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 ℒ,ℒ' 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 ℒX Zadeh [1].

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 [2] and Molaei [3]. We assume that μ as a function from X to ℒ is an observer of X on lattice ℒ and denote [μ]={λ∈ℒX:λ⊆μ}, where λ⊆μ implies that λ(x)≤μ(x) for all x∈X.

Definition 1.1.

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

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

if λ1, λ2∈[μ], then 𝒯μ(λ1∩λ2)≥𝒯μ(λ1)⋀𝒯μ(λ2), where ⋀ is the minimum operator in ℒ′;

𝒯μ(∪{λi:i∈I})≥⋀{𝒯μ(λi):i∈I}.

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 [4], and if we take ℒ=[0,1],ℒ′={0,1}, and μ=χX then the χX-STS coincides with the known definition of fuzzy topological space (X¯,τ) defined by Chang [5]. If ℒ=ℒ′={0,1}, and μ=χX then 𝒯χX is a classical topology.

Definition 1.3.

Let μ∈ℒX. A μ-smooth cotopological space is a triple (X,μ,ℱμ), where ℱμ:[μ]→ℒ′ is a mapping satisfying the following properties:

ℱμ(μ)=ℱμ(χϕ)=1;

if η1, η2∈[μ], then ℱμ(η1∪η2)≥ℱμ(η1)⋀ℱμ(η2);

ℱμ(∩{ηi:i∈I})≥⋀{ℱμ(ηi):i∈I}.

We call ℱμ 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 ℱμ:[μ]→ℒ′ be a mapping defined by ℱμ(η)=𝒯μ(η′), where η′=μ-η. Then ℱμ is a fuzzy family of μ-closed sets.

Proof.

It is clear.

It flows from
(η1∪η2)′=μ-(η1∪η2)=μ-sup{η1,η2}=inf{μ-η1,μ-η2}=η1′∩η2′.
So, ℱμ(η1∪η2)=𝒯μ((η1∪η2)′)=𝒯μ(η1′∩η2′).

It flows from
(∩i∈Iηi)′=μ-∩i∈Iηi=μ-inf{ηi:i∈I}=sup{μ-ηi:i∈I}=∪i∈Iηi′.
So, ℱμ(∩{ηi:i∈I})=𝒯μ(∪{ηi′:i∈I}).

Theorem 1.5.

Let ℱμ be a fuzzy family of μ-closed sets and define 𝒯ℱμ:[μ]→ℒ′ by 𝒯ℱμ(η)=ℱμ(η′). Then 𝒯ℱμ 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 ℱμ a fuzzy family of μ-closed sets. Then 𝒯ℱ𝒯μ=𝒯μ and ℱ𝒯ℱμ=ℱμ.

Proof.

Suppose λ,η∈[μ] then we have 𝒯ℱ𝒯μ(λ)=ℱ𝒯μ(λ′)=𝒯μ(λ) and ℱ𝒯ℱμ(η)=𝒯ℱμ(η′)=ℱμ(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 ℒ be the set of real-valued functions defined on (1,∞). Let μ:X→ℒ be defined by μ(f)=f′+Exp, where Exp is the exponential function. For nonnegative integer n define λn:X→ℒ by
(λn(f))(x)=f′(x)+∑i=1nxi-1(i-1)!.
If we take ℒ′=[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 λn∩λm=λm,λn∪λm=λn, where n>m and ∪λni=μ whenever ni tends to +∞, so
𝒯μ(λn∩λm)=𝒯μ(λm)≥𝒯μ(λn)⋀𝒯μ(λm),
and for I⊆ℕ we find
𝒯μ(∪{λn:n∈I})≥⋀{𝒯μ(λn):n∈I}.

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 𝒯μ1≻𝒯μ2 if 𝒯μ1(λ)≥𝒯μ2(λ) for every λ∈[μ].

Theorem 1.9.

Let {𝒯μi:i∈I} be a family of μ-STS on X. Then 𝒯μ=⋀i∈I𝒯μi is also μ-STS on X, where
(⋀i∈I𝒯μi)(λ)=⋀i∈I𝒯μi(λ).

Proof.

It is clear.

For every λ,η∈[μ],
𝒯μ(λ∩η)=⋀i∈I𝒯μi(λ∩η)≥⋀i∈I(𝒯μi(λ)⋀𝒯μi(η))=⋀i∈I(𝒯μi(λ))⋀(⋀i∈I𝒯μi(λ))=𝒯μ(λ)⋀𝒯μ(η).

For Γ,
𝒯μ(∪{λj:j∈Γ})=⋀i∈I𝒯μi(∪{λj:j∈Γ})≥⋀i∈I⋀{𝒯μi(λj):j∈Γ}=⋀(⋀i∈I{𝒯μi(λj):j∈Γ})=⋀{𝒯μ(λj):j∈Γ}.

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 A⊂X. Define a mapping 𝒯μ|A:[μ]→ℒ′ by 𝒯μ|A(λ)=∨{𝒯μ(η):η∈[μ],η|A=λ}. Then 𝒯μ|A is a μ-STS on A.

∀i∈I,𝒯μ|A{λi:i∈I}=∨{𝒯μ(ηi):ηi∈[μ],ηi|A=λi}. So
𝒯μ|A(∪{λi:i∈I})=∨{𝒯μ(∪{ηi:i∈I}):ηi∈[μ],∪ηi|A=∪λi}≤∨{⋀{𝒯μ(ηi):i∈I}:ηi∈[μ],∪ηi|A=∪λi}=⋀𝒯μ|A{λi:i∈I}.

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 λ∈ℒA,λ∈[μ]. Then

ℱ𝒯μ|A(λ)=∨{ℱ𝒯μ(η):η∈[μ],η|A=λ},

if B⊂A⊂X, then 𝒯μ|B=(𝒯μ|A)μ|B.

Proof.

we have
ℱ𝒯μ|A(λ)=𝒯μ|A(λ′)=∨{𝒯μ(η):η∈[μ],η|A=λ′}=∨{𝒯μ(η):η′∈[μ],η′|A=λ}=∨{ℱ𝒯μ(η′):η′∈[μ],η′|A=λ}=∨{ℱ𝒯μ(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:ℒ→ℒ′ be a linear isomorphism of vector lattices (or an order preserving one-to-one mapping when ℒ and ℒ′ are copies of [0,1]) and (X,μ,𝒯μ),(Y,γ,𝒯γ)μ-STS and γ-STS, respectively. A function f:X→Y is called (h,μ,γ)-smooth fuzzy continuous if 𝒯μ(f-1(η))≥𝒯γ(η) for all η∈𝒯γ, where f-1(η)(x)=(h∘η)(f(x))∩μ(x) for all x∈X. f-1(γ)∩μ is called the inverse image of γ relative to μ.

Remark 2.2.

When ℒ=ℒ′=[0,1] and μ=χX then the (X,χX,τχX), χX-RST coincides with the fuzzy topological space (X¯,τ) defined by Chang [5–7].

Theorem 2.3.

Let ℒ=ℒ′=[0,1] and f:X→Y be (I,χX,χY)-fuzzy continuous, where I:ℒ→ℒ′ 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-1∘γ∘f)∩χX)=𝒯χX(γ∘f)≥𝒯χX(γ).
So f is a fuzzy continuous function.

Theorem 2.4.

Let μ∈ℒX,γ∈ℒ'Y where ℒ and ℒ′ are copies of [0,1] and f:X→Y 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-1∘η∘f,μ}=sup{μ-h-1∘η∘f,0}=μ-h-1∘η∘f=μ-f-1(η).
Hence
𝒯μ(f-1(η′))=𝒯μ(f-1(γ-η))=𝒯μ(h-1∘(γ-η)∘f∩μ)=𝒯μ((h-1∘γ∘f-h-1∘η∘f)∩μ)=𝒯μ(f-1(γ)-f-1(η))=𝒯μ(μ-f-1(η))=ℱ𝒯μ(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:X1→X2 and g:X2→X3 are relative smooth continuous maps and μ1=f-1(μ2) then so is g∘f.

Proof.

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

Theorem 2.6.

Let (X,μ,𝒯μ) and (Y,ν,𝒯ν) be two relative smooth topological spaces, f:X→Y a relative smooth continuous map, A⊂X, 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∈ℒ′. 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 α1≤α2 implies 𝒯μα1≥𝒯μα2.

Proof.

It is clear that χ∅,μ∈𝒯μα. When λ,η∈𝒯μα, we have
𝒯μ(λ)≥α,𝒯μ(η)≥α,
and so
𝒯μ(λ∩η)≥𝒯μ(λ)⋀𝒯μ(η)≥α.
This implies that λ∩η∈𝒯μα. When λj∈𝒯μα for each j∈I we have
𝒯μ(∪λj)≥⋀𝒯μ(λj)≥α.
Hence ∪λj∈𝒯μα. So 𝒯μα is a relative topology.

The second part is trivial to verify, since for λ∈𝒯μα2, 𝒯μ≥α2≥α1,λ∈𝒯μα1, so 𝒯μα1≥𝒯μα2.

Theorem 3.2.

Let 𝒯μα,α∈(0,1] be a family of μ-fuzzy topology on X such that α1≥α2 implies 𝒯μα1≤𝒯μα2. Let τ be the ℒ′-fuzzy set built by τμ(λ)=∨{α:λ∈𝒯μα}. 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,η∈𝒯μα then λ∩η∈𝒯μα. Therefore
{α:λ∩η∈𝒯μα}⊃{α:λ∈𝒯μα,η∈𝒯μα}
implies that
τμ(λ∩η)≥τμ(λ)⋀τμ(η).

If every λi∈𝒯μα then ∪λi∈𝒯μα. Since
{α:∪λi∈𝒯μα}⊃{α:∪λi∈𝒯μα,∀i∈I},
then
τμ(∪i∈Iλi)≥⋀i∈Iτμ(λi).

For τμ being a relative ℒ′-fuzzy set, with ℒ′=[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:λ∈𝒯μα}. Then 𝒯1μ=𝒯μ.

Proof.

The proof is trivial from the preceding results and the well-known fact that
∨{α:λ∈𝒯μα}=∨{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={λ∈ℒX:𝒯μ(λ∩μ)>0}.

Example 3.5.

Let X be a nonempty set and 𝒯μ:[μ]→ℒ′ 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]→ℒ′ 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,ℬ⊂[μ].𝒜 is called a relative cover of λ, if ∨𝒜≥λ 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,ℬ is called a subcover of λ, if ℬ⊂𝒜 and ℬ 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 λ∈𝒜,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 𝒜0⊂𝒜 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:i∈I} be a ν-open α-cover of ν. Now consider the family {f-1(λi):i∈I}, since f is μ-smooth continuous, we have
λ∈𝒯να⇔𝒯ν≥α⇔𝒯μ(f-1(λ))≥α⇔f-1(λ)∈𝒯μα.
It follows that {f-1(λi):i∈I} 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):i∈I0} is a μ-open α-cover of (X,μ,𝒯μ). Since f is onto, then {λi:i∈I0} is a ν-open α-cover of (Y,ν,𝒯ν), which concludes the proof.

ZadehL. A.Fuzzy setsAnvariM. H.MolaeiM. R.Genomic from the viewpoint of observational modelingMolaeiM. R.Relative semi-dynamical systemsRamadanA. A.Smooth topological spacesChangC. L.Fuzzy topological spacesLowenlR.Fuzzy topological spaces and fuzzy compactnessYing-MingL.Mao-KangL.