IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 Some Types of Compactness in Bitopological Spaces* N .A. Jabbar. A. I. Nasir. Department of Mathematics, Ibn Al-Haitham, College of Education, University of Baghdad Abstract In this paper, we give the concept of N-open set in bitopological spaces, where N is the first letter of the name of one of the authors, then we used this concept to define a new kind of compactness, namely N-compactness and we define the N-continuous function in bitopological spaces. We study some properties of N-compact spaces, and the relationships between this kind and two other known kinds which are S-compactness and pair-wise compactness. 1- Introduction In 1963, the concept of "bitopological space" was introduced by Kelly[1]. A set equipped with two topologies is called a 'bitopological space" and denoted by (X,τ,τ), where (X,τ), ((X,τ) are two topological spaces. From that time many authors used the concept of bitopological space to define new concepts like seperation axioms, some types of connectedness and covering properties, for more details see [2] and [3]. In this paper, we introduce the concept of N-compactness, we study some properties of this kind with many examples, we also give some new properties about the S-compactness and pair-wise compactness which was introduced by Mrsevic and Reilly [4], where we give for example propositions 2.21, 2.23, 2.24, 2.27, 2.28, 2.40 and theorem 2.41. We also study the relationships between the three kinds of compactness, where we proved the valid directions and give counter examples for the invalid ones, and we put certain conditions to make the invalid direction true. * This paper is a part of an M.Sc. thesis by the second author and is supervised by the first author. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 2.2 Definition A subset A of a bitopological space (X,τ,τ) is called an "N-open set"if and only if it is open in the space (X,ττ), where ττ is the supremum topology on X contains τ and τ. 2.3 Definition The complement of an N-open set in a bitopological space (X,τ,τ) is called "N-closed set". 2.4 Remark Let (X,τ,τ) be a bitopological space, then: (i) Every open set in (X,τ) or in (X,τ) is an N-open 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 opposite direction of this remark 2.4 may be untrue as the following example 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-open subsets of (X,τ,τ). {1,2} is an N-open set in (X,τ,τ) but it is not open 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 Definition Let (X,τ,τ) be a bitopological space, let A be a subset of X. A subcollection of the family ττ is called an "N-open cover of A" if the union of members of this collection contains A. 2.7 Definition A bitopological space (X,τ,τ) is said to be an "N-compact space" if and only if every N- open cover of X has a finite subcover. 2.8 Proposition If (X,τ,τ) is an N-compact space, then both (X,τ) and (X,τ) are compact spaces. Proof: Follows from remark (2.4).  2.9 Note The implication in proposition (2.8) is not reversible, as the following example shows: Example Let � be the set of all natural numbers, τ={ � }  P(O + ) and τ={ � } P(E + ). Then ττ is the discrete topology on N, where P(O +) and P(E+) are the power sets of O+ and E+ respectively. Now, both (� ,τ) and (� ,τ) are compact spaces, but (� ,τ,τ) is not N-compact. Since the N-open cover {{n}n� } of � has no finite subcover. The opposite direction of proposition (2.8) becomes valid in a special case, when τ is a subfamily of τ, as the following proposition shows: 2.10 Proposition If τ is a subfamily of τ, then (X,τ,τ) is an N-compact space if and only if (X,τ) and (X,τ) are compact. Proof:     IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 Necessity , follows from proposition (2.8). Sufficiency, in view of τ is a subfamily of τ, then ττ = τ. So (X,τ,τ) is N- compact.  2.11 Proposition The N-closed subset of an N-compact space is N-compact. Proof: Let (X,τ,τ) be an N-compact space and let A be an N-closed subset of X To show that A is an N-compact set. Let {Ui : i  } be an N-open cover of A. Since A is N-closed subset of X, then X–A is N-open subset of X, so {X–A}  {Ui : i  } is an N-open cover of X, which is an N-compact space. Therefore, there exists i1, i2,,in  , such that { X–A, i1 U , i2 U , , in U } is a finite subcover of X. As A  X and X–A covers no part of A, then { i1 U , i2 U ,, in U } is a finite subcover of A. So A is N-compact set.  2.12 Definition 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-open subset of Y is an N-open subset of X. 2.13 Proposition The N-continuous image of an N-compact space is an N-compact space. Proof: Let (X,τ,τ) be an N-compact space, and let f: (X,τ,τ)  (Y,T,T) be an N- continuous, onto function. To show that (Y,T,T ) is an N-compact space. Let {Ui : i  } be an N-open cover of Y, then {f – 1 (Ui): i  } is an N-open cover of X, which is N-compact space. 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 Proposition If A and B are two N-compact subsets of a bitopological space (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-compact subsets of a bitopology space (X,τ,τ), then A  B need not be N-compact. For example, 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}, then both A and B are N-compact subsets of the bitopologycal space (X,τ,τ), but A  B = � is not N-compact set. In the following definition, we study another kind of open sets in bitopological spaces, namely "S-open set". 2.16 Definition [4] A subset A of a topological space (X,τ,τ) is said to be "S-open set" if it is -open or -open. The complement of the S-open set is called "S-closed set". IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 2.17 Remark (i) Every S-open set in a bitopological space (X,τ,τ) is an N-open set. (ii) Every S-closed set in a bitopological space (X,τ,τ) is an N-closed set. 2.18 Note The implication in remark (2.17) is not reversible. See the example of note (2.5), where the set {1,2} is N-open set which is not S-open set. So the set {3} is N-closed set which is not S-closed set. 2.19 Definition [4] Let (X,τ,τ) be a bitopological space, let A be a subset of X. A subcollection of the family  is called an "S-open 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-open sets in bitopological spaces inorder to expose another type of compactness in bitopological spaces, called S- compactness, which was introduced in the first time by Mrsevic and Reilly, (4). 2.20 Definition [4] A bitopological space (X,τ,τ) is called an "S-compact space" if and only if every S-open cover of X has a finite subcover. 2.21 Proposition If (X,τ,τ) is an S-compact space, then both (X,τ) and (X,τ) are compact. Proof: Clear.  2.22 Note The opposite direction of proposition (2.21) may be false. For example: Let X =[0,1] and let  = {,X,{0}} and ={,X,(0,1]} 1 {( ,1] n } n � . Then both (X,τ) and (X,τ) are compact spaces, but (X,τ,τ) is not S-compact, since the S-open cover {{0}}  1 {( ,1] n } n � of X has no finite subcover. The opposite direction of proposition (2.21) becomes valid in a special case, where  is a subfamily of , as the following proposition shows: 2.23 Proposition If τ is a subfamily of τ, then (X,τ,τ) is an S-compact space if and only if (X,τ) and (X,τ) are compact spaces. Proof: Clear.  2.24 Proposition An S-closed subset of an S-compact space is S-compact. Proof: Let (X,τ,τ) be an S-compact space, let A be an S-closed subset of X. To show that A is an S-compact set. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 Let {Ui : i  } be an S-open cover of A. Since A is an S-closed subset of X, then X– A is an S-open subset of X. Then {Ui : i  }{X–A} is an S-open cover of X, which is S- compact space. 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 part of A, then {Uij: j=1, 2, …,n} is a finite subcover of A. So A is an S-compact.  2.25 Definition [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 topologies on X and Y respectively. 2.27 Proposition A bicontinuous image of an S-compact space is an S-compact space. Proof: Let f: (X,τ,τ)  (Y,T,T) be a bicontinuous, onto function and let (X,τ,τ) be an S- compact space. To prove that (Y,T,T) is an S-compact. Let {Ui : i  } be an S-open cover of Y, then {f – 1(Ui) : i  } is an S-open cover of X, which is an S-compact space. 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-compact space.  2.28 Proposition If A and B are two S-compact subsets of a topological space (X,τ,τ), then AB is an S- compact subset of X. Proof: Clear.  2.29 Remark If A and B are two S-compact subsets of a bitopological space (X,τ,τ), then AB need not be S-compact set. For example: See the example of remark (2.15), both A and B are S-compact subsets of the bitopological space (X,τ,τ), A  B = � is not S-compact set. 2.30 Proposition Every N-compact space is an S-compact. Proof: Follows from remark (2.17).  2.31 Proposition Let (X,τ,τ) be a topological space. If τ is a subfamily of τ, then the concepts of S- compactness and N-compactness are coincident. Proof: Follows from propositions (2.10) and (2.23).  IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 The following diagram shows the relationships between N-compact and S-compact spaces: Now, we shall recall another kind of compactness on bitopological spaces called "pair- wise compact" to study this kind and compare it with the two above kinds of compactness in bitopological spaces. 2.32 Definition [4] Let (X,τ,τ) be a bitopological space, A  X, an S-open cover of A is called a "pair-wise open cover" if it contains at least one non-empty element from τ, and at least one non-empty 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 pair-wise open cover of X. 2.34 Remark Every pair-wise open cover of the bitopological space (X,τ,τ) is an S-open cover. 2.35 Note The implication in remark 2.34 is not reversible. For example: Let X = {1,2,3}, τ = { ,{1},X} and τ = { ,{2},{3},{2,3},{1,2},X}. Then the cover C = {{1,2},{3}} is an S-open cover of X, but it is not pair-wise open cover.. 2.36 Definition [4] A bitopological space (X,τ,τ) is called a "pair-wise compact space" if every pair-wise open cover of X has a finite subcover. 2.37 Remark Let (X,τ,τ) be a bitopological space. If τ = τI or τ = τI, then X is a pair-wise compact space. 2.38 Proposition Every S-compact space isa pair-wise compact space. Proof: Follows from remark 2.34. 2.39 Note The converse of proposition 2.38 may be false. For example: ( � ,τu, τI) is pair-wise compact space, but not S-compact. N-compact S-compact  – τ is a subfamily of τ   IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 The example of note 2.39 shows that, if (X,τ,τ) is a pair-wise compact space, then (X,τ) need not to be compact and the example of note 2.9 shows that, if (X,τ) and (X,τ) are compact space, then (X,τ,τ) need not to bea pair-wise compact. 2.40 Proposition If τ is a subfamily of τ and (X,τ) isa compact space, then (X,τ,τ) is a pair-wise compact space. Proof: Follows from proposition 2.10 and proposition 2.38. 2.41 Theorem If (X,τ) and (X,τ) are compact spaces, then (X,τ,τ) is S-compact if and only if it is a pair-wise compact. Proof: Necessity , follows from proposition 2.38. Sufficiency, suppose (X,τ,τ) is a pair-wise compact space, to prove it, is an S-compact space. Let W be an S-open cover of X, then there are three probabilities i) If W is a τ-open cover, since (X,τ) is compact, then W has a finite subcover of X, so the proof is over. ii) If W is a τ-open cover, since (X,τ) is compact space, then W has a finite subcover of X, so the proof is over. iii) If W is a pair-wise open cover, since (X,τ,τ) isa pair-wise compact space, then W has a finite subcover. iv) Therefore, (X,τ,τ) isan S-compact space. From proposition 2.38 and theorem 2.41, we get the following diagram:: 2.42 Corollary If τ is a subfamily of τ, and (X,τ) is a compact space, then (X,τ,τ) is a pair-wise compact if and only if it is an S-compact space. 2.43 Remark The following diagram shows the relations among the different types of compactness that are studied in this section: In a bitopological space (X,τ,τ) Refrences 1. J.C.Kelly, (1963), Proc. London Math. Soc. 13, 71-89. 2. B.Dvalishvili, (2003), MATEMAT. BECH., 55, 37-52 3. Ivan L.Reilly, (2005), Hacettepe Journal of Mathematics and Statistics, 345, 27-34. 4. Mrsevic and I.L.Reilly, (1996), Indian J.Pure Appl. Math., 27 (10), 995-1004, Oct 5. S.N.M aheshwari and S.S. Thakur, (1985), Bulletin of the Institute of Mathematics Academia Sinica, Vol. 13, No.4, Dece., 341-347. S-compact Pair-wise compact  – both (X,τ) and (X,τ) are compact space +   N-compact  S-compact  Pair-wise compact   – –   both (X,τ) and (X,τ) are compact space+ τ is a subfamily of τ+ للعلوم الصرفة والتطبیقیة 2010) 1( 23المجلد مجلة ابن الھیثم انواع الفضاءات ثنائیة الرص بعض أحمد إبراهیم ناصرنرجس عبد الجبار ، قسم الریاضیات ، كلیة التربیة ، ابن الهیثم ، جامعة بغداد الخالصة میناها ــات المفتوحــة فـــي الفضــاءات التبولوجیـــة الثنائیــة اســـ ـا فــي هـــذا البحــث بتعریـــف نــوع جدیـــد مــن المجموعـ قمنـ 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