Preconnectedness Degree of L-Fuzzy Topological Spaces

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Preconnectedness Degree of L-Fuzzy Topological Spaces

A. Ghareeb

Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt E-mail: [email protected]

Abstract

In this paper, we useL-fuzzy preopen operator to introduce the degree of pre-separatedness and the degree of preconnectedness inL-fuzzy topological spaces. Many characterizations of the degree of preconnectedness are presented inL-fuzzy topological spaces.

Key Words: L-fuzzy topology;L-fuzzy preclosure operator; preseparatedness; preconnectedness.

1. Introduction

It is well known that after the introduction of the L- fuzzy topological space by Kubiak [1] and ˇSostak [2] in 1985, a large number of mathematicians have taken great interests in generalizing and extending different concepts of set topology and Chang’s fuzzy topology [3] into L- fuzzy topology. The concept of connectedness along with some of its allied forms is one of the directions that have hitherto been ventured with meticulous attention. How- ever, the results obtained in connection with different con- texts like fuzzy connectedness, semi-connectedness, preconnectedness etc. inL-fuzzy topological spaces are seen to be quite parallel and analogous. This is chiefly due to the fact that the study of these variations of the concept of fuzzy connectedness has been effected only by replacing openL-subsets by ther-level cut of fuzzy openL-subset or the like. It can thus be conjectured that the use of a suitable operator should obviates the use ofr-level cut inL-fuzzy topological spaces.

In [4], Shi introduced the notion ofL-fuzzy preopen op- eratorTpinL-fuzzy topological spaces as a generalization of preopenL-subsets, whereLcompletely distributive De- Morgan algebra. Tp(A) can be regarded as the degree to whichAis preopen. So that, actuallyTpreflects the essence ofL-fuzzy topology.

In this paper, we introduce and characterize the degree of preseparatedness and the degree of preconnectedness inL- fuzzy topological spaces. The results in our paper is a generalization to the results of [5].

2. Preliminaries

Throughout this paper, (L, ≤,V ,W

,⁰) denotes a completely distributive DeMorgan algebra, X is a nonempty set. The smallest element and the largest element inLare denoted by 0and1, respectively. The set of all nonzero co-prime elements ofLis denoted byM (L).

We say thatais wedge belowbinL, denoted bya ¿ bif and only if for every subset D ⊆ L, the relation b ≤ W

D

always implies the existence of d ∈ Dwitha ≤ d[6]. A complete latticeLis completely distributive if and only if

b =W

{a ∈ L|a ¿ b}for eachb ∈ L. For anyb ∈ L, define

β(b) = {a ∈ L|a ¿ b}.

For a nonempty set X, the set of all nonzero coprime elements ofL^X is denoted byM (L^X). It is easy to see that

M (L^X)is exactly the set of all fuzzy pointsxλ(λ ∈ M (L)).

The smallest element and the largest element inL^X are denoted by0and1, respectively.

AnL-topological space is a pair(X, T), whereTis a subfam- ily ofL^Xwhich contains0;1and is closed for any suprema and finite infima.Tis called anL-topology onX. Members ofTare called openL-subsets and their complements are called closedL-subsets. The closure and the interior ofL- subsetAare denoted byCl(A)andInt(A), respectively. An

L-subsetAin(X, T)is called preopen [7] iffA ≤ Int(Cl(A)). The preclosure operator ofAin(X, T)is denoted bypCl(A). In an L-topological space (X, T), two L-subsets A, B are called preseparated [5] ifpCl(A) ∧ B = A ∧ pCl(B) = 0. AnL- subsetCis called preconnected ifCcan not be represented as a union of two preseparated non-nullL-subsets.

Definition 2.1. [1, 2]AnL-fuzzy topology on a setXis a mapT : L^X → Lsuch that

(O1) T (0) = T (1) = 1;

(O2) for allA,B ∈ L^X,T (A ∧ B) ≥ T (A) ∧ T (B); Manuscript received Nov. 4, 2010; revised Jan. 14, 2011;

Accepted Jan. 25, 2011.

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(O3) for allAj∈ L^X,j ∈ J,T (W

j∈JAj) ≥V

j∈JT (Aj).

T (A)can be interpreted as the degree to whichAis an open set. T^∗(A) = T (A⁰)will be called the degree of closedness ofA. The pair(X, T )is called anL-fuzzy topological space.

A mappingf : (X, T1) → (Y, T2)is said to be continuous with respect toL-fuzzy topologiesT1andT2ifT1(f_L^←(B)) ≥ T2(B)

holds for allB ∈ L^Y, wheref_L^← is defined byf_L^←(B)(x) = B(f (x)).

Theorem 2.1. [8] LetT : L^X → Lbe a function. Then the following conditions are equivalent:

(1) T is anL-fuzy topology onX;

(2) T_[a]is anL-topology onX, for eacha ∈ M (L).

Definition 2.2. [9] AnL-fuzzy closure operator onXis a mappingCl : L^X→ L^{M (LX )}satisfying the following conditions:

(C1) Cl(A)(xλ) =V

µ¿λCl(A)(xµ), for allxλ∈ M (L^X); (C2) Cl(0)(xλ) = 0for anyxλ∈ M (L^X);

(C3) Cl(A)(xλ) = 1for anyxλ≤ A; (C4) Cl(A ∨ B) = Cl(A) ∨ Cl(B); (C5) for alla ∈ L0,(Cl(W

(Cl(A))[a]))[a]⊂ (Cl(A))[a].

Cl(A)(xλ)is called the degree to whichxλ belongs to the closure ofA.

Lemma 2.1. [9] Let(X, T )be anL-fuzzy topological space and letCl be theL-fuzzy closure operator induced by T. Then for allxλ∈ M (L^X)andA ∈ L^X,

Cl(A)(xλ) = ^

xλ6≤D≥A

(T (D⁰))⁰.

Definition 2.3. [4] Let (X, T ) be an L-fuzzy topological space. ForA ∈ L^X, define the mappingTp: L^X→ Lby

Tp(A) = ^

xλ¿A

_

xλ¿B



T (B) ∧ ^

yµ¿B

^

yµ6≤D≥A

(T (D⁰))⁰



.

ThenTpis calledL-fuzzy preopen operator induced byT, whereT_p(A) can be regarded as the degree to whichAis preopen andT_p^∗(A) = Tp(A⁰)can be regarded as the degree to whichAis preclosed.

Theorem 2.2. [4] Let (X, T ) be an L-fuzzy topological space andA ∈ L^X. ThenA ∈ (Tp)[a]if and only ifAis preopen inT[a], wherea ∈ M (L)and(Tp)[a]= {A ∈ L^X| Tp(A) ≥ a}.

Lemma 2.2. LetTp: L^X→ Lbe AnL-fuzzy preopen operator induced byL-fuzzy topologyT. ThenTpsatisfies the following conditions:

(1) Tp(0) = Tp(1) = 1;

(2) for allAj∈ L^X,j ∈ J,Tp(W

j∈JAj) ≥V

j∈JTp(Aj). Proof. Straightforward.

Definition 2.4. AnL-fuzzy preclosure operator onX is a mappingpCl : L^X → L^{M (LX )} satisfying the following conditions:

(P1) pCl(A)(xλ) =V

µ¿λpCl(A)(xµ), for allxλ∈ M (L^X); (P2) pCl(0)(xλ) = 0for anyxλ∈ M (L^X);

(P3) pCl(A)(xλ) = 1for anyxλ≤ A; (P4) for alla ∈ L0,(pCl(W

(pCl(A))_[a]))_[a]⊂ (pCl(A))_[a].

pCl(A)(x_λ)is called the degree to whichx_λ belongs to the preclosure ofA.

Theorem 2.3. LetTpbe theL-fuzzy preopen operator on

Xand letpCl^Tpbe theL-fuzzy preclosure operator induced byTp. Then for eachxλ∈ M (L^X)andA ∈ L^X,

pCl^Tp(A)(xλ) = ^

x6≤D≥A

(Tp(D⁰))⁰.

Proof. Straightforward.

3. Preseparatedness Degree in L-Fuzzy Topological Spaces

Definition 3.1. Let(X, T )be anL-fuzzy topological space andA,B ∈ L^X. Define

P(A, B) =³ ^

xλ≤A

(pCl(B)(xλ))⁰

´

∧³ ^

yµ≤B

(pCl(A)(yµ))⁰

´ ,

ThenP(A, B)is said to be the preseparatedness degree ofA

andB.

Proposition 3.1. LetT : L^X → {0, 1}be anL-fuzzy topology onXandA,B ∈ L^X. ThenP(A, B) = 1if and only ifA

andBare preseparated in(X, T ).

Lemma 3.1. Let (X, T ) be an L-fuzzy topological space andA,B ∈ L^X. IfA ∧ B 6= 0, thenP(A, B) = 0.

Proof. Letzµ∈ M (L^X)such thatxµ≤ A ∧ B. Thus we have

P(A, B) = ³ ^

xλ≤A

(pCl(B)(x_λ))⁰´

∧³ ^

xλ≤B

(pCl(A)(x_λ))⁰´

≤ ¡

pCl(B)(zµ)¢₀

∧¡

pCl(A)(zµ)¢₀

= 1⁰∧ 1⁰= 0.

Lemma 3.2. Let (X, T ) be anL-fuzzy topological space, andA,B,C,D ∈ L^X. IfC ≤ AandD ≤ B, thenP(A, B) ≤ P(C, D).

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Proof. If C ≤ A and D ≤ B, then pCl(C) ≤ pCl(A) and

pCl(D) ≤ pCl(B). Hence we have

P(A, B) = ³ ^

xλ≤A

(pCl(B)(xλ))⁰

´

∧³ ^

yβ ≤B

(pCl(A)(yβ))⁰

´

≤ ³ ^

xλ≤A

(pCl(D)(xλ))⁰´

∧³ ^

yβ ≤B

(pCl(C)(yβ))⁰´

≤ ³ ^

xλ≤C

(pCl(D)(xλ))⁰

´

∧³ ^

yβ ≤D

(pCl(C)(yβ))⁰

´

= P(C, D).

Lemma 3.3. Let (X, T ) be anL-fuzzy topological space,

A,B ∈ L^X anda ∈ M (L). Then(P(A, B))⁰ ≥ aif and only if there existD,E ∈ L^X such thatD ≥ A,E ≥ B,D ∧ B = E ∧ A = 0and(Tp(D⁰))⁰∨ (Tp(E⁰))⁰6≥ a.

Proof. Suppose that(P(A, B))⁰≥ a. Then(P(A, B))⁰≥ bfor someb ∈ β^∗(a). Then

_

xλ≤A

pCl(B)(xλ) ∨ _

yβ ≤B

pCl(A)(yβ) 6≥ b.

Moreover, we have

_

xλ≤A

^

xλ6≤E≥B

(Tp(E⁰))⁰∨ _

yβ ≤B

^

yβ 6≤D≥A

(Tp(D⁰))⁰6≥ b.

Hence for any xλ ≤ A and for any yβ ≤ B, there exist

D_yβ, E_xλ ∈ L^X such thatxλ 6≤ E_xλ ≥ B, yβ 6≤ D_yβ ≥ A

and(Tp(D⁰_yβ))⁰∨ (Tp(E_xλ⁰ ))⁰ 6≥ b. Let E = V

xλ≤AE_xλ and

D = V

yβ ≤BD_yβ. Then, we have that D ≥ A, E ≥ B,

D ∧ B = E ∧ A = 0and

(Tp(D⁰))⁰∨ (Tp(E⁰))⁰ = (Tp( _

yβ ≤B

D⁰_yβ))⁰∨ (Tp( _

xλ≤B

E_xλ⁰ ))⁰

≤ _

yβ ≤B

(Tp(D⁰_yβ))⁰∨ _

xλ≤A

(Tp(E_xλ⁰ ))⁰

6≥ a.

Conversely if there existD,E ∈ L^Xsuch thatD ≥ A,E ≥ B,

D ∧ B = E ∧ A = 0and(Tp(D⁰))⁰∨ (Tp(E⁰))⁰6≥ a. Since

(P(A, B))⁰ = _

xλ≤A

pCl(B)(xλ) ∨_

yβ

pCl(A)(yβ)

= _

xλ≤A

^

xλ6≤G≥B

(Tp(G⁰))⁰

∨ _

yβ ≤B

^

yβ 6≤H≥A

(Tp(H⁰))⁰

≤ (T_p(D⁰))⁰∨ (T_p(E⁰))⁰.

Then(P(A, B))⁰6≥ a.

4. Preconnectedness Degree in L-Fuzzy Topological Spaces

Definition 4.1. Let(X, T )be anL-fuzzy topological space andG ∈ L^X. Define

PC(G) =^ n

P(A, B)⁰: A, B ∈ L^X\{0}, G = A ∨ Bo .

ThenPC(G)is said to be the preconnectedness degree ofG. From Definition 3.1, we have

PC(G) = ^

A,B∈LX \{0}, G=A∨B





 _

xλ≤A

pCl(B)(xλ) ∨ _

yβ ≤B

pCl(A)(yβ)





.

Proposition 4.1. LetT : L^X→ {0, 1}be anL-topology onX

andG ∈ L^X. ThenPC(G) = 1if and only ifGis preconnected in(X, T ).

Theorem 4.1. Let(X, T )be anL-fuzzy topological space andG ∈ L^X. Then

PC(G) = ^

G∧A6=0,G∧B6=0, G∧A∧B6=0,G≤A∨B

©(Tp(A⁰))⁰∨ (Tp(B⁰))⁰ª .

Proof. (⇒):PC(G)

= ^

A,B∈LX \{0}, G=A∨B

n _

xλ≤A

pCl(B)(x_λ) ∨ _

yβ ≤B

pCl(A)(yβ) o

= ^

A,B∈LX \{0}, G=A∨B

n _

xλ≤A

^

xλ6≤D≥B

(Tp(D⁰))⁰∨ _

yβ ≤B

^

yβ 6≤E≥A

(Tp(E⁰))⁰ o

= ^

G∧A6=0, G∧B6=0, G∧A∧B=0,

G=A∨B

n _

xλ≤G∧A

^

xλ6≤D D≥G∧B

(Tp(D⁰))⁰∨ _

yβ ≤G∧B

^

yβ 6≤E E≥G∧A

(Tp(E⁰))⁰ o

≤ ^

G∧A6=0,G∧B6=0, G∧A∧B=0,G=A∨B

n _

xλ≤G∧A

(Tp(D⁰))⁰∨ _

yβ ≤G∧B

(Tp(E⁰))⁰o

= ^

G∧A6=0,G∧B6=0, G∧A∧B=0,G=A∨B

n

(Tp(D⁰))⁰∨ (Tp(E⁰))⁰o

(⇐): Suppose thatPC(G) 6≥ awherea ∈ M (L). Then there existA,B ∈ L^X\{0}such thatG = A ∨ Band(P(A, B))] 6≥ a. By Lemma 3.3, there exist D, E ∈ L^X such that D ≥ A,

E ≥ B,D ∧ B = E ∧ A = 0and(Tp(D⁰))⁰∨ (Tp(E⁰))⁰6≥ a. Hence we have

^

G∧A6=0,G∧B6=0, G∧A∧B=0,G≤A∨B

{(Tp(B⁰))⁰∨ (Tp(A⁰))⁰} 6≥ a.

Therefore

PC(G) ≥ ^

G∧A6=0,G∧B6=0, G∧A∧B=0,G≤A∨B

{(Tp(B⁰))⁰∨ (Tp(A⁰))⁰}

and this completes the proof.

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Corollary 4.2. Let(X, T )be anL-fuzzy topological space.

Then

PC(1) = ^

A6=0,A∧B=0, A∨B=1

{(Tp(A))⁰∨ (Tp(B))⁰}.

Theorem 4.3. For anyxλ∈ M (L^X), it follows thatPC(xλ) = 1.

Proof. Straightforward.

Theorem 4.4. For anyG ∈ L^X, we have

_

b∈M (LX )

PC(_

(pCl(G))_[b]) ≥ PC(G).

Proof. Let a ≤ PC(G) where a ∈ M (L) and suppose that

W

b∈M (L)PC(W

(pCl(G))_[b]) 6≥ a. ThenPC(W

(pCl(G))_[a]) 6≥ a. By using Theorem 4.1, there exist A, B ∈ L^X such that

¡W(pCl(G))[a]

¢∧ A 6= 0,^¡W(pCl(G))[a]

¢∧ B 6= 0,^¡W(pCl(G))[a]

¢∧

A ∧ B = 0,^W(pCl(G))[a]≤ A ∧ Band(Tp(B⁰))⁰∨ (Tp(A⁰))⁰6≥ a. Since^¡W(pCl(G))_[a]¢

∧ A 6= 0, there existsxλ≤ Asuch that

pCl(G)(xλ) ≥ a. Furthermore, Since^¡W(pCl(G))_[a]¢

∧A∧B = 0, we havexλ6≤ B.

IfG ∧ A 6= 0, thenG ≤W

(pCl(G))_[a]≤ A ∨ Bwe haveG ≤ B, hence it follows that

a ≤ pCl(G)(xλ) = ^

xλ6≤E≥G

(Tp(E⁰))⁰≤ (Tp(B⁰))⁰

which is a contradiction. Analogously, we can proveG∧B 6=

0. Thus byG ∧ A 6= 0,G ∧ B 6= 0,G ∧ A ∧ B = 0,G ≤ A ∨ B,

(Tp(B⁰))⁰∨ (Tp(A⁰))⁰ 6≥ a and Theorem 4.1, we know that

PC(G) 6≥ a, contradictingPC(G) ≥ a. It is proved that

_

b∈M (LX )

PC(_

(pCl(G))[b]) ≥ PC(G).

Theorem 4.5. For anyG,H ∈ L^X, we have

PC(G ∨ H) ≥ (P(G, H))⁰∧ PC(G) ∧ PC(H).

Proof. Leta ≤ (P(G, H))⁰∧ PC(G) ∧ PC(H)wherea ∈ M (L)

and suppose thatPC(G∨H) 6≥ a. Then by using Theorem 4.1, there existA,B ∈ L^Xsuch thatG ∨ H) ∧ A 6= 0,(G ∨ H) ∧ B 6= 0,

(G ∨ H) ∧ A ∧ B = 0,G ∨ H ≤ A ∨ Band(Tp(B⁰))⁰∨ (Tp(A⁰))⁰6≥ a. Since(G ∨ H) ∧ A 6= 0, we haveG ∧ A 6= 0andH ∧ A 6= 0. Suppose that G ∧ A 6= 0(The case ofH ∧ A 6= 0is analogous). Then we haveG ∧ B = 0, otherwise ifG ∧ B 6= 0, then byG ∧ A 6= 0,G ∧ B 6= 0,G ∧ A ∧ B = 0,G ≤ A ∨ Band

(Tp(B⁰))⁰∨ (Tp(A⁰))⁰6≥ a, we know thatPC(G) 6≥ a, which is a contradiction. In this case by(G ∨ H) ∧ B 6= 0we know that

H ∧ B 6= 0. Analogously we can proveH ∧ A = 0. Thus by

G ∨ H ≤ A ∨ Bwe can obtain thatG ≤ AandH ≤ B. Hence byG ≤ A,H ≤ B,G ∧ B = H ∧ A = 0,(Tp(B⁰))⁰∨ (Tp(A⁰))⁰6≥ a

and Lemma 3.3, we have(P(G, H))⁰6≥ a, which is a contradiction. This shows thatPC(G ∨ H) ≥ aand this completes the proof.

Corollary 4.6. Let(X, T )be anL-fuzzy topological space andG,H ∈ L^X. IfA ∧ B 6= 0, then

PC(G ∨ H) ≥ PC(G) ∧ PC(H).

Theorem 4.7. Let(X, T )be anL-fuzzy topological space andG ∈ L^X. Then

PC(G) = ^

xλ,yβ ≤G

_{PC(D_xλ,yβ) : xλ, yβ≤ D_xλ,yβ≤ G}.

Proof. Suppose that ^V_{xλ,yβ ≤G}^W{PC(D_xλ,yβ) : xλ, yβ ≤ D_xλ,yβ≤ G} ≥ awherea ∈ M (L). Take axλ≤ Gfixed. Then for anyyβ≤ G, there existsD_xλ,yβ∈ L^X such thatxλ,yβ ≤ D_xλ,yβ ≤ GandPC(D_xλ,yβ) ≥ a. LetD_xλ =W

yβ ≤GD_xλ,yβ. ThenD_xλ = Gand^V_{yβ ≤G}D_xλ,yβ 6= 0. By using Corollary 4.6, we have PC(G) = PC(D_xλ) ≥ V

yβ ≤GPC(D_xλ,yβ) ≥ a. This shows that

PC(G) ≥ ^

xλ,yβ ≤G

Since

PC(G) ≤ ^

xλ,yβ ≤G

_{PC(D_xλ,yβ) : xλ, yβ≤ D_xλ,yβ≤ G}

is clear, then we have

PC(G) = ^

xλ,yβ ≤G

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A. Ghareeb received the B.S. degree in Mathematics from South Valley University, Qena, Egypt in 2001 and the M.S. and the Ph.D. degrees in Topology from South Valley University, Qena, Egypt in 2007 and 2009, respectively.

Heiscurrentlya/HFWXUHULQWKH'HSDUWPHQW

of Mathematics, South Valley University, Egypt. His current research interests include General Topology, Fuzzy Topology, Fuzzy Mathematics.