A Study On C*G-Closed Sets In Bitopological Spaces

A Study On C*G-Closed Sets In Bitopological Spaces

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

Vol. 2 Issue 4, April – 2013

A Study On C*G-Closed Sets In Bitopological Spaces

A. P ushpalatha

Professor Department of Mathematics, Government Arts College,

Udumalpet-642 126, Tirupur District, Tamil Nadu, India.

Abstract

In this paper, we have introduced the concept of c*g closed and some of their properties in bitopological space.

Key words: (I,j)-c*g-closed

1. Kavithamani

   Research Scholar Karpagam University Coimbatore District Tamil Nadu, India

   .

   1. INTRODUCTION 2. PRELIMINARIES

A triple (x,

, ) where X is

DEFINITION 2.1: A subset A of X is

1 2

called

non-empty set and 1 and 2 are

topologies on X is called a

1. (i,j)*-generalized closed (briefly (i,j)-

   bitopological space and Kelly [11]

   g-closed) [5] if

   j -cl(A) U

   initiated the study of such spaces. In 1985 Fukutake [5] introduced the concepts of g-closed sets in bitopological spaces and after that several authors turned their attention to the generalization of various concepts of topology by considering bitopological spaces instead of topological spaces. In 2004, P.Sundaram [12] introduced the concept of g*-closed sets in bitopological spaces.

   Throughout this chapter (X, , ) (or X)and (Y, , ) (or

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

2. (i,j)- regular generalized closed (briefly (i,j)-rg-closed) [1] if j – cl(A) U whenever A U and U is i – regular open in X.

3. (i,j)-generalized pre regular closed (briefly (i,j)-gpr-closed) [7] if pcl(A) U whenever A U and U is i – regular open in X.

iv)(i,j)-weakly generalized closed (briefly (i,j)-wg-closed) [6] if

1 2 1 2

Y) denote two non empty bitopological spaces. In this section we introduce the concept of (i,j)-c*g-closed sets and we obtain some interesting results in bitopological spaces.

j -cl(int(A)) U whenever A U

and U is i – open in X.

1. (i,j)-strongly generalized closed (briefly (i,j)-strongly g-closed) [12] if j -cl(A) U whenever A U

   and U is i -g-open in X.

2. (i,j)-weakly closed (briefly (i,j)-w-

   closed) [7] if

   j -cl(A) U

   whenever A U and U is

   i -semi

   Since A is j -closed,that is j -cl(A) = A.

   open in X.

   Therefore

   j -cl (A) U. Hence A is

3. (i,j) -generalized -closed (briefly (i,j)-g-closed) [3] if j -cl(A) U

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

   The converse of the above

   whenever A U and U is open in X.

   i – –

   theorem need not be true as seen from the following example.

4. (i,j) generalized semi-closed

EXAMPLE 3.3: Consider the

(briefly (i,j)-gs-closed) [3] if

j –

topological space X = {a,b,c} with the

scl(A) U whenever A U and U

topologies

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

is i – open in X.

3. c*g-CLOSED -CLOSED SETS IN

BITOPOLOGICAL SPACES

DEFINITION 3.1: A subset A of a

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

THEOREM 3.4: Union of two (i,j)- c*g-closed sets is (i,j)- c*g-closed.

Proof: – Let A and B be (i,j)- c*g-

bitopological space (X, i , j ) is said to

closed sets in X. Let U be a

i -c*-set

be an (i,j)-c*g-closed set if j -cl(A) U

in X. such that AUB U. Then A U

whenever A U and U is

i -c*-set in

and B U. Since A and B are (i,j)-

X. c*g-closed,

j -cl(A) U and j -cl(B)

We denote the family of all (i,j)-

U. Hence

j -cl(AUB) =

j -cl(A) U

c*g-closed sets in (X, i , j ) by C*(i,j).

j -cl(B) U. Therefore AUB is (i,j)-

c*g-closed.

THEOREM 3.2: Every j -closed set

in (X, i , j ) is (i,j)-c*g-closed set in (X, i , j ) but not conversely.

Proof: – Let A be a j -closed set in X.

REMARK 3.5: Intersection of two (i,j)

-c*g -closed sets in X need not be (i,j) – c*g closed sets in X is proved in the

following example.

Let U be a i

-c* -set such that A U.

EXAMPLE 3.6: Consider the

Conversely assume that

j –

topological space X = {a,b,c} with the topologies 1 = { ,X,{a}};

cl(A)/A contains no non empty i -c*- set. Let A U, U is i -c*- set. Suppose

2 = { ,X,{a}, {c}, {a,c}}. In this

that

j -cl(A) is not contained in U.

topology consider the set {b,c} and

{a,c}. Intersection of these two sets not contained in (1,2)-c*g-closed sets.

Then – cl(A) Uc is a non empty – c*- set and contained in j -cl (A)/A,

j i

j i

Therefore intersection of two (1,2) -c*g

which is contradiction. Therefore

j –

-closed sets in X is not (1,2) -c*g closed sets in X.

THEOREM 3.7: A subset A of X is

cl(A) U. Hence A is (i,j)-c*g -closed.

REMARK 3.8: The converse of the above two theorems is not true as seen

(i,j)-c*g -closed in X if and only if

j –

from the following example.

cl(A)/A does not contain any non empty i -c*- set in X.

Proof:- Suppose that A is a (i,j)-c*g –

closed set in X. We prove the result by

EXAMPLE 3.9: Consider

X = {a, b, c} with the topology

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

contradiction. Let U be i -c*- set such

and 2

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

that U

j -cl(A)/A and U . Then

A={a},then 2 -cl(A)/A={{a,c}/{a}} =

c

c

i

i

c

c

c

c

c

c

U j -cl(A) A . Therefore U j – cl(A) and U Ac is -c*- set and A is (i,j)-c*g-closed, j -cl(A) U . That is U [ j -cl(A)] . Hence U j –

cl(A)[ j -l(A)] .That is U .

{c}does not contain any non empty

1 -c*- set, but A={c} is not (1,2)-c*g – closed set in X.

THEOREM 3.10: Every (i,j)-strongly g-closed set in X is a (i,j)-c*g -closed

Which is contradiction. Hence

j –

set in X but not conversely.

cl(A)/A does not contain any non empty i -c*- set in X.

The converse of the above theorem need not be true as seen from the following example.

EXAMPLE 3.11: Consider the topological space X={a,b,c} with the topologies 1 = { ,X,{a,b}};

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

THEOREM 3.12: Every (i,j)- c*g – closed set in X is a (i,j)-gpr- closed set in X but not conversely.

The converse of the above theorem need not be true as seen from the following example.

EXAMPLE 3.13: Consider the

topological space X={a,b,c} with the

In the above diagram none of the implications can be reversed.

REMARK 3.15:

The concept of (i,j)-c*g -set is independent of the following classes of sets namely (i,j)– closed, (i,j)-g-

closed, (i,j)-wg- closed, (i,j)- -closed,

(i,j)-g -closed, (i,j)-w -closed, (i,j)-rw- closed, (i,j)-pre- closed and (i,j)- g closed.

EXAMPLE 3.16: Consider the

topologies

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

topological space X={a,b,c} with the

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

REMARKS 3.14: From the above theorem and example we get the following diagram.

j-closed

(i,j)-strongly gclosed

(i,j)-c*g -closed

(i,j)- gpr-closed.

Figure-3.1.1

topologies 1 = { ,X, {a}}and

2 = { ,X,{a},{c},{a,c}}.Then the set A= {c} is not (1,2)-c*g- closed set but (1,2)– closed, (1,2)- closed, (1,2)- pre- closed and (1,2)-semi-closed. In the same topologies the set A= {a} is (1,2)-c*g- closed set but not (1,2)–

closed, not (1,2)- closed, not (1,2)- pre- closed and not (1,2)-semi-closed.

EXAMPLE 3.17: Consider the topological space X={a,b,c} with the topologies 1 = { ,X, {b}}and

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

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

REMARK 3.20: From the above discussion and known results we have the following diagram.

EXAMPLE 3.18: Consider the topological space X={a,b,c} with the topologies 1 = { ,X, {a}}and

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

In this bitopologies the set A= {c} is not (1,2)-c*g- closed set but (1,2)- w- closed and (1,2)- wg- closed. For the same topologies,

The set A={a,b} is (1,2)-c*g- closed set but not (1,2)- w- closed.

The set A={b,c} is (1,2)-c*g- closed set but not (1,2)- wg- closed.

(i,j)-c*g-closed

REMARK 3.21:

(i,j)-pre-closed (i,j)-semi-closed (i,j)- -closed

(i,j)- -closed

(i,j)-g-closed

(i,j)-gs-closed

(i,j)-sg-closed

(i,j)-w-closed

(i,j)-wg-closed

EXAMPLE 3.19: Consider the topological space X={a,b,c} with the

topologies 1 ={ ,X, {b},{c},{b,c}}and

The concept of (i,j)-c*g closed set is independent of the following classes of sets namely j -strongly g- closed, j –

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

The set A= {c} is not (1,2)-c*g- closed set but (1,2)- gs- closed and (1,2)- sg- closed.

The set A={a,b} is (1,2)-c*g- closed set

but not (1,2)- gs- closed and (1,2)- sg- closed.

rg-closed and j -gpr closed.

EXAMPLE 3.22: Consider the topological space X={a,b,c} with the topologies 1 = { ,X,{a}}and

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

2 -strongly g- closed,

2 -gpr closed.

2 -rg-closed and

Consider another topologies 1 =

{ ,X, {b},{a,c}}and 2 = { ,X,{a,c}}. Then the set A= {a,b} is 2 -strongly g-

EXAMPLE 3.26: Consider the topological space X={a,b,c} with the

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

closed, 2 -rg-closed and 2 -gpr closed

{a,c}}and

2 = { ,X}.Then the set

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

REMARK 3.23: From the above discussion and known results we have the following diagram.

2 -strongly g- closed

A= {a,c} is (1,2)-c*g- closed set but not 1 -rg-closed. For the same topology the set A={a,b} is 1 -rg-closed but not (1,2)-c*g- closed set.

EXAMPLE 3.27: Consider the topological space X={a,b,c} with the

(1,2)-c*g-closed

2 -rg-closed

2 -gpr-closed

topologies

{a,c}}and

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

2 = { ,X,{a}}.Then the set

A= {a,c} is (1,2)-c*g- closed set but

REMARK 3.24:

not

1 -gpr-closed. For the same

The concept of (i,j)-c*g -set is

topology the set A={b} is

1 -gpr-

independent of the following classes of

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

sets namely

i -strongly g- closed,

j –

rg-closed and j -gpr closed.

EXAMPLE 3.25: Consider the topological space X={a,b,c} with the

topologies 1 = { ,X, {c},{a,b}}and

REMARK 3.28: From the above discussion and known results we have the following diagram.

1 -strongly g- closed

2 = { ,X,{b},{a,b},{b.c}}.Then the set A= {a} is (1,2)-c*g- closed set but

(1,2)-c*g-closed

1 -rg- closed

1 -gpr- closed

not 1 -g*-closed. For the same topology the set A={a,b} is 1 -g*- closed but not (1,2)-c*g- closed set.

REMARK 3.29: c*(i,j) is generally not equal to c*(j,i). Consider the following example.

EXAMPLE 3.30: Consider the topological space X={a,b,c} with the topologies 1 ={ ,X,{b},{c}, {b,c}}and

2 = { ,X,{b}}.Then the set A= {b} is

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

1. Di Maio. G and T.Noiri., 1987. On s- closed spaces. Indian J. Pure appl. Math., 18 : 226-233.

2. Fukutake.T., 1985.On generalized closed sets in bitopological spaces.Bull.Fukuoka Univ.Ed.Part- III., 35:19-28.

3. Fukutake.T, P.Sundaram and N.Nagaveni.,1999.Bull. Fukuoka

REMARK 3.31: If

1

2 in (X,1 , 2 )

Univ. Ed. Part III., 48: 33-40.

then c*(2,1) c*(1,2). The converse of this remark is not true as seen from the following example.

EXAMPLE 3.32: Let the topological space X={a,b,c} with the topologies

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

2 = { ,X,{b},{c},{b,c}}. In this case c*(2,1) c*(1,2) but1 2 .

*****

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