. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011  -Generalized b- Closed Sets in Topological Spaces A. K. Al-Obiadi Department of Mathematics, College of Basic Education University of Al- Mustansiryah Received in : 10 May 2011 Accepted in : 16 June 2011 Abstract In this paper we introduce a new class of sets called -  generalized b- closed (briefly  gb closed) sets. We study some of its basic properties. This class of sets is strictly placed between the class of  gp- closed sets and the class of  gsp- closed sets. Further the notion of  b- 2 1T space is introduced and studied. 2000 Mathematics Subject Classification: 54A05 Keywords: b- open set, regular open set,  -generalized b- closed set. 1. Introduction and Prelimimnaries. Park[1] introduced the class of  -generalized pre-closed(briefly  gp closed) sets and the class of  -generalized semipreopen closed (briefly  gsp closed) sets was introduced by Sarsak [2] as a generalization of closed sets. In this paper we define and study a new class of  - generalized closed sets, we denote by  -generalized b- closed (briefly  gb- closed) sets, which is strictly placed between the class of  gp- closed set and  gsp- closed sets. Moreover, we define  b- 2 1T space as the space in which every  gb- closed set is b- closed. Throughout this paper ),( X and ),( Y represent nonempty topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space ),( X , cl(A), int(A) and P(X) denote the closure , the interior and power set of A respectively. ),( X will be replaced by X if there is no confusion. Let us recall the following definitions which are useful in the sequel. Definition 1.1. A subset A of a space 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. SCI. 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 regular open sets, and  -closed if A is the intersection of regular closed sets. [9] The b- interior (briefly bint) of a subset A of X is the union of all b- open sets contained in A. The b- closure (resp. pre-closure, semipre- closure) of A is the intersection of all b-closed (resp. preclosed, semipre- closed) sets containing A, and is denoted by bcl(A) ( resp.pcl(A), spcl(A)). The collection of all b- open (resp. b- closed) sets is denoted by BO(X) (resp. BC(X)).[8] It is well known that: (1)  - open set preopen set b- open set  semi- preopen.[8] (2) The intersection of a b- open set with  - open set is b- open.[8] Definition 1.2. A subset A of a space X is called: (1) generalized closed ( briefly g- closed) if UAcl )( whenever UA  and U is open in X.[10] (2)  - generalized closed (briefly  g- closed) if UAcl )( whenever UA  and U is  -open.[11] (3)  - generalized pre cosed ( briefly  gp- closed) if UApcl )( whenever UA  and U is  - open.[1] (4)  -generalized semipre- closed (briefly  gsp- closed) if UAspcl )( whenever UA  and U is  - open.[2] Lemma 1.3. [12] Let AX then, (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- Closed Sets. Definition 2.1. A subset A of a space X is called  - generalized b- closed (breifly  gb closed) if bcl(A) U whenever A U and U is  - open. The complement of  gb- closed set is called  gb- open. The family of all  gb- closed (resp.  gb- open) subsets of the space X is denoted by  GBC(X) (resp.  GBO(X)). Definition 2.2. The  - kernel ( - ker (A)) of A is the intersection of all  - open sets containing A. . IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Remark 2.3. A subset A of a space X is  gb- closed if and only if bcl(A)   - ker(A). Remark 2.4. Every b- closed set is  gb- closed. Proposition 2.5. Every  gp- closed set is  gb- closed. Proof. Let A be  gp- closed subset of X and U be  - open such that AU. Then pcl (A)U. Since every preclosed set is b-closed. Therefore bcl(A)pcl (A). Hence A is  gb- closed. Proposition 2.6. Every  gb- closed set is  gsp- closed. Proof. Let A be  gb- cosed and U be  - open such that AU, then bcl(A) U. Since every b-closed set is  gsp-closed. Therefore spcl(A) bcl(A). Hence, A is  gsp- closed. The following diagram summarizes the implications among the introduced concept and other related concepts.  g- closed  gp- closed  b- closed  gb- closed   gsp- closed Diagram (1) The following three examples show that the converses of Remarks 2.4 and Proposition 2.5 are not true in general. Example 2.7. Let X= {a, b, c},  = {X, , {a}}and A = {a, b}. Then X is the only regular open ( - open) set containing A. Hence A is  gb- closed, but A is not b- closed, since bcl (A) = X. Example 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 regular open ( - open) and pcl(A)= {a,c}A. 3. Some Properties of  gb- Closed Sets. Proposition 3.1. If A is  - open and  gb- closed, then A is b- closed and hence gb- closed. Proof. Since A is  - open and  gb- closed. So bcl(A) A. But Abcl(A). So A= bcl(A). Hence A is b- closed. Hence gb- closed. Proposition 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. SCI. 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. Since A is  gb- closed and X\ F is  - open. 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. Then A is b- closed if and only if bcl(A)- A is  - closed. Proof. Let A be  gb- closed. By hypothesis bcl(A)= A and so bcl(A)\A= , which is  - closed. Conversely, suppose that bcl(A)\A is  - closed. Then by Theorem 3.2, bcl(A)\A=  , that is bcl(A)= A. Hence A is b- closed. Proposition 3.4. If A is  gb- closed and ABbcl(A). Then B is  gb- closed. Proof. Let BU, where U is  - open. Then AB implies A U. Since 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. Definition 3.5.[13] Let (X, ) be a topological space, A X and xX. Then x is said to be a b- limit point of A and only if every b- open set containing x contains a point of A different from x, and the set of all b- limit points 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 to the well known ones. Lemma 3.6. Let (X, ) be a topological space and A X. Then bcl(A) = A bD (A). Remark 3.7. The union of two  gb- closed sets is not necessarily a  gb- closed set as the following example shows. Example 3.8. Consider the space (X, ) in Example 2.8, the sets A= {a} and B= {b} are  gb- closed. But A B= {a, b} is not  gb- closed. Proposition 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  - open 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, bcl (A B) U and hence A B is  gb- closed. From the fact that bD (A)  D (A) and Lemma 3.6 we have the following, . IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Remark 3.10. For any subset A of X such that D (A)  bD (A). Then cl(A)= bcl(A). We get the 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. Proposition 3.12. For every xX its complement X\{x} is  gb- closed or  -open in (X, ). Proof. Suppose X\{x} is not  - open. Then X is the only  - open set containing X\{x}. This implies bcl (X\{x}) X. Hence X\{x} is  gb- closed. 4.  gb- Open Sets. The following result is analogous to well known corresponding ones. Lemma 4.1. bcl(X\ A)= X\ bint(A). By Lemma 4.1 and definition 2.1 we get the following which is similar to Corollary 4.1 of [2]. Corollary 4.2. A subset A of X is  gb- open if and only if Fbint(A) whenever F is  -closed in X and FA. Proposition 4.3. If bint(A) BA and A is  gb- open, then B is  gb- open. 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- open. Proposition 4.4. Let A be  gb- open in X and let B be  - open. Then AB is  gb- open 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- open and UA}. Then F  (U B), where U is a b- open set contained in A. Since U B is a b- open set contained in A B for each b- open set U contained in A, Fbint(AB), and by Theorem 4.2, AB is  gb- open 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 there 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. SCI. VOL.24 (3) 2011 Proposition 4.6. Let ABX and let bcl(A)\A be  gb- closed set. Then bcl(A)\B is also  gb- open. Proof. Suppose bcl(A)\A is  gb- open 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). Proposition 4.7. Let AX be a  gb-closed. Then bcl(A)\ A is  gb- open. Proof. Let F be a  - closed such that Fbcl(A)- A. Then by Theorem 3.2, F= . So F bint (bcl(A)\A). Therefore bcl(A)-A is  gb- open, by Theorem 4.2. 5.  B- 2 1T Spaces In this section we define a new class of spaces, named  b- 2 1T space which is a generalization of 2 1T [14]. Definition 5.1. A space (X, ) is called a  b- 2 1T space if every  gb- closed set is b- closed. Example 5.2. If X   be any set. Then (X, .ind ) is  b- 2 1T space. Recall that X is 2 1T space if every g- closed set is closed or equivalently if every singleton is open or closed. The notions of  b- 2 1T and 2 1T are independent as it can be seen through the following examples. Example 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 . Example 5.4. Consider (N, ) where N is the set of natural numbers and  = {UN: 1U} { }, then  is a topology on N, and (N, ) is 2 1T but not  b- 2 1T . Next, we recall the following, Definition 5.5. A space X is  gsp - 2 1T (or  gsp in [2]) if every  gsp- closed subset of X is semi- preclosed . . IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Remark 5.6. It seems that the notions of  gsp- 2 1T and  b- 2 1T are independent of each other, but we could not disprove it. The following result is analogous to Proposition 3.7 in [2]. Proposition 5.7. A space X is  b- 2 1T if and only if every singleton of X is either  - closed or b- open. Proof. Necessity: Let x X and assume that {x} is not  - closed, then X\ {x} is not  - open, so the only  - open set containing X\ {x} is X, hence X\ {x} is  gb- closed. By assumption X\ {x} is b- closed. Thus {x} is b- open. Sufficiency: Let A be a  gb- closed subset of X and xbcl(A). By assumption, we have the following two cases: (i) {x} is b- open. 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. Therefore in both cases xA. This shows that bcl(A) A or equivalently A is b-closed. Proposition 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- open. Then X- A is b- closed and so  gb- closed. Thus A is  gb- open. Therefore BO(X)  GBO(X). (ii) Necessity: Let X be  b- 2 1T . Let AGBO(X). Then X-A is  gb- closed. By hypothesis, 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- open. Hence X-A BO(X). Thus A is b- closed. Therefore X is  b- 2 1T . Acknowledgment. The author is grateful to the referees for their help in improving the quality of this paper. References 1- Park, J. H., (2006) “On  gp- closed sets in topological spaces” Indian J. Pure Appl. Math., Acta Mathematica Hungarica 112,(4), 257- 283. 2 -Sarsak, M. S. (2010) “ - Generalied semi- preclosed sets” Int. Math. Foram, 5, no. 12, 573- 578. 3- Levine N., (1963)" Some- open sets and semi continuity in topological spaces", Math. Monthly 70, 36-41. . IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 4- Njasted, O.,(1965), “On some classes of nearly open sets” Pacific J. Math., 15, 961- 970. 5- Mashhour, A. S. Abd El- Monsef M. E. and El. Deep, S. N., (1982), (1983), "On p recontinuous and weak precontinuous mappings, Proc, Phis. Soc. Egypt No. 52, 47- 53 6- Andrijevic D, (1986), "Semipreopen sets" Math. Vesnik 38 no.1, 24- 32. 7- Stone, M. (1937), "Application of theory of Boolean rings to general topology", Trans. Amer. Math. Soc. 41, 374- 481. 8- Andrijevic, D., ( 1996), "On b- open sets, Math.Vesnik 48no. 1-2, 59- 64. 9- Zaitsav V., (1968), “On certain classes of topological spaces and their bicompactifications” Dokl Akad SSSR 178, 778- 779. 10-Levine, N., (1970), " Generalized closed sets in topology, Rend. Gen. Math. Palermo (2) 19, 89- 96. 11- Dontchev, Z. and Noiri, T., (2000), “ Quasi- normal spaces and  g- closed sets", Acta Math.Hungar, 89, (3), 211- 219. 12- Adea, K.,2009 “On b- compactness and b*- compactness in topological spaces”.accepted in Journal of Basic Education, 12(2007). 13- Al- Omeri, A., and Noorani, MD. M. S.,(2009)," On generalized b0 closed sets" Bull. Math. Sci. (2), 19- 30. 14- Levine, N., (1970), " Generalized closed sets in topology" Rend. Gen. Math. 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 .