, j)− gα − open set, so xr ∈ C ≤ E, and since every fuzzy (i, j)−gα−open is fuzzy (i, j)−gs−open and (i, j)−gp−open, then ∃C is fuzzy (i, j) − gs − open and (i, j) − gp − open set, so xr ∈ C ≤ E, and hence E ∈ Ngs (i,j)(xr), and E ∈ Ngp (i,j)(xr). (4) Suppose that E ∈ Ngs (i,j)(xr), or E ∈ Ngp (i,j)(xr), thus ∃C is fuzzy (i, j)− gs− open or (i, j) − gp − open set, so xr ∈ C ≤ E, and since every fuzzy (i, j) − gs − open or (i, j)− gp− open is fuzzy (i, j)− gβ − open, then ∃C is fuzzy (i, j)− gβ − open set, so xr ∈ C ≤ E, and hence E ∈ Ngβ (i,j)(xr). Remark 3. In fbts (X, δ1, δ2) every fuzzy Ngs (i,j)(xr), and Ngp (i,j)(xr) are independents. The following example show that if X = {a, b, c}, δ1 = {0, 1, E}, and δ2 = {0, 1, C,D}. As Ea,b,c = {0.7, 0.5, 0.6}, Ca,b,c = {0.5, 0.4, 0.3}, and Da,b,c = {0.4, 0.3, 0.2}, then ∃ Sa,b,c = {0.5, 0.5, 0.6} ∈ Ngs (i,j)(xr), but S /∈ Ngp (i,j)(xr), as Ec ≤ S, but Ec ≰ δ2 − p − int(S) = C. On other hand for the same topologies above if Ea,b,c = {0.3, 0.5, 0.4}, Ca,b,c = {0.6, 0.8, 0.9}, and Da,b,c = {0.4, 0.3, 0.2}, then ∃ Sa,b,c = {0.8, 0.6, 0.5} ∈ Ngp (i,j)(xr), but S /∈ Ngs (i,j)(xr), as Ec ≤ S, but Ec ≰ δ2 − s− int(S) = Cc. The following Figure explaining the relation between nbds structures of all cases. Figure 1: Explain the relations between all types of fuzzy Ngφ (i,j)(xr), and all types of NgφQ (i,j) (xr). The reversal of the prior relationships in Figure (1) is wrong in fbts (X, δ1, δ2), as demonstrated by the instances that follow: Suppose X = {a, b, c}, δ1 = {0, 1, E}, and δ2 = {0, 1, H,R}. Example 2. If Ea,b,c = {0.7, 0.5, 0.4}, Ha,b,c = {0.7, 0.6, 0.5}, Ra,b,c = {0.2, 0.4, 0.3}, Sa,b,c = {0.6, 0.6, 0.5}. The conclusion is S ∈ Ng (1,2)(xr), but never S /∈ N2(xr). The next example clear that Ngα (1,2)(xr) ⇏ Ng (1,2)(xr). Example 3. Suppose Ea,b,c = {0.5, 0.4, 0.3}, Ha,b,c = {0.7, 0.5, 0.4}, Ra,b,c = {0.4, 0.3, 0.2}, Sa,b,c = {0.7, 0.6, 0.8}. The conclusion is S ∈ Ngα (1,2)(xr), but never S /∈ Ng (1,2)(xr). A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1985 In the coming example we show that Ngs (1,2)(xr) ⇏ Ngα (1,2)(xr). Example 4. Suppose Ea,b,c = {0.7, 0.5, 0.5}, Ha,b,c = {0.5, 0.4, 0.3}, Ra,b,c = {0.4, 0.3, 0.2}, Sa,b,c = {0.5, 0.5, 0.5}. The conclusion is S ∈ Ngs (1,2)(xr), but never S /∈ Ngα (1,2)(xr). The following example clear that Ngp (1,2)(xr) ⇏ Ngα (1,2)(xr). Example 5. Suppose Ea,b,c = {0.7, 0.5, 0.4}, Ha,b,c = {0.6, 0.8, 0.8}, Ra,b,c = {0.4, 0.3, 0.2}, Sa,b,c = {0.8, 0.6, 0.7}. The conclusion is S ∈ Ngp (1,2)(xr), but never S /∈ Ngα (1,2)(xr). The example follow indicates that Ngβ (1,2) ⇏ Ngs (1,2)(xr). Example 6. Suppose Ea,b,c = {0.5, 0.7, 0.6}, Ha,b,c = {0.6, 0.5, 0.4}, Ra,b,c = {0.4, 0.3, 0.2}, Sa,b,c = {0.5, 0.5, 0.6}. The conclusion is S ∈ Ngβ (1,2)(xr), but never S /∈ Ngs (1,2)(xr). As well, the coming example demonstrates that Ngβ (1,2) ⇏ Ngp (1,2)(xr). Example 7. Suppose Ea,b,c = {0.5, 0.7, 0.6}, Ha,b,c = {0.4, 0.6, 0.7}, Ra,b,c = {0.3, 0.4, 0.5}, Sa,b,c = {0.5, 0.5, 0.6}. The conclusion is S ∈ Ngβ (1,2)(xr), but never S /∈ Ngp (1,2)(xr). Definition 9. A fuzzy singleton set xr is named fuzzy (i, j) − generalizedφ − cluster point of fuzzy subset E in fbts (X, δ1, δ2) if and only if all H ∈ NgφQ (i,j) of xr, H q E. The following theorem examines some of the generalized neighborhood characteristics. Theorem 2. If (X, δ1, δ2) is fbts. Next, we find: (1) ∀xr ∈ X,Ngφ (i,j)(xr) ̸= ϕ. (2) ∀H ∈ Ngφ (i,j)(xr),xr ∈ H. (3) when H,R ∈ Ngφ (i,j)(xr), then H ∧R ∈ Ngφ (i,j)(xr). (4) when H ∈ Ngφ (i,j)(xr) and H ≤ R, then R ∈ Ngφ (i,j)(xr). (5) when H ∈ Ngφ (i,j)(xr), then there exists R ∈ Ngφ (i,j)(xr) such that R ≤ H and R ∈ Ngφ (i,j)(xh), ∀xh ∈ R. Proof. From Definition 8 we conclude the prove of (1) and (2). (3) Assume H,R ∈ Ngφ (i,j)(xr). Thus ∃S, T are fuzzy (i, j)− gφ−open, so xr ∈ S, xr ∈ T , then xr ∈ S ∧ T . As S, T are fuzzy (i, j)− gφ−open, and hence we find S ∧ T is fuzzy (i, j)− gφ−open, S ∧ T ≤ H ∧R. As a result of that, A ∧B ∈ Ngφ (i,j)(xr). A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1986 (4) Assume H ∈ Ngφ (i,j)(xr). So ∃S is fuzzy (i, j)− gφ−open and xr ∈ S ≤ H but H ≤ R, then xr ∈ S ≤ R. As a result of that, R ∈ Ngφ (i,j)(xr). (5) Assume H ∈ Ngφ (i,j)(xr), so there exists fuzzy (i, j)− gφ−open set R, xr ∈ R ≤ H. As R is fuzzy (i, j) − gφ−open, then R ∈ Ngφ (i,j)(xr). Since it is an (i, j) − gφ − nbd of each of it is point. As a result of that, B ∈ Ngφ (i,j)(xh),∀xh ∈ B. The following theorem examines some of the generalized Q-neighborhood properties. Theorem 3. When (X, δ1, δ2) is fbts. Then we have: (1) ∀xr q or ∈ X,NgφQ (i,j) (xr) ̸= ϕ. (2) ∀E ∈ NgφQ (i,j) (xr),xr q E. (3) when E, T ∈ NgφQ (i,j) (xr), then E ∧ T ∈ NgφQ (i,j) (xr). (4) when E ∈ NgφQ (i,j) (xr), and E ≤ T , then T ∈ NgφQ (i,j) (xr). (5) when E ∈ NgφQ (i,j) (xr), then ∃ T ∈ NgφQ (i,j) (xr) so T ≤ E, and T ∈ NgφQ (i,j) (xh) ∀xh ∈ T. Proof. It resembles the earlier Theorem 2 proof. Using the above-mentioned novel notion of fuzzy neighbourhood and quasi-neighborhood structure, we introduced the study of the degree of affiliation of a fuzzy element to fuzzy generalised closure in the subsequent theorem. Theorem 4. If E is fuzzy set and xr is fuzzy point of fbts (X, δ1, δ2), then the following propositions are correct: (1) xr ∈ (i, j)− gφ− cl(E) ⇐⇒ ∀T ∈ NgφQ (i,j) (xr), T q E. (2) xr ∈ (i, j)− gφ− cl(E) ⇐⇒ ∀T ∈ Ngφ (i,j)(x1−r), T q E. (3) If E is fuzzy (i, j)− gφ−closed, and hence δi − cl(xr) q E holds ∀xr q δj − φ− cl(E). Proof. (1) Let xr ∈ (i, j)− gφ− cl(E) ⇔ ∀F is fuzzy (i, j)− gφ− closed,E ≤ F, r ≤ F (x) ⇔ ∀F c is fuzzy (i, j)− gφ− open, F c ≤ Ec, F c(x) ≤ 1− r ⇔ ∀T is fuzzy (i, j)− gφ− open, T ≤ Ec, T (x) ≤ 1− r ⇔ ∀T is fuzzy (i, j)− gφ− open, 1− r < T (x) ⇒ T ̸≤ Ec ⇔ ∀T is fuzzy (i, j)− gφ− open, xr q T, T q E ⇔ ∀T ∈ NgφQ (i,j) (xr), T q E. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1987 (2) From (1) we have xr ∈ (i, j) − gφ − cl(E) ⇔ ∀T ∈ NgφQ (i,j) (xr), T q E. So, we need to show that T ∈ Ngφ (i,j)(x1−r) ⇔ T ∈ NgφQ (i,j) (xr). Assume T ∈ Ngφ (i,j)(x1−r). After that, ∃ fuzzy (i, j) − gφ − open set V so x1−r ∈ V ≤ T , then xr q V ≤ T . As a result of that, T ∈ NgφQ (i,j) (xr). In the opposite direction, assume T ∈ NgφQ (i,j) (xr). Thus ∃ fuzzy (i, j)− gφ− open set V so xr q V ≤ T , and hence x1−r ∈ V ≤ T . As a result of that, T ∈ Ngφ (i,j)(x1−r). (3) Assume E be fuzzy (i, j)− gφ−closed. Suppose there ∃ fuzzy point xr so xr q δj −φ− cl(E), but δi− cl(xr) q E. Thus E ≤ (δi − cl(xr)) c. As E is fuzzy (i, j)− gφ−closed, thus δj −φ− cl(E) ≤ (δi − cl(xr)) c, hence δj −φ− cl(E) q δi− cl(xr). As xr ∈ δi− cl(xr), thus xr q δj − φ − cl(E) that is a contradiction. As a result of that, δi − cl(xr) q E holds ∀xr q δj − φ− cl(E). Corollary 2. If E is fuzzy (i, j)−gφ−closed, and xr is fuzzy point in fbts (X, δ1, δ2), then δi − cl(xr) q E holds ∀xr q δj − β − cl(E). In the theory that follows, we studied the most fundamental generalized closure char- acteristics and demonstrated them using new neighborhood structure notions. Theorem 5. If E, and T are fuzzy subsets of fbts (X, δ1, δ2), thus the following arguments are correct: (1) 0, and 1 are fuzzy (i, j)− gφ− closed. (2) when E ≤ T , then (i, j)− gφ− cl(E) ≤ (i, j)− gφ− cl(T ). (3) E ≤ (i, j)− gφ− cl(E), ∀ fuzzy set E ∈ IX . (4) when E is fuzzy (i, j)−gφ−closed, then (i, j)−gφ−cl(E) = E. The converse is false, as the intersection of fuzzy (i, j)−gφ−closed sets need not be fuzzy (i, j)−gφ−closed. (5) (i, j)− gφ− cl((i, j)− gφ− cl(E)) = (i, j)− gφ− cl(E). (6) when v is (i, j)− gφ− open, then v q E ⇐⇒ v q (i, j)− gφ− cl(E). (7) (i, j)− gφ− cl(E) ∨ (i, j)− gφ− cl(T ) ≤ (i, j)− gφ− cl(E ∨ T ). Proof. By using Definition 7 and Theorem4 we can easily proved (1), (2), (3), and (4). (5) Assume xr is fuzzy point with xr ̸∈ (i, j) − gφ − cl(E). After that, ∃V ∈ NgφQ (i,j) (xr) so xr q V , V q E, then ∃U is fuzzy (i, j)− gφ− open so xr q U ≤ V and U q E. Thus from (6)U q (i, j)− gφ− cl(E). As ∃U is fuzzy (i, j)− gφ− open so xr q U and U q (i, j)− gφ− cl(E). Then xr ̸∈ (i, j)− gφ− cl((i, j)− gφ− cl(E)), after that (i, j)− gφ− cl((i, j)− gφ− cl(E)) ≤ (i, j)− gφ− cl(E). But (i, j)− gφ− cl(E) ≤ (i, j)− gφ− cl((i, j)− gφ− cl(E)). As a result of that, (i, j)− gφ− cl(E) = (i, j)− gφ− cl((i, j)− gφ− cl(E)). A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1988 (6) Sufficiency, assume V q E. After that, E ≤ V c, V c is fuzzy (i, j)−gφ−closed, then by applying (i, j)−gφ−clouser for all sides and from (5) we find V q (i, j)−gφ−cl(E). As a result of that, V q E ⇐⇒ V q (i, j)− gφ− cl(E). (7) As E ≤ (E ∨ T ), and T ≤ (E ∨ T ), then (i, j)− gφ− cl(E) ∨ (i, j)− gφ− cl(T ) ≤ (i, j)− gφ− cl(E ∨ T ). From the relationship between closure, interior, complement, and Theorem5 we con- clude the following: Theorem 6. If E and T are fuzzy subsets of fbts (X, δ1, δ2), then the coming statements are correct: (1) 0, and 1 are fuzzy (i, j)− gφ− open. (2) when E ≤ T , then (i, j)− gφ− int(E) ≤ (i, j)− gφ− int(T ). (3) (i, j)− gφ− int(E) ≤ E, ∀ fuzzy set E ∈ IX . (4) when E is fuzzy (i, j) − gφ − open, then (i, j) − gφ − int(E) = E. The converse is false, as the combination of fuzzy (i, j) − gφ − open sets not necessary to be fuzzy (i, j)− gφ− open. (5) (i, j)− gφ− int((i, j)− gφ− int(E)) = (i, j)− gφ− int(E). (6) when v is (i, j)− gφ− closed, then v q E ⇐⇒ v q (i, j)− gφ− int(E). (7) (i, j)− gφ− int(E ∧ T ) ≤ (i, j)− gφ− int(E) ∧ (i, j)− gφ− int(T ). Theorem 7. If xr is fuzzy point, and E is fuzzy subset of fbts (X, δ1, δ2), then xr ∈ (i, j)− gφ− int(E) ⇐⇒ ∃ fuzzy (i, j)− gφ− open set G, so xr ∈ G ≤ E. Theorem 8. Suppose E is fuzzy set in fbts (X, δ1, δ2). If E is fuzzy (i, j) − gφ − open, then E ∈ Ngφ (i,j)(xr) for each xr ∈ E. Theorem 9. If (X, δ1, δ2) is fbts, E is fuzzy (i, j)−gφ−closed, and E ≤ T ≤ δj−φ−cl(E), then T is fuzzy (i, j)− gφ− closed. Proof. Assume T ≤ U , and U is fuzzy open of δi. As E ≤ T , thus E ≤ U , after that δj − φ− cl(E) = δj − φ− cl(T ), which implies δj − φ− cl(T ) ≤ U . As a result of that, T is fuzzy (i, j)− gφ− closed. From the above we conclude the following: Corollary 3. Assume (X, δ1, δ2) is fbts, E is fuzzy (i, j)− gφ− closed, and E ≤ T ≤ δj − β − cl(E). Then T is fuzzy (i, j)− gφ− closed. Corollary 4. Assume (X, δ1, δ2) is fbts, E is fuzzy (i, j)− gφ− open, and δj − φ− int(E) ≤ T ≤ E. Then T is fuzzy (i, j)− gφ− open. A. A. Alharbi, A. Kilicman / Eur. J. Pure Appl. Math, 16 (3) (2023), 1980-1990 1989 Corollary 5. Assume (X, δ1, δ2) is fbts, E is fuzzy (i, j)− gφ− open, and δj − β − int(E) ≤ T ≤ E. Then T is fuzzy (i, j)− gφ− open. The following important study demonstrates when equivalence between the types of generalized closed sets in fuzzy bitopology and types of fuzzy sets from one topology is attained. Theorem 10. In fbts (X, δ1, δ2) the following statements are equivalents: (i) δi ⊆ Ffφ (X,δj) (ii) All fuzzy groups of X are fuzzy (i, j)− gφ− closed. Proof. (i) → (ii) Assume E is fuzzy subset of X, so E ≤ U ∈ δi. Then from (i) we find U ∈ Ffφ (X,δj) , after that δj − φ− cl(E) ≤ U . As a result of that, E is fuzzy (i, j)− gφ− closed. (ii) → (i) Let E be fuzzy (i, j)−gφ−closed, E ∈ δi. Since E ≤ E, then δj−φ−cl(E) ≤ E, thus E is fuzzy δj − φ− closed. Therefore δi ⊆ Ffφ (X,δj) . From the above and the complenent relation we conclude the following: Corollary 6. In fbts (X, δ1, δ2) the following statements are equivalents: (1) Fδi ⊆ Ofφ (X,δj) (2) All fuzzy subset of X is fuzzy (i, j)− gφ− open. Corollary 7. Assume E, and T are fuzzy (i, j)− gφ− closed sets in fbts (X, δ1, δ2) with E ∨ δi − int(T ) = T ∨ δi − int(E) = 1, then E ∧ T is fuzzy (i, j)− gφ− closed. 4. Conclusion In this study, we introduced and studied the definition of some types of generalized neighborhood and generalized quasi-neighborhood ideas fuzzy bitopology space, and we prove some relations and inclusion relation between them by listing some examples, then applied them to, closure, interior, and studied some key properties of them. Acknowledgements The authors would like to thank the referee(s) for their valuable comments. REFERENCES 1990 References [1] A. Kandil, Biproximities and fuzzy bitopological spaces. Simon Stevin, (1989), pp. 45-66. [2] Andal, M., and Thiripurasundari, V. Fuzzy Generalized π Closed Set in Fuzzy Topo- logical Spaces. Journal of Information and Computational Science, (2019), pp. 1548– 7741. [3] Benchalli, S. S., Patil, P. G., Toranagatti, J. B., and Vighneshi, S. R. A New Class of Generalized Closed Sets in Topological Spaces. Global Journal of Pure and Applied Mathematics, (2017), pp. 331–345. [4] C. Chang, Fuzzy topological spaces. J. Math. Anal Appl, (1968), pp. 182–190. [5] Das, B., Bhattacharya, B., Chakraborty, J., and Tripathy, B. C. Generalized fuzzy closed sets in a fuzzy bitopological space via γ-open sets. Afrika Matematika, (2021), pp. 333–345. [6] G. Balasubramanian and P. Sundaram, On some generalizations of fuzzy continuous functions. Fuzzy Sets and Systems, (1997), pp. 93–100. [7] Kandil, A., Tantawy, O., El-Sheikh, S., and Shalaby, E. Generalized Locally Pair- wise Closed Sets on Bitopological Spaces and Some of Its Properties. Journal of the Egyptian Mathematical Society, (2018), 116–126. [8] L. Zadeh, Fuzzy sets, Information and Control, (1965), pp. 338–353. [9] M. El-Shafei, Some applications of generalized closed sets in fuzzy topological space. Kyngpook Math, (2005), pp. 13–19. [10] N. Palaniappan, Fuzzy topology. Alpha Science International Ltd, (2002), pp. 1–177. [11] Ramaboopathi, M., and Dharmalingam, K. M. On (1, 2)∗-ˇg-closed sets in bitopo- logical spaces. Malaya Journal of Matematik, (2019), pp. 463–467. [12] Xuzhu Wang, Da Ruan, and Etienn E. Kerre. Mathematics of Fuzziness-Basic Issues. Springer Nature, (2009). [13] Zahran, A. M and El-Maghrabi, A. I. Generalized-Operations on Fuzzy Topological Spaces. Abstract and Applied Analysis, Hindawi, (2011), pp. 1–12.