EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5856 ISSN 1307-5543 – ejpam.com Published by New York Business Global Novel Categories of Spaces in the Frame of Generalized Fuzzy Topologies via Fuzzy gµ-Closed Sets Salem Saleh1,2, Fathea M. Osman Birkea3∗, Tareq M. Al-shami4,5, Murad Arar6, M. Omran7 1 Department of Mathematics, Hodeidah University, Hodeidah, Yemen 2 Department of Computer Science, Cihan University-Erbil, Erbil, Iraq 3 Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia 4 Department of Mathematics, Sana’a University, P.O.Box 1247 Sana’a, Yemen 5 Jadara University Research Center, Jadara University, Irbid, Jordan 6 Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia 7 Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Egypt Abstract. One of the known approaches to studying topological concepts is to utilize subclasses of topology, such as clopen sets and generalized closed sets. In this study, we apply the notion of fuzzy generalized µ-closed sets (Fgµ-closed sets) to establish and analyze novel categories of spaces, namely Fgµ-regular, Fgµ-normal, and Fµ-symmetric spaces in the frame of generalized fuzzy topology (GFT ). We investigate the fundamental properties of these classes, exploring their unique characteristics and preservation theorems under Fgµ-continuous maps. We estab- lish the interrelationships between these classes and the other separation axioms in this setting, and we demonstrate that Fµ-regular, Fµ-normal, and Fµ-symmetric spaces are special cases of Fgµ-regular, Fgµ-normal, and Fµ-T1 spaces, respectively. Additionally, we show that the equiv- alence for these cases hold when the GFT is Fµ-T 1 2 . The connections between these classes and their counterparts in the crisp GT are studied. Finally, we discuss these classes’ hereditary and topological properties, further enhancing our comprehension of their behavior and implications. 2020 Mathematics Subject Classifications: 54A40, 54C08, 54D10, 54D15 Key Words and Phrases: Fuzzy µ-closed set; fuzzy gµ-closed set; generalized fuzzy topology; fuzzy gµ-continuous map; fuzzy gµ-regular; fuzzy gµ-normal space ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5856 Email addresses: s wosabi@yahoo.com (S. Saleh), fathia.birkia@nbu.edu.sa (F.M.O. Birkea), tareqalshami83@gmail.com (T.M. Al-shami), muradshhada@gmail.com (M. Arar), Manar.Omran@f-eng.tanta.edu.eg (M. Omran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 2 of 15 1. Introduction Fuzzy sets (F -sets) were proposed by Zadeh [49] in 1965 as a suitable approach to address with uncertainty cases that we cannot be efficiently managed using classical tech- niques. Over the last decades, the researches of F -sets have a vital role in mathematics and applied sciences and garnered significant attention due to its ability to handle uncer- tain and vague information in various real-life applications such as artificial intelligence [47, 50], control systems [8, 28], decision-making [17, 23], image processing [1, 46], clas- sifications [22, 24], etc. Chang [16], in 1968, defined the fuzzy topology (FT ), allowing the study of topological properties within the frame of F -sets. This development has led to the expansion and investigation of many classical topological notions in the context of FT [2, 7, 9, 10, 42], providing more accurate and flexible models to address problems of uncertainty in various real life ears. Moreover, the hybridization of fuzzy topology with soft topology was introduced and studied by several authors [41, 43, 44]. Generalized closed sets, abbreviated as g-closed sets, is a fundamental notion in both topology and FT . It was proposed in general topology by Levine [29] in 1970. This notion has undergone extensive study in the fields of topology and FT by numerous authors, as in [15, 19, 30, 33, 36, 40, 48]. Since then, it has been widely used as a powerful tool to explore various concepts, including g-regular and g-normal spaces, which have been further generalized and investigated as in [11, 21, 25, 34, 35, 38], and others. These studies have also led to the introduction of new separation axioms that are weaker than T1. In the fuzzy context, Balasubramanian et al [12] proposed the notion of generalized fuzzy closed sets in 1997, sparking further research by authors like Saraf et al. [45] and Park et al. [37] who extensively studied different forms of generalized fuzzy closed sets. On the other hand, Császár [20] introduced the concept of generalized topology (or GT ), expanding the scope of general topology. Over time, many researchers have endeav- ored to extend the notion of g-closed sets to the broader framework of GT . Maragathavalli et al [16] notably explored g-closed sets and their fundamental properties within GTS. Prior to that, Chetty [18] extended the concept of GT into a fuzzy environment, leading to the development of GFT . Mandal et al [31] defined the notion of Fgµ-closed sets in GFTS and study the concepts of Fµ-regular and Fµ-normal in GFTS. Furthermore, Chakraborty et al [15] study some properties of Fgµ-closed sets in GFTS. They also, in- vestigated various concepts within GFTS as in [13, 14]. However, there are many research gaps and further developments that have not yet been achieved in the context of GFT . This article aims to contribute to developing the theoretical foundation for GFT by introducing and analyzing novel categories of spaces within the framework of GFT via Fgµ-closed sets. After introductory section, the rest of the article is systematized as follows: • In section 2. We have review some fundamental definitions and findings that will be utilized throughout this article. • In section 3. We apply the notion of Fgµ-closed sets to introduce and discuss novel categories of spaces such as Fgµ-regular, Fµ-G3, Fµ-T 1 2 , Fµ-T2 1 2 , and Fµ-symmetric S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 3 of 15 spaces in the context of GFT . We analyze their basic characteristics and properties. Some related theorems, relations, and implications are discussed. • In section 4. We introduce new classes of spaces named, Fgµ-normal and Fµ-G4 spaces in GFT via Fgµ-closed sets. We investigate some properties, related the- orems, implications, and results in this sequel. We explore the interrelationships between these classes and the other separation axioms with some supporting exam- ples. • In section 5. The connections of Fgµ-regular (Fgµ-normal) spaces and that in the crisp GT are presented. Moreover, we have explore the basic preservation theorems and discuss the hereditary and topological property of these classes. • In section 6. Conclusion and future works, we outline the article’s contributions and suggest some points to open up new avenues for further research in this area. 2. Basic definitions and results In this document, U refers to a universe set, IU (I = [0, 1]) is the class of all F -sets on U , (U , τ) means FTS, and (U , µ) means GFTS. In the following, let’s review a few fundamental definitions and findings that will be utilized throughout the rest of this study. Definition 1. [49] A fuzzy set (or F -set) H in U is a map H : U −→ I. It can be written as H = {(u,H(u)) : u ∈ U , H(u) ∈ I}. The fuzzy point (or F -point) uα is an F -set such that uα (v) = α > 0 if u = v and uα (v) = 0 if u ̸= v for all v ∈ U . uα ∈ H if α ≤ H(u). FP (U) refers to the family of all F -points in U . The constant F -sets 0 and 1 are given by 0(u) = 0 and 1 (u) = 1 for any u ∈ U . For H, G ∈ IU , we have the following properties of F -sets (see [16, 39, 49]): (i) H ∪G ∈ IU given by (H ∨G)(u) = max{H(u), G(u)} for every u ∈ U . (ii) H ∩G ∈ IU given by (H ∧G)(u) = min{H(u), G(u)} for every u ∈ U . (iii) Hc ∈ IU given by Hc(u) = 1−H(u) for all u ∈ U . (iv) For A ⊂ U , the characteristic function χA is an F -set on U . (v) The support of H ∈ IU is denoted by S(H) and given by S (H) = {u ∈ U : H (u) > 0}. (vi) For a map f : U −→ W and H ∈ IU , G ∈ IW , we have: (a) f(H) is an F -set on W given as f(H)(w) = sup{H(u) : u ∈ f−1(w)} if f−1(w) ̸= ∅ and f(H)(w) = 0 if f−1(w) = ∅. (b) f−1(G) is an F -set on U given as f−1(G)(u) = G(f(u)) for every u ∈ U . Definition 2. [16] An FTS is the pair (U , τ), where τ ⊆ IU which is closed under finite intersections, arbitrary union, and 0, 1 in τ . An F -set H is called F -open set if H ∈ τ and the complement of H is called F -closed set. For an F -set H in (U , τ), the F -complement, F -interior, and F -closure of H are written as Hc, int (H) , and cl(H) respectively. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 4 of 15 Definition 3. [39] An F -point uα is called quasi-coincident with a F -set H in U , symbol- ized by uαqH, if there is u ∈ U such that α+H(u) > 1. In general, HqG if H(u)+G(u) > 1 for some u ∈ U . If H is not quasi-coincident with G, then we write Hq̃G. Definition 4. [16, 27, 32, 39] For any two F -sets H,G in (U , τ) and uα ∈ FP (U), we have: (1) uαq̃H ⇐⇒ uα ∈ Hc,in general Hq̃G ⇐⇒ H ⊆ Gc (2) H ∩G = 0 =⇒ Hq̃G (3) Hq̃G, F ⊆ G =⇒ Hq̃F (4) H ⊆ G ⇐⇒ (uαqH =⇒ uαqG) for all uα ∈ FP (U) (5) uαq̃vβ ⇐⇒ u ̸= v or (u = v and α+ β > 1). For a map f : U −→ V, H ∈ IU , G ∈ IV , and uα ∈ FP (U), we have: (i) f(uα)qG =⇒ uαqf −1 (G), and uαqH =⇒ f(uα)qf (H). (ii) uαqf −1(G) if f(uα) ∈ G, and f(uα) ∈ f(H) if uα ∈ H. Definition 5. [18] A collection µ ⊆ IU is called GFT on U iff 0 ∈ µ and ∨ i∈J Hi ∈ µ for any class {Hi : i ∈ J} ⊂ µ. The structure (U , µ) is called an GFTS. Every member of µ is called a fuzzy µ-open set (in short, Fµ-open set) and the complement of a Fµ-open set is called Fµ-closed set. The family FµO(U) (resp. FµC(U) denotes to the class of all Fµ-open (resp. Fµ-closed) sets on U . For an GFTS (U , µ) and H∈IU . The Fµ-closure of H is the smallest Fµ-closed set containing H, it is symbolized by clµ(H) and the Fµ-interior of H, symbolized by intµ(H) is the largest Fµ-open set contained in H. Evidently, H ∈ IU is Fµ-open (resp. Fµ-closed) if and only if H = intµ(H) (resp. H = clµ(H)) It is clear that intµ and clµboth are monotonic and idempotent operators. Notation. For an GFTS (U , µ) and uα ∈ FP (U). Ouα refers to an Fµ-open set contain- ing uα and it is called an Fµ-open neighborhood (or Fµ-open nbd) of uα. In general, OH refers to an Fµ-open set containing H. Definition 6. Let (U , µ) be an GFTS and V ⊆ U . The family µV = {χV ∩H : H ∈ µ} is an GFT on V . The pair (V, µV ) is called an GFT -subspace (or GFTSS) of (U , µ). Lemma 1. [27] Let (U , µ) be an GFTS, H ∈ IU and uα ∈ FP (U), we have: (i) for any two Fµ-open sets F and G, if F q̃G, then clµ(F )q̃G and F q̃clµ(G). (ii) uαqclµ(H) if and only if OuαqH for all Ouα ∈ µ. Definition 7. [14] An F -set H in GFTS (U , µ) is called: (i) Fµ-regular closed (resp., Fµ-regular open) if H = clµ(intµ(H))(resp.,H = intµ(clµ(H)). (ii) Fµ-locally closed if there is F ∈ µ and G ∈ FµC(U) such that H = F ∩G. Definition 8. [31] An F -set H in GFTS (U , µ) is called fuzzy generalized µ-closed (or Fgµ-closed) if clµ(H) ⊆ G whenever H ⊆ G and G ∈ FµO(U). The class of all Fµg- closed sets in (U , µ) is symbolized by FgµC(U). The complement of Fgµ-closed set is called an Fgµ-open set. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 5 of 15 Remark 1. [15] In GFTS (U , µ), we have: (i) Every Fµ-closed (resp., Fµ-open) set is an Fgµ-closed (resp., Fgµ-open) set, (ii) Every Fµ-closed ( Fµ-open) set is an Fµ-locally closed set, but not conversely. Remark 2. The concepts of Fgµ-closed sets and Fµ-locally closed sets are generalizations of Fµ-closed sets but both are independent to each other. For examples see [15] . Proposition 1. [15] An Fgµ-closed set in an GFTS (U , µ) is an Fµ-closed set if and only if it is Fµ-locally closed. According to definition provided by Kandil et al. [26], the next definition is obtained by taking µ = δ and replacing F -open sets with Fµ-open sets. Definition 9. An GFTS (U , µ) is said to be: (i) Fµ-T0 iff for any uα, vβ ∈ FP (U) with uαq̃vβ implies uαq̃clµ(vβ) or clµ(uα)q̃vβ . (ii) Fµ-T1 iff for any uα, vβ ∈ FP (U) with uαq̃vβ implies uαq̃clµ(vβ) and clµ(uα)q̃vβ . (iii) Fµ-T2 iff for any uα, vβ ∈ FP (U) with uαq̃vβ, there are G,H ∈ µ such that uα ∈ G, vβ ∈ H and Gq̃H. (iv) Fµ-regular (or Fµ-R2) iff for any uα ∈ FP (U) and any H ∈ FµC(U) with uαq̃H, there are F,G ∈ µ such that uα ∈ F,H ⊆ G and F q̃G. (v) Fµ-normal (or Fµ-R3) iff for any Fµ-closed sets F1, F2 with F1q̃F2, there are H,G ∈ µ such that F1 ⊆ H,F2 ⊆ G and Hq̃G. (vi) Fµ-T3 (resp. Fµ-T4) iff it is both Fµ-R2 (resp. Fµ-R3) and Fµ-T1. Note. Evidently, Fµ-T4 =⇒ Fµ-T3 =⇒ Fµ-T2 =⇒ Fµ-T1. Definition 10. A map f : (U , µ1 ) −→ (V, µ2) is called: (i) Fµ-continuous iff f−1(H) ∈ FµC(U) for each H ∈ FµC(V)[31]. (ii) Fgµ-continuous iff f−1(H) ∈ FgµC(U) for each H ∈ FµC(V)[15]. (iii) Fgµ-closed(Fgµ-open) iff f(H) is Fgµ-closed(Fgµ-open) set in (V, µ2) for every Fµ-closed(Fµ-open) set H in (U , µ1) [14]. Note. Evidently, every Fµ-continuous is Fgµ-continuous. 3. Fuzzy gµ-regular spaces In this part, we introduce and discuss some characteristics and properties of a new class of spaces named, Fgµ-regular spaces in GFTS. First let’s give the next definition. Definition 11. An GFTS (U , µ) is named: (i) Fµ-T 1 2 iff every Fgµ-closed set in (U , µ) is an Fµ-closed set. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 6 of 15 (ii) Fµ-T2 1 2 iff for each uα,vβ ∈ FP (U) with uαq̃vβ, there are G,H ∈ µ such uα ∈ G, vβ ∈ H and clµ(G)q̃clµ(H). Proposition 2. For an GFTS (U , µ), the next items are equivalent: (1) (U , µ) is Fµ-T 1 2 . (2) Every Fgµ-closed set is an Fµ-locally closed set. Proof. (1) =⇒ (2). Let (U , µ) be an Fµ-T 1 2 space, then every Fgµ-closed set is Fµ- closed set. By Remark 1, every Fµ-closed set is Fµ-locally closed set. The result holds. (2) =⇒ (1). It follows directly from Proposition 1. Definition 12. An GFTS (U , µ) is called Fgµ-regular (or FG-µR2) iff for every Fgµ- closed set H with uαq̃H for each F -point uα, there are Fµ-open sets F,G containing uα, H respectively, such that F q̃G. Remark 3. Evidently, any FG-µR2 space is Fµ-R2 but not conversely. Example 1. Let U = {u, v} and µ = {0, 1, H,G}, where H = (u0.3, v0.5) , G = (u0.7, v0.5), then µ is an GFT on U . One can check that (U , µ) is Fµ-R2 but not FG-µR2. Indeed, for u0.5 ∈ FP (U) and Fgµ-closed set F = (u0.4, v0.7) with u0.5q̃F , there are Ou0.5 = G ∈ µ and OF = 1 ∈ µ but Ou0.5qOF . Hence (U , µ) is not FG-µR2. Theorem 1. An GFTS (U , µ) is FG-µR2 if and only if is both Fµ-R2 and Fµ-T 1 2 . Proof. Assume that (U , µ) is FG-µR2. By Remark 3, it is Fµ-R2. Let H be any Fgµ-closed set with uαq̃H for each uα ∈ FP (U) that is, uα ∈ Hc, there are F,G ∈ µ such that Ouα ∈ F , H ⊆ G and F q̃G implies that F q̃H. From Lemma 1, we have uαq̃clµ(H) that is, uα∈(clµ(H))c. Therefore, Hc⊆(clµ (H))c implies clµ(H)⊆H and so, H = clµ(H) this means that, any Fgµ-closed set in (U , µ) is an Fµ-closed set. Hence (U , µ) is Fµ-T 1 2 . Conversely, it is obvious. Theorem 2. Let (U , µ) be GFTS and uα ∈ FP (U). The next items are equivalent: (1) (U , µ) is FG-µR2, (2) For any Fgµ-open set Ouα containing uα, there is O∗ uα ∈ µ such that clµ(O ∗ uα ) ⊆ Ouα. Proof. (1) =⇒ (2). Suppose that (U , µ) is FG-µR2 and Ouα is an Fgµ-open set containing uα, we have O c uα = H ∈ FgµC(U). Clearly, Ouα q̃H that is, uαq̃H. Since (U , µ) is FG-µR2, there are O∗ uα , OH ∈ µ such that O∗ uα q̃OH implies that O∗ uα ⊆Oc H so that, clµ(O ∗ uα )⊆Oc H . Since H ⊆ OH , we have Oc H ⊆ Hc = Ouα . Therefore clµ(O ∗ uα )⊆Ouα . (2) =⇒ (1). Let G ∈ FgµC(U) with uαq̃G, then uα ∈ Gc = Ouα which is Fgµ-open set containing uα. By given, there is Fµ-open set O∗ uα such that clµ(O ∗ uα ) ⊆ Ouα = Gc that is, G ⊆ (clµ(O ∗ uα ))c = OG and clµ(O ∗ uα )q̃ (clµ(O ∗ uα ))c = OG. Therefore O∗ uα q̃O G . This completes the proof. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 7 of 15 Theorem 3. For an GFTS (U , µ) and uα ∈ FP (U). The next items are equivalent: (1) (U , µ) is FS-GR2, (2) For any Fgµ-closed set G with uαq̃G, there are Ouα , OG ∈ µ such that clµ(Ouα)q̃clµ(OG). Proof. Necessity. Let (U , µ) be FG-µR2 and G ∈ FgµC(U) with uαq̃G, there are O∗ uα and OG ∈ µ such that OGq̃O ∗ uα . By Lemma 1, we have clµ(OG)q̃O ∗ uα implies clµ(OG)q̃uα. In similar, since (U , µ) is FG-µR2, there are O ∗∗ uα and Oclµ(OG) ∈ µ such that O∗∗ uα q̃Oclµ(OG). By Lemma 1, we get clµ(O ∗∗ uα )q̃Oclµ(OG). TakeOuα = O∗ uα ∪O∗∗ uα ∈ µ. By the above theorem, there is Ouα ∈ µ such that clµ(Ouα)⊆O∗ uα . Since clµ(OG)q̃O ∗ uα , we have clµ(OG)q̃clµ(Ouα). Conversely, It follows by the hypothesis. Corollary 1. An GFTS (U , µ) is FS-GR2 if and only if for any H ∈ IUand any Fgµ- closed set G with Hq̃G, there are OH , OG ∈ FgµO(U) such that OH q̃OG. Proof. It can be obtained from Definition 12 and Remark 1. Definition 13. An GFTS (U , µ) is said to be Fµ-symmetric iff uαq̃clµ(vβ) implies vβ q̃clµ(uα) for any uα, vβ ∈ FP (U). Theorem 4. For an GFTS (U , µ). The next items are equivalent: (1) (U , µ) is Fµ-symmetric, (2) clµ(uα)q̃G for any G ∈ FµC(U) with uαq̃G. Proof. Necessity. Let G ∈ FµC(U) such that uαq̃G, then clµ(Vβ) ⊆ G for any vβ ∈ G. This implies that uαq̃clµ(vβ). Since (U , µ) is Fµ-symmetric, we have vβ q̃clµ(uα) for any vβ ∈ G and so, there is Ovβ ∈ µ, vβ ∈ Ovβ with uαq̃Ovβ . Now take H = ∪{Ovβ : vβ ∈ G and uαq̃Ovβ}, then H = OG and uαq̃H implies uα ∈ Hc and so, clµ(uα) ⊆ Hc that is, clµ(uα)q̃H. Therefore clµ(uα)q̃G. Conversely. It is obvious. Corollary 2. An GFTS (U , µ) is Fµ-symmetric iff uα is an Fgµ-closed set for any uα ∈ FP (U). Remark 4. Evidently, every Fµ-T1 space is Fµ-symmetric but not conversely. Example 2. Consider U = {u} and µ = {0, 1, u0.5}, then µ is an GFT on U . One can check that µ is Fµ-symmetric but not Fµ-T1. Further, µ is not Fµ-T 1 2 . Proposition 3. An GFTS (U , µ) is Fµ-T1 iff is both Fµ-symmetric and Fµ-T0. Proof. Evidently, if (U , µ) is Fµ-T1, then it is Fµ-symmetric and Fµ-T0. Conversely, let (U , µ) be Fµ-symmetric and Fµ-T0. Suppose uαq̃vβ, we have either uαq̃clµ(vβ) or vβ q̃clµ(uα). By Fµ-symmetric, we get uαq̃clµ(vβ) and vβ q̃clµ(uα) for any uα, vβ ∈ FP (U). This completes the proof. From the pervious results, one can verify the following proposition. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 8 of 15 Proposition 4. For an Fµ-symmetric space (U , µ). The next items are equivalent: (1) (U , µ) is Fµ-T0, (2) (U , µ) is Fµ-T 1 2 , (3) (U , µ) is Fµ-T1. Definition 14. An GFTS (U , µ) is called Fµ-G3 iff it is FG-µR2 and Fµ-symmetric. Theorem 5. Every Fµ-G3 space is Fµ-T2 1 2 . Proof. Let (U , µ) be Fµ-G3 and uα, vβ ∈ FP (U) with uαq̃vβ. Since (U , µ) is Fµ- symmetric and so, uα is a Fgµ-closed set for any uα ∈ FP (U). By Theorem 3, there are Ouα , Ovβ ∈ µ such that cl(Ouα)q̃cl(Ovβ ). Therefore (U , µ) is Fµ-T2 1 2 . Corollary 3. Clearly, every Fµ-G3 space is Fµ-T2 but not conversely. Example 3. Let U be an infinite set. For u, v ∈ U , u ̸= v, let Hu,v ∈ IU defined as: Hu,v(w) =  1, if w = u 0, if w = v 0.5, if w ̸= u and w ̸= v for all w ∈ U . Consider the GFT µ on U which is induced by the class {Hu,v : u, v ∈ U , u ̸= v}. One can verify that µ is Fµ-T2 but not FG-µR2 and so, is not Fµ-G3. Theorem 6. For an GFTS (U , µ). The next items are equivalent: (1) (U , µ) is Fµ-G3, (2) (U , µ) is Fµ-T3. Proof. (1) =⇒ (2). Assume that (U , µ) is Fµ-G3, we have it is both FG-µR2 and Fµ- symmetric. Clearly, every FG-µR2 is Fµ-R2 also, every Fµ-G3 is Fµ-T2. Hence (U , µ) is Fµ-R2 and Fµ-T1 that is, (U , µ) is Fµ-T3. (2) =⇒(1). Let (U , µ) be Fµ-T3, then it is both Fµ-R2 and Fµ-T1. This implies that (U , µ) is Fµ-T 1 2 and Fµ-symmetric. Thus, (U , µ) is Fµ-R2 and Fµ-T 1 2 implies that (U , µ) is FG-µR2 as well as, it is Fµ-symmetric. Therefore (U , µ) is Fµ-G3. 4. Fuzzy gµ-normal spaces In this part, we introduce a new class of spaces called, Fgµ-normal spaces in the frame of GFTS via Fgµ-closed sets and discuss some properties and related theorems in this sequel. Definition 15. An GFTS (U , µ) is called Fgµ-normal (or FG-µR3) iff for any G,H ∈ FgµC(U) with Gq̃H, there are OG, OH ∈ µ containing G,H respectively, such that OGq̃OH . S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 9 of 15 Remark 5. Evidently, every FG-µR3 space is Fµ-R3. Corollary 4. An GFTS (U , µ) is FG-µR3 if and only if for any two Fgµ-closed sets G,H with Gq̃H, there are OG, OH ∈ FgµO(U) such that OGq̃ OH . Proof. It can be obtained from Definition 15 and Remark 1. Theorem 7. For an GFTS (U , µ). The next items are equivalent: (1) (U , µ) is FG-µR3, (2) For any H ∈ FgµC(U) and any OH ∈ µ containing H, there is O∗ H ∈ µ such that clµ( O ∗ H) ⊆ OH . Proof. Necessity. Assume that (U , µ) be FG-µR3, H ∈ FgµC(U), and OH ∈ FµO(U) containingH, we have Oc H ∈ FµC(U). Clearly, OH q̃Oc H that impliesHq̃Oc H . Since (U , µ) is FG-µR3, there areO ∗ H , OOc H ∈ µ such thatO∗ H q̃OOc H implies thatO∗ H⊆(OOc H )c and so, clµ(O ∗ H)⊆(OOc H )c. Since Oc H⊆OOc H , we have (OOc H )c⊆OH and clµ(O ∗ H)⊆(OOc H )c⊆OH . The result holds. Conversely, it follows directly by the hypothesis. Theorem 8. For an GFTS (U , µ). The following items are equivalent: (1) (U , µ) is FG-µR3, (2) For any F,G ∈ FgµC(U) with F q̃G, there are OF , OG ∈ µ containing F,G respectively, such that clµ(OF )q̃clµ(OG). Proof. Necessity. Assume that (U , µ) is FG-µR3 and F,G ∈ FgµC(U) with F q̃G, there are O∗ F , OG ∈ µ such that O∗ F q̃OG implies that O∗ F q̃clµ(OG) (by Lemma 1). Again, (U , µ) is FG-µR3, there are O∗∗ F , Oclµ(OG) ∈ µ such that O∗∗ F q̃Oclµ(OG). This implies that clµ(O ∗∗ F )q̃Oclµ(OG) ( by Lemma 1). Take OF = O∗ F ∪ O∗∗ F ∈ µ. Since (U , µ) is FG-µR3 and O∗ F ∈ µ. So by the above theorem, there is OF ∈ µ such that clµ(OF ) ⊆ O∗ F . Since O∗ F q̃clµ(OG), we have clµ(OF )q̃clµ(OG). Conversely, it follows directly from the hypothesis. Definition 16. An GFTS (U , µ) is called Fµ-G4 iff it is FG-µR3 and Fµ-symmetric. Theorem 9. Every Fµ-G4 space is Fµ-G3. Proof. Suppose that (U , µ) is Fµ-G4, then it is FG-µR3 and Fµ-symmetric. Consider H is Fgµ-closed set with uαq̃H, then uα is an Fgµ-closed set (as (U , µ) is Fµ-symmetric). Since (U , µ) is FG-µR3, there are Ouα , OH ∈ µ such that Ouα q̃OH . Hence (U , µ) is FG-µR2. Therefore (U , µ) is Fµ-G3. Corollary 5. If (U , µ) is FG-µR3 and Fµ-symmetric space, then (U , µ) is FG-µR2 . Proposition 5. An GFTS (U , µ) is FG-µR3 iff it is both Fµ-R3 and Fµ-T 1 2 . S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 10 of 15 Proof. It is analogues to that of Theorem 1. Theorem 10. An GFTS (U , µ) is Fµ-G4 if and only if it is Fµ-T4. Proof. It is analogues to that of Theorem 6. From the definitions and discussions in section 3 and 4. The following implications hold. Corollary 6. The following implications hold. Fµ− T4 =⇒ Fµ− T3 =⇒ Fµ− T2 1 2 =⇒ Fµ− T2 =⇒ Fµ− T1 =⇒ Fµ− T0 ⇕ ⇕ Fµ−G4 =⇒ Fµ−G3 ⇐⇒ FG− µR2 ∧ Fµ− symmetric ⇕ FG− µR3 ∧ Fµ− symmetric =⇒ FG− µR2 =⇒ Fµ−R2 5. Further applications and relations In the following discussion, we will explore the basic preservation theorems and some relations of FG-µR2 and FG-µR3. Definition 17. For an GTS (U , θ). The class µθ = {χH : H ∈ θ} forms an GFT on U generated by θ. Theorem 11. (U , µθ) is FG-µR2 ⇐⇒ (U , θ) is µ-regular. Proof. Necessity. Suppose that (U , µθ) is FG-µR2 and G is any µ-closed set in (U , θ) such that u /∈ G, then χG = H ∈ FµC(U) which is also, Fgµ-closed set in (U , µθ) with u1q̃H. Since (U , µθ) is FG-µR2, there are Ou1 , OH ∈ µθ such that Ou1 q̃OH . Thus, there are Ou, OG ∈ θ such that Ou1 = χOu , OH = χOG and Ou ∩ OG = ∅. Hence (U , θ) is µ-regular. Conversely, let (U , θ) be µ-regular and H any Fgµ-closed set in (U , µθ) such that uαq̃H, there is µ-closed set B in (U , θ) such that H = χOB and u /∈ B. Since (U , θ) is µ-regular, there are Ou, OB ∈ θ such that Ou∩OB = ∅ and so, there are Ouα and OH ∈ µθ such that Ouα = χOu , OH = χOB with Ouα q̃OH . Therefore (U , µθ) is FG-µR2. Theorem 12. (U , µθ) is FG-µR3 ⇐⇒ (U , θ) is µ-normal. Proof. It can be obtained by a similar way of that in Theorem 11. Definition 18. For two GFTSs (U , µ1), (V, µ2). A map f :(U , µ1)−→(V, µ2) is called Fgµc-irresolute iff f−1(G) ∈ FgµC(U) for any G ∈ FgµC(V). Note. Evidently, every Fgµc-irresolute map is Fgµ-continuous. Theorem 13. Let f : (U , µ1)−→(V, µ2) be an Fµ-open and Fgµ-continuous bijection map. If H is Fgµ-closed set in (V, µ2), then f−1(H) is Fgµ-closed set in (U , µ1). S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 11 of 15 Proof. Consider H ∈ FgµC(V) and f−1(H) ⊆ G where G ∈ µ1, then H ⊆ f(G). Since f is Fµ-open, we have f(G) ∈ µ2. By given,H is a Fgµ-closed set in (V, µ2), then clµ(H) ⊆ f(G). So that f−1(clµ(H)) ⊆ G (as f is injective). Evidently, f is Fgµ-continuous, then f−1(clµ(H)) is Fgµ-closed set in (U , µ1). Hence f−1(clµ(H)) ⊆ clµ(f −1(clµ(H))) = f−1(clµ(H)) ⊆ G. Therefore, f−1(H) is an Fgµ-closed set in (U , µ1). Corollary 7. For two GFTSs (U , µ1), (V, µ2). If f : (U , µ1) −→ (V, µ2) is an Fµ-open and Fgµ-continuous bijection map, then f is Fgµc-irresolute. Theorem 14. Let f : (U , µ1)−→(V, µ2) be a Fµ-open and Fgµ-continuous bijection map. If (U , µ1) is FG-µR2, then (V, µ2) also, is FG-µR2. Proof. LetH ∈ FgµC(V) and vαq̃H. Since f is Fµ-open and Fgµ-continuous bijective, we have by Theorem 13, f−1(H) is Fgµ-closed. Put f (uα) = vα, then uαq̃f −1(H). Since (U , µ1) is FG-µR2, there are Ouα , Of−1(H) ∈ µ1 such that Ouα q̃Of−1(H). Since fup is Fµ-open and bijective, we get f(Ouα), f(Of−1(H)) ∈ µ2 such that vα ∈ f(Ouα), H ⊆ f(Of−1(H)) and f(Ouα)q̃f(Of−1(H)). Therefore, (V, µ2) is FG-µR2. Theorem 15. Consider f : (U , µ1)−→(V, µ2) is Fµ-open and Fgµ-continuous bijection. If (U , µ1) is FG-µR3, then (V, µ2) also, is FG-µR3. Proof. It can be obtained by a similar way of that in Theorem 14. Theorem 16. For an Fµ-continuous and Fgµ-closed injective map f : (U , µ1)−→(V, µ2). If (V, µ2) is FG-µR2, then (U , µ1) also, is FG-µR2. Proof. Let H ∈ FgµC(U) with uαq̃H. By Fµ-continuity and Fgµ-closedness, we have f(H) ∈ FgµC(V). In fact, if f(H) ⊆ G and G ∈ µ2, then H ⊆ f−1(G) and so, clµ(H) ⊆ f−1(G) implies that f(H) ⊆ f(clµ(H)) ⊆ ff−1(G) ⊆ G that is, f(H) ⊆ G. Hence f(H) is Fgµ-closed. Since f is injective, we have f (uα) q̃f(H). Given that (V, µ2) is FG-µR2, there areOf(uα), Of(H) ∈ µ2 such thatOf(uα)q̃Of(H). Since f is Fµ-continuous, we have f−1(Of(uα)), f −1(Of(H)) ∈ µ1 such that uα ∈ f−1(Of(uα)), H ⊆ f−1(Of(H)), and f−1(Of(uα))q̃f −1(Of(H)). Therefore, (U , µ1) is FG-µR2. Theorem 17. Let f : (U , µ1)−→(V, µ2) be Fµ-continuous, Fgµ-closed injective. If (V, µ2) is FG-µR3, then (U , µ1) is FG-µR3. Proof. Suppose thatH,G ∈ FgµC(U)) withHq̃G. As in the above theorem f (H) , f(G) ∈ FgµC (V) . Since fup is injective, then f(H)q̃f (G). By given (V, µ2) is FG-µR3, there are Of(H), Of(G) ∈ µ2 withOf(H)q̃Of(G). Since f is Fµ-continuous, we have f−1(Of(H)), f −1(Of(G)) ∈ µ1 andH ⊆ f−1(Of(H)), G ⊆ f−1(Of(G)) with f−1(Of(H))q̃f −1(Of(G)). Therefore, (U , µ1) is FG-µR2. Theorem 18. Let f : (U , µ1)−→(V, µ2) be Fgµc-irresolute and Fµ-open surjective. If (U , µ1) is FG-µR3, then (V, µ2) is FG-µR3. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 12 of 15 Proof. LetG,H ∈ FgµC(V) such thatGq̃H, then f−1 (G) , f−1(H) are Fgµ-closed sets in (U , µ1) with f−1 (G) q̃f−1(H). Since (U , µ1) is FG-µR3, there are Of−1(G), Of−1(H) ∈ µ1 containing f−1 (G) , f−1 (H) , respectively with Of−1(G)q̃Of−1(H). Since f is surjective, we have G ⊆ f(Of−1(G)), H ⊆ f(Of−1(H)) and f(Of−1(G)), f(Of−1(H)) ∈ µ2(As f is Fµ-open) with (Of−1(G))q̃f(Of−1(H)). Hence (V, µ2) is FG-µR3. From Theorems 14 and 15, one can verify the next theorem. Theorem 19. The property of being FG-µR2 (FG-µR3) is a Fgµ-topological property. In the following, we show that the being FG-µR2 (FG-µR3) are hereditary property. Theorem 20. Every Fµ-subspace (V, µV ) of FG-µR2 is FG-µR2. Proof. Let (U , µ) be FG-µR2, uα ∈ FP (V) and H ∈ FgµC(V) with uαq̃H, there is an Fgµ-closed set G in (U , µ) such that H = χV ∩G and uαq̃G. Since (U , µ) is FG-µR2, we have Ouα , OG ∈ µ such that Ouα q̃OG. Put O ∗ uα = χV ∩Ouα ∈ µV and O∗ G = χV ∩OG ∈ µV which are containing uα and G respectively, and O∗ uα q̃O∗ G. This completes the proof. Theorem 21. Every Fµ-closed subspace (V, µV ) of FG-µR3 is FG-µR3. Proof. It follows by a similar way of that in Theorem 20. 6. Conclusion and future work Topology is a branch of mathematics that studies the properties of space preserved under continuous transformations. It allows mathematicians to analyze and classify spaces, leading to applications in various fields, where understanding the fundamental structure of spaces is of great importance. This study focuses on the applications of Fgµ-closed sets in generalized fuzzy topol- ogy. Some classes, namely Fgµ-regular, Fgµ-normal, Fµ-symmetric, have been introduced and analyzed. The paper thoroughly have investigated the fundamental properties and unique characteristics of these classes. A comprehensive is framework has been established through the presentation of related theorems and relations, demonstrating their interrela- tionships with other separation axioms in this context. Additionally, the hereditary and topological properties of these classes have been explored. The present results in this article are very useful and contribute to the development of the theoretical and practical foundations of the GFT , and will open up the door for future works in this area such as: • We plan to further study for Fgµ-closed sets with some separation axioms in this settings. • We intend to discuss some weaker forms of Fgµ-closed sets with some applications of them in FGT -spaces. S. Saleh et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5856 13 of 15 • We plan to extend the characterizations of these classes within infra soft topological spaces [3, 4], and explore their potential applications. • Exploring the proposed concepts through well-known topological structures [5, 6] is an interesting idea for researchers and scholars interested in topological studies, which we have left for future investigations. Acknowledgments The authors extend their appreciation to the Deanship of Scientific Research at North- ern Border University, Arar, KSA for funding this research work through the project number “NBU-FFR-2025-1166-02” Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. References [1] N. Ahmed and P. Kumar. Fuzzy logic in image processing: A survey of methods and applications. Image and Vision Computing, 100:100–120, 2022. [2] T. M. Al-shami. Investigation and corrigendum to some results related to g-soft equality and gf -soft equality relations. Filomat, 33(11):3375–3383, 2019. [3] T. M. Al-shami. Infra soft compact spaces and application to fixed point theorem. Journal of Function Spaces, 2021:9 pages, 2021. [4] T. M. Al-shami. New soft structure: Infra soft topological spaces. Mathematical Problems in Engineering, 2021:12 pages, 2021. [5] T. M. Al-shami and M. E. El-Shafei. On supra soft topological ordered spaces. 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