# Onr-Generalized Fuzzy ℓ-Closed Sets: Properties and Applications

Authors: I. M. Taha
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present study, we introduce and characterize the class ofr-generalized fuzzy ℓ-closed sets in a fuzzy ideal topological space X,τ,ℓ in Šostak sense. Also, we show that r-generalized fuzzy closed set by Kim and Park (2002) ⟹r-generalized fuzzy ℓ-closed set, but the converse need not be true. Moreover, if we take ℓ=ℓ0, the r-generalized fuzzy ℓ-closed set and r-generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized ℓ-continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are studied with the help of examples. Finally, some separation axioms of r-generalized fuzzy ℓ-closed sets are introduced and studied. Also, the notion of r-fuzzy G∗-connected sets is defined and studied with help of r-generalized fuzzy ℓ-closed sets.


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## Body

## 1. Introduction

Zadeh [1] introduced the basic idea of a fuzzy set as an extension of classical set theory. The theory of fuzzy sets provides a framework for mathematical modeling of those real-world situations, which involve an element of imprecision, uncertainty, or vagueness in their description. This theory has found wide applications in engineering, information sciences, etc.; for details, the reader is referred to [2, 3].A fuzzy multifunction is a fuzzy set valued function [4–7]. The biggest difference between fuzzy functions and fuzzy multifunctions has to do with the definition of an inverse image. For a fuzzy multifunction, there are two types of inverses, upper and lower. These two definitions of the inverse then lead to two definitions of continuity; for more details, the reader is referred to [8, 9].In this work, we introduce and characterize the class ofr-generalized fuzzy ℓ-closed sets in a fuzzy ideal topological space in Šostak sense. Also, we show that r-generalized fuzzy closed set ⟹r-generalized fuzzy ℓ-closed set, but the converse need not be true. Moreover, if we take ℓ=ℓ0, the r-generalized fuzzy ℓ-closed set and r-generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized ℓ-continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are discussed with the help of examples. In the end, some separation axioms of r-generalized fuzzy ℓ-closed sets are introduced and studied. Also, the notion of r-fuzzy G∗-connected sets is defined and studied with help of r-generalized fuzzy ℓ-closed sets.


## 2. Preliminary Assertions

In this section, we present the basic definitions and results which we need in the next sections. Throughout this work,X refers to an initial universe, IX is the set of all fuzzy sets on X, and for μ∈IX, μca=1−μa for all a∈X (where I°=0,1 and I=0,1). For s∈I, s¯a=s for all a∈X. The difference between μ,λ∈IX [9] is defined as follows:(1)μ∧¯λ=0¯,ifμ≤λ,μ∧λc,otherwise.A fuzzy idealℓ on Y [10] is a map ℓ:IY⟶I that satisfies the following:(i)
∀μ,λ∈IY and μ≤λ⟹ℓλ≤ℓμ(ii)
∀μ,λ∈IY⟹ℓμ∨λ≥ℓμ∧ℓλThe simplest fuzzy ideals onY, ℓ0, and ℓ1 are defined as follows:(2)ℓ0μ=1,ifμ=0¯,0,otherwiseandℓ1μ=1∀μ∈IY.In a fuzzy topological spaceX,τ based on the sense of Šostak [11], the closure and the interior of μ∈IX are denoted by Cτμ,r and Iτμ,r. A fuzzy set λ∈IX is called a r-generalized fuzzy closed set [12] if Cτλ,r≤μ whenever λ≤μ and τμ≥r. Also, X,τ is called r-GF-regular [12] iff xtq¯λ for each r-generalized fuzzy closed set λ∈IX implies that there exist μi∈IX with τμi≥r for i∈1,2 such that xt∈μ1, λ≤μ2, and μ1q¯μ2. Moreover, X,τ is called r-GF-normal [12] iff λ1q¯λ2 for each of the r-generalized fuzzy closed sets λi∈IX for i∈1,2 implies that there exist μi∈IX with τμi≥r such that λi≤μi and μ1q¯μ2.Definition 1 (see [9]).
LetX,τ,ℓ be a fuzzy ideal topological space, λ∈IX, and r∈I°. Then, the r-fuzzy local function λr∗ of λ is defined as follows: λr∗=∧μ∈IX:ℓλ∧¯μ≥r,τμc≥r.Remark 1 (see [9]).
(i)
If we takeℓ=ℓ0, for each λ∈IX, we have(3)λr∗=∧μ∈IX:λ≤μ,τμc≥r=Cτλ,r.(ii)
If we takeℓ=ℓ1 (resp., ℓλ≥r), for each λ∈IX, we have λr∗=0¯.Definition 2 (see [9]).
LetX,τ,ℓ be a fuzzy ideal topological space. Then, for each λ∈IX and r∈I°, we define an operator Cτ∗:IX×I°⟶IX as follows:(4)Cτ∗λ,r=λ∨λr∗.
Now, ifℓ=ℓ0, then Cτ∗λ,r=λ∨λr∗=λ∨Cτλ,r=Cτλ,r for each λ∈IX.Theorem 1 (see [9]).
LetX,τ,ℓ be a fuzzy ideal topological space. Then, for any fuzzy sets λ,ν∈IX and r∈I°, the operator Cτ∗:IX×I°⟶IX satisfies the following properties:(i)
Cτ∗0¯,r=0¯(ii)
λ≤Cτ∗λ,r≤Cτλ,r(iii)
Ifλ≤ν, then Cτ∗λ,r≤Cτ∗ν,r(iv)
Cτ∗λ∨ν,r=Cτ∗λ,r∨Cτ∗ν,r(v)
Cτ∗λ∧ν,r≤Cτ∗λ,r∧Cτ∗ν,r(vi)
Cτ∗Cτ∗λ,r,r=Cτ∗λ,rDefinition 3 (see [13, 14]).
LetX,τ,ℓ be a fuzzy ideal topological space, λ∈IX, and r∈I°. Then, λ is said to be(i)
r-fuzzy ℓ-open iff λ≤Iτλr∗,r(ii)
r-fuzzy semi-ℓ-open iff λ≤Cτ∗Iτλ,r,r(iii)
r-fuzzy pre-ℓ-open iff λ≤IτCτ∗λ,r,r(iv)
r-fuzzy α-ℓ-open iff λ≤IτCτ∗Iτλ,r,r,r(v)
r-fuzzy β-ℓ-open iff λ≤Cτ∗IτCτ∗λ,r,r,r(vi)
r-fuzzy δ-ℓ-open iff IτCτ∗λ,r,r≤Cτ∗Iτλ,r,r
The complement ofr-fuzzy ℓ-open (resp., semi-ℓ-open, pre-ℓ-open, α-ℓ-open, β-ℓ-open, and δ-ℓ-open) set is a r-fuzzy ℓ-closed (resp., semi-ℓ-closed, pre-ℓ-closed, α-ℓ-closed, β-ℓ-closed, and δ-ℓ-closed) set.Definition 4 (see [15, 16]).
LetH:X⊸Y. Then, H is called a fuzzy multifunction iff Hx∈IY for each x∈X. The degree of membership of y in Hx is denoted by Hxy=GHx,y for any x,y∈X×Y. The domain of H is denoted by domH and the range of H is denoted by rngH, for any x∈X and y∈Y: domHx=∨y∈YGHx,y and rngHy=∨x∈XGHx,y.Definition 5 (see [15]).
LetH:X⊸Y be a fuzzy multifunction. Then, H is called(i)
Normalized iff for eachx∈X, there exists y0∈Y such that GHx,y0=1.(ii)
A crisp iffGHx,y=1 for each x∈X and y∈Y.Definition 6 (see [15]).
LetH:X⊸Y be a fuzzy multifunction. Then, we have the following:(i)
The image ofλ∈IX is a fuzzy set Hλ∈IY and defined by(5)Hλy=∨x∈XGHx,y∧λx.(ii)
The lower inverse ofμ∈IY is a fuzzy set Hlμ∈IX and defined by(6)Hlμx=∨y∈YGHx,y∧μy.(iii)
The upper inverse ofμ∈IY is a fuzzy set Huμ∈IX and defined by(7)Huμx=∧y∈YGHcx,y∨μy.
All properties of image, lower and upper, are found in [15].


## 3. Onr-Generalized Fuzzy ℓ-Closed Sets

In this section, we introduce and characterize the class ofr-generalized fuzzy ℓ-closed sets in a fuzzy ideal topological space X,τ,ℓ in Šostak sense. Also, we show that r-generalized fuzzy closed set ⟹r-generalized fuzzy ℓ-closed set, but the converse need not be true. Moreover, if we take ℓ=ℓ0, the r-generalized fuzzy ℓ-closed set and r-generalized fuzzy closed set are equivalent. Our fundamental definition is the following.Definition 7.
Letμ,λ∈IX and r∈I°. In X,τ,ℓ, μ is called a r-generalized fuzzy ℓ-closed set if Cτ∗μ,r≤λ whenever μ≤λ and τλ≥r. The complement of r-generalized fuzzy ℓ-closed set is called r-generalized fuzzy ℓ-open.Lemma 1.
Everyr-generalized fuzzy closed set [12] is r-generalized fuzzy ℓ-closed.Proof.
It follows from Theorem1 (2).
In general, the converse of Lemma1 is not true as shown by Example 1.Example 1.
Defineτ,ℓ:IX⟶I as follows:(8)τμ=1,ifμ∈0¯,1¯,13,ifμ=0.4¯,0,otherwise,ℓν=1,ifν=0¯,13,if0¯<ν≤0.4¯,0,otherwise.
Then,0.35¯ is 1/3-generalized fuzzy ℓ-closed set but it is not 1/3-generalized fuzzy closed.Lemma 2.
Ifℓ=ℓ0, the r-generalized fuzzy ℓ-closed set and r-generalized fuzzy closed set are equivalent.Proof.
It follows from Definition2.
The following implications are obtained:(9)r−fuzzy closed⇓r−generalized fuzzy closed⇓r−generalized fuzzyℓ−closed.Remark 2.
r-generalized fuzzy ℓ-closed and r-fuzzy α-ℓ-closed [13] are independent notions as shown by Examples 2 and 3.Example 2.
LetX=a,b,c,d be a set and λ1,λ2∈IX be defined as follows: λ1=a/0.9,b/0.9,c/0.4,d/0.5 and λ2=a/0.1,b/0.1,c/0.1,d/0.5. Define τ,ℓ:IX⟶I as follows:(10)τμ=1,ifμ∈0¯,1¯,23,ifμ=λ1,0,otherwise,ℓν=1,ifν=0¯,23,if0¯<ν<0.3¯,0,otherwise.
Then,λ2 is a 2/3-fuzzy α-ℓ-closed set, but it is not 2/3-generalized fuzzy ℓ-closed.Example 3.
Defineτ,ℓ:IX⟶I as follows:(11)τμ=1,ifμ∈0¯,1¯,13,ifμ=0.2¯,23,ifμ=0.8¯,0,otherwise,ℓν=1,ifν=0¯,23,if0¯<ν<0.4¯,0,otherwise.
Then,0.5¯ is a 1/3-generalized fuzzy ℓ-closed set, but it is not 1/3-fuzzy α-ℓ-closed.Remark 3.
r-generalized fuzzy ℓ-closed and r-fuzzy δ-ℓ-closed [14] are independent notions as shown by Examples 4 and 5.Example 4.
LetX=a,b,c,d be a set and λ1,λ2,λ3,λ4∈IX be defined as follows: λ1=a/1.0,b/0.0,c/0.0,d/0.0, λ2=a/0.0,b/1.0,c/0.0,d/0.0, λ3=a/0.0,b/0.0,c/1.0,d/0.0, and λ4=a/0.0,b/0.0,c/0.0,d/1.0. Define τ,ℓ:IX⟶I as follows:(12)τμ=1,ifμ∈0¯,1¯,13,ifμ=λ4,23,ifμ=λ1∨λ2∨λ3,0,otherwise,ℓν=1,ifν=0¯,13,ifν=λ3,23,if0¯<ν<λ3,0,otherwise.
Then,λ3∨λ4c is a 1/3-fuzzy δ-ℓ-closed set, but it is not 1/3-generalized fuzzy ℓ-closed.Example 5.
LetX=a,b,c,d be a set and λ1,λ2,λ3∈IX be defined as follows: λ1=a/1.0,b/0.0,c/0.0,d/0.0, λ2=a/0.0,b/1.0,c/0.0,d/0.0, and λ3=a/0.0,b/0.0,c/1.0,d/0.0. Define τ,ℓ:IX⟶I as follows:(13)τμ=1,ifμ∈0¯,1¯,23,ifμ=λ1∨λ2,0,otherwise,ℓν=1,ifν=0¯,0,otherwise.
Then,λ1∨λ3c is a 2/3-generalized fuzzy ℓ-closed set, but it is not 2/3-fuzzy δ-ℓ-closed.Definition 8.
InX,τ,ℓ, for any μ∈IX and r∈I°, we define GCτ∗:IX×I°⟶IX as follows: GCτ∗μ,r=∧λ∈IX:μ≤λ,λisr−generalized fuzzyℓ−closed.Theorem 2.
InX,τ,ℓ, for any μ,ν∈IX, the operator GCτ∗:IX×I°⟶IX satisfies the following:(i)
GCτ∗0¯,r=0¯(ii)
μ≤GCτ∗μ,r(iii)
Ifμ≤ν, then GCτ∗μ,r≤GCτ∗ν,r(iv)
GCτ∗GCτ∗μ,r,r=GCτ∗μ,r(v)
GCτ∗μ,r∨GCτ∗ν,r≤GCτ∗μ∨ν,r(vi)
Ifμ is r-generalized fuzzy ℓ-closed, then GCτ∗μ,r=μProof.
(i), (ii), (iii), and (vi) are easily proved from the definition ofGCτ∗.(iv) From (ii) and (iii),GCτ∗μ,r≤GCτ∗GCτ∗μ,r,r. Now, we show that GCτ∗μ,r≥GCτ∗GCτ∗μ,r,r. Suppose that GCτ∗μ,r≱GCτ∗GCτ∗μ,r,r. There exist x∈X and t∈0,1 such that(14)GCτ∗μ,rx<t<GCτ∗GCτ∗μ,r,rx.A.SinceGCτ∗μ,rx<t, by the definition of GCτ∗, there exists r−generalized fuzzyℓ−closedλ1 with μ≤λ1 such that GCτ∗μ,rx≤λ1x<t. Since μ≤λ1, we have GCτ∗μ,r≤λ1. Again, by the definition of GCτ∗, we have GCτ∗GCτ∗μ,r,r≤λ1. Hence, GCτ∗GCτ∗μ,r,rx≤λ1x<t; it is a contradiction for A. Thus, GCτ∗μ,r≥GCτ∗GCτ∗μ,r,r. Then, GCτ∗GCτ∗μ,r,r=GCτ∗μ,r.(v) However,μ and ν≤μ∨ν imply GCτ∗μ,r≤GCτ∗μ∨ν,r and GCτ∗ν,r≤GCτ∗μ∨ν,r. Thus, GCτ∗μ,r∨GCτ∗ν,r≤GCτ∗μ∨ν,r. Hence, the proof is complete.
In a similar way, one can prove the following theorem.Theorem 3.
InX,τ,ℓ, for any μ∈IX and r∈I°, we define GIτ∗:IX×I°⟶IX as follows: GIτ∗μ,r=∨λ∈IX:λ≤μ,λisr−generalized fuzzyℓ−open. For each μ,ν∈IX, the operator GIτ∗ satisfies the following:(i)
GIτ∗1¯,r=1¯(ii)
GIτ∗μ,r≤μ(iii)
GIτ∗μc,r=GCτ∗μ,rc(iv)
GIτ∗GIτ∗μ,r,r=GIτ∗μ,r(v)
GIτ∗μ,r∧GIτ∗ν,r≥GIτ∗μ∧ν,r(vi)
Ifμ is r-generalized fuzzy ℓ-open, then GIτ∗μ,r=μ


## 4. Fuzzy Upper and Lower Generalizedℓ-Continuous Multifunctions

In this section, we define fuzzy upper (resp., lower) generalizedℓ-continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are discussed with the help of examples. Moreover, we show that the notions of fuzzy upper (resp., lower) semi-ℓ-continuous multifunctions [17] and fuzzy upper (resp., lower) generalized ℓ-continuous multifunctions are independent.Definition 9.
A fuzzy multifunctionH:X,τ,ℓ⊸Y,η is called(i)
Fuzzy upper generalizedℓ-continuous at a fuzzy point xt∈domH iff xt∈Huμ for each μ∈IY with ημ≥r, there exists a r-generalized fuzzy ℓ-open set λ∈IX and xt∈λ such that λ∧domH≤Huμ(ii)
Fuzzy lower generalizedℓ-continuous at a fuzzy point xt∈domH iff xt∈Hlμ for each μ∈IY with ημ≥r, there exists a r-generalized fuzzy ℓ-open set λ∈IX and xt∈λ such that λ≤Hlμ(iii)
Fuzzy upper (resp., lower) generalizedℓ-continuous iff it is fuzzy upper (resp., lower) generalized ℓ-continuous at every xt∈domHRemark 4.
IfH is normalized, then H is fuzzy upper generalized ℓ-continuous at a fuzzy point xt∈domH iff xt∈Huμ for each μ∈IY with ημ≥r, there exists r-generalized fuzzy ℓ-open set λ∈IX and xt∈λ such that λ≤Huμ.Theorem 4.
A fuzzy multifunctionH:X,τ,ℓ⊸Y,η is fuzzy lower generalized ℓ-continuous if Hlμ (resp., Huμ) is r-generalized fuzzy ℓ-open (resp., r-generalized fuzzy ℓ-closed) for each μ∈IY with ημ≥r (resp., μ∈IY with ημc≥r), r∈I°.Proof.
Letxt∈domH, μ∈IY with ημ≥r, and xt∈Hlμ. Then, we have Hlμ as r-generalized fuzzy ℓ-open. Thus, H is fuzzy lower generalized ℓ-continuous. The other case follows similar lines and from Huμc=Hlμc.
In a similar way, one can prove the following theorem.Theorem 5.
A normalized fuzzy multifunctionH:X,τ,ℓ⊸Y,η is fuzzy upper generalized ℓ-continuous if Huμ (resp., Hlμ) is a r-generalized fuzzy ℓ-open (resp., r-generalized fuzzy ℓ-closed) set for each μ∈IY with ημ≥r (resp., μ∈IY with ημc≥r) and r∈I°.Lemma 3.
Every fuzzy lower (resp., upper) generalized continuous multifunction [18] is a fuzzy lower (resp., upper) generalized ℓ-continuous multifunction.Proof.
It follows from Lemma1.
In general, the converse of Lemma3 is not true as shown by Example 6.Example 6.
LetX=a1,a2, Y=b1,b2,b3, and H:X⊸Y be a fuzzy multifunction defined by GHa1,b1=0.2,GHa1,b2=0.8,GHa1,b3=0.3,GHa2,b1=0.5,GHa2,b2=0.3, and GHa2,b3=0.7. Define τ,ℓ:IX⟶I and η:IY⟶I as follows:(15)τμ=1,ifμ∈0¯,1¯,13,ifμ=0.4¯,0,otherwise,ηλ=1,ifλ∈0¯,1¯,13,ifλ=0.65¯,0,otherwise,ℓν=1,ifν=0¯,13,if0¯<ν≤0.4¯,0,otherwise.
Then,H:X,τ,ℓ⊸Y,η is fuzzy lower (resp., upper) generalized ℓ-continuous but not fuzzy lower (resp., upper) generalized continuous.Lemma 4.
LetH:X,τ,ℓ⊸Y,η be a fuzzy multifunction (resp., normalized fuzzy multifunction). If ℓ=ℓ0, fuzzy lower (resp., upper) generalized ℓ-continuous multifunction and fuzzy lower (resp., upper) generalized continuous multifunction are equivalent.Proof.
It follows from Lemma2.Remark 5.
Fuzzy lower (resp., upper) semi-ℓ-continuous multifunctions [17] and fuzzy lower (resp., upper) generalized ℓ-continuous multifunctions are independent notions as shown by Examples 7 and 8.Example 7.
LetX=a1,a2, Y=b1,b2,b3, and H:X⊸Y be a fuzzy multifunction defined by GHa1,b1=0.1,GHa1,b2=1.0,GHa1,b3=0.3,GHa2,b1=0.5,GHa2,b2=0.1, and GHa2,b3=1.0. Define τ,ℓ:IX⟶I and η:IY⟶I as follows:(16)τμ=1,ifμ∈0¯,1¯,13,ifμ=0.2¯,23,ifμ=0.8¯,0,otherwise,ℓν=1,ifν=0¯,13,if0¯<ν<0.4¯,0,otherwise,ηλ=1,ifλ∈0¯,1¯,13,ifλ=0.5¯,0,otherwise.
Then,H:X,τ,ℓ⊸Y,η is fuzzy lower (resp., upper) generalized ℓ-continuous, but it is not fuzzy lower (resp., upper) semi-ℓ-continuous.Example 8.
LetX=a1,a2,a3, Y=b1,b2,b3, and H:X⊸Y be a fuzzy multifunction defined by GHa1,b1=0.8,GHa1,b2=0.3,GHa1,b3=0.3,GHa2,b1=0.6,GHa2,b2=0.1, GHa2,b3=0.4,GHa3,b1=0.1,GHa3,b2=0.2,andGHa3,b3=1.0. Define λ1,λ2∈IX and λ3∈IY as follows: λ1=a1/0.3,a2/0.4,a3/0.8, λ2=a1/0.2,a2/0.3,a3/0.2, and λ3=b1/0.3,b2/0.4,b3/0.8. Define τ,ℓ:IX⟶I and η:IY⟶I as follows:(17)τμ=1,ifμ∈0¯,1¯,12,ifμ=λ2,0,otherwise.ℓν=1,ifν=0¯,23,if0¯<ν<0.2¯,0,otherwise.ηλ=1,ifλ∈0¯,1¯,34,ifλ=λ3,0,otherwise.
Then,H:X,τ,ℓ⊸Y,η is fuzzy lower (resp., upper) semi-ℓ-continuous, but it is not fuzzy lower (resp., upper) generalized ℓ-continuous.Theorem 6.
LetX,τ,ℓ be a fuzzy ideal topological space that satisfies the condition: for λ∈IX, μ∈IY, and r∈I°, G∗ If λ=GIτ∗λ,r, it implies that λ is r-generalized fuzzy ℓ-open.Then,H:X,τ,ℓ⊸Y,η is fuzzy lower generalized ℓ-continuous if one of the following statements is hold:(i)
HGCτ∗λ,r≤CηHλ,r(ii)
GCτ∗Huμ,r≤HuCημ,r(iii)
HlIημ,r≤GIτ∗Hlμ,rProof.
(i)⟹ (ii) For each μ∈IY and r∈I°, let λ=Huμ. Then, by (i), HGCτ∗Huμ,r≤CηHHuμ,r≤Cημ,r. Hence, we have GCτ∗Huμ,r≤HuCημ,r.(ii)⟹ (iii) Let μ∈IY. Then, by (ii), GIτ∗Hlμ,rc=GCτ∗Huμc,r≤HuCημc,r=HlIημ,rc. Hence, HlIημ,r≤GIτ∗Hlμ,r.
Suppose that (iii) holds. Letμ∈IY with ημ≥r. Then, by (iii), Hlμ≤GIτ∗Hlμ,r, and by G∗, Hlμ is r-generalized fuzzy ℓ-open. Thus, H is fuzzy lower generalized ℓ-continuous. Hence, the proof is complete.Corollary 1.
LetH:X,τ,ℓ⊸Y,η and F:Y,η⊸Z,γ be two fuzzy multifunctions. Then, we have the following:(i)
IfH is fuzzy lower generalized ℓ-continuous and F is fuzzy lower semicontinuous [12], then F°H is fuzzy lower generalized ℓ-continuous.(ii)
IfH is normalized fuzzy upper generalized ℓ-continuous and F is normalized fuzzy upper semicontinuous [15], then F°H is fuzzy upper generalized ℓ-continuous.


## 5. Some Applications ofr-Generalized Fuzzy ℓ-Closed Sets

In this section, some separation axioms ofr-generalized fuzzy ℓ-closed sets are introduced and studied. Also, the notion of r-fuzzy G∗-connected sets is defined and studied with help of r-generalized fuzzy ℓ-closed sets.Definition 10.
(i)
X,τ,ℓ is called r-GF∗-regular iff xtq¯λ for each r-generalized fuzzy ℓ-closed set, λ∈IX implies that there exist μi∈IX with τμi≥r for i∈1,2 such that xt∈μ1, λ≤μ2, and μ1q¯μ2.(ii)
X,τ,ℓ is called r-GF∗-normal iff λ1q¯λ2 for each r-generalized fuzzy ℓ-closed sets, λi∈IX for i∈1,2 imply that there exist μi∈IX with τμi≥r such that λi≤μi and μ1q¯μ2.Lemma 5.
Everyr-GF-regular (resp., r-GF-normal) space X,τ,ℓ [12] is r-GF∗-regular (resp., r-GF∗-normal) space.Proof.
It follows from Lemma1.Lemma 6.
Ifℓ=ℓ0, r-GF∗-regular (resp., r-GF∗-normal) space and r-GF-regular (resp., r-GF-normal) space are equivalent.Proof.
It follows from Lemma2.Theorem 7.
For anyX,τ,ℓ, the following are equivalent:(i)
X,τ,ℓ is r-GF∗-regular(ii)
Ifxt∈λ for each r-generalized fuzzy ℓ-open set λ∈IX, there exists μ∈IX with τμ≥r such that xt∈μ≤Cτμ,r≤λ(iii)
Ifxtq¯λ for each r-generalized fuzzy ℓ-closed set λ∈IX, there exist μi∈IX with τμi≥r for i∈1,2 such that xt∈μ1, λ≤μ2, and Cτμ1,rq¯Cτμ2,rProof.
(i) (⟹) (ii) Let xt∈λ for each r-generalized fuzzy ℓ-open λ. Then, xtq¯λc for r-generalized fuzzy ℓ-closed λc. Since X,τ is r-GF∗-regular, there exist μ,ν∈IX with τμ≥r and τν≥r such that xt∈μ, λc≤ν, and μq¯ν. It implies that xt∈μ≤νc≤λ. Since τν≥r, xt∈μ≤Cτμ,r≤λ.(ii) (⟹) (iii) Let xtq¯λ for each r-generalized fuzzy ℓ-closed λ. Then, xt∈λc for r-generalized fuzzy ℓ-open λc. By (ii), there exists μ∈IX with τμ≥r such that xt∈μ≤Cτμ,r≤λc. Since τμ≥r, μ is r-generalized fuzzy ℓ-open and xt∈μ. Again, by (ii), there exists μ1∈IX with τμ1≥r such that(18)xt∈μ1≤Cτμ1,r≤μ≤Cτμ,r≤λc.It impliesλ≤Cτμ,rc=Iτμc,r≤μc. Put μ2=Iτμc,r; then, τμ2≥r. So, Cτμ2,r≤μc≤Cτμ1,rc, that is, Cτμ1,rq¯Cτμ2,r.(iii) (⟹) (i) It is trivial.
In a similar way, one can prove the following theorem.Theorem 8.
For anyX,τ,ℓ, the following are equivalent:(i)
X,τ,ℓ is r-GF∗-normal.(ii)
Ifν≤λ for each r-generalized fuzzy ℓ-closed ν∈IX and r-generalized fuzzy ℓ-open λ∈IX, there exists μ∈IX with τμ≥r such that ν≤μ≤Cτμ,r≤λ(iii)
Ifλ1q¯λ2 for each r-generalized fuzzy ℓ-closed sets λi∈IX for i∈1,2, there exist μi∈IX with τμi≥r such that λi≤μi and Cτμ1,rq¯Cτμ2,rDefinition 11.
InX,τ,ℓ, μ and λ∈IX are called r-fuzzy G∗-separated iff μq¯GCτ∗λ,r and λq¯GCτ∗μ,r. Also, any fuzzy set which cannot be expressed as the union of two r-fuzzy G∗-separated sets is called r-fuzzy G∗-connected set.Theorem 9.
For anyX,τ,ℓ, we have the following:(i)
Ifμ, λ∈IX are r-fuzzy G∗-separated and ω,ν∈IX−0¯ such that ν≤μ and ω≤λ, then ω and ν are r-fuzzy G∗-separated(ii)
Ifμq¯λ and either both are r-generalized fuzzy ℓ-closed or both are r-generalized fuzzy ℓ-open, then μ and λ are r-fuzzy G∗-separated(iii)
Ifμ and λ are either both r-generalized fuzzy ℓ-closed or both r-generalized fuzzy ℓ-open, then μ∧λc and λ∧μc are r-fuzzy G∗-separatedProof.
(i) and (ii) are obvious.(iii) Letμ and λ be r-generalized fuzzy ℓ-open. Since μ∧λc≤λc, GCτ∗μ∧λc,r≤λc, and hence, GCτ∗μ∧λc,rq¯λ. Then, GCτ∗μ∧λc,rq¯λ∧μc. Again, since λ∧μc≤μc, GCτ∗λ∧μc,r≤μc, and hence, GCτ∗λ∧μc,rq¯μ. Then, GCτ∗λ∧μc,rq¯μ∧λc. Thus, μ∧λc and λ∧μc are r-fuzzy G∗-separated. The other case follows similar lines.Theorem 10.
For anyX,τ,ℓ, μ and λ∈IX−0¯ are r-fuzzy G∗-separated iff there exist two r-generalized fuzzy ℓ-open sets ω and ν such that λ≤ω, μ≤ν, λq¯ν, and μq¯ω.Proof.
(⟹) Let μ and λ∈IX−0¯ be r-fuzzy G∗-separated, λ≤GCτ∗μ,rc=ω, and μ≤GCτ∗λ,rc=ν, where ν and ω are r-generalized fuzzy ℓ-open; then, νq¯GCτ∗λ,r and ωq¯GCτ∗μ,r. Thus, μq¯ω and λq¯ν. Hence, we obtain the desired result.
(⇐) Let ω and ν be r-generalized fuzzy ℓ-open sets such that μ≤ν, λ≤ω, μq¯ω, and λq¯ν. Then, μ≤ωc and λ≤νc. Hence, GCτ∗μ,r≤ωc and GCτ∗λ,r≤νc. Then, GCτ∗μ,rq¯λ and GCτ∗λ,rq¯μ. Thus, μ and λ are r-fuzzy G∗-separated. Hence, we obtain the desired result.Theorem 11.
For anyX,τ,ℓ, if μ∈IX−0¯ is a r-fuzzy G∗-connected set such that μ≤λ≤GCτ∗μ,r, then λ is r-fuzzy G∗-connected.Proof.
Letλ be not r-fuzzy G∗-connected. Then, there exist r-fuzzy G∗-separated sets ν1 and ν2∈IX such that λ=ν1∨ν2. Let ω=μ∧ν2, ν=μ∧ν1, and μ=ν∨ω. Since ω≤ν2 and ν≤ν1, by Theorem 9 (i), ω and ν are r-fuzzy G∗-separated, and it is a contradiction. Thus, λ is r-fuzzy G∗-connected, as required.


## 6. Conclusion

In this article, we have continued to study the fuzzy sets on a fuzzy ideal topological spaceX,τ,ℓ in Šostak sense. First, we defined the class of r-generalized fuzzy ℓ-closed sets, and some basic properties are given. Moreover, we show that r-generalized fuzzy closed set ⟹r-generalized fuzzy ℓ-closed set, but the converse need not be true. Also, if we take ℓ=ℓ0, r-generalized fuzzy ℓ-closed set and r-generalized fuzzy closed set are equivalent. After that, we define fuzzy upper (lower) generalized ℓ-continuous multifunctions, and some properties of these multifunctions along with their mutual relationships are discussed with the help of examples. In the end, some separation axioms of r-generalized fuzzy ℓ-closed sets are introduced and studied. Also, the notion of r-fuzzy G∗-connected sets is defined and studied with help of r-generalized fuzzy ℓ-closed sets. We hope that the concepts initiated herein will find their applications in many fields soon.



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