On (gg)*- Closed Sets in Topological Spaces

- In this paper


I. INTRODUCTION
The concept of generalized closed sets [1] in Topological spaces was introduced by N. Levine in 1970. D. E. Cameron and M. Stone introduced regular semi open sets [2] and regular open sets [3] respectively. In 2017, Basavaraj M. Ittanagi and Govardhana Reddy introduced and studied generalization of generalized closed sets [4] in Topological spaces.
In this paper we introduce a new class of closed set called (gg)* -closed sets in Topological spaces. Section 1, gives the overall introduction to the paper , followed by section 2, where we recall some of the existing closed and open sets. Section 3, provides us with the introduction to the concept of (gg)* -closed set. In section 4, the independency of (gg)* -closed sets with some of the existing closed and generalized closed sets are studied and its outcome is shown in the form of a diagram. In section 5, some of the properties of (gg)* -closed sets are studied, analyzed and proved; which leads to section 6, the conclusion of the paper. After which, the references that were dealt with during the analyses are given at the end of the paper.

II. PRELIMINARIES
Throughout this paper (X, τ) represent the topological space on which no separation axioms are assumed unless otherwise mentioned. For a subset A of X, the closure of A and interior of A are denoted by cl (A) and int (A) respectively.
and U is open in X .
(2) regular semi open [2] if there is a regular open set U such that U  A  cl ( U ).
(3) regular open set [3] if A = int ( cl (A)) and a regular closed set if cl (int ( A)) = A.
(11) generalized semi -pre closed (briefly gsp -closed) [10 ] if spcl(A)  U whenever A  U and U is open in X.
(12) generalized pre -closed set (briefly gp -closed) [11] if pcl(A)  U whenever A  U and U is open in X.
(13) generalized semi -closed set (briefly gs -closed) [12 ] if scl(A)  U whenever A  U and U is open in X.
(15) regular generalized closed set (briefly rg -closed ) [14] if cl (A)  U whenever A  U and U is regular -open in X.
(16) generalized pre -regular closed set (briefly gpr -closed) [15 ] if pcl (A)  U whenever A  U and U is regular open in X.
(17) generalized semi -pre regular -closed set (briefly gspr -closed) [16] if spcl(A)  U whenever A  U and U is regular open in X.
(18) generalized star pre closed (briefly g*p -closed ) [17 ] if pcl (A)  U whenever A  U and U is g -open in X. (19) weakly closed set (briefly w -closed) [18 ] if cl (A)  U whenever A  U and U is semi-open in X. (20) tgr -closed set [19] if rcl (A)  U whenever A  U and U is a t -set. (25) R*-closed set [24] if rcl (A)  U whenever A  U and U is regular semi -open in X.
(26) R # -closed set [25] if gcl (A)  U whenever A  U and U is R* -open in X.
(27) βg* -closed set [26] if gcl (A)  U whenever A  U and U is β -open in X.
(28) r ^ g -closed set [27] if gcl (A)  U whenever A  U and U is regular -open in X.
(29) g**-closed set [28] if cl (A)  U whenever A  U and U is g*-open in X.
(30) g* -closed set [29] if cl(A)  U whenever A  U and U is g -open in X.
(31) generalized regular closed set (briefly gr -closed) [30] if rcl (A)  U whenever A  U and U is open in X.
(32) generalized regular star closed (briefly gr *-closed) [31] if rcl (A)  U whenever A  U and U is g -open in X.
The complements of the above closed sets are their open sets and vice versa.

Proposition 3.3 Every regular closed set is (gg)* -closed.
Proof: Let A be a regular closed set in X such that A  U and U is gg -open. Then rcl(A) = A. Hence rcl (A)  U. Therefore A is (gg)*-closed.

Remark 3.4
The converse of the above proposition need not be true as shown in the following example.  (1) Every (gg)* -closed set is g-closed.
(12) Every (gg)* -closed set is gr*-closed. Proof: (2) Let A be a (gg)* -closed set in X. Let U be a g*-open set in X such that A  U. Since

(11) Let A be a (gg)* -closed set in X. Let U be an open set in X such that A  U. Since every open set is gg-open [4] and since A is (gg)*-closed, rcl(A)  U. Hence A
is gr-closed.
(12) Let A be a (gg)* -closed set in X. Let U be a g-open set in X such that A  U.

Since every g-open set is gg-open [4] and since A is (gg)*-closed, rcl(A)  U. Hence
A is gr*-closed.

Remark 3.7
The converse of the above proposition need not be true as shown in the following example. (1) {b} is g -closed but not (gg)*-closed.
(3) {a,b} is (gg)*-closed but not regular semi -closed and {c} is regular semi -closed but not (g g)*-closed.
(11) {b,c} is (gg)*-closed but not tgr-closed and {c, d} is tgr-closed but not (gg)*-closed. Proof: Let A and B be the (gg)*-closed sets in X. Let U be a gg -open set in X such that A∪ B  U. Then A  U and B  U. Since A and B are (gg)*-closed sets in X, rcl(A)  U and rcl(B)  U. We have by [19], rcl(A∪B) = rcl(A) ∪ rcl(B)  U. This implies rcl (A∪B)  U. Hence A∪ B is (gg)*-closed.

Remark 5.2
Intersection of two (gg)*-closed sets need not be (gg)*-closed as shown in the following example. That is X-{ }  X .This implies rcl(X-{ })  rcl(X)  X. Therefore X -{ } is a (gg)*-closed set in X. Theorem 5. 9 (1) If A is β-open and βg*-closed set in X. Then A is (gg)*-closed.
(2) If A is R*-open and R # -closed set in X. Then A is (gg)*-closed.
(3) If A is regular open and r ^ g-closed set in X. Then A is (gg)*-closed.

Proof:
(1) Let A be a β-open and βg*-closed set in X. Let U be any gg-open set in X such that A  U.
By Definition 2.1 (27), gcl(A)  A. But we have rcl (A)  gcl (A)  A. Therefore rcl (A)  U. Thus we get A is (gg)*-closed.

VI. CONCLUSION
The class of (gg)*-closed sets in topological spaces is defined using regular closure and gg-open sets .We have studied the relation of this set with some other closed sets and some of the properties are investigated.