# Connectedness Degrees inL-Fuzzy Topological Spaces

Authors: Fu-Gui Shi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

The notion of separatedness degrees ofL-fuzzy subsets is introduced
in L-fuzzy topological spaces by means of L-fuzzy closure operators. Furthermore, the notion of
connectedness degrees of L-fuzzy subsets is introduced. Many properties of connectedness in
general topology are generalized to L-fuzzy topological spaces.


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## Body

## 1. Introduction

Since Chang [1] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. In a Chang I-topology, the open sets are fuzzy, but the topology comprising those open sets is a crisp subset of IX. However, in a completely different direction, Höhle [2] presented a notion of fuzzy topology being viewed as an I-fuzzy subset of 2X. Then Kubiak [3] and Šostak [4] independently extended Höhle's fuzzy topology to L-subsets of LX, which is called L-fuzzy topology (see [5, 6]). From a logical point of view, Ying [7] studied Höhle's topology and called it fuzzifying topology.Connectivity is one of the most important notions in general topology. It has been generalized toL-topology in terms of many forms (see [8–17], etc.). In a fuzzifying topological space, Ying [18] introduced a definition of connectivity and Fang [19] proved Fan's theorem. In a [0,1]-fuzzy topological space (X,𝒯), Šostak introduced a notion of connectedness degree by means of the level [0,1]-topological spaces (X,𝒯α) [20, 21], that is, it can be viewed as connectivity in a [0,1]-topological space. Although a definition of connectivity was also presented by Yue and Fang [22] in [0,1]-fuzzy topological spaces, it was defined for whole L-fuzzy topological space not for arbitrary L-fuzzy subset.In this paper, we first introduce the notion of separatedness degrees inL-fuzzy topological spaces by means of L-fuzzy closure operators. Furthermore, we present the notion of connectedness degrees of L-fuzzy subsets, which is a generalization of Yue and Fang's connectedness degree. Many properties of connectedness in general topology can be generalized to L-fuzzy topological spaces.


## 2. Preliminaries

Throughout this paper,(L,⋁,⋀,′) denotes a completely distributive DeMorgan algebra. The smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively. The set of all nonzero co-prime elements of L is denoted by J(L).We say thata is wedge below b in L, denoted by a≺b, if for every subset D⊆L, ⋁D≥b implies d≥a for some d∈D. A complete lattice L is completely distributive if and only if b=⋁{a∈L:a≺b} for each b∈L. For any b∈L, define β(b)={a∈L:a≺b}. Some properties of β can be found in [23].For a nonempty setX, the set of all nonzero coprime elements of LX is denoted by J(LX). It is easy to see that J(LX) is exactly the set of all fuzzy points xλ (λ∈J(L)). The smallest element and the largest element in LX are denoted by ⊥̲ and ⊤̲, respectively.For anyL-fuzzy set A∈LX and any a∈L, we use the following notations:
(2.1)A[a]={x∈X:A(x)≥a},a̲(x)=a,∀x∈X.Definition 2.1 (see [3–5]).
AnL-fuzzy topology on a set X is a map 𝒯:LX→L such that (LFT1)
𝒯(⊤̲)=𝒯(⊥̲)=⊤;(LFT2)
for allU,V∈LX, 𝒯(U⋀V)≥𝒯(U)⋀𝒯(V);(LFT3)
for allUj∈LX,  j∈J, 𝒯(⋁j∈JUj)≥⋀j∈J𝒯(Uj).𝒯(U) can be interpreted as the degree to which U is an open set. 𝒯*(U)=𝒯(U′) will be called the degree of closedness of U. The pair (X,𝒯) is called an L-fuzzy topological space.
A mappingf:(X,𝒯1)→(Y,𝒯2) is said to be continuous with respect to L-fuzzy topologies 𝒯1 and 𝒯2 if 𝒯1(fL←(U))≥𝒯2(U) holds for all U∈LY, where fL← is defined by fL←(U)(x)=U(f(x)) [24].Definition 2.2 (see [25]).
AnL-fuzzy closure operator on X is a mapping Cl:LX→LJ(LX) satisfying the following conditions: (LFC1)
Cl(A)(xλ)=⋀μ≺λCl(A)(xμ), for all xλ∈J(LX);(LFC2)
Cl(⊥̲)(xλ)=⊥ for any xλ∈J(LX);(LFC3)
Cl(A)(xλ)=⊤ for any xλ≤A;(LFC4)
Cl(A∨B)=Cl(A)∨Cl(B);(LFC5)
for alla∈L∖{⊥}, (Cl(⋁(Cl(A))[a]))[a]⊆(Cl(A))[a].Cl(A)(xλ) is called the degree to which xλ belongs to the closure of A.Lemma 2.3 (see [25]).
Let(X,𝒯) be an L-fuzzy topological space and let Cl be the L-fuzzy closure operator induced by 𝒯. Then for all xλ∈J(LX), for all A∈LX,
(2.2)Cl(A)(xλ)=⋀{(𝒯(D′))′:D∈LX,xλ≰D≥A}=⋀xλ≰D≥A(𝒯(D′))′.Definition 2.4 (see [17, 23]).
In anL-topological space (X,τ), two L-fuzzy sets A, B are called separated if A-⋀B=A⋀B-=⊥̲, where A- denotes the closure of A.Definition 2.5 (see [17, 23]).
In anL-topological space (X,τ), an L-fuzzy set D is called connected if D can not be represented as a union of two separated non-null L-fuzzy sets.


## 3. Separatedness Degrees inL-Fuzzy Topological Spaces

In this section, in order to generalize Definition2.5 to L-fuzzy topological spaces, we will introduce the concept of separatedness degrees in L-fuzzy topological spaces by means of L-fuzzy closure operators.Definition 3.1.
Let(X,𝒯) be an L-fuzzy topological space and A,B∈LX. Define
(3.1)Sep(A,B)=(⋀{(Cl(B)(xλ))′:xλ≤A})⋀(⋀{(Cl(A)(yμ))′:yμ≤B})=(⋀xλ≤A(Cl(B)(xλ))′)⋀(⋀yμ≤B(Cl(A)(yμ))′).
Then Sep(A,B) is said to be the separatedness degree of A and B.The following result is obvious.Proposition 3.2.
Let𝒯:LX→{⊥,⊤} be an L-topology on X and A,B∈LX. Then Sep(A,B)=⊤ if and only if A and B are separated in (X,𝒯).Lemma 3.3.
Let(X,𝒯) be an L-fuzzy topological space and A,B∈LX. If A⋀B≠⊥̲, then Sep(A,B)=⊥.Proof.
FromA⋀B≠⊥̲, we can take zγ∈J(LX) such that zγ≤A⋀B. Thus we have
(3.2)Sep(A,B)=(⋀xλ≤A(Cl(B)(xλ))′)⋀(⋀xλ≤B(Cl(A)(xλ))′)≤(Cl(B)(zγ))′⋀(Cl(A)(zγ))′=⊤′⋀⊤′=⊥.Lemma 3.4.
Let(X,𝒯) be an L-fuzzy topological space, and A,B,C,D∈LX. If C≤A and D≤B, then Sep(A,B)≤Sep(C,D).Proof.
IfC≤A and D≤B, then Cl(C)≤Cl(A) and Cl(D)≤Cl(B). Hence we have
(3.3)Sep(A,B)=(⋀xλ≤A(Cl(B)(xλ))′)⋀(⋀yμ≤B(Cl(A)(yμ))′)≤(⋀xλ≤A(Cl(D)(xλ))′)⋀(⋀yμ≤B(Cl(C)(yμ))′)≤(⋀xλ≤C(Cl(D)(xλ))′)⋀(⋀yμ≤D(Cl(C)(yμ))′)=Sep(C,D).Lemma 3.5.
Let(X,𝒯) be an L-fuzzy topological space, A,B∈LX and a∈J(L). Then (Sep(A,B))′≱a if and only if there exist D,E∈LX such that
(3.4)D≥A,E≥B,D⋀B=E⋀A=⊥̲,(𝒯(D′))′∨(𝒯(E′))′≱a.Proof.
Suppose that(Sep(A,B))′≱a. Then (Sep(A,B))′≱b for some b∈β*(a). This implies
(3.5)⋁xλ≤ACl(B)(xλ)∨⋁yμ≤BCl(A)(yμ)≱b.
Further more, we have
(3.6)⋁xλ≤A⋀xλ≰E≥B(𝒯(E′))′∨⋁yμ≤B⋀yμ≰D≥A(𝒯(D′))′≱b.
Hence for any xλ≤A and for any yμ≤B, there are Dyμ,Exλ∈LX such that xλ≰Exλ≥B, yμ≰Dyμ≥A and (𝒯(Dyμ′))′∨(𝒯(Exλ′))′≱b. Let E=⋀xλ≤AExλ and D=⋀yμ≤BDyμ. Then, obviously, we have that D≥A, E≥B, D⋀B=E⋀A=⊥̲ and
(3.7)(𝒯(D′))′∨(𝒯(E′))′=(𝒯(⋁yμ≤BDyμ′))′∨(𝒯(⋁xλ≤AExλ′))′≤⋁yμ≤B(𝒯(Dyμ′))′∨⋁xλ≤A(𝒯(Exλ′))′≱a.
Conversely if there existD,E∈LX such that
(3.8)D≥A,E≥B,D⋀B=E⋀A=⊥̲,(𝒯(D′))′∨(𝒯(E′))′≱a.
Then by
(3.9)(Sep(A,B))′=⋁xλ≤ACl(B)(xλ)∨⋁yμ≤BCl(A)(yμ)=⋁xλ≤A⋀xλ≰G≥B(𝒯(G′))′∨⋁yμ≤B⋀yμ≰H≥A(𝒯(H′))′≤(𝒯(D′))′∨(𝒯(E′))′
we can obtain that (Sep(A,B))′≱a.


## 4. Connectedness Degrees inL-Fuzzy Topological Spaces

Definition 4.1.
Let(X,𝒯) be an L-fuzzy topological space and G∈LX. Define
(4.1)Con(G)=⋀{(Sep(A,B))′:A,B∈LX∖{⊥̲},G=A∨B}=⋀{⋁xλ≤ACl(B)(xλ)∨⋁yμ≤BCl(A)(yμ):A,B∈LX∖{⊥̲},G=A∨B}.
Then Con(G) is said to be the connectedness degree of G.The following proposition shows that Definition4.1 is a generalization of Definition 2.5.Proposition 4.2.
Let𝒯:LX→{⊥,⊤} be an L-topology on X and G∈LX. Then Con(G)=⊤ if and only if G is connected in (X,𝒯).Theorem 4.3.
Let(X,𝒯) be an L-fuzzy topological space and G∈LX. Then
(4.2)Con(G)=⋀{(𝒯(A′))′∨(𝒯(B′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B}.Proof.
On one hand, we have the following inequality:(4.3)Con(G)=⋀{⋁xλ≤ACl(B)(xλ)∨⋁yμ≤BCl(A)(yμ):A,B∈LX∖{⊥̲},G=A∨B}=⋀{⋁xλ≤A⋀xλ≰D≥B(𝒯(D′))′∨⋁yμ≤B⋀yμ≰E≥A(𝒯(E′))′:A,B∈LX∖{⊥̲},G=A∨B}=⋀{⋁xλ≤G⋀A⋀xλ≰D≥G⋀B(𝒯(D′))′∨⋁yμ≤G⋀B⋀yμ≰E≥G⋀A(𝒯(E′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G≤A∨B}≤⋀{⋁xλ≤G⋀A(𝒯(B′))′∨⋁yμ≤G⋀B(𝒯(A′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B}=⋀{(𝒯(B′))′∨(𝒯(A′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B};
on the other hand, in order to prove the inverse, we suppose that Con(G)≱a  (a∈J(L)). Then there exist A,B∈LX∖{⊥̲} such that G=A∨B and (Sep(A,B))′≱a. By Lemma 3.5 we know that there exists D,E∈LX such that
(4.4)D≥A,E≥B,D⋀B=E⋀A=⊥̲,(𝒯(D′))′∨(𝒯(E′))≱a.
ObviouslyG⋀D≠⊥̲, G⋀E≠⊥̲, G⋀D⋀E=⊥̲, G≤D∨E. Hence we have
(4.5)⋀{(𝒯(B′))′∨(𝒯(A′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B}≱a.
Therefore,
(4.6)Con(G)≥⋀{(𝒯(B′))′∨(𝒯(A′))′:G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B}.
The proof is completed.Example 4.4.
LetX={x,y} and L=[0,1]. Define B∈[0,1]X by B(x)=0.5 and B(y)=0, and define C∈[0,1]X by C(y)=0.5 and C(x)=0, respectively. Let 𝒯:[0,1]X→[0,1] be defined as follows:
(4.7)𝒯(A)={1,A∈{⊤̲,⊥̲,0.5̲},0.5,A∈{B′,C′},0,others.
Then 𝒯 is an L-fuzzy topology on X. It is easy to verify that Con(a̲)=0.5 for any a∈(0,0.5] and Con(b̲)=1 for any b∈(0.5,1].Corollary 4.5.
Let(X,𝒯) be an L-fuzzy topological space. Then
(4.8)Con(⊤̲)=⋀{(𝒯(A′))′∨(𝒯(B′))′:A≠⊥̲,B≠⊥̲,A⋀B=⊥̲,⊤̲=A∨B}=⋀{(𝒯(A))′∨(𝒯(B))′:A≠⊥̲,B≠⊥̲,A⋀B=⊥̲,⊤̲=A∨B}.Remark 4.6.
Yue and Fang [22] introduced a definition of connectivity in a [0,1]-fuzzy topological space. It is easy to see that Yue and Fang's definition is a special case of our definition from Corollary 4.5.Theorem 4.7.
For anye∈J(LX), it follows that Con(e)=⊤.Proof.
From Theorem4.3 we have
(4.9)Con(e)=⋀{(𝒯(A′))′∨(𝒯(B′))′:e⋀A≠⊥̲,e⋀B≠⊥̲,e⋀A⋀B=⊥̲,e≤A∨B}=⋀∅=⊤.Theorem 4.8.
For anyG∈LX, one has
(4.10)⋁r∈J(L)Con(⋁(Cl(G))[r])≥Con(G).Proof.
Leta∈J(L) and a≤Con(G). Now we prove ⋁r∈J(L)Con(⋁(Cl(G))[r])≥a. Suppose that ⋁r∈J(L)Con(⋁(Cl(G))[r])≱a. Then Con(⋁(Cl(G))[a])≱a. By Theorem 4.3 we know that there exists A,B∈LX such that
(4.11)(⋁(Cl(G))[a])⋀A≠⊥̲,(⋁(Cl(G))[a])⋀B≠⊥̲,(⋁(Cl(G))[a])⋀A⋀B=⊥̲,⋁(Cl(G))[a]≤A∨B,(𝒯(B′))′∨(𝒯(A′))′≱a.
By (⋁(Cl(G))[a])⋀A≠⊥̲ we know that there exists xλ≤A such that Cl(G)(xλ)≥a. Furthermore by (⋁(Cl(G))[a])⋀A⋀B=⊥̲ we obtain xλ≰B.
Now we proveG⋀A≠⊥̲. In fact, if G⋀A=⊥̲, then by G≤⋁(Cl(G))[a]≤A∨B we have G≤B, hence it follows that
(4.12)a≤Cl(G)(xλ)=⋀xλ≰E≥G(𝒯(E′))′≤(𝒯(B′))′,
contradicting a≰(𝒯(B′))′. Analogously, we can prove G⋀B≠⊥̲. Thus by
(4.13)G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B,(𝒯(B′))′∨(𝒯(A′))′≱a,
and Theorem 4.3, we know that Con(G)≱a, contradicting Con(G)≥a. It is proved that ⋁r∈J(L)Con(⋁(Cl(G))[r])≥Con(G).Theorem 4.9.
For anyG,H∈LX, one has
(4.14)Con(G∨H)≥(Sep(G,H))′⋀Con(G)⋀Con(H).Proof.
Leta∈J(L) and a≤(Sep(G,H))′⋀Con(G)⋀Con(H). Now we prove Con(G∨H)≥a. Suppose that Con(G∨H)≱a. Then by Theorem 4.3 we know that there exist A,B∈LX such that
(4.15)(G∨H)⋀A≠⊥̲,(G∨H)⋀B≠⊥̲,(G∨H)⋀A⋀B=⊥̲,G∨H≤A∨B,(𝒯(B′))′∨(𝒯(A′))′≱a.
By (G∨H)⋀A≠⊥̲ we know that one of G⋀A≠⊥̲ and H⋀A≠⊥̲ must be true.
Suppose thatG⋀A≠⊥̲ (the case of H⋀A≠⊥̲ is analogous). Then we must have G⋀B=⊥̲, otherwise if G⋀B≠⊥̲, then by
(4.16)G⋀A≠⊥̲,G⋀B≠⊥̲,G⋀A⋀B=⊥̲,G≤A∨B,(𝒯(B′))′∨(𝒯(A′))′≱a
we know that Con(G)≱a, contradicting Con(G)≥a. In this case by (G∨H)⋀B≠⊥̲ we know that H⋀B≠⊥̲. Analogously we can prove H⋀A=⊥̲. Thus by G∨H≤A∨B we can obtain that G≤A and H≤B. Hence by
(4.17)G≤A,H≤B,G⋀B=H⋀A=⊥̲,(𝒯(B′))′∨(𝒯(A′))′≱a
and Lemma 3.5 we know that (Sep(G,H))′≱a, contradicting (Sep(G,H))′≥a. This shows that Con(G∨H)≥a. It is proved that Con(G∨H)≥(Sep(G,H))′⋀Con(G)⋀Con(H).By Lemma3.3 we can obtain the following result.Corollary 4.10.
Let(X,𝒯) be an L-fuzzy topological space and G,H∈LX. If A⋀B≠⊥̲, then Con(G∨H)≥Con(G)⋀Con(H).Theorem 4.11.
Let(X,𝒯) be an L-fuzzy topological space and G∈LX. Then
(4.18)Con(G)=⋀xλ,yμ≤G⋁{Con(Dxλyμ):xλ,yμ≤Dxλyμ≤G}.Proof.
It is obvious that(4.19)Con(G)≤⋀xλ,yμ≤G⋁{Con(Dxλyμ):xλ,yμ≤Dxλyμ≤G}.
Now we prove that
(4.20)Con(G)≥⋀xλ,yμ≤G⋁{Con(Dxλyμ):xλ,yμ≤Dxλyμ≤G}.
Suppose that⋀xλ,yμ≤G⋁{Con(Dxλyμ):xλ,yμ≤Dxλyμ≤G}≥a (a∈J(L)). Take a fixed xλ≤G. Then for any yμ≤G, there exists a Dxλyμ∈LX such that xλ,yμ≤Dxλyμ≤G and Con(Dxλyμ)≥a. Let Dxλ=⋁yμ≤GDxλyμ. Obviously Dxλ=G and ⋀yμ≤GDxλyμ≠⊥̲. By Corollary 4.10 we easily obtain Con(G)=Con(Dxλ)≥⋀yμ≤GCon(Dxλyμ)≥a. This shows
(4.21)Con(G)≥⋀xλ,yμ≤G⋁{Con(Dxλyμ):xλ,yμ≤Dxλyμ≤G}.Theorem 4.12.
IffL→:(X,𝒯1)→(Y,𝒯2) is continuous, then Con(fL→(G))≥Con(G).Proof.
This can be proved from Theorem4.3 and the following inequality:
(4.22)Con(fL→(G))=⋀{(𝒯1(fL←(A′)))′(𝒯2(A′))′∨(𝒯2(B′))′:fL→(G)⋀A≠⊥̲,fL→(G)⋀B≠⊥̲,fL→(G)⋀A⋀B=⊥̲,fL→(G)≤A∨B}≥⋀{(𝒯2(A′))′∨(𝒯2(B′))′:G⋀fL←(A)≠⊥̲,G⋀fL←(B)≠⊥̲,G⋀fL←(A)⋀fL←(B)=⊥̲,G≤fL←(A)∨fL←(B)(𝒯1(fL←(A′)))′}≥⋀{(𝒯1(fL←(A′)))′(𝒯1(fL←(A′)))′(𝒯1(fL←(A′)))′(𝒯1(fL←(A′)))′∨(𝒯1(fL←(B′)))′:G⋀fL←(A)≠⊥̲,G⋀fL←(B)≠⊥̲,G⋀fL←(A)⋀fL←(B)=⊥̲,G≤fL←(A)∨fL←(B)(𝒯2(A′))′}≥Con(G).



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