On frontier and exterior in intutionistic supra α- closed set

. The main aim of the study of this paper is to work with the prop- erties of frontier and exterior in intuitionistic supra topological spaces. Consid-ering this we have introduced intuitionistic supra (cid:11) -frontier and intuitionistic (cid:11) -exterior in intuitionistic supra topological space. We have also deliberated the properties of intuitionistic suppra (cid:11) -frontier and intuitionistic supra (cid:11) - exterior in intuitionistic supra topological space. The comparative study has been done with the use of intuitionistic supra (cid:11) -open set between Intuitionistic supra frontier, Intuitionistic supra exterior and intuitionistic supra (cid:11) -frontier, intuitionistic (cid:11) -exterior in intuitionistic supra topological space.


Introduction
In 1970, Levine [4] introduced the concept of generalized closed sets in topological spaces. Njastad.O [12] and Maki.H et al [6] introduced -closed sets and g -closed sets in topological spaces. In 1965 ,O.Njastad [12] introduced -open sets. The concept of intuitionistic set and intuitionistic topological spaces was introduced by Coker[1] [2]. Supra topology was introduced by A.S.Mashhour et.al [6] Intuitionistic supra -open set was introduced by the Author [8] on intuitionistic supra topological spaces and discussed the properties of Intuitionistic supra -open sets in supra topological spaces.
The purpose of this paper is to study the properties of -frontier and -exterior in intuitionistic supra topological spaces. Also to study the comparison between Intuitionistic supra frontier, Intuitionistic supra exterior and intuitionistic supra -frontier, -exterior in intuitionistic supra topological space.
De…nition 2.3 [6] An intuitionistic topology on a non-empty set X is a family of IS's in X satifying the following axioms: (ii) A 1 \ A 2 2 for any A 1 ; A 2 2 . The pair (X; ) is called an intuitionistic topological space (ITS in short) and IS in is known as an intuitionitic open set (IOS in short) in X, the complement of IOS is called an intuitionistic closed set (ICS in short).
De…nition 2.4 [6] The supra closure of a set A is denoted by cl (A), and is de…ned as, supra cl(A) = T fB : B is supra closed and A Bg. The supra interier of a set A is denoted by int (A), and is de…ned as supra int(A) = S fB : B is supra open and A Bg.
An Intuitionistic supra topology on a non-empty set X is a family of IS's in X satisfying the following axioms: The pair(X, ) is called intuitionistic supra topological space (ISTS in short) and IS in is known as an intuitionistic supra open set (ISOS in short) in X, the complement of ISOS is called intuitionistic supra closed set(ISCS in short).
De…nition 2.6 [1] Let (X, ) be an ISTS and let A = hX; A 1 ; A 2 i be an IS in X, then the supra closure and supra interior of A are de…ned by: De…nition 2.7 [8] Let (X, ) be an ISTS and let A = hX; A 1 ; A 2 i be an IS in X , then the supra closure and supra interior of A are de…ned by: I cl (A) = T fk : k is an IS CS in X and A kg I int (A) = S fk : k is an IS OS in X and A kg The complement of intuitionistic supra -closed set is intuitionistic supra -open set (IS OS in short).   Proof. Let A be a IS in ISTS X. The proof of the above theorem is shown in the following example:      Here I F r (A) IF r (A) is true but converse is not true.

Intuitionistic supra Frontier
Theorem 3.7 Let X be an ISTS then and for any a subset A of IS in ISTS X, the following statements holds: