Download regular pre semi i locally closed sets in ideal topological spaces

Asian Journal of Current Engineering and Maths 2 : 5 September – October (2013) 312 - 315.
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ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
REGULAR PRE SEMI I LOCALLY CLOSED SETS IN IDEAL TOPOLOGICAL SPACES
B. Sakthi Sathya*, S. Murugesan
Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti. Tamilnadu. India.
Department of Mathematics, S. Ramasamy Naidu Memorial College, Sattur. Tamilnadu. India.
ARTICLE INFO
ABSTRACT
Corresponding Author
B. Sakthi Sathya
Department of Mathematics,
Department of Mathematics, G.
Venkataswamy Naidu College,
Kovilpatti. Tamilnadu. India.
Key Words: rpsI-open, rpsI-closed,
rpsIlc-sets, rpsIlc*-sets, rpsIlc**-sets,
locally rpsI-connected.
In this paper, we introduce and study the new class of sets called regular pre
semi I locally closed sets (briefly. rpsIlc-closed) in ideal topological spaces and
established their relationships with some generalized closed sets in ideal
topological spaces. Also we define and study the notions of RPSILC-continuity
and locally rpsI-connectedness.
AMS Subject Classification: 54A05.
©2013, AJCEM, All Right Reserved.
1. INTRODUCTION
Ideals in topological spaces have been considered
since 1930. This topic has won its importance by the paper
of Vaidyanathaswamy[8]. A subset A of a space (X,) is
locally closed [1] if A is the intersection of an open set and a
closed set. Locally closed sets are further investigated by
Ganster and Reilly in [3]. In 1999, Dontchev[2] introduced
I-locally closed subsets in an ideal topological spaces. In
this paper we introduce new class of sets namely rpsIlc-set,
rpsIlc*-set and
rpsIlc**-set and discuss about their
properties. Finally we use the concepts of rpsIlc-set,
rpsIlc*-set and rpsIlc**-set to define RPSILC-continuous,
RPSILC*-continuous, RPSILC**-continuous functions also
we define locally rpsI-connected space and discuss about
their relationships with some other connected spaces.
2. PRELIMINARIES
Definition 2.1
An ideal is defined as a non-empty collection I of subsets of
X satisfying the following two conditions.
i. If A  I and B  A, then B  I
ii. If A  I and B  I then A  B  I
An ideal topological space is a topological space (X,
) with an ideal I on X and it is denoted by (X, , I). For A 
X, A*(I,) = {x  X / A  U  I, for every U  (x)}, where
(x) = {U   / x  U}. Note that cl*(A) = A  A* defines a
Kuratowski operator for a topology *( I ) (also denoted by
* when there is no chance for confusion), finer than . A
basis β(I, ) = {U \ I : U  and I  I}. β is not always a
topology[5]. cl*(A) and int*(A) will denote the closure and
interior of A in (X, *) respectively.
Definition 2.2
A subset A of an ideal topological space (X, , I) is called
i. regular generalized I-closed[6](rgI-closed) if cl*(A)
 U whenever A  U and U is regular I-open.
ii. pre generalized pre regular I-closed[6](pgprIclosed) if pIcl(A)  U whenever A  U and U is regular
generalized I-open.
iii. regular pre semi I-closed[6](rpsI-closed) if spIcl(A)
 U whenever A  U and U is regular generalized I-open.
The complement of the above mentioned generalized Iclosed sets are their respective I-open sets. rpsIcl(A) is the
smallest rpsI-closed set containing A.
Definition 2.3
A subset A of an ideal topological space (X, , I) is called
a)
*-perfect[4] if A = A*
b)
I-locally closed[2] if A = GV, where G is open and
V is *-perfect.
We will denote the collection of all I-locally closed
sets in (X, , I) by ILC(X,).
3.RPSILC*-SETS AND RPSILC**-SETS
In this section, we introduce rpsIlc-set, rpsIlc*-set
and rpsIlc**-set each of which is stronger than rpsIlc-set
and is weaker than I-locally closed set and study their
relations with existing ones.
Definition 3.1
A subset A of an ideal topological space (X, , I) is called
a)
rpsIlc-set if there exists a rpsI-open set U and a
rpsI-closed set F of X such that A = U  F.
b)
rpsIlc*-set if there exists a rpsI-open set U and a
closed set F of X such that A = U  F.
c)
rpsIlc**-set if there exists an open set U and a rpsIclosed set F of X such that A = U  F.
The collection of all rpsIlc-sets(resp. rpsIlc*-sets, rpsIlc**sets) of (X, , I) will be denoted by RPSILC(X, ) (resp.
RPSILC*(X, ), RPSILC**(X, )).
From the above definitions we have the following
results.
Theorem 3.2
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For an ideal topological space (X, , I) the following
implications hold:
a)
ILC(X,)  RPSILC*(X, )  RPSILC(X, )
b)
ILC(X,)  RPSILC**(X, )  RPSILC(X, )
The reverse implications need not be true as seen from the
following example.
Example 3.3
Let X = {a,b,c,d}  = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}}
and I = {, {a}}. Then we have
ILC(X,) = {, X, {a}, {b}, {c}, {d}, {a,b}, {b,c}, {c,d}, {a,b,c},
{b,c,d}}. RPSILC(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c},
{a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. In this
ideal space the set {a,c} is rpsIlc-set but not Ilc-set.
RPSILC*(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d},
{b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. The set
{a,d} is rpsIlc*-set but not Ilc-set.
Remark 3.4
RPSILC* and RPSILC** are independent of each other.
Example 3.5
Let X = {a,b,c,d}  = {, X, {a,c}, {d}, {a,c,d}} and I =
{, {c},{d},{c,d}}. Then we have RPSILC(X, ) = {, X, {a},
{b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c},
{a,b,d}, {a,c,d}, {b,c,d}}. RPSILC*(X,) = {, X, {a}, {b}, {d},
{a,b}, {a,c}, {a,d}, {b,d}, {a,b,c}, {a,b,d}, {a,c,d}}.
RPSILC**(X,) = {, X, {b}, {c}, {d}, {a,c}, {b,c}, {b,d}, {c,d},
{a,b,c}, {a,c,d}, {b,c,d}}.In this ideal space the set {c} is
rpsIlc**-set but not rpsIlc*-set and the set {a,b} is rpsIlc*set but not rpsIlc**-set. Also the set {b,c} is rpsIlc-set but
not rpsIlc*-set and the set {a} is rpsIlc-set but not rpsIlc**set.
From Theorem 3.2 to Example 3.5 we have the
following diagram.
Theorem 3.6
Let A and B be subsets of (X, , I). If A  RPSILC(X,)
and B  RPSILC(X,) then AB and AB  RPSILC(X,).
Proof:
The proof follows from Definition 3.1
Remark 3.7
The union and intersection of two rpsIlc*-sets need
not be a rpsIlc*-set.
Example 3.8
Let X = {a,b,c,d}  = {, X, {a}, {a,b}, {b,d}, {a,b,d}} and
I = {, {a}}. Then we have
RPSILC*(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {b,d},
{b,c}, {c,d}, {a,b,c}, {a,b,d}, {b,c,d}}. In this ideal space
{a}{c,d}= {a,c,d} which is not rpsIlc*-set.
Example 3.9
Let X = {a,b,c,d}  = {, X, {a,b,c}} and I = {, {a}}.
Then we have
RPSILC*(X, ) = {, X, {b}, {c}, {d}, {a,b}, {a,c}, {b,c}, {b,d},
{c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}}. In this ideal space {a,b}
 {a,c}={a} which is not rpsIlc*-set.
Remark 3.10
The union and intersection of two rpsIlc**-sets need
not be a rpsIlc**-set.
Example 3.11
Let X = {a,b,c,d}  = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}}
and I = {, {a}}. Then we have RPSILC**(X, ) = {, X, {a},
{b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {c,d}, {a,b,c}, {a,b,d},
{a,c,d}, {b,c,d}}. In this ideal space {a,b,d}  {b,c,d}={b,d},
{b}{d}= {b,d} which are not rpsIlc**-set.
Theorem 3.12
Let A be any subset of X, a) A is rpsI closed in X if and
only if A = rpsIcl(A) b) rpsIcl(A) is rpsI closed in X c) x 
rpsIcl(A) if and only if A  U   for every rpsI-open set U
containing x.
Proof:
(a)
and (b) are trivially true.
(c). Suppose that there exists a rpsI-open set U
containining x such that A  U = . Since X\U is rpsI-closed
and A  X\U, rpsIcl(A)  X\U hence x  rpsIcl(A).
Conversely, suppose that x  rpsIcl(A). Then U = X\
rpsIcl(A) is rpsI-open set containing x and A  U = .
Theorem 3.13
For a subset A of (X, , I), the following statements
are equivalent.
a)
A  RPSILC(X,).
b)
A = U  rpsIcl(A) for some rpsI-open set U.
c)
rpsIcl(A)\A is rpsI-closed.
d)
A  (X\ rpsIcl(A)) is rpsI-open.
Proof:
(a)  (b). Suppose A  RPSILC(X,). Then there exists
a rpsI-open subset U and rpsIclosed subset F such that A = U  F. Since A  U and A 
rpsIcl(A), A  U  rpsIcl(A). Conversely by Theorem
3.12(b) rpsIcl(A)  F and hence U  rpsIcl(A)  U  F = A.
Therefore A = U  rpsIcl(A).
(b)  (a). By Theorem 3.12(b) rpsIcl(A) is rpsI closed
and hence A = U  rpsIcl(A) 
RPSILC(X,).
(c)  (d). Let S = rpsIcl(A)\A. Then by the assumption
S is rpsI-closed which implies X\S is rpsI-open and X/S = X
 Sc = X  (rpsIcl(A)\A)c = A  (X\rpsIcl(A)). Thus A  (X\
rpsIcl(A)) is rpsI-open.
(d)  (c). Let W = A  (X\rpsIcl(A)). Then W is rpsIopen. This implies that X\W is rpsI-closed and X\W = X \ A
 (X\ rpsIcl(A)) = rpsIcl(A)  X\A = rpsIcl(A)\A. Thus
rpsIcl(A)/A is rpsI-closed.
(d)  (b). Let U = A  (X\ rpsIcl(A)). Then U is rpsIopen. Now U  rpsIcl(A) = (A  (X\rpsIcl(A)))  rpsIcl(A)
= (rpsIcl(A)  A)  (rpsIcl(A)  X\rpsIcl(A)) = A   = A.
Therefore A = U  rpsIcl(A) for some rpsI-open set U.
(b)
 (d). Let A = U  rpsIcl(A), for some rpsI-open
set U. A  (X\ rpsIcl(A)) = (U 
rpsIcl(A))  (X\ rpsIcl(A)) = U  (rpsIcl(A)  (X\
rpsIcl(A))) = U  X = U, is rpsI-open.
Theorem 3.14
For a subset A of (X, , I), the following statements
are equivalent.
a) A  RPSILC*(X,).
b) A = U  cl(A) for some rpsI-open set U.
c) cl(A)\A is rpsI-closed.
d) A  (X\cl(A)) is rpsI-open.
Proof:
The proof is similar to that of Theorem 3.13
Theorem 3.15
Let A be a subset of (X, , I). Then A  RPSILC**(X,)
if and only if A = U  rpsIcl(A) for some open set U.
Proof:
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Let A  RPSILC**(X,). Then A = U  F where U is
open and F is rpsI-closed. Since
A  F implies rpsIcl(A)  F. Now A = A  rpsIcl(A) = (U 
F)  rpsIcl(A) = U  rpsIcl(A). The converse part is
obvious.
Theorem 3.16
Let A be a subset of (X, , I). If A  RPSILC**(X,) then
rpsIcl(A)\A is rpsI-closed and
A  (X\ rpsIcl(A)) is rpsI-open.
Proof
Suppose A  RPSILC**(X,) then by Theorem 3.15
there exists a open set U such that A = U  rpsIcl(A). A 
(X\ rpsIcl(A)) = (U  rpsIcl(A))  (X\ rpsIcl(A)) = U 
(rpsIcl(A)  (X\ rpsIcl(A))) = U  X = U, which is open.
Since every open set is rpsI-open we have A  (X\
rpsIcl(A)) is rpsI-open.
Let W = A  (X\ rpsIcl(A)). Then W is rpsI-open. This
implies that X\W is rpsI-closed
and X\W = X \ A  (X\ rpsIcl(A)) = rpsIcl(A)  X\A =
rpsIcl(A)\A. Thus rpsIcl(A)\A is rpsI-closed.
The converse of the above theorem need not be true as
seen from the following example.
Example 3.17
Let X = {a,b,c,d}  = {, X, {a,b,c}} and I = {, {a}}.
Then we have {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,d},
{c,d}, {a,b,d}, {a,c,d}} is the set of all rpsI-closed sets in X
and RPSILC**(X, ) = {, X, {a}, {b}, {c}, {d}, {a,b}, {a,c},
{a,d}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}}. If A = {b, c} then
rpsIcl(A)\A = {a,d} is rpsI-closed and A  (X\ rpsIcl(A) =
{b,c} is rpsI-open but A  RPSILC**(X,).
4. RPSILC*-continuous and RPSILC**-continuous
In this section we define RPSILC*-continuous and
RPSILC**-continuous functions in ideal topological spaces
and discuss about their relations.
Definition 4.1
A function f : (X,,I)  (Y,) is called ILC-continuous[2] if f 1(A) is Ilc-set in X, for every open subset A of Y.
Definition 4.2
a) A function f : (X,,I)  (Y,) is called RPSILC-continuous
if f -1(A) is rpsIlc-set in X, for every open subset A of Y.
b) A function f : (X,,I)  (Y,) is called RPSILC*-continuous
if f -1(A) is rpsIlc*-set in X, for every open subset A of Y.
c) A function f : (X,,I)  (Y,) is called RPSILC**continuous if f -1(A) is rpsIlc**-set in X, for every open
subset A of Y.
Theorem 4.3
Let f:(X,,I)  (Y,) be a function.
a) If f is ILC-continuous, then f is RPSILC*-continuous.
b) If f is RPSILC*-continuous, then f is RPSILC-continuous.
c) If f is RPSILC**-continuous, then f is RPSILC-continuous.
d) If f is ILC-continuous, then f is RPSILC-continuous.
Proof:
The proof follows from Theorem 3.2, Converse of the above
theorem need not be true as seen from the following
example.
Example 4.4
Let X = Y = {a,b,c,d}.  = {, X, {a}, {b}, {a,b}, {b,c},
{a,b,c}} and I = {, {a}},  = {, Y, {a,c}, {d}, {a,c,d}}. Define
f : (X,, I)  (Y,) by f(a) = d, f(b) = c, f(c) = b, f(d) = d. Then
f is RPSILC*-continuous but not ILC-continuous.
Example 4.5
Let X = Y = {a,b,c,d}.  = {, X, {a,c}, {d}, {a,c,d}}. and I
= {, {c},{d},{c,d}},  = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}.
Define f : (X,, I)  (Y,) by f(a) = d, f(b) = b, f(c) = b, f(d) =
c.
Then f is RPSILC-continuous but not RPSILC*continuous.
Example 4.6
Let X = Y = {a,b,c,d}.  = {, X, {a,c}, {d}, {a,c,d}}. and I
= {, {c},{d},{c,d}},  = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}.
Define f : (X,, I)  (Y,) by f(a) = a, f(b) = c, f(c) = f(d) = d.
Then f is RPSILC-continuous but not RPSILC**-continuous.
Example 4.7
Let X = Y = {a,b,c,d}.  = {, X, {a}, {b}, {a,b}, {b,c},
{a,b,c}} and I = {, {a}},  = {, Y, {a,c}, {d}, {a,c,d}}. Define
f : (X,, I)  (Y,) by f(a) = f(c) = d, f(b) = b, f(d) = c. Then f
is RPSILC-continuous but not ILC-continuous.
Remark 4.8
RPSILC*-continuous
and
RPSILC**-continuous
functions are independent of each other.
Example 4.9
Let X = Y = {a,b,c,d}.  = {, X, {a,c}, {d}, {a,c,d}}. and I
= {, {c},{d},{c,d}},  = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}}.
a) Define f : (X,, I)  (Y,) by f(a) = d, f(b) = c, f(c)
= a, f(d) = c. Then f is RPSILC**-continuous but not
RPSILC*-continuous.
b) Define f : (X,, I)  (Y,) by f(a) = b, f(b) = b, f(c)
= d, f(d) = c. Then f is RPSILC*-continuous but not
RPSILC**-continuous.
Theorem 4.3 to Example 4.9 we have the following Diagram
5. RPSILC-irresolute Functions
Definition 5.1
a) A function f : (X,,I)  (Y,) is called RPSILC-irresolute if
f -1(A) is rpsIlc-set in X, for every rpsIlc-set A of Y.
b) A function f : (X,,I)  (Y,) is called RPSILC*-irresolute
if f -1(A) is rpsIlc*-set in X, for every rpsIlc*-set A of Y.
c) A function f : (X,,I)  (Y,) is called RPSILC**-irresolute
if f -1(A) is rpsIlc**-set in X, for every rpsIlc**-set A of Y.
Theorem 5.2
Let f:(X,,I)  (Y,) be a function.
a) If f is RPSILC* irresolute function then f is RPSILCirresolute function.
b) If f is RPSILC**-irresolute function, then f is RPSILCirresolute function.
Proof:
The proof follows from Theorem 3.2
The converse of the above theorem need not be true as
seen from the following example.
Example 5.3
Let X = Y = {a,b,c,d}.  = {, X, {a,c}, {d}, {a,c,d}}. and I = {,
{c},{d},{c,d}},  = {, Y, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and J =
{,{a}}. Define f : (X,, I)  (Y,, J) by f(a) = a, f(b) = b, f(c)
= c, f(d) = d. Then f is RPSILC-irresolute but not RPSILC*irresolute and RPSILC**-irresolute.
Remark 5.4
RPSILC*-irresolute and RPSILC**-irresolute functions
are independent of each other.
Example 5.5
Let X = Y = {a,b,c,d}.  = {, X, {a}, {a,b}, {b,d}, {a,b,d}},
I = {, {a}},  = {, Y,
{a}, {b}, {a,b}, {b,c}, {a,b,c}} and J = {,{a}}.
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a) Define f : (X,,I)  (Y,,J) by f(a) = b, f(b) = a, f(c) =
a, f(d) = d. Then f is RPSILC*-irresolute but not RPSILC**irresolute.
b) Define f : (X,, I)  (Y,, J) by f(a) = b, f(b) = c, f(c)
= d, f(d) = b. Then f is RPSILC**-irresolute but not RPSILC*irresolute.
Theorem 5.2 to Example 5.5 we have the following
Diagram.
6 Locally rpsI-connected Spaces:
In this section the idea of locally rpsI-connectedness
in an ideal topological space is introduced and obtain some
of its basic properties.
Definition 6.1
An ideal topological space (X,, I) is called semiIconnected if it can not be written as the union of two
disjoint non empty semiI open sets.
Definition 6.2
An ideal topological space (X,, I) is called rpsIconnected[7] if it can not be written as the union of two
disjoint non empty rpsI-open sets.
Definition 6.3
An ideal topological space (X,, I) is called locally
connected at x  X if for every open set U containing x,
there exists a connected, open set C such that x  C  U.
An ideal topological space (X,, I) is called locally
connected if it is locally connected at every point of X.
Definition 6.4
An ideal topological space (X,, I) is called locally
rpsI-connected at x  X if for every rpsI-open set E
containing x, there exists a rpsI-connected, open set C such
that x  C  E.
An ideal topological space (X,, I) is called locally
rpsI-connected if it is locally rpsI-connected at every point
of X.
Lemma 6.5[7]
Every rpsI-connected space is connected.
Remark 6.6
Every locally rpsI-connected space is locally
connected but the converse is not true as seen from the
following example, consider the ideal topological space
(X,, I), where X = {a,b,c,d},  = {, X, {a}, {b,d}, {a,b,d}}. and
I = {, {a}}. rpsI-open sets of X are , X, {a}, {b}, {d}, {a,b},
{a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}
and rpsI-closed sets are , X, {a}, {b}, {c}, {d}, {a,b}, {a,c},
{a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,c,d}, {b,c,d}.Here X is
locally connected but not locally rpsI-connected because
the sets X, {b,c,d}, {a,b,d}, {a,b,c}, {b,d}, {b,c}, {a,b}, {b} are
rpsI-open sets containing b, but there is no open subset of
{b} containing b and so X is not locally rpsI-connected.
Remark 6.7
Locally rpsI-connected and rpsI-connected spaces
are independent of each other. For example, let X ={a,b,c,d},
 = {, X, {a}, {b}, {a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then
we have {a,b} is locally rpsI-connected but not rpsIconnected and {b,c,d} is rpsI-connected but not locally rpsIconnected.
Remark 6.8
Locally rpsI-connected and semiI-connected spaces
are independent of each other. For example, consider the
ideal topological space in Remark 5.5, {b,c,d} is semiIconnected but not locally rpsI-connected and {a,b} is locally
rpsI-connected but not semiI-connected.
Theorem 6.9
Every rpsI-connected space is locally-connected.
Proof:
Proof follows from Lemma 6.5
But the converse need not be true as seen from the
following example. Consider X = {a,b,c,d},  = {, X, {a}, {b},
{a,b}, {b,c}, {a,b,c}} and I = {, {a}}. Then we have {a,b} is
locally connected but not rpsI-connected.
From Remark 6.5 to Theorem 6.9 we have the
following Diagram
REFERENCES
[1] N. Bourbaki, General Topology, Part I, Addison-Wesley,
Reading, Mass, 1966.
[2] J.Dontchev, Idealization of
Ganster Reilly
Decomposition Theorems, Math.GN/9901017,(5), Jan.1999.
[3] M. Ganster and I.L.Reilly, Locally closed sets and Lccontinuous
functions,
Internat.
,J.
Math
and
Sci.,12(1989),417-424.
[4] E. Hatir, A. Keskin and T. Noiri, On a new decomposition
of continuity via idealization, JP Jour. Geometry and
Topology, 3(1)(2003), 53-64.
[5] K. Kuratowski, Topology, Academic Press, New York,
Vol 1(1966).
[6] B. Sakthi alias Sathya and S. Murugesan, On regular pre
semi I-closed sets in ideal topological spaces, International
Journal of Mathematics and soft computing, vol
3,(1)(2013), 43-52.
[7] B. Sakthi alias Sathya and S. Murugesan, connectedness
and compactness in ideal topological spaces, Internat.,
J.Math Sci.,(communicated).
[8] R. Vaidyanathaswamy, The localization theory in set
topology, Proc. Indian Acad.Sci. Math. Sci., 20(1945), 51-61
.
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