729
Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 729-745
ON PAIRWlSE S-CLOSED BITOPOLOGICAL SPACES
M. N. MUKHERJEE
Department of Mathematics
Charu Chandra College
22 Lake Road
Calcutta, India 700 029
(Received August 5, 1982)
ABSTRACT.
The concept of pairwise S-closedness in bitopological spaces has been
introduced and some properties of such spaces have been studied in this paper.
EY WORDS AND PHRASES. Pairwise semi-open, Pairwise almost compact, Pairwise
S-closed, Pairwise regularly open and regularly closed, Pairwise extremal
disconnectedness, Pairwise semi-continuous and irresolute functions.
1980 AMS MATHEMATICS SUBJECT CLASSIFICATION CODS.
i.
INTRODUCTION.
[1]
Travis Thompson
in 1976 initiated the notion of S-closed topological spaces,
which was followed by its further study by Thompson
[2], T. Noiri [3,4] and others.
It is now the purpose of this paper to introduce and investigate the corresponding
concept, i.e., pairwise S-closedness in bitopological spaces.
To make the exposition
of this paper self-contained as far as possible, we shall quote some definitions and
epunciate
some theorems from [5,6,7].
DEFINITION 1.1.
(i)
i
[7] Let (X,
A subset A of
s.o.w.r.t.
j)
X
in
I’ 2
is called
X
(where B j denotes the
T.
be a bitopological space.
semi-open with respect to
if there exists a
j-closure
of
B
i
in
open set
.
(abbreviated as
B such that B C-A-B
X), where i, j
1,2 and
A is called pairwise semi-open (written as p.s.o) in X if A is 1
s.o.w.r.t. 1 in X.
as well as
s.o.w.r.t.
1
2
# j.
M.N. MUKHERJEE
730
(ii) A subset A of X
s.cl.w.r.t.
I
s.cl.w.r.t.
I
2
is called
A subset N of X
where
x e X, if there is a
N.
A point x of X
of
X
s.o. set w.r.t.
i
x
and contained in
i semi-accumulation point of a subset A
is said to be a
l-j, if every i
containing
j
semi-neighborhood of
x, where
x
w.r.t.
1,2 and
i,j
j
intersects
A
j.
i s.cl. sets w.r.t. j, each containing a subset
called the
i semi-closure of A w.r.t. j and will be denoted by
The intersection of all
X, is
of
2
i semi-neighborhood of x w.r.t. Tj,
is called a
in at least one point other than
(iv)
Definitions for
T2"
and p. s.cl. sets can be given similarly as in (i).
(iii)
w.r.t,
2 (denoted as
semi-closed with respect to
A is T1 s.o.w.r.t.
X
if
I
A
-i(j)’
where
i,j
1 2 and
# j
It has been proved in [7] that a subset A of a bitopological space
and moreover,
(X, I’ 2 is i s.cl. w.r.t. j if and only if A
if and only if x is either a point of A or a i semi-accumulation
x
A
-i ()
j
1,2.
j and i, j
point of A w.r.t. j, where
2
In [7] it was deduced that AC (X 1’ 2 is 1 s o.w r t 2 iff
__i(j)A
(A
1) 2
where
A 1 denotes the
use A 2 to mean the
2-interior
l-interior
of
A in X.
of
Similarly we shall
A in X.
It is very easy to see that every i open set in (X, 1’ 2 is i s.o.w.r.t.
j and the union of any collection of sets that are i s.o.w.r.t. j, is also so,
where i, j
1,2;
# j. It was shown in [5] that the intersection of two 1 s.o.
sets w.r.t.
THEOREM 1.2.
is not necessarily
[5] If A is i
AFt B is
1
s.o.w.r.t.
s.o.w.r.t.
j
in
2" But
(X,
we have,
I’ 2
and
B
IC 2’
s.o.w.r.t.
1,2 and
# j.
i
j, where i, j
The first part of the following theorem was proved in [7] and the converse part
then
in
2
[52.
Let AC-YC-- (X, ], 2). If A is i s.o.w.r.t. Tj, then A is
s.o.w.r.t. (j)y. Conversely, if A is (i)y s.o.w.r.t. (j)y and Y
THEOREI4 1.3.
(Ti) Y
A is i s.o.w.r.t. j, where i, j
DEFINITION 1.4. [6] (a) A bitopological space
i’
then
almost compact w.r.t.
j cluster point. (X,
j
(i, j
TI’ 2
1,2;
j)
1,2 and
j.
(X, 1’ 2
if every
is said to be
Ii
i open filterbase
is called pairwise almost compact if it is
has a
T
731
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
almost compact w.r.t.
and
T2
(b) A bitopological space
(X,
space
-
(X*,
TI"
is called an extension of a bitopological
I’ 2
X
XC_ X*
if
I’ 2
almost compact w.r.t.
T
*
(i)X
X* and
A pairwise Hausdorff bitopological space
(X,
T
1,2.
for
is called pairwise
2
I,
H-closed if the space cannot have any pairwise Hausdorff extension.
[6] (a) (X,
THEOREM 1.5.
each cover
X by i
of
I}
{G
is pairwise almost compact if and only if for
I’ 2
open sets
there exists a finite
,
n
G
{G
subcollection
I
such that
X
.l
where i, j
1,2
and
k=l
n
ij.
(b)
(X,
If
(X, i
then
i regular w.r.t. Tj and i almost compact w.r.t. Tj,
is
I’ 2
s compact, for
(c) A pairwise Hausdorff
1,2
i, j
and
# j.
and pairwise almost compact bitopological space is pairwise
H-closed.
(X, 1’ 2
In what follows, by
i.e.,
2.
we shall always mean a bitopological space,
X endowed with two opologies 1 and
PAIRWISE S-CLOSED SPACES.
set
DEFINIT.ON 2.1.
F
Let
{F}
T
2.
be a filterbase in
(X, 1’ 2
and
F
X.
x
is said to
(i) i
x
containing
(ii)
and each
.
x w.r.t.
S-accumulate to
Tj if for every i s.o. set V w.r.t. j
V Tj #
F, F
’
F
i S-converge w.r.t. j to x,
w.r.t.
.
x,
containing
there exists
In (i) and (ii) above,
S-converge to x
if
F is
S-convergent to x w.r.t.
# j
I
1"
-
if corresponding to each
and
F
F such that F
1,2.
i,j
s.o.set V
i
VTj.
F is said to pairwise
S-convergent to x w.r.t.
as well as
2
T2
lhe definition of pairwise S-accumulation point of
F is similar.
DEFINITION 2.2.
{V{V
I}
1,2
o
(X,
X
I’ 2
with
I
n}
is called
T1
S-closed w.r.t.
s.o. sets w.r.t.
such that
is called pairwise S-closed if it is
T2’
n
k.V 2
T
1
X
2
if for each cover
there is a finite subfamily
(where
is some index set).
S-closed w.r.t.
2
and
2
X
S-closed
w.r.t.
THEOREM 2.3.
Let F be an ultrafilter in X.
Then
F
1
S-accumulates to a point
M.N. MUKHERJEE
732
X w.r.t.
x
Let F be
PROOF"
1
S-accumulate w.r.t.
(containing x o)
Since
To
to
x
Then there exist a
Xo.
F
to
T2
F such that
V2
X
and let it not
.
1
1
V w.r.t.
s.o. set
Fa (’2
2"
2
-2V
F C_. X
Then
(2.1)
F
F is 1 S-convergent w.r.t. 2 to x o, corresponding to V there exists
2. Then
F such that
F
(2.2). Clearly (2.1) and (2.2) are
FBC_
F
Note that for this part we do not need maximality of F.
incompatible.
F does not
Conversely, if
s.o. set V w.r.t.
has
F
_
S-convergent to x o w.r.t.
S-convergent w.r.t.
and some
and hence
I
F is
if and only if
2
e
xo
as a
F.
Thus
1 S-converge w.r.t. 2
2
point w.r.t.
I S-accumulation
3
2
F g
and F I(X- g
g2
X
and
V
for each F
for each
F
F
e
e
,
2
32. Hence F f V
2)
maximal, this shows that
x o, there exists a
F T2,
x o, such that
containing
3
to
T
F.
1
But F
for each
F
Since
is
both belong to F, which is a
contradiction.
In the above theorem, the indices
NOTE 2.4.
THEOREM 2.5.
and 2 could be interchanged.
I’ 32)
In a bitopological space (X,
the following are equivalent"
(a) X is I S-closed w.r.t. 2"
(b) Every ultrafilterbase F is I S-convergent w.r.t. 2"
(c) Every filterbase I S-accumulates w.r.t. 32 to some point of X.
(d) For every family {F
of
sets w r t.
s cl
1
B
2’ with(F
there
n
}n
{F
exists a finite subcollection
of
ii=l
PROOF- (a) => (b)
Let F
S-converge w.r.t.
32
to any point of
{V(x)" x
Thus for every
32
(a), there
and by
{V(x)" x
Now,
F
X}
F(x)
is a cover of
X}
n
V(x i)
i=1
being a filterbase, there exists
n
F o__
’
i=1
F(xi)
x
e
X, there is a
F such that
31 s.o.
.
F(x)(
V-TT
set
2
X with sets that are 1 s.o.w.r.t.
exists a finite subcollection
such that
2
X, which does not 1
Then by Theorem 2.3, F has no 31
X.
V(x) w.r.t. 32 containing x and an
Evidently,
( (Fi)
i=1
be an ultrafilterbase in
{F
S-accumulation point w.r.t.
{F}
such that
{V
(xi)"
X.
F o z F such that
1,2
n} of
%.
2
.
F
Then
V
2
(xi)
}
for
F o (- X
(c) Every filterbase F
(b)
=>
I
S-convergent w.r.t.
S-accumulation point of
F w.r.t.
(d) Let F
F F*
Since
%2"
%2"
be a family of
{F
which is a contradiction
by (b), and hence
to some point x
2
F* w.r.t.
:>
Fo
:>
Zl
xo
s.cl. sets w.r.t.
i2
(F
n
positive integer,
F
F}
F*
and
is a
is
I
T1
with ( F
2
n
2
(F
(say),
{F
si i=1
n
xo
is also a
n
and be such that for every finite subfamily
F
F*
is contained in an ultrafilter base
S-accumulation point of
(c)
n.
1,2
n
1 V(xi :2)
F r i=1
:>
733
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
IB.
i
i=1
Thus
X and
forms a filterbase in
i=1
hep.ce
by hypothesis has a
1 s.o. set V(x o)
F
F.
(F a
0
(d)
F
Since
Fo
X
x
w.r.t.
,
a
which is
i2 (X
(F 4
2)
(X
Then
V
Fs
which is impossible
.
be a covering of
V
X
a
n
X
k=l
n
NOTE 2.6.
X with sets that are
(X
Vk)
k=l
j
2
k=l
V--
2
X and hence X
n
}
k=l
with corresponding alterations in
{V-- j },
1
F3
2
or,
So
s.o w r t
2
k
0, or
is
S-closed w.r.t.
1
2"
A subset Y of (X, 1’ 2
where
i, j
and
2
could have been
(a) can be replaced by "X is pairwise
(b), (c) and (d).
will be called
I}
that
Y
(X
Hence
0
k=1
j of X, there exists a finite set of indices
w.r.t.
Fa
for each
n
or
X if and only if for every cover {V
in
(X
iB, i.e.,
Obviously, in the above theorem, the indices
DEFINITION 2.7.
#
By (d), there exists finite number of indices
interchanged and hence the statement
S-closed"
2
n
such that
Vk ,
-
T
V(xo
2
Then for any
2"
F such that x O
O
n
I’ 2
i
2" Hence (Fo
s.o.w.r.t.
O
(a) Let {V (
=>
containing x o, (F) 2
2
there is some
I
x o w.r.t.
S-accumulation point
1
1,2 and
j.
of
S-closed w.r.t.
Y
I’ 2
by
s.o. sets
such
M.N. MUKHERJEE
734
THEOREM 2.8.
Y
if
is
A subset Y of (X,
S-closed w.r.t.
Ti
j
X and Y
in
PROOF- We prove the theorem by taking
when
2
and
(l)y
by sets that are
that are
indices
k=l
THEOREM 2.9.
is
T
I,
2))
Similar will be the proof
I}
o
is
S.O.w.r.t.
2
for each
s.o.w.r.t.
(2)y
s o w r.t
o
en
such that
=>
y
(closed) w.r.t.
i2)
j
X, for i, j
Let
2.
=>
y
1,
2
is
1,2
GO ( Y
2
# j.
and
Y,
be a cover of
{G}
Go (
Y is
is
T
(l)y
A
T
(A
2)
S-closed w.r.t.
1
K
if and only if
2
and
_L.,)2
k=l
[7] A subset A in (X,
T
in
Y
and
n
k=l
DEFINITION 2.10.
of
By hypothesis, there exists a finite number of
n
...) (G.--;(T2)y
-k
j
Then by Theorem
2
(Tj)y
S-closed w.r.t.
and hence by Theorem 1.3,
for each o.
el’ 2
Y
of
can be regarded as a cover of Y by sets
(i)y
is
S-closed w.r.t.
i
GO
A=(A
2.
# j.
k=l
We prove only the case when
indices
j
(j)y
1,2 and
"U
k
where each
y
(2)y
s.o.w.r.t.
Y (C(X,
If
T2, then Y
PROOF-
Ti’
where i, j
1 s.o.w.r.t. 2" Then by hypothesis, there is a finite number
o
I’ 2
n such that
n
n
T2
V-( T2 )Y and the theorem follows.
=>
Y
Y
t
_. o
TI
and
S-closed w.r.t.
By virtue of Theorem 1.3, every cover {V
1.
j
(i)y
will be
TI’ 2
is called
2
X.
in
regularly open
I
(respectively if and only if
). Similarly we define sets that are
regularly open (closed) w.r.t.
2
1"
It has been shown in [7] that a subset B of (X,
closed w.r.t.
j
iff
(X
TI’ 2
is
T
regularly
B) is i regularly open w.r.t. Tj, for i, j
1,2 and
ij.
LEMMA 2.11.
A of a bitopological space
A is i s.o.w.r.t.
Proof is done only in the case when
closed w.r.t.
PROOF"
If a subset
i’
then
A is 2 regularly closed w.r.t.
T1
(X
Tj,
(X,
where i, j
and
A)
1’ T2)
is
j
T
2
is
1,2 and
Tj regularly
j.
2.
regularly open w.r.t.
T
(2.3)
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
(-
X
Let 0
2
Then 0 is
-A
(X
2 and
Thus O---AC_L
LEMMA 2.12.
x
0
-A
A (by (2.3)).
Hence A is
1"
3
i,
where
is
2’ we
ATI2
A C A CZ
have
Then
It has been shown in [7] that a set A in
3j (i,
w.r.t
A
j)
1,2;
j
if it is
j,
PROOF-
We only take up the case
for
Let X be
X w.r.t
s.o.w.r.t.
F)
1
i12
X2
(A 2
(2.4)
i regularly closed
is
closure of some
31
32
1,2
i, j
and
3j open set.
Since
Let
Since
{VF is
s.o.w.r.t.
is
and
s.o. sets w.r.t.
32
is
S-closed
3
(X
1
regularly open set
32
F by sets that are
regularly closed w.r.t.
X is
Since
2"
F be a proper
be a cover of
and hence
32
X
of
3
3j if
2.
j
and
32
I}
X
S-closed w.r.t.
3
j.
S-closed w.r.t.
I
is
j regularly open set w.r.t.
w.r.t.
by
Sinc A is
2.
j
A bitopological space (X, 31 32
and only if every proper
(X
is
1,2.
(X, 31 32
3
J
j then
open, by virtue of (2.4) the result follows.
is
THEOREM 2.13.
of
and
1
32
s.o.w.r.t.
3
# j and i, j
As before we consider the case
s.o.w.r.t.
s.e.w.r.t.
I
A of (X, 31 2
If a subset
3j regularly closed w.r.t.
PROOF:
open and
I
(x
X
735
F).}{V
1
31
by Lemma 2.11,
I}
is a cover of
S-closed w.r.t.
32
X
there exists a
n
finite-number of indices
n
1’ 2
X
such that
F) 32
(X
[
k=l
Since
F is
2 open,
F FiX
(’k 2)].
n
F T2
and hence
LI(Vk 32),
F
proving that
k=l
F
is
31
S-closed w.r.t.
32
sets that are
Conversely, let
1 s.o.w.r.t. 32
proved. So, suppose X -2
Vg
for
subset of
X.
Since
V
X
g 32, for each
some
and
V
a e
IB
be a cover of
X by
I, then the theorem is
Then
31 s.o.w.r.t. 32 by Lemma 2.12, V B
B
is a proper
is
32
2 is
so that X
proper 32 regularly open w.r.t.
and by hypothesis, it is
S-closed
w.r.t.
31
2" Then there exists a finite
regularly closed w.r.t.
I
is
If
I}
{V
1’
1
M.N. MUKHERJEE
736
m
set of indices
a
a I, a 2,
m
(U va k 2)
VB2
x
and
is
2"
k=l
A subset A in (X,
THEOREM 2.14.
and only if every cover of A
We consider only the case
w.r.t.
2
in
w.r.t.
I’
which is a cover of
{Va
1,2
i, j
PROOF-
X and
is
I’ 2
1
j
2.
Let A be
2
regularly closed sets in X
Va
Then each
is
Val
2 LJ
.]V
A w.r.t.
V=
{V 2}
UV
al
number of indices
S-closed w.r.t.
THEOREM 2.15.
a
is a cover of
ar
and
B are
Then it is a cover of
be a
k=l
S-closed w.r.t.
j
1,2 and
i, j
is
2
s.o. cover of
1
I’ 2 )’
(X,
in
s.o.w.r.t.
i
S-closed w.r.t.
2
a2r
a22,
k
(C’
..
I
A is
showing that
then
j
in
X.
By hypothesis, there will exist a finite
ga
A is
a, by
for each
j.
AU B by sets that are
.
I
2
ak
A as well as of B.
U
If
a}
such that
n
Va.
(since each
an
{V
A
number of indices all’ 12’
alk and a21,
r
k
T.
J
Then d B
and BC
lk
k=l 2k
k=l
S-closed w.r.t.
hence A L] B is
THEOREM 2.16.
a
2
by
2,
A. Then by hypothesis, there exist a finite
such that
.I
be a cover of
{Va
a
1,
regularly closed w r t
2
:2"
is also so, where
Let
is
I, a 2,
If A
a
T
T
Then
2"
Lemma 2.12, and
PROOF"
V
Conversely, let the given condition hold and
closed).
ALDB
2
an
S-closed
1
s.o.w.r.t.
I
Lemma 2.11 and hence there exists a finite set of indices
A(.T_
in
i
# j.
and
and
X if
in
j
closed w.r.t.
j regularly
be a collection of
A.
S-closed w.r.t.
i
by sets that are
has a finite subcover, where
X,
S-closed w.r.t.
I
Hence
k
k=l
m
X
a2
-g2___ LJ
such that X
k=l
in
(X,
T.
such that
r T.
al’lR)J(k,.-j 2"lk
and
k=1
1’ 2
2 is also
then
SO.
PROOF-
Let
{V}
2
be a cover of
is also a cover of
A.
U .2
i=I
1
s.o.w.r.t.
Thus there exists a finite number of indices
2’
1’
n
n
such that A
by sets that are
=>
2
C
Va.
i=I
2 and the result follows.
From
then it
an
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
737
Theorem 2.9 anc Theorem 2.16 we get:
COROLLARY 2.]7.
(X,
A
If
.
.
Let AC(X,
THEOREM 2.19.
X.
1
in
PROOF"
I}
be a
Let
{V-
some index set.
s.o.w.r.t.
I
w.r.t.
X-B is
Since
which is
.
if there exists a
,l
X, where i,
dense in
J
is
1,2
.
S-closed w.r.t.
T
A(B is
-
I
cover of ACt B w.r.t.
S.O.
.
I’ 2
eLEI {v}
an,,
B
and
2
S-closed w.r.t.
regularly closed w.r.t.
2
%2"
Then there exist indices
S-closed w.r.t.
I
Then
Thus A
2
be
I’ 2
regularly open w.r.t.
j
(X-B)
I’
and
is
2
2"
where
2’
is
by Lemma 2.11,
A is
(X-B)
S-closed
I
finite in number, such that
n
n
L)
AC
i=I
i=I
A(I B
Thus
V2
--_
"=
regularly open w.r.t.
(a) B
is
1
(b) A 2 is
PROOF"
(a)
I’
T
be
1’ 2
1 S-closed w r t.
T1
S-closed w.r.t.
S-closed w.r.t.
1)
2
is
2
and
B
is
r
A
A
2
then
2
if
B_. A.
if A is 1
1 S-closed w.r.t. 2
Follows immediately from Theorem 2.19.
(b) Since (%
A 2
AI-IB is
and
Let A_ (X,
COROLLARY 2.20.
2
regularly open w.r.t.
closed in
I)
and
1
X.
2
A
the result follows by virtue of Theorem 2.19.
THEOREM 2.21.
then
J
If
(X,
(X, i
1’ 2
is
i
regular w.r.t.
is compact, where i, j
1,2;
i
almost compact w.r.t.
and
j
i
S-closed w.r.t.
j.
By virtue of Theorem 1.5(a), we see that every
Proof
is
for
(I)A (2)A)
j.
and
is
X,
in
J
.
is
1’ 2
A w.r.t.
S-closed subset
1,2
(X,
A space
COROLLARY 2 18
X,
is pairwise S-closed in
pairwise S-closed, then
(A,
is pairwise open and
I’ 2
.
S-closed space w.r.t.
j. Hence by Theorem 1.5(b) the result follows.
In Theorem 3.7 we shall prove a partial converse of the above theorem.
3.
PAIRWISE EXTP, EMALLY DISCONNECTEDNESS AND S-CLOSED SPACE.
DEFINITION 3.1.
A bitopological space
extremally disconnected w.r.t.
-j
(X,
1’ 2
is said to be
if and only if for every
i
i
open set A of X,
M.N. MUKHERJEE
738
X
J is
open, where i, j
t
1,2
disconnected if and only if it is
extremally disconnected w.r.t.
and
j.
X is called pairwise extremally
extremally disconnected w.r.t.
tl
and
2
2
%1"
Datta in [8] has defined pairwise extremally disconnected bitopological space
identically as above, we shall show (see Corollary 3.4) that the concept can be
defined by a weaker condition.
The conclusion of the following theorem was also derived in [8] under the
hypothesis that the space is pairwise Hausdorff and pairwise extremally disconnected.
We prove a much stronger result here.
THEOREM 3.2.
(X,
Let
be
I’ T2
extremally disconnected w.r.t.
X, where A
PROOF-
and
1
(X,
Suppose
B
I"
2 /-
is
T
B
2
AB
with
B.
x
then there exists
t2
Then
2
x
and
B
B
2
Hence
(I). Similarly the other case can be handled.
We prove a stronger converse of the above theorem.
(X,
THEOREM 3.3.
PROOF"
Suppose
there is a
(X
%
2)
is not
1’ 2
and
1
.T 2)
X
A
if
A
and
2
# B,
B
contradicting
).
and
B
2
X
TI"
Then
%
2
is
1
2
X
+2 /-
Hence by hypothesis,
X
X2 (r T1
t2’
holds.
extremally disconnected w.r.t.
T
open set A such that
1
A /’I (X
that
1’ 2
(X,
A, B in
is pairwise extremally disconnected if for every pair of
A and B, where A
disjoint sets
-
(I). Now,
I"
T2
(B.
extremally disconnected w.r.t.
I
or
2
Then for every pair of disjoint sets
one has
2’
I’ 2
extremally disconnected w.r.t.
I
closed.
Thus
2
_T
(X
%
A
2
2)
and
2"
Then
A
T1
}.
Then
such
T
is
-open.
1
A
contradiction.
Similarly,
(X,
2 extremally disconnected w.r.t.
From Theorems 3.2 and 3.3 we have,
COROLLARY 3.4.
either
1
w.r.t.
I"
LEMMA 3.5.
(X,
I’ z2
1’ 2
is
T
(X, 1, 2
I.
is pairwise extremally disconnected if and only if it is
extremally disconnected w.r.t.
If
T
2
or
2
extremally disconnected
is pairwise extremally disconnected, then for every
tl
PAIRWISE S-CLOSEO BITOPOLOGICAL SPACES
s.o. set V w.r.t.
PROOF"
Obviously,
V 2(1)
V
.
x
if
I,
V(3W
open and
(X, i,
Since
set W w r t
s o
2
and W 2
V
Then
open.
2
2
then there exists a
-2 (1)’
x such that
1
s.o. set U w.r.t
2
:
u
-I(O).
Now
and for every
-V2(l)
2’
739
containing
1’
are nonempty disjoint sets, respectively
2)
is pairwise extremally disconnected, we
have
T
2
i.e.,
(
Similarly the other part can be proved.
W 2
V
LEMMA 3.6.
},
(
V
x
Thus
-2(i)
(X, 1’ 2 )’ every i
In a pairwise extremally disconnected space
regularly open set w.r.t.
i open
is
j
and
V2
Hence V
j closed,
where
1,2
i, j
and
i#j.
Let A be a
PROOF"
(X
Now,
(X,
Since
(X
I
2
A )
A are disjoint sets, respectively
and
2
I,
y
Theorem 3.2.
2
Hence
-closed.
open and
2
2) I
(X
Then
2
i-open, so that
is
2)
1
A.
open.
1
2
X
2)
A is
2
X
and
I
is
open and
-closed.
Similarly, we can show that every
-open and
THEOREM 3.7.
PROOF"
(X,
(X,
exists a
I’ T2)
I}
X,
1
X
S-closed w.r.t.
2
xk
}.
{ 2
Hence X
k=l
T1
is
containing
x
such that
X
-2
0
a
Oa X
xk
But 0
Va X (
Vx
X
is
for each
x 1, x
x.
s o.w.r t
I
I.
x
a2
2
Hence
S-closed w.r.t.
2
Then there
X
Since
is pairwise
X
x e X.
xn
2
x
I
is
1
By compactness of
X
x
and
x, for some
open for each
is
(X,
2"
X by sets that are
there exists a finite set of points
{0
k=l
V
there is a
extremal!y isconnected,
1
be a cover of
open set 0a
1
is pairwise extremally disconnected and
is
I’ 2
Let {V(
For each x
X w.r.t.
regularly open set in
2
-closed.
1
If
compact, then
(X,
(%
so that
2’
is pairwise extremally disconnected, we have
1’ 2
2) TI
X w.r.t.
regularly open set in
i
2"
of
X such that
2x
M.N. MUKHERJEE
740
.
We have earlier observed that every
is always
almost compact w.r.t.
3
THEOREM 3.8.
32)
(X, i’
If
extremally disconnected, then
PROOF"
2"
_
for each
I’ 2 )"
(X,
almost compact w.r.t.
(X,
T
T2
2
u
"
follows
I,
it
(X,
Xl’ 2
is
U L)
U
being
2"
U
I)
which is
=-0 2
is
Again
U
is a
I
I
Since
-closed and hence,
2
almost compact w.r.t.
I
1
[(
V
{U L) V
(e2
U
2
that
and pairwise
2
I) of X with sets that are 1
U(ZI UL) VC3
2’ by Lemma 3.6,
Thus
Now we have"
# j.
S-closed w.r.t.
31
I, we consider the set
Then
2
1,
{V-
reoularly, open w.r.t.
i
r___ u_) v
U
=
For each
regularly open w.r.t
is
31
Let us consider a cover
s.o.w.r.t.
U=
is
w.r.t.
1’ 2
and
1,2
for i, j
3j
(X,
S-closed space
being
-open
I
-open cover of
there exists a
2’
finite subfamily
L__J
X
such that
of
o
J V }
{U
Now since U
U V__
V
X
L_J
0
(X,
Hence
T
I,
is
%2
2
UL V 2 C
I, we have
each
and hence
for each
S-closed w.r.t. 32.
31
for
2}
0
SEMI CONTINUITY, IRRESOLUTE FUNCTIONS AND S-CLOSEDNESS.
4.
[7] A
DEFINITION 4.1.
(Y’
into a bitopological space
if for each
A
f-1
Ol,
T2 2
semi-continuity of
f
11
is
LEMMA 4.2.
is called
I’ 2
(A)is
1
s.o.w.r.t.
f w.r.t.
T
f-
PROOF"
Let y
that
f(x)
Hence
f-I
y and x
(V)(A #
THEOREM 4.3.
)
:>
2
Similar goes the definition of
2"
-
2 2 semi-continuous w.r.t. TI"
(Y’ i’ 02) is T1 1 semi-continuous
32, then for any subset A of X, f
V
and y
II
and
2
(X, I’ 2
w.r.t.
f(A_ I(2
1’ 2
semi-continuous w.r.t.
is called pairwise semi-continuous if
f
1.
semi-continuous w.r.t.
If a function
(X,
from a bitopological space
f
function
o
(A.I( 2
I.
f-l(v) and f-I (V)
)
f (f-I (V)(A)
O
f-
I.
is
I
s.o.w.r.t.
V(’ f (a)
=>
A
_I(2
x
Then there exists
such
2"
}
=>
y
f(A)
I.
Pairwise semi-continuous surjection of a pairwise S-closed space onto a
pairwise Hausdorff space is pairwise H-closed.
PROOF"
where
Let f-
(X,
(Y,
31 32
X is pairwise S-closed.
compact w.r.t.
2" Let
{V
1’ 2)
be a pairwise semi-continuous surjection,
We first show that
I}
be a
1
(Y,
o
02
is
open cover of Y.
01
Then
almost
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
{f-1
is
(V)"
_
S-closed w.r.t.
T1
f-l(ve)
X
o
and
is a cover of
I}
W be any
2
U_. W
such that
[ LE)
o
.
s.o. set w.r.t.
U
and
2 open set must
f-1 (V)] }. Then
o
f(X)
[I
f
2"
of
I,
T
Then there exists
x.
f-I (V)
o
Since
I f-l(va)
intersect
such that
In fact, let x
X.
containing
2,
X
Since
is
U
dense in
2
X
2
X,
and hence
o
(
W
]I f-1
(V)) # )
o
and hence
Now,
]
(V)
f
s.o.w.r.t.
f-l(v)
We show that
f-1 (V) 2(i).
Y
I
there exists a finite subfamily
T2
every nonempty
u
X by sets that are
741
2(1)
0
f ("L_) f- (V ))o 2
0
C E
(
o
(using Lemma 4.2 and the fact that f
by Theorem 1.5(a),
Y
almost compact w.r.t,
1. Since
virtue of Theorem 1.5(c) that (Y,
A function f"
lute w.r.t.
if for every
s.o.w.r.t.
irresolute
Y
o
DEFINITION 4.4.
,.
o
2
2
semi-continuous w.r.t
(X,
o
Clearly, every
(Y’
I’ 2
continuous w.r.t.
j,
where
s.o. set
V w.r.t.
22
irresolute
converse is not true, in general.
addition, pairwise open
1,2
i, j
but
f j,
iOl -irreso02 f-1 (V) is 1
w.r.t. 1 and pairwise
. .
is
o.
semi-
D
but it can be shown that the
f
is, in
[7].
(Y,
A of X, f
every subset
is called
This converse is true if the function
LEMMA 4.5. A function f from a bitopological space
bitopological space
o
I’ 2)
irresolute function w.r.t.
T.o.
)" Thus
is pairwise H-closed.
be defined in the usual manner.
cap
1
2. Similarly, Y is orL
is pairwise Hausdorff, it finally follows by
1’ 2)
Functions that are
2"
T
almost compact w.r.t,
o
is
is
o
2)
(_A
tl(2
I, o
is
II
)_
(X,
I’ 2
irresolute w.r.t
2
to a
if and only if for
(A)
(o2)"
If
I’ 2 (Y’ I’ 2) be 1 1 -irresolute w.r.t. 2 and
AZX. Then f-1(f(A)
oi(o2)) is I s.cl.w.r.t. 2" Since A C_T f-l(f(A)) (1
f-1 (f(A) (o))’ we have A (2) f-I (f(A)l(O2)) an hence
PROOF"
Let f"
2
(X,
-
M.N. MIIKHERJEE
742
f(A 1 ())
2
f-I (f(A)o 1(2
f
B be
Conversely, let
f
01
f-i
COROLLARY 4.6.
w.r.t.
and
and then
2
If a function
(X,
f-
f-l(B)1(2)-
f
This shows that
1 1 irresolute
(Y’ 1’ 2) is
is
.
1’ 2
A of X,
then for any subset
j,
f(f-l(B),I(2))C
By hypott,sis,
in Y.
2
f-l(B
and hence
s.cl.w.r.t.
I
o
(02).
B.
f-l(B)
f-l(B). ()<_
2
(B) is
)C f(A) o
-I(2
s.cl.w.r.t.
f-1(Bl (o2)C B_ol (o2)
Then
f(A
e
I,
f(A_ () )__.
w.r.t.
where
2"
irresolute
o.
i, j
I,?
j.
For every subset B of a bitopological space
PROOF"
-()
C g 1,
for
1, 2
i, j
and
(X,
T
we always have
T2
1,
Hence by Lemma 4.5, the corollary
j.
fol lows.
,OTE 4.7.
Following a similar line of proof as in Lemma 4.2, we could also prove the
above corollary 4.6.
THEOREM 4.8.
f"
(X, %1’
(X,
Let
T2)
(Y’
bitopological space.
l’ 2)
Y.
{f-1(A
be pairwise irresolute, where
Let {A
I}
)-
2 2
2)
is a
X, then f(G)
is pairwise S-closed in
f(G) by sets that are
be a cover of
I}
G.
is a cover of
s.o.w r t
o
o
2
_
X there
is pairwise S-closed in
n
I’ 2
2
f-l(Aek
such that
G(
U (f-l(A k)2)"
f-l(A
for
k 2 (1)
we have by Lemna 4.5
1’
Ak
k
f( f-1 (A a
1,2
for k
1,2
n.
Since
f
2(i))
n.
l
n
21]c
L]
k=l
f(G)
Similarly,
wise S-closed in
G
k=l
irresolute w.r.t.
Hence f(G) f
Since
n
(f-i(Ak) o2(o Ak o2(oi)
Y.
o I, o
Then
By temma 3.5, we have
in
(Y,
Y.
exist a fi,ite number of indices
is
G of X
If a subset
is pairwise S-closed in
PROOF"
be pairwise extremally discon.ected and
I’ 2
Y.
n
02
and then f(G) is
w.r.t,
2 S-closed
This completes the proof.
is
o
I
S-closed w.r.t, o 2
k=1
o
in Y.
Hence f(G)
is pair-
743
PAIRWISE S-CLOSED BITOPOLOGICAL SPACES
X,
of Theorem 4.8 is the whole space
G
If the set
NOTE 4.9.
(X,
require the condition that
then we do not
is pairwise extremally disconnected.
I’ 2
In
fact, proceeding iF: a similar fashion as in Theorem 4.3 and using Corollary 4.6, we
can have
THEOREM 4.10.
If
surjective, where
1’ L
1’ 2 is
(X,
i’ c2)
(Y’
(X,
f"
is pairwise irresolute and
pairwise S-closed, then
(Y,
0
01
is also
2
pairwise S-closed.
Let
THEOREM 4.11.
f-
0
2
w.r.t.
PROOF-
(Y,
2)
(X,
T
2
in
Let
X,
o
continuous w.r.t.
1’
T2’
is
be
f-l(
LI
V
V-2 )w_f- (V T2
f-l(v
is
1
GC X
If
-
I
2 in Y.
by sets that are o I
f(G)
such that
o
semi-continuous w.r.t.
1 I
S-closed w.r.t.
01
be a cover of
V
is open, we have
2
f(G)
I}
{U
1’ 2)
(Y’
1’ T2
is continuous and open.
0
2
then
For each e, there is
(Y,
(X,
f"
Since
s.o.w.r.t.
S-closed
0
0
2
f
Since
is
s.o.w.r.t.
f-
I Ol
(X,
0
2
2)
semi-
and hence there exists
s2
such that
-
2
0
OCf-I(v)(ZI 2=> O_f-l(v) 2 Thus O(Zf-I(v)(Zf-I(u)-- f-l(- 2
f-l(v) (ZZ 2 That is, O C f-l(u)--2 and 0 1 Therefore,
I} is a cover
f-l(u) is 1 s.o.w.r.t. 2’ for each I, and {f-l(u)"
of G. Then there exists a finite number of indices 1
n such that
n
U
G
i=l
f
2
f-l(u
i
2,
f’l(uai
f(G)
Since
(X,
f"
for
1,2
2
n.
(Y, 0 2
is continuous,
Therefore,
f(G)
U
o
2
and then
2 in Y.
COROLLARY 4.12. Pairwise S-closedness is a bitopological invariant.
PROOF" Since every pairwise continuous function is pairwise semi-continuous, the
is
01
S-closed w.r.t.
0
corollary follows by virtue of Theorem 4.11.
2
I}
COROLLARY 4.13. Let
{(X,
(X,
2
1
each
PROOF"
(X
, )"
be their product space. If (X,
2
is also pairwise S-closed
be a family of bitopological spaces and
2
is pairwise S-closed, then
)
Since
and for each
P -(X,
(X
i
is an open, continuous suriection, for
I,, the corollary becomes evident because of Theorem 4.11.
1,2
744
M.N. MUKHERJEE
THEOREM 4.14.
The pairwise irresolute image of a pairwise S-closed and pairwise
extremally disconnected bitopological space in any pairwise Hausdorff bitopological
space is pairwise closed.
PROOF-
Let f be a pairwise irresolute function from a pairwise S-closed and
pairwise extremally disconnected space (X,
into
a
pairwise liausdorff space
T1, 2)
(Y,
o2).
0
at y
in
Since
X
2 and
Let y
(Y,
is
c2).
o
F
Then
S-closed w.r.t.
T2
Nl(Y
{f-1(V)-
denote the o -open neighborhood system
I
V Nl(Y)} is a filter-base in X.
31, F has a 2 S-accumulation
point
x
w.r.t.
%1"
We show that f(F)
f(x)
W
V
N
o
(y),
Then
2
f-1(W)
has
f-1(V)
f(x) as a
is
F and hence
2
o
2
accumulation point.
s.o.w.r.t.
I
Indeed, if
[f-l(w)]
2
i.e.,
.
’
N
[f-l(v)]
I(v)
f[(f-1(w)ilF’f-1(V))
.
32
2
0,
then
’
[f-l(w)]
2
B such that y
A, f(x)
(Y, 01
,
02),
_.
.
there exist
B and A F1 B
,
T
[f-l(v)]
o
Since
v.
in
.
is
I- [f- I( V)] 2 #
Now, B
2](Z f[f- 1( w)r f- (v)] w
#
This shows that f(x) is a o
2 accumulation point of f(F)
being finer than N (y), N (y) also o
accumulates to f(x).
open set
Now, for each
(X, Zl’ 2
Since
[f-l(w)]
which is not the case.
by pairwise Hausdorff property of
.
and contains
f-l(w)F f-l(v-----
pairwise extremally disconnected, we then must have
In fact, let
2
Hence W CIV
Y.
But f(F)
Now, if y f(x),
open set A and
A
Nl(Y), f(f-l(A
f(F) and hence B C f(f-l(A) #
because f(x) is a o accumulation point of
2
f(F). In other words B I A
which is a contradiction. Hence y
f(x) and
then y
f(X). Consequently f(X) is
02 closed in Y. Similarly f(X) is
closed in Y.
This completes the proof.
ACKNOWLEDGEMENT.
sincerely thank Dr. S. Ganguly, Reader, Department of Pure
Mathematics, Calcutta University, for his kind help in the preparation of this paper.
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