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        Almost Fuzzy Compactness in L-fuzzy Topological Spaces

        2015-11-03 11:42:46LiHongYananDCuiWei

        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 ??LX.Then

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

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

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

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

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

        (6)? 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 ??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.

        References

        [1]Chen S L.Almost F-compactness in L-fuzzy topological spaces.Northeastern Math.J.,1991,7(4):428-432.

        [2]Concilio A D,Gerla G.Almost compactness in fuzzy topological spaces.Fuzzy Sets and Systems,1984,13:187-192.

        [3]Haydar Es A.Almost compactness and near compactness in fuzzy topological spaces.Fuzzy Sets and Systems,1987,22:289-295.

        [4]Kudri S R T,Warner M W.Some good L-fuzzy compactness-related concepts and their properties I.Fuzzy Sets and Systems,1995,76:141-155.

        [5]Kaur C,Sharfuddin A.On almost compactness in the fuzzy setting.Fuzzy Sets and Systems,2002,125:163-165.

        [6]Meng H,Meng G W.Almost N-compact sets in L-fuzzy topological spaces.Fuzzy Sets and Systems,1997,91:115-122.

        [7]Mukherjee M N,Sinha S P.Almost compact fuzzy sets in fuzzy topological spaces.Fuzzy Sets and Systems,1990,38:389-396.

        [8]Mukherjee M N,Chakraborty R P.On fuzzy almost compact spaces.Fuzzy Sets and Systems,1998,98:207-210.

        [9]Wen G F,Shi F G,Li H Y.Almost S?-compactness in L-topological spaces.Iran.J.Fuzzy Syst.,2008,5(3):31-44.

        [10]Shi F G.A new approach to almost fuzzy compactness.Proyecciones,2009,28(1):75-87.

        [11]Liu Y M,Luo M K.Fuzzy Topology,Singapore:World Scientific Publishing,1997.

        [12]H¨ohle U,Rodabaugh S E eds.Mathematics of Fuzzy Sets:Logic,Topology,and Measure Theory,The Handbooks of Fuzzy Sets Series,vol.3.Boston-Dordrecht-London:Kluwer Academic Publishers,1999.

        [13]?Sostak A P.On Compactness and Connectedness Degrees of Fuzzy Sets in Fuzzy Topological Spaces.in:General Topology and its Relations to Modern Analysis and Algebra,Berlin:Heldermann Verlag,1988:519-532.

        [14]Shi F G,Li R X.Compactness in L-fuzzy topological spaces.Hacet.J.Math.Stat.,2011,40(6):767-774.

        [15]Shi F G.A new definition of fuzzy compactness.Fuzzy Sets and Systems,2007,158:1486-1495.

        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).

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