 Open Access
 Total Downloads : 324
 Authors : A. P. Ushpalatha, K. Kavithamani
 Paper ID : IJERTV2IS4974
 Volume & Issue : Volume 02, Issue 04 (April 2013)
 Published (First Online): 27042013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Study On C*GClosed Sets In Bitopological Spaces
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 2 Issue 4, April – 2013
A Study On C*GClosed Sets In Bitopological Spaces
A. P ushpalatha
Professor Department of Mathematics, Government Arts College,
Udumalpet642 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*gclosed

Kavithamani
Research Scholar Karpagam University Coimbatore District Tamil Nadu, India
.

INTRODUCTION 2. PRELIMINARIES

A triple (x,
, ) where X is
DEFINITION 2.1: A subset A of X is
1 2
called
nonempty set and 1 and 2 are
topologies on X is called a

(i,j)*generalized closed (briefly (i,j)
bitopological space and Kelly [11]
gclosed) [5] if
j cl(A) U
initiated the study of such spaces. In 1985 Fukutake [5] introduced the concepts of gclosed 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.

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

(i,j)generalized pre regular closed (briefly (i,j)gprclosed) [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)wgclosed) [6] if
1 2 1 2
Y) denote two non empty bitopological spaces. In this section we introduce the concept of (i,j)c*gclosed 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.

(i,j)strongly generalized closed (briefly (i,j)strongly gclosed) [12] if j cl(A) U whenever A U
and U is i gopen in X.

(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

(i,j) generalized closed (briefly (i,j)gclosed) [3] if j cl(A) U
(i,j) c*gclosed 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.

(i,j) generalized semiclosed
EXAMPLE 3.3: Consider the
(briefly (i,j)gsclosed) [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*gCLOSED 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*gclosed set but not 2 closed.
THEOREM 3.4: Union of two (i,j) c*gclosed sets is (i,j) c*gclosed.
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*gclosed 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*gclosed,
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*gclosed sets in (X, i , j ) by C*(i,j).
j cl(B) U. Therefore AUB is (i,j)
c*gclosed.
THEOREM 3.2: Every j closed set
in (X, i , j ) is (i,j)c*gclosed 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*gclosed 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*gclosed, 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 gclosed 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.
jclosed
(i,j)strongly gclosed
(i,j)c*g closed
(i,j) gprclosed.
Figure3.1.1
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)semiclosed. 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)semiclosed.
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*gclosed
REMARK 3.21:
(i,j)preclosed (i,j)semiclosed (i,j) closed
(i,j) closed
(i,j)gclosed
(i,j)gsclosed
(i,j)sgclosed
(i,j)wclosed
(i,j)wgclosed
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.
rgclosed 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 rgclosed 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 rgclosed 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 rgclosed. For the same topology the set A={a,b} is 1 rgclosed but not (1,2)c*g closed set.
EXAMPLE 3.27: Consider the topological space X={a,b,c} with the
(1,2)c*gclosed
2 rgclosed
2 gprclosed
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 gprclosed. 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 –
rgclosed 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*gclosed
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).

Di Maio. G and T.Noiri., 1987. On s closed spaces. Indian J. Pure appl. Math., 18 : 226233.

Fukutake.T., 1985.On generalized closed sets in bitopological spaces.Bull.Fukuoka Univ.Ed.Part III., 35:1928.

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: 3340.
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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