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 Total Downloads : 656
 Authors : R.Sudha, K.Sivakamasundari
 Paper ID : IJERTV1IS8366
 Volume & Issue : Volume 01, Issue 08 (October 2012)
 Published (First Online): 29102012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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) – gclosed 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) gclosed 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 gclosed 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 ElMonsef [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.

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

semiopen set [9] if A cl(int(A)).

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

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

Preopen set [11] if A int(cl(A)).
The complement of a semi open (resp. open, regular open, preopen) set is called semiclosed (resp. closed, regular closed, preclosed).
The semiclosure [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 semiclosed (resp.closed, pre closed) sets containing A. It is known that scl(A) (resp. cl (A) , pcl(A)) is a semiclosed (resp.
closed, preclosed) set.


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

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

(i, j) g*pclosed [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 gopen in i .

(i, j) wclosed [7] if j cl(A) U whenever
A U and U is semiopen in i .


Definition

(i, j) sg*closed [13] if
j cl(A) U
A subset A of (X,) is called

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

generalized closed (briefly gclosed) [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)gclosed set is

g* closed [19] if cl(A) U whenever A U and U is gopen in (X, ).
Throughout this paper by the spaces X and Y represent nonempty 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)gclosed 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 gopen
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.

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

(i, j) gclosed [5] if
j cl(A) U whenever
j
UGO X, i
A U and U is open in i .

(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 gopen in i .

(i, j) rgclosed [2] if j cl(A) U whenever
A U and U is regular open in i .

(i, j) wgclosed [6] if j cl( i int(A)) U


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 .

(i, j) gprclosed [6] if j pcl(A) U whenever A U and U is regular open in i .


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 gopen 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.

Proposition
For each element x of X, 1, 2 , x is
i gclosed or xc is (i, j) g*closed.
closed, then A is j –closed.
Proof: Let A be both i gopen and (i, j)g*
Proof: If x is i gclosed, then the proof is over. Assume x is not i gclosed. Then xc is not
closed. Since A is (i, j)g*closed, we have
i gopen. So the only i gopen containing xc
A U and U GO X, i which implies
j cl (A) U and since A is i gopen. 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.

Proposition
If A is (i, j) – g*closed, then

Proposition
j cl (A) / A
set.
contains no nonempty i gclosed
If A is both i gopen and (i, j) g*
Proof: Let A be (i, j)g*closed and F be a non empty i gclosed 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 gopen

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 gclosed 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 gclosed set. But A is not (1, 2) – g*
3.16. Proposition
If A is an (i, j)g*closed set of X, i , j

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 gopen set in
i j such
gclosed.
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 gclosed.
(Sufficiency) If j cl (A) / A is i gclosed, 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.

Proposition
j cl (A) A. Therefore

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 gopen
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 gopen, B is gopen in i . Since A is (i, j) g*closed, we get

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)gclosed. 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 gopen set. Then
j cl (A)
U, we
3.28. Proposition
know that j cl(U)
is (i, j) gclosed.

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 gopen, U
A (i, j)gclosed need not be (i, j) g* closed as shown in the following example.
is gopen 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

Example
Let X = {a, b, c},
1 = {X, , {a}},
2 =
j cl(i int(A)) U. Hence A is (i, j)wgclosed.
{X, , {a, b}}. Then the set {b} is (1, 2)gclosed but not (1, 2) – g*closed.

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 gopen set in i Then j cl (A) U, we

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

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)

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.

Proposition
Every (i, j) g*closed set is (i, j)rg
closed.
Proof: The proof follows from every regular open
closed.

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

Example
set is gopen.
Let X = {a, b, c},
1 = {X, , {a, b}},

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

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)gprclosed. Proof: Let A be (i, j) g*closed. Let A U and U be regular open. Since every regular open set is gopen, U is gopen. 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)gprclosed.
3.35. Remark
A (i, j)gprclosed 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

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

Proposition
Every (i, j) g*closed set is (i, j)g*p
closed.
Proof: Let A be (i, j) g*closed. Let

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

Example
A U and U is gopen 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*pclosed.

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

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

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

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

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) wclosed.
open, U is gopen. Then j cl (A)
U. We know
j cl(A) j cl (A),
which implies

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

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

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) gclosed but not
11
9
8
g* 1
2
Figure 1
1. (i, j) g*closed set, 2. (i, j) wgclosed set,

(i, j) g*closed set, 4. (i, j) wclosed set, 5. (i, j) gclosed set, 6. (i, j) sg*closed set, 7. (i, j) rg
(i, j)g*closed. Hence X, 1, 2
space.


Proposition
is not
g*T1 –
2
closed set, 8. (i, j) g*pclosed set, 9. (i, j) gpr
closed set, 10. (i, j) gclosed set, 11. (i, j) g* closed set,
1
g*T
2
Every (i, j) –
space.
g*T1
2
space is a (i, j) –

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

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

Example
A bitopological space X, 1, 2 is said to be a
1) (i, j) gT1 – space if every (i, j) gclosed 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)gclosed.
(i, j)
g*T
– space not (i, j)
g*T1
space. Since
1
2) (i, j)
T space if every (i, j)gclosed set is 2 2
g* 1
2
{b, c} is (i, j) gclosed 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.

Proposition
1
4) (i, j) g*T
2
space if every (i, j) gclosed set
Every (i, j)
g*T1
– space is (i, j)
gT1 –
is (i, j) g*closed.
2 2
space.


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) gclosed. Then A is (i, j) g*closed. Since every (i, j)g*closed set is (i, j) gclosed. We get
space.
Proof: Let X be a (i, j) –
g*T1
space and A be
A is (i, j) gclosed. Hence X is a (i, j)
space.
gT1 –
2
2
(i, j) g*closed. Since every (i, j) g*closed set is (i, j) gclosed. Then A is (i, j) gclosed. 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) gclosed but not (i, j)g*closed.
4.8. Proposition

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) gclosed 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

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)gclosed. Since
(i, j) g*T
space. But {a, b} is (i, j) gclosed 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

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) gclosed but
4.11. Example
Let X = {a, b, c}, 1 = {X, , {a}, {b, c}},
not (i, j)gclosed.

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) gclosed 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
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Njastad, O, On some classes of nearly open sets,
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2
(i, j) gT1

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2
1



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Figure 2

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