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Open Journal of Discrete Mathematics, 2012, 2, 142-144
http://dx.doi.org/10.4236/ojdm.2012.24028 Published Online October 2012 (http://www.SciRP.org/journal/ojdm)
A Note on Weakly-α-I-Functions and
Weakly-α-I-Paracompact Spaces*
Shi-Qin Liu
Department Mathematics and Computer, Hengshui College, Hebei, China
Email: [email protected]
Received August 9, 2012; revised September 3, 2012; accepted September 10, 2012
ABSTRACT
This paper introduces the new notion of weakly-α-I-functions and weakly-α-I-paracompact spaces in the ideal topo-
logical space. And it obtains that some properties of them.
Keywords: Weakly-α-I-Open Set; Weakly-α-I-Functions; Weakly-α-I-Paracompact Spaces
1. Introduction
Throughout this paper,
ClA and

I
ntA denote the
closure and interior of
A
, respectively. Let
,X
be
a topological space and let
I
be an ideal of subsets of
X
. An ideal topological space is a topological space
X
,
with an ideal
I
on
X
, and is denoted by
,,
X
I
. For a subset
A
X,

for
*each neighborhood of
A
IxXU x
AI
is called the local function of
A
with respect to
I
and
[1]. It is well known that defines a
Kuratowski closure operator for

**
=AA A

Cl
.
Let be a subset of a topological space
S
,X
.
The complement of a semi-open set is said to be semi-
closed [2]. The intersection of all semi-closed sets con-
taining , denoted by
S

s
Cl S is called the semi-clo-
sure [3] of . The semi-interior of , denoted by
S S

s
Int S, is defined by the union of all semi-open sets
contained in .
S
In recent years, E. Hatir and T. Noiri have extended
the study to α-I-open and semi-I-open sets. In this paper,
we introduce the new sets which are called weakly-
α-I-open and weakly-α-I-functions, then obtain some
properties of t h em.
First we recall some definitions used in the sequel.
Definition 1.1. [4] A subset
A
of an ideal topological
space
,,
X
I
is said to be weakly-α-I-open, if




AsClIntClIntA
.
Definition 1.2. [4] A subset of an ideal topological
space S
,,
X
I
is said to be a weakly-α-I-closed set if
its complement is a weakly-α-I-open set.
Theorem 1.1. [4] Let
,,
X
I
be an ideal top ological
space. Then all weakly-α-I-open sets constitute a topo-
logy of
X
. Then
(1)
and
X
are weakly-α-I-open sets.
(2) The finite intersection of weakly-α-I-open sets are
weakly-α-I-open sets.
(3) If
A
is weakly-α-I-open for each
, then
A
is weakly-α-I-open.
Theorem 1.2. [4] Let
,,
X
I
be an ideal topological
space and AU
. Then
A
is weakly-α-I-open if
and only if
A
is weakly-α-I-open in

,,
UU
UI
.
2. Weakly-α-I-Functions
Definition 2.1. A function

:,, ,fXI Y

1 is said
to be weakly-α-I-continuous , if
f
V
is weakly-
α-I-open in
,,
X
I
, for any

,VY

.
Definition 2.2. A mapping
,,:,,
f
XIYJ

U
is said weakly-α-I-open (resp weakly-α-I-closed) if for
any U is weakly-α-I-open (resp. is weakly-α-I-closed)
f
U is weakly-α-I-open (resp.

f
U is weakly-α-I-
closed).
Theorem 2.1. For a function

:,, ,fXI Y
,
the followings are equivalent:
(1)
f
is weakly-α-I-continuous.
(2) For any
x
X
and each V
containing
f
x,
there exists weakly-α-I-open containing
U
x
such that
f
UV.
(3) The inverse image of each closed set in is weakly-
α-I-closed.
Proof:
13 it is obviously.
12 For each
x
X
and each weakly-α-I-
-open
,VY
containing

f
x. Since
f
is weakly-
α-I-continuous, then
1
f
V
is a weakly-α-I-open set
containing
x
in
X
. Let
*Supported by Hebei Province of the Scientific Research in mentoring
p
rograms Z2010187.
1
=UfV
, then
f
UV.
C
opyright © 2012 SciRes. OJDM
S.-Q. LIU 143
 
21 For any
x
X and each V
contain-
ing

f
x, there exists weakly-α-I-open
x
U containing
x
such that

x
f
U

1
V and .

1V
x
Uf
We have

1
x
xf V
f
V
U
, where
x
U is weakly-
α-I-open for any

1
x
fG. Thus is weakly-
α-I-open by Theorem 1.1. And conclude that is weakly-
α-I-continuous.

fV
f
1
Theorem 2.2. Let

:,, ,fXI Y
be a fun-
ction and

U

an open cover of
X
. Then
is weakly-α-I-continuous if and only if the restriction
f
:,, ,
UUU
fU IY

is weakly-α-I-continuous
for each
.
Proof: Necessity. Let be any open set of V
,Y
.
Since
f
is weakly-α-I-continuous,

1
f
V
is a weakly-
α-I-open set of
,,
X
I
. Since U
, from Theorem
1.2 is weakly-α-I-open in

1
UfV
,,
UU
UI

.
On the other hand,
 
11
U
f
VU fV
and

1
U
f
V
is weakly-
α-I-open in
,,
UU
UI

. This shows that
U
f
is weakly-α-I-continuous for each
.
Sufficiency. Let be any open set of
V
,Y
. Since
U
f
is weakly-α-I-continuous for each
.

1
U
f
V
is weakly-α-I- open of
,,
UU
UI

and hence by Theorem 1.2.

1
U
f
V
is weakly open in
,,
X
I
for each
. Moreover,
we have
 


11
1
U
1
f
VUfVUfV
fV



 






.
Therefore

1
f
V
is a weakly open in

,,
X
I
by
Theorem 1.1. This shows that
f
is weakly-α-I-conti-
nuous.
Theorem 2.3. A function
f
is weakly-α-I-continuous
if and only if the graph function
:
g
XXY defined by

,
g
xxfx for each
x
X, is weakly-α-I-continuous.
Proof: Necessity. Suppose that
f
is weakly-α-I- con-
tinuous. Let
x
X and W be any open set of
X
Y
containing

g
x. Then there exists a basic open set
such that
UV


,
g
xxf UVWx. Since f
is weakly-α-I-continuous, there exists a weakly-α-I-open
set 0 of U
X
containing
x
such that

0
f
U
U
V. By
Theorem 1.1. is weakly-α-I-open and
0
U

0
g
UU UV W. This show that
g
is weakly-
α-I-continuous.
Sufficiency. Suppose that
g
is weakly-α-I-continuous.
Let
x
X
and V be any open set of containing Y
f
x. Then
X
V
is open in
X
Y and by the
weaklyα-I-continuous of
g
,there exists a weakly-α-I-
open set U containing
x
such that

g
UXV.
Therefore we obtain
f
UV. This shows that
f
is
weakly-α-I-continuous
Theorem 2.4. Let

:,, ,,
f
XIY J

is weakly-
α-I-open (resp weakly-α-I-closed) mapping. If
y
Y
and is a weakly-α-I-closed (resp weakly-α-I-open)
set of
U
X
containing
1
f
y
V
, then ther e exists a w ea kl y-
α-I-closed (resp weakly-α-I-open) subset of con-
taining such that Y
y
1
f
VU
.
Proof: Suppose that
f
is weakly-α-I- c l osed m a ppin g.
Given
y
Y
and U is a weakly-α-I-open subset of
X
containing
1
f
y
, then
X
U is a weakly-α-I-
closed set. Since
f
is weakly-α-I-closed,
f
XU
U
is
weakly-α-I-closed. Hence VY is weakly-
α-I-open. It follows fro m
X
1f

f
yU
that
UyfX
.
Therefore
y
YfXU V
 and

11 1
f
VfYfXU XffXUU
 
 .
Similar argument holds for a weakly-α-I-open map-
ping.
3. Weakly-α-I-Paracompact Spaces
Definition 3.1. A space
X
is said to be weakly-α-I-
Hausdorff, if for each pair of distinct points
x
and
in y
X
, there exist disjoint weakly-α-I-open sets and
in U
V
X
such that
x
U
and
y
V, and UV
.
Definition 3.2. A space
X
is said to be weakly-α-I-
regular space, if for every
x
X and every weakly-α-I-
closed set
F
X such that
x
F
, there exist weakly-
α-I-open sets , such that
1
U2
U1
x
U, 2
F
U and
12
UU
.
Definition 3.3. A space
X
is said to be weakly-α-I-
normal space, if for every pair of disjoint weakly-α-I-
closed sets
A
, , there exist weakly-α-I-op e n s e t s
, such that
BX
U V
A
U, and UVBV
.
Definition 3.4. An ideal topological space
,,
X
I
is said to be a weakly-α-I-compact space if every weakly-
α-I-open cover of
X
has a finite subcover.
Definition 3.5. An ideal topological space
,,
X
I
is said to b e weakly-α-I-paracompact space, if every w eak -
ly-α-I-open cover of
X
has a locally finite weakly-α-I-
open refinem ent .
Definition 3.6. Mapping

:,, ,,
f
XIY J

is
said to be weakly-α-I-perfect, if
f
is weakly-α-I-closed
and for any
y
Y
,
1
f
y
is a weakly-α-I-compact
subset of
X
.
Theorem 3.1. An ideal topological space

,,
X
I
is
Copyright © 2012 SciRes. OJDM
S.-Q. LIU
Copyright © 2012 SciRes. OJDM
144
s
s
S
U
, and we conclude that
X
is a weakly-α-I-
a weakly-α-I-compact space if and only if every family
of weakly-α-I-closed sets of X satisfying the finite in-
tersection property has nonempty intersection. compact space.
Lemma 3.1. Let
X
be a weakly-α-I-paracompact
space and
A
, a pair of weakly-α-I-closed sets of B
X
. If for every
x
B
there exist weakly-α-I-open sets
x
U,
x
V such that
x
A
U,
x
x
V and xx
UV
U V
.
Then there also exist weakly-α-I-open sets , such
that
A
U, and UV . BV
Proof: Necessity. If is any family of weakly-α-I-
closed sets which has finite intersection property, and
F
F

, then

FF
X
FXF

 


X.
Thus
X
F is a weakly-α-I-open set. Since
X
is a
weakly-α-I-compact space, hence there exist finite
Proof:
sets 12
,,,
n
F
FF, such that

1
n
i
i
X
FX

. So
1
n
i
i
X
FX

, and , a contradiction.
1
n
i
i
F

Sufficiency. If is any weakly-α-I-open cover for
X
, then is a weakly-α-I-closed
family that satisfies
:A
XA

FA A
FXAXA
 

 
 
. So dose not
satisfy finite intersection property, which means has
a finite subfamily
12
,,,
n
F
FF
,1
ii
which has intersection
empty. Suppose ,2,,
F
XAi
nn
n, where
i
A. We have . So

11 1
n
iii
ii i
FXAXA
 

 

,,,
12 n
A
AA is a finite cover of .
Theorem 3.2. Weakly-α-I-compactness is an inverse
invariant of weakly-α-I-perfect mapping.
Proof: Let :
f
XY be a weakly-α-I-perfect map-
ping onto a weakly-α-I-compact space Y. Given
a weakly-α-I-open cover

s
s
S
U of the space X. Since
f
is a weakly-α-I-perfect mapping and for every
y
Y choose a finite set , such that

y
SS


1
s
y
sS
y
f
yU

U.
y
U is a weakly-α-I-op en from
Theorem 1.1. And from Theorem 2.4 there exists a weakly-
α-I-open set
y
V containing such that y


11
y
y
f
yfV U


. Since is a weakly-α-I-com -
pac t space, the weakly-
Y
I
open cover

y
y
Y
V
of
has a finite subcover such that
Y

1,,k:=
yi
Vi
1
k
yi
i
YV
. Therefore

1
111
kkk
yy
ii
iiisS
yi
s
X
fV UU





 


1
k
, which means
s
isS
yi
X
U





 . Because the family of these
()
:,,,
sy
i
UsS iik
is a finite subfamily of
x
x
B
XB V
is a weakly-α-I-open cover
of the weakly-α-I-paracompact space
X
, so that it has a
locally finite weakly-α-I-open refinement

s
s
S
W
. Let
:,
s
SxBWV
1x
Ss ,, then for any
1
s
S
, s
AW
and
1
s
sS
BW
. Since
s
s
S
W
is
a locally finite family, it is a closure preserving family.
So
11
s
s
sS sS
WW


, then the set
1
s
sS
UXW

is open. Therefore U is a weakly-α-I-
open set.
1
s
sS
VW
UV
is also a weakly-α-I-open set from
theorem1.1. and
.
Theorem 3.3. Assuming
X
is a weakly-α-I-para-
compact space, if one-point sets of
X
are weakly-α-I-
closed sets, then
X
is weakly-α-I-normal space.
Proof: Substituting one-point sets for A in Lemma 4. 1.,
and if one-point sets are weakly-α-I-closed sets, we see
that every weakly-α-I-paracompact space is weakly-α-I-
regular. Applying Lemma 3.1. again we have the con-
clusion.
4. Summary
Combining the topological structure with other mathe-
matical features has provided many interesting topics in
the development of general topology. And this paper has
done much work on the ideal topological space. On
certain extent, it promotes the development of topology.
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