Connectedness Degrees in 𝐿-Fuzzy Topological Spaces

Abstract

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.

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 𝐼^𝑋. However, in a completely different direction, Höhle [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 Höhle'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 Höhle'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 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 that 𝑎 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 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: 𝐴_[𝑎]𝑎={𝑥∈𝑋∶𝐴(𝑥)≥𝑎},(𝑥)=𝑎,∀𝑥∈𝑋. (2.1)

Definition 2.1 (see [3–5]). An 𝐿-fuzzy topology on a set 𝑋 is a map 𝒯∶𝐿^𝑋→𝐿 such that (LFT1)𝒯(⊤)=𝒯(⟂)=⊤;(LFT2) for all 𝑈,𝑉∈𝐿^𝑋, 𝒯(𝑈⋀𝑉)≥𝒯(𝑈)⋀𝒯(𝑉);(LFT3) for all 𝑈_𝑗∈𝐿^𝑋, 𝑗∈𝐽, 𝒯(⋁_𝑗∈𝐽𝑈_𝑗)≥⋀_𝑗∈𝐽𝒯(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,𝒯₁)→(𝑌,𝒯₂) is said to be continuous with respect to 𝐿-fuzzy topologies 𝒯₁ and 𝒯₂ if 𝒯₁(𝑓^←_𝐿(𝑈))≥𝒯₂(𝑈) holds for all 𝑈∈𝐿^𝑌, where 𝑓^←_𝐿 is defined by 𝑓^←_𝐿(𝑈)(𝑥)=𝑈(𝑓(𝑥)) [24].

Definition 2.2 (see [25]). An 𝐿-fuzzy closure operator on 𝑋 is a mapping Cl∶𝐿^𝑋→𝐿^{𝐽(𝐿𝑋)} satisfying the following conditions: (LFC1)Cl(𝐴)(𝑥_𝜆)=⋀_𝜇≺𝜆Cl(𝐴)(𝑥_𝜇), for all 𝑥_𝜆∈𝐽(𝐿^𝑋);(LFC2)Cl(⟂)(𝑥_𝜆)=⟂ for any 𝑥_𝜆∈𝐽(𝐿^𝑋);(LFC3)Cl(𝐴)(𝑥_𝜆)=⊤ for any 𝑥_𝜆≤𝐴;(LFC4)Cl(𝐴∨𝐵)=Cl(𝐴)∨Cl(𝐵);(LFC5) for all 𝑎∈𝐿⧵{⟂}, (Cl(⋁(Cl(𝐴))_[𝑎]))_[𝑎]⊆(Cl(𝐴))_[𝑎].Cl(𝐴)(𝑥_𝜆) is called the degree to which 𝑥_𝜆 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 𝑥_𝜆∈𝐽(𝐿^𝑋), for all 𝐴∈𝐿^𝑋, Cl(𝐴)(𝑥_𝜆)={(𝒯(𝐷^′))^′∶𝐷∈𝐿^𝑋,𝑥_𝜆=≰𝐷≥𝐴}_{𝑥𝜆≰𝐷≥𝐴}(𝒯(𝐷^′))^′. (2.2)

Definition 2.4 (see [17, 23]). In an 𝐿-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 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. 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 Sep(𝐴,𝐵)=({(Cl(𝐵)(𝑥_𝜆))^′∶𝑥_𝜆≤𝐴})({(Cl(𝐴)(𝑦_𝜇))^′∶𝑦_𝜇≤𝐵})=(_{𝑥𝜆≤𝐴}(Cl(𝐵)(𝑥_𝜆))^′)(_{𝑦𝜇≤𝐵}(Cl(𝐴)(𝑦_𝜇))^′). (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).

Lemma 3.3. Let (𝑋,𝒯) be an 𝐿-fuzzy topological space and 𝐴,𝐵∈𝐿^𝑋. If 𝐴⋀𝐵≠⟂, then Sep(𝐴,𝐵)=⟂.

Proof. From 𝐴⋀𝐵≠⟂, we can take 𝑧_𝛾∈𝐽(𝐿^𝑋) such that 𝑧_𝛾≤𝐴⋀𝐵. Thus we have Sep(𝐴,𝐵)=(_{𝑥𝜆≤𝐴}(Cl(𝐵)(𝑥_𝜆))^′)(_{𝑥𝜆≤𝐵}(Cl(𝐴)(𝑥_𝜆))^′)≤(Cl(𝐵)(𝑧_𝛾))^′(Cl(𝐴)(𝑧_𝛾))^′=⊤^′⊤^′=⟂. (3.2)

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)=(_{𝑥𝜆≤𝐴}(Cl(𝐵)(𝑥_𝜆))^′)(_{𝑦𝜇≤𝐵}(Cl(𝐴)(𝑦_𝜇))^′)≤(_{𝑥𝜆≤𝐴}(Cl(𝐷)(𝑥_𝜆))^′)(_{𝑦𝜇≤𝐵}(Cl(𝐶)(𝑦_𝜇))^′)≤(_{𝑥𝜆≤𝐶}(Cl(𝐷)(𝑥_𝜆))^′)(_{𝑦𝜇≤𝐷}(Cl(𝐶)(𝑦_𝜇))^′)=Sep(𝐶,𝐷). (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 𝐷≥𝐴,𝐸≥𝐵,𝐷𝐵=𝐸𝐴=⟂,(𝒯(𝐷^′))^′∨(𝒯(𝐸^′))^′≱𝑎. (3.4)

Proof. Suppose that (Sep(𝐴,𝐵))^′≥𝑎̸. Then (Sep(𝐴,𝐵))^′≥𝑏̸ for some 𝑏∈𝛽^∗(𝑎). This implies _{𝑥𝜆≤𝐴}Cl(𝐵)(𝑥_𝜆)∨_{𝑦𝜇≤𝐵}Cl(𝐴)(𝑦_𝜇)≱𝑏. (3.5) Further more, we have _{𝑥𝜆≤𝐴𝑥𝜆≰𝐸≥𝐵}(𝒯(𝐸^′))^′∨_{𝑦𝜇≤𝐵𝑦𝜇≰𝐷≥𝐴}(𝒯(𝐷^′))^′≱𝑏. (3.6) Hence for any 𝑥_𝜆≤𝐴 and for any 𝑦_𝜇≤𝐵, there are 𝐷_𝑦𝜇,𝐸_𝑥𝜆∈𝐿^𝑋 such that 𝑥_𝜆≰𝐸_𝑥𝜆≥𝐵, 𝑦_𝜇≰𝐷_𝑦𝜇≥𝐴 and (𝒯(𝐷^′_𝑦𝜇))^′∨(𝒯(𝐸^′_𝑥𝜆))^′≥𝑏̸. Let 𝐸=⋀_{𝑥𝜆≤𝐴}𝐸_𝑥𝜆 and 𝐷=⋀_{𝑦𝜇≤𝐵}𝐷_𝑦𝜇. Then, obviously, we have that 𝐷≥𝐴, 𝐸≥𝐵, 𝐷⋀𝐵=𝐸⋀𝐴=⟂ and (𝒯(𝐷^′))^′∨(𝒯(𝐸^′))^′=(𝒯(_{𝑦𝜇≤𝐵}𝐷^′_𝑦𝜇))^′∨(𝒯(_{𝑥𝜆≤𝐴}𝐸^′_𝑥𝜆))^′≤_{𝑦𝜇≤𝐵}(𝒯(𝐷^′_𝑦𝜇))^′∨_{𝑥𝜆≤𝐴}(𝒯(𝐸^′_𝑥𝜆))^′≱𝑎. (3.7)
Conversely if there exist 𝐷,𝐸∈𝐿^𝑋 such that 𝐷≥𝐴,𝐸≥𝐵,𝐷𝐵=𝐸𝐴=⟂,(𝒯(𝐷^′))^′∨(𝒯(𝐸^′))^′≱𝑎. (3.8) Then by (Sep(𝐴,𝐵))^′=_{𝑥𝜆≤𝐴}Cl(𝐵)(𝑥_𝜆)∨_{𝑦𝜇≤𝐵}Cl(𝐴)(𝑦_𝜇)=_{𝑥𝜆≤𝐴𝑥𝜆≰𝐺≥𝐵}(𝒯(𝐺^′))^′∨_{𝑦𝜇≤𝐵𝑦𝜇≰𝐻≥𝐴}(𝒯(𝐻^′))^′≤(𝒯(𝐷^′))^′∨(𝒯(𝐸^′))^′ (3.9) we can obtain that (Sep(𝐴,𝐵))^′≥𝑎̸.

4. Connectedness Degrees in 𝐿-Fuzzy Topological Spaces

Definition 4.1. Let (𝑋,𝒯) be an 𝐿-fuzzy topological space and 𝐺∈𝐿^𝑋. Define Con(𝐺)={(Sep(𝐴,𝐵))^′∶𝐴,𝐵∈𝐿^𝑋⧵{⟂={},𝐺=𝐴∨𝐵}_{𝑥𝜆≤𝐴}Cl(𝐵)(𝑥_𝜆)∨_{𝑦𝜇≤𝐵}Cl(𝐴)(𝑦_𝜇)∶𝐴,𝐵∈𝐿^𝑋⧵{⟂},𝐺=𝐴∨𝐵}. (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 Con(𝐺)={(𝒯(𝐴^′))^′∨(𝒯(𝐵^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵}. (4.2)

Proof. On one hand, we have the following inequality: Con={(𝐺)_{𝑥𝜆≤𝐴}Cl(𝐵)(𝑥_𝜆)∨_{𝑦𝜇≤𝐵}Cl(𝐴)(𝑦_𝜇)∶𝐴,𝐵∈𝐿^𝑋⧵{⟂={},𝐺=𝐴∨𝐵}_{𝑥𝜆≤𝐴𝑥𝜆≰𝐷≥𝐵}(𝒯(𝐷^′))^′∨_{𝑦𝜇≤𝐵𝑦𝜇≰𝐸≥𝐴}(𝒯(𝐸^′))^′∶𝐴,𝐵∈𝐿^𝑋⧵{⟂={},𝐺=𝐴∨𝐵}_{𝑥𝜆⋀𝐴≤𝐺𝑥𝜆⋀𝐵≰𝐷≥𝐺}(𝒯(𝐷^′))^′∨_{𝑦𝜇⋀𝐵≤𝐺𝑦𝜇⋀𝐴≰𝐸≥𝐺}(𝒯(𝐸^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂≤{,𝐺≤𝐴∨𝐵}_{𝑥𝜆⋀𝐴≤𝐺}(𝒯(𝐵^′))^′∨_{𝑦𝜇⋀𝐵≤𝐺}(𝒯(𝐴^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂=,𝐺≤𝐴∨𝐵}{(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵}; (4.3) 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))^′≥𝑎̸. By Lemma 3.5 we know that there exists 𝐷,𝐸∈𝐿^𝑋 such that 𝐷≥𝐴,𝐸≥𝐵,𝐷𝐵=𝐸𝐴=⟂,(𝒯(𝐷^′))^′∨(𝒯(𝐸^′))≥𝑎̸. (4.4)
Obviously 𝐺⋀𝐷≠⟂, 𝐺⋀𝐸≠⟂, 𝐺⋀𝐷⋀𝐸=⟂, 𝐺≤𝐷∨𝐸. Hence we have {(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵}≥𝑎̸. (4.5) Therefore, Con(𝐺)≥{(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′∶𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵}. (4.6) 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: 𝒯(𝐴)=1,𝐴∈{⊤,⟂,0.5},0.5,𝐴∈{𝐵^′,𝐶^′},0,others. (4.7) 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 Con(⊤)={(𝒯(𝐴^′))^′∨(𝒯(𝐵^′))^′∶𝐴≠⟂,𝐵≠⟂,𝐴𝐵=⟂,⊤==𝐴∨𝐵}{(𝒯(𝐴))^′∨(𝒯(𝐵))^′∶𝐴≠⟂,𝐵≠⟂,𝐴𝐵=⟂,⊤=𝐴∨𝐵}. (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.

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^′))^′∶𝑒𝐴≠⟂,𝑒𝐵≠⟂,𝑒𝐴𝐵=⟂=,𝑒≤𝐴∨𝐵}∅=⊤. (4.9)

Theorem 4.8. For any 𝐺∈𝐿^𝑋, one has _{𝑟∈𝐽(𝐿)}Con((Cl(𝐺))_[𝑟])≥Con(𝐺). (4.10)

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 ((Cl(𝐺))_[𝑎])𝐴≠⟂,((Cl(𝐺))_[𝑎])𝐵≠⟂,((Cl(𝐺))_[𝑎])𝐴𝐵=⟂,(Cl(𝐺))_[𝑎]≤𝐴∨𝐵,(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′≥𝑎̸. (4.11) By (⋁(Cl(𝐺))_[𝑎])⋀𝐴≠⟂ we know that there exists 𝑥_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 𝑎≤Cl(𝐺)(𝑥_𝜆)=_{𝑥𝜆≰𝐸≥𝐺}(𝒯(𝐸^′))^′≤(𝒯(𝐵^′))^′, (4.12) contradicting 𝑎≰(𝒯(𝐵^′))^′. Analogously, we can prove 𝐺⋀𝐵≠⟂. Thus by 𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵,(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′≱𝑎, (4.13) 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 any 𝐺,𝐻∈𝐿^𝑋, one has Con(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. Suppose that Con(𝐺∨𝐻)≱𝑎. Then by Theorem 4.3 we know that there exist 𝐴,𝐵∈𝐿^𝑋 such that (𝐺∨𝐻)𝐴≠⟂,(𝐺∨𝐻)𝐵≠⟂,(𝐺∨𝐻)𝐴𝐵=⟂,𝐺∨𝐻≤𝐴∨𝐵,(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′≱𝑎. (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 𝐺⋀𝐵=⟂, otherwise if 𝐺⋀𝐵≠⟂, then by 𝐺𝐴≠⟂,𝐺𝐵≠⟂,𝐺𝐴𝐵=⟂,𝐺≤𝐴∨𝐵,(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′≱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 𝐺≤𝐴,𝐻≤𝐵,𝐺𝐵=𝐻𝐴=⟂,(𝒯(𝐵^′))^′∨(𝒯(𝐴^′))^′≱𝑎 (4.17) 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 Lemma 3.3 we can obtain the following result.

Corollary 4.10. 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).

Theorem 4.11. Let (𝑋,𝒯) be an 𝐿-fuzzy topological space and 𝐺∈𝐿^𝑋. Then Con(𝐺)=_{𝑥𝜆,𝑦𝜇≤𝐺}{Con(𝐷_{𝑥𝜆𝑦𝜇})∶𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺}. (4.18)

Proof. It is obvious that Con(𝐺)≤_{𝑥𝜆,𝑦𝜇≤𝐺}{Con(𝐷_{𝑥𝜆𝑦𝜇})∶𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺}. (4.19) Now we prove that Con(𝐺)≥_{𝑥𝜆,𝑦𝜇≤𝐺}{Con(𝐷_{𝑥𝜆𝑦𝜇})∶𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺}. (4.20)
Suppose that ⋀_{𝑥𝜆,𝑦𝜇≤𝐺}⋁{Con(𝐷_{𝑥𝜆𝑦𝜇})∶𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺}≥𝑎 (𝑎∈𝐽(𝐿)). Take a fixed 𝑥_𝜆≤𝐺. Then for any 𝑦_𝜇≤𝐺, there exists a 𝐷_{𝑥𝜆𝑦𝜇}∈𝐿^𝑋 such that 𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺 and Con(𝐷_{𝑥𝜆𝑦𝜇})≥𝑎. Let 𝐷_𝑥𝜆=⋁_{𝑦𝜇≤𝐺}𝐷_{𝑥𝜆𝑦𝜇}. Obviously 𝐷_𝑥𝜆=𝐺 and ⋀_{𝑦𝜇≤𝐺}𝐷_{𝑥𝜆𝑦𝜇}≠⟂. By Corollary 4.10 we easily obtain Con(𝐺)=Con(𝐷_𝑥𝜆)≥⋀_{𝑦𝜇≤𝐺}Con(𝐷_{𝑥𝜆𝑦𝜇})≥𝑎. This shows Con(𝐺)≥_{𝑥𝜆,𝑦𝜇≤𝐺}{Con(𝐷_{𝑥𝜆𝑦𝜇})∶𝑥_𝜆,𝑦_𝜇≤𝐷_{𝑥𝜆𝑦𝜇}≤𝐺}. (4.21)

Theorem 4.12. If 𝑓^→_𝐿∶(𝑋,𝒯₁)→(𝑌,𝒯₂) is continuous, then Con(𝑓^→_𝐿(𝐺))≥Con(𝐺).

Proof. This can be proved from Theorem 4.3 and the following inequality: Con(𝑓^→_𝐿(𝐺))={(𝒯₂(𝐴^′))^′∨(𝒯₂(𝐵^′))^′∶𝑓^→_𝐿(𝐺)𝐴≠⟂,𝑓^→_𝐿(𝐺)𝐵≠⟂,𝑓^→_𝐿(𝐺)𝐴𝐵=⟂,𝑓^→_𝐿≥(𝐺)≤𝐴∨𝐵}{(𝒯₂(𝐴^′))^′∨(𝒯₂(𝐵^′))^′∶𝐺𝑓^←_𝐿(𝐴)≠⟂,𝐺𝑓^←_𝐿(𝐵)≠⟂,𝐺𝑓^←_𝐿(𝐴)𝑓^←_𝐿(𝐵)=⟂,𝐺≤𝑓^←_𝐿(𝐴)∨𝑓^←_𝐿≥(𝐵)}{(𝒯₁(𝑓^←_𝐿(𝐴^′)))^′∨(𝒯₁(𝑓^←_𝐿(𝐵^′)))^′∶𝐺𝑓^←_𝐿(𝐴)≠⟂,𝐺𝑓^←_𝐿(𝐵)≠⟂,𝐺𝑓^←_𝐿(𝐴)𝑓^←_𝐿(𝐵)=⟂,𝐺≤𝑓^←_𝐿(𝐴)∨𝑓^←_𝐿≥(𝐵)}Con(𝐺). (4.22)

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (10971242).