# On Maximal and Minimal Fuzzy Sets in I-Topological Spaces

Authors: Samer Al Ghour
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

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.


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## Body

## 1. Introduction

In this paper, the unit interval[0,1] will be denoted by I. Let X be a nonempty set. A member of IX is called a fuzzy subset of X [1]. Throughout this paper, for A,B∈IX we write A≤B if and only if  A(x)≤B(x) for all x∈X. By A=B, we mean that A≤B and B≤A, that is, A(x)=B(x) 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 in X, then (⋁Aj)(x)=sup{Aj(x):j∈J},  x∈X, and (⋀Aj)(x)=inf{Aj(x):j∈J}, x∈X. If r∈[0,1], then rX denotes the fuzzy set given by rX(x)=r  for all x∈X. The complement Ac of a fuzzy set A in X is given by Ac(x)=1-A(x), x∈X. If U⊆X,  then 𝒳U denotes the characteristic function of U.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 set p defined by
(1.1)p(x)={tifx=xp,0ifx≠xp,
where 0<t≤1, is called a fuzzy point in X, xp∈X is called the support of p and p(xp)=t the value (level) of p [2]. Two fuzzy points p and q in X are said to be distinct if and only if their supports are distinct, that is, xp≠xq. A∈IX is called a crisp subset of X if A is a characteristic function of some ordinary subset of X [2]. A∈IX is called a fuzzy crisp point in X if it is a characteristic function of singleton [2].In this paper, we follow [3] for the definition of “belonging to”. Namely, a fuzzy point p in X is said to belong to a fuzzy set A in X (notation: p∈A) if and only if p(xp)≤A(xp).Let(X,τ) be a topological space and let U be a subset of X. U is called semiopen [4] if U⊆Cl(Int(U)) and U is called preopen [5] if U⊆Int(Cl(U)). A nonempty open subset U of X is called a minimal open set if the only nonempty open set which is contained in U is U. min(X,τ) will denote the family of all minimal open sets in X. (X,τ) is said to be homogeneous if for any two points x1,x2∈X, there exists an autohomeomorphism on (X,τ) takes x1 to x2. In 1997, Fora and Al-Bsoul [6] used minimal open sets to characterize and count finite homogeneous spaces. In 2001, Nakaoka and Nobuyuki [7] characterized minimal open sets and proved that any subset of a minimal open set is pre-open. A non-empty pre-open subset U of X is called a minimal pre-open set [8] if the only non-empty pre-open set which is contained in U is U. The author in [8] characterized minimal pre-open sets as pre-open singletons. A non-empty semi-open subset U of X is called a minimal semi-open set [9] if the only non-empty semi-open set which is contained in U is U. The authors in [9] proved that the set of minimal open sets and the set of minimal semi-open sets in a space are equal. An I-topological space (X,ℑ) is said to be fuzzy homogeneous [10] if for any two points x1,x2∈X, there exists a fuzzy autohomeomorphism on (X,ℑ) takes x1 to x2. The authors in [11] extended the concept of minimal open sets to include I-topological spaces as follows; a fuzzy open set A of an I-topological space (X,ℑ) is called minimal fuzzy open set [11] if A is nonzero and there is no nonzero proper fuzzy open subset of A. min(X,ℑ) will denote the family of all minimal fuzzy open sets in X. The authors in [11] obtained 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 in [12] generalized minimal fuzzy open sets by two methods. A proper non-empty open subset U of a topological space (X,τ) is called a maximal open set [13] if any open set which contains U is X or U. max(X,τ) will denote the family of all maximal open sets in X. The authors in [13] obtained 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 in [14] introduced 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 of X. If A is a fuzzy set in X, then the support of A is denoted by S(A) and defined by S(λ)=λ-1(0,1], and the set {x∈X:A(x)=1} will be denoted by 1(A).


## 2. Maximal Fuzzy Open Sets

Definition 2.1.
Let(X,ℑ) be an I-topological space. A nonzero fuzzy open subset A of X is said to be maximal fuzzy open set if A≠1X and for any fuzzy open set B in X with A≤B, B=A or B=1X.Throughout this paper, the set of all maximal fuzzy open subsets of theI-topological space (X,ℑ) will be denoted by max(X,ℑ).Theorem 2.2.
Let(X,ℑ) be an  I-topological space and  A∈max(X,ℑ)with 1(A)=∅.  Then for every  B∈ℑ-{1X} one has B≤A.Proof.
Suppose to the contrary that there existB∈ℑ-{1X} and x∘∈X such that B(x∘)>A(x∘). Since (A∨B)(x∘)=B(x∘)≠A(x∘) and A∈max(X,ℑ), A∨B=1X. Therefore, for each x∈X, A(x)<1=B(x) and hence B=1X, which is a contradiction.Corollary 2.3.
Let(X,ℑ) be an I-topological space.  If A∈max(X,ℑ) with 1(A)=∅,  then max(X,ℑ)={A}.Corollary 2.4.
Let(X,ℑ) be an  I-topological space.  If |max(X,ℑ)|>1,  then for every A∈max(X,ℑ), 1(A)≠∅.Example 2.5.
LetX=ℝ and τ be the usual topology on ℝ, then max(X,τ)={ℝ-{x}:x∈ℝ}. Let ℑ={𝒳U:U∈τ}. then max(X,ℑ)={𝒳U:U∈max(X,τ)} is uncountable. Therefore, the condition “1(A)=∅” in Corollary 2.3 cannot be dropped.Proposition 2.6 (see [13]).
Let(X,τ) be a topological space and let U∈max(X,τ), then Cl(U)=X or Cl(U)=U.The following example shows that the exact fuzzy version of Proposition2.6 is not true in general.Example 2.7.
LetX={x1,x2,x3} and let A,B be fuzzy sets in X defined as follows:
(2.1)A(x1)=0.3,A(x2)=0.3,A(x3)=1,B(x1)=0.3,B(x2)=0.3,B(x3)=0.
Letℑ={0X,1X,A,B}, then max(X,ℑ)={A} and Cl(A)={(x1,0.7),(x2,0.7),(x3,1)} but Cl(A) is neither A nor 1X.The following lemma will be used in the following main result.Lemma 2.8.
Let(X,ℑ) be an I-topological space and A∈max(X,ℑ).  If  B∈ℑ-{0X} such that A∧B=0X,  then A=𝒳S(A) and B=Ac.Proof.
Choosex∘∈S(B). Since A∧B=0X, A(x∘)=0. Thus, (A∨B)(x∘)=B(x∘)≠A(x∘). Since A∈max(X,ℑ), A∨B=1X. Since A∧B=0X, it follows that A=𝒳S(A) and B=Ac.The following result is a partial fuzzy version of Proposition2.6.Theorem 2.9.
Let(X,ℑ) be an I-topological space. If A=𝒳S(A)∈max(X,ℑ), then either Cl(A)=A or Cl(A)=1X.Proof.
Suppose thatCl(A)≠1X, then there exists x∘∈X such that (Cl(A))(x∘)<1. Let B=(Cl(A))c, then B∈ℑ,B≠0X, and A∧B=0X. Hence by Lemma 2.8, it follows that B=Ac. Therefore, Cl(A)=A.Proposition 2.10 (see [13]).
Let(X,τ) be a topological space. If U∈max(X,τ) and S is a non empty subset of X-U, then Cl(S)=X-U.The following result is the exact fuzzy version of Proposition2.10.Theorem 2.11.
Let(X,ℑ) be an I-topological space. If A∈max(X,ℑ) and B is a non zero fuzzy set in X with B≤Ac, then Cl(B)=Ac.Proof.
Suppose to the contrary thatCl(B)≠Ac. Since B≤Ac and Ac is a fuzzy closed set in X, then Cl(B)≤Ac. Therefore, there exists x∘∈Xsuch that (Cl(B))(x∘)<Ac(x∘). Since A(x∘)<(1X-Cl(B))(x∘) and A∈max(X,ℑ), then A∨(1X-Cl(B))=1X. We are going to show that Cl(B)=0X. Let x∈X=1(A)∪(X-1(A)). If x∈1(A), then Ac(x)=0 and hence (Cl(B))(x)≤Ac(x)=0. If x∈(X-1(A)), then A(x)<1. Since A∨(1X-Cl(B))=1X, we get (1X-Cl(B))(x)=1 and hence (Cl(B))(x)=0. Hence, we complete the proof that Cl(B)=0X. Therefore, B=0X, which is a contradiction.Recall that a fuzzy subsetA of an I-topological space (X,ℑ) is called fuzzy preopen if A≤int(Cl(A)).From now on the setPO(X,ℑ) will denote the set of all fuzzy preopen subsets of the I-topological space (X,ℑ).Corollary 2.12.
Let(X,ℑ) be an I-topological space and A∈max(X,ℑ), then {B∈IX:B≤int(Ac)}⊆PO(X,ℑ).Proof.
LetB∈IX such that B≤int(Ac). If B=0X, then B∈PO(X,ℑ). If B≠0X, then by Theorem 2.11, Cl(B)=Ac and hence B≤int(Ac)=int(Cl(B)), that is, B∈PO(X,ℑ).Corollary 2.13.
Let(X,ℑ) be an I-topological space and A∈max(X,ℑ) with A is a clopen fuzzy set, then {B∈IX:B≤Ac}⊆PO(X,ℑ).Proof.
LetB∈IX such that B≤Ac. If B=0X, then B∈PO(X,ℑ). If B≠0X, then by Theorem 2.11, Cl(B)=Ac and hence B≤Ac=int(Ac)=int(Cl(B)), that is, B∈PO(X,ℑ).Proposition 2.14 (see [13]).
Let(X,τ) be a topological space. If  U∈max(X,τ) and M is a subset of X with U⊊M, then Cl(M)=X.Corollary 2.15.
Let(X,τ) be a topological space. If U∈max(X,τ) and M is a closed subset of X with U⊊M, then M=X.The following example shows that the exact fuzzy version of each of Proposition2.14 and Corollary 2.15 is invalid in general.Example 2.16.
LetX be a non empty set with the I-topology ℑ={0X,1X,(0.3)X}, then max(X,ℑ)={(0.3)X} and (0.3)X<(0.4)X, but Cl((0.4)X)=(0.7)X. This shows that the exact fuzzy version of Proposition 2.14 is invalid in general. On the other hand, since (0.7)X is a fuzzy closed subset of (X,ℑ) and 0X<(0.7)X<1X, we get that the exact fuzzy version of Corollary 2.15 is invalid in general.The following result is a partial fuzzy version of Corollary2.15.Theorem 2.17.
Let(X,ℑ) be an I-topological space and A∈max(X,ℑ). If B is a fuzzy closed subset of X with A<B, then for every x∈X with A(x)<B(x), one has B(x)>0.5.Proof.
Suppose to the contrary that there exists a fuzzy closed subsetB of X and x∘∈X such that A(x∘)<B(x∘) and B(x∘)≤0.5. Since B is a fuzzy closed subset of X, Bc∈ℑ and so either Bc∨A=A or Bc∨A=1X. If Bc∨A=A, then 1-B(x∘)≤A(x∘) and so B(x∘)>0.5 which  is a contradiction because B(x∘)≤0.5. If Bc∨A=1X, then B(x∘)=0 or A(x∘)=1 which is also a contradiction because A(x∘)<B(x∘).For crisp maximal fuzzy sets, the exact fuzzy version of Corollary2.15 is valid as the following result shows.Theorem 2.18.
Let(X,ℑ) be an I-topological space. If A=𝒳S∈max(X,ℑ) and B is a fuzzy closed subset of X with A<B, then B=1X.Proof.
Suppose to the contrary that there existsx∘∈X such that B(x∘)<1. Since A<B, then x∘∈X-S. Therefore, (Bc∨A)(x∘)=Bc(x∘)≠0=A(x∘) and so Bc∨A≠A. Hence, we must have Bc∨A=1X. Choose x1∈X such that A(x1)<B(x1). Then Bc(x1)=1 and so B(x1)=0, but 0=A(x1)<B(x1), which is a contradiction.Corollary 2.19.
Let(X,ℑ) be an I-topological space. If A=𝒳S∈max(X,ℑ) and B is a fuzzy subset of X with A<B, then Cl(B)=1X.Corollary 2.20.
Let(X,ℑ) be an I-topological space. If A=𝒳S∈max(X,ℑ) and B is a fuzzy subset of X with A≤B, then B is a fuzzy preopen set.Proof.
IfB=A, then B is fuzzy open and so it is a fuzzy preopen set. On the other hand, if A<B, then by Corollary 2.19, it follows that B≤1X=int(1X)=int(Cl(B)).The following ordinary topological spaces result follows easily.Proposition 2.21.
Let(X,τ) be an ordinary topological space, then {U,Uc}⊆max(X,τ) for some U⊆X if and only if  τ={∅,X,U,Uc}.The exact fuzzy version of Proposition2.21 is true as the following result shows.Theorem 2.22.
Let(X,ℑ) be an I-topological space and let A∈IX with A≠Ac, then {A,Ac}⊆max(X,ℑ) if and only if  A=𝒳S(A) and ℑ={0X,1X,A,Ac}.Proof.
⇒) Suppose that {A,Ac}⊆max(X,ℑ). Choose x∘∈X such that A(x∘)≠Ac(x∘). Then (A∨Ac)(x∘)≠A(x∘) or (A∨Ac)(x∘)≠Ac(x∘). Since {A,Ac}⊆max(X,ℑ), then A∨Ac=1X. Thus for every x∈X with A(x)<1, we must have Ac(x)=1 and so A(x)=0. Therefore, A=𝒳S(A). Let B∈ℑ-{0X}. If B≤A, then Ac∧B=0X and, by Lemma 2.8, B=(Ac)c=A. If  BA, then A∨B=1X and so B=Ac.
⟸) Clear.


## 3. Images and Products of Maximal Sets

From now onπx and πy will denote the projections on X and Y, respectively, τprod will denote the product topology of τ1 and τ2, and ℑprod will denote the product I-topology of ℑ1 and ℑ2.We start by the following result.Proposition 3.1.
Let(X,τ1) and (Y,τ2) be two ordinary topological spaces. If f:(X,τ1)→(Y,τ2) is continuous, open, and surjective, then for every U∈max(X,τ1) with f(U)≠Y one has f(U)∈max(Y,τ2).Proof.
Suppose thatH∈τ2 with f(U)⊆H⊂Y. Then f-1(H)∈τ1 with U⊆f-1(H). Since U∈max(X,τ1), then f-1(H)=U or f-1(H)=X. If f-1(H)=X, then f(f-1(H))=f(X)=Y. Thus f-1(H)=U and so H=f(f-1(H))=f(U).The following result is the exact fuzzy version of Proposition3.1.Theorem 3.2.
Let(X,ℑ1) and (Y,ℑ2) be two I-topological spaces and let f:(X,ℑ1)→(Y,ℑ2) be fuzzy continuous, fuzzy open, and surjective function. If  A∈max(X,ℑ1), then either f(A)=1Y or f(A)∈max(Y,ℑ2).Proof.
Suppose thatf(A)≠1Y. It is sufficient to show that f(A)∈max(Y,ℑ2). Since f is fuzzy open and A is a fuzzy open set, then f(A) is a fuzzy open set. Since A≠0X, there exists x°∈X such that A(x°)>0 and so (f(A))(f(x°))=sup{A(x):f(x)=f(x°)}≥A(x°)>0,hencef(A)≠0Y. Suppose that B∈ℑ2  such that f(A)<B. We are going to show B=1Y which completes the proof. Choose y°∈Y such that (f(A))(y°)<B(y°). Since f is onto, there exists x1∈X  such that f(x1)=y°. Thus, A(x1)≤(f(A))(y°)<B(y°). Since f is fuzzy continuous, f-1(B)∈ℑ1. Thus, we have f-1(B)∨A∈ℑ1, A≤f-1(B)∨A, and (f-1(B)∨A)(x1)=max{A(x1),B(y°)} = B(y°)>A(x1). Since A∈max(X,ℑ1), then f-1(B)∨A=1X. To see that B=1Y, let y∈Y and choose x∈X such that f(x)=y, then 1=max{A(x),B(y)}≤max{f(A)(y),B(y)}=B(y) and so B(y)=1.Corollary 3.3.
Let(X,ℑ1) and (Y,ℑ2) be two I-topological spaces and let  f:(X,ℑ1)→(Y,ℑ2) be a fuzzy homeomorphism. If A∈max(X,ℑ1), then f(A)∈max(Y,ℑ2).Proof.
LetA∈max(X,ℑ1). According to Theorem 3.2, it is sufficient to see that f(A)≠1Y. Since A∈max(X,ℑ1), there exists x°∈X such that A(x°)<1. Thus, f(A)(f(x°))=A(x°)<1.Proposition 3.4.
Let(X,τ1) and (Y,τ2) be two ordinary topological spaces. If  U∈max(X,τ1) and V∈max(Y,τ2),then U×V∉max(X×Y,τprod).Proof.
Follows becauseU×Y∈τprod, but U×V⊂U×Y⊂X×Y.Proposition 3.5.
Let(X,τ1) and (Y,τ2) be two ordinary topological spaces and let G∈max(X×Y,τprod) such that πx(G)×πy(G)≠X×Y, then there exists U∈max(X,τ1) or V∈max(Y,τ2) such that G=U×Y or G=X×V.Proof.
SinceG⊆πx(G)×πy(G), πx(G)×πy(G)∈τprod, πx(G)×πy(G)≠X×Y, and G∈max(X×Y,τprod), it follows that G=πx(G)×πy(G). If πx(G)≠X and πy(G)≠Y, then by Proposition 3.1, πx(G)∈max(X,τ1) and πy(G)∈max(Y,τ2), which is impossible by Proposition 3.4. Therefore, either πx(G)=Xorπy(G)=Y. Hence, there exists U∈max(X,τ1) or V∈max(Y,τ2) such that G=U×Y or G=X×V.The following example shows that the condition “πx(G)×πy(G)≠X×Y” in Proposition 3.5 cannot be dropped.Example 3.6.
LetX=ℝ with the usual topology τu and let G=ℝ2-{(0,0)}, then G∈max(X×X,τprod), πx(G)×πy(G)=X×X, and there is no U∈max(X,τu) such that G=U×X or G=X×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∈max(X,ℑ1) and  B∈max(Y,ℑ2), then A×B∈max(X×Y,ℑprod) if and only if there exists 0<c<1 such that A=cX and B=cY.Proof.
Suppose thatA×B∈max(X×X,ℑprod). Since A×1Y, 1X×B∈ℑprod, A×B≤A×1Y<1X×Y, A×B≤1X×B<1X×Y, then A×B=A×1Y  and A×B=1X×B. Therefore, for every x∈X and y∈Y, min{A(x),B(y)}=min{A(x),1}=min{1,B(y)} and hence A(x)=B(y). Thus, there exists 0<c<1 such that A=cX and B=cY. Conversely, suppose for some 0<c<1 that A=cX and B=cY. Let M∈ℑprod-{1X×Y} such that cX×cY=cX×Y≤M. Choose families {Aα:α∈Λ}⊆ℑ1 and {Bα:α∈Λ}⊆ℑ2 such that M=⋁{Aα×Bα:α∈Λ}. Since M≠1X×Y, then for every α∈Λ, Aα≠1X or Bα≠1Y. We are going to show that Aα×Bα≤cX×Y for every α∈Λ. Let α∈Λ.Case 1.
Aα≠1X and Bα≠1Y, then Aα≤cX and Bα≤cY and hence Aα×Bα≤cX×Y.Case 2.
Aα≠1X and Bα=1Y, then for every (x,y)∈X×Y, (Aα×Bα)(x,y)=min{Aα(x),1}=Aα(x)≤cX(x)=cX×Y(x,y). Hence, Aα×Bα≤cX×Y.Case 3.
Aα=1X and Bα≠1Y. Similar to Case 2, we get Aα×Bα≤cX×Y.Therefore, we get thatM=⋁{Aα×Bα:α∈Λ}≤cX×Y and hence M=cX×Y. This ends the proof of this direction.


## 4. Maximal Sets andTc Property

We start this section by the following nice characterization ofT1 ordinary topological spaces.Proposition 4.1.
Let(X,τ) be an ordinary topological space with |X|>1, then (X,τ) is T1 if and only if max(X,τ)={X-{x}:x∈X}.Proof.
It is straightforward.Corollary 4.2.
If(X,τ) is a T1 ordinary topological space with |X|>1, then max(X,τ) covers X.Definition 4.3 (see [15]).
An I-topological space(X,ℑ) is said to be Tc if every fuzzy crisp point in X is fuzzy closed.Definition 4.4.
Let(X,ℑ) be an I-topological space and let x∈X. Denote the fuzzy set
(4.1)⋁{A∈ℑ:A(x)<1}
by x(ℑ).Proposition 4.5.
Let(X,ℑ) be a Tc I-topological space and x∈X, then for every y∈X-{x},(x(ℑ))(y)=1.Proof.
Since(X,ℑ) is Tc, then 𝒳X-{x}∈ℑ  with  (𝒳X-{x})(x)=0<1. Thus, for every y∈X-{x}, it follows that 1=(𝒳X-{x})(y)≤(x(ℑ))(y)≤1 and hence (x(ℑ))(y)=1.Theorem 4.6.
Let(X,ℑ) be a Tc  I-topological space and x∈X, then
(4.2)max(X,ℑ)={x(ℑ):x∈X,(x(ℑ))(x)<1}.Proof.
LetB∈max(X,ℑ). Choose x∈X such that B(x)<1. If (x(ℑ))(x)=1, then there exists A∈ℑ such that B(x)<A(x)<1 and hence we have B∨A∈ℑ with B<B∨A<1X which is impossible. Since B∨x(ℑ)∈ℑ and (B∨x(ℑ))(x)<1, then B∨x(ℑ)=B. Thus, x(ℑ)≤B. On the other hand, by the definition of x(ℑ) and that B(x)<1, we have B≤x(ℑ). Therefore, B=x(ℑ).
Conversely, letx∈X  such that (x(ℑ))(x)<1 and M∈ℑ-{1X} such that x(ℑ)≤M. By Proposition 4.5, it follows that M(y)=1 for every y∈X-{x}. Since M∈ℑ-{1X}, it follows that M(x)<1, and by the definition of x(ℑ), we must have M≤x(ℑ). Therefore, M=x(ℑ) and hence x(ℑ)∈max(X,ℑ).Corollary 4.7.
Let(X,ℑ) be a Tc I-topological space, then the following are equivalent:(a)
max(X,ℑ)=∅,(b)
(x(ℑ))(x)=1  for every  x∈X,(c)
x(ℑ)=1X  for every  x∈X.Corollary 4.8.
For every non empty setX and for the discrete I-topology ℑdisc on X, max(X,ℑdisc)=∅.Corollary4.8 shows that Corollary 4.2 is invalid in I-topological spaces.


## 5. Maximal Sets and Homogeneity

We start this section by the following ordinary topological spaces result.Proposition 5.1.
If(X,τ) is a homogeneous ordinary topological space which contains a maximal open set, then max(X,τ) covers X.Proof.
Letx∈X. Choose M∈max(X,τ) and y∈M. Since (X,τ) is homogeneous, then there exists a homeomorphism h:(X,τ)→(X,τ) such that h(y)=x. By Proposition 3.1, it follows that h(M)∈max(X,τ). Since x=h(y)∈h(M), the proof is ended.The following example shows that the condition “homogeneous”  in Proposition5.1 cannot be dropped.Example 5.2.
LetX=ℝ and τ={∅,X,ℚ}, then max(X,τ)={ℚ} and so max(X,τ) does not cover X.Theorem 5.3.
Let(X,ℑ) be a fuzzy homogeneous topological space. If A∈max(X,ℑ) such that 1(A)=∅, then A is a constant fuzzy set.Proof.
Suppose to the contrary that there existsx1,x2∈X such that A(x1)<A(x2). Since (X,ℑ) is fuzzy homogeneous, there exists a fuzzy homeomorphism f:(X,ℑ)→(X,ℑ) such that f(x1)=x2. Note that (A∨f-1(A))(x1)=max{A(x1),A(x2)}=A(x2)<1. Then A∨f-1(A)∈ℑ and A<A∨f-1(A)<1X, which contradicts the fuzzy maximality of A.The following example shows that the condition “fuzzy homogeneous” in Theorem5.3 cannot be dropped.Example 5.4.
LetX={2,3} with the I-topology ℑ={0X,1X,A} where A(2)=0.3 and A(3)=0.5, then max(X,ℑ)={A} and 1(A)=∅ while A is not constant.The following example shows that the condition “1(A)=∅” in Theorem 5.3 cannot be dropped.Example 5.5.
LetX={2,3} with the I-topology ℑ={0X,1X,A,B} where A(2)=0, A(3)=1, B(2)=1, and B(3)=0, then (X,ℑ) is fuzzy homogeneous and A∈max(X,ℑ) while A is not constant.Theorem5.3 shows that Proposition 5.1 is invalid in I-topological spaces in general. However, the following main result is a partial fuzzy version of Theorem 5.3.Theorem 5.6.
Let(X,ℑ) be a homogeneous I-topological space which consists a maximal fuzzy set, then ⋁{B:B∈max(X,ℑ)}=1X if and only if there exists A∈max(X,ℑ) such that 1(A)≠∅.Proof.
⇒) Suppose that ⋁{B:B∈max(X,ℑ)}=1X and suppose to the contrary that 1(B)=∅ for all B∈max(X,ℑ). Choose A∈max(X,ℑ). Then by Corollary 2.3, max(X,ℑ)={A}. Applying Theorem 5.3, it follows that A is a constant fuzzy set and hence ⋁{B:B∈max(X,ℑ)}=A≠1X, which is a contradiction.
⟸) Suppose A∈max(X,ℑ) such that 1(A)≠∅. Choose x∘∈X such that A(x∘)=1. To see that ⋁{B:B∈max(X,ℑ)}=1X, let x∈X such that x≠x∘.  Choose a fuzzy homeomorphism f:(X,ℑ)→(X,ℑ) such that f(x∘)=x. By Corollary 3.3, it follows that f(A)∈max(X,ℑ). Thus, (⋁{B:B∈max(X,ℑ)})(x)≥f(A)(x)=A(x∘)=1 and hence
(5.1)(⋁{B:B∈max(X,ℑ)})(x)=1.


## 6. Minimal Fuzzy Open Sets

Definition 6.1.
Let(X,ℑ) be an I-topological space. For each x∈X, we call the fuzzy set ⋀{A∈ℑ:x∈S(A)} the lower fuzzy set at x, and we denote it by Ax.The following is an example of a zero lower fuzzy set.Example 6.2.
LetX be any non empty set together with any I-topology ℑ such that {cx:0≤c≤1}⊆ℑ, then for every x∈X, Ax=0X.The following lemma will be used in the next main result.Lemma 6.3.
Let(X,ℑ) be an I-topological space and let A∈min(X,ℑ). If x∈S(A), then for every B∈ℑ with x∈S(B), one has A≤B.Proof.
Sincex∈S(A)∩S(B), 0X≠A∧B≤A. Since A∈min(X,ℑ), then A∧B=A and hence A≤B.Theorem 6.4.
Let(X,ℑ) be an I-topological space and let A∈min(X,ℑ), then for every x∈S(A), A=Ax.Proof.
Letx∈S(A). Then by Lemma 6.3, it follows that A≤B for all B∈ℑ with x∈S(B) and hence A≤⋀{B∈ℑ:x∈S(B)}=Ax. On the other hand, since A∈ℑ with x∈S(A), then Ax≤A.Corollary 6.5.
Let(X,ℑ) be an I-topological space. If A∈min(X,ℑ), then A is a lower fuzzy set in (X,ℑ).The following is an example of a non zero lower fuzzy set in anI-topological space that is not a minimal fuzzy set.Example 6.6.
LetX=ℝ with the I-topology ℑ={𝒳U:U∈τu}, then A0≤𝒳(-1/n,1/n) for each n∈ℕ and so A0≤⋀{𝒳(-1/n,1/n):n∈ℕ}=𝒳{0}. On the other hand, A0(0)=inf{A(0):A∈ℑand0∈S(A)}=1. Therefore, A0=𝒳{0}∉min(X,ℑ).In ordinary topological spaces, we have the following result.Proposition 6.7 (see [7]).
Let(X,τ) be a topological space. If U∈min(X,τ), then every set V⊆U  is preopen.The following example shows that the exact fuzzy version of Proposition6.7 is invalid in general.Example 6.8.
LetX be any nonempty set together with the I-topology ℑ={0X,1X,(0.4)X,(0.7)X}, then (0.4)X∈min(X,ℑ), (0.2)X≤(0.4)X, but int(Cl((0.2)X))=int((0.3)X)=0X, that is, (0.2)X is not a fuzzy preopen set.Theorem 6.9.
Let(X,ℑ) be an I-topological space and A∈min(X,ℑ) with A(x∘)>0.5 for some x∘∈X, then for every fuzzy set B with S(B)⊆S(A) and B(x∘)≥0.5 one has (Cl(B))(x)=1 for all x∈S(B).Proof.
Suppose to the contrary that there exists a fuzzy setB such that S(B)⊆S(A) and B(x∘)≥0.5 with Cl(B)(x1)<1 for some x1∈S(B), since S(B)⊆S(A), A(x1)>0. Thus, (A∧(Cl(B))c)(x1)=min{A(x1),1-(Cl(B))(x1)}>0 and hence A∧(Cl(B))c=A. Therefore, A≤(Cl(B))c and so A(x∘)+(Cl(B))(x∘)≤1, but A(x∘)+(Cl(B))(x∘)>0.5+0.5=1, which is a contradiction.Corollary 6.10.
Let(X,ℑ) be an I-topological space and A∈min(X,ℑ) with A(x∘)>0.5 for some x∘∈X, then for every B∈IX with S(B)=S(A) and B(x∘)≥0.5, one has A≤Cl(B).Proof.
LetB∈IX with S(B)=S(A) and B(x∘)≥0.5 and let x∈X. If x∈S(B), then by Theorem 6.9, (Cl(B))(x)=1 and so A(x)≤(Cl(B))(x). If x∉S(B), then x∉S(A) and so 0=A(x)≤(Cl(B))(x).The following corollary is a partial fuzzy version of Proposition6.7.Corollary 6.11.
Let(X,ℑ) be an I-topological space and A∈min(X,ℑ) with A(x∘)>0.5 for some x∘∈X, then for every B∈IX such that B(x∘)≥0.5, B≤A, and S(B)=S(A), B is a fuzzy preopen set.Proof.
LetB∈IX with B(x∘)≥0.5, B≤A, and S(B)=S(A), then by Corollary 6.10, A≤Cl(B) and so B≤A≤int(Cl(B)).



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*Source: 180196-2010-10-11.xml*