Microsoft Word - 204-213 204 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 On generalized b*-Closed Sets In Topological Spaces Zinah T. ALhawez Dept. of Mathematics/College of Education for Woman / University of Tikrit Received in:22/December/2014 , Accepted in:20/September/2015 Abstract In this paper, we introduce and study the concept of a new class of generalized closed set which is called generalized b*-closed set in topological spaces ( briefly .g b*-closed) we study also. some of its basic properties and investigate the relations between the associated topology. Keywords: gb* -closed set, gb -closed set,g-closed set. 205 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Introduction 1. Levine[9 ] introduced the concept of generalized closed sets (briefly ,g-closed) and studied their most fundamental properties in topological spaces . Arya and Nour[6], Bhattacharya and Lahiri[7], Levine[10], Mashhour[11], Njastad[13]and Andrijevic[3,4] introduced and investigated generalized semi-open sets,semi generalized open sets, generalized open sets, semi-open sets, pre-open sets and α - open sets, semi pre-open sets and b-open sets which are some of the weak forms of open sets and the complements of these sets are called the same types of closed sets. A.A.Omari and M.S.M.Noorani[14] introduced and studied the concept of generalized b-closed sets(briey gb-closed) in topological spaces. Recently Sundaram and Sheik John [15] introduced and studied w-closed sets. S.Muthuvel and R.Parimelazhagan [12] introduced and studied b*closed sets , A.Poongothai and R.Parimelazhagan [5] introduced and studied strongly b*-closed set in topological spaces. In this paper, we introduce a new class of sets, namely gb*- closed sets for topological spaces. this class lies between the class b*-closed set and strongly b*-closed set. 2.Preliminaries Let ( X,T) be topological spaces and A be a subset of X .The closure of A and interior of A are denoted by cl(A) and int(A) respectively, union of all b-open (semi-open , pre-open , α – open ) sets X contained in A is called b- interior (semi- interior, pre-interior, α – interior ,respectively) of A, it is denoted by b-int (A)(s-int(A),p-int(A), α-int(A), respectively),The intersection of all b-closed (semi- closed , pre- closed , α – closed) sets X containing A is called b- closure (semi- closure, pre- closure , α – closure ,respectively) of A and it is denoted by bcl(A) ( scl(A) ,pcl(A) , αcl(A) ,respectively).In this section , we recall some definitions of open sets in topological spaces. Definition 2-1[15]: A subset A of a topological space (X,T) is called a pre –open set if A ⊆ int cl A and pre-closed set if cl int A ⊆ A. Definition 2-2[10]:A subset A of a topological space (X,T) is called a semi –open set if A ⊆ cl int A and semi-closed set if int cl A ⊆ A. Definition 2-3[3]: A subset A of a topological space (X,T) is called a α –open set if A ⊆ int cl int A and α -closed set if cl int cl A ⊆ A. Definition 2-4[8]:A subset A of a topological space (X,T) is called a β–open set if A ⊆ cl int cl A and β-closed set if int cl int A ⊆ A. Definition 2-5[1]:A subset A of a topological space (X,T) is called a b –open set if A ⊆ cl int A ∪ int cl A and b-closed set if int cl A ∩ cl int A ⊆ A. Definition 2-6[9]:A subset A of a topological space (X,T) is called a generalized –closed set (briefly, g-closed) if cl A ⊆ U ,whenever A ⊆ U and U is open set . Definition 2-7[7]:A subset A of a topological space (X,T) is called a semi generalized closed set (briefly, sg-closed) if scl A ⊆ U ,whenever A ⊆ U and U is semi- open set . 206 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Definition 2-8[8]:A subset A of a topological space (X,T) is called a generalized α- closed set (briefly gα-closed) if αcl A ⊆ U ,whenever A ⊆ U and U is α- open set . Definition 2-9[2]:A subset A of a topological space (X,T) is called a generalized b- closed set (briefly gb -closed) if bcl A ⊆ U ,whenever A ⊆ U and U is open set . Definition 2-10[8]:A subset A of a topological space (X,T) is called a generalized β- closed set (briefly gβ-closed) if βcl A ⊆ U ,whenever A ⊆ U and U is open set . Definition 2-11[5]:A subset A of a topological space (X,T) is called weakly generalized closed set (briefly wg-closed) if cl int A ⊆ U ,whenever A ⊆ U and U is open set . Definition 2-12[15]:A subset A of a topological space (X,T) is called wekly-closed set (briefly w-closed) if cl A ⊆ U ,whenever A ⊆ U and U is semi- open set . Definition 2-13[12]:A subset A of a topological space (X,T) is called b*-closed set if int cl A ⊆ U ,whenever A ⊆ U and U is b- open set . Definition 2-14[5]:A subset A of a topological space (X,T) is called g* -closed set if cl A ⊆ U ,whenever A ⊆ U and U is g- open set . Definition 2-15[5]:A subset A of a topological space (X,T) is called a g*b -closed set if bcl A ⊆ U ,whenever A ⊆ U and U is g- open set. Definition 2-16[5] :A subset A of a topological space (X,T) is called strongly b*-closed set (briefly, sb*-closed) if cl int A ⊆ U ,whenever A ⊆ U and U is b- open set . Definition 2-17[5] :A subset A of a topological space (X,T) is called b**-open set if A⊆int(cl(int(A)))∪cl(int(cl(A))) and b**-closed set if cl(int(cl(A))∩int(cl(int(A)) ⊆ ). 3. Generalized b*-closed sets. In this section , we introduce and study the concept of generalized b*-closed set in topological spaces . Also we study the relationship between this set and the other types of sets. Definition 3-1: A subset A of a topological space (X,T) is called generalized b*- closed set (briefly, gb*-closed) if int cl A ⊆ U ,whenever A ⊆ U and U is gb- open set . Theorem 3-2:Every closed set is gb* -closed set. Proof : Assume that A is a closed set in X then cl (A)=A ,and U be any gb-open set where A ⊆ U . Since int A ⊆ A . implies that int cl A ⊆ U .Hence A is gb* -closed set in X. 207 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Remark 3-3: The converse of the Theorem [3- 2] need not be true as seen by the following example. Example3-4: let X={ a,b,c } with T={X ,  , {a} }.In this topological space , the sub set A={b} is gb*- closed set but not closed set . Theorem 3-5: A set A is gb*-closed set iff int cl(A)-A contains no non-empty gb-closed set. Proof : Necessity: Suppose that F is a non-empty gb-closed subset of int(cl(A)) such that F ⊆ int cl A A . then F ⊆ int cl A ∩ A .Therefore F ⊆ int cl A and F ⊆ A . Since F is gb-closed set and A is gb*-closed set , int cl A ⊆ F . thus F ⊆ int cl A . Therefore F ⊆ int cl A ∩ int cl A  .Therefore F=  and this implies that int(cl(A))-A contains no non-empty gb-closed set . Sufficiency : Assume that int(cl(A))-A contains no non-empty gb-closed. Let A ⊆ U ,U is gb-open set .Suppose that int(cl(A)) is not contained in U , then int cl A ∩ U is a non- empty gb-closed set of int(cl(A))-A which is a contradiction. Therefore int cl A ⊆ and hence A is gb*-closed set . Theorem 3-6:Let B ⊆ Y ⊆ X ,if B is gb*-closed set relative to Y and that Y is both gb-open and gb*‐closed set in X,T then B is gb*‐closed set in X,T . Proof: Let U ⊆ B and U be a gb-open set in (X,T).But Given that B ⊆ Y ⊆ X . Therefore B ⊆ Y and U ⊆ B . This implies that Y ∩ U ⊆ B. Since B is gb*-closed set relative to Y, Then Y ∩ U ⊆ int cl Y .(i.e) Y ∩ U ⊆ Y ∩ int cl Y .implies that U ⊆ Y ∩ int cl Y . thus U ∪ int cl B ⊆ Y ∩ int cl B ∪ int cl B . This implies that U ∪ int cl B ⊆ int cl Y ⊆ int cl B . Therefore U ⊆ int cl B . Since int(cl(B)) is not contained in int cl B . Thus B is gb*-closed set relative to X. Theorem 3-7:Let A ⊆ Y ⊆ X and suppose that A is gb* -closed set in X then A is gb* - closed set relative to Y. Proof : Assume that A ⊆ Y ⊆ X and A is gb* -closed set in X . To show that A is gb* - closed set relative to Y , let A ⊆ Y ∩ U where U is gb-open in X . Since A is gb* -closed set in X , A ⊆ U implies that int cl A ⊆ U , i.e) Y ∩ int cl A ⊆ Y ∩ U. where Y ∩ int cl A is interior of closure of A in Y . Thus A is gb* -closed set relative to Y. Theorem 3-8:If A is a gb* -closed set and A ⊆ B ⊆ int cl A then B is a gb* -closed set. 208 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Proof : Let U be a gb -open set of X , such that B ⊆ U . Then A ⊆ U . Since A is gb* - closed, Then int cl A ⊆ U .Now int cl B ⊆ int cl A ⊆ U .Therefore B is gb*- closed set in X . Theorem 3-9 : The intersection of a gb* -closed set and a closed set is a gb* -closed set. Proof: Let A be a gb* -closed set and F be a closed set . Since A is gb* -closed set , int cl A ⊆ U whenever A ⊆ U ,where U is agb-open set . To show that A ∩ Fis gb*-closed set ,it is enough to show that int cl A ∩ F ⊆ U whenever A ∩ F ⊆ U , where U is gb-open set . Let G =X – F then A ⊆ U ∪ G .Since G is open set , U ∪ G is gb-open set and A is gb* - closed set , int cl A ⊆ U ∪ G . Now int cl A ∩ F ⊆ int cl A ∩ int cl F ⊆ int cl A ∩ F ⊆ U ∪ G ∩ F ⊆ U ∩ F ∪ G ∩ F ⊆ U ∩ F ∪  ⊆ U. This implies that A ∩ F is gb* -closed set. Theorem 3-10: If A and B are two gb* -closed sets defined for a non –empty set X, then their intersection A ∩ B is gb* -closed set in X. Proof: Let A and B are two gb* -closed sets in X. Let A ∩ B ⊆ U , U is gp-open set in X. Since A is gb* -closed , int cl A ⊆ U ,whenever A⊆ U , U is g-open set in X .Since B is gb* -closed , int cl B ⊆ U ,whenever B⊆ U , U is g-open set in X . hence A ∩ B is gb* - closed set . Remark 3-11: The Union of two gb* -closed sets need not to be gb* -closed set. Example3-12: Let X ={a,b,c} with T={X ,  , {a},{c},{a,c} } .If A= {a}, B= {c} are gb* -closed set in X . then A ∪ B is not a gb* -closed set . Theorem 3-13:Every gb - closed set is gb* -closed set. Proof : Assume that A be a g b - closed set in X. and let U be an open set such that A ⊆ U. Since every open set is gb-open set . Then int cl A ⊆ bcl A ⊆ U. hence A is gb*-closed set . Remark 3-14: The converse of the Theorem [3-13] need not be true as seen by the following example. Example3-15: let X={ a,b,c } with T={X ,  , {a} }.In this topological space , the subset A={ a,b} is gb* -closed set, but not gb- closed set. Theorem 3-16::Every gb*-closed set is b-closed set . Proof: Assume that A is a gb* -closed set in X , and let U be an open set such that A ⊆ U .since every open set is b-open set and A is gb*-closed set, then int cl A ⊆ intcl A ⋃cl int A ⊆ U. Therefore A is b-closed set in X . 209 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Remark 3-17: The converse of the Theorem [3-16] need not be true as the following example shows. Example3-18: let X={ a,b,c } with T={X ,  , {a} ,{b} ,{a,b}}. In this topological space , the subset A={ a,c} is b -closed set but not gb*- closed set. Theorem 3- 19: Every w-closed set is gb* -closed set . proof: Assume that A is w-closed set in X , and U is semi-open set such that A ⊆ U, every semi- open set is gb-open set then cl A ⊆int(cl(A)) therefore A is gb* -closed set. Remark 3- 20: The converse of the Theorem [3-19] need not be true as seen by the following example Example 3-21: let X={ a,b,c } with T={X ,  , {a} ,{c} ,{a,c}}. In this topological spaces , the subset A={ a } is gb*- closed set but not w-closed set . Theorem 3- 22: Every b* -closed set is g b* -closed set . Proof: Assume that A is a b*-closed set in X , and U is b- open set such that A ⊆ U. every b-open set is g b-open set . Then int cl A ⊆ , Therefore A is g b* -closed set. Remark 3- 23: The converse of the Theorem 3‐22 need not be true as seen by the following example. Example3-24: let X={ a,b,c,d } with T={X , ,{b},{c,d},{ b,c,d} }. In this topological spaces the subset A={ c} is gb*-closed set ,but not b*-closed set . Theorem 3-25:Every gb*-closed set is g*b-closed set. Proof: Assume that A is a g b*-closed set in X .Then int(cl(A) ⊆ U ,U is gb-open set such that A ⊆ U. Then bcl A ⊆ . Since every g-open set is gb-open set .Then bcl A ⊆ , Uis g-open set .Therefore A is g*b-closed set. Remark 3-26: The converse of the Theorem [3-25] need not be true as seen by the following example. Example3-27: let X={ a,b,c } with T={X ,  }.In this topological spaces , the subset A={ a,b} is g*b -closed set but not gb*- closed set. Theorem 3- 28 : Every gb*-closed set is sg -closed set . proof: Assume that A is gb*-closed set in X , and U is open set such that A ⊆ U every open set is semi-open set, A is gb*-closed and U is gb-closed then int cl A ⊆ A ∪ scl A ⊆ U therefore A is sg -closed set. 210 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 Remark 3- 29: The converse of the Theorem [3-28] need not be true as seen by the following example. Example3-30: let X= {a ,b ,c} ,T={X,,{a,b }, {c}} In this example A={a,b} is sg- closed set but not gb* -closed set. Theorem 3- 31: Every gb* -closed set is gβ-closed set . proof: Assume that A is g*b* -closed set in X , and U is open set such that A ⊆ U, every open set is gb-open set then int cl A ⊆ A ∪ β closed ⊆ U Therefore A is g*b* -closed set. Remark 3- 32: The converse of the Theorem [3-31] need not be true as seen by the following example. Example3-33: let X= {a ,b ,c} ,T={X,,{b}, {b,c} }. In this example A={a,b} is gβ –closed set but not gb* -closed set. theorem 3- 34: Every gb* -closed set is b** -closed set . proof: Assume that A is g*b* -closed set in X , and U is open set such that A ⊆ U, every open set is gb-open set then int cl A ⊆ cl int cl A ∪ int cl int A ⊆ U.Therefore A is b**-closed set. Remark 3- 35: The converse of the Theorem [3-34] need not be true as seen by the following example. Example3-36: let X= {a ,b ,c} ,T={X,,{a , b}, { c} } . In this topological spaces the subset A={b, c} is b**-closed set but not g b* -closed set.                                                      gb‐closed closed set                                                                                                                                        w- closed   gβ-closed                                                                                                       sg-closed                                                 gb*-closed set                        b-closed                                                                                               g*b-closed                                b*-closed                      b**-closed  diagram (1) 211 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 4.gb*-closed set is independent of other closed sets In this section ,we explain independency of gb*-closed set with some other closed sets. Remark 4- 1: The following example shows that the concept of g-closed and gb*-closed sets are independent . Example4-2: let X= {a ,b ,c} ,T={X,,{ b}, { b,c} } , In this topological space ,the subset A={a, b}is g-closed set but not gb*-closed set. And , in this topological space , the subset B={c}is gb*-closed set but not g-closed set. Remark 4- 3: The following example shows that the concept of sb*-closed and gb*-closed sets are independent . Example4-4 : let X= {a ,b ,c} ,T={X,,{ a},{c} ,{ a,c} } , In this topological space ,the subset A={a,c}is sb*-closed set but not gb*-closed set. And , in this topological space ,the subset B={c}is gb*-closed set but not sb*-closed set. Remark 4- 5: The following example shows that the concept of g*-closed and gb*-closed sets are independent . Example4-6: let X= {a ,b ,c} ,T={X,,{ b}, { b,c} } , In this topological space ,the subset A={a, b}is g*-closed set but not gb*-closed set. And , in this topological space ,the subset B={c}is gb*-closed set but not g*-closed set. Remark 4- 7: The following example shows that the concept of gα-closed and gb*-closed sets are independent . Example4-8: let X= {a ,b ,c} ,T={X,,{ a},{c} ,{ a,c} } , In this topological space ,the subset A={b,c}is gα -closed set but not gb*-closed set. And , in this topological space ,the subset B={a}is gb*-closed set but not gα -closed set Remark 4- 9: The following example shows that the concept of gp-closed and gb*-closed sets are independent . Example4-10: let X= {a ,b ,c} with the topology ,T1={X,,{ a,b},{,c} } , In this topological space ,the subset A={a,c}is gp -closed set but not gb*-closed set. For the topology T2={X,,{ a },{b}, {a,b} } topological ,the subset B={b}is gb*-closed set but not gp-closed set. Remark 4- 11: The following example shows that the concept of wg-closed and gb*- closed sets are independent . Example4-12: let X= {a ,b ,c} with the topology ,T1={X,,{ a } } , In this topological space ,the subset A={a,b}is wg-closed set but not gb*-closed set. For the topology T2={X,,{ a },{c}, {a,c } topological ,the subset B={a}is gb*-closed set but not wg-closed set. 212 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015                                                                                 gp-closed                                            sb*-closed                                                                  gα-closed                                                                                gb*-closed set                                                                                                                            g-closed                             g*-closed                wg-closed  diagram (2) Reference 1. EL-Monsef, M.D.; Abd, El-Atik ,A. A. and El-Sharkasy, M. M., (2005), Some topologies induced by b-open sets, Kyungpook Math. J. 45 . 4, 539{547. 2. Ahmad Al-Omari and Mohd. Salmi, Md. Noorani, (2009), On Generalized b-closed sets, Bull. Malays. Math. Sci. Soc(2)32(1) .19-30. 3. Andrijevic.D, (1986) ,Semi-preopen sets, Mat. Vesink, 38, 24 - 32. 4. Andrijevic.D, (1996), On b-open sets, Mat. Vesink, 48 ,59 - 64. 5. 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O, (1965), On some classes of nearly open sets,Pacific J.Math., , 15 961-970. 14. Omari.A.A and Noorani .M.S.M, On Generalized b-closed sets, Bull. Malays. Math. Sci. Sco. (2) 32 (1) (2009), 19-36. 15. Sundaram,P. and Shiek John,M., (2000) On w-closed sets in Topology ,Acta ciencia Indica , 4,389-392. 213 | Mathematics 2015) عام 3العدد ( 28الھيثم للعلوم الصرفة و التطبيقية المجلد مجلة إبن Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 28 (3) 2015 *gb–حول المجموعات المغلقة بالنمط زينة طه الحويز قسم الرياضيات /كلية التربية للبنات /جامعة تكريت 2015/أيلول/20قبل البحث في:، 2014/كانون األول/22 :استلم البحث في الخالصة *gbدراسة مفھوم جديد من المجموعات المغلقة يسمى المجموعات المعممة المغلقة يعرض ھذا البحث كما نقوم بدراسة بعض الخصائص األساسية، ودراسة العالقات بينھا وبين المجموعات المغلقة ةفي الفضاءات التبولوجي في الفضاء التبولوجي. : والمجموعات المعممة المغلقة. -bوالمجموعات المعممة المغلقه -*bالمجموعات المعممة المغلقة :الكلمات ألمفتاحيه