Relative Smooth Topological Spaces
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.
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 .
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  and Molaei . 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 , and if we take and then the STS coincides with the known definition of fuzzy topological space defined by Chang . 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
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 .
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
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.
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