SOME RESULTS ON THE FUZZY SHEAF OF THE FUNDAMENTAL GROUPS OVER FUZZY TOPOLOGICAL SPACES

Let X be a fuzzy path connected topological space and (H,ψ) be the fuzzy sheaf of fundamental groups over X. Constructing the group of fuzzy sections, it is shown that there is a covariant functor from the category of fuzzy path connected topological spaces and fuzzy continuous mappings to the category of groups of fuzzy sections and homomorphisms. Furthermore, defining the direct sum of the fuzzy sheaves, it is proved that the mapping Pi = (pi, p ∗ i ) : (X1×X2, H1×H2)→ (Xi,Hi) is a homomorphism for i = 1, 2.


INTRODUCTION
The concept of a fuzzy set was discovered by Zadeh [5] and one of its earliest branches, the theory of fuzzy topology, was developed by Chang [1] and others.Recently, Zheng [6] introduced the concept of fuzzy path.Using this concept, Salleh and Md Tap [3] constructed the fundamental group of a fuzzy topological space.
Let X be a set and I the unit interval [0, 1].A fuzzy set X is characterized by a membership function µ A which associates with each point x ∈ X its"grade of membership"µ A (x) ∈ I. Definition 1.1.A fuzzy point in X is a fuzzy set with membership function µ a λ defined by µ a λ (x) = ½ λ, x = a 0, otherwise for all x ∈ X.Where 0 < λ ≤ 1.
We denoted by k λ the fuzzy set in X with the constant membership function µ k λ (x) = λ for all x ∈ X .Definition 1.2.A fuzzy topology on a set X is a family τ of fuzzy sets in X which satisfies the following conditions: The pair (X, τ ) is called a fuzzy topological space.Every member of τ be called an open fuzzy set.The complement of an open fuzzy set is called a closed fuzzy set.
Definition 1.3.Let (X, τ 1 ), (Y, τ 2 ) be two fuzzy topological spaces.A mapping f of (X, Definition 1.4.A bijective mapping f of fuzzy topological space (X, τ 1 ) into a fuzzy topological space (Y, τ 2 ) is called a fuzzy topological mapping if it is fuzzy continuous and fuzzy open.Definition 1.5.Let τ be a fuzzy topology on a set X. A subfamily B of τ is called a base for τ if each member of τ can be expressed as the union of members of B [4] .Definition 1.6.Let (X, τ ) be a fuzzy topological space.If α : (I, e ε I ) → (X, τ ) is a fuzzy continuous function and a fuzzy set A is connected in (I, e ε I ) with A(0) > 0 and A(1) > 0, then the fuzzy set α(A) in (X, τ ) is called a fuzzy path in (X, τ ).
The fuzzy point (α(0)) A(0) = α(0 A(0) ) and (α(1)) A(1) = α(1 A(1) ) are called the initial point and the terminal point of the fuzzy path α(A), respectively [2] .Definition 1.7.Let F be a fuzzy set in a fuzzy topological space (X, τ ).If for any two fuzzy points a λ and b µ in F , there is a fuzzy path from a λ to b µ contained in F , then F is said to be fuzzy path connected in (X, τ ).
If F = X in the above definition, we call (X, τ ) a fuzzy path connected space [3] .
Let X be a fuzzy path connected topological space and H a λ be the fundamental group of X based for any a λ ∈ X, that is H a λ = π 1 (X, a λ ) [4] .Let X = (X, x p ) be a pointed fuzzy topological space for an arbitrary fixed fuzzy point x p ∈ X.Let H denotes the disjoint union of all fundamental groups obtained for each a λ ∈ X, by H is a set over X and the mapping ψ : H → X defined by If B is a fuzzy base for X, then B * = {s(W ) : W ∈ B, s ∈ Γ(W, H)} is a fuzzy base for H.The mappings ψ and s are fuzzy continuous in this topology.Moreover ψ is a locally fuzzy topological mapping.Then (H, ψ) is a fuzzy sheaf over X. (H, ψ) or only H is called "The fuzzy sheaf of the fundamental groups" over X.
The fundamental group π 1 (X, a λ ) = ψ −1 (a λ ) is called the stalk of the fuzzy sheaf (H, ψ) over X and denoted by H a λ for every a λ ∈ X.For any open fuzzy set W ⊂ X, an element s of Γ(W, H) is called a fuzzy section of the fuzzy sheaf H over W .

SOME RESULTS ON THE FUZZY SHEAF OF THE FUNDAMENTAL GROUPS OVER FUZZY TOPOLOGICAL SPACES
In this section,constructing the group of fuzzy sections, it is shown that there is a covariant functor from the category of fuzzy path connected topological spaces and fuzzy continuous mappings to the category of groups of fuzzy sections and homomorphisms.
We begin by giving the following theorem.
Theorem 2.1.The set Γ(W, H) is a group with the pointwise operation of multiplication. Then, It follows from this definition that the operation of multiplication is well-defined and closed.Clearly, the operation of multiplication is associative.The mapping I : W → H is the identity element which is obtained by the identity element of π 1 (X, x p ).The mapping I is defined by Thus, the set of fuzzy sections Γ(W, H) is a group with the pointwise operation of multiplication.Definition 2.2.Let f * : H 1 → H 2 be a mapping.If f * is a fuzzy continuous, a homomorphism on each stalk of fuzzy sheaf H 1 and maps every stalk of fuzzy sheaf H 1 into a stalk of fuzzy sheaf H 2 , then it is called a fuzzy sheaf homomorphism.
Let f : (X 1 , x p ) → (X 2 , f(x p ) = y q ) be a fuzzy continuous mapping.We know that the mapping f * : H 1 → H 2 is a fuzzy sheaf homomorphism.Also, each element between (H 1 ) x p and (H 2 ) y q gives the correspondence s 1 ←→ s 2 between Γ(X 1 , H 1 ) and Γ(X 2 , H 2 ).If we denote this correspondence as Therefore, ). Hence we have the following theorem.
Theorem 2.3.Let f : X 1 → X 2 be a fuzzy continuous mapping.Then there exists a homomorphism f * : Γ(X 1 , H 1 ) → Γ(X 2 , H 2 ).Theorem 2.4.Let f 1 : X 1 → X 2 , f 2 : X 2 → X 3 be fuzzy continuous mappings.Then, there exists a homomorphism Proof.Since the mappings f 1 , f 2 are fuzzy continuous, the mapping f = f 2 • f 1 is fuzzy continuous.Therefore, there exists a homomorphism In fact, for any s 1 ∈ Γ(X 1 , H 1 ) and a λ ∈ X. ¤ Let C be the category of fuzzy path connected topological spaces and fuzzy continuous mappings and D be the category of fuzzy section groups and homomorphisms.Let us define a mapping F : C → D with F (X) = Γ(X, H) and F (f ) = f * for any element X ∈ C and morphism f : X 1 → X 2 .Then F is a covariant functor.In fact, the followings are satisfied. 1 Moreover, Hence, ).We then have the following theorem.
Theorem 2.5.There is a covariant functor from the category of fuzzy path connected topological spaces and fuzzy continuous mappings to the category of groups of fuzzy sections and homomorphisms.Now, let f : X 1 → X 2 be a fuzzy topological mapping.Then there is the mapping So, we can give the following corollary.
Now, let H 1 and H 2 be fuzzy sheaves of fundamental groups over X 1 , X 2 , respectively.Then, and so H 1 × H 2 is a fuzzy set over the fuzzy topological space X 1 × X 2 .Moreover, H 1 × H 2 is also a fuzzy topological space, since H 1 , H 2 are fuzzy topological spaces.
Let us now define a mapping as follows: We assert that Φ = (ψ 1 , ψ 2 ) is a locally fuzzy topological mapping.In fact, since the mappings ψ 1 : H 1 → X 1 , ψ 2 : H 2 → X 2 are locally fuzzy topological mappings, there exist open fuzzy neighborhoods U 1 ((σ 1 ) are fuzzy topological mappings.Therefore are open fuzzy neighborhoods of the fuzzy points ((σ 1 ) a λ 1 , (σ 2 ) a λ 2 ) and (a λ 1 , a λ 2 ), respectively.Finally, it is clearly seen that Φ| U : U → W is a fuzzy topological mapping.Thus, Definition 2.6.Let H 1 and H 2 be fuzzy sheaves of fundamental groups over X 1 and X 2 , respectively.Then the fuzzy sheaf H 1 × H 2 is called the direct sum of the fuzzy sheaves H 1 and H 2 .
Definition 2.7.Let the pairs (X 1 , H 1 ) and (X 2 , H 2 ) be given.It is said that there is a homomorphism between these pairs and it is written f * preserves the stalks with respect to f , (4) For every a ) is a homomorphism.Now, we can give the following theorem.
Theorem 2.8.Let the pairs (X 1 , H 1 ) and (X 2 , H 2 ) be given.Then the mapping Proof.Let us first show that p * i is a stalk preserving mapping with respect to p i .In fact, The mappings p * i : H 1 × H 2 → H i , p i : X 1 × X 2 → X i are fuzzy continuous, since they are fuzzy projections.
Thus P i is a homomorphism between the pairs (X 1 × X 2 , H 1 × H 2 ) and (X i , H i ), i = 1, 2.