CONTENTS

J. Duhok Univ., Vol.14, No.1 (Pure and Eng. Sciences), Pp 9-16, 2011
ON CP-OPEN SETS AND TWO CLASSES OF FUNCTIONS
ALIAS B. KHALAF * and SHILAN A. MOHAMMAD **
* Dept. of Math., Faculty of Science, University of Duhok, Kurdistan Region-Iraq
** Dept. of Math., Faculty of Science, University of Zakho, Kurdistan Region-Iraq
(Received: October 1, 2009; Accepted for publication: November 3, 2010)
ABSTRACT
In this paper we introduce the concept of cp-open set and study some of its properties. Further we introduce the
cp-continuous and cp-open functions and investigate the basic propertied. Necessary and sufficient condition of P1 -
closed graph and cp-continuous function were found.
KEYWORDS: c-open set, cp-open set, cp-continuous and cp-open functions, P1 -closed graph.
T
1. INTRODUCTION AND
PRELIMINARIES
hroughout the present paper, X and Y
denote topological spaces in which no
separation axiom is assumed. Let A be a subset
of X, we denote the interior and the closure of a
set A by int(A) and cl(A) respectively. A subset
A is said to be preopen [10] (resp. semi open[7])
set if A � intcl(A)( resp. A � clint(A)). The
complement of a preopen set is called preclosed.
The intersection of all preclosed sets containing
A is called the preclosure of A and is denoted by
pcl(A). The preinterior of A is defined as the
union of all preopen sets contained in A and is
denoted by pint(A). The family of all preopen
sets of X is denoted by PO(X) and the set of all
preopen set containing x�X is denoted by
PO(X, x). The union of any family of preopen
sets is preopen. A subset N of X is
preneighbourhood[12] of a point x of X if there
exists a preopen set U containing x with U � N
and it is denoted by N p (x).
As stated by Mashhour et al.[10], Katetov
made some comments on the paper [9] to find
conditions under which the intersection of any
two preopen sets is preopen.
Mashhour together with others offered an
answer to this remark in the form of a
theorem[10, Theorem 2.3].
Definition 1.1[16]. A space (X, � ) will be said
to have the property P if the closure is preserved
under finite intersection or equivalently, if the
closure of intersection of any two subsets equals
the intersection of their closures.
Lemma 1.2[16]. From the above definition it
readily follows that if a space X has the property
P, then the intersection of any two preopen sets
is preopen. As a consequence of this PO(X) is a
topology for X and it is finer than � .
Lemma 1.3[4]. Let A and X 0 be subsets of a
space X. If A�PO(X) and X 0 is semi open in X,
then A � X 0 �PO(X 0 ),
Lemma 1.4[2]. If A � Y � X and Y is a preopen
set in X, then A�PO(X) if and only if
A�PO(Y).
Definition 1.5. A function f : X �Y is called:
1- preirresolute[15] if f
� 1
(V)�PO(X) for each
preopen set V of Y.
2- M-preopen[16] if the image of every preopen
set in X is a preopen set in Y,
3- M-preclosd[13] if the image of every
preclosed set in X is a preclosed set in Y.
Definition 1.6[1]. Let f :X � Y be any
function, the graph of the function f is denoted
by G( f ) and is said to be P1 -closed if for each
(x, y)�G( f ), there exist U�PO(X,x) and
V�PO(Y,y) with (U� V) � G( f )= � .
We proved if the graph of f is P1 -closed
and X has the property P, then the inverse image
of a strongly compact subset in Y is preclosed
set in X.
Definition 1.7[4]. A space X is said to be
submaximal if every dense subset of X is open.
Lemma 1.8[4]. A space X is submaximal if and
only if every preopen set is open.
Definition 1.9. A space X is:
1- pre- T 1 [11] if for every distinct points x, y of
X, there exists U�PO(X, x) not containing y
and V� PO(X, y) not containing x,
2- T 2 [17](resp. pre- T 2 [11]) if for every distinct
points x, y of X, there exist two disjoint
open(resp. preopen) sets each containing one of
them,
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