It is known that fuzziness within the concept of openness of a fuzzy
set in a Chang's fuzzy topological space (fts) is absent. In this
paper we introduce a gradation of openness for the open sets of a
Chang jts (X, $\mathcal{T}$) by means of a map $\sigma\;:\; I^{x}\longrightarrow I\left(I=\left[0,1\right]\right)$,
which is at the same time a fuzzy topology on X in Shostak 's sense.
Then, we will be able to avoid the fuzzy point concept, and to introduce
an adeguate theory for $\alpha$-neighbourhoods and $\alpha-T_{i}$
separation axioms which extend the usual ones in General Topology.
In particular, our $\alpha$-Hausdorff fuzzy space agrees with $\alpha${*}
-Rodabaugh Hausdorff fuzzy space when (X, $\mathcal{T}$) is interpreservative
or $\alpha$-locally minimal

Gregori, Valentín

Vidal, Anna

OpenstarTs

Rend. Istit. Mat. Univ. TriesteSuppl. Vol. XXX, 111{121 (1999)Fuzziness in Chang's FuzzyTopological SpacesValentn Gregori and Anna Vidal ()Summary. - It is known that fuzziness within the concept of open-ness of a fuzzy set in a Chang's fuzzy topological space (fts)is absent. In this paper we introduce a gradation of opennessfor the open sets of a Chang fts (X; T ) by means of a map : IX  ! I (I = [0; 1]), which is at the same time a fuzzytopology on X in Shostak's sense. Then, we will be able to avoidthe fuzzy point concept, and to introduce an adequate theory for-neighbourhoods and  Ti separation axioms which extend theusual ones in General Topology. In particular, our -Hausdorfuzzy space agrees with -Rodabaugh Hausdor fuzzy space when(X;T ) is interpreservative or -locally minimal.1. IntroductionIn 1968 C. Chang [1] introduced the concept of a fuzzy topology on aset X as a family T  IX , where I = [0; 1], satisfying the well-knowaxioms, and he referred to each member of T as an open set. So, inhis denition of a fuzzy topology some authors notice fuzziness in the() Authors' addresses: Valentn Gregori, Departamento de Matematica Apli-cada, Universidad Politecnica de Valencia Escola Universitaria de Gandia Car-retera Nazaret-Oliva S.N. 46730-Grau de Gandia, Valencia, Spain, e-mail: vgre-gori@mat.upv.es,avidal@mat.upv.esAnna Vidal, Departamento de Matematica Aplicada, Universidad Politecnica deValencia Escola Universitaria de Gandia Carretera Nazaret-Oliva S.N. 46730-Grau de Gandia, Valencia, SpainAMS Classication: 54A40Keywords: fuzzy topology, gradation of openness, fuzzy point.While working on this paper the rst-listed author has been partially supportedby a grant from Ministerio de Educacion y Ciencia DGES PB 95-0737.112 V. GREGORI and A. VIDALconcept of opennes of a fuzzy set has not been considered. Keepingthis in view, A.P. Shostak [8], began the study of fuzzy structures oftopological type.The idea of this paper is to allow open sets of a Chang's fuzzy topol-ogy to be open to some degree by means of a particular Shostak'sfuzzy topology (or gradation of openness [2]) on X (Proposition 3.1).This gradation of openness will enable us to introduce fuzzy topo-logical concepts which are generalitation of the corresponding onesin General Topology and to work with points of X instead of fuzzypoints (the idea of a fuzzy point and fuzzy point belonging is a ratherproblematic, see Gottwald [4] for a discussion). After preliminarysection, in section 3 we dene the concept of an -set and studysome denitions and properties relative to it. In particular we showthe family of all -neighborhoods of x 2 X, have similar propertiesto the classic cases. In section 4 we dene and study the familiesof interpreservative and -locally minimal spaces. In section 5 wedene the concept of an -Ti space (i = 0; 1; 2) and show that theconcept of an -T2 space coincides with the -Hausdor conceptdue to S.E. Rodabaugh [7] in the spaces mentioned in section 4. Ourstudy may be thought to be just the beginning of this subjet whichis far from being completed.2. Preliminary notionsLet X be a nonempty set and I the closed unit interval. A fuzzy setof X is a map M : X  ! I. M(x) is interpreted as the degree ofmembership of a point x 2 X in a fuzzy set M , while an ordinarysubset A  X is identied with its characteristic function and, inconsequence ; and X are identied with the constant functions onX, 0 and 1 respectively. As usual in fuzzy sets, we write A B if A(x)  B(x); x 2 X. We dene the union, intersection andcomplement of fuzzy sets as follows: [iAi!(x) =_iAi(x); x 2 X \iAi!(x) =^iAi(x); x 2 X.Ac(x) = 1 A(x); x 2 XFUZZINESS IN CHANG'S FUZZY TOPOLOGICAL SPACES 113A.P. Shostak [8] dened a fuzzy topology on X as a function : IX  ! I satisfying the following axioms:(i) (0) = (1) = 1(ii) ;  2 IX implies ( \ )  () ^ ()(iii) i 2 IX for all i 2 J implies ([ii) ^i(i)K.C. Chattopadhyay et al. [2] rediscovered the Shostak's fuzzytopology concept and called gradation of openness the function  .Also, they called gradation of closedness on X [2], a function F :IX  ! I satisfying the above axioms (i)-(iii) but interchanging theintersection with the union and vice-versa. From now, a fuzzy topol-ogy in Shostak's sense will be called gradation of openness, and wedene a fuzzy topological space, or fts for short, as a pair (X; T )where T is a fuzzy topology in Chang's sense, on X, i.e., T is acollection of fuzzy sets of X, closed under arbitrary unions and -nite intersections. A set is called open if it is in T , and closed if itscomplement is in T . The interior of a fuzzy set A is the largest openfuzzy set contained in A. If confusion is not possible we say X is aspace instead of a fts. We will denote inf B, the inmum of a set Bof real numbers.Recall the support of a fuzzy set A is supp A = fx 2 X : A(x) >0g. We denote x 2̂ A whenever x 2 supp A, and we say A containsthe point x or that x is in A.The next denition was given by Pu Pao-Ming et al. [6].Definition 2.1. A fuzzy point is a fuzzy set px which takes thevalue 0 for all y 2 X except one, that is x 2 X. The fuzzy point px issaid to belong to the fuzzy set A, denoted by px~2A, i px(x)  A(x).We notice x2̂A if px~2A.3. Gradation of opennessThe proof of the following proposition can be seen in [5]Proposition 3.1. Let X be a nonempty set. Then the map  :IX  ! I given by (0) = 1 and (A) = inffA(x) : x 2 supp Agif A 6= 0, satises both the axioms of gradation of openness and theaxioms of gradation of closedness.114 V. GREGORI and A. VIDALThe real number (A) is the degree of openness [8] of the fuzzyset A; clearly, (A) =  implies the degree of membership of eachpoint in the support of A, in the fuzzy set A, is at least . We notice(A) = 1 i A is an ordinary subset of X and (A) = 0 i there is asequence fxng in X such that A(xn) > 0, 8n 2 N and limnA(xn) = 0.With this terminology we give the following denitions.Definitions 3.2. The fuzzy set A of X is an -set if (A)  ;moreover, if A was open (closed) we will say A is -open (-closed).Clearly, each A 2 IX is a 0-set and the 1-sets are only the ordi-nary subsets of X.Since  is a gradation of openness and closedness, we have thefollowing proposition.Proposition 3.3. The union and intersection of -sets is an -set.The following example shows that if A is an -set and A  B,then B is not an -set necesarily.Example 3.4. Let X be a set with at least two points and  2]0; 1].Let fM;Ng be a partition of X. We dene the following fuzzy setsA and B:A(x) =  if x 2M and A(x) = 0 if x 2 NB(x) =  if x 2M and B(x) = =2 if x 2 NWe have that A is an -set and A  B, but B is not.Nevertheless we have the following proposition.Proposition 3.5. Let A;B be fuzzy sets. If A is an -set, A  Band supp B supp A, then B is an -set.Proof. It is obvious.Definitions 3.6. Let (X;T ) be a fts and let  2 I. The fuzzy topol-ogy T = fA 2 T : (A)  g is called the  -level of openness ofthe fuzzy topology T .Clearly fT :  2 Ig is a descending family, (i.e.,  >  impliesT  T), where T0 = T and T1 is an ordinary topology on X.FUZZINESS IN CHANG'S FUZZY TOPOLOGICAL SPACES 115We call -interior of the fuzzy set A, denoted int(A), the largest-open contained in A, i.e.,int(A) = [fG 2 T : G  Ag.Clearly, int(A) is welldened, since 0 2 T 8 2 I, and int(A) A for each A 2 IX . Note, int0(A) is the interior of A in Chang'ssense and the -interior of a fuzzy set A is just its interior in the-level fuzzy topology T.We say that the fuzzy set A is an -neighborhood, or -nbhdfor short, of p 2 X if there exists G 2 T such that p 2̂ G  A.Equivalently, a point in (the support of) int(A) will be called an-interior point of A. The -nbhd system of a point p 2 X, isthe family N(p) of all -nbhd's of the point p. Obviously, if  > then N(p)  N(p). With this notation we have the followingproposition.Proposition 3.7. Let (X; T ) be a fts, A 2 IX ,  2 I and p 2 X.Then,(i) A is -open if and only if A = int(A).(ii) p 2̂ int(A) if and only if A 2 N(p).Proof. It is obvious.If A is an -open set, then A is an -neighbourhood of all pointsof its support, but the converse is not true as shows the followingexample.Example 3.8. Let (X;T ) be a topological space, 2]0; 1[ and let T =f0;1g [ f  U : U 2 Tg. Then any U 2 T is an -neighbourhoodfor any point x 2 U , i.e. for any point of its support, but obviouslyU fails to be -open in (X; T ).In the next proposition we show that the family N(p) satisessimilar properties to the corresponding ones in General Topology.Proposition 3.9. Let (X; T ) be a fts and let  2 I. For each pointp 2 X let N(p) be the family of all -nbhd's of p. Then1. If M 2 N(p), then p 2̂ M .2. If M;N 2 N(p), then M \N 2 N(p).116 V. GREGORI and A. VIDAL3. If M 2 N(p) and M  N , then N 2 N(p).4. If M 2 N(p), then there is N 2 N(p) such that (N)  ,N M and N 2 N(q); 8q 2̂ N .Proof. 1. If M 2 N(p), then there exists G 2 T such thatp2̂G M , and therefore p 2̂ M .2. If M;N 2 N(p), then there are two -open fuzzy sets G1 andG2 such that p 2̂ G1  M , p 2̂ G2  N . Since G1 \G2 2 Tand p 2̂ G1 \G2 M \N , we have M \N 2 N(p).3. If M 2 N(p), there exists G 2 T such that p 2̂ G M  Nand therefore N 2 N(p).4. Suppose M 2 N(p). Then there exists G 2 T such thatp 2̂ G  M . Let N = G. We have (N)   and N 2N(q); 8q 2̂ N .Proposition 3.10. Let  2 I. If N is a function which assignsto each p 2 X a nonempty family N(p) of fuzzy sets satisfyingproperties 1, 2 and 3 of the above proposition, then the familyT = fM 2 IX : (M)  ;M 2 N(p); 8p 2̂ Mgis a fuzzy topology on X. If property 4 of the above proposition isalso satised, then N(p) is precisely the -nbhd system of p relativeto the topology T.Proof. First, we will show that T is a fuzzy topology on X.Obviously 0 2 T and since M  1, 8M 2 N(p), according toproperty 3, 1 2 N(p), 8p 2 X, i.e, 1 2 T.LetM;N 2 T. If p 2̂M\N , clearly p 2̂M and p 2̂ N , thereforeM;N 2 N(p) and according to property 2 we have M \N 2 N(p).Now, by Proposition 3.3, (M \N)   and then M \N 2 T.Let fMigi2J be a family of sets of T and let M =[iMi. Ifp 2̂ M , then we have 0 < [iMi!(p), and therefore there existsFUZZINESS IN CHANG'S FUZZY TOPOLOGICAL SPACES 117j 2 J such that 0 < Mj(p), i.e, p 2̂ Mj . Therefore, Mj 2 N(p)and according to property 3, M 2 N(p). Now, by Proposition 3.3,(M)   and then M 2 T.Now, we suppose property 4 is also satised. We will see that the-neighbourhood system of p, V(p), relative to the fuzzy topologyT, is the family N(p).If M 2 V(p), then there exists an -open, G of T with p2̂G M . Therefore, G 2 N(p) and, according to property 3, we haveM 2 N(p).If M 2 N(p), according to property 4, there exists N 2 N(p),with N M and such that (N)   and N 2 N (q), 8q2̂N . Thenwe have N 2 T such that p 2̂ N M , i.e., M 2 N(p).Observetion 3.11: In [6], the authors dened the concept of neigh-borhood of a fuzzy point and they showed similar results, but in [9]Shostak remarks that there are inaccuracies in the formulation ofthese authors. In fact, the family constructed by the authors is abase for a fuzzy topology, but it is not a fuzzy topology.4. Interpreservative and locally minimal ftsWe begin with the following denitions.Definitions 4.1. Let (X;T ) be a fts. We say X is interpreser-vative if the intersection of each family of open sets is an open set,or equivalently, if the family of closed sets is a fuzzy topology on X.We say X is locally minimal if \fG 2 T : x 2̂ Gg is open foreach x 2 X, i.e., each x 2 X admits a smallest nbhd. We say X is-locally minimal,  2]0; 1], if \fG 2 T : x 2̂ Gg is -open, foreach x 2 X.Clearly, for  >  > 0, -locally minimal implies -locally min-imal. Also, an -locally minimal space is locally minimal but theconverse is false as shows the next example.Example 4.2. Let X the real interval ]1;+1[ with the fuzzy topologyT = f0;1; Gg where G(x) = 1=x; x 2 X. Obviosly G is the smallestnbhd of each point of X and so, X is locally minimal but (G) = 0and then X is not -locally minimal for any  2]0; 1].118 V. GREGORI and A. VIDALIn the following proposition we study the relationship betweeninterpreservative and locally minimal spaces.Proposition 4.3. Let  2]0; 1] and let (X;T ) be an interpreserva-tive fts where (G)  , for each G 2 T . Then T is - locallyminimal (therefore locally minimal).Proof. Let x 2 X and Ax = fG 2 T : x2̂Gg. We consider Gx =\G2AxG. Since T is interpreservative we have Gx 2 T . It is sucientto prove Gx 6= 0. Now, for each G 2 Ax we have G(x)   > 0.Therefore Gx(x) =^G2AxG(x)   > 0 and x 2̂ Gx, i.e., Gx 6= 0.Then the -open Gx is the smallest nbhd of x. In the followingexample we will see that we cannot remove the condition (G)  > 0, for each G 2 T , in the above proposition.Example 4.4. Let X be the unit interval [0,1]. For each h 2]0; 1],we consider the following functionsfh(x) =2hx; x 2 [0; 1=2]2h(1   x); x 2 [1=2; 1]The family A = ffh : 0 < h  1g [ f0g [ f1g is an interpreservativefuzzy topology, however there does not exist the smallest nbhd for anyx 2 X.Also, we can nd a locally minimal space that is not an interpreser-vative space.Example 4.5. Consider in the real line R the laminated indiscretefuzzy topology L, i.e., L is constituted by the constant functions fromR to the unit interval I. We denote fc : R  ! I the constantfunction fc(x) = c for each x 2 R. Take  2]0; 1] and consider thefuzzy topology T = C [ ff=2g with C = ffc 2 L : c > g [ f0g.Then, f=2 is the smallest nbhd of x, for all x 2 R and therefore(R; T ) is locally minimal. However\c>fc = f =2 T :FUZZINESS IN CHANG'S FUZZY TOPOLOGICAL SPACES 119We have seen that in general the two concepts interpreservativeand locally minimal are not equivalent, but they are for ordinarytopologies.Proposition 4.6. Let (X;T ) be a topological space. Then X is in-terpreservative if and only if it is locally minimal.Proof. It is obvious.Proposition 4.7. Let (X;T ) be a locally minimal fts. Then eachnonempty intersection of open sets contains a nonempty open set.Proof. Let G =\i2JGi with Gi 2 T ; 8i 2 J . If G 6= 0, then thereexists x 2̂ G and therefore x 2̂ Gi; 8i 2 J . For this x 2 X let Gxbe the smallest nbhd of x. We have Gx  Gi; 8i 2 J and thereforeGx \i2JGi. Gx is the required open set.5. Separation axioms in ftsWe will dene new separation axioms for fts.Definition 5.1. Let  2 I. We say the fts (X; T ) is -Hausdor,or -T2, if for all points of space x; y 2 X with x 6= y, there areG;H 2 T such that x2̂G, y2̂H and G \ H = 0. -T1 if for allx; y 2 X with x 6= y there are G;H 2 T such that x2̂G, y2̂H,x =2supp H and y =2 supp G. -T0 if for all x; y 2 X with x 6= y thereis G 2 T such that x2̂G, and y =2 supp G.Clearly, the following implications are satised.-T2  ! -T1  ! -T0Also, for  >  we have -Ti  ! -Ti, for i = 0; 1; 2. The followingdenition is due to S.E. Rodabaugh [7].Definition 5.2. A fts (X; T ) is -Hausdor if for all x; y 2 Xwith x 6= y, there are G;H 2 T such that G(x)  , H(y)   andG \H = 0.120 V. GREGORI and A. VIDALClearly an -Hausdor space is -Hausdor.We will see in the next two propositions that the -Hausdorand -Hausdor concepts agree in interpreservative and -locallyminimal spaces.Proposition 5.3. Let (X;T ) be an interpreservative fts and let  2]0; 1]. Then (X;T ) is -Hausdor if and only if it is -Hausdor.Proof. We only see the converse. Assume that (X;T ) is interp-reservative and -Hausdor and let x; z 2 X; x 6= z. Further,let the open fuzzy sets Ux = ^fU : U 2 T ; U(x)  g,Vz = ^fV : V 2 T ; V (z)  g. Then, obviously, Ux ^ Vz = 0and (Ux)  ; (Vz)   (notice that supp Ux = fxg and supp Vz= fzg, since X is -Hausdor ) and hence X is -Hausdor.Proposition 5.4. Let  2]0; 1] and let (X; T ) be an -locally min-imal space. Then (X;T ) is -Hausdor if and only if it is -Hausdor.Proof. We only see the converse.Suppose X is -Hausdor. For each a 2 X we denote Ga thesmallest nbhd of a, which is -open by the hypothesis. Now considerUa = ^fU : U 2 T ; U(a)  g. We have Ga  Ua, and thereforesupp Ga = fag, since X is -Hausdor. Finally, let x; z 2 X withx 6= z. Then, Gx \Gz = 0 and thus (X; T ) is -Hausdor.There are some denitions of Hausdorfness depending on fuzzypoints. One of these was given by D. Adnajevic.Definition 5.5. The fts (X;T ) is Hausdor (denoted Adn-H2, here)if for all fuzzy points px; qy 2 IX with x 6= y, there are G;H 2 T ,such that px 2̂ G, qy 2̂ H and G \H = 0.Observetion 5.6: In [3] there is the following diagram which re-lates various fuzzy Hausdor conditions:Adn-H2 () GSW-H =) SLS-H () LP-FT2 =) -HausdorClearly a fts X is 1-Hausdor if and only if it is Adn-H2. Now,as a consequence of Propositions 5.3 and 5.4 we can complete andparticularize the above diagram. In fact, the conditions Adn-H2,FUZZINESS IN CHANG'S FUZZY TOPOLOGICAL SPACES 121GSW-H, SLS-H, LP-FT2, 1-Hausdor and 1-Hausdor are equiva-lent for interpreservative fts , 1-locally minimal fts or locally minimal(ordinary) topological space.Acknowledgement The authors are grateful to the referee forhis valuable suggestions.References[1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968),182{190.[2] K.C. Chattopadhyay, R.N. Hazra, and S.K. Samanta, Grada-tion of openness: fuzzy topology, Fuzzy sets and Systems 49 (1992),237{242.[3] D.R. Cutler and I.L. Reilly, A comparison of some Hausdornotions in fuzzy topological spaces, Computers Math. Applic. 19 (1990),no. 11, 97{104.[4] S. Gottwald, Fuzzy points and local properties of fuzzy topologicalspaces, Fuzzy Sets and Systems 5 (1981), 199{201.[5] V. Gregori and A. Vidal, Gradations of openness and Chang's fuzzytopologies, to appear.[6] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology. I. Neighbourhoodstructure of a fuzzy point and Moore-Smith convergence, Journal ofMathematical Analysis and Applications 76 (1980), 571{599.[7] S.E. Rodabaugh, The Hausdor separation axiom for fuzzy topologi-cal spaces, Topology and its Applications 11 (1980), 319{334.[8] A.P. Shostak, On a fuzzy topological structure, Rend. Circ. Mat.Palermo Ser II (1985), no. 11, 89{103.[9] A.P. Shostak, Two decades of fuzzy topology: basic ideas, notions,and results, Uspeki Mat. Nau. 44 (1989), no. 6, 99{147.Received October 24, 1997.

Fuzziness in Chang's fuzzy topological spaces

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Fuzziness in Chang's fuzzy topological spaces

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