on δg*-Closed Sets in Bitopological Spaces

on g*-Closed Sets in Bitopological Spaces

R.Sudha Assistant professor,

SNS College of Technology, Coimbatore

K.Sivakamasundari Associate professor,

Avinashilingam Deemed University for Women

Abstract

1

The aim of this paper is to introduce the concept of

g*-closed sets and we discuss some basic properties of (i, j)-g*-closed sets in bitopological spaces. Applying these sets, we obtain the new

(i, j) – g*-closed sets in bitopological spaces and study their properties. We prove that this class lies

spaces called (i, j)- g*T1

2

– space, (i, j)- g*T*

2

-space,

between the class of (i, j) – -closed sets and the class of (i, j) – g-closed sets. Also we discuss some basic properties and applications of (i, j)- g*- closed sets, which defines a new class of spaces

(i, j)- g*T – space and (i, j)- gT1 – space.

1

2 2

2. Preliminaries

namely (i, j)-

g*T1

2

-spaces, (i, j)-

*

T

g* 1

2

-spaces,

If A is a subset of X with the topology,

(i, j)- g*T1 -spaces and (i, j)- gT1 -spaces.

then the closure of A is denoted by -cl(A) or cl(A),

2 2 the interior of A is denoted by -int(A) or int(A) and

Keywords: (1, 2)- g-closed set, (1, 2)- -closed set,

(1, 2)- g*-closed set.

Ams subject classification: 54E55, 54C55.

1. Introduction

A triple X, 1, 2 where X is a non- empty set and 1 and 2 are topologies on X is

called a bitopological space and Kelly [8] initiated the study of such spaces. Njastad[12], Velicko [20] introduced the concept of -open sets and -closed sets respectively. Dontchev and Ganster [4] studied -generalized closed set in topological spaces. Levine [10] introduced generalization of closed sets and discussed their properties. In 1985, Fukutake [5] introduced the concepts of g-closed sets in bitopological spaces and after that several authors turned their attention towards generalizations of various concepts of topology by considering bitopological spaces. Also M. E. Abd El-Monsef [1] et al investigated -closed sets in topological spaces. Sheik John et al [14] introduced g*-closed sets in bitopological spaces. Sudha et al. [16] introduced the concept of g*-closed sets in topological spaces and investigated its relationship with the other types of closed sets. The purpose of the present paper is to define a new class of closed sets called (i, j) –

the complement of A in X is denoted by Ac.

1. Definition

   A subset A of a topological space (X, ) is called a

   1. semi-open set [9] if A cl(int(A)).

   2. -open set [12] if A int(cl(int(A))).

   3. regular open set [16] if A = int (cl(A)).

   4. Pre-open set [11] if A int(cl(A)).

      The complement of a semi open (resp. -open, regular open, pre-open) set is called semi-closed (resp. -closed, regular closed, pre-closed).

      The semi-closure [3] (resp. -closure [12], pre- closure [11]) of a subset A of (X, ), denoted by scl(A) (resp. cl (A) , pcl(A)) is defined to be the

      intersection of all semi-closed (resp.-closed, pre- closed) sets containing A. It is known that scl(A) (resp. cl (A) , pcl(A)) is a semi-closed (resp.-

      closed, pre-closed) set.

2. Definition

   The -interior [20] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by int (A) .The subset A is called

   -open [20] if A = int (A) . i.e., a set is -open if it is the union of regular open sets, the complement of a -open is called -closed. Alternatively, a set A X is called -closed [20] if A = cl (A) , where

   1. (i, j) g*-closed [18] if j -cl(A) U whenever A U and U is -open in i .

   2. (i, j) g*p-closed [17] if j -pcl(A) U

      cl (A) x X; int (cl(U)) A , U and x U.

      Every -closed set is closed [20].

      whenever A U and U is g-open in i .

   3. (i, j) w-closed [7] if j -cl(A) U whenever

      A U and U is semi-open in i .

3. Definition

1. (i, j) sg*-closed [13] if

j -cl(A) U

A subset A of (X,) is called

1. -generalized closed (briefly g-closed) [4] if cl (A) U whenever A U and U is open in (X, ).

2. generalized closed (briefly g-closed) [10] if

   whenever A U and U is g*-open in i .

   2.5. Definition

   A bitopological space X, 1 , 2 is called

   cl(A) U whenever A U and U is open in (X, ).

   1) (i, j)- T1

   2

   -space [5] if every (i, j)-g-closed set is

3. g*- closed [19] if cl(A) U whenever A U and U is g-open in (X, ).

Throughout this paper by the spaces X and Y represent non-empty bitopological spaces on which

j -closed.

1

2) (i, j)- T* -space [14] if every (i, j)-g*-closed set

2

is j -closed.

no separation axioms are assumed, unless otherwise mentioned and the integers i, j {1, 2}.

3) (i, j)- * T1

2

-space [14] if every (i, j)-g-closed set

For a subset A of X,

i cl(A) (resp.

is (i, j)-g*-closed.

i int(A) , i pcl(A) ) denote the closure (resp.

interior, pre closure) of A with respect to the topology i . We denote the family of all g-open

3. (i, j) – g*-closed sets in bitopological spaces

subsets of X with respect to the topology i by

GO(X, i ) and the family of all j -closed sets is denoted by the symbol Fj. By (i, j) we mean the pair of topologies i , j .

2.4. Definition

A subset A of a bitopological space X, ,

In this section we introduce the concept of (i, j) – g*-closed sets in bitopological spaces and discuss the related properties.

1. Definition

   A subset A of a bitopological space

   X, 1 , 2 is said to be an (i, j) – g*-closed

   1 2 set if

   cl (A) U, whenever A U and

   is called

   1. (i, j) g-closed [5] if

      j -cl(A) U whenever

      j

      UGO X, i

      A U and U is open in i .

   2. (i, j) g*-closed [14] if j -cl(A) U whenever

      We denote the family of all (i, j) – g*- closed sets in X, 1 , 2 by D* (i, j).

      A U and U is g-open in i .

   3. (i, j) rg-closed [2] if j -cl(A) U whenever

      A U and U is regular open in i .

   4. (i, j) wg-closed [6] if j -cl( i -int(A)) U

2. Remark

   By setting 1 2 in Definition 3.1., a (i, j) – g*-closed set is g*-closed.

   whenever A U and U is open in i .

   1. (i, j) gpr-closed [6] if j -pcl(A) U whenever A U and U is regular open in i .

3. Proposition

If A is

j –closed subset of X, 1 , 2 ,

then A is (i, j)-g*-closed.

Proof: Let A be a

j –closed subset of X, 1 , 2 .

3.8. Remark

Then j cl (A) A.

Let UGO X, i such that

The intersection of two (i, j)-g*-closed

A U, then

j cl (A) A U which implies A

need not be (i, j)-g*-closed as seen from the

is (i, j) – g*-closed.

The converse of the above proposition is not true as seen from the following example.

3.4. Example

following example.

3.9. Example

Let X = {a, b, c},

1 = {X, , {a}},

Let X = {a, b, c}, 1 = {X, , {a}},

2 = {X ,{b}, {c},{a, b},{b, c}}. Then the subset

{b, c} is (1, 2) – g*-closed but not 2 – -closed set.

3.5. Proposition

If A is both i -g-open and (i, j) – g*-

2 = {X, , { b}, {c},{a, b},{b, c}}. Then {a, b} &

{b, c} are (1, 2) – g*-closed sets but {a, b} {b, c}

= {b} is not (1, 2) – g*-closed.

1. Proposition

   For each element x of X, 1, 2 , x is

   i -g-closed or xc is (i, j)- g*-closed.

   closed, then A is j –closed.

   Proof: Let A be both i -g-open and (i, j)-g*-

   Proof: If x is i -g-closed, then the proof is over. Assume x is not i -g-closed. Then xc is not

   closed. Since A is (i, j)-g*-closed, we have

   i -g-open. So the only i -g-open containing xc

   A U and U GO X, i which implies

   j cl (A) U and since A is i -g-open. Put

   in X. Hence xc is (i, j) – g*-closed.

   A = U, then we have j cl (A) A , implies A is a j –closed set.

2. Proposition

If A is (i, j) – g*-closed, then

1. Proposition

   j cl (A) / A

   set.

   contains no non-empty i -g-closed

   If A is both i -g-open and (i, j)- g*-

   Proof: Let A be (i, j)-g*-closed and F be a non empty i -g-closed subset of j cl (A) / A .

   closed, then A is j -closed.

   Proof: Since closedness closedness, the result follows the above Proposition 3.5.

   Now F j cl (A) / A j cl (A) A c

   which implies F j cl (A) and F Ac .

   Therefore A Fc . Since Fc is i -g-open

2. Proposition

and A is (i, j)-g*-closed in X, we

* * have

j cl (A) Fc

which implies that

j

j

If A, B D (i, j), then A B D (i, j).

Proof: Let A and B be (i,j)-g*-

F cl (A) cl (A) c .

Therefore

closed. Let A B U where U GO X, i .

F = . Hence j cl

(A) / A

contains no non-

Now A B U implies A U and B U.

empty -g-closed set.

Since A, B D* (i, j), implies cl (A) U i

and

j

j cl (B) U. Then ( j cl (A)

The following example shows that the reverse implication of the above theorem is not true.

j cl (B)) U. That is Hence A B D* (i, j),

j cl (A B) U.

3.12. Example

Let X = {a, b, c},

1 = {X, , {a}, {a, c}},

2 = {X, , {a, b}}. If A = {a}, then

j cl (A) / A = {b, c} does not contain any non-

empty closed.

1 -g-closed set. But A is not (1, 2) – g*-

3.16. Proposition

If A is an (i, j)-g*-closed set of X, i , j

1. Corollary

   such that A B j cl (A) , then B is also an

   If A is (i, j)-g*-closed in X, 1, 2 , then

   (i, j)- g*-closed set of X, i , j .

   X, ,

   A is j –closed if and only if j cl (A) / A is i –

   Proof: Let U be a i -g-open set in

   i j such

   g-closed.

   that B U and hence A U. Since A is (i, j)-g*-

   Proof: (Necessity) Let A

   D* (i, j)

   and let A be

   closed,

   j cl (A)

   U. Since A B

   j –closed.

   Then j cl (A) A. i.e., j cl (A) / A = and

   j cl (A) , j cl (B)

   j cl ( j cl (A) ) j cl (A) U.

   Hence

   hence j cl (A) / A is i -g-closed.

   (Sufficiency) If j cl (A) / A is i -g-closed, then

   j cl (B) U which implies that B is a (i, j)- g*-closed set of X, i , j .

   by Proposition 3.11, j cl (A) / A = , since A is

   (i, j)-g*-closed. Hence A is j –closed.

2. Proposition

j cl (A) A. Therefore

1. Proposition

   Let A Y X and suppose that A is (i, j)- g*-closed in X. Then A is (i, j)- g*-closed relative to Y.

   Proof: Let A D* (i, j) & A Y U, U is g-open

   If A is an (i, j)-g*-closed set, then

   i cl (x) A holds for each x j cl (A)

   in X. A Y U implies A U and since A

   D* (i, j) , j cl (A) U. That is j cl (A)

   Proof: Let A be (i, j)- g*-closed and we know

   Y U Y. Hence

   j cl

   (A)

   U Y.

   i GO(X, i ). Suppose

   i cl (x) A

   for Y

   some x

   j cl

   (A) , then A X j

   cl (x) B,

   Therefore A is (i, j) – g*-closed relative to Y.

   say. Then B is a i –open set. Since a -open set is an open set and a open set is g-open, B is g-open in i . Since A is (i, j)- g*-closed, we get

2. Theorem

In a bitopological space X, 1 , 2 ,

j cl (A) B X j cl (x). Then

GO(X, i )

F if and only if every subset of X is

j

j cl (A) j cl (x)

which implies that

an (i, j) – g*-closed set, where F

j

is the collection

j cl (A) (x) . Hence x j cl (A) , which is a contradiction.

The converse of the above proposition is

not true as seen in the following example.

of -closed sets with respect to j .

j

Proof: Suppose that GO(X, i ) F . Let A be a subset of X, 1 , 2 such that A U where

UGO(X, i ). Then j cl (A) j cl (U) = U.

3.15. Example

Let X = {a, b, c}, 1 = {X, , {a}}, 2 =

{X, , {a}, {b, c}}. The subset A = {b} in

Therefore A is (i, j)-g*-closed set.

Conversely, suppose that every subset of X is (i, j)- g*-closed. Let U GO(X, i ). Since

j

X, 1 , 2 is not (1, 2) – g*-closed. However

U is (i, j) – g*-closed, we have j cl (U) U.

1 cl (x) A holds for each x 2 cl (A).

Therefore U F

j

and hence GO(X, i ) F .

3.19. Proposition

Every (i, j)- g*-closed set is (i, j)-g-closed. Proof: Let A be (i, j) – g*-closed. Let A U and U be a open set in i . Since every open set is g-

open, U is a g-open set. Then

j cl (A)

U, we

3.28. Proposition

know that j cl(U)

is (i, j)- g-closed.

1. Remark

   j cl (U) U. Hence A

   Every (i, j)- g*-closed set is (i, j)-wg-

   closed.

   Proof: Let A be (i, j) – g*-closed. Let A U and U be a open set in i Since every open set is g-open, U

   A (i, j)-g-closed need not be (i, j)- g*- closed as shown in the following example.

   is g-open in i Now i int(A) A, implies

   j cl(i int(A)) j cl(A) j cl (A) . Since

   A is (i, j) – g*-closed, j cl (A) U . Therefore

2. Example

   Let X = {a, b, c},

   1 = {X, , {a}},

   2 =

   j cl(i int(A)) U. Hence A is (i, j)-wg-closed.

   {X, , {a, b}}. Then the set {b} is (1, 2)-g-closed but not (1, 2) – g*-closed.

3. Proposition

Every (i, j) – g*-closed set is (i, j) – g*-

closed.

Proof: Let A be (i, j) – g*-closed. Let A U and U be a g-open set in i Then j cl (A) U, we

1. Remark

   A (i, j)-wg-closed need not be (i, j) – g*- closed as shown in the following example.

2. Example

Let X = {a, b, c}, 1 = {X, , {a}},

2 = {X, , {b, c}}. Then the set {b} is (1, 2)-wg-

know that

j cl(U)

j cl (U) U. Hence A

closed but not (1, 2)- g*-closed.

is (i, j)- g*-closed.

3.23. Remark

A (i, j)-g*-closed need not be (i, j) – g*- closed as shown in the following example.

3.31. Proposition

Every (i,j)- g*-closed set is (i, j)-g*-closed.

Proof: Let A be (i, j)- g*-closed. Let A U

GO(X, i ), since i GO(X, i ).Then j cl (A)

1. Example

   U. We know

   j -cl(A)

   j cl (A) which

   Let X = {a, b, c}, 1 ={X, , {a, b}},

   implies j cl(A) U Therefore A is (i, j)-g*-

   2 ={X, , {b, c}}. Then the set {a} is (1, 2)-g*- closed but not (1, 2) – g*-closed.

2. Proposition

Every (i, j)- g*-closed set is (i, j)-rg-

closed.

Proof: The proof follows from every regular open

closed.

1. Remark

   A (i, j)-g*-closed need not be (i, j) – g*- closed as shown in the following example.

2. Example

set is g-open.

Let X = {a, b, c},

1 = {X, , {a, b}},

1. Remark

   A (i, j)-rg-closed need not be (i, j)- g*- closed as shown in the following example.

2. Example

Let X = {a, b, c}, 1 = {X, , {a}, {a, b}},

2 = {X, , {a, b}}. Then the set {a, b} is (1, 2)-rg- closed but not (1, 2)- g*-closed.

2 = {X, , {a},{b},{a, b}}. Then the set {a} is (1, 2)-g*-closed but not (1, 2)- g*-closed.

3.34. Proposition

Every (i,j)-g*-closed set is(i,j)-gpr-closed. Proof: Let A be (i, j)- g*-closed. Let A U and U be regular open. Since every regular open set is g-open, U is g-open. Since A is (i, j)- g*-closed, j cl (A) U, We

know that

j pcl(A) j cl (A).

That is,

3.42. Example

j pcl(A) j cl (A) U Therefore A is (i, j)-gpr-closed.

3.35. Remark

A (i, j)-gpr-closed need not be (i, j)- g*- closed as shown in the following example.

Let X = {a, b, c}, 1 = {X, , {a, b}},

2 = {X, , {a},{b},{a, b}}. Then the set {b} is (1, 2)-sg*-closed but not (1, 2)- g*-closed.

3.43. Proposition

1. Example

   closed.

   Every (i, j)- g*-closed set is (i, j)-g-

   Let X = {a, b, c}, 1 = {X, , {a, b}},

   2 =

   Proof: The proof follows from the fact that every open set is g-open.

   {X, , {a},{b, c}}. Then the set {b} is (1, 2)-gpr- closed but not (1, 2)- g*-closed.

2. Proposition

Every (i, j)- g*-closed set is (i, j)-g*p-

closed.

Proof: Let A be (i, j)- g*-closed. Let

1. Remark

   A (i, j)- g -closed need not be (i, j)- g*- closed as shown in the following example.

2. Example

A U and U is g-open in

i .

Then j cl (A)

Let X = {a, b, c},

1 = {X, , {a}},

U. We know

j pcl(A) j cl (A). Therefore

2 = {X, , {a, b}}. Then the set {b} is (1, 2)- g –

j pcl(A) U. Hence A is (i, j)- g*p-closed.

1. Remark

   A (i, j)-g*p-closed need not be (i, j) – g*- closed as shown in the following example.

2. Example

closed but not (1,2)- g*-closed.

1. Remark

   The following examples show that (i, j)- w- closed and (i, j)- g*-closed are independent to each other.

2. Example

Let X = {a, b, c}, 1 = {X, , {a, b}},

Let X = {a, b, c}, = {X, , {a}},

2 = {X, , {a},{b, c}}. Then the set {b} is (1, 2)- g*p-closed but not (1, 2)- g*-closed.

3.40. Proposition

1

2 = {X, , {a},{a, b}}. Then the set {a} is (1, 2)- w-closed but not (1,2)- g*-closed.

1. Example

   closed.

   Every (i, j)- g*-closed set is (i, j)-sg*-

   Let X = {a, b, c},

   1 = {X, , {a}, {a, b}},

   Proof: Let A be (i, j) – g*-closed. Let A U and U is g*-open set in i . Since every g*-open set is g-

   2 = {X, , {a, b}}. Then the set {a} is (1, 2) – g*- closed but not (1,2)- w-closed.

   open, U is g-open. Then j cl (A)

   U. We know

   j cl(A) j cl (A),

   which implies

2. Remark

   j cl(A) U. Therefore A is (i, j)-sg*-closed.

   3.41. Remark

   A (i, j)-sg*-closed need not be (i, j)- g*- closed as shown in the following example.

   The following diagram has shown the relationship of (i, j)- g*-closed sets with other known existing sets. A B represents A implies B but not conversely and A B represents A and B are independent to each other.

   5

   3

   2

1. assumption, we get A is (i, j)- g*-closed. Hence X

is a (i, j) – T* -space.

g* 1

2

1

The converse of the above proposition is not true as seen by the following example.

6

7

1. Example

   Let X = {a, b, c}, 1 = {X, , {a, b}},

   2 = {X, , {b}, {a, b}}. Then X, 1 , 2 is (i, j)-

   10

   T*

   -space. But {a, b} is (i, j)- g-closed but not

   11

   9

   8

   g* 1

   2

   Figure 1

   1. (i, j)- g*-closed set, 2. (i, j)- wg-closed set,

   1. (i, j)- g*-closed set, 4. (i, j)- w-closed set, 5. (i, j)- g-closed set, 6. (i, j)- sg*-closed set, 7. (i, j)- rg-

      (i, j)-g*-closed. Hence X, 1, 2

      space.

2. Proposition

is not

g*T1 –

2

closed set, 8. (i, j)- g*p-closed set, 9. (i, j)- gpr-

closed set, 10. (i, j)- g-closed set, 11. (i, j)- g*- closed set,

1

g*T

2

Every (i, j) –

-space.

g*T1

2

-space is a (i, j) –

1. Applications

   Proof: Let X be a (i, j) –

   g*T1 -space and A be

   2

   In this section we introduce the new closed

   (i, j)- g-closed. Since every (i, j)- g -closed set is (i, j)- g-closed. Then A is (i, j)- g-closed. By

   spaces namely (i, j)-

   T -space, (i, j)-

   T* –

   assumption, we get A is (i, j)- g*-closed. Hence X

   g* 1

   2

   g* 1

   2

   is a (i, j)- T

   -space.

   space, (i, j)- g*T

   -space and (i, j)- gT1

   -space in

   g* 1

   2

   1

   2

   bitopological spaces.

   1. Definition

      2 The converse of the above proposition is not true as seen by the following example.

      1. Example

         A bitopological space X, 1, 2 is said to be a

         1) (i, j)- gT1 – space if every (i, j)- g-closed set is

         2

         Let X = {a, b, c}, 1 = {X, , {b}, {c}, {b,

         c}, {a, b}}, 2 = {X, , {a}}. Then X, 1, 2 is

         (i, j)-g-closed.

         (i, j)-

         g*T

         – space not (i, j)-

         g*T1

         -space. Since

         1

         2) (i, j)-

         T -space if every (i, j)-g-closed set is 2 2

         g* 1

         2

         {b, c} is (i, j)- g-closed but not (i, j)-g*-closed.

         (i, j)- g*-closed.

         Hence X, 1, 2 is not g*T1 -space.

         3) (i, j)- T* -space if every (i, j)-g*-closed set 2

         g* 1

         2

         is (i, j)- g*-closed.

      2. Proposition

      1

      4) (i, j)- g*T

      2

      -space if every (i, j)- g-closed set

      Every (i, j)-

      g*T1

      – space is (i, j)-

      gT1 –

      is (i, j)- g*-closed.

      2 2

      space.

   2. Proposition

   Proof: Let X be a (i, j)-

   g*T1 – space and A be

   2

   Every (i, j)-

   g*T1 -space is a (i, j) –

   2

   g*T* –

   1

   2

   (i, j)- g-closed. Then A is (i, j)- g*-closed. Since every (i, j)-g*-closed set is (i, j)- g-closed. We get

   space.

   Proof: Let X be a (i, j) –

   g*T1

   -space and A be

   A is (i, j)- g-closed. Hence X is a (i, j)-

   space.

   gT1 –

   2

   2

   (i, j)- g*-closed. Since every (i, j)- g*-closed set is (i, j)- g-closed. Then A is (i, j)- g-closed. By

   The converse of the above proposition is not true as seen by the following example.

   4.7. Example

   Let X = {a, b, c},

   = {X, , {a}},

   4.12. Remark

   The following examples show that (i, j)-

   1

   = {X, , {a, b}}. Then X, ,

   is (i, j)-

   g*T

   1

   2

   and (i, j)-

   gT1

   2

   are independent to each

   2 1 2

   gT1

   2

   -space not (i, j)-

   g*T1

   2

   • space. Since {b} is

     other.

     (i, j)- g-closed but not (i, j)g*closed.

     4.8. Proposition

     1. Example

        Let X = {a, b, c}, 1 = {X, , {a}},

        2 = {X, , {a, b}}. Then X, 1, 2

        is (i, j)-

        X, 1, 2

        is both (i, j)-

        gT1

        2

        -space and

        gT1

        -space. But {b} is (i, j)- g-closed but not

        (i, j)- T -space if and only if it is a (i, j)- 2

        g* 1

        2

        (i, j)g*closed.

        g*T1 – space.

        2

        Proof : (Necessity): Let X, 1, 2 be (i, j)- gT1 –

        2

     2. Example

        Let X = {a, b, c},

        1 = {X, , {a, b}},

        space and (i, j)-

        g*T

        -space. Consider A is

        1

        1

        2

        2 = {X, , {a}, {b}, {a, b}}. Then X, 1, 2 is

        (i, j)-g*-closed. Then A is (i, j)-g-closed. Since

        (i, j)- g*T

        -space. But {a, b} is (i, j)- g-closed but

        X, 1, 2

        be (i, j)-

        gT1

        2

        -space, A is (i, j)- g-

        2

        not (i, j)gclosed.

        1

        closed. Since X, 1, 2 be a (i, j)- g*T -space, A

        2

     3. Remark

     is (i, j)- g*-closed. Therefore X, 1, 2 is a (i, j)-

     g*T1

   • space.

   The following examples shows that (i, j)-

   2 T*

   and (i, j)- T are independent to each

   (Sufficiency): It satisfies by Proposition 4.4 and Proposition 4.6.

   4.9. Remark

   g* 1

   2

   other.

   4.16. Example

   g 1

   2

   The following examples show that (i, j)-

   Let X = {a, b, c}, 1 = {X, , {a}},

   T and (i, j)- T* are independent to each

   = {X, , {a, b}}. Then X, ,

   is (i, j)-

   g* 1

   2

   g* 1 2 1 2

   2

   other.

   gT1

   2

   -space. But {c} is (i, j)- g*-closed but not

   4.10. Example

   Let X = {a, b, c}, 1 = {X, , {b}, {c}, {b,

   1

   c}, {a, b}}, 2 = {X, , {a}}. Then X, 1, 2 is

   (i, j)-g*-closed.

   4.17. Example

   Let X = {a, b, c},

   1 = {X, , {a, b}},

   (i, j)-

   g*T

   -space. But {b, c} is (i, j)- g*-closed

   2 = {X, , {a}, {b}, {a, b}}. Then X, 1, 2 is

   2

   but not (i, j)g*closed.

   (i, j)-

   *

   T

   g* 1

   2

   -space. But {a, b} is (i, j)- g-closed but

   4.11. Example

   Let X = {a, b, c}, 1 = {X, , {a}, {b, c}},

   not (i, j)gclosed.

   1. Remark

      2 = {X, }. Then X, 1, 2

      is (i, j)- g*T* -space.

      1

      2

      The following diagram has shown the relationship of (i, j)g*closed spaces with other

      But {b} is (i, j)- g-closed but not (i, j)g*closed.

      known existing space. A B represents A implies B but not conversely and A B represents A and B are independent to each other.

      (i, j) g*T1

      2

      1

      (i, j) g*T*

      2

      (i, j) g*T

      pre continuous mappings, Proc. Math and Phys. Soc. Egypt 55, 1982, pp. 47 53.

      1. Njastad, O, On some classes of nearly open sets,

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         2

         1

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Figure 2

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