#### Abstract

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 be a nonempty set and let be two lattice which will be copies of or . The family of all fuzzy sets on will be denoted by 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 see Anvari and Molaei [2] and Molaei [3]. We assume that as a function from to is an observer of on lattice and denote where implies that for all .

*Definition 1.1. *Let . A relative smooth topological space or smooth topological space or STS for short is a triple , where is a mapping satisfying the following properties:(i), where is the characteristic function;(ii)if , , then , where is the minimum operator in ;(iii). We call a smooth topology from view point of or a smooth topology or a fuzzy family of open sets on .

*Remark 1.2. *If then the STS coincides with the smooth topological space defined by Ramadan [4], and if we take and then the STS coincides with the known definition of fuzzy topological space defined by Chang [5]. If , and then is a classical topology.

*Definition 1.3. *Let . A smooth cotopological space is a triple , where is a mapping satisfying the following properties:(i);(ii)if , , then ;(iii). We call a smooth co-topology or a fuzzy family of closed sets on .

Theorem 1.4. *Let be a STS and be a mapping defined by , where . Then is a fuzzy family of closed sets. *

*Proof. *(i)It is clear.(ii)It flows from
So, (iii)It flows from
So,

Theorem 1.5. *Let be a fuzzy family of closed sets and define by . Then is a STS on . *

*Proof. *The proof is similar to the previous theorem.

Corollary 1.6. *Let be a STS and a fuzzy family of closed sets. Then and . *

*Proof. *Suppose then we have and

*Example 1.7. *Let be the set of all differentiable real-valued functions on with positive derivative of order one and let be the set of real-valued functions defined on . Let be defined by where is the exponential function. For nonnegative integer define by
If we take and define by for Then is a STS. Since where and whenever tends to , so
and for we find

*Definition 1.8. *Let and be two smooth topological spaces on . We say that is finer than or is coarser than and denoted by if for every .

Theorem 1.9. *Let be a family of STS on . Then is also STS on , where
*

*Proof. *(i)It is clear.(ii)For every ,
(iii)For ,

Let be a subset of and . The restriction of on is denoted by .

Theorem 1.10. *Let be a STS and . Define a mapping by . Then is a STS on .*

*Proof. *(i)It is clear that . (ii). (iii) So

*Definition 1.11. *The STS is called a subspace of and is called the induced STS on from .

Theorem 1.12. *Let be a smooth subspace of and . Then*(a)(b)*if , then *

*Proof. *(a)we have
(b)we have

#### 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 be a linear isomorphism of vector lattices (or an order preserving one-to-one mapping when and are copies of ) and STS and STS, respectively. A function is called smooth fuzzy continuous if for all , where for all . is called the inverse image of relative to .

*Remark 2.2. *When and then the , RST coincides with the fuzzy topological space defined by Chang [5–7].

Theorem 2.3. *Let and be -fuzzy continuous, where is the identity function. Then is continuous in Chang's view.*

*Proof. *In Remark 2.2 we considered and as fuzzy topological spaces. Now let be an open set of smooth topology . Then
So is a fuzzy continuous function.

Theorem 2.4. *Let where and are copies of and a fuzzy continuous functions where . Then for every closed fuzzy set is a closed fuzzy set.*

*Proof. *Let be closed set. Then is a open set and we have
Hence
So is a closed fuzzy set.

Theorem 2.5. *Let be relative smooth topological spaces for . If and are relative smooth continuous maps and then so is .*

*Proof. *Using the relative smooth continuity of and it follows that
Since for every

Theorem 2.6. *Let and be two relative smooth topological spaces, a relative smooth continuous map, and . Then the is also relative smooth continuous.*

*Proof. *For each

#### 3. The Representation of a Relative Smooth Topology

Now we study the representation of a relative smooth topology .

Let be a STS, . Then we define

Theorem 3.1. *Let be a STS. Then for every is a relative topological space. Moreover implies *

*Proof. *It is clear that . When , we have
and so
This implies that When for each we have
Hence So is a relative topology.

The second part is trivial to verify, since for , , so .

Theorem 3.2. *Let be a family of fuzzy topology on such that implies . Let be the fuzzy set built by Then is a smooth topology.*

*Proof. *(a) by the definition.(b)For every and if then . Therefore
implies that
(c)If every then . Since
then

For being a relative fuzzy set, with we can state a representation theorem.

Theorem 3.3. *Let be a relative smooth topology and the cut of . From the families of relative fuzzy topologies one built . Then .*

*Proof. *The proof is trivial from the preceding results and the well-known fact that

*Definition 3.4. *Let be a Chang fuzzy topology on . Then a smooth topology on is said to be compatible with if .

*Example 3.5. *Let be a nonempty set and be a mapping defined by for every .

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

*Example 3.6. *Let be a nonempty set and define a mapping by for every .

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

#### 4. Relative Smooth Compactness

*Definition 4.1. *Let be a STS. is called a relative cover of , if particularly, is called a cover if is a cover of is called a open cover of , if is a family of open and is a cover of .

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

*Definition 4.2. *Let be a STS. For every , a family is called an cover, if for every ; is called a open cover if is a family of open set and is a cover; is called a sub-cover of if and is an cover.

*Definition 4.3. *Let . A STS is called compact if every open cover has a finite sub-cover.

Theorem 4.4. *Let be an onto smooth continuous mapping and . If is compact then so is .*

*Proof. *Let be a open cover of . Now consider the family , since is smooth continuous, we have
It follows that is a open cover of . Since is compact there exists a finite subset of such that is a open cover of . Since is onto, then is a open cover of which concludes the proof.