# On Convergence inL-Valued Fuzzy Topological Spaces

Authors: A. A. Ramadan; M. El-Dardery; Hu Zhao
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We introduce the concept ofL-fuzzy neighborhood systems using complete MV-algebras and present important links with the theory of L-fuzzy topological spaces. We investigate the relationships among the degrees of L-fuzzy r-adherent points (r-convergent, r-cluster, and r-limit, resp.) in an L-fuzzy topological spaces. Also, we investigate the concept of LF-continuous functions and their properties.


---

## Body

## 1. Introduction

Šostak [1–3] introduced a new definition of L-fuzzy topology as the concept of the degree of the openness of fuzzy set. It is an extension of  I=[0,1]-fuzzy topology defined by Chang [4]. It has been developed in many directions [5–11]. The study of neighborhood systems and convergence of nets in Chang fuzzy topology was initiated by Pao-Ming and Ying-Ming [11] and Liu and Luo [12]. In [13] Ying introduced the degree to which a fuzzy point xt belongs to a fuzzy subset λ by m(xt,λ)=min⁡(1,1-t+λ(x)) and gave the idea of graded neighborhood on fuzzy topological spaces. This plays an important role in the theory of convergence in Chang fuzzy topology see also [14–18]. Following Ying [13], Demirci [5] introduced the idea of graded neighborhood systems in smooth toplogical spaces [19] (a smooth topology is similar to fuzzy topology as defined by Šostak [1], Hazra and Samanta [6]) in a different approach but restricted himself to the I-valued fuzzy sets.In this paper, we study the concept ofL-fuzzy neighborhood systems and present important links with the theory of L-fuzzy topological spaces and investigate some of their properties. We investigate the relationships among the degrees of L-fuzzyr-adherent points (r-convergent,r-cluster, andr-limit, resp.) nets in an L-fuzzy topological spaces. Also, we give some related examples to illustrate some of the introduced notions. In the end, we characterize LF-continuous functions in terms of some of the various notions introduced in this paper.


## 2. Preliminaries

Throughout the text we consider(L,≤,∧,∨,0,1) as a completely distributive lattice with 0 and 1, respectively, being the universal upper and lower bound and L0=L-{0}. A lattice L is called order dense if for each a,b∈L such that a<b, there exist c∈L such that a<c<b. If L is a completely distributive lattice and x⊲⋁i∈Γyi, then there must be i0∈Γ such that x⊲yi0, where x⊲a means K⊂L,  a≤⋁K⇒∃y∈K such that x≤y. If a⊲b and c⊲d, we always assume a∧c⊲b∧d [20] and some properties of ⊲ can be found in [12].A completely distributive latticeL=(L,≤,∧,∨,⊙,→,0,1) (or L, in short) is called a residuated lattice [9, 21–23] if it satisfies the following conditions: for each x,y,z∈L,(R1)
(L,⊙,1) is a commutative monoid,(R2)
ifx≤y, then x⊙z≤y⊙z (⊙ is isotone operation),(R3)
(Galois correspondence)x≤y→z⇔x⊙y≤z.In a residuated latticeL, x′=x→0 is called complement of x∈L.A residuated latticeL is called a BL-algebra [9, 21, 23] if it satisfies the following conditions: for each x,y,z∈L,(B1)
x∧y=x⊙(x→y),(B2)
x∨y=[(x→y)→y]∧[(y→x)→x],(B3)
(x→y)∨(y→x)=1.ABL-algebra is called an MV-algebra if x=x′′, for each x∈L.Lemma 1 (see [9, 21, 23]).
LetL be a completeMV-algebra. For each x,y,z∈L, {yi,xi∣i∈Γ}⊂L, one has the following properties:(1)
x⊙y≤x∧y≤x∨y,(2)
x⊙y≤x,y,(3)
Ify≤z, (x⊙y)≤(x⊙z), x→y≤x→z and z→x≤y→x,(4)
x⊙y=(x→y′)′,(5)
x≤y iff x′≥y′,(6)
x→y=y′→x′,(7)
⋀i∈Γ(x⊙yi)=x⊙(⋀i∈Γyi),(8)
⋁i∈Γ(x⊙yi)=x⊙(⋁i∈Γyi),(9)
x→1=1,0→x=1,x→x=1,(10)
x≤y⇔x→y=1 and 1→x=x,(11)
x→⋀i∈Γyi=⋀i∈Γ(x→yi),(12)
(⋁i∈Γyi)→x=⋀i∈Γ(yi→x),(13)
x→⋁i∈Γyi=⋁i∈Γ(x→yi),(14)
⋀i∈Γyi→x=⋁i∈Γ(yi→x),(15)
⋀i∈Γyi′=(⋁i∈Γyi)′ and ⋁i∈Γyi′=(⋀i∈Γyi)′.In this paper, we always assume thatL is a completeMV-algebra. Let X be a nonempty set, and the family LX denotes the set of all L-fuzzy subsets of a given set X. For α∈L,λ∈LX, we denote (α→λ), (α⊙λ), and αX∈LX as (α→λ)(x)=α→λ(x), (α⊙λ)(x)=α⊙λ(x), and αX(x)=α.A fuzzy pointxt for t∈L0 is an element of LX such that
(1)xt(y)=t,ify=x,0,ify≠x.
The set of all fuzzy points in X is denoted by Pt(X). For λ∈LX and xt∈Pt(X),xt∈λ if and only if t≤λ(x).Given a mappingϕ:X→Y, we write ϕ← for the mapping LY→LX defined by ϕ←(μ)=μ∘ϕ; we write ϕ→ for the mapping LX→LY defined by ϕ→(μ)(y)=⋁{μ(x)∣ϕ(x)=y} for all μ∈LX,y∈Y.For a given setX, define a binary mapping S(,):LX×LX→L as
(2)S(λ,μ)=⋀x∈X(λ(x)⟶μ(x)),∀(λ,μ)∈LX×LX.
For each λ,μ∈LX, S(λ,μ) can be interpreted as the degree to which λ is fuzzy included in μ. It is called the L-fuzzy inclusion order [24].Lemma 2 (see [24]).
For eachλ,μ,ρ,μi∈LX, i∈Γ and e,xt∈Pt(X), the following properties hold:(1)
λ≤μ⇔S(λ,μ)=1,(2)
λ≤μ⇒S(ρ,λ)≤S(ρ,μ) and S(λ,ρ)≥S(μ,ρ), for any ρ∈LX,(3)
Sx,λ=λx, for any λ∈LX,(4)
S(xt,λ)=0 if and only if t=1 and λ(x)=0,(5)
S(e,λ)∧S(e,μ)=S(e,λ∧μ),(6)
S(xt,⋀i∈Γμi)=⋀i∈ΓS(xt,μi), for any {μi}i∈Γ⊂LX,(7)
S(xt,⋁i∈Γμi)=⋁i∈ΓS(xt,μi), for any {μi}i∈Γ⊂LX.Lemma 3 (see [16]).
Letf:X→Y be a mapping. Then the following statement hold:(1)
S(λ,μ)≤S(f→(λ),f→(μ)), for each λ,μ∈LX(2)
S(ρ,ν)≤S(f←(ρ),f←(ν)), for each ρ,ν∈LY.In particular, if the mappingf:X→Y is bijective, and then the equalities hold.Definition 4 (see [1, 9]).
A mapT:LX→L is called an L-fuzzy topology on X if it satisfies the following conditions:(LO1)
T(1X)=T(0X)=1,(LO2)
T(μ1∧μ2)≥T(μ1)∧T(μ2), for all μ1,μ2∈LX,(LO3)
T(⋁i∈Λμi)≥⋀i∈ΛT(μi), for any {μi}i∈Λ⊂LX.The pair (X,T) is called an L-fuzzy topological space.LetT1 and T2 be L-fuzzy topologies on X. We say that T1 isfiner than T2  (T2 iscoarser than T1), denoted by T2≤T1, if T2(λ)≤T1(λ) for all λ∈LX. Let (X,T1) and (Y,T2) be L-fuzzy topological space spaces. A map f:(X,T1)→(Y,T2) is L-fuzzy continuous (LF-continuous, for short) if T2(λ)≤T1(f←(λ)),∀λ∈LY.Theorem 5 (see [7, 9]).
Let(X,T) be an L-fuzzy topological space. For each r∈L0 and λ∈LX, one defines operators IT,CT:LX×L0→LX as follows:
(3)ITλ,r=⋁ρ∈LX∣ρ≤λ,Tρ≥r,CT(λ,r)=⋀{ν∈LX∣λ≤ν,T(ν′)≥r}.For eachλ,μ∈LX and r,s∈L0, one has the following properties:(I1)
IT(1X,r)=1X,(I2)
IT(λ,r)≤λ,(I3)
ifλ≤μ and r≤s, then IT(λ,s)≤IT(μ,r),(I4)
IT(λ∧μ,r∧s)≥IT(λ,r)∧IT(μ,s),(I5)
IT(IT(λ,r),r)=IT(λ,r),(I6)
IT(λ′,r)=(CT(λ,r))′.Definition 6 (see [12]).
LetD be a directed set. A function T:D→Pt(X) is called a fuzzy net in X. Let λ∈LX, and one says that T is a fuzzy net in λ if T(n)∈λ for every n∈D.Definition 7 (see [12, 25]).
LetT be a fuzzy net and λ∈LX.(1)
T is often in λ if for each n∈D, there exists n0∈D such that n0≥n and T(n0)∈λ.(2)
T is finally in λ if there exists n0∈D such that for each n∈D with n≥n0, one has T(n)∈λ.Definition 8 (see [12, 25]).
LetT:D→Pt(X) and U:E→Pt(X) be two fuzzy nets. A fuzzy net U is called a subnet of T if there exists a function N:E→D, called by a cofinal selection on T, such that(1)
U=T∘N;(2)
for everyn0∈D, there exists m0∈E such that N(m)≥n0, for m≥m0.


## 3.L-Fuzzy Neighborhood Systems

Definition 9.
Letλ∈LX and xt∈Pt(X). Then the degree to which xt belongs to λ is
(4)S(xt,λ)=⋀x∈X(t⟶λ(x)).Definition 10.
Let(X,T) be an L-fuzzy topological space, λ∈LX, e∈Pt(X), and r∈L0. The degree to which λ is ar-neighborhood of e is defined by
(5)NTe(λ,r)=⋁{S(e,μ)∣μ≤λ,r⊲T(μ)}.
A mapping (NT)e:LX×L0→L is calledtheL-fuzzy neighborhood system of e.Theorem 11.
Let(X,T) be an L-fuzzy topological space and let (NT)e be the fuzzy neighborhood system of e. For all λ,μ∈LX and r,s∈L0, the following properties hold:(1)
(NT)e(0X,r)=S(e,0X) and (NT)e(1X,r)=1,(2)
(NT)e(λ,r)≤S(e,λ),(3)
(NT)e(λ,r)≥(NT)e(λ,s), if r≤s,(4)
(NT)e(λ,r)≤(NT)e(μ,r), if λ≤μ,(5)
(NT)e(λ1,r)∧(NT)e(λ2,s)≤(NT)e(λ1∧λ2,r∧s),(6)
(NT)e(λ,r)≤⋁{(NT)e(μ,r)∣μ≤λ,S(d,μ)≤(NT)d(μ,r)∀d∈Pt(X)},(7)
(NT)xt(λ,r)=⋀x∈X(t→(NT)x1(λ,r)).Proof.
(
1
), (3), and (4) are easily proved.
(
2
) is proved from the following:
(6)NTeλ,r=⋁Se,μi∣μi≤λ,r⊲τμ≤⋁Se,⋁μi∣μi≤λ,r⊲τμkkkkkkkkkkkkkkkbyLemma22≤Se,⋁μi∣⋁μi≤λ,r≤τ⋁μi≤S(e,λ).
In(5) if a⊲(NT)e(λ1,r)∧(NT)e(λ2,s), then a⊲(NT)e(λ1,r) and a⊲(NT)e(λ2,s), and there exists ρ1∈LX with ρ1≤λ1 and r⊲T(ρ1) such that a⊲S(e,ρ1). Again, there exists ρ1∈LX with ρ2≤λ2 and r⊲T(ρ2) such that a⊲S(e,ρ2). So, ρ1∧ρ2≤λ1∧λ2, r∧s⊲T(ρ1)∧T(ρ2), and a≤S(e,ρ1)∧S(e,ρ2)=S(e,ρ1∧ρ2)≤(NT)e(λ1∧λ2,r∧s). Hence,
(7)NTe(λ1∧λ2,r∧s)≥NTe(λ1,r)∧NTe(λ2,s).
In(6) if r⊲T(μ), then S(d,μ)=(NT)d(μ,r), for each d∈Pt(X). It implies
(8)NTλ,r=⋁Se,μ∣μ≤λ,r⊲Tμ=⋁NTeμ,r∣μ≤λ,kkkkkkkkkkkkkklkkS(d,μ)=NTdμ,r,kkkkkkkkkkkkkkkklNTeμ,r∣μ≤λ,∀d∈Pt(X)≤⋁NTeμ,r∣μ≤λ,kkkkkkkkkkkkkkklkS(d,μ)≤NTd(μ,r),kkkkkkkkkkkkkklkkNTeμ,r∣μ≤λ,∀d∈Pt(X).
(
7
) is proved from
(9)NTxtλ,r=⋁{S(xt,μ)∣μ≤λ,T(μ)≥r}=⋁⋀x∈Xt⟶μx∣μ≤λ,Tμ≥r=⋀x∈X{t⟶⋁{μ(x)∣μ≤λ,T(μ)≥r}}kkkkkkkkkkkkkkkkkklkbyLemma27=⋀x∈Xt⟶NTx1λ,r.Theorem 12.
LetX be a nonempty set. Let for each e∈Pt(X), and Ne:LX×L0→L satisfying the above conditions (1)–(5). Define TN:LX→L by
(10)TN(λ)=⋁{r∈L0∣S(e,λ)=Ne(λ,r),∀e∈Pt(X)}.
Then one has the following:(a)
TN is an L-fuzzy topology on X;(b)
if(NT)e is the L-fuzzy neighborhood system of e induced by (X,T), then TNT=T;(c)
ifNe’s satisfy the conditions (6) and (7), then
(11)TN(λ)=⋁r∈L0Sx,λ=Nxλ,r,∀x∈X;(d)
NTN=N.Proof.
(a) (LO1) It is easily proved from Theorem11(1).
(LO2) It is proved from the following:(12)TNλ1∧TNλ2=⋁r∈L0∣Se,λ1=Neλ1,r∧⋁s∈L0∣Se,λ2=Neλ2,s=⋁r∧s∈L0∣Se,λ1∧Se,λ2kkkkkkkkkkkkkkkk=Neλ1,r∧Neλ2,s≤⋁r∧s∈L0∣Se,λ1∧Se,λ2kkkkkkkkkkkkkkkkl≤Neλ1∧λ2,r∧s≤⋁r∧s∈L0∣Se,λ1∧λ2≤Neλ1∧λ2,r∧skkkkkkkkkkkkkkkkkkkkkkkkkkkkkk(byLemma2(5))≤TN(λ1∧λ2).
(LO3) Ifa⊲⋀i∈ΓTN(λi), then a⊲TN(λi) for each i∈Γ, and note that
(13)TNλi=⋁ri∈L0∣Se,λi=Neλi,ri,kkkkkkkkkkkk∀e∈PtX,
so there exists ri∈L0, with S(e,λi)=Ne(λi,ri) such that a⊲ri. Put r=⋀i∈Γri, and then a≤r. By Theorem 11, we have
(14)S(e,λi)≤Ne(λi,ri)≤Ne(λi,r)≤S(e,λi).
It implies S(e,λi)=Ne(λi,r). Furthermore, by Lemma 2(7), we have
(15)Se,⋁i∈Γλi=⋁i∈ΓSe,λi=⋁i∈ΓNeλi,ri≤⋁i∈ΓNeλi,r≤Ne⋁i∈Γλi,r≤Se,⋁i∈Γλi.
So Ne(⋁i∈Γλi,r)=S(e,⋁i∈Γλi). Hence, TN(⋁i∈Γλi)≥r≥a. Therefore, TN(⋁i∈Γλi)≥⋀i∈Γλi(λi).
(b) Ifa⊲TN(λ), then there exists r0∈L0 with S(e,λ)=Ne(λ,r0) such that r0⊲T(λ). Since
(16)S(e,λ)=Ne(λ,r0)=⋁Se,μi∣μi≤λ,r0⊲Tμi,
then, for each x1∈Pt(X),
(17)λx=Sx1,λ=⋁Sx1,μi∣μi≤λ,r0⊲Tμi=Sx1,⋁i∈Γμi=⋁i∈Γμi(x).
Thus,λ=⋁μi. So T(λ)≥r0≥a. Hence, TN(λ)≤T(λ). We can easily obtain TN(λ)≥T(λ).
(c) We only show thatS(xt,λ)=Nxt(λ,r),∀xt∈Pt(X)
if and only ifS(x,λ)=λ(x)=Nx(λ,r),∀x∈X.
(
⇒
) It is trivial.
(
⇐
) From condition (7),
(18)Nxtλ,r=⋀x∈Xt⟶Nx1λ,r=⋀x∈Xt⟶Sx1,λ=⋀x∈Xt⟶λx=S(xt,λ).
(d) From the proof of Theorem11(6), we easily obtain NTN≥N.
Ifa⊲(NTN)e(λ,r)=⋁{S(e,μ)∣μ≤λ,r⊲TN(μ)}, there exists μ0 with μ0≤λ,  r⊲TN(μ0) such that a⊲S(e,μ0). Note that
(19)TN(μ0)=⋁{t∈L0∣S(e,μ0)=Ne(μ0,t),∀e∈Pt(X)},
and there exists t0∈L0 with S(e,μ0)=Ne(μ0,t0) such that r⊲t0 (thus r≤t0). So a⊲Ne(μ0,t0)≤Ne(μ0,r)≤Ne(λ,r). Therefore, NTN≤N.By Theorem12, we have the following corollary.Corollary 13.
The set of allL-fuzzy topologies on X and the set of all L-fuzzy neighborhood systems on X are in one to one correspondence.Example 14.
LetL=[0,1],  X={a,b} be a set, x→y=min⁡(1-x+y,1), and let μ∈LX be defined as follows:
(20)μ(a)=0.3,μ(b)=0.4.
We define an L-fuzzy topology on X as
(21)T(λ)=1,ifλ=0Xor1X,12,ifλ=μ,0,otherwise.
From Definition 10, Na1,Nb2:LX×L0→L as follows:
(22)Na1(λ,r)=1,ifλ=1X,r∈L0,0.3,if1X≠λ≥μ,0<r≤12,0,otherwise,Nb1λ,r=1,ifλ=1X,r∈L0,0.4,if1X≠λ≥μ,0<r≤12,0,otherwise.
From Theorem 12(c), we have
(23)TN(λ)=1,ifλ=0Xor1X,12,ifλ=μ,0,otherwise.


## 4. R-Convergence

Definition 15.
Let(X,T) be an L-fuzzy topological space, λ∈LX,e∈Pt(X), and r∈L0. The degree to which a fuzzy net T in X isr-convergent to e and T isr-cluster to e are defined, respectively, as follows:
(24)Cone(T,r)=⋀Ne′λ,r∣Tisofteninλ′,Cle(T,r)=⋀Ne′λ,r∣Tisfinallyinλ′.Definition 16.
Let(X,T) be be an L-fuzzy topological space, λ∈LX,e∈Pt(X), and r∈L0. The degree to which e isr-adherent point of e is defined by
(25)Ade(λ,r)=Ne′(λ′,r).Proposition 17.
Let(X,T) be an L-fuzzy topological space. For each λ∈LX,e,xt∈Pt(X) and r∈L0, one has(1)
S(e,IT(λ,r))=Ne(λ,r),(2)
S(e,CT′(λ,r))=Ade′(λ,r),(3)
Adxt(λ,r)=⋁x∈X(t⊙Adx(λ,r)).Proof.
(1) From Lemma2(7), we have
(26)Se,ITλ,r=Se,⋁μi∣μi≤λ,Tμi≥r=⋁Se,μi∣μi≤λ,Tμi≥r=Ne(λ,r).
(2) From Theorem5, we have
(27)Se,CT′λ,r=Se,ITλ′,r=Neλ′,rby1=Ade′(λ,r).
(3) From Theorem11(7), we have
(28)Adxtλ,r=Nxt′λ′,r=⋀x∈Xt⟶Nxt(λ′,r)′=⋁x∈Xt⟶Nxtλ′,r′=⋁x∈Xt⊙Nx1′λ′,rkkkk(byLemma2(4))=⋁x∈X(t⊙Adx(λ,r)).Theorem 18.
Let(X,T) be an L-fuzzy topological space. Let T:D→Pt(X) be fuzzy net and let U:E→Pt(X) be a subnet of S. For r,s∈L0, the following properties hold:(1)
ifr1≤r2, Cone(T,r1)≤Cone(T,r2), and Cle(T,r1)≤Cle(T,r2),(2)
Cone(T,r)≤Cle(T,r),(3)
Cle(U,r)≤Cle(T,r),(4)
ConeT,r≤ConeU,r,(5)
Conxt(T,r)=⋁x∈X(t⊙Conx(T,r)), and Clxt(T,r)=⋁x∈X(t⊙ClxT,r).Proof.
(1) is easily proved.
In (2) ifT is finally in λ′, T is often in λ′. Hence
(29)ConeT,r=⋀Ne′λ,r∣Tisofteninλ′≤⋀Ne′λ,r∣Tisfinallyinλ′=Cle(T,r).
In (3) ifT is finally in λ′, U is finally in λ′. Hence
(30)CleU,r=⋀Ne′λ,r∣Uisfinallyinλ′≤⋀Ne′λ,r∣Tisfinallyinλ′=Cle(T,r).
In (4) letU be often in λ′. We will show that T is often in λ′. Let n∈D. Since U:E→Pt(X) is a subnet of T, there exists a cofinal selection N:E→D. For each n∈D, there exists m∈E such that N(k)≥n for k≥m. Since U is often in λ′, for m∈E, there exists m0∈E such that m0≥m for U(m0)∈λ′. Put n0=N(m0). Then n0≥n and T(n0)=T(N(m0))=T(n0)∈λ′. Thus, U is often in λ′. Hence
(31)ConeT,r=⋀Ne′λ,r∣Tisofteninλ′≤⋀Ne′λ,r∣Uisofteninλ′=Cone(U,r).
In (5) one has(32)ConxtT,r=⋀Nxt′λ,r∣Tisofteninλ′=⋀⋀x∈Xt⟶Nx1λ,r′∣kkkkkkk⋀x∈Xt⟶Nx1λ,r′Tisfinallyinλ′kkkkkkbyTheorem117=⋁x∈X⋀t⟶Nx1λ,r′∣kkkkkkkkkt⟶Nx1λ,r′Tisfinallyinλ′=⋁x∈X⋀t⊙Nx1′λ,r∣Tisfinallyinλ′kkkkkkkkkkkkkkkkkklkbyLemma14=⋁x∈Xt⊙⋀Nx1′λ,r∣Tisfinallyinλ′=⋁x∈X(t⊙Conx(T,r)).
The other case is the same.Proposition 19.
Let(X,T) be an L-fuzzy topological space, let T be a fuzzy net, e∈Pt(X), and r∈L0. Then one has
(33)Adeλ,r=⋁ConeT,r∣Tisafuzzynetinλ=⋁CleT,r∣Tisafuzzynetinλ.Proof.
SinceT is finally in λ, T is often in λ. We easily show that
(34)Adeλ,r=Ne′λ′,r≥⋁CleT,r∣Tisafuzzynetinλ≥⋁ConeT,r∣Tisafuzzynetinλ.
We only show that
(35)Ade(λ,r)≤⋁{Cone(T,r)∣Tisafuzzynetinλ}.
Let Ade(λ,r)=t. If t>0, then Ne′(λ′,r)=t. Put D={μ∈LX∣Ne(μ,r)>t′}. Define a relation on D by
(36)μ1⪯μ2iffμ1≥μ2,∀μ1,μ2∈D.
For each μ1,μ2∈D, since by Theorem 11(5),
(37)Ne(μ1∧μ2,r)≥Ne(μ1,r)∧Ne(μ2,r)>t′.
Hence, μ1∧μ2∈D and μ1,μ2⪯μ1∧μ2. Thus, (D,⪯) is a directed set. For each μ∈D, that is, Ne(μ,r)>t′, we have μ≰λ′; that is, there exists x∈X such that λ(x)>μ′(x). Thus, we can define a fuzzy net T0:D→Pt(X) by T0(μ)=xλ(x) where T0(μ)∈λ and λ(x)=T0(μ)(x)>μ′(x).
We will show that ifμ∈D, then T0 is not often in μ′. Suppose that T0 is often in μ′. For μ∈D, there exists ρ∈D such that μ⪯ρ such that
(38)T0ρ=yλy∈μ′,
and λ(y)=T0(ρ)(y)>ρ′(y). Since μ⪯ρ implies μ≥ρ, it implies
(39)λy≤μ′y≤ρ′y,
It is contradiction for the definition of T0. Thus, if T0 is often in μ′, then μ∉D; that is, Ne(μ,r)≤t′. Therefore,
(40)⋁ConeT,r∣Tisafuzzynetinλ≥ConeT,r=⋀Ne′μ,r∣T0isofteninμ′≥t=Ade(λ,r).Theorem 20.
Let(X,T) be L-fuzzy topological space and let T,U:D→Pt(X) be fuzzy nets such that T(n)∨U(n),T(n)∧U(n)∈Pt(X) for each n∈D. Define fuzzy nets T∨U,T∧U:D→Pt(X) by, for each n∈D,
(41)T∨Un=Tn∨Un,(T∧U)(n)=T(n)∧U(n).
For each r∈L0, the following properties hold:(1)
ifT(n)≤U(n) for all n∈D, then
(42)CleT,r≤CleU,r,ConeT,r≤ConeU,r,(2)
CleT∧U,r≤CleT,r∧CleU,r,(3)
ConeT∨U,r≥ConeT,r∨ConeU,r,(4)
ConeT∧U,r≤ConeT,r∧ConeU,r,(5)
ifL is order dense, then Cle(T∨U,r)=Cle(T,r)∨Cle(U,r).Proof.
In(1) let U be finally (often) in λ. Then let T be finally (often) in λ, respectively. Thus it is trivial. (2), (3), and (4) are easily proved.
In(5) since T≤T∨U and U≤T∨U, by (1), we have
(43)Cle(T∨U,r)≥Cle(T,r)∨Cle(U,r).
Suppose thatCle(T∨U,r)≱Cle(T,r)∨Cle(U,r). Since L is order dense, then there exist t∈L0 and a fuzzy point e∈Pt(X) such that
(44)Cle(T∨U,r)>t>Cle(T,r)∨Cle(U,r).
Since Cle(T,r)<t and Cle(U,r)<t, by the definition Cle, there exist λ,μ∈LX such that T and U are finally in λ′ and μ′, respectively, with
(45)Cle(T,r)∨Cle(U,r)≤Ne′(λ,r)∨Ne′(μ,r)<t.
Since T is finally in λ′, there exists n1∈D such that T(n)∈λ′ for every n∈D with n≥n1. Since U is finally in μ′, there exists n2∈D such that T(n)∈μ′ for every n∈D with n≥n2. Let n3∈D such that n3≥n1 and n3≥n2. For n≥n3, we have
(46)(T∨U)(n)≤λ′∨μ′=λ∧μ′.
Thus, (T∨U) is finally in (λ∧μ)′. It implies
(47)CleT∨U,r≤Ne′λ∧μ,r≤Ne′(λ,r)∨Ne′(μ,r)<t.
It is a contradiction. Hence, we have
(48)Cle(T∨U,r)≤Cle(T,r)∨Cle(U,r).Example 21.
Let(L=[0,1],→) be defined as Example 14. Let X={a,b} be a set and μ∈IX as follows:
(49)μ(x)=0.3,μ(y)=0.4.
We define L-fuzzy topology T:IX→I as follows:
(50)T(λ)=1,ifλ=0Xor1X,12,ifλ=μ,0,otherwise.
(
1
) In general, Cle(T∧U,r)≠Cle(T,r)∧Cle(U,r).
LetN be a natural numbers. Define fuzzy nets T,U:N→Pt(X) by
(51)T(n)=xan,an=0.8+-1n0.2.Un=xbn,bn=0.8+-1n+10.2.
From Theorem 20, (T∧U)(n)=x0.6  is a fuzzy net. Let e=x0.3. From Definition 15, we have for 0<r≤1/2,
(52)Cle(x0.6,r)=1-Ne(μ,r)=1-m(x0.3,μ)=0.
Since T or U is finally in 1X,
(53)Cle(T,r)=1-Ne(0X,r)=1-m(x0.3,0X)=0.3.
Similarly, Cle(U,r)=0.3. For 0<r≤1/2,
(54)0=Cle(T∧U,r)≠Cle(T,r)∧Cle(U,r)=0.3.
(
2
) In general, Cone(T∨U,r)≠Cone(T,r)∨Cone(U,r).
Define fuzzy netsT,U:N→Pt(X) by
(55)T(n)=xan,an=0.6+-1n0.2.U(n)=xbn,bn=0.6+-1n+10.2.
From Theorem 20,  (T∨U)(n)=x0.8  is a fuzzy net. Let e=x0.3. For all r∈I0,
(56)Ade(x0.8,r)=1-Ne(0X,r)=1-m(x0.3,0X)=0.3.
Since T or U is often in μ′, for 0<r≤1/2,
(57)Cle(T,r)=1-Ne(μ,r)=1-m(x0.3,μ)=0.
Similarly, Cle(U,r)=0. For 0<r≤1/2(58)0.3=Cone(T∨U,r)>ConeT,r∨ConeU,r=0.


## 5. Fuzzyr-Limit Nets and LF-Continuous Mappings

Definition 22.
Let(X,T) be an L-fuzzy toplogical space. Let T:D→Pt(X) be fuzzy net in X, e∈Pt(X), and r∈L0. Then the degree to which T isr-limit to e is defined, denoted by lim⁡e⁡(T,r)=t, if Cle(T,r)=Cone(T,r)=t.Theorem 23.
Let(X,T) be L-fuzzy topological space and let T,U:D→Pt(X) be fuzzy nets such that T(n)∨U(n)∈Pt(X) for each n∈D. If L is order dense, Cle(T,r)=Cone(T,r), and Cle(U,r)=Cone(U,r), then
(59)lim⁡e(T∨U,r)=lim⁡e(T,r)∨lim⁡e(U,r).Proof.
From Theorem20, T∨U is a fuzzy net. We easily proved it from the following:
(60)CleT∨U,r=CleT,r∨CleU,rbyTheorem202(sinceCleT,r=ConeT,r,CleU,r=ConeU,r)=ConeT,r∨ConeU,r≤Cone(T∨U,r)(byTheorem20(4))≤Cle(T∨U,r)(byTheorem20(2)).Theorem 24.
Let(X,T) be L-fuzzy topological space. Let T be a fuzzy net and H={U∣UisasubnetofT}. Then, if L is an order dense, the following statements hold:(1)
Cone(T,r)=⋀T∈HCle(U,r);(2)
Cle(T,r)=⋁T∈HCone(U,r).Proof.
(
1
) For each U∈H, by Theorem 18, we have
(61)Cone(T,r)≤Cone(U,r)≤Cle(U,r)≤Cle(T,r).
Hence
(62)Cone(T,r)≤⋀U∈HCle(U,r).
Suppose
(63)Cone(T,r)≱⋀U∈HCle(U,r).
Then there exist xp∈Pt(X) and t∈L0 such that
(64)Conxp(T,r)<t<⋀U∈HClxp(U,r).
Since Conxp(T,r)<t, there exists μ∈LX with T is often in μ′ such that
(65)Conxp(T,r)≤Nxp′(μ,r)<⋀U∈HClxp(U,r).
Since T is often in μ′, for each n∈D there exists N(n)∈D with N(n)≥n and T(N(n))∈μ′. Hence there exists a cofinal selection N:E→D such that U=T∘N. Thus U is a subnet of T and U is finally in μ′. It is a contradiction.
(
2
) From (1), we have
(66)⋁U∈HCone(U,r)≤Cle(T,r).
Conversely, let Cle(T,r)=t>0. Then Ne(λ,r)≤t′, for T is finally in λ′. Let F={μ∣Ne(μ,r)>t′}. Define a relation on E=D×F by
(67)m,μ1≤n,μ2iffm≤n,μ1≥μ2.
Then (E,≤) is a directed set. If μ∈F, then T is not finally in μ′. For each (n,μ)∈E, there exists N(n,μ)∈D with N(n,μ)≥n such that T(N(n,μ))≰λ′. So, we can define N:E→D. For each n0∈D and μ0∈F, there exists N(n0,μ0)∈D with N(n0,μ0)≥n0 such that T(N(n0,μ0))≰μ0′. Hence for every (n,μ)≥(n0,μ0), since n≥n0, we have N(n,μ)≥n≥n0. Therefore N is a cofinal selection on T. So U=T∘N is a fuzzy subnet of T and U is finally to every member of F. If U is often in λ′, then U is not finally of λ; that is, λ∉F. Thus
(68)⋁U∈HCone(T,r)=⋀{Ne′(λ,r)∣Uisofteninλ′}≥t.
Since t is arbitrary, we complete the proof.Theorem 25.
LetL be an order dense, let (X,T) be L-fuzzy topological space, and let T be a fuzzy net. If every subnet U of T has a subnet K of  U such that lim⁡e(K,r)=t, then lim⁡e(T,r)=t.Proof.
LetH={U∣UisasubnetofT}. For each U∈H, since U has a subnet K with lim⁡T(K,r)=t, by Theorem 18(4), we have
(69)Cone(U,r)≤Cone(K,r)=Cle(K,r)=t.
Hence, by Theorem 24(2),
(70)Cle(T,r)=⋁U∈HCone(U,r)≤t.
Conversely, by Theorem 18(2),
(71)t=Cone(K,r)=Cle(K,r)≤Cle(U,r).
Hence, by Theorem 24(1),
(72)t≤⋀U∈HCle(U,r)=Cone(T,r).
Hence, Cle(T,r)≤Cone(T,r). Since Cone(T,r)≤Cle(T,r) from Theorem 18(2), Cle(T,r)=Cone(T,r); that is, lim⁡e(T,r)=t.Example 26.
Let (L=[0,1],→) be defined as in Example 21. Let N be a natural number set. Define a fuzzy net T:N→Pt(X) by
(73)T(n)=xan,an=0.6+-1n0.2.
Let e=x0.3. Since T is often in μ′, for  0<r≤1/2,
(74)Cone(T,r)=1-Ne(μ,r)=1-m(x0.3,μ)=0.
Since T is finally in 1X, for each r∈I0,
(75)Cle(T,r)=1-Ne(0X,r)=1-m(x0.3,0X)=0.3.
Thus, since Cone(T,r)≠Cle(T,r) for 0<r≤1/2, lim⁡e(T,r) does not exists.
SinceCone(T,r)=Cle(T,r)=0.3 for 1/2<r≤1, lim⁡e(T,r)=0.3.Theorem 27.
Let(X,T1) and (Y,T2) be L-fuzzy topological spaces. For every fuzzy net T in X, xt∈Pt(X), r∈L0, and λ∈LX, the following statements are equivalent:(1)
f:(X,T1)→(Y,T2) is LF-continuous;(2)
Nf→(e)(μ,r)≤⋁{Ne(λ,r)∣f→(λ)≤μ};(3)
Cle(T,r)≤Clf→(e)(f∘T,r);(4)
Cone(T,r)≤Conf→(e)(f∘T,r);(5)
f→(CT1(λ,r))≤CT2(f→(λ),r);(6)
CT1(f←(μ),r))≤f←(CT2(μ),r);(7)
f←(IT2(μ,r))≤IT1(f←(μ),r).Proof.
(
1
)
⇒
(
2
) For any ρ∈LY such that T2(ρ)≥r  and ρ≤μ. Since f is LF-continuous, then T1(f←(ρ))≥T2(ρ)≥r, and we have by Lemma 3(2)(76)Sf→e,ρ≤S(e,f←(ρ))(e=xt,f→(e)=fxt)=Ne(f←(ρ),r)(T1(f→(f←)(ρ))≥r)≤⋁Neλ,r∣f→λ≤μkkkkkkk(f→(f←(ρ))≤ρ≤μ).
Thus, Nf→(e)(μ,r)≤⋁{Ne(λ,r)∣f→(λ)≤μ}.
(
2
)
⇒
(
3
) If f→(λ)≤μ and f∘T is finally in μ′, there exists n0∈D such that, for all n≥n0, f(T(n))∈μ′. Let T(n)=xt. Then
(77)t≤μ′(f(x))≤fλ′(f(x))≤λ′(x).
It implies T(n)∈λ′. Therefore, T is finally in λ′. One has
(78)CleT,r=⋀{Ne′(λ,r)∣Tisfinallyinλ′}≤⋀Ne′λ,r∣∃μ,f→λ≤μ,kkkkkkkkkklkklf∘Tisfinallyinμ′=⋀⋁Neλ,r∣f→λ≤μ′,kkkkkkkkkkkkkkklllkkf∘Tisfinallyinμ′≤⋀{Nf→e′(μ,r),f∘Tisfinallyinμ′}=Clef∘T,rby2.
(
3
)
⇒
(
4
) Every subnet U:E→Pt(Y) of f(T), and there exists a cofinal selection N:E→D such that U=f(T)∘N=f∘(T∘N). Put K=T∘N. Then K is a subnet of T. We can prove it from the following:
(79)ConeT,r≤Cone(K,r)(byTheorem18(5))≤CleK,rbyTheorem182≤Clf→ef∘K,rby3=Clf→ef∘T∘N,r=Clf→(e)(U,r).
From Theorem 18(2), we have Cone(T,r)≤Conf→(e)(f∘T,r).
(
4
)
⇒
(
5
) From Theorem 5 and Proposition 17(2),
(80)Sx1,CT1′λ,r=CT1′(λ,r)(x)=Adx′(λ,r).
It implies
(81)CT1(λ,r)(x)=Adx(λ,r).
Thus, we have
(82)f→CT1λ,ry=⋁CT1λ,rx∣fx=y=⋁Adxλ,r∣fx=y(by(81))=⋁fx=y⋁ConxT,r∣TisfuzzynetinλkkkkkkkkkkkkkkkkkkklkkbyProposition19≤⋁fx=y⋁Conyf∘T,r∣Tisfuzzynetinλkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkby4=⋁Conyf∘T,r∣Tisfuzzynetinλ≤⋁ConyT,r∣Tisfuzzynetinf→λ=Adyf→λ,rbyProposition19=CT2f→λ,ryby81.
(
5
)
⇒
(
6
) and (6)⇒(7) are easily proved.
(
7
)
⇒
(
1
) We will show that T1(f←(μ))≥T2(μ), for all μ∈LY.
LetT2(μ)=0. It is trivial.
LetT2(μ)=r>0. Since TN=T2 from Theorem 12(b), we have for all y∈Y,
(83)S(y,μ)=Ny(μ,r).
It implies, for all x∈X,
(84)S(f(x),μ)=S(x,f←(μ))=Nf(x)(μ,r).
Since f←(IT2(μ,r))=f←(μ),
(85)Sx,f-1μ=Sx,f←IT2μ,rsincef←IT2μ,r≤IT1f←μ,r≤Sx,IT1f←μ,r=Nxf←μ,rbyProposition171.
Thus, by Theorem 11(2), we have
(86)Sx,f←μ=Nxf←μ,r.
Hence, T1(f←(μ))≥r.



---

*Source: 730940-2015-02-11.xml*