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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 ],ifcl ( 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 . 997 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 clp ( 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)) ). 998 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. 999 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 1000 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. 1002 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 ). 1004 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. References .Andreijevic D,Some properties of the topology of -sets,Math.Vesnik36(1984)1-10 Bhattacharya P. and B.K .Lahiri ,Semi generalized closed sets in topology, Indian J. Math. ,29(3)(1987) 375-382. Devi R., A. Selvakumar and M. Paramili , -closed sets in topological spaces (submitted) . R.Devi and M.Parimala, Some Applications of P-Open Sets , International Journal of Computer Applications, Volume 5- No. 1(2010) 6-12. M.Jelic ,A decomposition of pair wise continuity,J.Inst.Math.Sci.Ser.3(1990)25-29 N.Levine , semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,70(1963), 36-41. Mashhour A.S., M.E. Abd El-monsef and S.N. El-deeb , On pre continuous and weak pre continuous mappings ,Proc. Math. Phys. Soc.,Egypt , 53(1982) 47-35. .Maheshwar S.N i and R.Prasad,Semi-open sets and semi-continuous functions in bitopological spaces,Math.Notae26(1977/78)29-37 Njastad, O. On some classes of nearly open sets Pacific J.Math., 15(1965),961-970. . Raychaudhuri S and M.N. Mukherijee , On -almost continuity and -preopen sets , Bull. Inst. Math. Acad, Sinica, 21(1993) ,357-366. .Sampath S Kumar,On a decomposition of pair wise continuity ,Bull. Cal. Math .Soc. 89(1997)441-446. Thamizharasi G. and P. Thangavelu, Remarks on Closure and Interior Operators in Bitopological Spaces , KBM Journal of Mathematics Sciences and Computer Applications 1(1) (2010) 1-8. . EL-Tantawy O.A and H.M. Abu-Donia , Some bitopological concepts based on the alternative effects of closure and interior operator, Chaos, Solitons and Fractals 19(2004) 1119-1129. . Velicko N.V, H-closed topological spaces Amer. Math. Soc .Transl. , 78(1968), 103 -118. 1006