Almost Fuzzy Compactness in L-fuzzy Topological Spaces

2015-11-03 11:42LiHongYananDCuiWei

Li Hong-YananD Cui Wei

(1.School of Mathematics and Information Science,Shandong Institute of Business and Technology,Yantai,Shandong,264005)

(2.Department of Modern Science and Technology,Shenyang Municipal Party Committee Party School,Shenyang,110036)

Communicated by Lei Feng-chun

Almost Fuzzy Compactness in L-fuzzy Topological Spaces

Li Hong-Yan1anD Cui Wei2

(1.School of Mathematics and Information Science,Shandong Institute of Business and Technology,Yantai,Shandong,264005)

(2.Department of Modern Science and Technology,Shenyang Municipal Party Committee Party School,Shenyang,110036)

Communicated by Lei Feng-chun

In this paper,the notion of almost fuzzy compactness is defined in L-fuzzy topological spaces by means of inequality,where L is a completely distributive DeMorgan algebra.Its properties are discussed and many characterizations of it are presented.

L-fuzzy topological space,L-fuzzy almost compactness,L-fuzzy compactness,almost fuzzy compactness

2010 MR subject classification:54A40,54D30,03E72

Document code:A

Article ID:1674-5647(2015)03-0267-07

1 Introduction

Almost compactness is a very important concept.Many researchers have tried successfully to generalize the compactness theory of general topology to L-topology(see[1-9]).Recently,Shi[10]introduced new definitions of almost fuzzy compactness in L-topological spaces with the help of inequality,where L is a completely distributive DeMorgan algebra.The aim of this paper is to generalize the notion of almost compactness in[10]to L-fuzzy topological spaces,thus some properties and characterizations are researched.

2 Preliminaries

In this paper,(L,∨,∧,′)is a completely distributive DeMorgan algebra(i.e.,completely distributive lattice with order-reversing involution,see[11]).The largest element and thesmallest element in L are denoted by⊤and⊥,respectively.

Definition 2.1[12]An L-fuzzy topology on a set X is a map τ:LX→L such that

For a subfamily Φ⊆LX,2(Φ)denotes the set of all finite subfamilies of Φ.

Definition 2.2[13]is an L-fuzzy inclusion on X,it is defined asFor simplicity,it is denoted by[A~⊂B]instead of~⊂(A,B),

3 Definitions and Properties of L-fuzzy Almost Compactness

Definition 3.1Let(X,τ)be an L-fuzzy topological space and A∈LX.For all r∈L,

is called the r-interiors of A with respect to τ.The r-closures of A with respect to τ is defined as

An L-topology T can be regarded as a map χT:LX→L defined by

In this way,(X,χT)is a special L-fuzzy topological space and

This shows that Definition 3.1 can be regarded as the generalization in L-fuzzy topological space of the interiors and closures in L-topological space.

By Definition 3.1,we have the following theorem.

Theorem 3.1Let(X,T)be an L-topological space and A∈LX.Then,for all r,s∈L,

(6)A∈τr⇔τ(A)≥r.

Definition 3.2Let(X,τ)be an L-fuzzy topological space.G ∈LXis called L-fuzzy almost compact,if U⊆LX,it follows that

In an L-topological space(X,T),G∈LXis almost fuzzy compact(see[10])if and only if for all U⊆T,the inequality

is satisfied.

At the same time,(X,χT)is a special L-fuzzy topological space,and for all U⊆T,we have r=χT(U)=⊤.So,G∈LXis almost fuzzy compactness if and only if for all U⊆T,it follows that

Thus,the following theorem can be obtained.

Theorem 3.2Let(X,T)be an L-topological space and G∈LX.Then G is almost fuzzy compact in(X,T)if and only if G is L-fuzzy almost compact in(X,χT).

The definition of L-fuzzy compactness(see[14])in an L-fuzzy topological space can be described as follows:Let(X,τ)be an L-fuzzy topological space.G∈LXis called L-fuzzy compact if U⊆LX,then

In this way,the proposition“L-fuzzy compactness⇒L-fuzzy almost compactness”can be proved by Theorem 3.1.

By Definitions 2.1,2.2,3.1,3.2 and Theorem 3.1,we can prove the following theorem.

Theorem 3.3Let(X,τ)be an L-fuzzy topological space and G∈LX.Then G is L-fuzzy almost compact if and only if for all P⊆LX,one has

Theorem 3.4Let(X,τ)be an L-fuzzy topological space and G∈LX.Then G is L-fuzzy almost compact if and only if for all r∈L and for τ(U)≤r,U⊆τr,one has

Proof.(1)Necessity.Suppose that G is L-fuzzy almost compact,and for all r∈L and for τ(U)≤r,U⊆τr.As U⊆τr,by Theorem 3.1,we have τ(A)≥r for all A∈U.Then we get τ(U)≥r.Therefore,τ(U)=r.By L-fuzzy almost compactness of G,we have

(2)Sufficiency.Suppose that for all r∈L and for τ(U)≤r,U⊆τr.We have

For all U⊆LX,let r=τ(U).Then for all A∈U,τ(A)≥r.So we get U⊆τrby Theorem 3.1.Thus

Therefore,G is L-fuzzy almost compact.

Theorem 3.5Let(X,τ)be an L-fuzzy topological space and G∈LX.If both G and H are L-fuzzy almost compact,then G∨H is L-fuzzy almost compact.

Proof.For all U⊆LX,let r=τ(U).Since G and H are L-fuzzy almost compact,we have

and

By the facts

and

we know that

Therefore,G∨H is L-fuzzy almost compact.

Theorem 3.6Let(X,τ)be an L-fuzzy topological space and G∈LX.If G is L-fuzzy almost compact,and H ∈LXwith τ(H)=τ∗(H)=⊤,then G∧H is L-fuzzy almost compact.

Proof.For all U⊆LX,let r=τ(U),W=U∪H′.Then we get W ⊆LX.It is easy to check

Then we obtain

from the following facts

and

Therefore,G∧H is L-fuzzy almost compact.

4 Characterizations of L-fuzzy Almost Compactness

Definition 4.1Let(X,τ)be an L-fuzzy topological space,G∈LXand Ω⊆LX.Then

(1)Ω is r-cover(see[9],r-shading in[15])of G if and only if

(2)Ω is r+-cover(see[9],strong r-shading in[15])of G if and only if

(3)Ω is almost r-cover of G with respect to τ if and only if

(4)Ω is almost r+-cover of G with respect to τ if and only if

(5)Ω is an r-remote family(see[15])of G if and only if

(6)Ω is a strong r-remote family(see[15])of G if and only if

By Definitions 3.2,4.1 and Theorem 3.3,we can get the following theorem.

Theorem 4.1Let(X,τ)be an L-fuzzy topological space and G∈LX.Then the following conditions are equivalent:

(1)G is L-fuzzy almost compact;

(2)For any r∈L{⊤},each r+-cover U of G with τ(U)≰r has a finite subfamily V which is an almost r+-cover of G;

(3)For any r∈L{⊤},each r+-cover of G with τ(U)≰r has a finite subfamily which is an almost r-cover of G;

(4)For any r∈P(L),each r+-cover of G with τ(U)≰r has a finite subfamily which is an almost r-cover(r+-cover)of G;

(5)For any r∈P(L),each r+-cover U of G with τ(U)≰r has b∈α∗(r)and a finite subfamily V such that V is an almost b-cover of G;

(6)For any r∈L{⊤},each strong r-remote family P of G with τ∗(P)≰r′has a finite subfamily Q such thatis a strong r-remote family of G;

(7)For any r∈L{⊤},each strong r-remote family P of G with τ∗(P)≰r′has a finite subfamily Q such thatis a r-remote family of G;

(8)For any r∈M(L),each strong r-remote family P of G with τ∗(P)≰r′has a finite subfamily Q and b∈β∗(r)such thatis a strong b-remote family of G;

(9)For any r∈M(L),each strong r-remote family P of G with τ∗(P)≰r′has a finite subfamily Q and b∈β∗(r)such thatis a b-remote family of G.

Definition 4.2[15]Let(X,τ)be an L-fuzzy topological space,G∈LXand Ω⊆LX.We define that

It is easy to prove the following theorem.

Theorem 4.2Let(X,τ)be an L-fuzzy topological space and G∈LX.Then the following conditions are equivalent:

(1)G is L-fuzzy almost compact;

(2)For any r∈L{⊥}(r∈M(L)),each strong βr-cover U of G with r∈β(τ(U))has a finite subfamily V such thatis a(strong)βa-cover of G;

(3)For any r∈L{⊥}(r∈M(L)),each strong βr-cover U of G with r∈β(τ(U))has a finite subfamily V of U and b∈L(b∈M(L))with a∈β(b)such thatis a(strong) βb-cover of G;

(4)For any r∈L{⊥}(r∈M(L))and any b∈β∗(r),each Qr-cover U of G with τ(A)≥r for any A∈U has a finite subfamily V such thatis a Qb-cover of G;

(5)For any r∈L{⊥}(r∈M(L))and any b∈β∗(r),each Qr-cover U of G with τ(A)≥r for any A∈U has a finite subfamily V such thatis a(strong)βb-cover of G.

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10.13447/j.1674-5647.2015.03.09

date:Oct.28,2014.

The NSF(11471297)of China.

E-mail address:lihongyan@sdibt.edu.cn(Li H Y).