# I. Introduction
N. Levine [1] introduced the concept of generalized closed sets in topological spaces. Later many authors introduced new types closed sets in topological spaces and established their properties. They also studied the relationship with other types of closed sets in topological spaces. The concept of Grill was first introduced by Choquet [2] in the year 1947.Some authors introduced the concept of generalized closed set in Grill topological spaces in later years. In 2012, Dhananjoy Mandal and M. N. Mukherjee [3], introduced the concept of Gg-closed set in Grill topological spaces and studied their properties. In the year 2017, M. Kaleswari and others [4] introduced the concept of Gg*-closed sets in Grill topological spaces and established some of their properties. With their inspiration the concept of Gg**-closed set in a Grill topological space was introduced in the present work and studied some of their properties.
# II. Preliminaries
Definition 2.1 A Grillon a topological space ( , ) X ? is a nonempty collection G of nonempty subsets of X such that (i) , A G A B X B G ? ? ? ? ? (ii) , , A X B X A B G A G ? ? ? ? ? ? or B G ? .
If G is a Grill on a topological space ( , ) X ? , then it is called a Grill topological space denoted with ( , , ) X G ? .
Let ( , , ) X G ? be a Grill topological space and A is any subset of X .The operator
: ( ) ( ) P A P A ? ? is defined as ( ) / , ( ) { } A x X U A G U x ? ? = ? ? ? ? ? where ( ) x
? denotes the neighbourhood of x in the space X . Definition 2.3:
A subset A of a Grill topological space ( , , ) X G ? is said to be Gg-closed if ( ) A U ? ? whenever A U ? and U is open in X .
The complement of a Gg-closed set is a Gg-open set.
# Definition 2.4: [6]
A subset A of a Grill topological space ( , , )
X G ? is said to be Gg*-closed if ( ) A U ? ? whenever A U ? and U is g-open in X .The complement of a Gg*-closed set is a Gg*-open set.
Throughout the paper, by a space X we always mean a topological space ( , ) X ? with no separation axioms assumed. For any subset A of the space X , the closure of A is denoted with ( ) cl A and interior of the subset A is denoted with ( ) int A . Theorem 2.5: [5] Let ( , , ) X G ? be a Grill topological space. Then for any , A X B X ? ? the following hold:
( ) ( ) (a) ( ) ( ) (b) ( ) ( ) ( ) (c) ( ) ( ) ( ) ( ) A B A B A B A B A A cl A cl A ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? = ? III. Gg**-Closed Sets in Grill Topological Spaces
In this section a new type of closed set was defined in a Grill topological space with an example.
# Definition 3.1:
A subset A of a Grill topological space ( , , ) X G ? is said to be if ( )
A U ? ? whenever A U ? and U is g*-open in X . The complement of a Gg**-closed set is a Gg**-open set. Example 3.2: Consider { , , } X a b c = , { , ,{ },{ , },{ , },{ }} X a a b a c b ? = ? and { ,{ },{ , } G X a a c = . Then, ( , , ) X G ? is a Grill topological space. In this space, g*-closed sets are { , ,{ },{ },{ , },{ , }} X b c b c a c ? . Take the set { , } A a c = in the space. Then, ( ) { ,{ },{ , },{ , } a X a a b a c ? = , ( ) { ,{ },{ , }} b X b a b ? = and ( ) { ,{ , }} c X a c ? = . Now, { } { } ,{ , } { } ,{ , } { , } , a A a G a b A a G a c A a c G X A A G ? = ? ? = ? ? = ? ? = ? . This shows that ( ) a A ? ? .
In a similar way we can check
# Notes
IV.
# Properties of Gg**-Closed Sets
This section is dedicated to study some simple properties of Gg**-closed sets.
Theorem 4.1:
In a Grill topological space ( )
, , X G ? , every non-member of G is Gg**-closed.
Proof:
Let A be any non-member of G and U be a g*-open set containing A . Then,
A U A G ? = ? . This shows that ( ) { } A U ? = ? and hence A is Gg**-closed set.
Remark:
The converse of the above theorem need not be true. This can be seen from the following example.
# Example 4.2:
Consider the Grill topological space ( )
, , X G ? defined by the sets { , , }, X a b c = { , ,{ },{ , },{ , },{ }}, X a a b a c b ? = ? { ,{ },{ , }} G X a a c = .In this space { , } A a c = is a Gg**-closed
set but it is a member of the grill G . Remark:
The converse of the above theorem need not be true. This can be seen from the following example. Proof:
Let A be ag*-closed set in the Grill topological space ( )
, , X G ? and U be any g*-open set containing A .Then, ( ) A U ? ? . Hence, A is a Gg**-closed set.
Remark:
The converse of the above theorem need not be true. This can be seen from the following example.
![, ( ) {a, c} A ? = . Also, the g*-open sets containing A are { ,{ , }} X a c and each of the sets contain ( ) A ? . Hence, the set { , } A a c = is a Gg**-closed set in the Grill topological space ( , , )](image-2.png "")
![set is a Gg**-closed set.Proof:Let A be any closed set in the Grill topological space ( ) be any g*-open set containing A . Then, U is a g*-open set containing ( ) cl A . We claim that ( )](image-3.png "")
![g ? closed set is a Gg**-closed set.](image-4.png "")
![Sets in Grill Topological SpacesNotes](image-5.png "")
Then, ( )
, , X G ? is a Grill topological space. In the space, {a, } A b = is a Gg**-closed set, but it is not a g*-closed set.
In a Grill topological space ( ) , , X G ? , every Gg*-closed set is a Gg**-closed set.
Proof:
Let A be any Gg*-closed set in the Grill topological space ( )
, , X G ? and U be any g*-open set containing A .Then, ( )
Remark:
The converse of the above theorem need not be true. This can be seen from the following example.
Example 4.8:
Then, ( )
This shows that the set A B ? is a Gg**-closed set.
Remark:
In a Grill topological space ( )
, , X G ? , intersection of two Gg**-closed sets need not be a Gg**-closed set. This can be seen in the following example. If , A B are two subsets of a grill topological space ( )
Proof:
Let U be any g*-open set containing B in the Grill topological space ( )
This shows that ( ) B U ? ? and hence B is a Gg**-closed set.
## V. Conclusion
In this paper an attempt was made to introduce the concept of Gg**-closed sets in a Grill topological space. Some basic properties of these sets were discussed. In continuation to this, continuity using these closed sets can be studied in future work.
*
Generalized closed sets in topological spaces
NLevine
Rend. Circ. Mat. Palermo
19
1970
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