Smooth Neighborhood Structures in a Smooth Topological Spaces

Abstract: Problem Statement: Various concepts related to a smooth topological sp ces have been introduced and relations among them studied by seve ral authors (Chattopadhyay, Ramadan, etc). Conclusion/Recommendations: In this study, we presented the notions of three so rts of neighborhood structures of a smooth topological spaces and give some of their properties which are results by Ying extended to smooth topological spaces.


INTRODUCTION
introduced the fuzzy topology as an extension of Chang (1968) fuzzy topology. It has been developed in many directions (Ramadan, 1992;Chattopadhyay and Samanta, 1993;EL Gayyar et al., 1994;Höhle and Rodabaugh, 1998;Kubiak and Šostak, 1997;Demirici, 1997;Ramadan et al., 2001;2009;Abdel-Sattar, 2006). Ying (1994) studied the theory of neighborhood systems in fuzzy topology with the method used to develop fuzzifying topology (Ying, 1991) by treating the membership relation as a fuzzy relation. In this study, we generate the structures of neighborhood systems in a smooth topology with the method used in (Ying, 1991), by using fuzzy sets and fuzzy points.

Notions and preliminaries:
The class of all fuzzy sets on a universal set X will be denote by L X , where L is the special lattice and L = ([0,1], ≤). Also, L 0 = (0,1] and L 1 = [0, 1).
Definition 1: Pu and Liu (1980) a fuzzy set in X is called a fuzzy point iff it takes the value 0 for all y ∈ X, except one, say x ∈ X. If its value at x is λ (0 < λ≤1) we denote this fuzzy point by x λ , where the point x is called its support. The fuzzy point is said to be contained in a fuzzy set A, or belong to A, denoted by x λ ∈ A, iff λ≤A(x). Evidently, every fuzzy set A can be expressed as the union of all fuzzy points which belong to A.
Definition 2: Ying (1991) Let X be a non-empty set. Let x λ be a fuzzy point in X and let A be a fuzzy subset of X. Then the degree to which x λ belongs to A is: Obviously, we have the following properties: Definition 3: Ying (1991) let (X,τ) be a fuzzy topological space (fts, for short), let e be a fuzzy point in X and let A be a fuzzy subset of X. Then the degree to which A is a neighborhood of e is defined by: is called the fuzzy neighborhood system of e in (X, τ). Ying (1991) let (X, τ) be a fts, e a fuzzy point in X and A a fuzzy subset of X.

Definition 4:
Then the degree to which e is an adherent point of A is given as: where, A c is the complement of A.
Definition 5: Ramadan (1992) A smooth topological space (sts, for short) is an ordered pair (X, τ), where X is a non-empty set and τ: L X → L is a mapping satisfying the following properties: Gayyar et al. (1994) let (X,τ) be a sts and α∈L o . Then the family: which is clearly a fuzzy topology Chang (1968) sense.
Definition 7: Demirici (1997) Let (X,τ) be a sts and A∈L X . Then the τ-smooth interior of A, denoted by: Remark 1: Demirici (1997) let τ be a Chang's fuzzy topology (CFT, for short) on the non-empty set X. Then the smooth topology and smooth cotopology τ s , τ s * : L X →L, defined by: Smooth neighborhood systems of a fuzzy set: Here, we build a smooth neighborhood systems of a fuzzy set in a sts and we give some of its properties.
For a mapping M: L X → L LX and A∈ L X , α∈ [0; 1); let us define the family M A α = {B ∈ L X : M A (B) > α}; which will play an important role in this part. where, τ α = {A∈L X : τ(A) > α} the strong αlevel of τ.

Remark 2:
• The real number N A (B) is called the degree of nbdness of the fuzzy set B to the fuzzy set A. If the smooth nbd system of a fuzzy set A has the following property: N A (L X ) ⊆ {0, 1}, then N A is called the fuzzy nbd system of A • We say that the family (N A ) α = {B: N A (B)>α} is a fuzzy nbd system of A for each α∈[0,1) and (N A ) α is called the strong α -level fuzzy nbd of A Proposition 1: Let (X, τ) be a sts and A ∈ L X . Then a mapping N A : L X → L LX is the smooth nbd system of A with respect to the st τ iff: (1) Suppose that the mapping N A : L X → L LX is the smooth nbd systems of A with respect to the st τ. Consider the following two cases: • For the case A⊄B, suppose that N A⊆C ⊆B} = λ > 0, then ∃C∈L X such that τ (C) >0 and A⊆C⊆B: We obtain N A (B) > 0, a contradiction Therefore: the mapping: N A : L X → L LX is a smooth nbd system of A, ∃C∈L X such that C∈τ λ−ε and A⊆C⊆B, i.e., sup{τ(C): A⊆C⊆B}>λ-ε. Since ε> 0 is arbitrary we have: On the other hand, let sup{τ (C): A⊆C⊆B}=γ >0. Then for every 0<ε≤γ, ∃C∈ L X such that τ(C)>γ-ε and A⊆C⊆B. Therefore B∈N A γ-ε , i.e., N A (B)>γ-ε. Since ε is an arbitrary we have: Hence the inequality follows: (2) For α∈ [0, 1), let B ∈ A N α , i.e., N A (B)>α: Then we can write α < N A (B)= sup{τ (C): A⊆C⊆B},i.e., ∃C∈ L X such that τ(C)>α, A ⊆ C ⊆ B. Then we have: By the same way we can show that: Remark 3: In Proposition 3, the fuzzy subsets A of X can be replaced by the fuzzy points on X, that is, by the special fuzzy subsets e, in this case: Proposition 2: Let (X,τ) be a sts and A∈L X . If the mapping X X L A N : L L → is the smooth nbd system of A with respect to the st τ, then the following properties hold: Proof: (N1) and (N2) follows directly from Definition 1 and Proposition 3. (N3) Suppose that N A (B 1 ) = α 1 >0 and N A (B 2 )>α 2 >0. Then for a fixed ε>0 such that: ε≤ Definition 1, it is clear that there exists C 1 , C 2 ∈L X such that: A⊆C⊆B}. From Proposition 3, we obtain τ(C)≤ N A (C) and τ(C)≤ N C (B).
Thus, sup{τ (C): A⊆C⊆B}≤ sup{N A (C)∧N C (B)}. Hence: Smooth neighborhood systems of a fuzzy points: Definition 9: Let (X,τ) be a sts, e a fuzzy point in X and A be a fuzzy subset of X. Then the degree to which A is a NBD of e is defined by: is called the smooth NBD system of e in (X, τ).
Definition 10: Let (X,τ) be a sts, e a fuzzy point in X and A a fuzzy subset of X. Then the degree to which e is an adherent point of A is given as: Fuzzy smooth r-neighborhood: Definition 11: Let (X,τ) be a sts, A∈L X , e a fuzzy point in X and r∈L 0 . Then the degree to which A is a fuzzy smooth r-nbd system of e is defined by: A mapping N e : L X × L 0 →L is called the fuzzy smooth r-nbd system of e.
Theorem 2: Let (X,τ) be a sts and N e the fuzzy smooth r-nbd system of e. For A, B∈L X and r, s∈L 0 , it satisfies the following properties: Since N e (A,r)>t and N e (B,r)>t, there exist C 1 ,C 2 ∈L X with: Such that: (1) We will show that τ N (B 1 ∩ B 2 )≥τ N (B 1 )∧ τ N (B 2 ), for any B 1 , B 2 ∈ L X .
Suppose there exists B = ∪ i∈Γ B i ∈L X and r 0 ∈L 0 such that: Thus, N e (∪ i ∈ Γ B i , r) = m(e, ∪ i ∈ Γ B i ), i.e., τ N (∪ i ∈ Γ B i ) ≥ r 0 . It is a contradiction for the Eq. III.
(2) Suppose there exists A∈L X such that: