Intuitionistic Fuzzy Proximity Spaces

We introduce the concept of the intuitionistic fuzzy proximity as a generalization of fuzzy proximity, and investigate its properties. Also we investigate the relationship among intu-itionistic fuzzy proximity and fuzzy proximity, and intuitionistic fuzzy topology. 1. Introduction. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was introduced by Atanassov [1]. Recently, Çoker and his colleagues [2, 3, 4] introduced the concept of intuitionistic fuzzy topology which is a generalization of fuzzy topology. Katsaras [5, 6] introduced the concept of fuzzy proximity, and studied the relationship between fuzzy topology and fuzzy proximity. Liu [9] introduced the concept of L-fuzzy proximity for a lattice L, and Liu and Luo [10] studied the relation between L-fuzzy proximity and L-fuzzy uniformity. Also, Khare [7] studied the relationship be


Introduction.
As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was introduced by Atanassov [1].Recently, Çoker and his colleagues [2,3,4] introduced the concept of intuitionistic fuzzy topology which is a generalization of fuzzy topology.
Katsaras [5,6] introduced the concept of fuzzy proximity, and studied the relationship between fuzzy topology and fuzzy proximity.Liu [9] introduced the concept of L-fuzzy proximity for a lattice L, and Liu and Luo [10] studied the relation between L-fuzzy proximity and L-fuzzy uniformity.Also, Khare [7] studied the relationship between classical and fuzzy proximities.
In [8], we studied the relationship between fuzzy topology and intuitionistic fuzzy topology.
In this paper, we introduce the concept of the intuitionistic fuzzy proximity as a generalization of fuzzy proximity, and investigate its properties.Also we investigate the relationship among intuitionistic fuzzy proximity and fuzzy proximity, and intuitionistic fuzzy topology.Moreover, we find an adjunction between intuitionistic fuzzy proximity spaces and fuzzy proximity spaces.

Preliminaries.
In this section, we recall some of the definitions and theorems related to fuzzy proximity and intuitionistic fuzzy topology.
Let X be a nonempty set and I the unit interval [0, 1].An intuitionistic fuzzy set A is an ordered pair where the functions µ A : X → I and γ A : X → I denote the degree of membership and the degree of nonmembership, respectively, and µ A +γ A ≤ 1.Let I(X) denote the set of all intuitionistic fuzzy sets in X.
Obviously every fuzzy set µ A in X is an intuitionistic fuzzy set of the form (µ A , 1−µ A ).
Definition 2.1 [1].Let A = (µ A ,γ A ) and B = (µ B ,γ B ) be intuitionistic fuzzy sets in Let f be a map from a set X to a set Y .Let A = (µ A ,γ A ) be an intuitionistic fuzzy set in X and B = (µ B ,γ B ) an intuitionistic fuzzy set in Y .Then f −1 (B) is an intuitionistic fuzzy set in X defined by and f (A) is an intuitionistic fuzzy set in Y defined by Definition 2.2 [3].An intuitionistic fuzzy topology on X is a familyof intuitionistic fuzzy sets in X which satisfies the following properties: ( The pair (X, -) is called an intuitionistic fuzzy topological space.Any element ofis called an intuitionistic fuzzy open set in X and the complement, an intuitionistic fuzzy closed set.Definition 2.3 [2,3].Let (X, -) be an intuitionistic fuzzy topological space and A an intuitionistic fuzzy set in X.Then the fuzzy closure of A is defined by and the fuzzy interior of A is defined by Theorem 2.4 [3].For any intuitionistic fuzzy set A in an intuitionistic fuzzy topological space (X, -), there exist Theorem 2.5 [2].Let (X, -) be an intuitionistic fuzzy topological space and cl : I(X) → I(X) the fuzzy closure in (X, -).Then for A, B ∈ I(X), the following properties hold: (1) cl(0 Definition 2.6 [3].Let α, β ∈ [0, 1] and α+β ≤ 1.An intuitionistic fuzzy point x (α,β) of X is an intuitionistic fuzzy set in X defined by (2.6) In this case, x is called the support of x (α,β) , α the value of x (α,β) and β the nonvalue of x (α,β) .An intuitionistic fuzzy point x (α,β) is said to belong to an intuitionistic fuzzy set A = (µ A ,γ A ) in X, denoted by x (α,β) ∈ A, if α ≤ µ A (x) and β ≥ γ A (x).

Intuitionistic fuzzy proximity spaces.
We are going to introduce the concept of intuitionistic fuzzy proximity spaces and continuous maps between them.Definition 3.1.An intuitionistic fuzzy proximity on X is a relation δ on I(X) satisfying the following properties: (1) AδB implies BδA; (2) (A ∪ B)δC if and only if AδC or BδC; (3) AδB implies A = 0 ∼ and B = 0 ∼ ; (4) Aδ /B implies that there exists an E ∈ I(X) such that Aδ /E and E c δ /B; (5) A ∩ B = 0 ∼ implies AδB.The pair (X, δ) is called an intuitionistic fuzzy proximity space.
First, we will show thatis an intuitionistic fuzzy topology on X.
Next, we will show that cl Finally, we will show that such ais unique.Suppose - * is an intuitionistic fuzzy topology on Theorem 3.6.Let (X, δ) be an intuitionistic fuzzy proximity space and define a map cl : I(X) → I(X) by for each A ∈ I(X).Then the following properties hold: (2) It is sufficient to show that cl(A)δ /B if and only if Aδ /B.If AδB, then cl(A)δB obviously.Conversely, suppose that Aδ /B and cl(A)δB.Then there exists an E ∈ I(X) such that Bδ /E and E c δ /A.Since cl(A)δB and Eδ /B, cl(A) ⊆ E and hence µ cl(A) ≤ µ E or γ cl(A) ≥ γ E .So there exists an x ∈ X such that Then G ∈ I(X) and G ⊆ E c .If Gδ /A then cl(A) ⊆ G c by the definition of closure and hence . This is a contradiction.Thus GδA.Since G ⊆ E c , AδE c .This is a contradiction to the fact that

If Hδ /A then cl(A) ⊆ H c by the definition of closure and hence γ
. This is a contradiction.Thus we have HδA.
Since H ⊆ E c , AδE c .This is a contradiction.In any case, we have a contradiction.So Aδ /B implies cl(A)δ /B.
(3) It is easy to show that cl(A ∪ B) ⊇ cl(A) ∪ cl(B).On the other hand, suppose ).Then there exists an x ∈ X such that (3.10) This is a contradiction.
Then a < γ cl(A) (x) and hence there exists an > 0 such that a + < γ cl(A) (x).Since γ cl(A) (x) = {µ C (x) | Cδ /A}, there exists an intuitionistic fuzzy set C ∈ I(X) such that Cδ /A and a + < µ C (x).Note that, This is a contradiction.In any case, we have a contradiction.Therefore cl Theorem 3.7.For an intuitionistic fuzzy proximity space (X, δ), the family is an intuitionistic fuzzy topology on X.
Proof.By Theorems 3.5 and 3.6, the proof follows.
Definition 3.8.The topology -(δ) defined in Theorem 3.7 is called the intuitionistic fuzzy topology on X induced by the fuzzy proximity δ.Theorem 3.9.Let (X, δ 1 ) and (Y , δ 2 ) be two intuitionistic fuzzy proximity spaces and ) is continuous with respect to the corresponding intuitionistic fuzzy topologies -(δ 1 ) and -(δ 2 ). (3.17) 4. The δ-neighborhood in the intuitionistic fuzzy proximity.In this section, we will introduce the notion of the δ-neighborhood in the intuitionistic fuzzy proximity.Definition 4.1.Let (X, δ) be an intuitionistic fuzzy proximity space.For A, B ∈ I(X), the intuitionistic fuzzy set B is said to be a δ-neighborhood of A (in symbols Clearly, we know that if A B, then A ⊆ B. Theorem 4.2.Let (X, δ) be an intuitionistic fuzzy proximity space and A, B ∈ I(X).Then the following properties hold: (

Category of intuitionistic fuzzy proximity spaces.
We knew the relationship between fuzzy topological spaces and fuzzy proximity spaces (see [5,6]).The relationship between fuzzy topological spaces and intuitionistic fuzzy topological spaces had been studied in [8].Also we have had the relationship between intuitionistic fuzzy proximity spaces and intuitionistic fuzzy topological spaces in Theorems 3.7 and 3.9.Now, we are going to find a categorical relationship between fuzzy proximity spaces and intuitionistic fuzzy proximity spaces.
Let FProx be the category of all fuzzy proximity spaces and proximity maps and IFProx the category of all intuitionistic fuzzy proximity spaces and continuous maps.
where for A, B ∈ I(X), AδB if and only if Then F is a functor.
Proof.First, we will show that δ is an intuitionistic fuzzy proximity on X.