IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Some Types o f Compactness in Bitopological Spaces* N .A. Jabbar. A. I. Nasir. Department of Mathematics, Ibn Al-Haitham, College of Education, Unive rsity of Baghdad Abstract In this p aper, we give the concept of N-op en set in bitop ological sp aces, where N is the first letter of the name of one of the authors, t hen we used this concep t to define a new kind of comp actness, namely N-compactness and we define the N-continuous function in bitop ological sp aces. We st udy some prop erties of N-compact sp aces, and the relationships between this kind and two other known kinds which are S-comp actness and p air-wise comp actness. 1- Introduction In 1963, the concep t of "bitop ological sp ace" was introduced by Kelly [1]. A set equipped with two top ologies is called a 'bitop ological sp ace" and denoted by (X,τ,τ), where (X,τ), ((X,τ) are two top ological sp aces. From that time many authors used the concept of bitop ological sp ace to define new concepts like sep eration axioms, some ty p es of connectedness and covering p rop erties, for more details see [2] and [3]. In this p aper, we introduce the concept of N-compactness, we study some prop erties of this kind with many examp les, we also give some new p rop erties about the S-comp actness and p air-wise comp actness which was introduced by M rsevic and Reilly [4], where we give for examp le p rop ositions 2.21, 2.23, 2.24, 2.27, 2.28, 2.40 and theorem 2.41. We also st udy the relationship s between the three kinds of comp actness, where we p roved the valid directions and give counter examp les for the invalid ones, and we p ut certain conditions to make the invalid direction true. * This p aper is a p art of an M .Sc. thesis by the second author and is sup ervised by the first author. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 2.2 De fini tion A subset A of a bitop ological sp ace (X,τ,τ) is called an "N-op en set"if and only if it is op en in the sp ace (X,ττ), where ττ is the sup remum top ology on X contains τ and τ. 2.3 De fini tion The comp lement of an N-op en set in a bitop ological sp ace (X,τ,τ) is called "N-closed set". 2.4 Remark Let (X,τ,τ) be a bitop ological sp ace, then: (i) Every op en set in (X,τ) or in (X,τ) is an N-op en set in (X,τ,τ). (ii) Every closed set in (X,τ) or in (X,τ) is an N-closed set in (X,τ,τ). 2.5 Note The opp osite direction of this remark 2.4 may be untrue as the following examp le shows: Example Let X={1,2,3}, τ={,{1},X} and τ={,{2},X} then ττ={,{1},{2},{1,2},X} is the family of all N-op en subsets of (X,τ,τ). {1,2} is an N-op en set in (X,τ,τ) but it is not op en in both (X,τ) and (X,τ). So {3} is an N-closed set in (X,τ,τ) which is not closed in both (X,τ) and (X,τ). 2.6 De fini tion Let (X,τ,τ) be a bitop ological sp ace, let A be a subset of X. A subcollection of the family ττ is called an "N-op en cover of A" if the union of members of this collection contains A. 2.7 De fini tion A bitop ological sp ace (X,τ,τ) is said to be an "N-compact sp ace" if and only if every N- op en cover of X has a finite subcover. 2.8 Proposi tion If (X,τ,τ) is an N-comp act sp ace, then both (X,τ) and (X,τ) are comp act sp aces. Proof: Follows from remark (2.4).  2.9 Note The implication in p rop osition (2.8) is not reversible, as the following examp le shows: Example Let � be the set of all natural numbers, τ={ � }  P(O+) and τ={ � } P(E+). Then ττ is the discrete top ology on N, where P(O + ) and P(E + ) are the p ower sets of O + and E + resp ectively. Now, both ( � ,τ) and ( � ,τ) are comp act sp aces, but ( � ,τ,τ) is not N-compact. Since the N-op en cover {{n}n� } of � has no finite subcover. The opp osite direction of p rop osition (2.8) becomes valid in a sp ecial case, when τ is a subfamily of τ, as the following p rop osition shows: 2.10 Proposi tion If τ is a subfamily of τ, then (X,τ,τ) is an N-compact sp ace if and only if (X,τ) and (X,τ) are comp act. Proof:     IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Necessity , follows from prop osition (2.8). Sufficiency, in view of τ is a subfamily of τ, then ττ = τ. So (X,τ,τ) is N- comp act.  2.11 Proposi tion The N-closed subset of an N-compact sp ace is N-comp act. Proof: Let (X,τ,τ) be an N-compact sp ace and let A be an N-closed subset of X To show that A is an N-comp act set. Let {Ui : i  } be an N-op en cover of A. Since A is N-closed subset of X, t hen X–A is N-op en subset of X, so {X–A}  {Ui : i  } is an N-op en cover of X, which is an N-comp act sp ace. Therefore, there exists i1, i2,,in  , such that { X–A, i1 U , i2U , , inU } is a finite subcover of X. As A  X and X–A covers no p art of A, then { i1 U , i2U ,, inU } is a finite subcover of A. So A is N-compact set.  2.12 De fini tion A function f: (X,τ,τ)  (Y,T,T) is said to be an "N-continuous function" if and only if the inverse image of each N-op en subset of Y is an N-op en subset of X. 2.13 Proposi tion The N-continuous image of an N-compact sp ace is an N-compact sp ace. Proof: Let (X,τ,τ) be an N-compact sp ace, and let f: (X,τ,τ)  (Y,T,T) be an N- continuous, ont o function. To show that (Y,T,T) is an N-comp act sp ace. Let {Ui : i  } be an N-op en cover of Y, then {f – 1 (Ui): i  } is an N-op en cover of X, which is N-compact sp ace. So, there exists i1, i2,,in  , such that the family {f – 1 (Uij): j=1, 2, …,n} covers X and since f is onto, then {Uij: j=1, 2, …,n} is a finite subcover of Y.  2.14 Proposi tion If A and B are two N-compact subsets of a bitop ological sp ace (X,τ,τ), then AB is an N-compact subset of X. Proof: Clear.  2.15 Remark If A and B are two N-comp act subsets of a bitop ology sp ace (X,τ,τ), then A  B need not be N-comp act. For examp le, let X= �  {0,-1} and let =P( � ){HX-1,0H(X–H) is finite}. Let =  {HX(-1H or 0 H)(X – H) finite}. Now, let A = �  {0} and B = �  {-1}, t hen both A and B are N-compact subsets of the bitop ology cal sp ace (X,τ,τ), but A  B = � is not N-compact set. In the following definition, we st udy another kind of op en sets in bitop ological sp aces, namely "S-op en set". 2.16 De fini tion [4] A subset A of a top ological sp ace (X,τ,τ) is said to be "S-op en set" if it is -op en or -op en. The complement of the S-op en set is called "S-closed set". IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 2.17 Remark (i) Every S-op en set in a bitop ological sp ace (X,τ,τ) is an N-op en set. (ii) Every S-closed set in a bitop ological sp ace (X,τ,τ) is an N-closed set. 2.18 Note The imp lication in remark (2.17) is not reversible. See the examp le of note (2.5), where the set {1,2} is N-op en set which is not S-op en set. So the set {3} is N-closed set which is not S-closed set. 2.19 De fini tion [4] Let (X,τ,τ) be a bitop ological sp ace, let A be a subset of X. A subcollection of the family  is called an "S-op en cover" of A if the union of members of this collection contains A. In the definitions (2.16) and (2.19), we use the concept of S-op en sets in bitop ological sp aces inorder to exp ose another ty p e of comp actness in bitop ological sp aces, called S- comp actness, which was introduced in the first time by M rsevic and Reilly, (4). 2.20 De fini tion [4] A bitop ological sp ace (X,τ,τ) is called an "S-comp act sp ace" if and only if every S-op en cover of X has a finite subcover. 2.21 Proposi tion If (X,τ,τ) is an S-comp act sp ace, then both (X,τ) and (X,τ) are comp act. Proof: Clear.  2.22 Note The opp osite direction of prop osition (2.21) may be false. For examp le: Let X =[0,1] and let  = {,X,{0}} and ={,X,(0,1]} 1 {( ,1] n } n � . Then both (X,τ) and (X,τ) are comp act sp aces, but (X,τ,τ) is not S-comp act, since the S-op en cover {{0}}  1 {( ,1] n } n � of X has no finite subcover. The opp osite direction of p rop osition (2.21) becomes valid in a sp ecial case, where  is a subfamily of , as the following p rop osition shows: 2.23 Proposi tion If τ is a subfamily of τ, then (X,τ,τ) is an S-comp act sp ace if and only if (X,τ) and (X,τ) are comp act sp aces. Proof: Clear.  2.24 Proposi tion An S-closed subset of an S-comp act sp ace is S-compact. Proof: Let (X,τ,τ) be an S-comp act sp ace, let A be an S-closed subset of X. To show that A is an S-comp act set. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Let {Ui : i  } be an S-op en cover of A. Since A is an S-closed subset of X, t hen X– A is an S-op en subset of X. Then {Ui : i  }{X–A} is an S-op en cover of X, which is S- comp act sp ace. Therefore, there exists i1, i2,,in  , such that {Uij: j=1, 2, …,n}{X–A} is a finite subcover of X. Since AX and X–A covers no p art of A, t hen {Uij: j=1, 2, …,n} is a finite subcover of A. So A is an S-comp act.  2.25 De fini tion [5] Let f: (X,τ,τ)  (Y,T,T) be a function, then f is said to be a "bicontinuous function" if and only if f – 1 (U)  τ, for each UT, and f – 1 (v)  τ, for each v  T. 2.26 Example Let X={1,2,3}, τ={,X,{1},{2},{1,2}} and τ = τD. And let Y={a,b,c}, T ={,Y,{a}} and T= τI. Define f: (X,τ,τ)  (Y,T,T), such that f(1) = a, f(2) = b and f(3) = c. Then f is bicontinuous function. Where τD and τI are the discrete and indiscrete top ologies on X and Y resp ectively. 2.27 Proposi tion A bicontinuous image of an S-comp act sp ace is an S-comp act sp ace. Proof: Let f: (X,τ,τ)  (Y,T,T) be a bicontinuous, onto function and let (X,τ,τ) be an S- comp act sp ace. To p rove that (Y,T,T) is an S-comp act. Let {Ui : i  } be an S-op en cover of Y, then {f – 1 (Ui) : i  } is an S-op en cover of X, which is an S-comp act sp ace. Therefore, there exists i1, i2,,in  , such that {f – 1 (Uij): j=1, 2, …,n} is a finite subcover of X and since f is onto, then we get {Uij: j=1, 2, …,n} is a finite subcover of Y. So Y is an S-comp act sp ace.  2.28 Proposi tion If A and B are two S-compact subsets of a top ological sp ace (X,τ,τ), then AB is an S- comp act subset of X. Proof: Clear.  2.29 Remark If A and B are two S-comp act subsets of a bitop ological sp ace (X,τ,τ), then AB need not be S-comp act set. For examp le: See the examp le of remark (2.15), both A and B are S-comp act subsets of the bitop ological sp ace (X,τ,τ), A  B = � is not S-comp act set. 2.30 Proposi tion Every N-compact sp ace is an S-comp act. Proof: Follows from remark (2.17).  2.31 Proposi tion Let (X,τ,τ) be a top ological sp ace. If τ is a subfamily of τ, then the concepts of S- comp actness and N-compactness are coincident. Proof: Follows from p rop ositions (2.10) and (2.23).  IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 The following diagram shows the relationships between N-comp act and S-comp act sp aces: Now, we shall recall another kind of comp actness on bitop ological sp aces called "p air- wise comp act" to st udy this kind and comp are it with the two above kinds of comp actness in bitop ological sp aces. 2.32 De fini tion [4] Let (X,τ,τ) be a bitop ological sp ace, A  X, an S-op en cover of A is called a "p air-wise op en cover" if it contains at least one non-emp ty element from τ, and at least one non-emp ty element from τ. 2.33 Example Let X = {1,2,3}, τ = {,X,{1}} and τ = {,X,{2},{3},{2,3}}. Then the cover C = {{1},{2},{3}} is a p air-wise op en cover of X. 2.34 Remark Every p air-wise op en cover of the bitop ological sp ace (X,τ,τ) is an S-op en cover. 2.35 Note The implication in remark 2.34 is not reversible. For examp le: Let X = {1,2,3}, τ = {,{1},X} and τ = {,{2},{3},{2,3},{1,2},X}. T hen the cover C = {{1,2},{3}} is an S-op en cover of X, but it is not p air-wise op en cover.. 2.36 De fini tion [4] A bitop ological sp ace (X,τ,τ) is called a "p air-wise comp act sp ace" if every p air-wise op en cover of X has a finite subcover. 2.37 Remark Let (X,τ,τ) be a bitop ological sp ace. If τ = τI or τ = τI, then X is a p air-wise comp act sp ace. 2.38 Proposi tion Every S-comp act sp ace isa p air-wise comp act sp ace. Proof: Follows from remark 2.34. 2.39 Note The converse of prop osition 2.38 may be false. For examp le: ( � ,τu, τI) is p air-wise comp act sp ace, but not S-comp act. N-compact S-compact  – τ is a subfamily of τ   IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 The examp le of note 2.39 shows that, if (X,τ,τ) is a pair-wise comp act sp ace, then (X,τ) need not to be comp act and the examp le of note 2.9 shows that, if (X,τ) and (X,τ) are comp act sp ace, then (X,τ,τ) need not to bea p air-wise comp act. 2.40 Proposi tion If τ is a subfamily of τ and (X,τ) isa compact sp ace, then (X,τ,τ) is a pair-wise comp act sp ace. Proof: Follows from p rop osition 2.10 and p rop osition 2.38. 2.41 The orem If (X,τ) and (X,τ) are comp act sp aces, then (X,τ,τ) is S-comp act if and only if it is a p air-wise comp act. Proof: Necessity , follows from prop osition 2.38. Sufficiency, sup p ose (X,τ,τ) is a pair-wise comp act sp ace, to p rove it, is an S-comp act sp ace. Let W be an S-op en cover of X, t hen there are three probabilities i) If W is a τ-op en cover, since (X,τ) is comp act, then W has a finite subcover of X, so the p roof is over. ii) If W is a τ-op en cover, since (X,τ) is comp act sp ace, then W has a finite subcover of X, so t he proof is over. iii) If W is a p air-wise op en cover, since (X,τ,τ) isa p air-wise comp act sp ace, then W has a finite subcover. iv) Therefore, (X,τ,τ) isan S-comp act sp ace. From p rop osition 2.38 and theorem 2.41, we get t he following diagram:: 2.42 Corollary If τ is a subfamily of τ, and (X,τ) is a comp act sp ace, then (X,τ,τ) is a p air-wise comp act if and only if it is an S-comp act sp ace. 2.43 Remark The following diagram shows the relations among the different ty p es of comp actness that are studied in this section: In a bitop ological sp ace (X,τ,τ) Refrences 1. J.C.Kelly, (1963), Proc. London M ath. Soc. 13, 71-89. 2. B.Dvalishvili, (2003), M AT EM AT . BECH., 55, 37-52 3. Ivan L.Reilly , (2005), Hacettepe Journal of M athematics and Statist ics, 345, 27-34. 4. M rsevic and I.L.Reilly , (1996), Indian J.Pure Ap p l. M ath., 27 (10), 995-1004, Oct 5. S.N.M aheshwari and S.S. Thakur, (1985), Bulletin of the Inst itut e of M athematics Academia Sinica, Vol. 13, No.4, Dece., 341-347. S-compact Pair-wise c ompact  – both (X,τ) and (X,τ) are compact space +   N-compact  S-comp act  Pair-wise compact   – –   both (X,τ) and (X,τ) are comp act sp ace+ τ is a subfamily of τ+ 2010) 1( 23المجلد مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة انواع الفضاءات ثنائیة الرص بعض أحمد إبراهیم ناصرنرجس عبد الجبار ، قسم الریاضیات ، كلیة التربیة ، ابن الهیثم ، جامعة بغداد الخالصة قمنــا فــي هـــذا البحــث بتعریـــف نــوع جدیـــد مــن المجموعـــات المفتوحــة فـــي الفضــاءات التبولوجیـــة الثنائیــة اســـمیناها هــذا المفهــوم فـــي مــن ثــم أســتعملنا هــو الحــرف االول الســم أحــد البــاحثین و Nان إذ N –ن نــوع المجموعــات المفتوحــة مــ .في الفضاءات الثنائیة N –وكذلك عرفنا الدالة المستمرة من نوع N –تعریف نوع جدید من التراص وهو التراص من نوع ذا النـوع بنـوعین آخـرین معـروفین همـا التـراص كما درسنا عالقـة هـ N –ولقد درسنا بعض الخواص للتراص من نوع .والتراص الثنائي S –من نوع