L-fuzzy interiors and L-fuzzy closures

Fuzzy Sets and Systems160(2009)1218–1232/locate/fss L-fuzzy interiors and L-fuzzy closuresFu-Gui Shi∗Department of Mathematics,School of Science,Beijing Institute of Technology,Beijing100081,PR ChinaReceived26October2007;received in revised form8September2008;accepted12September2008Available online26September2008AbstractIn this paper,we introduce new definitions of L-fuzzy neighborhood systems,L-fuzzy interior operators and L-fuzzy closure operators.Three characterizations of the category L-FTOP of L-fuzzy topological spaces and their L-fuzzy continuous mappings are presented by means of the category L-FNS of L-fuzzy neighborhood spaces and their continuous mappings,the category L-FIS of L-fuzzy interior spaces and their L-fuzzy continuous mappings,and the category L-FCS of L-fuzzy closure spaces and their L-fuzzy continuous mappings.©2008Elsevier B.V.All rights reserved.Keywords:L-fuzzy topology;L-fuzzy neighborhood system;L-fuzzy interior;L-fuzzy closure1.IntroductionSince Chang[1]introduced[0,1]-topology in fuzzy set theory,many researchers have tried to generalize the theory of general topology to the fuzzy setting.A Chang topology is a crisp subset of I X.However,in a completely different direction,Höhle[5]presented the notion of a fuzzy topology being viewed as an I-subset of a powerset2X.Then Kubiak [9]and˘Sostak[14]independently extended Höhle’s fuzzy topology to L-subsets of L X,which is called L-fuzzy topology (see[6,7,12]).It is well known that neighborhood systems,interior operators and closure operators play an important role in topology and they are very good ways to characterize topology.Many authors[2,4,7,8]have studied(quasi-)neighborhood systems,interior operators and closure operators in L-fuzzy topological spaces.The notions of quasi-neighborhood systems and remote-neighborhood systems play an important role in L-topology [10].In[4],Fang generalized the quasi-neighborhood systems from L-topological spaces to L-fuzzy topological spaces and proved that the category of L-fuzzy quasi-neighborhood spaces and their continuous mappings is isomorphic to the category of L-fuzzy topological spaces and their continuous mappings.However,the notion of neighborhoods is also important in L-topology.In some case,we can use neighborhoods and quasi-neighborhoods at the same time.For example,in[13]we defined a T2axiom by means of neighborhoods and remote-neighborhoods.Therefore it is necessary to generalize neighborhood systems to L-fuzzy topology.∗Tel.:+861088583137.E-mail addresses:fuguishi@,f.g.shi@(F.-G.Shi).0165-0114/$-see front matter©2008Elsevier B.V.All rights reserved.doi:10.1016/j.fss.2008.09.002F .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–12321219In [7],a notion of L -fuzzy neighborhood systems was introduced as follows:Definition 1.1(Höhle and ˘Sostak [7]).Let X be a set.A map N :X ×L X ×L →L is called an L -fuzzy neighborhoodsystem iff N satisfies the following axioms:(N1)N (x , ,a )= ;(N2)u v,b a ⇒N (x ,u ,a ) N (x ,v,b );(N3)N (x ,u 1,a )∧N (x ,u 2,b ) N (x ,u 1∧u 2,a ∧b )(where ⊗is replaced by ∧);(N4)N (x ,u ,a ) u (x );(N5)N (x ,u ,a ){N (x ,v,b ):v (y ) N (y ,u ,a )∀y ∈X };(N6)N (x ,u ,⊥)=u (x );(N7)For any nonempty subset {a i :i ∈ }of L the implication holds:u (x ) N (x ,u ,a i )∀i ∈ ,∀x ∈X ⇒u (x ) Nx ,u ,i ∈a i ∀x ∈X .In [2,7,8],two notions of interior operators and closure operators were defined,respectively,as follows:Definition 1.2(Höhle and ˘Sostak [7]).A map I :L X ×L →L X is called an L -fuzzy interior operator on X iff itsatisfies the following conditions:(I1)I ( ,a )= ∀a ∈L ;(I2)I (v,b ) I (u ,a )whenever v u ,a b ;(I3)I (u ,a )∧I (v,b ) I (u ∧v,a ∧b );(I4)I (u ,a ) u ;(I5)I (u ,a ) I (I (u ,a ),a );(I6)I (u ,⊥)=u ;(I7)If ∅ K ⊆L ,I (u ,a )=u 0∀a ∈K ,then I (u ,∨K )=u 0.Definition 1.3(Chattopadhyay and Samanta [2],Kim [8]).A map C :L X ×L 0→L X is called an L -fuzzy closure operator on X iff it satisfies the following conditions:for all , ∈L X ,r ,s ∈L 0,(C1)C (⊥,a )=⊥;(C2)C ( ,r ) ;(C3) ⇒C ( ,r ) C ( ,r );(C4)C ( ∨ ,r )=C ( ,r )∨C ( ,r );(C5)r s ⇒C ( ,r ) C ( ,s );(C6)C ( ,r )=C (C ( ,r ),r );(C7)If r = {s ∈L :C ( ,s )= },then C ( ,r )= .Definitions 1.2and 1.3actually rely on the level L -topology T [r ]of the L -fuzzy topology T ,i.e.,I ( ,r )and C ( ,r )can be viewed as the interior and the closure of in L -topology T [r ][15].Moreover the conditions of Definitions 1.1–1.3are also more complicated.Therefore their applications are restricted to some degree.The aim of this paper is to introduce new definitions of L -fuzzy neighborhood systems,L -fuzzy interior operators and L -fuzzy closure operators in L -fuzzy topological spaces in different ways from [2,7,8].Our definitions do not rely on the level L -topology T [r ]of the L -fuzzy topology T .Three characterizations of the category L -FTOP of L -fuzzy topological spaces and their L -fuzzy continuous mappings are presented by means of the category L -FNS of L -fuzzy neighborhood spaces and their continuous mappings,the category L -FIS of L -fuzzy interior spaces and their L -fuzzy continuous mappings,and the category L -FCS of L -fuzzy closure spaces and their L -fuzzy continuous mappings.2.PreliminariesThroughout this paper,L denotes a completely distributive DeMorgan algebra.The smallest element and the largest element in L are denoted by ⊥and ,respectively.X is a nonempty set.The set of all nonzero co-prime elements of L1220F .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–1232is denoted by J (L ).The set of all nonzero co-prime 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 ( ∈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 [3].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 (b )={a ∈L :a ≺b }.It is easy to see that for all a ∈L , (a )= b ≺a (b ).Hence the wedge below relation in a completely distributive lattice has the interpolation property,this means a ≺b ⇒∃c ∈L such that a ≺c ≺b .Moreover we can see that a ≺ i ∈ b i ⇒∃b i such that a ≺b i .Some properties of the map can be found in [10].For any L -fuzzy set A ∈L X and any a ∈L ,we use the following notations:A [a ]={x ∈X :A (x ) a },A (a )={x ∈X :A (x ) a },a (x )=a ,∀x ∈X .Definition 2.1(Höhle and ˘Sostak [7],Kubiak [9],Rodabaugh [12],˘Sostak [14]).An L -fuzzy topology on a set X isa map T :L X →L such that(LFT1)T ( )=T (⊥)= ;(LFT2)∀U ,V ∈L X ,T (U ∧V ) T (U )∧T (V );(LFT3)∀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 L -set.T ∗(U )=T (U )will be called the degree ofclosedness of U ,where U is the L -complement 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 1and T 2if T 1(f ←L (U )) T 2(U )holds for all U ∈L Y ,where f ←L is defined by f ←L(U )(x )=U (f (x ))[11].The category of L -fuzzy topological spaces and their continuous mappings is denoted by L -FTOP .In an L -topological space (X , ),an open L -set U is said to be a quasi-neighborhood of a fuzzy point x ,if x U or U (x )=(U (x )) .In [4],Fang generalized the concept of quasi-neighborhood systems to L -fuzzy topological spaces as follows:Definition 2.2(Fang [4]).An L -fuzzy quasi-neighborhood system on X is a set Q ={Q x :x ∈J (L X )}of maps {Q x :L X →L }satisfying the following conditions:(LFQ1)Q x ( )= and Q x (⊥)=⊥;(LFQ2)Q x (A ) ⊥⇒x A ;(LFQ3)Q x (A ∧B )=Q x (A )∧Q x (B );(LFQ4)Q x (A )=x D A y D Q y (D ).A set X equipped with an L -fuzzy quasi-neighborhood system Q ={Q x :x ∈J (L X )},denoted by (X ,Q ),is called an L -fuzzy quasi-neighborhood space.A mapping f :X →Y between two L -fuzzy quasi-neighborhood spaces (X ,Q 1)and (Y ,Q 2)is called continuous if ∀x ∈J (L X ),∀U ∈L Y ,(Q 2)f →L (x)(U ) (Q 1)x (f ←L (U )),where f →L (x )=f (x ).The category of L -fuzzy quasi-neighborhood spaces and their continuous mappings is denoted by L -FQN .3.L -fuzzy neighborhood systemsIn this section,we shall present a new definition of L -fuzzy neighborhood systems,and study the relation between L -fuzzy neighborhood systems and L -fuzzy topologies.In an L -topological space (X , ),an open L -set U is said to be a neighborhood of a fuzzy point x ,if x U or U (x ).It is easy to see that the neighborhood systems of a fuzzy point x is different from theF.-G.Shi/Fuzzy Sets and Systems160(2009)1218–12321221 quasi-neighborhood systems of x .Now we generalize the concept of neighborhood systems to L-fuzzy topological spaces as follows:Definition3.1.An L-fuzzy neighborhood system on X is defined to be a set N={N x:x ∈J(L X)}of maps{N x:L X→L}satisfying the following conditions:(LFN1)N x( )= and N x(⊥)=⊥;(LFN2)N x(U)=⊥for any x U;(LFN3)N x(U∧V)=N x(U)∧N x(V);(LFN4)N x(U)=x V Uy ≺VN y (V).N x(U)is called the degree to which U is a neighborhood of x .A set X equipped with an L-fuzzy neighborhood system N={N x:x ∈J(L X)},denoted by(X,N),is called anL-fuzzy neighborhood space.A mapping f:X→Y between two L-fuzzy neighborhood spaces(X,N1)and(Y,N2)is called continuous if ∀x ∈J(L X),∀U∈L Y,(N2)f→L (x )(U) (N1)x (f←L(U)).It is easy to check that L-fuzzy neighborhood spaces and their continuous mappings form a category,denoted by L-FNS.Theorem3.2.Let T:L X→L be an L-fuzzy topology on X.∀x ∈J(L X),define a mapping N T x:L X→L byN T x(U)=x V UT(V).Then N T={N T x:x ∈J(L X)}is an L-fuzzy neighborhood system,which is called the L-fuzzy neighborhood systeminduced by T.Proof.First we prove the following:(LFN0)N T x(U)=≺ NTx(U).It is obvious that N T x(U)≺ NTx(U).Now we prove N T x(U)≺ NTx(U).Let a≺≺ NTx(U).Thena≺N T x(U)for any ≺ .Hence there exists V x∈L X such that x V x U and a T(V x )for any ≺ .LetW=≺ V x .Then we have that x W U anda≺ T(V x ) T⎛⎝≺V x⎞⎠=T(W)x V UT(V)=N T x(U).This shows N T x(U)≺ NTx(U).(LFN1)and(LFN2)are true trivially.(LFN3)From the definition of N T x,it follows that∀U,V∈L X,U V⇒N T x(U) N T x(V).Hence∀U,V∈L X,we always haveN T x(U∧V) N T x(U)∧N T x(V).In order to prove N T x(U∧V) N T x(U)∧N T x(V),let a≺N T x(U)∧N T x(V).Then a≺N T x(U)and a≺N T x(V).This implies thata≺x W U T(W)and a≺x W VT(W).1222F.-G.Shi/Fuzzy Sets and Systems160(2009)1218–1232 Hence there are W1and W2such thatx W1 U,x W2 V,a T(W1)and a T(W2).Thus we have thatx W1∧W2 U∧V and a T(W1)∧T(W2) T(W1∧W2). Thereforea T(W1∧W2)x W V∧U T(W)=N T x(U∧V).This shows N T x(U∧V) N T x(U)∧N T x(V).(LFN4)∀V∈L X with x V U and∀ ≺ ,obviously we haveT(V)y ≺V N T y(V) N T x(V) N T x(U).This impliesN T x(U)=x V UT(V)x V Uy ≺VN T y(V) N T x(U).Hence by(LFN0)we obtainN T x(U)x V Uy ≺VN T y(V)≺N T x(U)=N T x(U).ThereforeN T x(U)=x V Uy ≺VN T y(V).çThe following lemma shows that an L-fuzzy topology can be denoted by its L-fuzzy neighborhood system.Lemma3.3.For an L-fuzzy topology T on X and for any U∈L X,it follows that T(U)=x ≺UN T x(U).Proof.It is obvious that T(U)x ≺UN T x(U).Now we prove T(U)x ≺UN T x(U).Let a≺x ≺UN T x(U).Then for any x ≺U,there exists V x ∈L X such that x V x U and a T(V x ).Obviously U=x ≺U V x .Thuswe haveT(U)=T ⎛⎝x ≺UV x⎞⎠x ≺UT(V x) a.This shows T(U)x ≺UN T x(U).çTheorem3.4.If f:(X,T1)→(Y,T2)is continuous with respect to L-fuzzy topologies T1and T2,then f:(X,N T1)→(Y,N T2)is also continuous with respect to L-fuzzy neighborhood systems N T1and N T2.Proof.If f:(X,T1)→(Y,T2)is continuous with respect to L-fuzzy topologies T1and T2,then∀U∈L Y, T2(U) T1(f←L(U)),this implies that∀x ∈J(L X),N T2f→L (x )(U)=f→L(x ) V UT2(V)x f←L(V) f←L(U)T1(f←L(V)) N T1x(f←L(U)).Therefore f:(X,N T1)→(Y,N T2)is continuous.çF .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–12321223Theorem 3.5.Let N ={N x :x ∈J (L X )}be an L-fuzzy neighborhood system on X.Define T N :L X →L byT N (U )=x ≺UN x (U ).Then T N is an L-fuzzy topology on X and N (TN )=N .Proof.Check that T N is an L -fuzzy topology on X as follows:(LFT1)is trivial from (LFN1).(LFT2)can be proved by the following fact:∀U ,V ∈L X ,T N(U ∧V )=x ≺U ∧VN x (U ∧V )=x ≺U ∧VN x (U )∧N x (V )⎛⎝ x ≺UN x (U )⎞⎠∧⎛⎝ x ≺VN x (V )⎞⎠=T N (U )∧T N (V ).(LFT3)holds by the following fact:∀{U i :i ∈ }⊆L X ,T N i ∈U i = x ≺i ∈U iN x i ∈U i = i ∈ x ≺U iN x i ∈U ii ∈x ≺U iN x (U i )= i ∈T N (U i ).By (LFN4)and Theorem 3.2we can obtain N (T N )=N .çBy Lemma 3.3we can prove the next result.Corollary 3.6.For an L-fuzzy topology T on X ,it follows that T =T (NT).Theorem 3.7.If f :(X ,N 1)→(Y ,N 2)is continuous with respect to L-fuzzy neighborhood systems N 1and N 2,then f :(X ,T N 1)→(Y ,T N 2)is continuous with respect to L-fuzzy topologies T N 1and T N 2.Proof.If f :(X ,N 1)→(Y ,N 2)is continuous with respect to L -fuzzy neighborhood systems N 1and N 2,then ∀U ∈L Y ,∀x ∈J (L X ),(N 2)f →L (x)(U ) (N 1)x (f ←L (U )),this impliesTN 2(U )=y ≺U(N 2)y (U )f →L (x)≺U (N 2)f →L(x)(U )=x ≺f ←L(U )(N 2)f →L(x)(U )x ≺f ←L(U )(N 1)x (f ←L (U ))=T N 1(f ←L (U )).Therefore f :(X ,T N 1)→(Y ,T N 2)is continuous.çThe next result follows from Theorems 3.4,3.5,3.7and Corollary 3.6.Theorem 3.8.L -FTOP is isomorphic to L -FNS .The above results show that L -fuzzy topologies and L -fuzzy neighborhood systems come to the same thing.It maybe a convenient way to investigate local properties of an L -fuzzy topology by means of L -fuzzy neighborhood systems or L -fuzzy quasi-neighborhood systems.1224F .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–12324.L -fuzzy interior operatorsIn this section,we shall introduce a new definition of L -fuzzy interior operators,and study the relation between them and L -fuzzy topologies.For an L -topological space (X , ),its L -interior operator i :L X →L X satisfies the following conditions:(1)i ( )= ;(2)i (A ) A ;(3)i (A ∧B )=i (A )∧i (B );(4)i (A ) i (i (A )).Now we consider such a function I (A ):J (L X )→2defined byI (A )(x )=1,x i (A ),0,x i (A ).It is easy to check the following conditions:(LI1)I (A )(x )=1⇔∀ ≺ ,I (A )(x )=1,∀x ∈J (L X );(LI2)I ( )(x )=1for any x ∈J (L X );(LI3)I (A )(x )=0for any x A ;(LI4)I (A ∧B )=I (A )∧I (B );(LI5)(I (A ))(0)⊆(I ( (I (A ))(0)))(0).Based on the above fact,we can generalize L -interior operators to L -fuzzy topological spaces as follows:Definition 4.1.An L -fuzzy interior operator on X is a map Int :L X →L J (LX )satisfying the following conditions:(LFI1)Int(A )(x )=≺ Int(A )(x ),∀x ∈J (L X );(LFI2)Int( )(x )= for any x ∈J (L X );(LFI3)Int(A )(x )=⊥for any x A ;(LFI4)Int(A ∧B )=Int(A )∧Int(B );(LFI5)∀a ∈L \{ },(Int(A ))(a )⊆(Int((Int(A ))(a )))(a ).Int(A )(x )is called the degree to which x belongs to the interior of A .A set X equipped with an L -fuzzy interior operator Int (or Int X ),denoted by (X ,Int),is called an L -fuzzy interior space.A mapping f :X →Y between two L -fuzzy interior spaces (X ,Int X )and (Y ,Int Y )is called continuous if ∀x ∈J (L X ),∀U ∈L Y ,Int Y (U )(f →L (x )) Int X (f ←L (U ))(x ).It is easy to check that L -fuzzy interior spaces and their continuous mappings form a category,denoted by L -FIS .The following theorem explains the relation between the L -fuzzy neighborhood system and the L -fuzzy interior operator of an L -fuzzy topology.Theorem 4.2.Let T be an L-fuzzy topology on X and let N T ={N T x:x ∈J (L X )}be the L-fuzzy neighborhood system induced by T .Define a mapping Int T :L X →L J (L X )byInt T (A )(x )=N Tx(A ).Then Int T is an L-fuzzy interior operator on X ,which is called the L-fuzzy interior operator induced by T .Proof.(LFI1)–(LFI3)are obvious.(LFI4)can be obtained from (LFN3).In order to prove (LFI5),let x ∈(Int T (A ))(a ).ThenInt T (A )(x )=N Tx(A )= x U Ay ≺UN Ty(U ) a .F .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–12321225Hence there exists U ∈L X such that x U A andy ≺UInt T (U )(y )=y ≺UN T y(U ) a .This implies that y ∈(Int T (U ))(a )holds for any y ≺U .By (LFI4)we havex U = {y :y ≺U } (Int T (U ))(a )(Int T (A ))(a ).ThereforeIntT(Int T(A ))(a )(x )=N Tx(Int T(A ))(a )=x U(Int T (A ))(a )y ≺UN T y(U ) a .This shows x ∈(Int T ((Int T (A ))(a )))(a ).The proof of (LFI5)is completed.çThe next corollary give a representation of the L -fuzzy interior operator by means of an L -fuzzy topology.It can be obtained from Theorems 3.2and 4.2.Corollary 4.3.Let T be an L-fuzzy topology on X and let Int T be the L-fuzzy interior operator induced by T .Then ∀x ∈J (L X ),∀A ∈L X ,Int T (A )(x )=x V AT (V ).Theorem 4.4.If f :(X ,T 1)→(Y ,T 2)is continuous with respect to L-fuzzy topologies T 1and T 2,then f :(X ,Int T 1)→(Y ,Int T 2)is continuous with respect to L-fuzzy interior operators Int T 1and Int T 2.Proof.If f :(X ,T 1)→(Y ,T 2)is continuous with respect to L -fuzzy topologies T 1and T 2,then ∀U ∈L Y ,T 2(U ) T 1(f ←L (U )),this implies that ∀x∈J (L X ),Int T 2(U )(f →L (x ))= f →L (x) V U T 2(V )x f ←L (V ) f ←L(U )T 1(f ←L (V )) Int T 1(f ←L (U ))(x ).Therefore f :(X ,Int T 1)→(Y ,Int T 2)is continuous.çTheorem 4.5.Let Int :L X →L J (LX )be an L-fuzzy interior operator on X .∀x ∈J (L X ),define N Int x:L X →L by N Int x(A )=Int(A )(x ).Then N Int ={N Int x:x ∈J (L X )}is an L-fuzzy neighborhood system on X .Proof.(LFN1)–(LFN3)holds trivially from (LFI2)–(LFI4).We need to prove (LFN4).First we check N Int x (U ) x V U y ≺V N Int y (V ).Suppose that b ∈L and N Int x(U ) b .Then there exists a ∈ (b )such that N Int x (U ) a .This implies Int(U )(x ) a .By (LFI5)we havex ∈(Int(U ))(a )⊆ Int (Int(U ))(a )(a ).Let V =(Int(U ))(a ).Then x V U and there exists y ∈(Int(U ))(a )such that y y for any y ≺V .Furtherwe havey ∈(Int(U ))(a )⊆ Int (Int(U ))(a )(a )=(Int(V ))(a ).1226F.-G.Shi/Fuzzy Sets and Systems160(2009)1218–1232 This implies Int(V)(y ) a.By(LFI1)we know Int(V)(y ) Int(V)(y ).Hence N Int y(V)=Int(V)(y ) a.Thereforey ≺VN Int y(V) b.This showsx V Uy ≺VN Int y(V) b.The proof of N Int x(U)x V Uy ≺V N Int y(V)is completed.Second we check N Int x(U)x V Uy ≺VN Int y(V).Suppose that b∈L andx V Uy ≺VN Int y(V) b.Then there exists a∈ (b)such thatx V Uy ≺VN Int y(V) a.Further there exists V∈L X such that x V U andy ≺VN Int y(V) a.Hence N Int x(V) a for any≺ .By(LFI1)we haveN Int x(U)=Int(U)(x )=≺Int(U)(x )≺Int(V)(x )=≺N Int x(V) b.This shows N Int x(U)x V Uy ≺VN Int y(V).çBy Theorems3.5,4.2and4.5we can obtain the following result.Corollary4.6.Let Int:L X→L J(L X)be an L-fuzzy interior operator on X.Define T Int:L X→L byT Int(U)=x ≺UInt(U)(x ).Then T Int is an L-fuzzy topology on X and Int(T Int)=Int.By Corollaries3.6,4.3and Theorem4.5we can obtain the following result.Corollary4.7.For an L-fuzzy topology T on X,T(Int T)=T.Theorem4.8.If f:(X,Int X)→(Y,Int Y)is continuous with respect to L-fuzzy interior operators Int X and Int Y,then f:(X,T Int X)→(Y,T Int Y)is continuous with respect to L-fuzzy topologies T Int X and T Int Y.Proof.If f:(X,Int X)→(Y,Int Y)is continuous with respect to L-fuzzy interior operators Int X and Int Y,then ∀U∈L Y,∀x ∈J(L X),Int Y(U)(f→(x )) Int X(f←(U))(x ),this impliesT Int Y(U)=y ≺U Int Y(U)(y )f→L(x )≺UInt Y(U)(f→L(x ))=x ≺f←L(U)Int Y(U)(f→L(x ))x ≺f←L(U)Int X(f←L(U))(x )=T Int X(f←L(U)).Therefore f:(X,T Int X)→(Y,T Int Y)is continuous.çThe next result follows from Theorems4.4,4.8and Corollaries4.6,4.7.Theorem4.9.L-FTOP is isomorphic to L-FIS.The above results show that L-fuzzy topologies and L-fuzzy interior operators come to the same thing.F .-G.Shi /Fuzzy Sets and Systems 160(2009)1218–123212275.L -fuzzy closure operatorsIn this section,we shall introduce a new definition of L -fuzzy closure operators,and study the relation between them and L -fuzzy topologies.For an L -topological space (X , ),its L -closure operator c :L X →L X satisfies the following conditions:(1)c (⊥)=⊥;(2)c (A ) A ;(3)i (A ∨B )=c (A )∨c (B );(4)c (A ) c (c (A )).Now we consider such a function C (A ):J (L X )→2defined byC (A )(x )=1,x c (A ),0,x c (A ).It is easy to check the following conditions:(LC1)C (A )(x )=1⇔∀ ≺ ,C (A )(x )=1,∀x ∈J (L X );(LC2)C (⊥)(x )=0for any x ∈J (L X );(LC3)C (A )(x )=1for any x A ;(LC4)C (A ∨B )=C (A )∨C (B );(LC5)(C ((C (A ))[1]))[1]⊆(C (A ))[1].Based on the above fact,we can generalize L -closure operators to L -fuzzy topological spaces as follows:Definition 5.1.An L -fuzzy closure operator on X is a mapping Cl :L X →L J (L X )satisfying the following conditions:(LFC1)Cl(A )(x )=≺ Cl(A )(x ),∀x ∈J (L X );(LFC2)Cl(⊥)(x )=⊥for any x ∈J (L X );(LFC3)Cl(A )(x )= for any x A ;(LFC4)Cl(A ∨B )=Cl(A )∨Cl(B );(LFC5)∀a ∈L \{⊥},(Cl((Cl(A ))[a ]))[a ]⊆(Cl(A ))[a ].Cl(A )(x )is called the degree to which x belongs to the closure of A .A set X equipped with an L -fuzzy closure operator Cl (or Cl X ),denoted by (X ,Cl),is called an L -fuzzy closure space.A mapping f :X →Y between two L -fuzzy closure spaces (X ,Cl X )and (Y ,Cl Y )is called continuous if ∀x ∈J (L X ),∀A ∈L X ,Cl X (A )(x ) Cl Y (f →L (A ))(f →L (x )).It is easy to check that L -fuzzy closure spaces and their continuous mappings form a category,denoted by L -FCS .The next theorem is obvious.Theorem 5.2.A mapping f :X →Y between two L-fuzzy closure spaces (X ,Cl X )and (Y ,Cl Y )is continuous if and only if ∀x ∈J (L X ),∀B ∈L Y ,Cl X (f ←L (B ))(x ) Cl Y (B )(f →L (x )).The following theorem explains the relation between the L -fuzzy quasi-neighborhood system and the L -fuzzy closure operator of an L -fuzzy topology.Theorem 5.3.Let T be an L-fuzzy topology on X and let Q ={Q x :x ∈J (L X )}be the quasi-neighborhood systeminduced by T .Define a mapping Cl T :L X →L J (L X)byCl T (A )(x )=(Q x (A )) .Then Cl T is an L-fuzzy closure operator on X ,which is called the L-fuzzy closure operator induced by T .Proof.(LFC1)–(LFC3)are obvious.(LFC4)holds from(LFQ3).Now we prove(LFC5). (LFC5)∀a∈L\{⊥},let x /∈(Cl T(A))[a].Then by(LFQ4)we knowx D Ay DQ y (D )=Q x(A )=(Cl T(A)(x )) a .Hence there exists D∈L X such that x D A andy D (Cl T(D)(y )) =y DQ y (D ) a .This implies that y /∈(Cl T(D))[a]holds for any y D.By(LFC4)we havex D(Cl T(D))[a](Cl T(A))[a].ThereforeCl T(Cl T(A))[a](x )=Q x(Cl T(A))[a]=x D(Cl T(A))[a]y DQ y (D ) a .This shows x /∈(Cl T((Cl T(A))[a]))[a].The proof of(LFC5)is completed.çThe next corollary give a representation of the L-fuzzy closure operator by means of an L-fuzzy topology. Corollary5.4.Let T be an L-fuzzy topology on X and let Cl T be the L-fuzzy closure operator induced by T.Then ∀x ∈J(L X),∀A∈L X,Cl T(A)(x )=x D A(T(D )) .Proof.By Theorem5.3and Q y(A )=x D AT(D )we can prove the corollary.çTheorem5.5.If f:(X,T1)→(Y,T2)is continuous with respect to L-fuzzy topologies T1and T2,then f:(X,Cl T1)→(Y,Cl T2)is continuous with respect to L-fuzzy closure operators Cl T1and Cl T2.Proof.If f:(X,T1)→(Y,T2)is continuous with respect to L-fuzzy topologies T1and T2,then∀B∈L Y, T2(B) T1(f←L(B)),this implies that∀x ∈J(L X),Cl T2(B)(f→L(x ))=f→L(x ) D B (T2(D ))x f←L(D) f←L(B)T1(f←L(D ))Cl T1(f←L(B))(x ).Therefore f:(X,Cl T1)→(Y,Cl T2)is continuous.çTheorem5.6.Let Cl:L X→L J(L X)be an L-fuzzy closure operator on X.∀x ∈J(L X),define Q x :L X→L by Q x(A)=(Cl(A )(x )) .Then Q={Q x:x ∈J(L X)}is an L-fuzzy quasi-neighborhood system on X.Proof.(LFQ1)–(LFQ3)hold trivially from(LFC2)–(LFC4).We need to prove(LFQ4).First we check Q x(A)x D Ay DQ y (D ).Suppose that b ∈L and Q x (A ) b .Then there exists a ∈ (b )such that Q x (A ) a .This implies Cl(A )(x ) a .By (LFC5)we havex /∈(Cl(A))[a ]⊇ Cl (Cl(A))[a ][a ].Let D =(Cl(A ))[a ].We shall check x D A .In fact,if x D ,then for any ≺ ,there exists x ∈(Cl(A ))[a ]such that x x .By (LFC1)we know Cl(A )(x ) Cl(A )(x ) a .Further Cl(A )(x )=≺ Cl(A )(x ) a ,this contradicts to x /∈(Cl(A ))[a ].There-fore x D A .By(Cl(A))[a ]⊇ Cl (Cl(A))[a ][a ]=(Cl(D ))[a ].we know Cl(D )(y ) a for any y D .HenceQ y (D )=(Cl(D )(y )) a .Therefore y D Q y (D ) b .This shows x D A y D Q y (D ) b .The proof of Q x(A ) x D A y D Q y (D)is completed.Second we check Q x (A ) x D Ay D Q y (D ).Suppose that b ∈L and x D Ay D Q y (D ) b .Then there exists D ∈L X such that x D A and y D Q y (D ) b .Hence Q x (D) b .Q x (A )=(Cl(A )(x )) (Cl(D )(x )) =Q x (D ) b .This shows Q x (A ) x D Ay D Q y (D ).çTheorem 5.7.Let Cl :L X →L J (LX )be an L-fuzzy closure operator on X.Define T Cl :L X →L byTCl(U )=x U(Cl(U )(x )) .Then T Cl is an L-fuzzy topology on X and Cl (TCl )=Cl.Proof.By Proposition 3.9in [4]and Theorem 5.6we know that T Cl is an L -fuzzy topology on X .Cl (T Cl )=Cl can beproved from Theorems 5.3and 5.6.çAnalogously we have the following result.Corollary 5.8.For an L-fuzzy topology T on X ,T (ClT)=T .Theorem 5.9.If f :(X ,Cl X )→(Y ,Cl Y )is continuous with respect to L-fuzzy closure operators Cl X and Cl Y ,then f :(X ,T Cl X )→(Y ,T Cl Y )is continuous with respect to L-fuzzy topologies T Cl X and T Cl Y .Proof.If f :(X ,Cl X )→(Y ,Cl Y )is continuous with respect to L -fuzzy closure operators Cl X and Cl Y ,then ∀B ∈L Y ,∀x ∈J (L X ),Cl Y (B )(f →(x )) Cl X (f ←(B ))(x ),this impliesT Cl Y (B )=y B(Cl Y (B )(y ))f →L (x) B (Cl Y (B )(f →L (x )))=x f ←L (B )(Cl Y (B )(f →L (x )))x f ←L(B )(Cl X (f ←L (B ))(x )) =T Cl X (f ←L (B )).Therefore f :(X ,T Cl X )→(Y ,T Cl Y )is continuous.çThe next result follows from Theorems 5.5,5.7and Corollary 5.8.Theorem 5.10.L -FTOP is isomorphic to L -FCS .The above results show that L -fuzzy topologies and L -fuzzy closure operators come to the same thing.6.The relation between Int T and Cl TIn this section,we give the relation between the L -interior operator Int T and the L -fuzzy closure operator Cl T .For an L -topological space (X , ),as we know,Cl (A )=(Int (A )) for any A ∈L X .But for an L -fuzzy topological space (X ,T ),Cl T (A )=(Int T (A )) is not true generally,i.e.,it is possible that there exists x ∈J (L X )such that (Cl T (A )(x )) Int T (A )(x ).This can be seen from the following example.Example 6.1.Let X ={x }be a single point set and L ={1,0.7,0.5,0.3,0},where 1 =0,0.7 =0.3,0.5 =0.5.Define an L -fuzzy topology on X byT (1)=T (0)=1,T (0.7)=0.7,T (0.5)=0.5,T (0.3)=0.3.ThenInt T (0.7)(x )=V 0.7T (V )=⎧⎪⎪⎨⎪⎪⎩0, =1,0.7, =0.7,0.7, =0.5,0.7, =0.3,(Cl T (0.7 )(x )) = <V 0.7T (V )=⎧⎪⎪⎨⎪⎪⎩0.7, =1,0.7, =0.7,0.7, =0.5,0, =0.3.Obviously (Cl T (A )) Int T (A ).However,we have the following relation theorem.Theorem 6.2.Let T be an L-fuzzy topology on X ,and let Int T and Cl T be ,respectively ,the L-fuzzy interior operatorand the L-fuzzy closure operator induced by T .Then ∀x ∈J (L X ),∀A ∈L X ,(1)Cl T (A )(x )=(Int T (A )(x )) ;(2)Int T (A )(x )=(Cl T (A )(x )) .Proof.(1)By Corollaries 4.3and 5.4we know that ∀x ∈J (L X ),∀A ∈L X ,Int T (A )(x )= x V AT (V ),(Cl T (A )(x )) =x D AT (D ).。