We introduce the concept of L-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.

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 of L-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-fuzzy r-adherent points (r-convergent, r-cluster, and r-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 lattice L=(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,

(L,⊙,1) is a commutative monoid,

if x≤y, then x⊙z≤y⊙z (⊙ is isotone operation),

(Galois correspondence) x≤y→z⇔x⊙y≤z.

In a residuated lattice L, x′=x→0 is called complement of x∈L.

A residuated lattice L is called a BL-algebra [9, 21, 23] if it satisfies the following conditions: for each x,y,z∈L,

x∧y=x⊙(x→y),

x∨y=[(x→y)→y]∧[(y→x)→x],

(x→y)∨(y→x)=1.

A BL-algebra is called an MV-algebra if x=x′′, for each x∈L.

Let L be a complete MV-algebra. For each x,y,z∈L, {yi,xi∣i∈Γ}⊂L, one has the following properties:

x⊙y≤x∧y≤x∨y,

x⊙y≤x,y,

If y≤z, (x⊙y)≤(x⊙z), x→y≤x→z and z→x≤y→x,

x⊙y=(x→y′)′,

x≤y iff x′≥y′,

x→y=y′→x′,

⋀i∈Γ(x⊙yi)=x⊙(⋀i∈Γyi),

⋁i∈Γ(x⊙yi)=x⊙(⋁i∈Γyi),

x→1=1,0→x=1,x→x=1,

x≤y⇔x→y=1 and 1→x=x,

x→⋀i∈Γyi=⋀i∈Γ(x→yi),

(⋁i∈Γyi)→x=⋀i∈Γ(yi→x),

x→⋁i∈Γyi=⋁i∈Γ(x→yi),

⋀i∈Γyi→x=⋁i∈Γ(yi→x),

⋀i∈Γyi′=(⋁i∈Γyi)′ and ⋁i∈Γyi′=(⋀i∈Γyi)′.

In this paper, we always assume that L is a complete MV-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 point xt 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 mapping ϕ: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 set X, 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 [<xref ref-type="bibr" rid="B6">24</xref>]).

For each λ,μ,ρ,μi∈LX, i∈Γ and e,xt∈Pt(X), the following properties hold:

λ≤μ⇔S(λ,μ)=1,

λ≤μ⇒S(ρ,λ)≤S(ρ,μ) and S(λ,ρ)≥S(μ,ρ), for any ρ∈LX,

Sx,λ=λx, for any λ∈LX,

S(xt,λ)=0 if and only if t=1 and λ(x)=0,

S(e,λ)∧S(e,μ)=S(e,λ∧μ),

S(xt,⋀i∈Γμi)=⋀i∈ΓS(xt,μi), for any {μi}i∈Γ⊂LX,

S(xt,⋁i∈Γμi)=⋁i∈ΓS(xt,μi), for any {μi}i∈Γ⊂LX.

Lemma 3 (see [<xref ref-type="bibr" rid="B7">16</xref>]).

Let f:X→Y be a mapping. Then the following statement hold:

S(λ,μ)≤S(f→(λ),f→(μ)), for each λ,μ∈LX

S(ρ,ν)≤S(f←(ρ),f←(ν)), for each ρ,ν∈LY.

In particular, if the mapping f:X→Y is bijective, and then the equalities hold.

Definition 4 (see [<xref ref-type="bibr" rid="B25">1</xref>, <xref ref-type="bibr" rid="B15">9</xref>]).

A map T:LX→L is called an L-fuzzy topology on X if it satisfies the following conditions:

T(1X)=T(0X)=1,

T(μ1∧μ2)≥T(μ1)∧T(μ2), for all μ1,μ2∈LX,

T(⋁i∈Λμi)≥⋀i∈ΛT(μi), for any {μi}i∈Λ⊂LX.

The pair (X,T) is called an L-fuzzy topological space.

Let T1 and T2 be L-fuzzy topologies on X. We say that T1 is finer than T2 (T2 is coarser 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 [<xref ref-type="bibr" rid="B13">7</xref>, <xref ref-type="bibr" rid="B15">9</xref>]).

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 each λ,μ∈LX and r,s∈L0, one has the following properties:

IT(1X,r)=1X,

IT(λ,r)≤λ,

if λ≤μ and r≤s, then IT(λ,s)≤IT(μ,r),

IT(λ∧μ,r∧s)≥IT(λ,r)∧IT(μ,s),

IT(IT(λ,r),r)=IT(λ,r),

IT(λ′,r)=(CT(λ,r))′.

Definition 6 (see [<xref ref-type="bibr" rid="B18">12</xref>]).

Let D 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 [<xref ref-type="bibr" rid="B18">12</xref>, <xref ref-type="bibr" rid="B17">25</xref>]).

Let T be a fuzzy net and λ∈LX.

T is often in λ if for each n∈D, there exists n0∈D such that n0≥n and T(n0)∈λ.

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 [<xref ref-type="bibr" rid="B18">12</xref>, <xref ref-type="bibr" rid="B17">25</xref>]).

Let T: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

U=T∘N;

for every n0∈D, there exists m0∈E such that N(m)≥n0, for m≥m0.

Let λ∈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 a r-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 called the L-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:

(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.

Let X 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:

TN is an L-fuzzy topology on X;

if (NT)e is the L-fuzzy neighborhood system of e induced by (X,T), then TNT=T;

if Ne’s satisfy the conditions (6) and (7), then
(11)TN(λ)=⋁r∈L0Sx,λ=Nxλ,r,∀x∈X;

NTN=N.

Proof.

(a) (LO1) It is easily proved from Theorem 11(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) If a⊲⋀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) If a⊲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 that S(xt,λ)=Nxt(λ,r),∀xt∈Pt(X)

if and only if S(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 Theorem 11(6), we easily obtain NTN≥N.

If a⊲(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 Theorem 12, we have the following corollary.

Corollary 13.

The set of all L-fuzzy topologies on X and the set of all L-fuzzy neighborhood systems on X are in one to one correspondence.

Example 14.

Let L=[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-ConvergenceDefinition 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 is r-convergent to e and T is r-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 is r-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

S(e,IT(λ,r))=Ne(λ,r),

S(e,CT′(λ,r))=Ade′(λ,r),

Adxt(λ,r)=⋁x∈X(t⊙Adx(λ,r)).

Proof.

(1) From Lemma 2(7), we have
(26)Se,ITλ,r=Se,⋁μi∣μi≤λ,Tμi≥r=⋁Se,μi∣μi≤λ,Tμi≥r=Ne(λ,r).

(2) From Theorem 5, we have
(27)Se,CT′λ,r=Se,ITλ′,r=Neλ′,rby1=Ade′(λ,r).

(3) From Theorem 11(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:

if r1≤r2, Cone(T,r1)≤Cone(T,r2), and Cle(T,r1)≤Cle(T,r2),

Cone(T,r)≤Cle(T,r),

Cle(U,r)≤Cle(T,r),

ConeT,r≤ConeU,r,

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) if T 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) if T 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) let U 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.

Since T 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 if μ∈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:

if T(n)≤U(n) for all n∈D, then
(42)CleT,r≤CleU,r,ConeT,r≤ConeU,r,

CleT∧U,r≤CleT,r∧CleU,r,

ConeT∨U,r≥ConeT,r∨ConeU,r,

ConeT∧U,r≤ConeT,r∧ConeU,r,

if L 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 that Cle(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).

Let N 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 nets T,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.

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 is r-limit to e is defined, denoted by lime(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)lime(T∨U,r)=lime(T,r)∨lime(U,r).

Proof.

From Theorem 20, 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:

Cone(T,r)=⋀T∈HCle(U,r);

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.

Let L 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 lime(K,r)=t, then lime(T,r)=t.

Proof.

Let H={U∣UisasubnetofT}. For each U∈H, since U has a subnet K with limT(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, lime(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, lime(T,r) does not exists.

Since Cone(T,r)=Cle(T,r)=0.3 for 1/2<r≤1, lime(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:

f:(X,T1)→(Y,T2) is LF-continuous;

Nf→(e)(μ,r)≤⋁{Ne(λ,r)∣f→(λ)≤μ};

Cle(T,r)≤Clf→(e)(f∘T,r);

Cone(T,r)≤Conf→(e)(f∘T,r);

f→(CT1(λ,r))≤CT2(f→(λ),r);

CT1(f←(μ),r))≤f←(CT2(μ),r);

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.

Let T2(μ)=0. It is trivial.

Let T2(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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous refree for his comments, which helped us to improve the final version of this paper.

ŠostakA. P.On a fuzzy topological structureŠostakA. P.On the neighborhood structure of a fuzzy topological spacesŠostakA. P.Basic structures of fuzzy topologyChangC. L.Fuzzy topological spacesDemirciM.Neighborhood structures of smooth topological spacesHazraR. N.SamantaS. K.ChattopadhyayK. C.Fuzzy topology redefinedChattopadhyayK. C.HazraR. N.SamantaS. K.Gradation of openness: fuzzy topologyHöhleU.HöhleU.RodabaughS. E.HöhleU.ŠostakA. P.Axiomatic foundations of fixed-basis fuzzy topologyPao-MingP.Ying-MingL.Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergenceLiuY.-M.LuoM.-K.YingM. S.On the method of neighborhood systems in fuzzy topologyChenS. L.ChengJ. S.On convergence of nets of L-fuzzy setsChenS. L.ChengJ. S.θ-convergence of nets of L-fuzzy sets and its applicationsFangJ.Relationships between L-ordered convergence structures and strong L-topologiesGeorgiouD. N.PapadopoulosB. K.Convergences in fuzzy topological spacesYaoW.On many-valued stratified L-fuzzy convergence spacesRamadanA. A.Smooth topological spacesGierzG.HofmannK. H.HájekP.TurunenE.Algebraic structures in fuzzy logicTurunenE.JinmingF.Stratified L-ordered convergence structuresLaiH.ZhangD.Fuzzy preorder and fuzzy topology