A Type-2 Fuzzy Sonewhere Dense Set In General Type-2 Fuzzy Topological Space

The multiplicity of connotations in any paper does not mean that there is no main objective for that paper and certainly one of these papers is our research the main objective is to introduce a new connotation which is type-2 fuzzy somewhere dense set in general type-2 fuzzy topological space and its relationship with open sets of the connotation type-2 fuzzy set in the same space topology and theories of this connotation.


Introduction
The ability to provide suitable solutions to mathematical problems, including the conversion of nonlinear equations into linear equations, has made type-2 fuzzy set a wide field of scientific studies of the branches of mathematics because its appearance in the hands of L. Zada in 1975 [1] was not surprising after he introduced in 1965 fuzzy set [2] type-1 fuzzy set (fuzzy set), but this discovery became a necessary need to complete modern scientific research and the entry of fuzzy logic in its formulation. In 1976, Mizumoto and Tanaka introduced the properties of type-2 fuzzy set and the various methods of finding union and intersection [3]. Mendel and Karnik then introduced new methods in the formation of operation of type-2 fuzzy set [4] and then Mendel introduced an important connotation which is Interval type-2 fuzzy set [5] to become the study of type-2 fuzzy set into two parts the first part is interval type-2 fuzzy set and the second part general type-2 fuzzy set all these studies opened the way to study the topology spaces of each part , Zhang introduced the concept of interval type-2 fuzzy topology space [6] and Hussan and AL-Khafaji general type-2 fuzzy topology space [7]. This paper complements the topology construction path by introducing the connotation of type-2 fuzzy somewhere dense set in the general type-2 fuzzy topology space and its relationship to some type-2 fuzzy open sets.

Preliminaries
Everything in this section is an important preamble of the main definitions and features that reflect the nature of type-2 fuzzy set work [1], [3], and [5].We assume that X  and I [0,1]  be closed unit interval.

Definition 1.1
A type-2 fuzzy set, denoted by A  characterized by a type-2 membership function,where x X    which is universal and and ∑∑ enotes the union in discrete sets and ∑ is replaced by ∫ is continuous universes are set.
The class of all type-2 fuzzy set of x X    denoted by 2 T F (x).

Definition 1.3 [5]:
When all the A (x ,u ) l     then type-2 fuzzy set is called interval type-2 fuzzy set.

Definition 1.3 [3]:
A The union of two type-2 fuzzy sets is defined as The intersection of two type-2 fuzzy sets is defined as The containment type-2 fuzzy sets are defined as The complement of type-2 fuzzy set defined as

General type-2 fuzzy topological space 1.4 [6]:
Let T  be the collection of type-2 fuzzy sets over X  then T  is called to be general type-2 fuzzy topology on X  if i.
The pair (X ,T)  is said to type-2 fuzzy topological space over X ( Simply   . We must note that all the type-2 fuzzy sets is normal type-2 fuzzy sets so as to complete the topological construction and especially check identity law ( A

Theorem 1.4.(1):
Let (X, T)  be a generaltype-2 fuzzy topology space over X , and let A , B  are type-2 fuzzy sets in X, then

Theorem 1.4.(2):
Let (X, T)  be a general type-2 fuzzy topology space over X, and let A , B  are type-2 fuzzy sets in X, then

Some type-2 Fuzzy Open Sets in General type-2 fuzzy Topological Space.
In a concise manner all lines of this section include a presentation of open set for type-2 fuzzy set and how it was first formed in general type-2 fuzzy topological space.

Definition (1.2):
A subset type-2 fuzzy set A  of a general type-2 fuzzy topological space is called: ).

Remark (4.3):
We denote the collection of all type-2 fuzzy somewhere dense set in a general type-2 fuzzy topology

Theorem (7.3):
A non-empty T s 2f The converse the theorem (7.3) is not true.

Theorem (11.3):
A non-empty T s 2f    is T 2f SD S   .