Download On Continuous Functions In Bitopological Spaces by

Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
On
  p  Continuous
Functions In Bitopological
Spaces by
Zahir D. AL-Nafie
University of Babylon ,Dept. Math. Collage of Education Ibn- Hayan
Abstract
In this paper we introduce some new continuous functions in bitopological spaces
by the notion of   p  open sets and   p  open sets ,and studied the
relationships of these sets from side and between these function from the other side
,many result are discussed in this paper.
Keywords
  p  open,   p  open,   p  continuous
‫ألخالصه‬
‫م ن خ نالل البحننق نا ننر سة اعننا سو ن الننة ال الاسننرا الجةيننة فننا الاءننرااث الي ر ننا الرننب سبلننب ا الاو فننا‬
‫ الااربحنا ح نق نا نر سة اعنا الوالننا سن ن‬  p - ‫ الااربحنا مجابعنرث‬  p  ‫سباعطا مجابعنرث‬
.‫الاجابعرث م ها الوالنا س الة ال ألاو فه سباعطرهر م ها أخ ى‬
1.Introduction
[A.S. Mashhour, M.E. Abd El-monsef and S.N. El-deeb ,1982] introduced the
notion of pre open sets and pre continuous functions in topological spaces ,[ R. Devi ,
A. Selvakumar and M. Paramili,submitted] ,introduced the notion of  - open sets
which are weaker than open sets, [R.Devi and M. Parimala,2010],introduced the
notion of   p  open sets and   p  continuity in topological spaces.
A sub set A of a topological space ( X , ) is called semi-open ( resp. semi-closed) if
A  cl (int( A)) (resp. int( cl ( A))  A ) [ N.Levine,1963 ],  -open(resp  closed)if A  int( cl (int( A))) (resp. cl (int( cl ( A)))  A )[ S.N.Maheshwari and
R.Prasad,1977/78 ],pre open(resp .pre closed)if A  int( cl ( A)) (resp. cl (int( A))  A )
[A.S. Mashhour, M.E. Abd El-monsef and S.N. El-deeb ,1982 ],regular-open ,if
A  int( cl ( A)) and it is called  -open if for each x  X ,there exist a regular open
set G such that x  G  A [ N.V. Velicko,1968].
A sub set A of a topological space ( X , ) is called a semi-generalized-closed(briefly
sg-closed)set [P. Bhattacharya and B.K .Lahiri,1987 ] if scl ( A)  U whenever
A  U and U is semi-open in ( X , ) .the complement of sg-closed is called sg-open
,and it is called  -closed set
[R. Devi , A. Selvakumar and M. Paramili , submitted ] if scl ( A)  U whenever
A  U and U is sg-open in ( X , ) [ R. Devi , A. Selvakumar and M. aramili,submitted
],the complement of -closed set is called  -open ,and it is called   closed set
[R. Devi and M. Parimala,2010 ],ifcl ( A)  U whenever A  U and U is  -open
in ( X , ) ,the complement of   closed is called   open where cl ( A) be the
intersection of all -closed sets containing A .
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A sub set A of a topological space ( X , ) is called a   p -open[R.Devi and M.
Parimala,2010 ] if A  int( cl ( A)) ,the complement of   p -open is called
  p -closed ,where (cl ( A) ) be the intersection of all   p -closed set
containing A ,in this paper we shall recall all these concepts in bitopological spaces
and introduce the notion of  -p-open sets and  -p-continuous function in
bitopological spaces.
2- Preliminaries
Definition(2.1)
For i  j and i, j  {1,2}, a sub set A of a bitopological space(X,  1 , 2 ) is called:
i) ij - semi open set if A  i  cl ( j  int( A) , and ij - semi closed set if
i  int( j  cl ( A))  A .[ S.N.Maheshwari and R.Prasad,1977/78 ]
ii) ij -   open set if A  i  int( j  cl (i  int( A))) , and ij -   closed set if
i  cl ( j  int( i  int( A)))  A .[ M.Jelic,1990 ],[ D.Andreijevic,1984]
iii) ij - pre open set if A  i  int( j  cl ( A)) , and ij - pre closed set if
i  cl ( j  int( A)  A .[ M.Jelic,1990 ],[ S.Sampath Kumar,1997]
iv) ij    open set if , for all x  A ,there exist  j - regular open set G such that
x  G  A, A  i ,and it is complements called ij    closed.[ M.Jelic,1990 ]
v) ij -pre regular p -open set if A  i  p int( j  pcl ( A)) , and ij -pre regular p -closed
set if A  i  pcl ( j  p int( A)). [ M.Jelic,1990 ]
vi) ij - semi generalized closed set(briefly ij - sg-closed ) if A  U  ij  semi open ,
 j  scl ( A)  U , and it is complement is called ij - sg-open.[ G. Thamizharasi and P.
Thangavelu,2010]
vii) ij   closed set if A  U  ij  sg-closed,  j  scl ( A)  U ,it is complement is
called ij   open set.
viii) ij    closed set if  j  cl ( A)  U , A  U  ij    open set, and it is
complement is called ij    open set.
ix) ij    pre open set if A  i  int( j  cl ( A)) , and it is complement is called
ij    pre closed set[ S. Raychaudhuri and M.N. Mukherijee,1993]
xii) ij    closed set if  j  cl ( A)  U , A  U  ij   -open set, and it is
complement is called ij    open set
The intersection of all ij - semi closed (resp. ij -   closed, ij - pre closed,
ij    closed, , ij -pre regular p -closed, ij - sg-closed, ij   closed,
ij    closed, ij    pre closed, , ij    closed) sets containing A is called ij semi closure (resp. ij -   closure, ij - pre closure, ij    closure, , ij -pre regular
p - closure, ij - sg- closure, ij   closure, ij    closure, ij    pre closure,
ij    closure) and it is denoted by ij - scl( A )(resp. ij -  cl( A ), ij - pcl( A ),
ij   cl( A ), , , ij - cl sg ( A) , ij  cl( A ), ij  cl ( A) , ij  clp ( A) ).
The family of all ij - semi open (resp. ij -   open, ij - pre open, ij    open, , ij pre regular p - open, ij - sg- open, ij   open, ij    open, ij    pre open,
ij    open) sets is denoted by ij -SO(X) (resp. ij -  O(X), ij -PO(X), ij -  O(X),
ij -PRO(X), ij -SGO(X), ij - O(X), ij -  O(X), ij -  PO(X), ij -  O(X)) ).
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
3- Some types of ij   p- open sets
Definition(3.1)
A sub set A of a bitopological space (X , 1 , 2 ) is called ij    p- closed set if
i  cl ( j  int  ( A))  A . The complement of ij    p- closed set is called
ij    p- open set, the family of all ij    p- closed set (resp. ij    p- open
set ) is denoted by ij  PC ( X ) (resp. ij  PO( X ) ) , For i  j and i, j  {1,2}
Definition(3.2)
A sub set A of a bitopological space (X , 1 , 2 ) is called ij    p- closed set
if i  cl ( j  int  ( A))  A . The complement of ij    p- closed set is called
ij    p- open set, the family of all ij    p- closed set (resp. ij    p- open
set ) is denoted by ij  PC ( X ) (resp. ij  PO( X ) ) , For i  j and i, j  {1,2}
The intersection of all ij    p- closed (resp. ij    p- closed ) sets containing
A is called ij    p- closure (resp. ij    p- closure ) and it is denoted by
ij  pcl ( A) (resp. ij  pcl ( A) ).
Proposition(3.3)
Let A, B are two sub sets in bitopological space (X , 1 , 2 ) , then the following
properties hold:
i) A is ij    p- closed (resp. ij    p- closed ) if and only if A = ij  pcl ( A)
(resp. ij  pcl ( A) ).
ii) if A  B , then ij  pcl ( A)  ij  pcl (B) (resp. ij  pcl ( A)  ij  pcl (B) )
iii) ij  pcl ( A) (resp. ij  pcl ( A) ) is ij    p- closed (resp. ij    pclosed , ij    p- closed ) and that is ij  pcl ( ij  pcl ( A) )= ij  pcl ( A)
(resp., ij  pcl ( ij  pcl ( A) )= ij  pcl ( A) ) .
Proposition(3.4)
Every ij -pre open is ij    p- open set.
Proof :
it follows from definition , notes that the converse of this proposition need not be
true by the following example.
Example(3.5)
Let X={a,b,c} and τ 1 = τ 2 = { X,Ø{a},{c},{a,c},{b,c} },here {b} and {a,b} are
12    p- open set but not 12-pre open
Proposition(3.6)
i)
Every ij    open ij -pre open set .
ii)
Every ij    open is ij    p- open set.
Proof:
(ii) follows from (i) and proposition (3.4)above. And the converse of (i) and(ii) need
not be true by the following example.
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Example(3.7)
Let X={a,b,c} and τ 1 = τ 2 = { X,Ø{a},{c},{a,c},{b,c} },here {a,c} is12-pre open but
not12    open and {b},{a,b}and{a,c} are 12    p- open set but not
12    open.
Proposition(3.8)
i)
Every ij    open is ij    pre open set.
ii)
Every ij    closed is ij    p- closed.
iii)
Every ij    closed is ij    p- closed.
Proof:
Easy by definition of each one ,but the converse is not true for any one by the
following example .
Example(3.9)
Let X={a,b,c} and τ 1 = τ 2 = { X,Ø{a},{c},{a,c},{b,c} },here {a,c} is 12    pre
open but not12    open,{c} is 12    p- closed but not12    closed,{a,b}
is 12    p- closed but not12    closed set.
Proposition(3.10)
Every ij    p- open set is ij    p- open set.
Proof :
by definition and the fact that every ij    open set is ij    open set , the converse
of this proposition need not true by the following example.
Example(3.11)
Let X={a,b,c} and τ 1 = τ 2 = { X,Ø{a},{c},{a,c},{b,c} },here {b} is 12    p- open
but not ij    p- open set
The following diagram shows the relationships between these concepts.
set but it is not 12    p- open
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
ij    closed
ij    closed
ij    p  closed
ij    closed
ij    open
ij    closed
ij    p  closed
ij  pre  closed
ij    p  closed
ij  pre  open
ij    p  open
ij    open
ij    open
ij    open
ij    p  open -
Where none of these implication is reversible and(
) means independent.
1001
ij    p  open
4- Some types of functions defending by ij   and ij   sets
Definition(4.1)
For i  j and i, j  {1,2} a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
i) ij    closed if ,for all F  i  C ( X ), f ( F )  ij  C (Y ) [13].
ii) ij    closed if ,for all F  i  C ( X ), f ( F )  ij  C (Y ) .
iii) ij    closed if ,for all F  i  C ( X ), f ( F )  ij  C (Y ) .
iv) ij    closed if ,for all F  i  C ( X ), f ( F )  ij  C (Y ) .
v) ij    p  closed if ,for all F  i  C ( X ), f ( F )  ij  pC (Y ) .
vi) ij    p  closed if ,for all F  i  C ( X ), f ( F )  ij  pC (Y ) .
Where i  C ( X ) be the set of all closed sets in  i , i  1 or 2
Proposition(4.2)
i) Every ij    closed set is ij    closed and the last is ij    p  closed .
ii) Every ij    closed set is ij    closed and the last is ij    p  closed .
Proof:
Easy by definition of each one ,but the converse is not true by the following example.
Example(4.3)
Let X={a,b,c} and τ 1 = { X,Ø{a}}, τ 2 ={ X,Ø{b,c}},here {a}is ij    closed and
ij    p  closed but it is not ij    closed also {b,c} is ij    closed and
ij    p  closed but it is not ij    closed .
Proposition(4.5)
i) Every ij    closed function ij    closed function and any
ij    closed function is ij    p  closed function.
ii) Every ij    closed function is ij    closed function and
any ij    closed function is ij    p  closed function
proof:
(i)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) be ij    closed function, then ,for all
F  i  C ( X ), f ( F )  ij  C (Y ) ,by (i) of proposition( 4.2) above, f (F ) is
ij    closed and it is ij    p  closed ,hence the result,
(ii)
same of (i) and using (ii) of proposition( 4.2) above.
Proposition(4.6)
Every ij    closed is ij    closed set.
Proposition(4.7)
i)Any ij    closed function is ij    closed function
ii)Any ij    closed function is ij    closed function.
Proof:
ii)It follows by definition and proposition (4.6) above.
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
5- Some types of continuous functions defending by ij   , and ij   sets
Definition (5.1)
For i  j and i, j  {1,2} , a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
i) ij    continuous if ,for each x  X and each V  i  O(Y ), there exists
U  ij  O ( X ) such that f (U )  V .
ii) ij    continuos if ,for each x  X and each V  i  O(Y ), there exists
U  ij  O( X ) such that f (U )  V .
Where i  O(Y ), be the set of all open sets in  i , i  1 or 2 .
Proposition(5.2)
For i  j and i, j  {1,2} , a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
i) ij    continuous if ,for all F  i  O(Y ), f 1 ( F )  ij  O( X ) .
ii) ij    continuos if ,for all F  i  O(Y ), f 1 ( F )  ij  O( X ) .
Where i  O(Y ), be the set of all open sets in  i , i  1 or 2 .
Proposition(5.3)
For i  j and i, j  {1,2} , a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
i) ij    continuous if ,for all F  i  C (Y ), f 1 ( F )  ij  C ( X ) . Where C (Y ) be
the set of all closed sets in Y. [ O.A. EL-Tantawy and H.M. Abu-Donia,2004]
ii) ij    continuos if ,for all F  i  C (Y ), f 1 ( F )  ij  C ( X ) .
Where i  C (Y ), be the set of all closed sets in  i , i  1 or 2
,
Proposition(5.4)
Any ij    continuos function is ij    continuous function .
Proof:
It follows by definition and the fact that any ij    closed set is ij    closed set
,and the converse is not true by the following example.
Example(5.5
Let X={a,b,c} and τ 1 = { X,Ø{a}}, τ 2 ={ X,Ø{b,c}},and Y={1,2,3} and 1 ={Y, Ø
,{1} },  2 ={ Y, Ø ,{2,3} }and f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is defined by f (a)=1,
f (b)=2 and f (c)=3.
Definition (5.6)
For i  j and i, j  {1,2} , a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
i) ij    continuos if ,for all F  i  C (Y ), f 1 ( F )  ij  C ( X ) .
ii) ij    continuos if ,for all F  i  C (Y ), f 1 ( F )  ij  C ( X ) .
Proposition(5.7)
Any ij    continuos function is ij    continuous function
Proof:
It follows by definition and the fact that any ij    closed set is ij    closed
set.
Definition (5.8)
For i  j and i, j  {1,2} a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called:
1003
i) ij    p  continuos if ,for each x  X and each all such
V  i  O(Y ), thereexistsU  ij  pO( X ) such that f (U )  V .
ii) ij    p  continuos if ,for all V  i  O(Y ), thereexistsU  ij  pO( X ) such
that f (U )  V .
Proposition(5.9)
For i  j and i, j  {1,2} a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called be:
i) ij    p  continuos if ,for all F  i  O(Y ), f 1 ( F )  ij  pO( X ) .
ii) ij    p  continuos if ,for all F  i  O(Y ), f 1 ( F )  ij  pO( X ) .
Proposition(5.10)
For i  j and i, j  {1,2} a function f : ( X , 1 , 2 )  (Y , 1 ,  2 ) is called be:
i) ij    p  continuos if ,for all F  i  C (Y ), f 1 ( F )  ij  pC ( X ) .
ii) ij    p  continuos if ,for all F  i  C (Y ), f 1 ( F )  ij  pC ( X ) .
Proposition(5.11)
Any ij    p  continuos function is ij    p  continuous function .
Proof:
It follows by definition and the fact that any ij    p-closed set is ij    pclosed set ,and the converse is not true by the following example.
Proposition(5.12)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) ,then the following are equivalent.
i) ij    p  continuos .
ii) B  i  C (Y ), f 1 ( B)  ij  pC ( X )
iii) f ( pcl ( A))  pcl ( f ( A)) ,for each sub set A of X .
iv) pcl ( f 1 ( B))  f 1 ( pcl (( B)) ,for each sub set B of Y .
Proof: (i)  (ii): obvious
(iii)  (iv) , let B be any sub set of Y ,then by (iii) , we have
f ( pcl ( f 1 ( B))  ( pcl ( f ( f 1 (( B))) ,this implies that
f 1 ( f ( pcl ( f 1 ( B)))  f 1 (( pcl ( f ( f 1 (( B)))) . And hence (iv)
Conversely ,let B  f ( A) where A is a sub set of X , then by (iv),we have
pcl ( A)  pcl ( f 1 ( f ( A))) thus f ( pcl ( A))  pcl ( f ( A)) .
(ii)  (iv) let B  i  C (Y )thenf 1 ( B)  ij  pC ( X ) and
f 1 ( B)  f 1 ( pcl ( B) then pcl ( f 1 ( B))  f 1 ( pcl (( B)) .
Conversely ,let K  i  C (Y ) ,by (iv)
pcl ( f 1 ( K ))  f 1 ( pcl (( K )) = f 1 ( K ) thus , f 1 ( K )  ij  pC ( X )
Proposition(5.13)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) ,then the following are equivalent.
i) ij    p  continuos .
ii) B  i  C (Y ), f 1 ( B)  ij  pC ( X )
iii) f ( pcl ( A))  pcl ( f ( A)) ,for each sub set A of X .
iv) pcl ( f 1 ( B))  f 1 ( pcl (( B)) ,for each sub set B of Y .
Proof:
same of proposition( 5.12 ).
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
Proposition(5.14)
i) Every ij    continuous function is ij    continuous and the last
is ij    p  continuous .
ii) Every ij    continuous function is ij    continuous and the last is
ij    p  continuous. .
Proof:
it follows from definition and proposition( 4.2)
The following diagram shows the relationships between these functions where none of
these implication is reversible
ij    continuous
ij    continuous
ij    p  continuous
ij    continuous
ij    continuous
ij    p  continuous
6. ij   (ij   , ij   , ij   ) Connectivity
Definition(6.1)
A bitopological space ( X , 1 , 2 ) is said to be ij   (resp ij   , ij , ij ) connected
if it is not union of two disjoint ij   (resp ij   , ij , ij ) open non-empty
subsets A and B of X such that X=A  B, i  j and i, j  {1,2}.
Proposition(6.2)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) be an ij    continuous function and ( X , 1 , 2 ) is
ij    connected then f ( X ) is ij    connected .
Proof:
Assume that f ( X ) is not connected then there exists G, H both ij    open in Y
,such that G  f ( X )   , H  f ( X )   and (G  f ( X ))  ( H  f ( X ))   and
(G  f ( X ))  ( H  f ( X ))  f ( X )
Now,   f 1 ( )  f 1 ( (G  f ( X ))  ( H  f ( X )))  f 1 (
( G  H )  f ( X )) = f 1 ( f ( X ))  f 1 (G  f ( X ))  ( H  f ( X )) =
f 1 ((G  H )  f ( X ))  ( f 1 (G)  f 1 ( H ))  f 1 ( f ( X ))) 
f 1 (G)  f 1 ( H )  X  f 1 (G)  f 1 ( H ) , since f is ij    continuous then
f 1 (G), f 1 ( H ) are ij    open in X which is mean that X is
not ij    connected ,which is not contradiction.
Proposition(6.3)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) be an ij    continuous function and ( X , 1 , 2 ) is
ij    connected then f ( X ) is ij    connected .
Proposition(6.4
1005
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) be an ij    continuous function and ( X , 1 , 2 ) is
ij    connected then f ( X ) is ij    connected .
Proposition(6.5)
let f : ( X , 1 , 2 )  (Y , 1 ,  2 ) be an ij    continuous function and ( X , 1 , 2 ) is
ij    connected then f ( X ) is ij    connected .
Remark(6.6)
The proof of propositions(6.3),(6.4) and (6.5) is the same manner of proposition(6.2)
and the definition of each one.
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