Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Paracompact Like Spaces
Received: 22/2/2004
Accepted: 11/5/2004
Talal Ali Al- Hawary *
‫ملخص‬
I ‫[ بدراسة ما يسمى بالفضاءات المتراصة المعدودة من الفئة‬1] ‫قام الباحث في المرجع‬
‫ هذا‬.‫( حيث أعطى عدة أوصاف لهذة الفضاءات‬Countably I –compact spaces)
‫ حيث نقدم مفهوم الفضاءات شبة المتراصة المعدودة من‬-[1] ‫مشابة لما في المرجع‬-‫البحث‬
‫ ( التي تكون‬X , ) ‫( وهي الفضاءات‬Countably I – paracompact spaces) I ‫الفئة‬
X ‫( ل‬Regular closed) ‫ وأي غطاء معدود مكون من مجموعات منتظمة االغالق‬T2
‫( مكونة من مجموعات منتظمة االغالق دواخلها‬Local refinement) ‫يملك تنقية محلية‬
‫ أذا وفقط أذا كان شبة‬I ‫ نبين أيضا أن الفضاء يكون شبة متراص معدود من الفئة‬.X ‫تغطي‬
‫ ثم نعطي عدة أوصاف‬.(Extremally disconnected) ‫ ومنفصال نهائيا‬S ‫معدود من الفئة‬
.‫لهذه الفضاءات ونلقي الضوء على االقترانات بين هذة الفضاءات‬
Abstract
In [1] the concept of countably I-compact spaces was introduced. A
decomposition and several characterizations of this class of spaces were also
given. In this paper-analogous to [1]- we introduce the relatively new notion of
( X , )
countably I-paracompact space as the space
that is T2 and every
countable cover by regular closed subsets of X has a countable locally finite
refinement by regular closed subsets whose interiors cover X. We prove that a
space is countably I-paracompact if and only if it is countably S-paracompact
and extremally disconnected. Several characterizations of countably Iparacompactness are also given. Moreover, we study maps of countably Iparacompact spaces.
Keywords: Countably I-paracompact, Regular closed set, Cover.
*
Department of Mathematics, Mu’tah University
Al-Manarah, Vol. 12, No. 2, 2006 .
199
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
1 Introduction
Let ( X , ) be a topological space (or simply, a space). A subset
A  X is called semi-open (simply, SO) if there exists an open set O  
such that O  A  O , where O denotes the closure of O in X. Clearly A is
o
a semi-open set if and only if
A A.
A complement of a semi-open set is
o
called semi-closed (simply, SC). A is called preopen (simply, PO) if
A A.
o
o
A is called regular-open (simply, RO) if A  A . Clearly, A is RO if A  F
for some closed subset F. Complements of regular-open sets are called
o
regular-closed (simply, RC). Clearly A is regular-closed if and only if A  A
or if A  O for some open subset O. A is called   open if A  A . A is
called a regularly semi-open (simply, RSO) if there exists an RO set U
such that U  A  U . Finally, A is called semi-preopen (simply, SPO) if
o
o
o
which is equivalent to say that A is an RC set. The collection of all
SO (resp., RO, RC, PO and  -open) subsets of X is denoted by SO ( X , )
A  A,


(resp., RO ( X , ) , RC ( X , ) , PO( X , ) and T ). We remark that T is a

topology on X and T  SO( X , )  PO( X , ) . A space ( X , ) is called
extremally disconnected (simply e.d.) if U is open for every open subset
U, which is equivalent to say that every disjoint open subsets, have disjoint
closures.
In [6], a space ( X , ) is called S-closed if every SO cover of X (i.e.
every cover of X by SO subsets) contains a finite subfamily whose union is
dense in X. In [2], ( X , ) is called I-compact if every RC cover of X (i.e.
every cover of X by RC subsets) contains a finite subfamily whose
interiors cover X. In [3], ( X , ) is called countably S-closed if every
countable RC cover of X contains a finite subcover for X. In [1], ( X , ) is
called countably I-compact if every countable RC cover of X contains a
finite subfamily whose interiors cover X. Also in [3], ( X , ) is called
feebly compact if every countable open cover of X contains a finite
subfamily whose union is dense in X.
Al-Manarah, Vol. 12, No. 2, 2006 .
200
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
By  , we mean the set of all natural numbers. Recall that a space
( X , ) is compact if and only if every open cover of X has a finite open
refinement that covers X, see for example [4]. Thus we have the following
result:
Lemma 1 A space ( X , ) is countably S-closed if and only if every RC
countable cover of X has a finite RC refinement that covers X.
Proof. Let   { An : n } be an RC cover of X. Then it has a finite RC
  {Bi : i  1,2,..., k}
refinement
that covers X. Thus for every i=1,2,...,k,
Bi  Ai   and as  covers X, { Ai : i  1,2,..., k} is a finite subcover for
X.
The converse is obvious. ■
Another immediate result is next.
Lemma 2 A space ( X , ) is countably I-compact if and only if every
countable RC cover of X has a finite RC refinement  such that
X
o
B

.
Proof. Follows from the fact that every finite collection is locally finite. ■
Analogous to [1,2,3,6], we study several classes of paracompact like
spaces, namely: S-paracompact, countably S-paracompact, countably Iparacompact and feebly paracompact spaces and characterize them. Our
main goal is to focus on countably I-paracompact spaces and show that a
space is countably I-paracompact if and only if it is countably Sparacompact and extremally disconnected. Moreover, we study maps of
countably I-paracompact spaces.
2 Countably I-paracompact spaces
We begin this section by characterizing S-paracompact spaces for
better understanding of countably I-paracompact spaces.
Definition 1 A space ( X , ) is called S-paracompact if it is T2 and for
every semiopen cover  of X, there exists a locally finite semiopen
Al-Manarah, Vol. 12, No. 2, 2006 .
201
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
X  B
 .


refinement
of
such that
Theorem 1 A space ( X , ) is S-paracompact if and only if it is T2 and for
every RC cover  of X, there exists a locally finite RC refinement  of
 that covers X.
  { A :  Δ} be an RC cover of the T2 space X. Then as
Proof. Let
every RC-set is an SO,  is an SO cover of X. Thus as X is S  {B :   }
paracompact, there exists a locally finite SO refinement
of  such that
X 
 B
B 
o
  {B : B  }
. Since  is locally finite,
is
o
B
locally finite. As for every    ,  is an RC-set (being the closure of an
o
o
B  A  A
open set) and as
for some
in  , 
. Thus  is a
locally finite RC refinement of  . Moreover as for every    ,
B  A
o
B  B
A
X 
o
, we have
B  B
and hence
 B



o
 B


. That is 
covers X.
Conversely, let   { A :  Δ} be an SO cover of the T2 space X.
o
  { A :   }
is an RC cover of X. Thus  has an RC locally
X   B
  {B :   }
 
finite refinement
such that
. Clearly  is a
Then
locally finite refinement of  that covers X. Hence
therefore X is S-paracompact. ■
X 
 B
 
and
Next, countably S-paracompactness, countably I-paracompactness and
feebly paracompactness are introduced.
Definition 2 A space ( X , ) is called countably S-paracompact if it is T2
and for every countable RC cover  of X, there exists a locally finite RC
refinement  of  that covers X.
Al-Manarah, Vol. 12, No. 2, 2006 .
202
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Definition 3 A space ( X , ) is called countably I-paracompact if it is T2
and for every countable RC cover  of X, there exists a countable locally
finite RC refinement  of  such that
X 
o
B

.
Definition 4 A space ( X , ) is called feebly paracompact if it is T2 and
for every countable open cover  of X, there exists a locally finite open
refinement  of  such that
X 
B

.
Next, we show that every countably S-paracompact space is feebly
paracompact.
Lemma 3 A countably S-paracompact space
paracompact.
( X , )
is feebly
  { An : n  }
Proof. Let
be an open cover of X. Then   {An : n }
is an Rc cover of X and hence there exists a countable RC locally finite
X
o
B

refinement   {Bm : m } of  such that
. Thus
  {Bm  An : m, n }
is a countable RC locally finite refinement of  .
X
o
B  A

Moreover,
feebly paracompact.
n
n

(A
n
n , m
o
 B)   B

. Therefore, ( X , ) is
■
The following result is a direct consequence of Lemma 1 and the fact
that every finite collection is locally finite.
Corollary 1 A countably S-compact space that is T2 is countably Sparacompact.
The following result follows directly from Lemma 2.
Corollary 2 A countably I-compact space that is T2 is countably Iparacompact.
Al-Manarah, Vol. 12, No. 2, 2006 .
203
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Next, we prove one of our main results that will be a useful tool
throughout this paper. In this result, we give a decomposition of countably
I-paracompact spaces in terms of countably S-paracompact spaces.
Theorem 2 A space ( X , ) is countably I-paracompact if and only if it is
countably S-paracompact and e.d.
Proof. Let ( X , ) be countably I-paracompact space. Obviously, X is
countably S-paracompact. Suppose X is not e.d. Then there exists an open
o
subset U of X such that U is not open. So U  U   and hence {U , X  U } is
a countable RC cover of X. Then it has a countable RC locally finite
refinement  such that
X 
o
B
B
o
o
. Thus
X U X  U .
and
X U  (X U)  X U
hence
and
o
o
But X  U is open
o
o
so
X U  X U
.
That
is
o
U  X U  X  U X U
and so
U U .
Therefore,
U
is open, a
contradiction.
Conversely, let ( X , ) be a countably S-paracompact and e.d. space
  { An : n  }
and let
be an RC cover of X. Then there exists an RC
  {Bm : m }
locally finite refinement
of  that covers X. Since for
each m , B m is an RC-set, we have Bm  U m for some open U m  X .
As X is e.d.,
X
B
m
m

o
U
m
m
Bm

is open which is also an RC-set. Now
U
o
{U m  U m : m }
m
m
. Therefore
finite RC refinement of  that covers X. ■
is a locally
For the proof of the following Lemma, see for example [4,pp.246].
Lemma 4 Let
space X. Then
  { A :   }

 
A 

 
be a locally finite collection of subsets of a
A .
Using Lemma 4, we can show that in an e.d. space, feebly
paracompactness is stronger than countably S-paracompactness.
Al-Manarah, Vol. 12, No. 2, 2006 .
204
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Lemma 5 Every feebly paracompact e.d. space is countably Sparacompact.
o
Proof. Let   { An : n  } be an RC cover of X. As X is e.d. and as An
o
is open, then An is open for each n  and hence  is a countable open
cover of X. Then there exists a locally finite open refinement  of 
X  B
B
such that
. Thus   {B : B  } is a locally finite RC refinement
of  such that
X 
B  B
B
B
by Lemma 4. Therefore,  covers X. ■
A stronger result than that in Theorem 2 is given next.
Theorem 3 In an e.d. space ( X , ) , the following are equivalent:
(1) ( X , ) is countably I-paracompact.
(2) ( X , ) is countably S-paracompact.
(3) ( X , ) is feebly paracompact.
Proof. (1)  ( 2) is the content of Theorem 2.
(2)  ( 3) : Let ( X , ) be a countably S-paracompact space and let
  { An : n }
be an open cover of X. Then   {An : n } is a
countable RC cover of X. Thus it has a locally finite RC refinement  that
o
covers X. For every B   , B is an RC-set and so B  B which is open as
{B  An : B  , n }
X is e.d. Then
is a locally finite open refinement of
 such that
X  (  B)  (  An ) 
B
n
 (B  A
n
B , n
)
 (B  A
n
).
B , n
Therefore, X is feebly paracompact.
(3)  (1) : As X is feebly paracompact and e.d., by Lemma 5, X is
countably S-paracompact and by Theorem 2, it is countably I-paracompact ■
Al-Manarah, Vol. 12, No. 2, 2006 .
205
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
The following Lemma follows easily by definitions.
Lemma 6 (1) A subset A of a space X is semipreopen if and only if A is
RC.
(2) RC  set  RSO  set  SO  set  SPO  set .
Theorem 4 For a space ( X , ) , the following are equivalent:
(1) ( X , ) is countably I-paracompact.
(2) For every SPO countable cover   { An : n } of X, there exists a
countable locally finite SPO refinement  of  such that
X 
o
 B.
B
(3) For every SO countable cover   { An : n } of X, there exists a
countable locally finite SO refinement  of  such that
(4) For every RSO countable cover
  { An : n }
X 
o
 B.
B
of X, there exists a
countable locally finite RSO refinement  of  such that
X 
o
 B.
B
Proof. (1)  ( 2)  ( 3)  ( 4)  (1) follow easily from Lemma 6. ■
3 On constructions of countably I-paracompact spaces
A property P of a space ( X , ) is a semiregular property if: ( X , ) has
property P if and only if ( X ,  S ) has property P, where  S is the topology
generated by the set of all RO-subsets in X.
Lemma 7 [3] e.d. is a semiregular property.
Theorem 5 Countably S-Paracompact is a semiregular property.
Proof. If ( X ,  S ) is a T2 space and x and y are two distinct elements in X,
then there exist disjoint RO-sets A and B such that x  A and y  B . Since
A and B are both open, ( X , ) is T2 .
Al-Manarah, Vol. 12, No. 2, 2006 .
206
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Conversely, if ( X , ) is T2 and x and y are two distinct elements in X,
then there exist disjoint open sets U and V such that x U and y V . As
o
o
o
o
x U  U and y V  V , x U and y V . As U and V are two
o
disjoint RO-sets (otherwise, there exists
o
z U  V
which implies the
o
existence of an open set W such that z W  U and as z V  V so
U  V   . Thus there exists a U  V , that is there exists an open set T
a  T  U . Hence as a V , U V   , which is a
contradiction). Therefore, ( X ,  S ) is T2 . The rest of the proof is similar to
that of Proposition 2.6 in [3]. ■
such that
Combining Lemma 7 and Theorem 5 together with Theorem 3, we
obtain the following result:
Corollary 3 Countably I-paracompactness is a semiregular property.
Lemma 8 [3] e.d. is a semiopen hereditary property.
The proof of the following result is similar to that of Proposition 2.9 in
[3] and thus omitted.
Lemma 9 Let ( X , ) be a space. If (U , |U ) is countably S-paracompact
and Y is a regular semiopen subset of X such that U  Y  U , then
(Y , |Y ) is countably S-paracompact.
Since every RO-set and every RC-set is an RSO-set, we have the
following result:
Corollary 4 Let ( X , ) be a countably I-paracompact space. Then
(1) (Y , |Y ) countably I-paracompact for every RO-subset Y of X.
(2) (Y , |Y ) countably I-paracompact for every RC-subset Y of X.
Lemma 10 [3] Let ( X , ) be a space and A is a PO subset of X. Then
RC ( A,  | A)  {F  A : F  RC ( X , )} .
Al-Manarah, Vol. 12, No. 2, 2006 .
207
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Theorem 6 If X i is an RO-set and ( X i , | X i ) is a countably Im
paracompact subspace of X such that
paracompact.
X 
X
i 1
i
, then X is countably I-
Proof. Let   { An : n } be an RC cover of X. Then by Lemma 10,
 i  {X i  An : n } is an RC cover of X i . Thus for each i=1,2,...,m,
 i such that
i
there exists a locally finite RC refinement
of
oi
X   (B )   (B )
B
where i means the interior of B i in the
oi
i
o
i
Bi  i
i
Bi  i

m

i
i 1
subspace topology ( X i , | X i ) . Then
is an RC locally finite
X   B.
refinement of A such that
Therefore, X is countably Iparacompact. ■
i
B
4 Maps of countably I-paracompact spaces
A map f : ( X ,  X )  (Y , Y ) is called irresolute (resp., semicontinuous)
if the inverse image of every SO (resp., open) subset of Y is an SO-set in
1
1
X. f is called almost open if f ( B)  f ( B) for all B Y .
Lemma 11 Let f : ( X ,  X )  (Y , Y ) be an irresolute map from the
countably S-paracompact space ( X ,  X ) onto (Y , Y ) . Then (Y , Y ) is
countably S-paracompact.
Proof. Let   { An : n } be an RC cover of Y. By Lemma 6, each An is
1
1
SO and as f is irresolute, f ( An ) is SO. Again by Lemma 6, f ( An ) is
1
1

SPO and thus f ( An ) is an RC-set. Then   { f ( An ) : n  } is a
Y 
A
n
, X 
f
1
( An ) 

f
1
( An )
countable RC cover of X (as
)


and so it has an RC locally finite refinement
that covers X. Hence
  { f ( B ) : B    } is an RC locally finite refinement of  that covers Y
n
n
n
Y  f (  Bn )   f ( Bn )
n
n
(as
) and therefore (Y , Y ) is countably Sparacompact. ■
Al-Manarah, Vol. 12, No. 2, 2006 .
208
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
Combining Lemma 11 together with Theorem 3, we obtain the
following result:
Theorem 7 Let f : ( X ,  X )  (Y , Y ) be an irresolute map from the
countably I-paracompact space ( X ,  X ) onto the e.d. space (Y , Y ) . Then
(Y , Y ) is countably I-paracompact.
By A , we mean the semi-closure of A. That is the smallest SC-set
containing A. We next recall the following Lemma to prove our final
result.
Lemma 12 (1) A map f : X  Y is semicontinuous if and only if
f ( A)  f ( A) , see [7].
(2) For an SO-set A in an e.d. space, A  A , see [5].
We end this section by showing that the semicontinuous almost open
image of a countably I-paracompact space is countably I-paracompact.
Theorem 8 Let f : ( X ,  X )  (Y , Y ) be a semicontinuous almost open
map from the countably I-paracompact space ( X ,  X ) onto (Y , Y ) . Then
(Y , Y ) is countably I-paracompact.
Proof. Using Theorem 2, we need only show (Y , Y ) is countably Sparacompact and e.d.
To show (Y , Y ) is e.d., we show that given any two disjoint open
subsets V1 and V 2 of Y, V1  V2   . If V1 and V 2 are two disjoint open
1
1
1
subsets of Y, then f (V1  V 2 )  f (V1 )  f (V 2 ) and as f is almost
1
1
1
1
open, f (V1 V2 )  f (V1 )  f (V1 ) . As f is semicontinuous, f (V1 )
1
and f (V2 ) are SO-sets and thus there exist open sets U,V such that
(V1 )  U and V  f 1 (V 2 )  V . Since f 1 (V1 )  f 1 (V2 )   ,
1
U V   and as ( X ,  X ) is e.d., U V   . Now as U  f (V1 ) and
U f
1
Al-Manarah, Vol. 12, No. 2, 2006 .
209
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
V f
1
(V2 ) ,
f
1
(V1 V2 )  f
1
(V1 )  f
1
(V1 )   . Thus as f is
surjective, V1  V 2   . Therefore, (Y , Y ) is e.d.
Let   { An : n } be an RC cover of Y. Then since X is e.d. by
Theorem 2, for all n  there exists Vn Y such that An  Vn . Now as f
is almost open
X f
1
1
(  An )  f
n
( Vn )   f
n
1
n
(Vn )   f
1
(Vn ).
n
   { f 1 (Vn ) : n  }
Thus
is a countable RC cover of X and as X is
countably S-paracompact, there exists an RC locally finite refinement  
of   that covers X. Thus
Y  f (  B )  f (  f
B 
and by Lemma 12 part (2),
Y   f(f
1
1
(Vn )).
n
Y  f ( f
1
(Vn ))
n
(Vn ))   Vn   An .
and hence by Lemma 12
Therefore,   {Vn : n } is
an RC locally finite refinement of  that covers Y. Thus (Y , Y ) is
countably S-paracompact. ■
part (1),
n
Al-Manarah, Vol. 12, No. 2, 2006 .
n
n
210
Paracompact like Spaces ……………………………………....… Talal Ali Al- Hawary
References
[1] Al-Nashef, B., Countably I-compact spaces, IJMMS, 26:12, 745-751,
2001.
[2] Cameron, D., Some maximal topologies which are QHD, Proc. Amer.
Math. Soc., 75, no. 1, 149-156, 1979.
[3] Dlaska, K., Ergun, N. and Ganster, M., Countably S-closed spaces,
Math. Slovaca, 44, no. 3, 337-348, 1994.
[4] Munkres, J., Topology a first course , Prentice-Hall inc., New Jersey,
1975.
[5] Sivaraj, D., A note on S-closed spaces, Acta Math. Hungar., 44, no. 34, 207-213, 1984.
[6] Thompson, T., S-closed spaces, Proc. Amer. Math. Soc., 60, 335-338,
1976.
[7] Thompson, T., Semicontinuous and irresolute images of S-closed
spaces, Proc. Amer. Math. Soc., 66, no. 2, 359-362, 1977.
Al-Manarah, Vol. 12, No. 2, 2006 .
211