INTUITIONISTIC FUZZY SEMI δ-PREOPEN SETS AND INTUITIONISTIC FUZZY SEMI δ-PRECONTINUITY

The purpose of this paper is to introduce the concepts of fuzzy semi δ-preopen sets and fuzzy semi δ-precontinuous mappings in intuitionistic fuzzy topological spaces and obtain some of their properties and characterizations.


INTRODUCTION
The concept of fuzzy sets was introduced by Zadeh [26]. Using the concept of fuzzy set Chang [2] introduced the concept of fuzzy topological spaces.
In the present paper, we introduced the concept of intuitionistic fuzzy semi δ-preopen sets and intuitionistic fuzzy semi δ-precontinuous mappings and study some of the basic properties.

PRELIMINARIES
This section contains some basic denitions and preliminary results which will be needed in the sequel.
Definition 2.1. [1] Let X be a nonempty xed set. An intuitionistic fuzzy set A is an object having the form A = {< x, µ A (x), ν A (x) >: x ∈ X} where the functions µ A : X → I and ν A : X → I denote the degree of membership namely µ A (x) and the degree of nonmembership (namely ν A (x)) of each element x ∈ X to the set A, respectively, and 0 ≤ µ A (x) + ν A (x) ≤ 1 for each x ∈ X.
Obviously, every fuzzy set A on a nonempty set X is an intuitionistic fuzzy set having the form For the basic properties of intuitionistic fuzzy set the reader should refer [1,5].
Definition 2.2. [5] Let X and Y be two nonempty sets and f : X → Y be a mapping.
x ∈ X} is an intuitionistic fuzzy sets in X, then the image of A under f denoted and defined by and Definition 2.3. [5] An intuitionistic fuzzy topology on a nonempty set X is a family τ of intuitionistic fuzzy sets in X satisfy the following axioms: In this case the pair (X, τ ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in τ is known as an intuitionistic fuzzy open set in X.

Definition 2.4. [5]
The Complement of A c of an intuitionistic fuzzy open set A is an intuitionistic fuzzy topological space (X, τ ) is called an intuitionistic fuzzy closed set in X.
Definition 2.5. [5] Let (X, τ ) be an intuitionistic fuzzy topological space and let A =< x, µ A (x), ν A (x) > be an intuitionistic fuzzy set in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A are dened by int(A) = ∪{ G | G is an intuitionistic fuzzy open set in X and G ⊆ A}, cl(A) = ∩{ K | K is an intuitionistic fuzzy closed set in X and A ⊆ K}.

Definition 2.24. [25]
The δ-pre closure of an intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, τ ) is the intersection of all intuitionistic fuzzy δ-pre closed sets which contain A and it is denoted by δpcl(A)).
Remark 2.27. [8] Every intuitionistic fuzzy continuous mappings is intuitionistic fuzzy α-continuous, Every intuitionistic fuzzy α-continuous mapping is intuitionistic fuzzy semi continuous and intuitionistic fuzzy pre continuous and every intuitionistic fuzzy semi continuous (resp. intuitionistic fuzzy pre continuous) mapping is intuitionistic fuzzy semi pre continuous. But the converse may not be true.
Remark 2.28. [10] Every intuitionistic fuzzy semi continuous (resp. intuitionistic fuzzy pre continuous) mapping is intuitionistic fuzzy γ-continuous and intuitionistic fuzzy γ-continuous mapping is intuitionistic fuzzy semi pre continuous but the separate converses may not be true.
Remark 2.29. [23] Every intuitionistic fuzzy pre continuous mapping is intuitionistic fuzzy δ-pre continuous but the converse may not be true.
Remark 2.30. [11] The concepts of intuitionistic fuzzy semi continuous and intuitionistic fuzzy pre continuous mappings are independent.

INTUITIONISTIC FUZZY SEMI δ-PREOPEN SETS
In this section, we introduce the concept of intuitionistic fuzzy semi δ-preopen set and study some of their properties in intuitionistic fuzzy topological spaces.
The result now follows from the fact that any union of intuitionistic fuzzy δ-preopen sets is intuitionistic fuzzy δ-preopen.  Lemma 3.14. Let Y be an intuitionistic fuzzy subspace of intuitionistic fuzzy topological space (X, τ ) and A be an intuitionistic fuzzy set in Y .
Theorem 3. 16. Let X and Y be intuitionistic fuzzy topological space, such that X is product related to Y .
. Now the result follows from (a).
Definition 3.17. Let (X, τ ) be an intuitionistic fuzzy topological space and A be an intuitionistic fuzzy set of X. Then the intuitionistic fuzzy semi δ-preinterior (denoted by sδpint) and intuitionistic fuzzy semi δ-preclosure (denoted by sδpcl) of A respectively defined as follows: The following theorem can be easily verified.

INTUITIONISTIC FUZZY SEMI δ-PRECONTINUOUS MAPPINGS
Remark 4.2. Every intuitionistic fuzzy δ-pre continuous (resp. intuitionistic fuzzy semi precontinuous) mappings is intuitionistic fuzzy semi δ-precontinuous but the converse may not be true. Then the mapping f : (X, τ 1 ) → (Y, τ 2 ) defined by f (a) = p, f (b) = q is an intuitionistic fuzzy semi δ-pre continuous but not intuitionistic fuzzy semi precontinuous and the mapping g : (X, τ 1 ) → (Y, τ 3 ) defined by g(a) = p, g(b) = q is an intuitionistic fuzzy semi δ-precontinuous but not intuitionistic fuzzy δ-pre continuous.        (e) ⇒ (c) Let x (α,β) be an intuitionistic fuzzy point of X and A be an intuitionistic fuzzy open set such that f (x (α,β) ) ∈ A. Then A is a neighborhood of f (x (α,β) ). So there is intuitionistic fuzzy semi δ-pre neighborhood U of x (α,β) in X such that x (α,β) ∈ U and f (U ) ⊆ A. Hence there is an intuitionistic fuzzy set O ∈ IF SδP O(X) such that x (α,β) ∈ O ⊆ U and so f (O) ⊆ f (U ) ⊆ A.  Theorem 4.8. Let X i and X * i (i = 1, 2) be an intuitionistic fuzzy topological spaces such that X 1 is product related to X 2 . If f i : X i → X * i (i = 1, 2) is an intuitionistic fuzzy semi δ-pre continuous, then f 1 × f 2 : X 1 × X 2 → X * i × X * 2 is an intuitionistic fuzzy semi δ-pre continuous.