Volume 2010, Article ID 180196,11pages doi:10.1155/2010/180196
Research Article
On Maximal and Minimal Fuzzy Sets in I-Topological Spaces
Samer Al Ghour
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan
Correspondence should be addressed to Samer Al Ghour,[email protected] Received 1 May 2010; Accepted 22 August 2010
Academic Editor: Naseer Shahzad
Copyrightq2010 Samer Al Ghour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The notion of maximal fuzzy open sets is introduced. Some basic properties and relationships regarding this notion and other notions of I-topology are given. Moreover, some deep results concerning the known minimal fuzzy open sets concept are given.
1. Introduction
In this paper, the unit interval0,1will be denoted byI. LetXbe a nonempty set. A member ofIX is called a fuzzy subset ofX1. Throughout this paper, forA, B∈IX we writeA≤ B if and only ifAx ≤ Bxfor allx ∈ X. ByA B, we mean thatA ≤ BandB ≤ A, that is, Ax Bx for all x ∈ X. Also we write A < B if and only if A ≤ B and A /B. If {Aj :j ∈J}is a collection of fuzzy sets inX, thenAjx sup{Ajx :j ∈J},x ∈X, and
Ajx inf{Ajx:j ∈J},x∈X. Ifr∈0,1, thenrXdenotes the fuzzy set given by rXx rfor allx∈X.The complementAcof a fuzzy setAinXis given byAcx 1−Ax, x∈X.IfU⊆X, thenXUdenotes the characteristic function ofU.
In this paper, we will follow 2 for the definitions of I-topology, the product I- topology, the direct and the inverse images of a fuzzy set under maps and their notations, fuzzy continuity, and fuzzy openness. A fuzzy setpdefined by
px
⎧⎨
⎩
t ifxxp,
0 ifx /xp, 1.1
where 0< t≤1, is called a fuzzy point inX,xp∈Xis called the support ofpandpxp tthe valuelevelofp2. Two fuzzy pointspandqinXare said to be distinct if and only if their supports are distinct, that is,xp/xq.A∈IXis called a crisp subset ofXifAis a characteristic function of some ordinary subset ofX 2.A ∈IXis called a fuzzy crisp point inXif it is a characteristic function of singleton2.
In this paper, we follow3for the definition of “belonging to”. Namely, a fuzzy point pinXis said to belong to a fuzzy setAinXnotation:p∈Aif and only ifpxp≤Axp.
LetX, τbe a topological space and letUbe a subset ofX.Uis called semiopen4 ifU ⊆ ClIntUandUis called preopen5ifU ⊆ IntClU. A nonempty open subset Uof X is called a minimal open set if the only nonempty open set which is contained in UisU. minX, τwill denote the family of all minimal open sets inX.X, τis said to be homogeneous if for any two pointsx1, x2∈X, there exists an autohomeomorphism onX, τ takesx1tox2. In 1997, Fora and Al-Bsoul6used minimal open sets to characterize and count finite homogeneous spaces. In 2001, Nakaoka and Nobuyuki7characterized minimal open sets and proved that any subset of a minimal open set is pre-open. A non-empty pre-open subsetUofXis called a minimal pre-open set8if the only non-empty pre-open set which is contained inUisU. The author in8characterized minimal pre-open sets as pre-open singletons. A non-empty semi-open subsetUofXis called a minimal semi-open set9if the only non-empty semi-open set which is contained inUisU. The authors in9proved that the set of minimal open sets and the set of minimal semi-open sets in a space are equal. An I-topological spaceX,Iis said to be fuzzy homogeneous10if for any two pointsx1, x2 ∈ X, there exists a fuzzy autohomeomorphism on X,Itakes x1 tox2. The authors in 11 extended the concept of minimal open sets to include I-topological spaces as follows; a fuzzy open setAof an I-topological spaceX,Iis called minimal fuzzy open set11ifAis nonzero and there is no nonzero proper fuzzy open subset ofA. minX,Iwill denote the family of all minimal fuzzy open sets inX. The authors in11obtained many results concerning minimal fuzzy open sets, and they proved that homogeneity in I-topological spaces forces the shape of minimal fuzzy open sets. Then the author in12generalized minimal fuzzy open sets by two methods. A proper non-empty open subsetUof a topological space X, τis called a maximal open set13if any open set which containsUisXorU. maxX, τwill denote the family of all maximal open sets inX. The authors in13obtained fundamental properties of maximal open sets such as decomposition theorem for a maximal open set and established basic properties of intersections of maximal open sets, such as the law of radical closure. By a dual concepts of minimal open sets and maximal open sets, the authors in14introduced the concepts of minimal closed sets and maximal closed sets and obtained results easily by dualizing the known results regarding minimal open sets and maximal open sets. This paper proposes mainly maximal fuzzy open sets in I-topological spaces. It also gives some deep results concerning the known minimal fuzzy open sets concept.
Throughout this paper, for any setX,|X|will denote the cardinality ofX. If Ais a fuzzy set inX, then the support ofAis denoted bySAand defined bySλ λ−10,1, and the set{x∈X:Ax 1}will be denoted by 1A.
2. Maximal Fuzzy Open Sets
Definition 2.1. LetX,Ibe an I-topological space. A nonzero fuzzy open subsetAofXis said to be maximal fuzzy open set ifA /1X and for any fuzzy open setBinXwithA≤B,BA orB1X.
Throughout this paper, the set of all maximal fuzzy open subsets of the I-topological spaceX,Iwill be denoted by maxX,I.
Theorem 2.2. LetX,Ibe an I-topological space andA∈maxX,Iwith 1A ∅. Then for every B∈I− {1X}one hasB≤A.
Proof. Suppose to the contrary that there existB∈I−{1X}andx◦∈Xsuch thatBx◦> Ax◦. SinceA∨Bx◦ Bx◦/Ax◦andA∈maxX,I,A∨B1X. Therefore, for eachx∈X, Ax<1Bxand henceB1X, which is a contradiction.
Corollary 2.3. Let X,I be an I-topological space. If A ∈ maxX,I with 1A ∅, then maxX,I {A}.
Corollary 2.4. Let X,I be an I-topological space. If |maxX,I| > 1, then for every A ∈ maxX,I, 1A/∅.
Example 2.5. LetXRandτbe the usual topology onR, then maxX, τ {R− {x}:x∈R}.
LetI{XU:U∈τ}. then maxX,I {XU:U∈maxX, τ}is uncountable. Therefore, the condition “1A ∅” inCorollary 2.3cannot be dropped.
Proposition 2.6see13. LetX, τbe a topological space and letU∈maxX, τ, then ClU Xor ClU U.
The following example shows that the exact fuzzy version ofProposition 2.6is not true in general.
Example 2.7. LetX{x1, x2, x3}and letA, Bbe fuzzy sets inXdefined as follows:
Ax1 0.3, Ax2 0.3, Ax3 1,
Bx1 0.3, Bx2 0.3, Bx3 0. 2.1
LetI {0X,1X, A, B}, then maxX,I {A}and ClA {x1,0.7,x2,0.7,x3,1}
but ClAis neitherAnor 1X.
The following lemma will be used in the following main result.
Lemma 2.8. LetX,Ibe an I-topological space and A ∈ maxX,I. IfB ∈ I− {0X}such that A∧B0X, thenAXSAandBAc.
Proof. Choosex◦∈SB. SinceA∧B0X,Ax◦ 0. Thus,A∨Bx◦ Bx◦/Ax◦. Since A∈maxX,I,A∨B1X. SinceA∧B0X, it follows thatAXSAandBAc.
The following result is a partial fuzzy version ofProposition 2.6.
Theorem 2.9. LetX,Ibe an I-topological space. IfAXSA∈maxX,I, then either ClA A or ClA 1X.
Proof. Suppose that ClA/1X, then there existsx◦ ∈ X such thatClAx◦ < 1. LetB ClAc, thenB ∈ I, B /0X, andA∧B 0X. Hence byLemma 2.8, it follows thatB Ac. Therefore, ClA A.
Proposition 2.10see13. LetX, τbe a topological space. IfU ∈ maxX, τandS is a non empty subset ofX−U, then ClS X−U.
The following result is the exact fuzzy version ofProposition 2.10.
Theorem 2.11. LetX,Ibe an I-topological space. IfA∈maxX,IandBis a non zero fuzzy set inXwithB≤Ac, then ClB Ac.
Proof. Suppose to the contrary that ClB/Ac. SinceB ≤ AcandAcis a fuzzy closed set in X, then ClB ≤ Ac. Therefore, there existsx◦ ∈ X such thatClBx◦ < Acx◦. Since Ax◦ < 1X−ClBx◦andA ∈ maxX,I, thenA∨1X−ClB 1X. We are going to show that ClB 0X. Letx∈X 1A∪X−1A. Ifx∈1A, thenAcx 0 and hence ClBx≤Acx 0. Ifx∈X−1A, thenAx<1. SinceA∨1X−ClB 1X, we get 1X−ClBx 1 and henceClBx 0. Hence, we complete the proof that ClB 0X. Therefore,B0X, which is a contradiction.
Recall that a fuzzy subsetAof an I-topological spaceX,Iis called fuzzy preopen if A≤intClA.
From now on the setPOX,Iwill denote the set of all fuzzy preopen subsets of the I-topological spaceX,I.
Corollary 2.12. Let X,Ibe an I-topological space and A ∈ maxX,I, then{B ∈ IX : B ≤ intAc} ⊆POX,I.
Proof. LetB ∈ IX such that B ≤ intAc. IfB 0X, thenB ∈ POX,I. IfB /0X, then by Theorem 2.11, ClB Acand henceB≤intAc intClB, that is,B∈POX,I.
Corollary 2.13. LetX,Ibe an I-topological space andA∈maxX,IwithAis a clopen fuzzy set, then{B∈IX:B≤Ac} ⊆POX,I.
Proof. Let B ∈ IX such that B ≤ Ac. If B 0X, then B ∈ POX,I. If B /0X, then by Theorem 2.11, ClB Acand henceB≤AcintAc intClB, that is,B∈POX,I.
Proposition 2.14see13. LetX, τbe a topological space. IfU∈maxX, τandMis a subset ofXwithUM, then ClM X.
Corollary 2.15. LetX, τbe a topological space. IfU∈maxX, τandMis a closed subset ofX withUM, thenMX.
The following example shows that the exact fuzzy version of each ofProposition 2.14 andCorollary 2.15is invalid in general.
Example 2.16. Let X be a non empty set with the I-topology I {0X,1X,0.3X}, then maxX,I {0.3X}and0.3X <0.4X, but Cl0.4X 0.7X. This shows that the exact fuzzy version of Proposition 2.14is invalid in general. On the other hand, since 0.7X is a fuzzy closed subset ofX,Iand 0X < 0.7X < 1X, we get that the exact fuzzy version of Corollary 2.15is invalid in general.
The following result is a partial fuzzy version ofCorollary 2.15.
Theorem 2.17. LetX,Ibe an I-topological space andA∈maxX,I. IfBis a fuzzy closed subset ofXwithA < B, then for everyx∈XwithAx< Bx, one hasBx>0.5.
Proof. Suppose to the contrary that there exists a fuzzy closed subsetBofXandx◦∈Xsuch thatAx◦< Bx◦andBx◦≤0.5. SinceBis a fuzzy closed subset ofX,Bc∈Iand so either Bc∨AAorBc∨A1X. IfBc∨AA, then 1−Bx◦≤Ax◦and soBx◦>0.5 which is a contradiction becauseBx◦≤0.5. IfBc∨A1X, thenBx◦ 0 orAx◦ 1 which is also a contradiction becauseAx◦< Bx◦.
For crisp maximal fuzzy sets, the exact fuzzy version ofCorollary 2.15is valid as the following result shows.
Theorem 2.18. LetX,Ibe an I-topological space. IfAXS∈maxX,IandBis a fuzzy closed subset ofXwithA < B, thenB1X.
Proof. Suppose to the contrary that there existsx◦ ∈Xsuch thatBx◦<1. SinceA < B, then x◦∈X−S. Therefore,Bc∨Ax◦ Bcx◦/0Ax◦and soBc∨A /A. Hence, we must haveBc∨A1X. Choosex1∈Xsuch thatAx1< Bx1. ThenBcx1 1 and soBx1 0, but 0Ax1< Bx1, which is a contradiction.
Corollary 2.19. LetX,Ibe an I-topological space. IfAXS∈maxX,IandBis a fuzzy subset ofXwithA < B, then ClB 1X.
Corollary 2.20. LetX,Ibe an I-topological space. IfAXS∈maxX,IandBis a fuzzy subset ofXwithA≤B, thenBis a fuzzy preopen set.
Proof. IfB A, thenBis fuzzy open and so it is a fuzzy preopen set. On the other hand, if A < B, then byCorollary 2.19, it follows thatB≤1Xint1X intClB.
The following ordinary topological spaces result follows easily.
Proposition 2.21. LetX, τbe an ordinary topological space, then{U, Uc} ⊆maxX, τfor some U⊆Xif and only ifτ{∅, X, U, Uc}.
The exact fuzzy version ofProposition 2.21is true as the following result shows.
Theorem 2.22. LetX,Ibe an I-topological space and letA ∈ IX with A /Ac, then {A, Ac} ⊆ maxX,Iif and only ifAXSAandI{0X,1X, A, Ac}.
Proof. ⇒Suppose that{A, Ac} ⊆maxX,I. Choosex◦∈Xsuch thatAx◦/Acx◦. Then A∨Acx◦/Ax◦orA∨Acx◦/Acx◦. Since{A, Ac} ⊆maxX,I, thenA∨Ac 1X. Thus for everyx ∈X withAx < 1, we must haveAcx 1 and soAx 0. Therefore, AXSA. LetB ∈I− {0X}. IfB≤ A, thenAc∧B 0X and, byLemma 2.8,B Acc A.
IfB A, thenA∨B1Xand soBAc.
⇐Clear.
3. Images and Products of Maximal Sets
From now onπxandπywill denote the projections onXandY, respectively,τprodwill denote the product topology ofτ1andτ2, andIprodwill denote the product I-topology ofI1andI2.
We start by the following result.
Proposition 3.1. LetX, τ1andY, τ2be two ordinary topological spaces. Iff:X, τ1−→Y, τ2 is continuous, open, and surjective, then for everyU∈maxX, τ1withfU/Y one hasfU∈ maxY, τ2.
Proof. Suppose thatH ∈τ2 withfU⊆ H ⊂Y. Thenf−1H∈ τ1withU⊆ f−1H. Since U∈maxX, τ1, thenf−1H Uorf−1H X. Iff−1H X, thenff−1H fX Y.
Thusf−1H Uand soHff−1H fU.
The following result is the exact fuzzy version ofProposition 3.1.
Theorem 3.2. LetX,I1and Y,I2 be two I-topological spaces and letf:X,I1 → Y,I2be fuzzy continuous, fuzzy open, and surjective function. IfA∈maxX,I1, then eitherfA 1Y or fA∈maxY,I2.
Proof. Suppose thatfA/1Y. It is sufficient to show thatfA∈maxY,I2. Sincefis fuzzy open and A is a fuzzy open set, thenfAis a fuzzy open set. Since A /0X, there exists x◦ ∈ X such thatAx◦ > 0 and sofAfx◦ sup{Ax : fx f x◦} ≥ Ax◦ >
0, hencefA/0Y. Suppose thatB ∈I2 such thatfA< B. We are going to showB 1Y which completes the proof. Choose y◦ ∈ Y such thatfAy◦ < By◦. Sincef is onto, there existsx1 ∈X such thatfx1 y◦. Thus,Ax1≤fAy◦< By◦. Sincef is fuzzy continuous,f−1B∈I1. Thus, we havef−1B∨A∈I1,A≤f−1B∨A, andf−1B∨Ax1
max{Ax1, By◦}By◦> Ax1. SinceA∈maxX,I1, thenf−1B∨A1X. To see that B 1Y, let y ∈ Y and choose x ∈ X such that fx y, then 1 max{Ax, By} ≤ max{fAy, By}Byand soBy 1.
Corollary 3.3. LetX,I1andY,I2be two I-topological spaces and let f:X,I1 → Y,I2be a fuzzy homeomorphism. IfA∈maxX,I1, thenfA∈maxY,I2.
Proof. Let A ∈ maxX,I1. According to Theorem 3.2, it is sufficient to see that fA/1Y. SinceA∈maxX,I1, there existsx◦ ∈Xsuch thatAx◦<1. Thus,fAfx◦ Ax◦<
1.
Proposition 3.4. LetX, τ1andY, τ2be two ordinary topological spaces. IfU∈maxX, τ1and V ∈maxY, τ2,thenU×V /∈maxX×Y, τprod.
Proof. Follows becauseU×Y ∈τprod, butU×V ⊂U×Y ⊂X×Y.
Proposition 3.5. LetX, τ1andY, τ2be two ordinary topological spaces and letG ∈ maxX × Y, τprodsuch thatπxG×πyG/X×Y, then there existsU∈maxX, τ1orV ∈maxY, τ2such thatGU×Y orGX×V.
Proof. Since G ⊆ πxG × πyG, πxG × πyG ∈ τprod, πxG × πyG/X × Y, and G ∈ maxX × Y, τprod, it follows that G πxG×πyG. If πxG/X and πyG/Y, then by Proposition 3.1,πxG ∈ maxX, τ1and πyG ∈ maxY, τ2, which is impossible by Proposition 3.4. Therefore, either πxG X or πyG Y. Hence, there exists U ∈ maxX, τ1orV ∈maxY, τ2such thatGU×YorGX×V.
The following example shows that the condition “πxG × πyG/X × Y” in Proposition 3.5cannot be dropped.
Example 3.6. LetXRwith the usual topologyτuand letGR2−{0,0}, thenG∈maxX× X, τprod,πxG×πyG X×X, and there is noU∈ maxX, τusuch thatG U×X or GX×U.
The following is the main result of this section.
Theorem 3.7. Let X,I1 and Y,I2 be two I-topological spaces. If A ∈ maxX,I1 andB ∈ maxY,I2, thenA×B∈maxX×Y,Iprodif and only if there exists 0 < c <1 such thatA cX andBcY.
Proof. Suppose thatA×B∈maxX×X,Iprod. SinceA×1Y, 1X×B∈Iprod,A×B≤A×1Y <1X×Y, A×B ≤1X×B <1X×Y, thenA×BA×1Y andA×B1X×B. Therefore, for everyx∈X andy ∈Y, min{Ax, By} min{Ax,1} min{1, By}and henceAx By. Thus, there exists 0 < c <1 such thatA cXandBcY. Conversely, suppose for some 0< c <1 thatAcXandBcY. LetM∈Iprod− {1X×Y}such thatcX×cY cX×Y ≤M. Choose families {Aα :α∈Λ} ⊆I1and{Bα :α∈Λ} ⊆I2such thatM
{Aα×Bα:α∈Λ}. SinceM /1X×Y, then for everyα∈Λ,Aα/1X orBα/1Y. We are going to show thatAα×Bα≤cX×Y for every α∈Λ. Letα∈Λ.
Case 1. Aα/1XandBα/1Y, thenAα≤cXandBα≤cY and henceAα×Bα≤cX×Y.
Case 2. Aα/1XandBα1Y, then for everyx, y∈X×Y,Aα×Bαx, y min{Aαx,1}
Aαx≤cXx cX×Yx, y. Hence,Aα×Bα≤cX×Y.
Case 3. Aα1XandBα/1Y. Similar to Case2, we getAα×Bα≤cX×Y.
Therefore, we get thatM{Aα×Bα:α∈Λ} ≤cX×Y and henceMcX×Y. This ends the proof of this direction.
4. Maximal Sets and T
cProperty
We start this section by the following nice characterization ofT1ordinary topological spaces.
Proposition 4.1. LetX, τbe an ordinary topological space with|X| > 1, thenX, τisT1if and only if maxX, τ {X− {x}:x∈X}.
Proof. It is straightforward.
Corollary 4.2. IfX, τis aT1ordinary topological space with|X|>1, then maxX, τcoversX.
Definition 4.3see15. An I-topological spaceX,Iis said to beTcif every fuzzy crisp point inXis fuzzy closed.
Definition 4.4. LetX,Ibe an I-topological space and letx∈X. Denote the fuzzy set
{A∈I:Ax<1} 4.1
byxI.
Proposition 4.5. Let X,I be a Tc I-topological space and x ∈ X, then for every y ∈ X − {x},xIy 1.
Proof. SinceX,IisTc, thenXX−{x}∈Iwith XX−{x}x 0<1. Thus, for everyy∈X−{x}, it follows that 1 XX−{x}y≤xIy≤1 and hencexIy 1.
Theorem 4.6. LetX,Ibe aTcI-topological space andx∈X, then
maxX,I {xI:x∈X, xIx<1}. 4.2
Proof. LetB∈maxX,I. Choosex∈Xsuch thatBx<1. IfxIx 1, then there exists A∈Isuch thatBx< Ax< 1 and hence we haveB∨A∈IwithB < B∨A < 1X which is impossible. SinceB∨xI ∈IandB∨xIx <1, thenB∨xI B. Thus,xI≤ B.
On the other hand, by the definition ofxIand thatBx<1, we haveB≤xI. Therefore, BxI.
Conversely, letx∈Xsuch thatxIx<1 andM∈I−{1X}such thatxI≤M. By Proposition 4.5, it follows thatMy 1 for everyy∈X− {x}. SinceM∈I− {1X}, it follows thatMx<1, and by the definition ofxI, we must haveM≤xI. Therefore,M xI and hencexI∈maxX,I.
Corollary 4.7. LetX,Ibe aTcI-topological space, then the following are equivalent:
amaxX,I ∅,
b xIx 1 for everyx∈X, cxI 1X for everyx∈X.
Corollary 4.8. For every non empty setXand for the discrete I-topologyIdisconX, maxX,Idisc
∅.
Corollary 4.8shows thatCorollary 4.2is invalid in I-topological spaces.
5. Maximal Sets and Homogeneity
We start this section by the following ordinary topological spaces result.
Proposition 5.1. IfX, τis a homogeneous ordinary topological space which contains a maximal open set, then maxX, τcoversX.
Proof. Letx∈X. ChooseM∈maxX, τandy∈M. SinceX, τis homogeneous, then there exists a homeomorphismh:X, τ−→X, τsuch thathy x. ByProposition 3.1, it follows thathM∈maxX, τ. Sincexhy∈hM, the proof is ended.
The following example shows that the condition “homogeneous” in Proposition 5.1 cannot be dropped.
Example 5.2. LetX Randτ {∅, X,Q}, then maxX, τ {Q}and so maxX, τdoes not coverX.
Theorem 5.3. Let X,I be a fuzzy homogeneous topological space. IfA ∈ maxX,Isuch that 1A ∅, thenAis a constant fuzzy set.
Proof. Suppose to the contrary that there exists x1, x2 ∈ X such thatAx1 < Ax2. Since X,Iis fuzzy homogeneous, there exists a fuzzy homeomorphismf :X,I−→X,Isuch thatfx1 x2. Note thatA∨f−1Ax1 max{Ax1, Ax2} Ax2 < 1. ThenA∨ f−1A∈IandA < A∨f−1A<1X, which contradicts the fuzzy maximality ofA.
The following example shows that the condition “fuzzy homogeneous” inTheorem 5.3 cannot be dropped.
Example 5.4. LetX {2,3}with the I-topologyI{0X,1X, A}whereA2 0.3 andA3 0.5, then maxX,I {A}and 1A ∅whileAis not constant.
The following example shows that the condition “1A ∅” inTheorem 5.3cannot be dropped.
Example 5.5. LetX {2,3}with the I-topologyI{0X,1X, A, B}whereA2 0,A3 1, B2 1, andB3 0, thenX,Iis fuzzy homogeneous andA∈maxX,IwhileAis not constant.
Theorem 5.3shows thatProposition 5.1 is invalid in I-topological spaces in general.
However, the following main result is a partial fuzzy version ofTheorem 5.3.
Theorem 5.6. LetX,Ibe a homogeneous I-topological space which consists a maximal fuzzy set, then
{B:B∈maxX,I}1Xif and only if there existsA∈maxX,Isuch that 1A/∅.
Proof. ⇒Suppose that{B:B∈maxX,I}1Xand suppose to the contrary that 1B ∅ for all B ∈ maxX,I. Choose A ∈ maxX,I. Then by Corollary 2.3, maxX,I {A}.
Applying Theorem 5.3, it follows that A is a constant fuzzy set and hence
{B : B ∈ maxX,I}A /1X, which is a contradiction.
⇐SupposeA ∈ maxX,Isuch that 1A/∅. Choosex◦ ∈ Xsuch thatAx◦ 1.
To see that
{B : B ∈ maxX,I} 1X, let x ∈ X such that x /x◦. Choose a fuzzy homeomorphismf : X,I −→ X,Isuch that fx◦ x. ByCorollary 3.3, it follows that fA∈maxX,I. Thus,
{B:B∈maxX,I}x≥fAx Ax◦ 1 and hence
{B:B∈maxX,I} x 1. 5.1
6. Minimal Fuzzy Open Sets
Definition 6.1. Let X,I be an I-topological space. For eachx ∈ X, we call the fuzzy set {A∈I:x∈SA}the lower fuzzy set atx, and we denote it byAx.
The following is an example of a zero lower fuzzy set.
Example 6.2. LetX be any non empty set together with any I-topologyIsuch that{cx : 0 ≤ c≤1} ⊆I, then for everyx∈X,Ax 0X.
The following lemma will be used in the next main result.
Lemma 6.3. LetX,Ibe an I-topological space and letA∈minX,I. Ifx∈SA, then for every B∈Iwithx∈SB, one hasA≤B.
Proof. Sincex∈SA∩SB, 0X/A∧B≤A. SinceA∈minX,I, thenA∧BAand hence A≤B.
Theorem 6.4. LetX,Ibe an I-topological space and letA∈minX,I, then for everyx∈SA, AAx.
Proof. Letx∈SA. Then byLemma 6.3, it follows thatA≤Bfor allB∈Iwithx∈SBand henceA≤
{B ∈I :x∈ SB} Ax. On the other hand, sinceA∈Iwithx ∈SA, then Ax≤A.
Corollary 6.5. LetX,Ibe an I-topological space. IfA∈minX,I, thenAis a lower fuzzy set in X,I.
The following is an example of a non zero lower fuzzy set in an I-topological space that is not a minimal fuzzy set.
Example 6.6. LetXRwith the I-topologyI{XU :U∈τu}, thenA0 ≤ X−1/n,1/nfor each n∈Nand soA0 ≤{X−1/n,1/n :n∈N} X{0}. On the other hand,A00 inf{A0:A∈ Iand 0∈SA}1. Therefore,A0X{0}/∈minX,I.
In ordinary topological spaces, we have the following result.
Proposition 6.7see7. LetX, τbe a topological space. IfU∈minX, τ, then every setV ⊆U is preopen.
The following example shows that the exact fuzzy version ofProposition 6.7is invalid in general.
Example 6.8. Let X be any nonempty set together with the I-topology I {0X,1X,0.4X,0.7X}, then 0.4X ∈ minX,I, 0.2X ≤ 0.4X, but intCl0.2X int0.3X 0X, that is,0.2Xis not a fuzzy preopen set.
Theorem 6.9. LetX,Ibe an I-topological space andA ∈ minX,Iwith Ax◦ > 0.5 for some x◦∈X, then for every fuzzy setBwithSB⊆SAandBx◦≥0.5 one hasClBx 1 for all x∈SB.
Proof. Suppose to the contrary that there exists a fuzzy set B such that SB ⊆ SAand Bx◦ ≥ 0.5 with ClBx1 < 1 for somex1 ∈ SB, sinceSB ⊆ SA,Ax1 > 0. Thus, A∧ClBcx1 min{Ax1,1−ClBx1}>0 and henceA∧ClBcA. Therefore, A≤ClBcand soAx◦ ClBx◦≤1, butAx◦ ClBx◦>0.5 0.51, which is a contradiction.
Corollary 6.10. LetX,Ibe an I-topological space andA∈minX,IwithAx◦>0.5 for some x◦∈X, then for everyB∈IXwithSB SAandBx◦≥0.5, one hasA≤ClB.
Proof. LetB ∈ IX withSB SAand Bx◦ ≥ 0.5 and letx ∈ X. Ifx ∈ SB, then by Theorem 6.9,ClBx 1 and soAx ≤ ClBx. Ifx /∈SB, then x /∈SA and so 0Ax≤ClBx.
The following corollary is a partial fuzzy version ofProposition 6.7.
Corollary 6.11. LetX,Ibe an I-topological space andA∈minX,IwithAx◦>0.5 for some x◦∈X, then for everyB∈IXsuch thatBx◦≥0.5,B≤A, andSB SA,Bis a fuzzy preopen set.
Proof. LetB∈IXwithBx◦≥0.5,B≤A, andSB SA, then byCorollary 6.10,A≤ClB and soB≤A≤intClB.
Acknowledgment
The author is grateful to Jordan University of Science and Technology for granting him a sabbatical leave in which this research is accomplished.
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