CONTENTS

J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011
We only prove part(3), the proof of the other
parts is obvious.
Proof.(3). Let G be any cp-open set in X
containing x, by Corollary 2.10,A is a cpneighborhood
of x, this shows that x is a cpinterior
of A. Hence x�cp-int(A), so x�G � cpint(A)
� A. We get cp-int(A) contains all cpopen
subset of A and it is therefore, the largest
cp-open subset of A.
In general cp-int(A) � cp-int(B) � cpint(A
� B), and cp-int(A � B) � cp-int (A) � cpint(B)
as shown by the following two examples:
Example 2.13. Let X={a, b, c, d} with topology
� ={ � , X, {c}, {a, d}, {a, c, d}} Take A={a, d}
and B={b, c}. cp-int(A � B)=X cp-int(A)={a,
d}and cp-int(B)={c}, so cp-int(A) � cpint(B)
� cp-int(A � B)
Example 2.14. Let X={a, b, c, d} and � = { � ,
X,{a, b}, {a, b, c}}. Take A={b, c} and B={a, c,
d}, then A � B={c}, cp-int(A � B)= � but cpint(A)={b,
c} and cp-int(B)={a, c, d}. So cpint(A)
� cp-int(B)={b} and hence cp-int(A � B)
� cp-int(A) � cp-int(B) .
It follows that, cp-int (A)=cp-int(B) does not
imply that A=B. This is shown by Example 4.9
in [12].
Definition 2.15. Let A be a subset of a
topological space(X, � ). A point x�X is said to
be a cp-limit point of A if every cp-open set
containing x contains a point of A different from
x. The set of all cp-limit points of A is called cpderived
set and denoted by cp-d(A).
Lemma 2.16. Let A be a subset of a topological
space (X, � ). A is cp-closed if and only if A
contains all of its cp-limit points.
Proof. Assume that A is cp-closed set and let p
is a cp-limit point of A which belongs to X\A.
Then X\A is a cp-open set containing the cplimit
point of A. Therefore, we get X\A contains
an element of A which is contradiction.
Conversely. Assume that A contains all of its
cp-limit points. For each x�X\A there exists a
cp-open set U containing x such that U � A= � ,
that is, x�U � X\A, by Theorem 2.9, X\A is a
cp-open set. Hence A is a cp-closed set.
Some properties of cp-derived set are
mentioned in the following results:
Proposition 2.17. For any subsets A and B of a
topological space X, then we have the following
properties.
1. If A � B, then cp-d(A) � cp-d(B),
2. x�cp-d(A) implies x�cp-d(X\A),
3.cp-d(A) � cp-d(B) � cp-d(A � B),
4.cp-d(A � B) � cp-d(A) � cp-d(B).
Proof. (3) and (4) follow from (1).
The equality does not hold in (3) and (4) as
shown by Examples 4.12 and 4.13 in [12].
Definition 2.18. The intersection of all cp-closed
sets containing A is called the cp-closure of A
and it is denoted by cp-cl(A).
Here we give some properties of cp-closure
of the set.
Proposition 2.19. For any subset E and F of a
topological space X. The following statements
are true.
1. E� cp-cl(E),
2. cp-cl(E) is cp-closed set in X containing E,
3. cp-cl(E) is the smallest cp-closed set
containing E,
4. E is cp-closed if and only if cp-cl(E)=E, then
cp- cl(cp-cl(E)) =cp-cl(E),
5.If E� F, then cp-cl(E) � cp-cl(F),
6.cp-cl(A) � cp-cl(B) � cp-cl(A � B),
7.cp-cl(A � B) � cp-cl(A) � cp-cl(B).
Generally, equality in (6) and (7) does not
holds as shown by Examples 4.22 and 4.23 in
[12].
Icp-cl(A)=cp-cl(B) does not imply A=B this is
shown in Example 4.18 in [12].
Theorem 2.20. Let X be a space and A be any
subset of X, then A � cp-d(A) is cp-closed.
Proof. Let x�A � cp-d(A). This implies that
x�A and x�cp-d(A). Since x�cp-d(A) there
exists a cp-open set G x of x which contains no
point of A other than x but x�A. So G x
contains no point of A, which implies
G is a cp-open set of each
G x � X\A. Again x
of its points. But G x does not contain any point
of A, no point of G x can be a cp-limit point of
A. Then no point of G x can belong to cp-d(A),
so G x � X\cp-d(A) implies
x� x
G � X\A � X\cp-d(A) = X\(A � cp-d(A)),
by Theorem 2.9, X\(A � cp-d(A)) is a cp-open
set, so A � cp-d(A) is a cp-closed set.
Corollary 2.21. For any subset A of a
topological space X. we have cp-cl(A) =A � cpd(A).
Proof. Since by Theorem 2.20, A � cp-d(A) is
cp-closed containing A and cp-cl(A) is the
smallest cp-closed containing A implies that cpcl(A)
� A � cp-d(A). Hence cp-cl(A) =A � cpd(A).
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