Intuitionistic fuzzy digital CS -filtered structured spaces

: The motive of this article is to introduce a new version of Intuitionistic fuzzy digital CS -filtered structure spaces and Intuitionistic fuzzy digital Hausdorff CS -filtered structure spaces in the Euclidean plane. Also we defined and studied about D * structure saturated sets, D * compact structure spaces and D * structure filtered family of sets. Moreover some of the properties are exhibited related to the above said spaces.


Introduction
In 1965, L. A. Zadeh [10] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty in real physical world. In 1983, K. T. Atanassov [2] published the concept of intuitionistic fuzzy sets and many works by him and his colleagues appeared in the literature [1,3].
Classical digital topology primarily concerns itself in the study of black white images in the digital plane [4,7]. A. Rosenfeld [9] represented the gray scale level images by the concept of fuzzy sets.
The two characteristic functions, namely, the membership and the non-membership functions, are used to define an intuitionistic fuzzy set (IFS) which describe, respectively, the belongingness or non-belongingness of an element. Because of this nature, the brighter and "non-brighter" parts of a digital image can be analyzed efficiently. In image processing, it is proved that the results using IFS is better than the fuzzy set theory. This paper introduces the concepts of intuitionistic fuzzy digital CS-filtered structure spaces and intuitionistic fuzzy digital Hausdorff CS-filtered structure spaces in the Euclidean plane and discusses some of its properties.

Preliminaries
To understand the theme of this paper, some definitions and results are recalled in this section. Throughout this paper E denotes the Euclidean plane and J denotes the index set.

Definition 2.1 [8]:
Let Σ be a rectangular array of integer-coordinate points or lattice points in the Euclidean plane. Thus the point P = (x, y) of Σ has four horizontal and vertical neighbors, namely (x ± 1, y) and (x, y ± 1) and it also has four diagonal neighbors, namely (x ± 1, y ± 1) and (x ± 1, y ∓ 1). We say that former points are 4-adjacent to, or 4-neighbors of P and we say that both types of neighbors are 8-adjacent to, or 8-neighbors of P. Note that if P is on the border of Σ, some of these neighbors may not exist.  G1~ ∩ G2~ ∈ D * for any G1~, G2~ ∈ D * ; iii. ∪ Gi~ ∈ D * for arbitrary family {Gi~ | i ∈ J} ⊆ D * .
Then the ordered pair (E, D * ) is called an intuitionistic fuzzy digital structure space or D * structure space. Each element of D * structure space is said to be a D * open set in E. The complement of a D * open set is said to be a D * closed set in E.

Definition 2.5 [6]:
A directed family of a set S is a family (Di), i ∈ I of subsets of S, such that for every i, j ∈ I there is some k ∈ I such, that Di ⊆ Dk and Dj ⊆ Dk.
3 Intuitionistic fuzzy digital compact structure spaces     ii.
Every D * open set which contains the intersection ( ∩ B) ∩ ( ∩ A) will also contain some IFD subset A~ ∩ B~ ≠ 0~. iii.
of nonempty D * open sets. Since (E, D * ) is a D * CS-filtered structure space, there exists some A~ ∈ A, such that A~ ∈ ∪ {B : B~ ∈ B}. Also there exists some B~ ∈ B such that A~ ⊆B , because A~ is D * structure compact. Hence A~ ∩ B~ = 0~, which contradicts the fact that which is a contradiction to or assumption.
(iii) Assume that F is a D * structure open cover of ( ∩ B) ∩ ( ∩ A). From proof (i), there exists A~ ∈ A and B~ ∈ B such that B~ ∩ A~ ⊆ ∪ F. Also there exists a finite subfamily