 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
 -CONTINUOUS LIKE MAPPINGS
Received: 5/12/2004
Accepted: 17/10/2005
Talal Al-Hawary*
&
Ahmed Al-Omari**
‫ملخص‬
‫ا د ددن انات ااد د د‬
‫د د د ا اس د د د‬
‫د ددا‬
‫اد د د‬
‫دن ا د د د‬
‫إن الهد د د ه ا د ددن أ د ددما الرسد د د‬
‫( كد ن اناتد ان اددن‬co--continuous) ‫الات د اددن الدداا‬
(open) ‫الا ت س د د د‬
‫اد د د ددن الد د د دداا‬
‫ادن الداا‬
‫ال د د د م العكس د د د ل ا ا د د د‬
‫الات د‬
‫( إما ك ا د د د‬co--continuous) ‫الد د دداا‬
‫اس د د د د انات اا د د د د‬
‫د د د ددن مل د د د د‬
‫ددن‬
‫( كددمل اا اش د رعددء ا اش د يا‬co--continuous)
‫ه للات اا‬
‫يددا انات اا د‬
‫ر لتس د انات ااد‬
‫د‬
‫ان األ‬
‫الت‬
‫ د د د ددل م‬. (-closed) ‫أ د د د د اد د د ددن الد د د دداا‬
‫( اا اش‬contra-continuous), (-continuous)
.(co--continuous)
‫ان الاا‬
‫الات‬
Abstract
The aim of this paper is to introduce and study a relatively new class of
continuous maps, namely co-  -continuous maps where a map is co-  continuous if the inverse image of every open subset is  -closed. Moreover,
connections to contra-continuity and  -continuity are discussed, several
characterizations of co-  -continuity are provided and some constructions
and applications via co-  -continuity are discussed.
KEYWORDS:  -closed,  -continuous, contra-continuous, co-  continuous 2004 AMS Subject Classification: 54C05, 54C08, 54C10.
1. INTRODUCTION:
Let ( X ,  ) be a topological space (or simply, a space) and A  X . Then the
closure of A and the interior of A will be denoted by cl ( A) and int( A) ,
respectively.
A subset A  X is called semi-open [8] if there exists an open set O  such
that O  A  cl (O) . Clearly A is semi-open if and only if A  cl (int( A)) . The
complement of a semi-open set is called semi-closed [8]. A is called preopen [10]
*
**
Associate Professor, Department of Mathematics & Statistics, Mu'tah University.
Full time Lecturer, Department of Mathematics & Statistics, Mu'tah University.
Al-Manarah, Vol. 13, No. 6, 2007 .
135
 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
if A  int( cl ( A)) . A is called  -open [11] if A  int( cl (int( A))) . A space
( X ,  ) is called anti-locally countable if every non-empty open subset is
uncountable. In [7], the concept of  -closed subsets was explored where a subset
A of a space ( X , ) is  -closed if it contains all of its condensation points (a
point x is a condensation point of A if A U is uncountable for every open U
containing x). The complement of an  -closed set is called  -open or
equivalently for each x  A , there exists U  such that x U and U  A is
countable.  -closure and  -interior, that can be defined in an analogous
manner to cl ( A) and int( A) , will be denoted by cl (A) and int  ( A) ,
respectively. For several characterizations of  -closed subsets, see [1,2,4].
Contra-continuity was introduced in [6] and  -continuity was introduced in
[7]. The aim of this paper is to introduce and study a relatively new class of
continuous maps, namely co-  -continuous maps. This is the content of Section
2. Moreover, connections to contra-continuity and  -continuity are discussed and
several characterizations of co-  -continuity are provided. Section 3 is devoted to
studying some constructions and applications via co-  -continuity, while in
Section 4 we introduce several relatively new variations of continuous maps as
well as new decompositions of contra-continuity via these notions.
2. CO-  -CONTINUITY:
In this section, co-  -continuity is introduced and explored. Several
characterizations of this notion are provided as well as connections to some other
well-known continuities are studied. We begin by recalling the following two
definitions:
Definition 2.1 [6] A map f : X  Y is called contra-continuous if the inverse
image of each open subset of Y is closed in X .
Definition 2.2 [7] A map f : X  Y is called  -continuous if the inverse image
of each open subset of Y is  -open in X .
Contra-continuity and continuity are independent notions, see [6]. Obviously,
every continuous map is  -continuous, but the converse need not be true, see [7].
Now as the identity map on the real line with the standard topology is continuous,
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 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
it is  -continuous which is not contra-continuous. In the following example we
show that contra-continuity need not imply  -continuity.
Example 2.1 Consider the map f from the real line  with the standard topology into
1, x  0
. Then f is
0, x  0
the space Y  0,1 and   {Y ,  , {0}} defined by f ( x)  
contra-continuous but not  -continuous since f
1
({0})  (,0] is not  -open.
Thus  -continuity and contra-continuity are independent notions. Next, we
introduce a relatively new notion of continuity related to both continuities in
Definitions 2.1 and 2.2.
Definition 2.3 A map f : X  Y is called co-  -continuous if the inverse image
of each open subset of Y is  -closed.
Since every closed set is  -closed, every contra-continuous map is co-  continuous, but the converse need not be true as shown next.
Example 2.2 Let X  a, b, c,    , X , b, c, b, c and   { X ,  , {a}} .
Then the identity map f : ( X , )  ( X , ) is co-  -continuous but not contracontinuous.
Next, we show that  -continuity and co-  -continuity are independent
notions.
Example 2.3 Consider the maps f and g from the real line  with the standard
topology into the space Y  0,1 and   {Y ,  , {0}} defined by
0, x    Q
1, x    Q
and g ( x)  
, where Q is the set of all
f ( x)  
1, x  Q
0, x  Q
rational numbers. Then f is  -continuous but not co-  -continuous while g is co -continuity but not  -continuity.
To characterize co-  -continuity, we recall the following definition and result
from [9].
Definition 2.4 Let A be a subset of a space ( X , ) . The Kernel of A is the set
ker( A)  {U  | A  U } .
Lemma 2.1 The following properties hold for subsets A and B of a space X:
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137
 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
(1) x  ker( A) if and only if A  F   for any closed subset F containing x.
(2) A  ker( A) and A  ker( A) if A is open in X.
(3) A  B , then ker( A)  ker( B ).
Our first characterization of co-  -continuity is as follows:
Theorem 2.1 The following are equivalent for a map f : X  Y :
f : X  Y is co-  -continuous.
1)
2) For every closed subset F of Y, f
1
( F ) is  -open.
3) For each x  X and each closed subset F  X containing
4)
f (x ) , there exists an  - open U such that f (U )  F .
5)
f (cl ( A)  ker( f ( A)) for each A  X .
6) cl ( f
1
( B))  f
1
(ker( B)) for each B  Y .
Proof (1)  (2): For each closed F  Y , Y-F is open and since f co-  continuous, we have f
f
1
1
(Y  F )  X  f 1 ( F ) is  -closed in X. Hence
( F ) is  -open.
(2)  (3): For each x  X and each closed subset F  X containing f (x ) ,
we have
x f
1
(F )
f (U )  f ( f
1
which
is
 -open.
U  f 1 ( F ) .
Set
Then
( F ))  F .
(3)  (4): Suppose that there exists y  f (cl ( A)) and y  ker( f ( A)) for
some A  X . Then y  f (x) for some x  cl (A) and by Lemma 2.1, there
exits
a
closed
set
F  Y such
that
y  F and
F  f ( A)   .
( F )  A   and as x  f ( F ) which is  -open, by (3) x  cl (A)
and y  f ( x)  f (cl ( A)) which is a contradiction .
Thus f
1
1
(4)  (5): Let B  Y . Then by Lemma 2.1 and the assumption, we
have f (cl ( f
1
( B)))  (ker B) and cl ( f
1
( B))  f
1
(ker B) .
(5)  (1): Let V be any open subset of Y. By Lemma 2.1 and the
assumption,
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 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
cl ( f
thus f
1
1
(V ))  f
1
(ker(V ))  f
1
(V ) . Hence cl f
1
(V )  f
1
(V ) and
(V ) is  -closed and f : X  Y is co-  -continuous .
Another characterization of co-  -continuity is as follows:
Corollary 2.1 The following are equivalent for a map f : X  Y :
1)
f : X  Y is co-  -continuous.
2) int  (cl ( f
3) cl (int  ( f
1
(V )))  int  ( f
1
( F )))  cl ( f
1
1
(V )) for every open subset V of Y.
( F )) for every closed subset F of Y.
Theorem 2.2 A map f : ( X , )  ( X , ) is co-  -continuous if and only if
f : ( X ,  )  ( X , ) is contra-continuous .
Proof. If f : ( X , )  ( X , ) is co-  -continuous and V is any open subset of
( X , ) , then f 1 (V ) is  -closed in ( X , ) and hence f
( X ,  ) . Thus f : ( X ,  )  ( X , ) is contra- continuous.
1
(V ) is closed in
Conversely, let V be any open subset of ( X , ) . Since f : ( X ,  )  ( X , )
(V ) is closed in ( X ,  ) and thus  -closed
in ( X , ) . Hence f : ( X , )  ( X , ) is co-  -continuous.
is contra-continuous, then f
1
The notion of perfectly continuous maps was first introduced by [13]. Next, a
relative definition is given.
Definition 2.5 A map f : X  Y is perfectly-  -continuous if the inverse image
of every open subset of Y is  -open and  -closed in X (simply,  -clopen).
Clearly, every perfectly-  -continuous map is both  -continuous and co-  continuous, but the converse need not be true as shown next. In spite of that and
the fact that both  -continuity and co-  -continuity are independent notions, we
show that  -continuity and co-  -continuity together characterize perfectly-  continuity.
Example 2.4 Consider the maps f and g from Example 2.3. The map f is  continuous which is not perfectly-  -continuous while the map g is co-  continuous which is not perfectly-  -continuous.
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139
 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
Theorem 2.3 A map f : X  Y is perfectly-  -continuous if and only if f co-  continuous and  -continuous.
Proof. Let V be any open subset of Y. Since f is  -continuous, f
open in X and since f is co-  -continuous, f
1
1
(V ) is  -
(V ) is  -closed in X. Hence f is
perfectly  -continuous .
The converse is obvious.
Definition 2.6 [4] A subset A of a space ( X , ) is semi-  -open if
A  cl (int  ( A)) .
Definition 2.7 [3] A map f : X  Y is semi   -continuous if the inverse
image of every open subset of Y is semi-  -open .
The following relatively new notion of continuity was given in [2].
Definition 2.8 A subset A of a space
( X , )
is
 -  -open if
A  cl (int  (cl ( A))) . A map f : X  Y is    -continuous if the inverse
image of every open subset of Y is    -open.
Now, we show that neither contra-continuity nor    - continuity implies
semi-  - continuity, but they both together imply semi-  - continuity.
Example 2.5 The map g in Example 2.3 is    -continuous but not semi-  continuous since
cl (int  ( g 1 ({0})))  cl (int  (Q))  cl ( )  
does
not
contain Q.
Example 2.6 Consider the map f from the real line  with the standard topology
0, x  0
. Then
1, x  0
into the space Y  0,1 and   {Y ,  , {0}} defined by f ( x)  
f
is
cl (int  ( f
but
not
semi-  -continuous
({0})))  cl ( )   does not contain {0}
contra-continuous
1
since
Theorem 2.4 Every contra-continuous map that is also    -continuous is
semi-  -continuous.
Proof. Let V be any open subset of Y . Since f is co-continuous, f
closed in X and hence cl ( f
1
(V ))  f
Al-Manarah, Vol. 13, No. 6, 2007 .
1
1
(V ) is
(V ) . Since f is    -continuous, then
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 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
f
f
1
1
(V ) is    -open
(V )  cl (int  ( f
1
and
thus f
1
(V )  cl (int  (cl ( f
1
(V )))) .
Hence
(V ))) . Therefore, f (V ) is semi-  -open.
1
3. ON CONSTRUCTIONS VIA CO-  -CONTINUITY:
In this section, several constructions involving co-  -continuous maps and
other interesting results involving connectedness notion are discussed. We begin
by showing that the composition of two co-  -continuous maps need not be co -continuous.
Example 3.1 Consider the map f from the real line  with the standard topology
0, x  0
and
1, x  0
into the space Y  0,1 and   {Y ,  , {1}} defined by f ( x)  
1, x  0
. Then f and g are co-  0, x  1
the map g : Y  Y defined by g ( x)  
continuous but g  f is not co-  -continuous since ( g  f ) 1 ({1})  (0, ) is
not  -closed.
The proof of the following result is obvious:
Theorem 3.1 If f : X  Y is a co-  -continuous map and g : Y  Z is a
continuous map, then g  f is co-  -continuous.
Corollary 3.1 If f : ( X , ) 
X  is co-  -continuous, then



P  f is co-
 -continuous for all    , where P is the  -projection.
Note that Theorem 3.1 need not hold when g is only  -continuous and not
continuous as shown next.
Example 3.2 Consider the map f from the real line  with the standard topology
into the space Y  0,1,2 and   {Y ,  , {0},{0,1}} defined by
2, x  0
f ( x)  
and the map g : Y  ( Z  {a, b},   {Z ,  ,{a}}) defined
1, x  0
a, x  0,2
by g ( x)  
. Then f is co-  -continuous and g is  -continuous but
b, x  1
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 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
g  f is not co-  -continuous since ( g  f ) 1 ({a})  (0, ) is not  -closed.
The restriction of a co-  -continuous need not be co-  -continuous as shown
next.
Example 3.3 Consider the map g from Example 2.3 and the subset
A    [0,1) . Then ( g A) 1 ({0,1})  g 1 ({0,1})  A    [0,1) which is not
 -closed and hence g A is not co-  -continuous.
The following results follow from the fact that the collection of all  -open
subsets of X is a topology on X that is finer than the original topology on X, see [5].
Lemma 3.1
1) If A is an open set and B is an  -open set, then A  B is  -open.
2) If ( A,  A ) is a subspace of ( X , ) and B  A such that B is  -open
3) in X, then B is  -open in ( A,  A ) .
Corollary 3.2 If f : ( X , )  (Y ,  ) is a co-  -continuous map and A is  open subset of X, then f | A : ( X ,  A )  (Y ,  ) is co-  -continuous.
Proof. Let V be a closed subset of Y, then ( f | A) 1 (V )  f
1
(V )  A is  -
open in X by Theorem 2.1 (2) and Lemma 3.1 (1). By Lemma 3.1 (2),
( f | A) 1 (V ) is  -open in ( A,  A ) . Again by Theorem 2.1 (2), f | A is co-  continuous.
Theorem 3.2 Let f : X  Y be a map and X i : i  I  be a cover of X such
that X i is  -open for each i  I . If f | X i : X  Y is co-  -continuous for each
i  I , then f is co-  -continuous.
Proof. Suppose that F is a closed subset of Y. Then we have
f 1 ( F )  ( f 1 ( F )  X i )  ( f | X i ) 1 ( F ) and since f | X i is co-  -

iI

iI
continuous for each i, it follows that ( f | X i ) 1 ( F ) is  -open, which mean that f is
co-  -continuous by Theorem 2.1 (1).
Next, a stronger notion than connectedness is given.
Definition 3.1 A space ( X , ) is  -connected if ( X ,  ) is connected.
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Since every open set is  -open, then every  -connected space is connected,
but the converse need not be true as shown next.
Example 3.4 Let X  a,b, and    , X , a . Then ( X , ) is connected but
not  -connected.
It is well-known that a continuous image of a connected space is connected, a
similar result is given next .
Theorem 3.3 If a map f : X  Y is a surjection co-  -continuous and X is
 -connected, then Y is connected.
Proof. Suppose that Y is not connected. Then there exist nonempty disjoint clopen
sets V and U such that Y  V U . Since f is co-  -continuous, then f 1 (V )
(U ) are  -closed and  -open sets such that f 1 (V ) and f 1 (U ) are
nonempty disjoint and X  f 1 (V )  f 1 (U ) . Therefore X is not  -connected.
and f
1
To prove our final result, we recall the following lemma:
Lemma 3.2 [12] Let X be an anti- locally countable space. Then for every  open subset A of X, cl ( A)  cl ( A) .
As an immediate consequence of the preceding lemma, we have the
following result:
Corollary 3.3 Let X be an anti-locally countable space. Then for every  closed subset A of X, int( A)  int  ( A) .
Theorem 3.4 Let X be a connected anti- locally countable space and Y be a T1 space. If f : X  Y is a co-  -continuous surjective map, then f is constant.
Proof. Assume Y   . Since Y is T1 and f is co-  -continuous, then by Theorem
2.1 (2)   { f
1
( y) : y  Y } is a disjoint  -open partition of X. If   2 , then
there exists a proper  -clopen subset A   . By Lemma 3.2 and Corollary 3.3, A
is clopen. This is a contradiction. Therefore   1 and thus f is a constant map.
4. ON VARIATIONS OF CONTINUITY:
In this section, several new variations of continuous maps are introduced. In
Al-Manarah, Vol. 13, No. 6, 2007 .
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 -Continuous Like Mappings……....………..Talal Al-Hawary & Ahmed Al-Omari
addition, contra-continuity is decomposed via these notions. We begin by
recalling the following definition from [1]:
Definition 4.1 A subset A of space ( X ,  ) is called  g-closed if cl A  U
whenever A  U and U  T .
Another relatively new notion of closed maps is given next:
Definition 4.2 A subset A of space ( X ,  ) is called  -c-closed provided that
there is a proper subset B of X such that A  cl (B) . A map f : X  Y is  gc-closed if f ( A) is  g- closed in Y for every  -c-closed subset A of X.
Theorem 4.1 If f : X  Y is continuous and  g-c-closed, then f ( A) is an
 g- closed subset of Y for every proper  g- closed subset A of X.
Proof Let A be any proper  g- closed subset of X and f ( A)  V where V is an
(V ) which is open and since A is  g-closed,
then cl ( A)  f (V ) and f (cl ( A))  V . Now since cl ( A)  cl ( A) , then
cl (A) is  -c-closed and by assumption f (cl ( A)) is  g- closed. Therefore
cl ( f (cl ( A)))  V , but cl ( f ( A))  cl ( f (cl ( A)))  V which proves
that f ( A) is  g- closed .
open subset of Y. Then A  f
1
1
By a similar argument to that in the preceding result, we can prove the
following:
Theorem 4.2 If f : X  Y is co-  -continuous map that is also  g-c-closed,
then f ( A) is a  g- closed subset of Y for every subset A of X.
Next, several new notions of maps are introduced and connections to co-  continuity are discussed. Moreover, we provide new decompositions of contracontinuity.
Definition 4.3 [6] A map f : X  Y is called contra -  -(semi-pre-  respectively) continuous if the inverse image of every open subset of Y is  (semi-pre-  -respectively) closed .
Definition 4.4 A subset A of space ( X , ) is called a c -set if
cl ( A)  cl ( A) . A map f : X  Y is called c -continuous if the inverse
image of every open subset of Y is a c -set.
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Let C ( X ) (resp. C ( X ) , Q ( X ) ) denote the set of all closed (resp.  closed, c -closed) sets in X. Then clearly for a subset A of a space X,
A  C ( X ) if and only if C ( X )  Q ( X ) .
Corollary 4.1 For a map f : X  Y , f is contra-continuous if and only if f is
contra-  -continuous and c - continuous.
For the following definition, see for example [8,10,11].
Definition 4.5 Tthe intersection of all semi-closed (pre-closed,  -closed and  closed) sets containing A is called the semi-closure of A, dented by scl ( A) (resp.
pre-closure of A, pcl ( A) ,  -closure of A, cl ( A) and  -closure of A,
cl ( A) ).
By a similar argument to that of the preceding Corollary, we may prove the
following result:
Definition 4.6 A subset A of space ( X , ) is called an if cl ( A)  cl ( A) . A
map f : X  Y is called c -continuous if the inverse image of every open
subset of Y is  -set.
Corollary 4.2 For a map f : X  Y , f is contra-continuous if and only if f is
contra-  -continuous and c - continuous.
Definition 4.7 A subset A of space ( X , ) is called semi-set if scl ( A)  cl ( A) . A
map f : X  Y is called cs -continuous if the inverse image of every open
subset of Y is a semi-set .
Corollary 4.3 A map f : X  Y is contra-continuous if and only if f is contrasemi-continuous and cs- continuous.
Definition 4.8 A subset A of space ( X , ) is called pre-set if pcl ( A)  cl ( A) . A
map f : X  Y is called cp-continuous if the inverse image of every open subset
of Y is a pre -set .
Corollary 4.4 A map f : X  Y is contra-continuous if and only if f is contrapre-continuous and cp- continuous.
Definition 4.9 A subset A of space ( X , ) is called  -set if cl ( A)  cl ( A) . A
map f : X  Y is called c  -continuous if the inverse image of open subset of Y
is  -set .
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Corollary 4.5 A map f : X  Y is contra-continuous if and only if f is contra -continuous and c  - continuous.
ACKNOWLEDGMENT:
The author would like to thank the referees for useful comments and
suggestions.
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