Download on pairwise hyperconnected spaces

SOOCHOW JOURNAL OF MATHEMATICS
Volume 27, No. 4, pp. 391-399, October 2001
ON PAIRWISE HYPERCONNECTED SPACES
BY
BISWANATH GARAI AND CHHANDA BANDYOPADHYAY
Abstract. This paper deals with pairwise hyperconnected and pariwise maximal
hyperconnected spaces. Pairwise door spaces have been introduced and spaces
which are both pairwise hyperconnected and pairwise door have been studied.
Introduction
E. Hewitt 2] dened a space (X ) to be irresolvable if each pair of dense sets
has a non-empty intersection otherwise resolvable. Many authors including M.
Ganster 6], D. R. Anderson 5] etc. have investigated interesting results of resolvable and irresolvable spaces. In 1993 C. Chattopadhyay and C. Bandyopadhyay
3] have introduced and studied resolvability and irresolvability in a bitopological
space. According to Chattopadhyay and Bandyopadhyay 3] a bitopological space
(X ) is pairwise hyperconnected if A \ B 6= for every non-empty -open set
A and -open set B . In the same paper the authors have called a bitopological
space to be pairwise irresolvable if intersection of a -dense set with a -dense
set is non-empty. The purpose of this paper is to study pairwise hyperconnected
and pairwise irresolvable spaces and to introduce pairwise door and pairwise submaximal spaces. In section 2 we have some interesting properties of a pairwise
hyperconnected bitopological space in terms of lter and then in section 3 and in
section 4 these spaces have been related with pairwise submaximal, pairwise door
and pairwise irresolvable spaces. Pairwise maximal hyperconnected and pairwise
minimal door spaces have also been studied.
Received January 13, 2000
revised April 16, 2001.
AMS Subject Classication. 54A10.
Key words. hyperconnected, submaximal, resolvable, irresolvable, maximal, door, lter, ultralter.
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BISWANATH GARAI AND CHHANDA BANDYOPADHYAY
1. Preliminaries
If B X , (X ) being a topological space we shall denote int B and cl B as
the interior and closure of B respectively in (X ). We shall require the following
known denitions.
Denition 1.1.(7]) A topological space (X ) is hyperconnected if inter-
section of any two non-empty open sets is non-empty.
Denition 1.2.(2]) A space (X ) is submaximal if every dense set is open
in (X ).
Denition 1.4.(4]) A subset A of a space (X ) is semi-open if there exists
an open set U such that U A cl U , where cl U denotes the closure of U in
(X ).
The following denition is due to P. M. Mathew 1].
Denition 1.5.(1]) A space (X ) is door if for each A X , either A or
X n A is open in (X ).
2. Pairwise Hyperconnected Spaces
Denition 2.1. A subset B of a bitopological space (X ) is said to be -
semi-open with respect to (resp. -semi-open with respect to ) if B cl int B
(resp. B cl int B ).
SO( ) will denote the collection of all -semi-open sets with respect to (resp. SO( )).
Theorem 2.1. Let (X ) be pairwise hyperconnected and be hypercon-
nected. Then SO( ) n fg is a lter on X .
Proof. Let A, B 2 SO( ) n fg. Then A cl int A and B cl int B .
Hence A \ B (cl int A) \ (cl int B ). Since is hyperconnected int A \
int B 6= . Now as (X ) is pairwise hyperconnected, 6= int (A \ B ) is
-dense and so A \ B cl int (A \ B ). Therefore A \ B 2 SO( ) n fg.
Now let A B 2 SO( ) n fg. Then B cl int B and so int A 6= . Thus
ON PAIRWISE HYPERCONNECTED SPACES
393
A cl int A. Hence A 2 SO( ) n fg. Consequently SO( ) n fg is a lter
on X .
Corollary 2.1. Let (X ) be pairwise hyperconnected and be hypercon-
nected. Then SO( ) n fg is a lter on X .
Corollary 2.2. Let (X ) be pairwise hyperconnected and be hypercon-
nected. Then SO( ) is a topology on X .
Observation 2.1. Let (X ) be pairwise hyperconnected. Then is hy-
perconnected if and only if SO( ) n fg is a lter on X .
The rst part follows from Theorem 2.1. Conversely if SO( ) n fg is
a lter on X , then for any two nonempty open sets U , V 2 we have U ,
V 2 SO( ) n fg. Since SO( ) n fg is a lter, U \ V 2 SO( ) n fg and
so U \ V 6= . Hence is hyperconnected.
However the following example shows that the converse of Theorem 2.1 holds
only in one way.
Example 2.1. Let
X = fa b cg
= f X fag fbg fa bgg
= f X fbg fb cg fa bgg:
Then SO( ) n fg = fX fbg fb cg fa bgg. Clearly is hyperconnected and
SO( ) n fg forms a lter on X but (X ) is not pairwise hyperconnected.
Denition 2.2. A bitopological property P is called pairwise contractive
(resp. expansive) if (X ) has the property P and (resp. ), (resp. ) then (X ) has also the property P .
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Remark 2.1. Pairwise hyperconnectedness is a pairwise contractive bitopo-
logical property.
Theorem 2.2. Let (X ) be a bitopological space and SO( )SO( )]
nfg be a lter on X . Then (X ) is pairwise hyperconnected and , are
hyperconnected.
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BISWANATH GARAI AND CHHANDA BANDYOPADHYAY
Proof. Let 6= U 2 and 6= V 2 . Then V 2 SO( ) n fg and
U 2 SO( ) n fg. Since SO( ) SO( )] n fg is a lter on X , U \ V 6= and hence (X ) is pairwise hyperconnected. Also if U V 2 or U , V 2 then as U , V 2 SO( ) SO( )] n fg, we have U \ V 6= . Consequently
both and are hyperconnected.
Theorem 2.3. Let one of and be irresolvable. Then (X ) is pairwise
hyperconnected and , are hyperconnected if and only if SO( ) SO( )] n
fg is a lter on X .
Proof. The suciency part follows from Theorem 2.2.
Let (X ) be pairwise hyperconnected, , be hyperconnected and be irresolvable. Let A, B 2 SO( ) SO( )] n fg.
Case 1. Let A, B 2 SO( ) n fg. Then from Theorem 2.1, A \ B 2
SO( ) n fg.
Case 2. Let A, B 2 SO( ) n fg. Then from Corollary 2.1, A \ B 2
SO( ) n fg.
Case 3. Let A 2 SO( ) n fg and B 2 SO( ) n fg. Then A cl int A
and B cl int B . Hence int A 6= , int B 6= . Therefore by pairwise hyperconnectedness of (X ), int A \ int B 6= , i.e., A \ B 6= . Since (X )
is pairwise hyperconnected, cl int B = X , i.e., int B is dense in and so by
irresolvability of , B has a -interior. Now cl int (A \ B ) = cl (int A \ int B )
and since is hyperconnected, int A \ int B is a non-empty -open set. So, by
pairwise hyperconnectedness of (X ), cl int (A \ B ) = X . Hence A \ B cl int (A \ B ). Thus A \ B 2 SO( ) n fg. Similarly if is irresolvable
then A \ B 2 SO( ) n fg, i.e., A \ B 2 SO( ) SO( )] n fg. Let
A B 2 SO( ) SO( )] n fg. If B 2 SO( ) n fg then from Theorem
2.1 it follows that A 2 SO( ) n fg and if B 2 SO( )] n fg then it follows
from Corollary 2.1 that A 2 SO( )] n fg. Hence SO( ) SO( )] n fg
is a lter on X . Thus the necessity is proved.
Theorem 2.4. Let (X ) be pairwise hyperconnected and both SO( ),
SO( ) are topologies on X . Then (X SO( ) SO( )) is pairwise hyperconnected.
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395
Proof. The proof is easy and hence omitted.
The equivalence class of all topologies on a set X which have the same collection of semi-open sets as that of a topology is denoted by ]. If (X ) is a
bitopological space, we shall denote by ( )] all those ordered pairs of topologies
( ) such that for the bitopological space (X ), SO( ) = SO( ).
Recall that a function f : (X ) ! (Y ) is feebly continuous if preimages
of nonempty open sets have nonempty interior. In this connection the following
theorem is interesting.
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Theorem 2.5. Let (X ) be pairwise hyperconnected and ( ) 2 ( )]
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be such that the identity function I : (X ) ! (X ) is feebly continuous. Then
(X ) is also pairwise hyperconnected.
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Proof. Let 6= U 2 and 6= V 2 . Then int U 6= since the identity
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map is feebly continuous. Again V 2 SO( ) = SO( ). So V cl int V .
Hence int V 6= . Since (X ) is pairwise hyperconnected, (int U )\(int V ) 6=
, i.e., U \ V 6= . Hence (X ) is pairwise hyperconnected.
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Theorem 2.6. Let (X ) be pairwise hyperconnected and both , be
hyperconnected. Let be the topology generated from the subbasis f g. Then
(X ) is hyperconnected.
Proof. Suppose (X ) is not hyperconnected. Then there exist two non
empty -open sets, say O1 and O2 such that O1 \ O2 = .
Now
O1 = P Q(O \ O ) and
O2 = I J (O \ O )
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2
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2
where O , O 2 , O , O 2 and P , Q, I , J are index sets. But then
= O1 \ O2 = (Q \ O \ O \ O ):
Hence for all , , , we have
O \ O \ O \ O = :
(1)
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BISWANATH GARAI AND CHHANDA BANDYOPADHYAY
Since , are hyperconnected, O \ O = O is a non-empty -open set and
O \ O = O is a non-empty -open set. Hence by pairwise hyperconnectedness
of (X ), O \ O 6= , which contradicts (1). Thus (X ) is hyperconnected.
3. Pairwise Maximal Hyperconnected Spaces
Denition 3.1. A bitopological space (X ) is called a pairwise maximal
P space (resp. a pairwise minimal P space) with a property P if (X ) has
the property P with (resp. ) and (resp. ) then = and = .
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Theorem 3.1. Let (X ) be pairwise maximal hyperconnected and be
hyperconnected. Then = SO( ).
Proof. Clearly SO( ). From Corollary 2.2, SO( ) is a topology
on X . Also (X SO( )) is pairwise hyperconnected. Since SO( ), by
pairwise maximal hyperconnectedness of (X ), we have = SO( ).
Theorem 3.2. Let (X ) be pairwise maximal hyperconnected and be
hyperconnected and irresolvable. Then SO( ) n fg is an ultralter on X .
Proof. From Theorem 2.1, SO( ) n fg is a lter on X . Let 6= A X
and A 62 SO( ) nfg. Then A 62 . Consider (A) = fU (V \ A) : U V 2 g,
the simple extension of by A. Then 6 (A) and so (X (A)) is not pairwise
hyperconnected. Hence there exist 6= U 2 and 6= V 2 (A) such that
U \ V = . Now V = V1 (V2 \ A), where V1, V2 2 . So U \ (V1 (V2 \ A)) = ,
i.e., (U \ V1 ) (U \ V2 \ A) = . Since (X ) is pairwise hyperconnected,
V1 = and U \ V2 \ A = . Since is hyperconnected and (X ) is pairwise
hyperconnected, int A = . Hence X n A is -dense and so by irresolvability
of , X n A has a nonempty -interior. Thus X n A 2 SO( ) n fg. Hence
SO( ) n fg is an ultralter on X .
Denition 3.2. A bitopological space (X ) is said to be pairwise sub-
maximal if each -dense set is -open and each -dense set is -open.
Following the denition of pairwise irresolvable space (3]) we get the following theorem.
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Theorem 3.3. Every pairwise submaximal space (X ) is pairwise ir-
resolovable.
Proof. Since (X ) is paiwise submaximal, every -dense set is -open
and hence the proof follows.
Theorem 3.4. Every pairwise hyperconnected and pairwise submaximal
space is pairwise maximal hyperconnected.
Proof. Let (X ) be pairwise hyperconnected and pairwise submaximal.
Now let (X ) be pairwise hyperconnected such that and . Let
6= O 2 . Then O \ U 6= , 8U 2 . Hence O is -dense. Since (X ) is
pairwise submaximal, O is -open. Thus = . Similarly = . Consequently
(X ) is pairwise maximal hyperconnected.
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The converse of Theorem 3.4 is not true. An example is cited below.
Example 3.1. Let
X = fa b cg
= f X fa bgg
= f X fag fbg fa bg fa cgg:
Then (X ) is pairwise maximal hyperconnected but not pairwise submaximal
as fb cg is -dense but not -open.
4. Pairwise Door Spaces
Denition 4.1. A bitopological space (X ) is said to be pairwise door
if for any A X , either A 2 (resp. A 2 ) or X n A 2 (resp. X n A 2 ).
Theorem 4.1. Let (X ) be pairwise hyperconnected and pairwise door.
Then \ n fg is a lter on X .
Proof. Let A, B 2 \ n fg. Then A, B 2 n fg and A, B 2 n fg.
Since (X ) is pairwise hyperconnected, A \ B 6= . Also A \ B 2 \ n fg.
Let A B 2 \ nfg. If possible let A 62 \ nfg. Then either A 62 nfg or
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BISWANATH GARAI AND CHHANDA BANDYOPADHYAY
A 62 nfg. If A 62 nfg then X n A 2 nfg, as (X ) is pairwise door. Now
B 2 n fg and B \ (X n A) = , contradicts the pairwise hyperconnectedness of
(X ). Also if A 62 nfg, we get a similar contradiction. Hence A 2 \ nfg.
Thus \ n fg is a lter on X .
Theorem 4.2. Let (X ) be a bitopological space such taht , are
hyperconnected. Then (X ) is pairwise hyperconnected, pairwise door and
both , are door if and only if \ n fg is an ultralter on X .
Proof. Suppose (X ) is pairwise hyperconnected, pairwise door and
both , are door. Then by Theorem 4.1, \ n fg is a lter on X . Now let
6= A X and A 62 \ n fg. Then either A 62 n fg or A 62 n fg. If
A 62 n fg then X n A 2 , as is a door topology. Since (X ) is pairwise
hyperconnected, A 62 n fg. Thus X n A 2 n fg, as is a door topology.
Hence X n A 2 \ n fg. Similary A 62 n fg implies X n A 2 \ n fg.
Therefore X n A 2 \ n fg. Hence \ n fg is an ultralter on X .
Conversely, let \ nfg be an ultralter on X and 6= A X . Then either
A 2 \ nfg or X n A 2 \ nfg and so (X ) is pairwise door and both ,
are door. Let 6= U 2 and 6= V 2 . If possible let U \ V = . Then V 62 as is hyperconnected. Again as (X ) is pairwise door, X n V 2 . But then
V \ (X n V ) = , contradicts the hyperconnectedness of . Hence U \ V 6= .
Thus (X ) is pairwise hyperconnected.
Theorem 4.3. Let (X ) be pairwise hyperconnected and pairwise door.
Then (X ) is pairwise maximal hyperconnected and pairwise minimal door.
Proof. Suppose (X ) is pairwise hyperconnected such that 0
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and . Let 6= O 2 be such that O 62 . Then X n O 2 as
(X ) is pairwise door and so X n O 2 , which contradicts the pairwise
hyperconnectedness of (X ). Thus O 2 and = . Similarly = and consequently (X ) is pairwise maximal hyperconnected. Let (X )
be pairwise door such that and . Let 6= O 2 and O 62 . Then
X n O 2 as (X ) is pairwise door and so X n O 2 , which contradicts
the pairwise hyperconnectedness of (X ). Thus = . Similarly = and
hence (X ) is pairwise minimal door.
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ON PAIRWISE HYPERCONNECTED SPACES
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Theorem 4.4. If (X ) is pairwise hyperconnected and \ n fg is an
ultralter on X then (X ) is pairwise minimal door and pairwise maximal
hyperconnected.
Proof. Since \ n fg is an ultralter on X , (X ) is pairwise door.
Hence by Theorem 4.3, (X ) is pairwise minimal door and pairwise maximal
hyperconnected.
Acknowledgment
Authors are thankful to the referee for helpful suggestions to improve the
exposition of the paper.
References
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3] C. Chattopadhyay and C. Bandyopadhyay, Resolvability and irresolvability in bitopological
spaces, Soochow Journal of Mathematics, 19:4(1993), 435-442.
4] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,
70(1963), 36-41.
5] D. R. Anderson, On connected irresolvable hausdor spaces, Proc. Amer. Math. Soc.,
16(1965), 463-466.
6] M. Ganster, Preopen sets and resolvable spaces, Kyungpook Math. J., 27(1987).
7] L. A. Steen and J. A. Seebach (Jr.), Counterexamples in topology, Holt, Rinchart and
Winston, New York, 1970.
Department of Mathematics, University of Burdwan, Burdwan-713 104, India.