Connectedness Degrees in L -Fuzzy Topological Spaces

The notion of separatedness degrees of L -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.


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 I X .However, in a completely different direction, H öhle 2 presented a notion of fuzzy topology being viewed as an I-fuzzy subset of 2 X .Then Kubiak 3 and Šostak 4 independently extended H öhle's fuzzy topology to L-subsets of L X , which is called L-fuzzy topology see 5, 6 .From a logical point of view, Ying 7 studied H öhle's topology and called it fuzzifying topology.
Connectivity is one of the most important notions in general topology.It has been generalized to L-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, T α 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 in L-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

International Journal of Mathematics and Mathematical Sciences
Fang's connectedness degree.Many properties of connectedness in general topology can be generalized to L-fuzzy topological spaces.

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 that a is wedge below b in L, denoted by a ≺ b, if for every subset For any b ∈ L, define β b {a ∈ L : a ≺ b}.Some properties of β can be found in 23 .
For a nonempty set X, the set of all nonzero coprime elements of L X is denoted by J L X .It is easy to see that J L X is exactly the set of all fuzzy points x λ λ ∈ J L .The smallest element and the largest element in L X are denoted by ⊥ and , respectively.
For any L-fuzzy set A ∈ L X and any a ∈ L, we use the following notations: LFT3 for all U j ∈ L X , j ∈ J, T j∈J U j ≥ j∈J T U j .
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 mapping f : X, T 1 → Y, T 2 is said to be continuous with respect to L-fuzzy topologies T 1 and Definition 2.2 see 25 .An L-fuzzy closure operator on X is a mapping Cl : L X → L J L X satisfying the following conditions: Cl A x λ is called the degree to which x λ belongs to the closure of A.
International Journal of Mathematics and Mathematical Sciences 3 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 x λ ∈ J L X , for all A ∈ L X ,

Separatedness Degrees in L-Fuzzy Topological Spaces
In this section, in order to generalize Definition 2.5 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 ∈ L X .Define

3.1
Then Sep A, B is said to be the separatedness degree of A and B.
The following result is obvious.
Proposition 3.2.Let T : L X → {⊥, } be an L-topology on X and A, B ∈ L X .Then Sep A, B if and only if A and B are separated in X, T .

3.3
Lemma 3.5.Let X, T be an L-fuzzy topological space, A, B ∈ L X and a ∈ J L .Then Sep A, B / ≥ a if and only if there exist D, E ∈ L X such that Proof.Suppose that Sep A, B / ≥a.Then Sep A, B / ≥b for some b ∈ β * a .This implies Cl A y μ / ≥ b.

3.5
Further more, we have Hence for any x λ ≤ A and for any y μ ≤ B, there are

3.7
International Journal of Mathematics and Mathematical Sciences 5 Conversely if there exist D, E ∈ L X such that

Then by
Sep A, B we can obtain that Sep A, B / ≥a.

Connectedness Degrees in L-Fuzzy Topological Spaces
Definition 4.1.Let X, T be an L-fuzzy topological space and G ∈ L X .Define

4.1
Then Con G is said to be the connectedness degree of G.
The following proposition shows that Definition 4.1 is a generalization of Definition 2.5.Proposition 4.2.Let T : L X → {⊥, } be an L-topology on X and G ∈ L X .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 ∈ L X .Then

International Journal of Mathematics and Mathematical Sciences
Proof.On one hand, we have the following inequality: on the other hand, in order to prove the inverse, we suppose that Con G / ≥ a a ∈ J L .Then there exist A, B ∈ L X \ {⊥} such that G A ∨ B and Sep A, B / ≥a.By Lemma 3.5 we know that there exists D, E ∈ L X such that The proof is completed.
Example 4.4.Let X {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: Then T is an L-fuzzy topology on X.It is easy to verify that Con a 0.5 for any a ∈ 0, 0.5 and Con b 1 for any b ∈ 0.5, 1 .
Corollary 4.5.Let X, T be an L-fuzzy topological space.Then

4.8
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.

4.9
Theorem 4.8.For any G ∈ L X , one has Proof.Let a ∈ 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 ∈ L X such that

4.15
By G ∨ H ∧ A / ⊥ we know that one of G ∧ A / ⊥ and H ∧ A / ⊥ must be true.Suppose that G∧A / ⊥ the case of H ∧A / ⊥ is analogous .Then we must have  Proof.This can be proved from Theorem 4.3 and the following inequality: 4.22

2 . 2 Definition 2 . 4
see 17, 23 .In an L-topological space X, τ , two L-fuzzy sets A, B are calledseparated if A − ∧ B A ∧ B − ⊥, where A − denotes the closure of A.Definition 2.5 see 17, 23 .In an L-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 . 2 International
Journal of Mathematics and Mathematical Sciences Lemma 3.4.Let X, T be an L-fuzzy topological space, and A, B, C, D ∈ L X .If C ≤ A and D ≤ B, then Sep A, B ≤ Sep C, D .Proof.If C ≤ A and D ≤ B, then Cl C ≤ Cl A and Cl D ≤ Cl B .Hence we have Sep A, B

Theorem 4 . 7 .
For any e ∈ J L X , it follows that Con e .Proof.From Theorem 4.3 we have Con e T A ∨ T B : e ∧ A / ⊥, e ∧ B / ⊥, e ∧ A ∧ B ⊥, e ≤ A ∨ B ∅ .

4 . 20 Suppose 21 Theorem 4 . 12 .
that x λ ,y μ ≤G {Con D x λ y μ : x λ , y μ ≤ D x λ y μ ≤ G} ≥ a a ∈ J L .Take a fixed x λ ≤ G. Then for any y μ ≤ G, there exists aD x λ y μ ∈ L X such that x λ , y μ ≤ D x λ y μ ≤ G and Con D x λ y μ ≥ a.Let D x λ y μ ≤G D x λ y μ .Obviously D x λ G and y μ ≤G D x λ y μ / ⊥.By Corollary 4.10 we easily obtain Con G Con D x λ ≥ y μ ≤G Con D x λ y μ ≥ a.This shows Con G ≥ x λ ,y μ ≤G Con D x λ y μ : x λ , y μ ≤ D x λ y μ≤ G .4.If f → L : X, T 1 → Y, T 2 is continuous, then Con f → L G ≥ Con G .
∧ A / ⊥ we know that there existsx λ ≤ A such that Cl G x λ ≥ a. Furthermore by Cl G a ∧ A ∧ B ⊥ we obtain x λ / ≤ B. Now we prove G ∧ A / ⊥.In fact, if G ∧ A ⊥, then by G ≤ Cl G a ≤ A ∨ B we have G ≤ B, hence it follows that Suppose that Con G ∨ H / ≥ a.Then by Theorem 4.3 we know that there exist A, B ∈ L X such that X , one hasCon G ∨ H ≥ Sep G, H ∧ Con G ∧ Con H . 4.14 Proof.Let a ∈ J L and a ≤ Sep G, H ∧ Con G ∧ Con H . Now we prove Con G ∨ H ≥ a.
Let X, T be an L-fuzzy topological space and G, H ∈ L X .If A ∧ B / ⊥, then Con G ∨ H ≥ Con G ∧ Con H . Let X, T be an L-fuzzy topological space and G ∈ L X .ThenCon D x λ y μ : x λ , y μ ≤ D x λ y μ ≤ G .λ ,y μ ≤G Con D x λ y μ : x λ , y μ ≤ D x λ y μ ≤ G .
a 4.16 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 that Con G ∨ H ≥ a.It is proved that Con G ∨ H ≥ Sep G, H ∧ Con G ∧ Con H .By Lemma 3.3 we can obtain the following result.Corollary 4.10.x