EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 180-191 ISSN 1307-5543 – ejpam.com Published by New York Business Global On g-Regularity and g-Normality in Fuzzy Soft Topological Spaces S. Saleh1,2,∗, Jawaher Al-Mufarrij3 1Computer Science Department, Cihan University-Erbil, Kurdistan region, Iraq 2Department of Mathematics, Hodeidah University, Hodeidah, Yemen. 3Department of Mathematics, Women section, King Saud University, Riyadh 12372, KSA. Abstract. The main aim of this work is to introduce and study the notions of generalized regu- larity, normality, and symmetric in fuzzy soft topological spaces via fuzzy soft generalized closed sets. Some of their basic properties are investigated. Many related theorems and relations of these notions are presented. Moreover, the hereditary property and some preservation theorems are discussed. 2020 Mathematics Subject Classifications: 54A40, 54C08, 54D15 Key Words and Phrases: Fuzzy soft set, fuzzy soft g-closed set, quasi coincident, fuzzy soft g-continuous maps, fuzzy soft g-regular, fuzzy soft g-normal space 1. Introduction and Preliminaries Levine [18] introduced the notion of generalized closed set, briefly g-closed in general topology. A subset B of a topological space (X, τ) is called g-closed, if cl(B) ⊆ U whenever B ⊆ U and U is open in (X, τ). This notion has been studied extensively in topology and fuzzy topology by many authors as in ([3, 5, 7, 8, 13, 17, 25–27, 32]). The investigation of g-closed sets has led to several new and interesting concepts, e.g. g-regular, g-normal spaces, their generalizations which are studied in ([12, 15, 21–24]), and new separation axioms weaker than T1 are presented. In recent time, the topological structures play an important role in many applications of complex real-life problems in various field, specially the fields that concerned with handling all cases that contain uncertainties such as medical diagnosis and decision making,...etc see e. g. ([10, 11]). After the discovery of fuzzy set theory by Zadeh [33], many authors generalized and applied this idea in different aspects see e. g. ([1, 2, 14, 19]. The concept of fuzzy topo- logical space was introduced by Chang in [6]. Balasubramanian et. al [5] introduced the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4597 Email addresses: salem.saleh@cihanuniversity.edu.iq (S. Saleh), jmufarij@ksu.edu.sa (J. Al-Mufarrij) https://www.ejpam.com 180 © 2023 EJPAM All rights reserved. S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 181 concept of fuzzy generalized closed sets. Then M. El-Shafei [12] introduced and studied some applications of fuzzy generalized closed sets. Tanay et. al. [30] defined and studied the notion of topological structure for fuzzy soft sets and studied many related concepts. Recently, Tarrannum et. al. [31] introduced the concept of fuzzy soft generalized closed sets, fuzzy soft generalized continuous maps, and studied some properties for them. In this paper, we define and study the notions of fuzzy soft generalized regular spaces, generalized normal spaces, and symmetric spaces by utilizing fuzzy soft generalized closed sets. We obtain some characterizations. Several related theorems and relationships of them are discussed. In addition, the hereditary property and some preservation theorems are presented. Throughout this work, U refers to an universe set, E is the set of all parameters, P (U) is the power set of U , IU is the set of all fuzzy sets on U , where I = [0, 1], FS- refers to fuzzy soft, and (U, δ,E) means fuzzy soft topological space. In the next, we recall some basic definitions and notations which are used in this sequel. A fuzzy set(or F -set) A in U is a mapping A : U −→ I assigns the value A(x) ∈ I for all x ∈ U . An F -point xα is an F -set such that xα(y) = α > 0 if x = y and xα (y) = 0 otherwise for all y ∈ U. We write xα ∈ A if α ≤ A(x). The class of all F -points of U is denoted by FP (U) [33]. A fuzzy soft set(or FS-set) fE = (f,E) on U is a mapping f : E −→ IU where f (e) = fe is an F -set on U . Thus fE can be written as the set of ordered pairs fE = {(e, f(e)) : e ∈ E, f(e) ∈ IU}. The class of all FS-sets on U is denoted by FSS(U) [19]. For two FS-sets fE and gE on U , we have[19]: 1) fE is called a null (resp. universal) FS-set, symbolized by 0̃E(resp.1̃E) if f(e) = 0(resp.f(e) = 1) for all e ∈ E. 2) fE is a subset of gE if f (e) ≤ g (e) ∀ e ∈ E, symbolized by f ⊑ g. 3) fE and gE are equal if fE ⊑ gE and gE ⊑ fE . It is symbolized by fE = gE . 4) The union of fE and gE is an FS-set hE defined by h (e) = f (e) ∨ g (e) for all e ∈ E. hE is symbolized by fE ⊔ gE . 5) The intersection of fE and gE is an FS-set lE defined by l (e) = f (e) ∧ g (e) for all e ∈ E. lE is symbolized by fE ⊓ gE . An FS-point xeα on U is an FS-set (xeα, E) given by xeα(e ′ ) = xα if e ′ = e and xeα(e ′ ) = 0 otherwise, where xα is an F -point in U with the support x and the value α, α ∈ (0, 1]. An FS-point xeα∈̃fE if α ≤ f(e)(x). The set of all FS-points in U is denoted by FSP (U). We can write xeα ̸= yeβ if x ̸= y [4, 9]. The triple (U, δ, E) is called a fuzzy soft topological space ( or FSTS) where E is a S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 182 fixed set of parameters and δ is the class of FS-sets on U which is closed under a finite intersection, an arbitrary union, and 0E , 1E belong to δ. The family FSOS(U) (resp. FSCS(U)) refers to the set of all FS-open(resp. FS-closed) sets on U [4, 30]. Notation. [29] For xeα ∈ FSP (U), Oxe α refers to an FSO-set contains xeα and is called FSO-nbd of xeα , NE (xeα) refers to the set of all FSO-nbds of xeα. In general OfE refers to an FSO-set contains fE . An FS-closure of an FS-set hE in (U, δ,E) denoted by cl(hE) is the smallest FSC- set on U which contains hE , and an FS-interior of hE denoted by int(hE) is the largest FSO-set contained in hE . It is clear that xeα∈̃int(hE) if and only if there exists Oxe α ∈ δ such that Oxe α ⊑ hE [4]. Definition 1. [16] Let FSS(U) and FSS(V ) be two classes of all FS-sets on U , V respectively, and let p : U −→ V and u : E −→ K be two maps, then the map fup : FSS (U) −→ FSS (V ) is called an FS-map for which: i) If hE ∈ FSS(UE), then the image of hE denoted by fup(hE) is an FS-set on V given by fup(hE) (k) = sup{p(h (e)) : e ∈ u−1 (k)} if u−1 (k) ̸= ∅ and fup (hE) (k) = 0K , otherwise ∀k ∈ K. ii) If gK ∈ FSS(V ), then the preimage of gK denoted by f−1 up (gK) is an FS-set on U defined by f−1 up (gK)(e) = p−1(g (u (e))) for all e ∈ E. An FS-map fup is called one-one(onto) if u and p are one-one(onto). For more details about the properties of image and preimage of the FS-sets see [16]. Definition 2. [4] The FS-sets hE and gE on U are called FS-quasi coincident, denoted by hEqgE if there is e ∈ E and x ∈ U such that h(e)(x) + g(e)(x) > 1. If hE is not quasi coincident with gE, we write hE q̃gE . In particular, xeαqgE if α+ g(e)(x) > 1. Proposition 1. [4, 29] (i) fE q̃gE ⇔ fE ⊑ gcE . (ii) fE ⊓ gE = 0E ⇒ fE q̃gE . (iii) fE q̃gE , hE ⊑ gE ⇒ fE q̃hE . (iv) xeαq̃fE ⇔ xeα∈̃f c E . (v) fE ⊑ gE ⇔ (xeαqfE ⇒ xeαqgE). (vi) fE q̃f c E . Lemma 1. [29] For an FSTS (U, δ, E) and xeα ∈ FSP (U), we have: (i) gE q̃fE if and only if gE q̃cl(fE) ∀ gE ∈ δ, (ii) xeαq̃cl(fE) if and only if Oxe α q̃fE ∀ Oxe α ∈ δ. Definition 3. [20] An FS-set hE in (U, δ,E) is said to be regular open (resp. regular closed) if hE = int(cl(hE)) (resp. hE = cl(int(hE)). The family of all FS-regular open (resp. all FS-regular closed) on U is denoted by FSRO(U) ( resp. FSRC(U) ). S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 183 Definition 4. [31] An FS-set fE in (U, δ,E) is said to be fuzzy soft generalized closed (or FSg-closed) if cl(fE) ⊑ hE for all fE ⊑ hE and hE ∈ FSOS(U). The collection of all FSg-closed sets in (U, δ, E) is denoted by FSgCS(U). The complement of an FSg-closed set is called an FSg-open set. Note. Clearly, every FSC-set is an FSg-closed set. Definition 5. [28] An FST (U, δ,E) is said to be: (i) FST0 iff for any xeα, y e β ∈ FSP (U) with xeαq̃y e β implies xeαq̃cl(y e β) or cl(xeα)q̃y e β . (ii) FST1 iff for any xeα, y e β ∈ FSP (U) with xeαq̃y e β implies xeαq̃cl(y e β) and cl(xeα)q̃y e β. (iii) FST2 iff for any xeα , yeβ ∈ FSP (U) with xeαq̃y e β, there are Oxe α , Oyeβ ∈ δ such that Oxe α q̃Oyeβ . Definition 6. [28] An FSTS (U, δ,E) is said to be: (i) FSR2(or FS-regular) iff for any xeα ∈ FSP (U) with xeαq̃fE , fE is an FSC-set, there are Oxe α , OfE ∈ δ such that Oxe α q̃OfE . (ii) FSR3 (or FS-normal) iff for any FSC-sets fE , gE with fE q̃gE, there are OfE , OgE ∈ δ such that OfE q̃OgE . (iii) FST3 (resp. FST4) iff it is FSR2 (resp. FSR3) and FST1. Theorem 1. [28] FST4 ⇒ FST3 ⇒ FST2 ⇒ FST1 ⇒ FST0. Definition 7. [29] Let (U, τ) be a topological space. The family δ = {χ̃A : A ∈ τ} defines an FST on U induced by τ. Definition 8. [31] An FS-map fup : (U, δ,E) −→ (V, ϑ,K) is said to be: (i) FSg-continuous if f−1 up (hE) ∈ FSgCS(U) for any hE ∈ FSCS(V ). (ii) FSgc-irresolute if f−1 up (gE) ∈ FSgCS(U) for any gE ∈ FSgCS(V ). Note. Clearly, every FSgc-irresolute map is FSg-continuous. 2. Fuzzy soft g-regular spaces Definition 9. An FSTS (U, δ, E) is said to be: (i) FST 1 2 iff any FSg-closed set on U is an FSC-set. (ii) FST2 1 2 iff for any xeα , yeβ ∈ FSP (U) with xeαq̃y e β, there are Oxe α , Oyeβ ∈ δ such that clOxe α q̃clOyeβ . Definition 10. An FSTS(U, δ, E) is said to be FSg-regular (or FS-GR2) if for any FSg-closed set hE and any FS-point xeα with xeαq̃hE, there are Oxe α , OhE ∈δ such that Oxe α q̃OhE . S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 184 Remark 1. Clearly, every FS-GR2 space is FSR2. The next example shows that the converse is not necessarily true. Example 1. Let U = {a, b}, E = {e, t}, and δ = {0E , 1E , fE = {(e, {a0.5, b0.3}) , (t, {a0.5, b0.7})}, gE = {(e, {a0.5, b0.3}) , (t, {a0.5, b0.7})} }, then δ is FST on U . One can easy to verify that (U, δ, E) is FSR2 but not FS-GR2. Theorem 2. An FSTS (U, δ, E) is FS GR2 if and only if it is FSR2 and FST 1 2 . Proof. If (U, δ, E) is FS-GR2, then by Remark 1 it is FSR2. For any FSg-closed set fE and any FS-point xeα with xeαq̃fE i.e. xeα∈̃fC E , there are Oxe α , OfE∈δ such that Oxe α q̃OfE =⇒ Oxe α q̃fE =⇒ xeαq̃cl (fE) implies that xeα∈̃[cl (fE)] C . Thus fC E⊑[cl (fE)] C =⇒ cl (fE)⊑fE and so, fE = cl (fE) this means every FSg-closed set in (U, δ, E) is an FSC- set. The result holds. Conversely, it is clear. Theorem 3. An FSTS(U, δ, E) is FS-GR2 if and only if for any FS-point xeα and any FSg-open set Oxe α , there is an FSO-set O∗ xe α such that cl(O∗ xe α )⊑Oxe α . Proof. Let (U, δ,E) be FS-GR2 and Oxe α be any FSg-open set containing FS-point xeα, then Oc xe α = fE which is an FSg-closed set. Since Oxe α q̃Oc xe α we have, xeαq̃O c xe α . Since (U, δ, E) is FS-GR2, there are O∗ xe α , OOc xeα ∈ δ such that O∗ xe α q̃OOc xeα = OfE implies O∗ xe α ⊑ Oc fE and so, cl(O∗ xe α ) ⊑ Oc fE . Since Oc xe α ⊑ OOc xeα = OfE , we obtain Oc fE ⊑ Oxe α . Hence cl(O∗ xe α ) ⊑ Oxe α . Conversely, let xeα be any FS-point and gE be any FSg-closed set with xeαq̃gE , then xeα ∈ gcE = Oxe α which is an FSg-open set containing xeα. So by hypothesis, there exists an FSO-set O∗ xe α such that cl(O∗ xe α ) ⊑ Oxe α = gcE implies gE ⊑ [cl(O∗ xe α )]C = OgE and cl(O∗ xe α )q̃[cl(O∗ xe α )]C = OgE . Therefore O∗ xe α q̃O gE . Hence the result holds. Theorem 4. An FSTS (U, δ, E) is FS-GR2 if and only if for any FSg-closed set gE and any FS-point xeα with xeαq̃gE, there are Oxe α , OgE ∈ δ such that cl(Oxe α )q̃cl(OgE ). Proof. Let (U, δ, E) be an FS-GR2 space and gE be any FSg-closed set with xeαq̃gE , there are O∗ xe α , OfE ∈ δ such that OfE q̃O ∗ xe α . From Lemma 1, we get cl(OfE )q̃O ∗ xe α that is, cl(OfE )q̃x e α. A gain, since (U, δ, E) is FS-GR2, there are O∗∗ xe α , Ocl(OfE ) ∈ δ such that O∗∗ xe α q̃Ocl(OfE ) implies that cl(O∗∗ xe α )q̃Ocl(OfE ) (by Lemma 1 ). Take Oxe α = O∗ xe α ⊓ O∗∗ xe α and by the above theorem, there exists Oxe α ∈ δ such that cl(Oxe α ) ⊑ O∗ xe α . Since cl(OfE )q̃O ∗ xe α , we get cl(OfE )q̃cl(Oxe α ). Conversely, It follows directly from hypothesis. Definition 11. An FSTS (U, δ, E) is said to be FS-symmetric iff for any FS-points xeα, y e β ∈ FSP (U) with xeαq̃cl(y e β) implies yeβ q̃cl(x e α). Theorem 5. An FSTS (U, δ,E) is FS-symmetric if and only if cl(xeα)q̃gE for any FSC- set gE with xeαq̃gE. S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 185 Proof. Suppose that gE is an FSC-set on U with xeαq̃gE . Clearly cl (yet ) ⊑ gE for all yet ∈̃gE and so, xeαq̃cl(y e t ). Since (U, δ, E) is FS-symmetric, we have yeβ q̃cl(x e α) for all yet ∈̃gE and so, for all yet ∈̃gE there is an FSO-set Oyet containing yet such that xeαq̃Oyet . Put hE = ⊔ { Oyet : yet ∈̃gE and xeαq̃Oyet } , then hE = OgE and xeαq̃hE . Thus x e α∈̃hCE and so, cl(xeα) ⊑ hCE implies cl(xeα)q̃hE . Therefore cl(xeα)q̃gE . Conversely, it is obvious. Corollary 1. An FSTS (U, δ, E) is said to be FS-symmetric if and only if every FS-point xeα ∈ FSP (U) is an FSg-closed set. Remark 2. Clearly, every FST 1 space is FS-symmetric. The next example shows that the converse may not be true. Example 2. Let U = {x} , E = {e}, and δ = {0E , 1E , xe0.5}, then one can verify δ is FS-symmetric but not FST 1. Moreover, δ is not FT 1 2 . Proposition 2. An FSTS (U, δ, E) is FST 1 if and only if it is FS-symmetric and FST 0. Proof. Clearly, if (U, δ,E) is FST 1, then it is FS-symmetric and FST 0. Conversely, let (U, δ, E) be FS-symmetric and FST 0. Suppose xeαq̃y e t . Then either xeαq̃cl(y e t ) or yet q̃cl(x e α). By FS-symmetry, we have xeαq̃cl(y e t ) and yet q̃cl(x e α) for any xeα , yet ∈ FSP (U). The result holds. Theorem 6. Every FS-GR2 space is FST 2 1 2 . Proof. Let (U, δ,E) be FS-GR2 and xeα, y e t ∈ FSP (U) with xeαq̃y e t . Then (U, δ, E) is FS-symmetric and so xeα is an FSg-closed set. From Theorem 4 there are FSO-sets Oxe α and Oyet such that cl(Oxe α )q̃cl(Oyet ). Hence the result holds. Proposition 3. For an FS-symmetric space (U, δ,E). The next properties are equivalent: (1) (U, δ, E) is FST 0, (2) (U, δ, E) is FST 1 2 , (3) (U, δ, E) is FST 1. Proof. It is obvious. Definition 12. An FSTS (U, δ, E) is called FSG3 iff it is FS-GR2 and FS-symmetric. Proposition 4. Every FSG3 space is FST 2. Proof. It follows directly from the above Definition, Definition 10, and Corollary 1. The next example shows that the converse of the above proposition may not be true. S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 186 Example 3. Let U be an infinite set and E = {e}. For x, y ∈ U , x ̸= y, let hE be an FS-set on U given by h(e)(z) = 1 if z = x , h(e)(z) = 0 if z = y, and h (e) (z) = 0.5 if z ̸= x, z ̸= y. Now for any z ∈ U . Consider the FST δ on U generated by the class {(hE)x,y : x, y ∈ U, x ̸= y}. Then one can check that δ is FST 2 but not FS-GR2 and so not FSG3. Theorem 7. An FSTS(U, δ, E) is FSG3 if and only if it is FST 3 . Proof. Let (U, δ, E) be an FSG3-space, then it is FS-GR2 and FS-symmetric. Now, every FS-GR2 is FSR2 and every FSG3 is FST 2. Thus (U, δ,E) is FSR2 and FST 2. So the result holds. Conversely, let (U, δ, E) be FST 3, then it is FSR2 and FT 1 and so, it is FST 1 2 and FS- symmetric. Thus (U, δ,E) is FSR2 and FST 1 2 which implies that (U, δ, E) is FS-GR2. Since (U, δ, E) is FS-symmetric. Hence (U, δ,E) is FSG3. 3. Fuzzy soft g-normal spaces Definition 13. An FSTS (U, δ, E) is said to be FSg-normal (or FS-GR3) if for every FSg- closed sets fE and hE with fE q̃hE, there are FSO-sets OfE and OhE containing fE and hE respectively, such that OfE q̃ OhE . Remark 3. Clearly, every FS-GR3 space is FSR3. Theorem 8. An FSTS (U, δ, E) is FS-GR3 if and only if for any FSg-closed set fE and for any FSO-set OfE containing fE, there is O∗ fE ∈ δ such that cl(O∗ fE ) ⊑ OfE . Proof. Let (U, δ,E) be an FS-GR3 space, hE be any FSg-closed set, and let OhE be any FSO-set containing hE , then Oc hE is an FSC-set. It is known that OhE q̃Oc hE and so, hE q̃O c hE . Since (U, δ, E) is FS-GR3, there are FSO-sets O∗ hE and OOc hE such that O∗ hE q̃OOc hE and so,O∗ hE ⊑ Oc Oc hE and cl(O∗ hE ) ⊑ Oc Oc hE . Since Oc hE ⊑OOc hE we get Oc Oc hE ⊑OhE and cl(O∗ hE )⊑Oc Oc hE ⊑O hE . Hence the result holds. Conversely, It follows directly from hypothesis. Theorem 9. An FSTS (U, δ, E) is FS-GR3 if and only if for any FSg-closed sets fE and gE with fE q̃gE, there are FSO-sets OfE and OgE containing fE and hE respectively, such that cl(OfE )q̃ cl(OgE ). Proof. Let (U, δ, E) be FS-GR3 and fE , gE be any FSg-closed sets with fE q̃gE , there exist O# fE , OgE ∈ δ such that O# fE q̃OgE =⇒ O# fE q̃cl(OgE ) ( by Lemma 1 ). A gain, since (U, δ, E) is FS-GR3, then there are O∗ fE , Ocl(OgE ) ∈ δ such that O∗ fE q̃Ocl(OgE ) =⇒ cl(O∗ fE )q̃Ocl(OgE ) ( by Lemma 1 ). Now we put OfE = O# fE ⊓ O∗ fE . Since (U, δ,E) is FS- GR3 and O# fE ∈ δ, by the above theorem there is OfE ∈ δ such that cl(OfE ) ⊑ O# fE . Since O# fE q̃cl(OgE ), we get cl(OfE )q̃cl(OgE ). Conversely, It follows directly from hypothesis. S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 187 Definition 14. An FSTS(U, δ, E) is called FSG4 iff it is FS-GR3 and FS-symmetric. Theorem 10. Every FSG4 space is FSG3. Proof. Let (U, δ, E) be FSG4, then it is FS-GR3 and FS-symmetric. Let hE be an FSg-closed set with xeαq̃hE . Then xeα is an FSg-closed set, because U is FS-symmetric. Since (U, δ,E) is FS-GR3, there are FSO-sets Oxe α and OhE such that Oxe α q̃OhE . Thus U is FS-GR2 and so, (U, δ, E) is FSG3. Corollary 2. Every FS-GR3 and FS-symmetric space is FS-GR2 . Proposition 5. An FSTS(U, δ, E) is FS-GR3 if and only if it is FSR3 and FST 1 2 . Proof. By similar way as that of Theorem 2. Theorem 11. An FSTS(U, δ,E) is FSG4 if and only if it is FST 4. Proof. By similar way as that of Theorem 7. 4. Some properties and relations Here we shall investigate some preservation theorems and relationships of FS-GR2 and FS-GR3 spaces. Definition 15. [31] An FS-map fup : (U, δ, E)−→(V, σ,K) is said to be: (i) FSg-closed if fup(hE) is FSg-closed in (V, σ,K) for any FSC-set hE in (U, δ,E). (ii) FSg-open if fup(hE) is FSg-open in (V, σ,K) for any FSO-set hE in (U, δ, E). Lemma 2. If fup : (U, δ,E) −→ (V, σ,K) is an FS-open, FSg-continuous bijection map, then fup is FSgc-irresolute. Proof. Let hE ∈ FSgC(V ) and f−1 up (hE) ⊑ gE , where gE ∈ FSO(U), then hE ⊑ fup(gE). Since fup is FS-open, we have fup(gE) ∈ FSO(V ). Since hE is an FSg-closed set on V , we obtain cl(hE) ⊑ fup(gE). Hence f−1 up (cl(hE) ⊑ gE (because fup is one- one). Since fup is FSg-continuous, we have f−1 up (cl(hE) is an FSg-closed set in U and so, cl(f−1 up (hE )) ⊑ cl(f−1 up (cl (hE))) ⊑ gE . Hence f−1 up (hE) is an FSg-closed set on V . Theorem 12. If fup : (U, δ, E)−→(V, σ,K) is an FS-open, FSg-continuous bijection map and (U, δ,E) is FS-GR2, then (V, σ,K) is FS-GR2. Proof. Let hE ∈ FSgC(V ) and yeαq̃hE . Since fup is FS-open, FSg-continuous bijec- tive, by the above lemma, fup is FSgc-irresolute and so, f−1 up (hE) is FSg-closed. Take fup (x e α) = yeα, then xeαq̃f −1 up (hE)). Since U is FS-GR2, there are FSO-sets Oxe α and Of−1 up (hE) such thatOxe α q̃Of−1 up (hE). Since fup is FS-open and bijective, we have yeα∈̃fup(Oxe α ), hE ⊑ fup(Of−1 up (hE)) and fup(Oxe α )q̃fup(Of−1 up (hE)). The result holds. S. Saleh, J. Al-Mufarrij / Eur. J. Pure Appl. Math, 16 (1) (2023), 180-191 188 Theorem 13. If fup : (U, δ,E)−→(V, σ,K) is an FSg-continuous, FSg-closed one-one map and (V, σ,K) is FS-GR2, then (U, δ, E) is FS-GR2. Proof. Let hE ∈ FSgc(U) and xeαq̃hE . By FS-continuity and FSg-closedness we have fup(hE) ∈ FSgc(V ). Indeed, if fup(hE) ⊑ gE and gE is an FSO-set in (V, σ,K), we have hE ⊑ f−1 up (gE), and so cl(hE) ⊑ f−1 up (gE). Thus fup(hE) ⊑ fup(cl (hE)) ⊑ fupf −1 up (gE) ⊑ gE . So cl(hE) ⊑ gE . Thus fup(hE) is FSg-closed. Since fup is one-one, we get fup (x e α) q̃fup(hE). Since (V, σ,K) is FS-GR2, there exist FSO-sets Ofup(xe α) and Ofup(hE) such that Ofup(xe α) q̃Ofup(hE). So, we get xeα∈̃f−1 up (Ofup(xe α) ) , hE ⊑ f−1 up (Ofup(hE)) and f−1 up (Ofup(xe α) )q̃f−1 up (Ofup(hE)). Since fup is FS-continuous, we get f−1 up (Ofup(xe α) ) and f−1 up (Ofup(hE)) are FSO-sets in (U, δ, E). The result holds. Theorem 14. If fup : (U, δ, E)−→(V, σ,K) is FS-continuous, FSg-closed one-one and (V, σ,K) is FS-GR3, then (U, δ, E) is FS-GR3. Proof. Let hE , gE ∈ FSgCS(U) with hE q̃gE . As in the above theorem fup (hE) and fup (gE) ∈ FSgC(V ). Since fup is one-one, we have fup(hE)q̃fup(gE). Since (U, δ,E) is FS-GR3, there are FSO-sets Ofup(hE), Ofup(gE) such that Ofup(hE)q̃ Ofup(gE). So we get, hE ⊑ f−1 up (Ofup(hE)), gE ⊑ f−1 up (Ofup(gE)) and f−1 up (Ofup(hE))q̃f −1 up (Ofup(gE)). Since fup is FS-continuous, we get f−1 up (Ofup(hE)) and f−1 up (Ofup(gE)) are FSO-sets in (U, δ, E). The proof is complete. Theorem 15. If fup : (U, δ,E)−→(V, σ,K) is FS-open, FSg-continuous bijection, and (U, δ, E) is FS-GR3, then (V, σ,K) is FS-GR3. Proof. It is analogous to that of the above theorem. Theorem 16. If fup : (U, δ, E)−→(V, σ,K) is FSgc-irresolute, FS-open onto and (U, δ, E) is FS-GR3, then (V, σ,K) is FS-GR3. Proof. It is similar to that of Theorem 14. The next two theorems show that FS-GR2 and FS-GR3 are hereditary property. Theorem 17. Every FS-subspace (ṼE , δV , E) of FS-GR2 is FS-GR2 . Proof. Let (U, δ,E) be FS-GR2. Suppose that hE any FSg-closed set in (ṼE , δV , E) with xeαq̃hE for any FS-point in (ṼE , δV , E), then there is an FSC-set and so FSg-closed set fE in (U, δ,E) with hE = ṼE ⊓ fE and xeαq̃fE . Since (U, δ,E) is FS-GR2, there are Oxe α , OfE ∈ δ such that Oxe α q̃OfE . Now take O∗ xe α = ṼE ⊓Oxe α ∈ δV and O∗ fE = ṼE ⊓OfE ∈ δV , then O∗ xe α and O∗ fE are FSO-sets in (ṼE , δV , E) containing xeα and fE respectively, such that O∗ xe α q̃O∗ fE . The result holds. Theorem 18. Every FSC-subspace (ṼE , δV , E) of FS-GR3 is FS-GR3. Proof. It is similar to that of the above theorem. REFERENCES 189 Theorem 19. (U, δτ , E) is FS-GR2 if and only if (U, τ) is g-regular. Proof. Let (U, δτ , E) be FS-GR2 and B any closed set in (U, τ) with x /∈ B, then B is a g-closed set and there is an FSC-set fE such that fE = χ̃B. Clearly fE is an FSg-closed set with xe1q̃fE . Since (U, δτ , E) is FS-GR2, there are Oxe 1 , OfE ∈ δτ such that Oxe 1 q̃OfE . Thus there are Ox, OB ∈ τ such that Oxe 1 = χ̃Ox , OfE = χ̃OB and Ox ∩OB = ∅. Therefore (U, τ) is g-regular. Conversely, let (U, τ) be g-regular and hE any closed set in (U, δτ , E) with xeαq̃hE . Then hE is an FSg-closed set and there is a closed set F in (U, τ) such that hE = χ̃OF and x /∈ F . Clearly F is g-closed and (U, τ) is g-regular, then there are Ox, OF ∈ τ with Ox ∩OF = ∅ and so, there are Oxe α and OhE ∈ δτ such that Oxe α = χ̃Ox , OhE = χ̃OF and Oxe α q̃OhE . Hence (U, δτ , E) is FS-GR2. Theorem 20. (U, δτ , E) is FS-GR3 if and only if (U, τ) is g-normal. Proof. It is similar to that of the above theorem. From the obtained results in section 2, 3. we conclude the next relations. Corollary 3. For An FSTS (U, τ, E), the next implications hold. 1) FSG4 ⇔ FST4 ⇒ FST3 ⇔ FSG3 ⇔ FS-GR2 ∧ FS-symmetric ⇒ FSR2. 2)FSG3 ⇒ FST2 1 2 ⇒ FST2 ⇒ FST1 ⇒ FST0. 5. Conclusion The topological structures play an important role in many applications of complex real- life problems in various field, specially the fields that concerned with handling all cases that contain uncertainties such as medical diagnosis , economic, and decision making,..etc. In this work, we introduced and studied the new classes of spaces namely, FSg-regular and FSg-normal space via fuzzy soft generalized closed sets. We investigated some characteri- zations for them. Some related theorems and relations are presented with some necessary examples. In addition, the hereditary property and some preservation theorems. In the future work we will try to present some applications for fuzzy soft generalized sets in different aspects. 6. Conflicts of interest The authors declare no conflict of interest. References [1] M. E. Abd El-Monsef, M. A. El-Gayar, and R. M. Aqeel. A comparison of three types of rough fuzzy sets based on two universal sets. International Journal of Machine Learning and Cybernetics, 8(1):343–353, 2017. REFERENCES 190 [2] ME Abd El-Monsef, MA El-Gayar, and RM Aqeel. On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces. International Journal of Granular Computing, Rough Sets and Intelligent Systems, 3(4):257–271, 2014. [3] S. P. Arya and M. P. Bhamini. A generalisation of normal spaces. Matematički Vesnik, 35(81):1–10, 1983. [4] S. Atmaca and I. Zorlutuna. On fuzzy soft topological spaces. Ann. Fuzzy Math. Inform, 5(2):377–386, 2013. [5] G. Balasubramanian and P. Sundaram. On some generalizations of fuzzy continuous functions. Fuzzy sets and systems, 86(1):93–100, 1997. [6] C. Chang. Fuzzy topological spaces. Journal of mathematical Analysis and Applica- tions, 24(1):182–190, 1968. [7] Á. Császár. Generalized open sets. Acta mathematica hungarica, 75(1-2):65–87, 1997. [8] Á. Császár. Normal generalized topologies. Acta Mathematica Hungarica, 115(4):309– 313, 2007. [9] İ. Demir and O. Özbakır. Some properties of fuzzy soft proximity spaces. The Scientific World Journal, 2015, 2015. [10] M. K. El-Bably and E. A. Abo-Tabl. A topological reduction for predicting of a lung cancer disease based on generalized rough sets. Journal of Intelligent & Fuzzy Systems, 42(2):3045–3060, 2021. [11] M. A. El-Gayar and A. A. El Atik. Topological models of rough sets and decision making of covid-19. Complexity, 2022, 2022. [12] M. E. El-Shafei. Some applications of generalized closed sets in fuzzy topological spaces. Kyungpook Mathematical Journal, 45(1):13–19, 2005. [13] R. Engelking. General Topology, PWN-Polish Sci. 1977. [14] El-Bably M. K., Ali M. I., and E. A. Abo-Tabl. New topological approaches to gener- alized soft rough approximations with medical applications. Journal of Mathematics, 2021, 2021. [15] L. Kalantan. Results about-normality. Topology Appl., 125:47–62, 2002. [16] A. Kharal and B. Ahmad. Mappings on fuzzy soft classes. Advances in fuzzy systems, 2009, 2009. [17] N. Levine. Semi-open sets and semi-continuity in topological spaces. The American mathematical monthly, 70(1):36–41, 1963. REFERENCES 191 [18] N. Levine. Generalized closed sets in topology. Rendiconti del Circolo Matematico di Palermo, 19(1):89–96, 1970. [19] P. K. Maji, R. Biswas, and A. Roy. Fuzzy soft sets. J. Fuzzy math., 9(3), 2001. [20] P. Mukherjee, R. P. Chakraborty, and C. Park. On fuzzy soft δ-open sets and fuzzy soft δ-continuity. Ann. Fuzzy Math. Inform, 11(2):327–340, 2016. [21] B. M. Munshi. Separation axioms. Acta Ciencia Indica, 12(2):140–145, 1986. [22] M. Navaneethakrishnan and J. P. Joseph. g-closed sets in ideal topological spaces. Acta Mathematica Hungarica, 119(4):365–371, 2008. [23] M. Navaneethakrishnan, J. P. Joseph, and D Sivaraj. I g-normal and i g-regular spaces. Acta Mathematica Hungarica, 125(4):327–340, 2009. [24] T. Noiri and V. Popa. On g-regular spaces and some functions. Mem. Fac. Sci. Kochi Univ. Math, 20:67–74, 1999. [25] R. Parimelazhagan and V. Subramonia. Strongly g* closed sets in topological spaces. Int. Jou. Of Math. Analy, 6(30):1481–1489, 2012. [26] J. K. Park and J. H. Park. Mildly generalized closed sets, almost normal and mildly normal spaces. Chaos, Solitons & Fractals, 20(5):1103–1111, 2004. [27] T. Rajendrakumar and G. Anandajothi. On fuzzy strongly g-closed sets in fuzzy topological spaces. Intern. J. Fuzzy Mathematical Archive, 3:68–75, 2013. [28] S. Saleh, A. M. Abd EL-Latif, and AL-Salemi Amany. On separation axioms in fuzzy soft topological spaces. South Asian J. Math., 8(2):92–102, 2018. [29] S. Saleh and Amani Al-Salemi. The r 0 and r 1 properties in fuzzy soft topological spaces. Journal of new theory, (24):50–58, 2018. [30] B. Tanay and M. B. Kandemir. Topological structure of fuzzy soft sets. Computers & Mathematics with Applications, 61(10):2952–2957, 2011. [31] Z. Tarrannum and A. P. Narappanavar. Fuzzy soft generalized closed sets in fuzzy soft topological spaces. International Journal of Applied Engineering Research, 14(3):633– 636, 2019. [32] M.K.R.S. Veera Kumar. Between closed sets and g-closed sets. Mem. Fac. Sci. Kochi Univ. Ser. A Math., 21:1–19, 2000. [33] L. A. Zadeh. Information and control. Fuzzy sets, 8(3):338–353, 1965.