Intuitionistic Fuzzy semiboundary and intuitionistic Fuzzy Product Related spaces

In this paper intuitionistic fuzzy semi boundary is introduced and its properties are investigated. Intuitionistic fuzzy semi continuous functions are characterized via intuitionistic fuzzy semi boundary.


Introduction
In the year 1965, Zadeh [10] introduced the concept of fuzzy sets. After that there have been a number of generalizations of this fundamental concepts. In the year 1986 Atanassov [1] introduced intuitionistic fuzzy set as a generalization of fuzzy set. Using the concept of intuitionistic fuzzy set, Coker [5] extended the concept of fuzzy topological spaces introduced by C.L.Chang [4] to intuitionistic fuzzy topological spaces. Pu and Liu [8] defined the notion of fuzzy boundary in fuzzy topological spaces in 1980. Properties of fuzzy boundary were investigated by Ahmed and Athar [2].
In this paper we introduce intuitionistic fuzzy semi boundary and investigate some of their properties. Further intuitionistic fuzzy semi boundary in product related spaces is analysed. Finally necessary conditions for intuitionistic fuzzy semi continuous functions are obtained via intuitionistic fuzzy semi boundary. Throughout this paper X,Y are non-empty sets.

Preliminaries
Definition 2.1: [1] An intuitionistic fuzzy set (IFS) A in X is an object having the form where the functions and denote the degree of membership (namely, µ A (x) ) and the degree of non-membership (namely,  A (x ) ) of each element x  X to the set A respectively, and for each xX The IFS's are respectively the empty set and the whole set of X. For the sake of simplicity, we will use the notation A= instead of  [5] An intuitionistic fuzzy topology (IFT) on X is a family  of IFS's in X satisfying the following axioms: In this case, the pair (X,) is called an intuitionistic fuzzy topological space(IFTS) and any IFS in  is known as an intuitionistic fuzzy open set (IFOS) in X. The complement Ā of an IFOS A in IFTS (X ) is called an intuitionistic fuzzy closed set (IFCS) in X. Definition 2.5: [5] Let (X,) be an IFTS and let A= be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A are defined by Properties of closure and interior of a intuitionistic fuzzy set which are needed in the sequel, are summarized in the following lemma.
Definition 2.7: [9] Let A be a IFS in an IFTS (X, Then the intuitionistic fuzzy boundary of A is defined as IBdAis a intuitionistic fuzzy closed set(IFCS). 58 Volume 2

Definition 3.1: [6] An IFS A in an IFTS (X,) is called an intuitionistic fuzzy semi open set(IFSOS) if A cl (int(A)) (or) if there exists V such that VAClV.
An IFS A is called intuitionistic fuzzy semi-closed set, if the complement of ̅ is an IFSOS. Definition 3.2: [6] Let A be a IFS in an IFTS (X,). Then Semi interior(SInt) and semi closure (SCl) of A are given as  Then, Hence, Definition 3.5: Let A be a IFS in an IFTS (X,). Then the intuitionistic fuzzy semiboundary of A is defined as ISBd A= SCl A  SCl Ā. ISBd A is a intuitionistic fuzzy semiclosed set(IFSCS).

Remark 3.6:
In classical topology, for an arbitrary set A of a topological space X, but for an arbitrary intuitionistic fuzzy set A in an IFTS (X,), where the equality may not hold as is seen in the following example. Hence, Hence, The converse of (2) , (3) and the equality may hold in (6) , (7) of Proposition 3.8 is not true as seen in the following example.
The following example shows that the equality may not hold in(2), (3) and (4) of Proposition 3.10.

Example 3.11: Choose IFS
in the IFTS (X ) defined in example 3.7.

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Hence, (2) Hence,

Remark 3.12:
In general topology, the following conditions hold.
In intuitionistic fuzzy topology, these may not hold as seen in the following example. Choose IFS Example 3.13: Choose IFS in the IFTS ( X) defined in example 3.7. (1) Hence, (3)

Proof:
Hence, In theorem 3.14, the equality may not hold as seen in the following example.  The equality in the above theorem may not hold as seen in the following example. Hence,

Intuitionistic Fuzzy product related spaces
K.K.Azad [3] introduced the concept of fuzzy product related spaces. H.M.Hanafy [7] extend this concept to intuitionistic fuzzy topological spaces as follows:

Definition 4.1: [7]
If and be IFSs of X and Y respectively. Then the product of intuitionistic fuzzy sets A and B be defined by, where and It may be notice that The following lemma is needed in sequel. where A  X and B  Y there exist A 1   X and B 1   Y such that

Lemma 4.4: [7] If A is an IFS of X and B is an IFS of Y, then
Proof: (1) Since
The following theorem gives a necessary condition of intuitionistic fuzzy irresolute functions in terms of intuitionistic fuzzy semi boundary.