. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011  -Generalized b- Closed Sets in Topological Spaces A. K. Al-Obiadi Departme nt of Mathematics, College of Basic Education Unive rsity of Al- Mustansiryah Received in : 10 May 2011 Accepte d in : 16 June 2011 Abstract In this p aper we introduce a n ew class of sets called -  generalized b- closed (brief ly  gb closed) sets. We st udy some of its basic p rop erties. This class of sets is st rictly p laced between the class of  gp - closed sets and the class of  gsp - closed sets. Further the notion of  b- 2 1T sp ace is introduced and studied. 2000 M athematics Subject Classification: 54A05 Keywords: b- open se t, regular open set,  -generalized b- closed set. 1. Introduction and Prelimimnaries . Park[1] introduced the class of  -generalized p re-closed(briefly  gp closed) sets and the class of  -gener alized semip reopen closed (briefly  gsp closed) sets was introduced by Sarsak [2] as a generalization of closed sets. In this p aper we define and st udy a new class of  - generalized closed sets, we denote by  -gener alized b- closed (briefly  gb- closed) sets, which is st rictly p laced between the class of  gp - closed set and  gsp - closed sets. M oreover, we define  b- 2 1T sp ace as t he sp ace in which every  gb- closed set is b- closed. Throughout this p ap er ),( X and ),( Y represent nonempty top ological sp aces on which no sep aration axioms are assumed unless otherwise mentioned. For a subset A of a sp ace ),( X , cl(A), int(A) and P(X) denote the closure , the interior and p ower set of A resp ectively. ),( X will be rep laced by X if there is no confusion. Let us recall the followin g defin itions which are useful in the sequel. De finition 1.1. A subset A of a sp ace X is called : (1) semi- open if ))(int( AclA  and semi- closed if AAcl ))(int( .[3] (2)  - open if )))(int(int( AclA  and  - closed if AAclcl )))((int( .[4] . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 (3) preopenif ))(int( AclA  and preclosed set if AAcl ))(int( .[5] (4) semi- preopenif )))((int( AclclA  and a semi- preclosed if .)))(int(int( AAcl  [6] (5) regular openif ))(int( AclA  and a regular closed set if ))(int( AclA  .[7] (6) b- open if ))(int())(int( AclAclA  and b- closed if .))(int())(int( AAclAcl  [8] (7)  - open if A is the union of regu lar op en sets, and  -closed if A is the intersection of regular closed sets. [9] The b- interior (br iefly bint) of a subset A of X is the union of all b- op en sets contained in A. The b- closure (resp . p re-closure, semip re- closure) of A is the intersection of all b-closed (resp . p reclosed, semip re- closed) sets containing A, and is denoted by bcl(A) ( resp .p cl(A), sp cl(A)). The collection of all b- op en (resp . b- closed) sets is denoted by BO(X) (resp . BC(X)).[8] It is well known t hat: (1)  - op en set p reop en set b- op en set  semi- p reopen.[8] (2) The intersection of a b- op en set with  - op en set is b- op en.[8] De finition 1.2. A subset A of a sp ace X is called: (1) generalized closed ( briefly g- closed) if UAcl )( whenever UA  and U is op en in X.[10] (2)  - generalized closed (briefly  g- closed) if UAcl )( whenever UA  and U is  -op en.[11] (3)  - generalized p re cosed ( briefly  gp - closed) if UApcl )( whenever UA  and U is  - op en.[1] (4)  -generalized semip re- closed (briefly  gsp - closed) if UAspcl )( whenever UA  and U is  - op en.[2] Lemma 1.3. [12] Let A X t hen, (1) A B bcl (A) bcl(B). (2) A is b- closed  bcl (A) =A. (3) Let xX, then xbcl (A) if and only if every UBO(X) such that xU, U  A  . 2.  - Generalized b- Close d Sets. De finition 2.1. A subset A of a sp ace X is called  - gener alized b- closed (breif ly  gb closed) if bcl(A)  U whenever A U and U is  - op en. The complement of  gb- closed set is called  gb- op en. The family of all  gb- closed (resp .  gb- op en) subsets of the sp ace X is denoted by  GBC(X) (resp .  GBO(X)). De finition 2.2. The  - kernel (  - ker (A)) of A is t he intersection of all  - op en sets containing A. . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Remark 2.3. A subset A of a sp ace X is  gb- closed if and only if bcl(A)   - ker(A). Remark 2.4. Every b- closed set is  gb- closed. Proposi tion 2.5. Every  gp - closed set is  gb- closed. Proof. Let A be  gp - closed subset of X and U be  - op en such that A U. Then pcl (A) U. Since every p reclosed set is b-closed. Therefore bcl(A) pcl (A). Hence A is  gb- closed. Proposi tion 2.6. Every  gb- closed set is  gsp - closed. Proof. Let A be  gb- cosed and U be  - op en such that A U, then bcl(A) U. Since ev ery b-closed set is  gsp -closed. Therefore spcl(A) bcl(A). Hence, A is  gsp - closed. The followin g diagram summar izes the imp lications amon g the introduced concept and other related concep ts.  g- closed  gp - closed  b- closed   gb- closed   gsp - closed Di agram (1) The following three e xamples show t hat t he converses of Remarks 2.4 and Prop osition 2.5 are not true in general. Exam ple 2.7. Let X= {a, b, c},  = {X, , {a}}and A = {a, b}. Then X is the only regular open ( - op en) set containin g A. Hence A is  gb- closed, but A is not b- closed, since bcl (A) = X. Exam ple 2.8. Let X= {a, b, c},  = {X, , {a}, {b}, {a, b}}. Let A= {a}. Then A is b- closed. Hence A is  gb- closed, but A is not  gp - closed, since A is re gu lar op en ( - op en) and pcl(A)= {a,c} A. 3. S ome Properties of  gb- Cl ose d S ets. Proposi tion 3.1. If A is  - op en and  gb- closed, then A is b- closed and hence gb- closed. Proof. Since A is  - op en and  gb- closed. So b cl(A) A. But A bcl(A). So A= bcl(A). Hence A is b- closed. Hence gb- closed. Proposi tion 3.2. Let A be a  gb- closed in X. Then bcl(A)\ A does not contain any nonempty  - closed set. . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Proof. Let F be a  - closed set such that F bcl(A)\ A, so F  X\A. Hence A X\F. Sin ce A is  gb- closed and X\ F is  - op en. So bcl(A) X\ F. That is FX\ bcl(A). Therefore Fbcl(A)  (X\ bcl(A)) =  . Thus F = . Corollary 3.3. Let A be  gb- closed set in X. T hen A is b- closed if and only if bcl(A)- A is  - closed. Proof. Let A be  gb- closed. By hyp othesis bcl(A)= A and so bcl(A)\A= , which is  - closed. Conversely , supp ose that bcl(A)\A is  - closed. Then by Theorem 3.2, bcl(A)\A=  , that is bcl(A)= A. Hence A is b- closed. Proposi tion 3.4. If A is  gb- closed and A B bcl(A). T hen B is  gb- closed. Proof. Let B U, where U is  - op en. Then A B imp lies A U. Sin ce A is  gb- closed, so bcl(A) U and since Bbcl(A), then bcl(B) bcl(bcl(A))= bcl(A). Therefore bcl(B) U. Hence B is  gb- closed. De finition 3.5.[13] Let (X, ) be a top ological sp ace, A X and xX. T hen x is said to be a b- limit p oint of A and only if every b- op en set containing x contains a point of A differ ent from x, and the set of all b- limit p oints of A is said to be the b- derived set of A and is denoted by bD (A). Usual derived set of A is denoted by D (A). The proof of the following result is analogous t o the well known ones. Lemma 3.6. Let (X, ) be a top ological sp ace and A X. T hen bcl(A) = A  bD (A). Remark 3.7. The union of two  gb- closed sets is not necessarily a  gb- closed set as the followin g example shows. Exam ple 3.8. Consider the sp ace (X, ) in Example 2.8, the sets A= {a} and B= {b} are  gb- closed. But A  B= {a, b} is not  gb- closed. Proposi tion 3.9. Let A and B be  gb- closed sets in (X, ) such that cl(A)= bcl(A) and cl(B)= bcl(B). Then A  B is  gb-closed. Proof. Let (A  B) U and U is  - op en in (X, ). Then bcl (A) U and bcl(B) U. Now, cl (A  B) = cl (A)  cl (B) = bcl (A)  bcl(B) U. But bcl (A  B)  cl (A  B). So, b cl (A  B) U and hence A  B is  gb- closed. From the fact t hat bD (A)  D (A) and Lemma 3.6 we have the followin g, . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Remark 3.10. For any subset A of X such that D (A)  bD (A). T hen cl(A)= bcl(A). We get t he following, Corollary 3.11. Let A and B be  gb- closed sets in (X,  ) such that D (A)  bD (A) and D (B)  bD (B). Then A B is  gb- closed. Proposi tion 3.12. For every xX its complement X\{x} is  gb- closed or  -op en in (X, ). Proof. Sup p ose X\{x} is not  - op en. Then X is t he only  - op en set containing X\{ x}. T his implies bcl (X\{x}) X. Hence X\{x} is  gb- closed. 4.  gb- Open S ets. The following r esult is analo gous t o well known corresp onding ones. Lemma 4.1. bcl(X\ A)= X\ bint(A). By Lemma 4.1 and definition 2.1 we get t he following which is similar to Corollary 4.1 of [2]. Corollary 4.2. A subset A of X is  gb- op en if and only if Fbint(A) whenever F is  -closed in X and F A. Proposi tion 4.3. If bint(A) B A and A is  gb- op en, then B is  gb- op en. Proof. Since bint(A) B A. Hence X\ A X\ B bcl(X\ A), by Lemma 4.1. Since X\ A is  gb- closed, so by Theorem 3.4, X\ B is  gb- closed. Thus B is  gb- op en. Proposi tion 4.4. Let A be  gb- op en in X and let B be  - op en. Then A B is  gb- op en in X. Proof. Let F be any  - closed subset of X such that F A  B. Hence F A and by Theorem 4.2, F bint(A)=  {U: U is b- op en and UA}. Then F  (U  B), where U is a b- op en set contained in A. Since U  B is a b- op en set contained in A  B for each b- op en set U contained in A, F bint(A  B), and by Theorem 4.2, A B is  gb- op en in X. Lemma 4.5. For any AX, bint(bcl(A)\ A)=  . Proof. If bint (bcl(A)\ A)   . Then there is an element xbint (bcl(A)-A), so t here is UBO(X) such that xU bcl(A)-A. Therefore Ubcl (A) and U A. Thus U  bcl(A) and U X-A. Hence there is UBO(X), U A= , a contradiction, since xbcl(A). . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Proposi tion 4.6. Let A B X and let bcl(A)\A be  gb- closed set. Then bcl(A)\B is also  gb- op en. Proof. Sup p ose bcl(A)\A is  gb- op en and let F be a  - closed subset of X with F bcl(A)\B. Then F bcl(A)\A. By Theorem 2.4 and Lemma 4.5, F bint(bcl(A)\A)=  . So, F=  .Consequently , F bint(bcl(A)\B). Proposi tion 4.7. Let A X be a  gb-closed. Then bcl(A)\ A is  gb- op en. Proof. Let F be a  - closed such that Fbcl(A)- A. Then by Theorem 3.2, F= . So F  bint (bcl(A)\A). T herefore bcl(A)-A is  gb- op en, by Theorem 4.2. 5.  B- 2 1T S paces In this section we define a new class of sp aces, named  b- 2 1T sp ace which is a generalization of 2 1T [14]. De finition 5.1. A space (X, ) is called a  b- 2 1T sp ace if every  gb- closed set is b- closed. Exam ple 5.2. If X   be any set. Then (X, .in d ) is  b- 2 1T sp ace. Recall that X is 2 1T sp ace if every g- closed set is closed or equivalently if every singleton is op en or closed. The notions of  b- 2 1T and 2 1T are indep endent as it can be seen through the followin g examples. Exam ple 5.3. Let X= {a, b, c},  ={X, , {c}, {a, b}}. Then RO(X) = , BO(X) = P(X) = BC(X) =  GBC(X). Then X is  b- 2 1T but not 2 1T . Exam ple 5.4. Consider (N, ) where N is the set of natural numbers and  = {U N: 1 U}  { }, t hen  is a top ology on N, and (N, ) is 2 1T but not  b- 2 1T . Next, we recall the following, De finition 5.5. A space X is  gsp - 2 1T (or  gsp in [2]) if every  gsp - closed subset of X is semi- p reclosed . . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 Remark 5.6. It seems t hat t he notions of  gsp - 2 1T and  b- 2 1T are ind ependent of each other, but we could not disp rove it. The following r esult is analo gous t o Prop osition 3.7 in [2]. Proposi tion 5.7. A space X is  b- 2 1T if and only if every singleton of X is either  - closed or b- op en. Proof. Necessity: Let x X and assume that {x} is not  - closed, then X\ { x} is not  - op en, so the only  - op en set containing X\ { x} is X, hence X\ { x} is  gb- closed. By assump tion X\ {x} is b- closed. Thus {x} is b- op en. Sufficiency: Let A be a  gb- closed subset of X and xbcl(A). By assumption, we have the followin g two cases: (i) {x} is b- op en. Since xbcl (A), So {x}  A   . Thus xA. (ii) {x} is  - closed. Then by Theorem 3.2, x  (bcl(A)- A). But xbcl(A), so xA. T herefore in both cases xA. T his shows that bcl(A) A or equivalently A is b-closed. Proposi tion 5.8. (i) BO(X)  GBO(X). (ii) A space X is  b- 2 1T if and only if BO(X) =  GBO(X). Proof. (i) Let A be a b- op en. Then X- A is b- closed and so  gb- closed. Thus A is  gb- op en. Therefore BO(X)   GBO(X). (ii) Necessity: Let X be  b- 2 1T . Let AGBO(X). Then X-A is  gb- closed. By hyp othesis, X-A is b-closed. Thus ABO(X). Hence  GBO(X) = BO(X). Suficiency: Let BO(X) =  GBO(X) and A be  gb- closed. Then X-A is  gb- op en. Hence X-A BO(X). Thus A is b- closed. Therefore X is  b- 2 1T . Acknowle dgment. The author is grateful to t he referees for their h elp in improving the quality of this p aper. References 1- Park, J. H., (2006) “On  gp - closed sets in top ological sp aces” Indian J. Pure App l. M ath., Acta M athematica Hungarica 112,(4), 257- 283. 2 -Sarsak, M . S. (2010) “  - Generalied semi- p reclosed sets” Int. M ath. Foram, 5, no. 12, 573- 578. 3- Levine N., (1963)" Some- op en sets and semi continuity in top ological sp aces", M ath. Mont hly 70, 36-41. . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (3) 2011 4- Njasted, O.,(1965), “On some classes of nearly op en sets” Pacific J. M ath., 15, 961- 970. 5- M ashhour, A. S. Abd El- M onsef M . E. and El. Deep, S. N., (1982), (1983), "On p recontinuous and weak p recontinuous mapp ings, Proc, Phis. Soc. Egy pt No. 52, 47- 53 6- Andrijevic D, (1986), " Semip reopen sets" M ath. Vesnik 38 no.1, 24- 32. 7- Stone, M . (1937), "App lication of theory of Boolean rin gs t o general top ology ", Trans. Amer. M ath. Soc. 41, 374- 481. 8- Andrijevic, D., ( 1996), "On b- op en sets, M ath.Vesnik 48no. 1-2, 59- 64. 9- Zaitsav V., (1968), “On certain classes of top ological sp aces and their bicompactifications” Dokl Akad SSSR 178, 778- 779. 10-Levine, N., (1970), " Gener alized closed sets in top ology , Rend. Gen. M ath. Palermo (2) 19, 89- 96. 11- Dont chev, Z. and Noiri, T., (2000), “ Quasi- normal sp aces and  g- closed sets", Acta M ath.Hungar, 89, (3), 211- 219. 12- Adea, K.,2009 “On b- compactness and b * - comp actness in top ological sp aces”.accepted in Journal of Basic Edu cation, 12(2007). 13- Al- Omeri, A., and Noorani, M D. M . S.,(2009)," On generalized b0 closed sets" Bull. M ath. Sci. (2), 19- 30. 14- Levine, N., (1970), " Gener alized closed sets in top ology " Rend. Gen. M ath. Palermo (2) 19 89- 96. . 2011) 3( 24المجلد مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة gbمن النمط المجموعات المغلقة العبیديعذیة خلیفة الجامعة المستنصریة ، كلیة التربیة االساسیة ، قسم الریاضیات 2011 یارآ 10: استلم البحث في 2011 حزیران 16 : قبل البحث في المقدمة gb(المجموعات المغلقة من النمط في هذا البحث قدمنا صنفا جدیدا من المجموعات اسمیناها  ودرسنا بعض ) gp (هما المجموعات المغلقة من النمط ان هذا النوع یقع بین صنفین من المجموعات إذ.الخواص االساسیة لها  ( gsp( والمجموعات المغلقة من النمط فضاء . ) كما عرفنا ودرسنا نوعا من الفضاءات اسمیناه ال b- 2 1T. gb المجموعة المغلقة من النمط ، المجموعة المفتوحة المنتظمة، bلمجموعة المفتوحة من النمط ا :الكلمات المفتاحیة .