Microsoft Word - 33 523 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 On bg**- Connected Spaces AfrahM. Ibraheem Department of Mathematics/ College of Education/ University of Al-Mustansiriyah Receved in : 29 January 2014 , Acceptde in : 8 July 2014 Abstract In this paper, we define the bg**-connected space and study the relation between this space and other kinds of connected spaces .Also we study some types of continuous functions and study the relation among (connected space, b-connected space, bg-connected space and bg**-connected space) under these types of continuous functions. Key words: bg**-closed set, bg**-connected space. 524 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 1.Introduction The notion of b-open set was introduced in 1996 [1], since then it has been widely investigated in the literature (see [1], [8]) .A.M. [7] introduce the concepts (bg**-open set, bg**-continuous function and bg**-irresolute function). The concepts (b-connected space and bg-connected space) were introduced in [9] and [6] respectively. In this work,we introduce the concept of bg**-connected space and study its relations with (b-connected and bg-connected space) .Also we study some types of continuous function which are: (b-continuous function, bg-continuous function and bg**-continuous function), and study the image of (connected space, b-connected space, bg-connected space and bg**-connected space) under these types of functions. 2. Preliminaries Throughout the paper X and Y represent topological spaces on which no separation axioms are assumed unless otherwise mentioned. We recall the following definitions, which are useful in the sequel. Definition 2.1: [1] A subset A of a topological space X is said to be b-open if A⊆cl(int (A)) U int(cl(A)). And A is said to be b-closed set if int(cl(A))∩cl(int(A)) ⊆A . Definition 2.2 : [5] A subset A of a topological space X is said to be bg-closed if bcl(A) ⊆ U whenever A ⊆U and U is open. A will be called bg-open if its complement is bg-closed. Definition 2.3 : [2] A subset A of a topological space X is said to be g**-open if and only if there exists an open set U of X such that U  A  cl**(U), and A is said to be g**- closed if its complement is g**-open set, where cl**(U) = { F: F is g-closed and U  F}. Definition 2.4 : [7] A subset A of a topological space X is said to be bg**-closed if bcl(A) ⊆ U whenever A ⊆ U and U is g**-open. The set of all bg**-closed sets of X denoted by bG**C(X). A subset A of X is called bg**-open if X – A is bg**-closed in X. Example : If X= {a, b, c} , ={, X, {a}} , then the set of all bg**-closed sets of X bG**C(X) are {, X,{b},{c},{b, c}}. Definition 2.5: A map f : X  Y from a topological space X into a topological space Y is called: 1) a b-continuous if f −1(V) is b-closed set in X for every closed set V of Y. [3] 2) a bg-continuous if f−1(V) is bg-closed in X for every closed set V of Y . [6] [7]for every closed set V in Y. closed set in X-(V) is a bg**1-if f continuous-bg**3) a [7].closed set V in Y-for every bg** closed set in X-(V) is a bg**1-if f irresolute-bg**4) a : Remark 2.6 1- Every open set is b-open (bg-open) [5]. 2- Every bg-open set is b-open [6]. 3- Every bg-open set is bg**-open [7]. 4- Every bg**-open set is b-open [7]. 525 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014  3. On bg**- Connected Space In this section we introduce the concept of bg**-connected space, and study some of their properties .Also we study the relation between it and (b-connected and bg-connected space). if X can not be connected -b[1] A topological space X is said to be Definition 3.1: expressed as a disjoint union of two non-empty b-open sets. A subset of X is b-connected if it is b-connected as a subspace. if X can not be connected-bg[6] A topological space X is said to be :Definition 3.2 expressed as a disjoint union of two non-empty bg-open sets. A subset of X is bg-connected if it is bg-connected as a subspace. if X can not be connected-bg**A topological space X is said to be :Definition 3.3 expressed as a disjoint union of two non-empty bg**-open sets, otherwise X is called (bg**-disconnected space). A subset of X is bg**-connected if it is bg**-connected as a subspace. connected.-. Then X is bg**}}a{X, φ, {= and let τ ,c} a, b{= : Let X Example Remark 3.4: 1- Every b-connected space is connected. 2- Every bg-connected space is connected. Proof: By remark 2.6. m 3.5: Theore (i) Every b-connected space is bg**-connected. (ii) Every bg**-connected space is bg-connected. Proof : (i) Let X be b-connected space .Suppose that X is not bg**-connected. Then there exist disjoint non-empty bg**-open sets A and B such that X= A B . By Remark 2.6(4), A and B are b-open sets . This is a contradiction with X is b-connected .Therefore X is bg**- connected. (ii) Its clear from Remark 2.6(3), and by the same way of proof (i). From Theorems 3.5 and Remarks 3.4, we have diagram (1) . Remark 3.6. Connected space b-Connected space bg-Connected space bg**-Connected space Diagram (1): The relationships between connected space ,b-connected space ,bg-connected space and bg**-connected space . For a topological space X, the following statements are equivalent. Theorem 3.7: 1- X is bg**-connected 2- The only subsets of X which are both bg**-open and bg**-closed are the empty set and X. 3- Each bg**-continuous map of X into a discrete space Y with at least two points is a constant map 526 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 Proof: (1) (2) Let U be a bg** -open and bg** closed subset of X. then X- U is both bg** -open and bg** -closed . Since X is the disjoint union of bg** - open sets U and X-U, then one of these must be empty, that is U =  or X – U = . (2) (1) Suppose that X = A B where A and B are disjoint non empty bg** - open sets of X, then A is both bg**-open and bg**-closed subset of X. By assumption, A= or A = X. This implies X is bg**-connected. (2)(3)Let f:X  Y be a bg**-continuous map, then X is covered by bg**-open and bg**– closed covering {f-1(y):yY}. By assumption f-1(y) =  then f fails to be bg**-continuous. Therefore f-1(y) = X . This implies f is a constant map. (3) (2) Let U be both bg**-open and bg**-closed in X. Suppose U   .Let f: X  Y be bg**-continuous map defined by f(U) = {y} and f(X-U) = {w} for some distinct points y and w in Y. By assumption, f is a constant map. Therefore we have U = X Theorem 3.8: (i)If f:XY is a bg**-continuous surjection map and X is bg**-connected, then Y is connected. (ii)If f:X Y is a bg**-irresolute surjection map and X is bg**-connected, then Y is bg**–connected. Proof:(i) Suppose that Y is not connected , then Y= A B where A and B are disjoint non- empty open sets in Y. Since f is bg**-continuous and onto , X = f -1(A)f -1(B) where f -1(A) and f -1(B) are disjoint non empty bg**-open sets which is a contradiction to our assumption that X is bg**-connected . Hence Y is connected. (ii) It follows from the definition of bg**-irresolute map. 4. On Some Types of Continuous Functions & bg**-Connected Space In this section we study some types of continuous functions, and study the relations between (connected space, b-connected space, bg-connected space and bg**-connected space) under these types of continuous functions. [8] Continuous image of connected space is connected. Theorem 4.1: : Theorem 4.2 (i) Continuous image of b-connected space is connected. (ii) Continuous image of bg-connected space is connected. connected space. To prove Y -Y be continuous function ,and let X be b f : X(i) Let Proof: is connected.Suppose that Y is disconnected space ,then Y= AB, where A and B are disjoint empty open -(B) are disjoint non1-(A) and f 1-empty open sets in Y.Since f is continuous f -non (B) (by Remark 1-f (A)1-open sets in X such that X = f -(B) are b1-(A) and f 1-sets in X ,and f 2.6(1)). This contradicts the fact that X is b-connected. Hence Y is connected. (ii) Its clear from Remark 2.6(1), and by the same way of proof (i). 527 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 connected space,Diagram (2) shows the relationships between ( Remark 4.3: b-connected space, bg-connected space and bg**-connected space) under the continuous function. Continuous f : X Y Connected  Connected b- Connected  Connected bg- Connected  Connected Connected  b- Connected Connected  bg- Connected Connected bg**- Connected Diagram (2):The relationships between (connected space, b-connected space, bg- connected space and bg**-connected space) under the continuous function. connected space is connected.-continuous image of b-b: Theorem 4.4. connected space. To prove Y is -continuous function ,and let X be b-Y be b f : XLet Proof: connected. Suppose that Y is disconnected space ,then Y= AB, where A and B are disjoint -empty b-(B) are disjoint non1-(A) and f 1-continuous f -empty open sets in Y.Since f is b-non connected. -This contradicts the fact that X is b (B) .1-f (A)1-open sets in X such that X = f Hence Y is connected. connected -connected space, bDiagram (3) shows the relationships between ( Remark 4.5: space, bg-connected space and bg**-connected space) under the b-continuous function. b-Continuous f : X Y Connected Connected b- Connected Connected bg- Connected Connected Connected b- Connected Connected bg- Connected Connected bg**- Connected Diagram (3):The relationships between (connected space, b-connected space, bg- connected space and bg**-connected space) under the b-continuous function. 528 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 6:Theorem 4. (i) bg-Continuous image of b-connected space is connected. (ii) bg-Continuous image of bg-connected space is connected. (iii) bg-Continuous image of bg**-connected space is connected. connected space. To prove -continuous function ,and let X be b-Y be bg f : X(i) Let Proof: Y is connected.Suppose that Y is disconnected space ,then Y= AB, where A and B are -(B) are disjoint non1-(A) and f 1-continuous f -empty open sets in Y.Since f is bg-disjoint non , such that by Remark 2.6(2) open-(B) are b1-(A) and f 1-open sets in X ,and f -empty bg connected. connected. Hence Y is-hat X is bThis contradicts the fact t (B) .1-f (A)1-X = f (ii) and (iii) by the same way of proof (i) ,and Remark 2.6(3) . connected -connected space, bDiagram (4) shows the relationship between ( Remark 4.7: space, bg-connected space and bg**-connected space) under the bg-continuous function. bg-Continuous f : X Y Connected Connected b- Connected Connected bg- Connected Connected bg**- Connected Connected Connected b- Connected Connected bg- Connected Diagram (4):The relationships between (connected space, b-connected space, bg- connected space and bg**-connected space) under the bg-continuous function. : Theorem 4.8 (i) bg**-Continuous image of b-connected space is connected. (ii) bg**-Continuous image of bg**-connected space is connected. connected space. To prove Y is -continuous ,and let X be b-**Y be bg f : X(i) Let Proof: connected. Suppose that Y is disconnected space, then Y= AB, where A and B are disjoint non-empty empty -(B) are disjoint non1-(A) and f 1-continuous, f -open sets in Y.Since f is bg** , such that by remark 2.6(4) open-(B) are b1-(A) and f 1-n sets in X, and f ope-bg** connected. connected. Hence Y is-This contradicts the fact that X is b (B) .1-f (A)1-X = f (ii) By Theorem 3.8(i). connected -connected space, bDiagram (5) shows the relationships between ( Remark 4.9: space, bg-connected space and bg**-connected space) under the bg**-continuous function. 529 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 bg**-Continuous f : X Y Connected Connected b- Connected Connected bg- Connected Connected bg**- Connected Connected Connected b- Connected Connected bg- Connected Diagram (5):The relationships between (connected space, b-connected space, bg- continuous function.-) under the bg**connected space-connected space and bg** References [1] Andrijevic D. (1996), On b-open sets, Math. Vesnik, 48, 59-64. [2] Dunham W. (1982),A new closure operator for non T1 topologies, Kungpook math. J., 22, 55-60. [3] Ekici E.and Caldas M. (2004), Slightly γ-continuous functions, B0l.Soc. Parana.Mat. (3)22 .2, 63-74. [4] El-Etik A.A. (1997), A study of some types of mappings on topological spaces,M.Sc thesis, Tanta University, Egypt. [5] Fukutake T., Nasef A.A.and El-Maghrabi A.I. (2003), Some topological concepts via γ-generalized closed sets, Bull.Fukuoka Univ. Edu. 52(3), 1-9. [6] Ganster M.and Steiner M. (2007), On bτ-closed sets, Appl. Gen.Topol.,8 .2, 243-247. [7] Ibraheem A.M.(2014), On a new class of closed sets in topological spaces, Journal of College of Education .1. [8] Mustafa H.L. (2001), On Connected Functions ,M.Sc.thesis,University of Al-Mustansirya. [9] Park J. H. (2006), Strongly γ -b-continuous functions, Acta Math. Hungar, 110(4), .347–359. 530 | Mathematics 2014) عام 3(العدد 27المجلد مجلة إبن الھيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (3) 2014 **bg-الفضاءات المترابطة افراح محمد ابراھيم الجامعة المستنصرية/ كلية التربية / الرياضياتقسم 2014تموز 8، قبل في : 2014كانون الثاني 29استلم البحث في: الخالصة ، ودرسنا العالقة بينه وبين انواع اخرى من الفضاءات . ودرسنا **bg–في ھذا البحث قمنا بتعريف الفضاء المترابط ، الفضاءات b-بعض االنواع من الدوال المستمرة ايضا ودرسنا العالقة بين ( الفضاءات المترابطة ،الفضاءات المترابطة ) تحت تأثير تلك االنواع من الدوال المستمرة . **bg-والفضاءات المترابطة bg - المترابطة . **bg -، الفضاء المترابط **bg -: المجموعة المغلقةالكلمات المفتاحية