AFSAdvances in Fuzzy Systems1687-711X1687-7101Hindawi Publishing Corporation17291710.1155/2009/172917172917Research ArticleRelative Smooth Topological SpacesGhazanfariB.1LeeZne-JungDepartment of MathematicsFaculty of ScienceLorestan UniversityP.O. Box 465Khoramabad 68137-17133Iranlu.ac.ir2009722010200922062009101120092009Copyright © 2009This 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.

In 1992, Ramadan introduced the concept of a smooth topological space and relativeness between smooth topological space and fuzzy topological space in Chang's (1968) view points. In this paper we give a new definition of smooth topological space. This definition can be considered as a generalization of the smooth topological space which was given by Ramadan. Some general properties such as relative smooth continuity and relative smooth compactness are studied.

1. Introduction

Let X be a nonempty set and let ,' be two lattice which will be copies of [0,1] or {0,1}. The family of all fuzzy sets on X will be denoted by X Zadeh .

In the consideration of the nature an observer can be modeled by an operator which evaluates each proposition by a number in the closed interval [0,1]; see Anvari and Molaei  and Molaei . We assume that μ as a function from X to is an observer of X on lattice and denote [μ]={λX:λμ}, where λμ implies that λ(x)μ(x) for all xX.

Definition 1.1.

Let μX. A relative smooth topological space or μ-smooth topological space or μ-STS for short is a triple (X,μ,𝒯μ), where 𝒯μ:[μ] is a mapping satisfying the following properties:

𝒯μ(μ)=𝒯μ(χϕ)=1, where χ is the characteristic function;

if λ1, λ2[μ], then 𝒯μ(λ1λ2)𝒯μ(λ1)𝒯μ(λ2), where is the minimum operator in ;

𝒯μ({λi:iI}){𝒯μ(λi):iI}.

We call 𝒯μ a smooth topology from view point of μ or a μ-smooth topology or a fuzzy family of μ-open sets on X.

Remark 1.2.

If μ=χX then the χX-STS (X,χX,𝒯χX) coincides with the smooth topological space (X,τ) defined by Ramadan , and if we take =[0,1],={0,1}, and μ=χX then the χX-STS coincides with the known definition of fuzzy topological space (X¯,τ) defined by Chang . If =={0,1}, and μ=χX then 𝒯χX is a classical topology.

Definition 1.3.

Let μX. A μ-smooth cotopological space is a triple (X,μ,μ), where μ:[μ] is a mapping satisfying the following properties:

μ(μ)=μ(χϕ)=1;

if η1, η2[μ], then μ(η1η2)μ(η1)μ(η2);

μ({ηi:iI}){μ(ηi):iI}.

We call μ a μ-smooth co-topology or a fuzzy family of μ-closed sets on X.

Theorem 1.4.

Let (X,μ,𝒯μ) be a μ-STS and μ:[μ] be a mapping defined by μ(η)=𝒯μ(η), where η=μ-η. Then μ is a fuzzy family of μ-closed sets.

Proof.

It is clear.

It flows from (η1η2)=μ-(η1η2)=μ-sup{η1,η2}=inf{μ-η1,μ-η2}=η1η2. So, μ(η1η2)=𝒯μ((η1η2))=𝒯μ(η1η2).

It flows from (iIηi)=μ-iIηi=μ-inf{ηi:iI}=sup{μ-ηi:iI}=iIηi. So, μ({ηi:iI})=𝒯μ({ηi:iI}).

Theorem 1.5.

Let μ be a fuzzy family of μ-closed sets and define 𝒯μ:[μ] by 𝒯μ(η)=μ(η). Then 𝒯μ is a μ-STS on X.

Proof.

The proof is similar to the previous theorem.

Corollary 1.6.

Let 𝒯μ be a μ-STS and μ a fuzzy family of μ-closed sets. Then 𝒯𝒯μ=𝒯μ and 𝒯μ=μ.

Proof.

Suppose λ,η[μ] then we have 𝒯𝒯μ(λ)=𝒯μ(λ)=𝒯μ(λ) and 𝒯μ(η)=𝒯μ(η)=μ(η).

Example 1.7.

Let X be the set of all differentiable real-valued functions on (1,), with positive derivative of order one and let be the set of real-valued functions defined on (1,). Let μ:X be defined by μ(f)=f+Exp, where Exp is the exponential function. For nonnegative integer n define λn:X by (λn(f))(x)=f(x)+i=1nxi-1(i-1)!. If we take =[0,1] and define 𝒯μ:[μ][0,1] by 𝒯μ(χ)=𝒯μ(μ)=1;  𝒯μ(λn)=1-1/n for n=1,2,. Then (X,μ,𝒯μ) is a μ-STS. Since λnλm=λm,λnλm=λn, where n>m and λni=μ whenever ni tends to +, so 𝒯μ(λnλm)=𝒯μ(λm)𝒯μ(λn)𝒯μ(λm), and for I we find 𝒯μ({λn:nI}){𝒯μ(λn):nI}.

Definition 1.8.

Let 𝒯μ1 and 𝒯μ2 be two μ-smooth topological spaces on X. We say that 𝒯μ1 is finer than 𝒯μ2 or 𝒯μ2 is coarser than 𝒯μ1 and denoted by 𝒯μ1𝒯μ2 if 𝒯μ1(λ)𝒯μ2(λ) for every λ[μ].

Theorem 1.9.

Let {𝒯μi:iI} be a family of μ-STS on X. Then 𝒯μ=iI𝒯μi is also μ-STS on X, where (iI𝒯μi)(λ)=iI𝒯μi(λ).

Proof.

It is clear.

For every λ,η[μ], 𝒯μ(λη)=iI𝒯μi(λη)iI(𝒯μi(λ)𝒯μi(η))=iI(𝒯μi(λ))(iI𝒯μi(λ))=𝒯μ(λ)𝒯μ(η).

For Γ, 𝒯μ({λj:jΓ})=iI𝒯μi({λj:jΓ})iI{𝒯μi(λj):jΓ}=(iI{𝒯μi(λj):jΓ})={𝒯μ(λj):jΓ}.

Let A be a subset of X and λ[μ]. The restriction of λ on A is denoted by λ|A.

Theorem 1.10.

Let (X,μ,𝒯μ) be a μ-STS and AX. Define a mapping 𝒯μ|A:[μ] by 𝒯μ|A(λ)={𝒯μ(η):η[μ],η|A=λ}. Then 𝒯μ|A is a μ-STS on A.

Proof.

It is clear that 𝒯μ|A(χ)=𝒯μ|A(μ)=1.

λ1,λ2A,λ1,λ2μ. 𝒯μ|A(λ1)𝒯μ|A(λ2)={𝒯μ(η1):η1[μ],η1|A=λ1}{𝒯μ(η2):η2[μ],η2|A=λ2}={𝒯μ(η1)𝒯μ(η2):η1,η2[μ],η1η2|A=λ1λ2}{𝒯μ(η1η2):η1,η2[μ],η1η2|A=λ1λ2}=𝒯μ|A(λ1λ2).

iI,𝒯μ|A{λi:iI}={𝒯μ(ηi):ηi[μ],ηi|A=λi}. So 𝒯μ|A({λi:iI})={𝒯μ({ηi:iI}):ηi[μ],ηi|A=λi}{{𝒯μ(ηi):iI}:ηi[μ],ηi|A=λi}=𝒯μ|A{λi:iI}.

Definition 1.11.

The μ-STS (A,μ,𝒯μ|A) is called a subspace of (X,μ,𝒯μ) and 𝒯μ|A is called the induced μ-STS on A from 𝒯μ.

Theorem 1.12.

Let (A,μ,𝒯μ|A) be a μ-smooth subspace of (X,μ,𝒯μ) and λA,λ[μ]. Then

𝒯μ|A(λ)={𝒯μ(η):η[μ],η|A=λ},

if BAX, then 𝒯μ|B=(𝒯μ|A)μ|B.

Proof.

we have 𝒯μ|A(λ)=𝒯μ|A(λ)={𝒯μ(η):η[μ],η|A=λ}={𝒯μ(η):η[μ],η|A=λ}={𝒯μ(η):η[μ],η|A=λ}={𝒯μ(ξ):ξ[μ],ξ|A=λ}.

we have 𝒯μ|B(λ)={𝒯μ(η):η[μ],η|B=λ}={{𝒯μ(η):η[μ],η|A=ξ}:ξ[μ]|A,ξ|B=λ}={𝒯μ|A(ξ):ξ[μ]|A,ξ|B=λ}=(𝒯μ|A)μ|B.

2. Relative Smooth Continuous Maps

The concept of continuity has been studied by Chang, Ramadan [4, 5] but here we shall study this concept from a different point of view.

Definition 2.1.

Let h: be a linear isomorphism of vector lattices (or an order preserving one-to-one mapping when and are copies of [0,1]) and (X,μ,𝒯μ),(Y,γ,𝒯γ)  μ-STS and γ-STS, respectively. A function f:XY is called (h,μ,γ)-smooth fuzzy continuous if 𝒯μ(f-1(η))𝒯γ(η) for all η𝒯γ, where f-1(η)(x)=(hη)(f(x))μ(x) for all xX. f-1(γ)μ is called the inverse image of γ relative to μ.

Remark 2.2.

When ==[0,1] and μ=χX then the (X,χX,τχX), χX-RST coincides with the fuzzy topological space (X¯,τ) defined by Chang .

Theorem 2.3.

Let ==[0,1] and f:XY be (I,χX,χY)-fuzzy continuous, where I: is the identity function. Then f is continuous in Chang's view.

Proof.

In Remark 2.2 we considered (X,χX,𝒯χX) and (Y,χY,𝒯χY) as fuzzy topological spaces. Now let γ be an open set of smooth topology 𝒯χY. Then 𝒯χX(f-1(γ))=𝒯χX((I-1γf)χX)=𝒯χX(γf)𝒯χX(γ). So f is a fuzzy continuous function.

Theorem 2.4.

Let μX,γ'Y where and are copies of [0,1] and f:XY a (h,μ,γ)-fuzzy continuous functions where f-1(γ)=μ. Then for every γ-closed fuzzy set η,f-1(η) is a μ-closed fuzzy set.

Proof.

Let η be γ-closed set. Then η is a γ-open set and we have (f-1(η))=μ-inf{h-1ηf,μ}=sup{μ-h-1ηf,0}=μ-h-1ηf=μ-f-1(η). Hence 𝒯μ(f-1(η))=𝒯μ(f-1(γ-η))=𝒯μ(h-1(γ-η)fμ)=𝒯μ((h-1γf-h-1ηf)μ)=𝒯μ(f-1(γ)-f-1(η))=𝒯μ(μ-f-1(η))=𝒯μ(f-1(η)). So f-1(η) is a μ-closed fuzzy set.

Theorem 2.5.

Let (Xi,μi,𝒯μi) be relative smooth topological spaces for i=1,2,3. If f:X1X2 and g:X2X3 are relative smooth continuous maps and μ1=f-1(μ2) then so is gf.

Proof.

Using the relative smooth continuity of g and f it follows that 𝒯μ1((gf)-1(η))=𝒯μ1(f-1(g-1(η)))𝒯μ2(g-1(η))𝒯μ3(η). Since for every xX,(gf)-1(η)(x)=η(gf)(x)μ1(x)=η(gf)(x)μ1(x)μ2(f(x))=g-1(η(f(x))μ1(x))=f-1(g-1(η)(x))=(f-1g-1)(η)(x).

Theorem 2.6.

Let (X,μ,𝒯μ) and (Y,ν,𝒯ν) be two relative smooth topological spaces, f:XY a relative smooth continuous map, AX, and f-1(ν)=μ. Then the f|A:(A,μ|A,𝒯μ|A)(Y,ν,𝒯ν) is also relative smooth continuous.

Proof.

For each η[ν],𝒯μ|A((f|A)-1(η))={𝒯μ(λ):λ[μ],λ|A=(f|A)-1(η)}𝒯μ(f-1(η))𝒯ν(η).

3. The Representation of a Relative Smooth Topology

Now we study the representation of a relative smooth topology 𝒯μ.

Let (X,μ,𝒯μ) be a μ-STS, α. Then we define

𝒯μα={λ[μ]:𝒯μ(λ)α}.

Theorem 3.1.

Let (X,μ,𝒯μ) be a μ-STS. Then for every α>0,𝒯μα is a relative topological space. Moreover α1α2 implies 𝒯μα1𝒯μα2.

Proof.

It is clear that χ,μ𝒯μα. When λ,η𝒯μα, we have 𝒯μ(λ)α,𝒯μ(η)α, and so 𝒯μ(λη)𝒯μ(λ)𝒯μ(η)α. This implies that λη𝒯μα. When λj𝒯μα for each jI we have 𝒯μ(λj)𝒯μ(λj)α. Hence λj𝒯μα. So 𝒯μα is a relative topology.

The second part is trivial to verify, since for λ𝒯μα2, 𝒯μα2α1,λ𝒯μα1, so 𝒯μα1𝒯μα2.

Theorem 3.2.

Let 𝒯μα,α(0,1] be a family of μ-fuzzy topology on X such that α1α2 implies 𝒯μα1𝒯μα2. Let τ be the -fuzzy set built by τμ(λ)={α:λ𝒯μα}. Then τμ is a μ-smooth topology.

Proof.

τμ(χ)=τμ(μ)=1 by the definition.

For every λ,η[μ] and α>0 if λ,η𝒯μα then λη𝒯μα. Therefore {α:λη𝒯μα}{α:λ𝒯μα,η𝒯μα} implies that τμ(λη)τμ(λ)τμ(η).

If every λi𝒯μα then λi𝒯μα. Since {α:λi𝒯μα}{α:λi𝒯μα,iI}, then τμ(iIλi)iIτμ(λi).

For τμ being a relative -fuzzy set, with =[0,1], we can state a representation theorem.

Theorem 3.3.

Let 𝒯μ be a relative smooth topology and 𝒯μα the α cut of 𝒯μ. From the families of relative fuzzy topologies 𝒯μα one built 𝒯1μ(λ)={α:λ𝒯μα}. Then 𝒯1μ=𝒯μ.

Proof.

The proof is trivial from the preceding results and the well-known fact that {α:λ𝒯μα}={α:𝒯μ(λ)α}=𝒯μ(λ).

Definition 3.4.

Let τ be a Chang fuzzy topology on X. Then a μ-smooth topology 𝒯μ on X is said to be compatible with τ if τ={λX:𝒯μ(λμ)>0}.

Example 3.5.

Let X be a nonempty set and 𝒯μ:[μ] be a mapping defined by 𝒯μ(μ)=𝒯μ(χ)=1,𝒯μ(λ)=0 for every λ[μ]{χ,μ}.

It is clear that 𝒯μ is the only relative smooth topology on X compatible with the indiscrete fuzzy topology of Chang.

Example 3.6.

Let X be a nonempty set and define a mapping 𝒯μ:[μ] by 𝒯μ(μ)=𝒯μ(χ)=1,𝒯μ(λ)=α for every λ[μ]{χ,μ}.

It is clear that 𝒯μ is a μ-smooth topology on X compatible with the discrete fuzzy topology of Chang.

4. Relative Smooth CompactnessDefinition 4.1.

Let (X,μ,𝒯μ) be a μ-STS. λ[μ],𝒜,[μ].𝒜 is called a relative cover of λ, if 𝒜λ particularly, 𝒜 is called a cover (X,μ,𝒯μ) if 𝒜 is a cover of μ.𝒜 is called a μ-open cover of λ, if 𝒜 is a family of μ-open and 𝒜 is a cover of λ.

For a cover 𝒜 of λ, is called a subcover of λ, if 𝒜 and is still a cover of λ.

Definition 4.2.

Let (X,μ,𝒯μ) be a μ-STS. For every α[0,1), a family 𝒜[μ] is called an α-cover, if for every λ𝒜,𝒯μ(λ)α; 𝒜 is called a μ-open α-cover if 𝒜 is a family of μ-open set and 𝒜 is a α-cover; 𝒜0[μ] is called a sub-α-cover of 𝒜 if 𝒜0𝒜 and 𝒜0 is an α-cover.

Definition 4.3.

Let α[0,1). A μ-STS (X,μ,𝒯μ) is called α-compact if every μ-open α-cover has a finite sub-α-cover.

Theorem 4.4.

Let f:(X,μ,𝒯μ)(Y,ν,𝒯ν) be an onto μ-smooth continuous mapping and f-1(ν)=μ. If (X,μ,𝒯μ) is α-compact then so is (Y,ν,𝒯ν).

Proof.

Let {λi:iI} be a ν-open α-cover of ν. Now consider the family {f-1(λi):iI}, since f is μ-smooth continuous, we have λ𝒯να𝒯να𝒯μ(f-1(λ))αf-1(λ)𝒯μα. It follows that {f-1(λi):iI} is a μ-open α-cover of μ. Since (X,μ,𝒯μ) is α-compact there exists a finite subset I0 of I such that {f-1(λi):iI0} is a μ-open α-cover of (X,μ,𝒯μ). Since f is onto, then {λi:iI0} is a ν-open α-cover of (Y,ν,𝒯ν), which concludes the proof.

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