Download Sequencial Bitopological spaces

Sequential Bitopological spaces
K.Chandrasekhara Rao  and T.Indra 
Key words:
Pairwise sequential separation axioms, pairwise sequential weak
Hausdorff space:
Mathematics Subject Classification: 54E55
The present work is a continuation of [2].
Introduction
The notion of a sequential set and a sequential topological space is
introduced in [2] which we explain first.
Let X be non empty set. Any sequence An of subsets of X is called
a sequential set in X. For each fixed n, An is called the nth component of An  .
The sequential sets are denoted by A(s), B(s), C(s) and so on.
The following
operations hold
A(s)  B(s) = An  Bn 
n 1
A(s)  B(s) = An  Bn 
n 1
An 
is the empty sequential set if An =   n
An  is the universal sequential set if An = X


n
Department of Mathematics, Mohamed Sathak Engineering College, Kilakarai-623 806. (T.N), India.
Department of Mathematics, Seethalakshmi Ramaswamy College, Tiruchirapalli - 620 001.(T.N), India.
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The empty sequential set in denoted by (s)
The universal sequential set is denoted by X(s)
A(s)  B(s) if An  Bn  n
A(s) = B(s) where A(s)  B(s) and
B(s)  A(s).
The complement
 As c
is the set X(s) – A(s) = X  An n 1

Let N be the set of all positive integers. A sequential set p(s)= Pn 
is called a sequential point in X if 
a non empty set M  N and point x  X
such that Pn = x for n  M
=  for n  N  M.
The point x is called the support and M is called the base of the sequential point.
The sequential point with support x and base M is denoted by (x, M). If M is the
singleton n, the sequential point (x, M) is called a simple sequential point and is
denoted by (x, n). The sequential point (x, N) is called a complete sequential point.
Let p = (x, M)
p  A(s) if (x, M)  A(s). p  wA (s), that is p belongs to weakly to A(s) if x 
An for atleast one n  M.
We extend the results of [2] to the bitopological space and discuss some
other results.
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Definition
Let X be a non empty set. A sequential topology on X is a collection
 of sequential sets in X with the following properties.
(i)
(s)  
(ii)
X(s)  
(iii)
Arbitrary union of members of
(iv)
Finite intersection of members of  is a member of 
(X,
called
 is a member of 
 ) is called a sequential topological space. The members of  are
 - open sequential sets.
(X,
 1,  2), where  1 and  2 are sequential topologies on X, is called a
sequential bitopological space.
Any sequential open set G containing a sequential point
p is called a
neighbour hood of p.
Any sequential set F(s) = Fn n 1 is called a sequential closed set if its

complement X(s)  F(s) is open.
Let A(s) be a sequential set. The sequential set As  =  {F(s): F(s) is a
closed sequential set with F(s)  A(s)} is called the closure of A(s)
A sequential open set A(s) is called a weak neighbour hood of a sequential
point p if p  wA(s).
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Definition
Let
D be a topology on X.
The collection of sequences of D –
opensets forms a sequential topology on X. This is called the sequential topology
generated by the topology D on X. It is denoted by
 (D).
Definition
 is a sequential topology X. Let Dn ( ) be the collection
of the nth component of the sequential sets in  . Dn ( ) forms a topology on X.
For each fixed n= 1, 2 … … Dn (  ) is called the nth component topology on X.
Suppose
Separation axioms are discussed in [1].
Main results
Theorem 1
(X, D1, D2) is pairwise To  (X,  (D1)  (D2)) is pairwise To
Proof
Suppose (X, D1, D2) is pairwise To
Let p = (x, P) and q= ( y, Q) be two distinct sequential points
Case (i)
Suppose x y
.  V  D1 such that x  V and y  V
 p  V(s) and q  w V(s) where V(s) = Vn  and Vn = V, n = 1, 2, …….
 (X,
 (D1),  (D2)) is pairwise To
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Case (ii)
Suppose that x = y
  V  D1 such that x  V
Choose i  P-Q. Then p  w V(s) but
q  w V(s) where V(s) = {Vn} and Vi = V
Vn =   n = 1, 2 … … (n i)
 {X,  (D1),  (D2)} is pairwise To
Conversely suppose that (X,  (D1),  (D2) is pairwise To Let x, y X. Then 
V(s)   (D1) such that (x,1)  V(s) and (y, 1) V(s)
where V(s) = Vn n 1

 x  V1 and y  V1
 (X, D1, D2) is pair wise To
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Theorem 2
(X,
 1 ,  2)
 X , Dn 1 , Dn  2 
is pair wise To 
its component space
are also pairwise To
Proof
Let x and y be two distinct points of X.
The for any n  N, the simple sequential points
p = (x, n) and q= ( y, n) are distinct
since (X,
 1,  2) is pairwise To,  a open  1 – sequential set V(s) = Vn n 1
such that p w V (s) and q  w V(s)
 x  Vn and y  Vn
 (X, Dn (  1), Dn (  2)) is pairwise To.
Definition
(X,  1,  2) is called pairwise To if for every pair of distinct sequential
points p and q,
 a  1 – open sequential set V(s) containing weakly p
but not q.
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Definition
 1 ,  2)is pairwise T1 if for
every pair of distinct sequential points p and q  a  1 – open sequential set U(s)
and  a  2 – open sequential set V(s) such that p w U(s) and q  w U(s)
A sequential bitopological space (X,
q  w V(s) and p w V(s)
Note Every pairwise T1 – space is pairwise To
Theorem 3
 1,  2) is pairwise T1  every pair of sequential points pand q
in X such that p is  2 – closed and q is  1 – closed.
(X,
Proof
Step 1
 1,  2) is pairwise T1 – space. Let p and q be any pair
of distinct sequential points. Then  a  2 – open sequential set V(s) such that q 
Suppose that (X,
w
V(s) and p w V(s)
 q is not a  2 - limit point of p
 p is a
 2 – closed sequential set similarly we can show that q is  1 – closed
sequential set.
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Step 2
Suppose that each pair of sequential points p and q is such that p is
closed and q is
2-
 1 closed in X is a  2 - closed sequential set.
Let p and
q be any two distinct sequential points in X.
U(s) = X(s) - q and
Consider
V s   X (s)  p . Then p w U (s), q  w U (s) and
q w V S , p w V S 
Since q is
 1 - closed U(s) in  1 - open
Since p is  2 - closed V(s) in
 2 - open
Theorem 4
(X, D1, D2) in pair wise T1
 (X,  (D1),
 (D2) is pairwise T1
Proof is similar to that of Theorem 1.
Theorem 5
(X,
 1,  2) is T1  its component spaces (X, Dn ( 1), Dn ( 2)) are
also pair wise T1
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Definition
A sequential bitopological space (X,
 1 ,  2) is said to be a Pairwise
Hausdorff or T2 - space if for any two distinct sequential points p and q  1 open sequential set U(s) and a
 2 - open sequential set V(s) such that
p w U (s), q w V (s) and p w 1  Cl V s  q w  2  cl U ( s)
Note
A pair wise T2 - space is pairwise T1
Theorem 6
The following conditions are equivalent.
(6.1) The space (X, 
1,
 2) is pairwise Hausdorff
(6.2) For any two distinct sequential points p and q
sequential set G(s) and  a
 a 1 - open
 2 - open sequential set H(s) such that p  G(s) ,
q w H (s) Gs   H s    s  and  a  1 - open sequential set D(s) and  a 
2
- open sequential set E(s) such that p w D(s), q  E (s), D(s)  E (s)   (s)
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Proof
Step 1
Suppose that 6.1 is true
  X ,1, 2  is pairwise Hausdorff
Let p and q be two distinct sequential points in X.
Then
 a
 1 - open
sequential set U (s ) and  a  2- open sequential set V(s) such that
p w U S , q w V S , p w 1  clV S , q w  2  clU S 
p w 1  Cl V s   p  X (s)  1clV (s)  G(s) say, Then G(s)
and
H(s) = V(s)
and
are the required for 6.2
Similarly
D(s) = U(s)
E(s) = X(s) -  2 cl U (s) have the properties required on (6.2)
Hence (6.2) holds.
Step 2
Suppose (6.2) holds
consider U(S) = G(s)  D(s)
and V(s) = H(s)  E(s)
 U(s) is 1 - open sequential set and V(s) is  2 - open sequential set
such that p w U (s), q w V (s) and p w 1  Cl V s  and q w  2  Cl U s 
 (X , 1,  2 ) is pairwise Hausdorff.
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Definition
A space (X,  1 , 2 ) is said to be pairwise weak Hausdorff if for any two
distinct sequential points p and q there exists 1 open sequential set U(s) and  a
 2 - open sequential set V(s) such that p w U (s), q w V (s) and U(s)  V(s)
=  (s)
Note
( X , 1 ,  2 )
is pairwise Hausdorff  ( X , 1,  2 ) is pairwise weak
Hausdorff
Theorem 7
The following assertions are equivalent.
(7.1) ( X , 1,  2 ) is sequential pairwise Hausdorff
(7.2)
p   2 cl N (s), N (s) is a 1 - open neighbour hood of p }
for each sequential point p in X.
Proof
Step 1
Suppose that (7.1) bolds, Then ( X , 1,  2 ) is pair wise Hausdorff.. Let p be
any sequential point in X. Let q be a simple sequential point in X. Let q be a simple
sequential point distinct from p. Since ( X , 1 , 2 ) is pairwise Hausdorff ,  a
2
open sequencial set V(s) such that q V(s) and p w 1  Cl V s 
 p  X (s)  1cl V (s) U (s) say
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  2  clU (s)  1  clV ( s)  
  2  clU (s)  X (s)  1  clV (s)  X (s)  V (s)
 q  2  clU (s)
 p  { 2  clN (s) : N (s) is a  1 - open n bd of p }
 (7. 2) holds
Step 2
Suppose that (7.2) holds. Let pand q be two distinct sequential points in X.
By (7.2)  a  1 - open nbd S(s) of p such that  q  2 clS (s)
Put V S   X S    2  clS s 
Then q w V(s), p w  2 clV (s)
Also V(s) is  2 - open
Similarly
 a  1 - open sequential set
U(s) such that p w U (s)
and
q w 1  clU (s)
 X , 1 ,  2 
in pairwise Hausdorff
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Theorem 8
(X, D1 ,D2) is pairwise Hausdorff
 (X, (D1) , 2 (D2) is paiwise Hausdorff.
Proof is obvious.
Theorem 9
((X,  1 ,  2) is pairwise weakly Hausdorff  the component spaces
(X, Dn (1), Dn ( 2)) are Hausdorff for n = 1, 2, … …
Proof is obvious.
Definition
A sequential bitopological space (X,  1, 2) is said to be pairwise
regular if for any sequential point p and a 1 - closed sequential set F(s) with
p  F(s) there exist 1 - open sequential set U(s) with p w U(s) and 
2 – open sequential set V(s) with
F(s)  w V(s) such that p 1 – cl V(s)
and F(s)  X (s)   2 –cl U(s)
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Theorem 10
(X, 1, 2 ) is pairwise regular  its components (X, Dn (1), Dn(2))
are pairwise regular.
Proof
Let F be a Dn (1) - closed in
(X, Dn (1), Dn (2)). Let x  X be any point with x  F . Put Fn = F n
Then F(s) = Fn n 1 be a 1 – closed sequential set.

Let p = (x, n) . Then p  F (s ) (Since X,  1,  2)
is regular,  a  1 - open
sequential set U(s) = { Un } such that p w U (s) and
F (s)  X (s)   2  clU (s)  V (s)  Vn  say
Then x U n , F  Vn ,
U n  Vn  
U n  Dn (1 ) and Vn  Dn ( 2 )
  X , Dn 1 , Dn  2  are pairwise regular for n = 1, 2 … …
Theorem 11
 X , D1, D2  is pairwise regular  the generated sequential bitopological
space  X , Dn 1 , Dn  2  is pairwise regular.
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Definition
A Sequential bitopological space  X , 1,  2  is said to be a weak Hausdorff
space, if for any two distinct Sequential points p and q  a Sequential  1 - open
weak neighbourhood U(S) of p and  a sequential
 2 open weak neighbourhood
of V(s) of q such that U s   V s  =  s 
Definition
A sequence of sequential points xn  is said to converge to a sequential point
p if every weak neighbourhood of p contains x n eventually. This situation is
represented by xn  p .
Theorem 12
Let
 X ,1, 2  be a pairwise sequentially weak Hausdorff space.
If xn  p w.r.t.  1 and xn  q w.r.t.  2 then p = q.
Proof
Assume that p  q . But  X , 1,  2  is pairwise weakly Hausdorff Hence
 a sequential  1 - open weak neighbourhood U (s) of p and  a sequential  2
- open weak neighbourhood V(s) of q such that U(s)  V(s) =  (s)
Now xn  p w.r.t.  1
Hence xn  U (s)  n  n1 for some n1
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xn  q w.r.to  2 . So xn V s , n  n2 ,
for some n2 . Take m = max
(n1,n2). Then xm  U (s) and xm  V(s), a contradiction. This contradiction shows
that p = q.
Theorem: 13
Every pairwise sequentially weak Hausdorff space is bi- T1
Proof:
Let
 X ,1, 2  be a pairwise sequentially weak
be a pair of distinct sequential points in X. Take
with respect to  1 .
Consequently
Hausdorff space. Let p,q
pn  p n . Then pn  p
But X is pairwise sequentially Hausdorff. So,
pn 
 q.
 a  2 - weak nbd U s  of p such that q U s . Similarly by
considering the sequence
qn  q n ,  a 1 - weak nbd V(s) of q such that
p V s  . Hence  X ,1, 2  is a bi- T1 space.
Theorem 14
The property of being a pairwise sequential weak Hausdorff space is a
hereditary property.
Proof
Let ( Y , 1 y ,  2 y ) be a subspace of  X , 1,  2  . Let p q in Y. Then p q
in X .
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But  X , 1,  2  is pairwise sequential Hausdorff space. Hence  a weak
 1 - open neighbourhood U(s) of p and  weak  2 open neighbourhood V(s)
of q such that U(s)  V(s) =  (s). Take G(s) = U (s)  Y and H(s) = VS  Y.
Then G(s) is weakly  1 y - open neighbourhood of p and H(s) is a weakly
2y
open neighbourhood of q such that G (S)  H(s) = (s). Hence ( Y , 1 y ,  2 y ) is
pairwise sequential Hausdorff space.
Theorem 15
 X ,1, 2  is
weakly  1 regular with respect to
 2 if for any sequential
point p and any  1 – closed sequential set F(s) with p  F(s) there exists  1 – open
set U(s) with p  w U(s) and  a
 2 open set V(s) such that F(s)  w V(s) and
U(s)  V(s) =  (s).
Similarly pairwise weakly  2 - regular with respect to  1 is defined. If X is
weakly  1 - regular with respect to  2 and vice versa, X is called pairwise weakly
regular.
A pairwise weakly regular, pairwise T1 space is called a pair wise weak
T3 – space.
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Definition
 X ,1, 2  is pairwise weakly normal if for any  1 -
open sequential set
A(s) and  2 - open sequential set B(s) with A(s) B(s)= (s). 
U( s) with A (s) 
w
U (s) and 
 1 -open set
 2 - open set V(s) with B (s)  w V(s) such
that U(s)  V(s) =  (s).
A pair wise weakly normal, pairwise T1 - space is called a pairwise weakly
T4 - space.
Note: Pairwise weakly
T4 space  pairwise weakly T3 – space.
Theorem:16
Product of an arbitrary family of pairwise sequentially weak Hausdorff
spaces is a sequentially weak Hausdorff space.
Proof:
Let I be an index set.
Let
 X i ,1i , 2i :i  I  be a family of pairwise weak Hausdorff spaces.
Let
and
 X ,1, 2  be their product space.
Let
pn  p with respect to 1
pn  q with respect to  2 .
Then
pni  pi with respect to 1i and
pni  qi with respect to  2i for each i  I .
18
But
Therefore
 X i ,1i , 2i 
is a pairwise weak Hausdorff space for each
iI .
pi  qi i . Consequently p  q .
Hence
 X ,1, 2  is a pairwise weak Hausdorff space.
Theorem 17
The property of being a sequentially weak Hausdorff space is preserved
under one-to-one, onto and pairwise open maps.
Proof:
Let
f :  X ,1, 2   Y , 1,  2  have the properties of the hypothesis.
Let qn  q with respect to  1 and q n  s with respect to  2 .
So
But f is onto.
 pn in X such that f  pn   qn  n . Also  p and r in X such that that
f  p   q , f r   s .
Thus
f  pn   f  p 
with respect to  1 and
f  pn   f r  with respect to  2 . But f is pairwise open and one-to-one .
Hence
p n  p with respect to  1 , p n  r with respect to  2 .
 X ,1, 2  is pairwise weakly Hausdorff. Hence p  r .
Consequently f  p   f r . Therefore q  s . Thus, Y ,  1,  2  is a pairwise
But
weak Hausdorff space.
19
Theorem 18
Let
 X ,1, 2  be a bitopological space.
respect to  2  for each point
1 - open set
Then
1
is weakly regular with
x  X , and every  1 open set G containing x,  a
H such that x  H   2  clH  G .
Proof:
Let
G s  be a 1 open set and let x  Gs  . Then x  X  G s  . Also
 X  G  is 1 closed. By hypothesis,  a 1 open set U s  and  2
V s  such that x  U s ,  X  G   V s ,U s   V s    s  .
Thus
U s   X  V s 
so
open set
that
 2  clU s    2  cl X  V s   X  V s   Gs  .
Hence U s  is a
1 open set such that x U s    2  cl U s   Gs  .
Theorem 19
Every subspace of pairwise
weakly regular space is pairwise
weakly
regular.
Proof:
 X ,1, 2  be a pairwise weakly regular space and let Y , 1,  2  be
a subspace of  X , 1,  2  . Let y  Y and let U s    2 such that y  U . .
Let
20
Then U s   U
*
s   Y , where
pairwise weakly regular
U *  2 .
Since y  U
*
and
 X , 1 ,  2 
is
 a  2 - open set V * such that
y  V * s   1  clV * s   U * .
Let V s   V
Then
*
s   Y .
V s    2
and


y  V s    1  cl V s    1  cl V * s   Y  Y


 1  clV * s   1  cl Y  Y 
 1  cl V * s   Y  U * s   Y  U
Thus
 a V s    2 such that
y V s   1  cl V s   U .

Y , 1,  2  is pairwise
weakly regular.
21
Theorem 20
Product of an arbitrary family of pairwise weakly regular spaces is pairwise
weakly regular.
Proof:
Let
 X  ,1 , 2  :   
be a family of pairwise
weakly regular
spaces.
Let
 X , 1 ,  2 
Let
x   x   X .
be their producct space.
Let U s  be a
1
open set containing x.
Then
U s    U  , where U  1 for each    and U  s   X  for all except
finitely
 ’s say 1,...., n .
Since x U  for each
  , 
a
1
- open
set V such that x V   2  cl V  U  . Let V   V ,    such that
V  X  except for   1 ,...., n .
Then V 1 and x  V   2  cl V  
  2  cl V    U   U .
 
 
Thus there exists a  1 -open set V such that x V   2  cl V   U .
Hence
 X ,1, 2  is pairwise
weakly regular.
22
Theorem 21
Every pairwise closed, pairwise continuous image of a pairwise normal
space is pairwise normal.
Proof:
 X ,1, 2  be a pairwise
f :  X ,1, 2   Y , 1,  2  be a pairwise
Let
mapping.
is
2
normal
space.
closed, pairwise
Let A and B be two disjoint subsets of Y where A is
- closed. Then
Let
continuous
 1 -closed and B
f 1  A is  1 - closed and f 1 B  is  2 -closed. Also
A  B    f 1  A  B   f 1      f 1  A  f 1 B    .
But X is pairwise normal. Hence
 disjoint sets G A and GB such that
f 1  A  G A , f 1 B   GB , G A is  1 - open GB is  2 - open.
Let


*
*
Then G A  GB
and

  , A  G *A , B  GB* . Also G*A  Y  f  X  G A 
GB*  Y  f  X  GB  . Hence
Hence

G*A  y : f 1  y   G A and GB*  y : f 1  y   GB .
Y , 1,  2  is pairwise
G *A is  2 - open and G B* is  1 -open.
normal.
23
Theorem 22
Every bi-closed subspace of a pairwise normal space is pairwise normal.
Proof:
Y , 1,  2  be a bi-closed subspace of a pairwise normal space
 X ,1, 2 . Let A be a  1 - closed set and B a  2 - closed set, disjoint from A.
Let
But the space is bi-closed. Therefore A is  1 - closed and B is
pairwise normality of
 X , 1 ,  2  ,
 2 -closed.
By
 a  2 - open set U and a 1 - open set V
such that A  U , B  V , U  V   .
Thus A  A  Y  U  Y
B  B Y V Y
with U  Y   V  Y    .
Also U  Y is  2 -open, and V  Y is  1 - open .
Thus
A U Y
 a  1 - open set V  Y
and
B  V  Y , with
Y , 1,  2  is pairwise
 2 -open U  Y
such that
U  Y   V  Y    .
Hence
and a
normal.
24
REFERENCES
1.
K.Chandrasekhara
Rao
and
K.S.Narayanan,
General
Topology,
S.Viswanathan (Printers & Publishers) Private Ltd., Chennai, 1986.
2.
M.K.Bose and Indrajit Lahiri, Sequential Topological Spaces and
Separation Axioms, Bulletin of the Allahabad Mathematical Society,
Volume 17, 2002, p.23 - 37.
25