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Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013 bc-continuous function Neeran Tahir Al-Khafagy , Hanan Ali Hussein Dept. of Math. / College of Education for Girls / Kufa University Abstract The main aim of this paper to study new class of continuous function is called bc-continuous function, For this aim , the nation of b-open and α-open sets and b-compact space introduced, and we shall study the relationship between bc-continuous and αc-continuous . الخالصة والجل ذلك فاننا قدمناbc- الهدف االساسي من هذا العمل هو دراسة نوع جديد من الدوال المستمرة والتي تسمى الدوال المستمرة bc - وكذلك درسنا العالقة بين الدوال المستمرةb - والفضاء المرصوصα - والمجموعة المفتوحةb - تعريف المجموعة المفتوحة . αc - والدوال المستمرة Introduction Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide . Indeed a significant theme in General Topology and Real analysis concerns the various modified forms of continuity , separation axioms etc. by utilizing generalized open sets.D. Andrijevic introduce a class of generalized open sets in a topological space , so called b- open sets .The class of b- open sets is contained in the class of semi open sets and pre-open sets.This paper includes two sections . In the first section we have dealt with the concepts " b- compact set " ," α-compact set " and " αc- continuous function " . In the second section , we have dealt with " bc-continuous function" , " C1 –space " and some theorems related to these . 1. Basic Definitions 1.1 Definition [Al Omari A. and M.S.Noorani, D.Andrijevic,2008] 0 0 A subset A of a topological space X is called b-open set if A A A .The complement of b-open set is defined to b-closed set . 1.2 Remark [Gaber,Sama Kadhim,2010] Every open set is b-open , but the converse is not true in general . 1.3 Definition [ Njastad O.,1965] 0 0 A subset A of a topological space X is called α-open set if A A .The complement of α-open set is defined to α-closed set . 1.4 Theorem [ T.Noiri,Al-Omari A. and M.S.M Noorani ,2009] Every α-open set is b-open set in any topological space X . 1994 1.5 Definition [K.R.Geentry and H.B.Hoyle.III,1970] A topological space X is called C1 space if and only if every infinite subsets of X has a non-empty interior . 1.6 Definition [Gaber,Sama Kadhim,2010] A subset A of a topological space X is called b- compact set relative to X , if every cover of A by b- open sets of X has a finite sub cover . 1.7 Theorem Every b-compact set is compact set in any topological space . Proof :The proof is complete by remark ( 1.2 ) . 1.8 Definition [Maheshwari S.N. and Thakur S.S,1980] A subset A of a topological space X is called α- compact set relative to X , if every cover of A by α- open sets of X has a finite sub cover . 1.9 Theorem Every b-compact set is α- compact set in any topological space . Proof :The proof is complete by theorem (1.4). 1.10 Definition [W.Fleissner , J.Kulesza , and DR.Levy,1991] A topological space ( X , T ) is said to be almost –compact if for each open cover {W : } ,there exist finite subsets i such that X {W : i } . 1.11 Definition [Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T,2010] A function f : X Y is called αc-continuous if and only if for each x X and each open set V containing f (x) , such that V c is α-compact relative to Y , there exists an open set U containing x such that f (U ) V . 1.12 Definition [P.E.Long and T.R.Hamlett ," H-continuous function,1975] A function f : X Y is called H-continuous if and only if for each x X and each open set V containing f (x) , such that V c is almost - compact relative to Y , there exists an open set U containing x such that f (U ) V . 1.13 Definition [Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T,2010] 1995 Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013 The family of open sets having the complement α-compact relative to X may be used as a base for a topology Tc . 2. bc-continuous function 2.1 Definition A function f : X Y is called bc-continuous if and only if for each x X and each open set V containing f (x) , such that V c is b-compact relative to Y , there exists an open set U containing x such that f (U ) V . 2.2 Theorem Every continuous function is bc-continuous . Proof :Let f : X Y be a continuous function , and let x X , V be an open set containing f (x ) such that V c is b-compact relative to Y . Since x X , V is an open set containing f (x) and f is continuous ,then there exists open set U containing x such that f (U ) V . Then f is bc-continuous function . 2.3 Theorem For a function f : X Y , the following are equivalent 1) f is bc-continuous . 2) if V is open in Y and V c is b-compact relative to Y , then f 1 (V ) is open in X . 3) if F is closed in Y and b-compact relative to Y , then f 1 ( F ) is closed in X . Proof :1 2 Let V be an open set in Y such that V c is b-compact relative to Y , and let x f 1 (V ) Then f ( x) V and there exists an open set U containing x such that f (U ) V ( from 1). Then x U f 1 (V ) . Then f 1 (U ) is an open set in X . 23 Let F be closed and b-compact set relative to Y . c Then F c is open and since F F c is b-compact relative to Y . Then f 1 ( F c ) is open in X ( from 2) . Then ( f 1 ( F c )) c is closed in X , but ( f 1 ( F c )) c ( f 1 (Y F )) c c ( f 1 ( F ))c f 1 ( F ) Then f 1 ( F ) is closed set in X . 3 1 Let x X and V be open set containing f (x) such that V c is b-compact relative to Y. 1996 Since V is an open set , then V c is closed set in Y , and since V c is b-compact relative to Y Then f 1 (V c ) is closed set in X ( from 3) . Then U ( f 1 (V c )) c is open set containing x such that f (U ) V . Then f is bc- continuous function . 2.4 Proposition If A1 and A2 are b- compact sets relative to X , then A1 A2 is b-compact relative to X. Proof:Let F W : is an open cover of A1 A2 . Then A1 A2 W . Then F is an open cover of A1 , A2 ,so for each i=1,2 there exists a finite subset i of , such that Ai So, we have W . A A W 1 i 2 , i=1,2 i Then A1 A2 is b-compact relative to X . Let ( X , T ) be a topological space , it follows from proposition (2.4) that the family of open sets having the complement b-compact relative to X may be used as a base for a topology Tbc . 2.5 Remark For a space X , we have Tc Tbc T . 2.6 Theorem A function f : ( X , T ) (Y , T ) is bc-continuous if and only if f : ( X , T ) (Y , T bc ) is continuous. Proof :Suppose f : ( X , T ) (Y , T ) is bc-continuous function , and let A Tbc Then A is an open set in Y and Ac is b-compact relative to Y , so from theorem ( 2.3) f 1 ( A) is an open set in X . Then f is continuous . Conversely Let x X and V be open set containing f (x) such that V c is b-compact relative to Y. Since f : ( X , T ) (Y , T bc ) is continuous function . Then f 1 (V ) is an open set in X containing x . Let f 1 (V ) U ,then f (U ) V Then f is bc-continuous . 1997 Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013 The following theorems are immediate consequences of theorems (2.6) and the proofs are omitted . 2.7 Theorem If f : X Y is bc-continuous and A is a subset of X , then the restriction f / A : A Y is bc-continuous . 2.8 Theorem If f : X Y continuous function and g : Y Z is αc-continuous function ,then the composition g f : X Z is bc-continuous function . 2.9 Theorem If X is Hausdorff space and A is b-compact relative to X , then A is closed . Proof :Let A be b-compact set relative to a Hausdorff space X . So A is compact ( from theorem 1.7 ) Since X is Hausdorff , so A is closed set . 2.10 Theorem Let Y is Hausdorff space ,then a function f : X Y is bc-continuous if and only if f 1 ( K ) is closed set in X for each set K is b-compact set relative to Y . Proof :Let f : X Y is bc-continuous , and let K is b-compact set relative to Y Since Y is Hausdorff space , and f : X Y is bc-continuous ,then K is closed set in Y. Then by theorem (2.9) f 1 ( K ) is closed in X . Conversely Since f 1 ( K ) is closed set in X for each set K is b-compact relative to Y Then by theorem (2.3) f is bc-continuous . 2.11 Theorem Let X is C1 - space , a subset A is b-compact relative to X if and only if A is almost compact relative to X . Proof:Let A is b-compact relative to X , and F W : be a cover of A , such that W is an open set in X . Then W is an b-open set in X from (1.2). Since A is b-compact relative to X . n n i 1 i 1 _ Then there exists 1 , 2 ,..., n in such that A Wi W i 1998 n_ Then A W i i 1 So A is almost-compact relative to X . Conversely Let W : be b-open cover of X . _ Then W : is an open cover of X . Since X is almost –compact , so there exists a finite subset i such that _ X W : i 0 _ Since X is C1 - space ,then W W is a finite for each i . Hence there is a finite subset F of X such that X W : i F ,therefore X is b-compact. So A is b-compact set . 2.12 Theorem Let Y be C1 - space , then a function f : X Y is bc-continuous if and only if f is H-continuous . Proof :Let f : X Y is bc-continuous, such that Y is C1 - space , and let x X and V be open set containing f (x) such that V c is almost –compact relative to Y . Since Y is C1 - space, then V c is b-compact relative to Y , and since f is bccontinuous , Then there exists open set U containing x such that f (U ) V . Then f is H-continuous . Conversely Suppose f is H-continuous , and let x X and V be open set containing f (x) such that V c is b-compact relative to Y . Then V c is almost –compact from theorem (2.11) . Since f is H-continuous ,then there exists open set U containing x such that f (U ) V . Then f is bc-continuous. 2.13 Theorem Evry αc- continuous bc – continuous function is function . The proof is complete from theorem (1.9). 1999 Journal of Babylon University/Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013 References Al Omari A. and M.S.Noorani ,2008, " Decomposition of Continuity Via b-open set " Bol.Soc.Paran.Math.V.26(1-2),pp.53-64. Andrijevic, D.1996, "On b-open sets ", Math.Vesnik,48(1996),pp.59-64. Gaber,Sama Kadhim,2010,"On b-Dimension Theory",M.S.C,University of AlQadisiya . Geentry K.R. and H.B.Hoyle.III ,1970," C-continuous function" , Yokohama. Math. J. 18 , pp.71-76 . Maheshwari S.N. and Thakur S.S ,1980, " On α-irresolute Mappings ",Tamkang J.Math.11,pp.209-214. Mustafa H.J. and Muhammad S.K. and Al-Kafaji N.T. ,2010," αc-continuous function " ,Journal of kufa for Mathematics and Computer, Vol.1,No.1,April,pp.104-111. 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