On Intuitionistic sgp-closed sets in Intuitionistic Topological Space

Accepted 10/Aug/2018, Online 30/Aug/2018 AbstractIn 2007, Navalagi and Bhat introduced and study a new class of sets, namely, sgp-closed sets in Topological Spaces. He observed that this class is properly placed between the class of closed sets and generalized preclosed sets. In this paper we have introduced intuitionistic sgp-closed sets and intuitionistic sgp-open sets in intuitionistic toplogical spaces and obtained its significant properties. We have constructed some examples which are quite useful in theory of intuitionistic sgp-closed and intuitionistic sgp-open sets.


I. INTRODUCTION
The notation of intuitionistic set was introduced by Coker [4] in 1996, and also he [5] has introduced the concept of intuitionistic topological spaces.Bajpai and Thakur [11] have introduced the concept of intuitionistic fuzzy sgpclosed sets in intuitionistic fuzzy topological spaces in 2017.In this paper we have obtained some significant properties of intuitionistic sgp-closed sets and intuitionistic sgp-open sets in intuitionistic toplogical spaces.

II. PRELIMINARIES
In this section we have studied set theoretical results of intuitionistic sets.Futher we have studied some generalized forms of intuitionistic open and intuitionistic closed set in intuitionistic topological space.Definition 2.1 [4] Let X is a non empty set.An intuitionistic set (IS for short) A is an object having the form A = < X, A 1 , A 2 >, where A 1 and A 2 are subsets of X satisfying A 1 ∩ A 2 = Φ.The set A 1 is called the set of members of A, while A 2 is called the set of nonmembers of A. Definition 2.2 [4] Let X be a non empty set and let A, B are intuitionistic sets in X of the form A = < X, A 1 , A 2 >, B = < X, B 1 , B 2 > respectively.Then Further if {A i : i ∈ J} is an arbitrary family of intuitionistic sets in X, where A i = < X, A i (1) , A i (2) >.Then 7. ∩A i = < X, ∩A i , ∪A i >; 8.A i = < X, A i , ∩A i >; Definition 2.3 [5] An intuitionistic topology (for short IT) on a non empty set X is a family of ISs in X satisfying the following axioms.
3. G i ∈ τ for any arbitrary family {G i : i ∈ J} ⊆ τ .In this case the pair (X,τ ) is called an intuitionistic topological space(for short ITS) and any intuitionistic set in τ is known as an intuitionistic open set (for short IOS) in X.The complement A c of an Intuitionistic open set A is known as an intuitionistic closed set (for short ICS) in X.
Definition 2.4 [4] Let X be a non empty set and p∈X.Then the IS P defined by P = < X, {p}, {p} c > is called an intuitionistic point (IP for short) in X.The intuitionistic point P is said to be contained in A =< X, A Theorem 2.1 [8] Let (X,τ) be an ITS and {A α : α ∈ I} be a family of intuitionistic semiopen sets in X.Then ∪ α ∈ I A α is also an Intuitionistic semiopen set.
Lemma 2.1 [8] Let (X,τ) be an ITS.If A and B be are two intuitionistic semiopen sets in X, then A ∩ B is not necessarily to be an intuitionistic semiopen set.
Theorem 2.2 [9] Let (X,τ) be an ITS and {A α : α ∈ I} be a family of intuitionistic preopen sets in X.Then ∪ α ∈ I A α is also an Intuitionistic preopen set.
Lemma 2.2 [9] Let (X,τ) be an ITS.If A and B be are two intuitionistic preopen sets in X, then A ∩ B is not necessarily to be an intuitionistic preopen set.But if one of the set is an IO set.Then A ∩ B is an intuitionistic preopen set.
Theorem 2.3 [10] Let (X,τ) be an ITS and { A α : α ∈ I} be a Theorem 2.5 [1] In ITS every intuitionistic closed set is intuitionistic α-closed set.Remark 2.1 The converse of the above theorem need not be true we have following example.

Remark 2.2[1]
The converse of the above theorem need not be true we have following example.
Remark 2.3 [1] The converse of the above theorem need not be true we have following example.
Remark 2.4 [1] The converse of the above theorem need not be true we have following example.
Remark 2.5 [1] The converse of the above theorem need not be true we have following example.
Remark 2.6 [1] The converse of the above theorem need not be true we have following example.

III. INTUITIONISTIC sgp-CLOSED SET
Definition 3.1 Let (X,τ) be an ITS and A be an intuitionistic set is said to be intuitionistic sgp-closed (for short Isgpclosed) set if Ipcl(A) ⊆ U whenever A ⊆ U and U is intuitionistic semiopen set in X.We note that the concept of intuitionistic sgp-closed is a generalized form of intuitionistic closed set.We have following results

Theorem 3.1
In ITS every intuitionistic closed set is intuitionistic sgp-closed set.Proof Let (X,τ) be an ITS and A be an intuitionistic closed set.Let A ⊆ V and V is an intuitionistic semiopen set in X.
Since A be an intuitionistic closed set this implies that Icl(A) = A. Thus we have Icl(A) ⊆ V. Since Ipcl(A) ⊆ Icl(A), therefore Ipcl(A) ⊆ V whenever A ⊆ V and V is intuitionistic semiopen set in X. Hence A be an intuitionistic sgp-closed set.
Remark 3.1 the converse of the above theorem need not be true we have following example.
Definition 3.2 Let (X,τ) be an ITS and A be an intuitionistic set in X.Then A is said to be intuitionistic sgp-open set if its compliment is intuitionistic sgp closed set.

Remark 3.2
From the above definition we note that in a intuitionistic topological space, each intuitionistic open set is intuitionistic sgp-open set.However the converse may not true.

Remark 3.3
In ITS the intersection of two intuitionistic sgpclosed sets need not be necessarily to sgp-closed set.We have following example.

Theorem 3.2
In ITS every intuitionistic preclosed set is intuitionistic sgp-closed set.Proof Let (X,τ) be an ITS and A be an intuitionistic set.Let A⊆ V and V is an intuitionistic semiopen set in X.Since A be an intuitionistic preclosed set this implies that Ipcl(A) = A. Thus we have Ipcl(A) ⊆ V whenever A ⊆ V and V is intuitionistic semiopen set in X. Hence A be an intuitionistic sgp-closed set.
Remark 3.4 the converse of the above theorem need not be true we have following example.

IV. CONCLUSION
We conclude that the concept of intuitionistic sgp-closed set is a generalization of intuitionistic preclosed set.Also further we conclude that the concept of intuitionistic sgpclosed set and intuitionistic semiclosed set, intuitionistic sgp-closed set and intuitionistic β-closed set are independent of each other.

Lemma 2 . 3 [ 10 ]
family of intuitionistic α−open sets in X.Then ∪ α ∈ I A α is also an Intuitionistic α−open set.Let (X,τ) be an ITS.If A and B be are two intuitionistic α-open sets in X, then A ∩ B is an intuitionistic α−open set.Theorem 2.4 [2] Let (X,τ) be an ITS and { A α : α ∈ I} be a family of intuitionistic β−open sets in X.Then ∪ α ∈ I A α is also an Intuitionistic β−open set.Lemma 2.4 [1] Let (X,τ) be an ITS.If A and B be are two intuitionistic β-open sets in X, then A ∩ B is not necessarily to be an intuitionistic β−open set.

Figure 3 . 1 Theorem 3 . 3 Theorem 3 . 4
Figure 3.1 Relations between Intuitionistic closed sets and Intuitionistic sgp-closed set.In this diagram A B" we mean A implies B but not conversely and "A B" means A and B independent of each other Theorem 3.3 Let (X,τ) be an ITS and if A be an intuitionistic sgp-closed set and A ⊆ B ⊆ Ipcl(A) then B be an intuitionistic sgp-closed set.Proof Let (X,τ ) be an ITS and let U be an intuitionistic