2_289_kilicman.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 3, 2009, (325-337) ISSN 1307-5543 – www.ejpam.com A Note On Pairwise Continuous Mappings and Bitopo- logical Spaces Adem Kılıçman1∗ and Zabidin Salleh2 1 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia 2 Department of Mathematics, Faculty of Science and Technology, University Malaysia Terengganu, 21030 Kuala Terengganu, Terengganu, Malaysia Abstract. We shall continue the study of bitopological separation axioms that was begun by Kelly and obtained some results. Furthermore, we introduce a concept of pairwise Lindelöf bitopological spaces, namely, p2-Lindelöf spaces and their properties are established. We also show that a p2-Lindelöf space is not a hereditary property. Finally, we show that a p2-Lindelöf space is a p2-topological property. 2000 Mathematics Subject Classifications: 54E55 Key Words and Phrases: Bitopological space, p2-compact, p2-Lindelöf, p1-regular, p2-normal, p2-continuity, p2-homeomorphism. ∗Corresponding author. Email addresses: akili man�putra.upm.edu.my (A. Kılıçman), zabidin�umt.edu.my (Z. Salleh) http://www.ejpam.com 325 c© 2009 EJPAM All rights reserved. A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 326 1. Introduction A bitopological space (X ,P ,Q) is known as a set X together with two arbitrary topologies P and Q which are defined on X , for the detail definitions, terminologies and notations we refer to [1]. In [3] and [4], the authors introduced and studied the idea of pairwise continuity and pairwise Lindelöfness in bitopological spaces and give some results concerning these ideas. Recently the authors in [5], [6] introduced and studied the notion of pairwise almost Lindelöf and pairwise weakly Lindelöf spaces in bitopological spaces. In this paper, we are concerned with another concept of pair- wise regular Lindelöfness and pairwise continuity in bitopological spaces. In section 3, we shall introduce another concept of pairwise regular and pairwise normal bitopological spaces, i.e., p1-regular spaces and p2-normal spaces and obtain a result. Furthermore, some examples will be given to describe its properties. In section 4, we shall define another concept of pairwise Lindelöf spaces, i.e., p2- Lindelöf spaces. We obtain a result about subset of such spaces. The main result we are obtain here is every p1-regular and p2-Lindelöf bitopological space is p2-normal. In section 5, we extend idea of continuity to a bitopological space, namely, p2- continuity and study their properties where the purpose is to study the effect of map- ping and p2-continuity on p2-Lindelöf bitopological spaces. We also show that this mapping preserve p2-Lindelöf property. The main result here is that the image of a p2-Lindelöf space under a p2-continuous function is p2-Lindelöf. A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 327 2. Preliminaries Throughout this paper, all spaces (X ,P ) and (X ,P ,Q) (or simply X ) are always meant topological spaces and bitopological spaces, respectively. In this paper, we shall use p- to denote pairwise. For instance, p-Lindelöf stands for pairwise Lindelöf. While p1- and p2- are used to denote other concepts of pairwise. Sometimes the au- thors write the term “pairwise Lindelöf spaces" which means that pairwise Lindelöf bitopological spaces. Kelly [1] was the first one who introduced the idea of p-regular spaces and p- normal spaces. Later these spaces will be generalized to p1-regular spaces and p2- normal spaces respectively. Definition 2.1 (Kelly). In a space (X ,P ,Q), P is said to be regular with respect to Q if, for each point x ∈ X , there is a P -neighbourhood base ofQ-closed sets, or, as is easily seen to be equivalent, if, for each point x ∈ X and each P -closed set P such that x /∈ P, there are a P -open set U and a Q-open set V such that x ∈ U , P ⊆ V, and U ∩ V = ;. Similarly, (X ,P ,Q) is, or P and Q are, p-regular if P is regular with respect to Q and vice versa. Definition 2.2 (Kelly). A bitopological space (X ,P ,Q) is said to be p-normal if, given a P -closed set A and a Q-closed set B with A∩ B = ;, there exist a Q-open set U and a P -open set V such that A⊆ U, B ⊆ V , and U ∩ V = ;. Further, in [3] the authors introduced the concept of p1-normal spaces. Moreover in the same paper, the authors also introduced the concept of p-Lindelöf spaces and p1-Lindelöf spaces as the following. A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 328 Definition 2.3 (see [3] ). A bitopological space (X ,P ,Q) is said to be p-Lindelöf if the topological space (X ,P ) and (X ,Q) are both Lindelöf. Equivalently, (X ,P ,Q) is p-Lindelöf if every P -open cover of X can be reduced to a countable P -open cover and every Q-open cover of X can be reduced to a countable Q-open cover. Definition 2.4 (see [3] ). In a bitopological space (X ,P ,Q), P is said to be Lindelöf with respect toQ if, everyP -open cover of X can be reduced to a countableQ-open cover and similarly, (X ,P ,Q) is, or P and Q are, p1-Lindelöf if P is Lindelöf with respect to Q and vice versa. 3. Bitopological Separation Axioms In this section, we shall introduce the concept of p1-regular spaces and p2-normal spaces. Before that we need the following definition. Definition 3.1. Let (X ,P ,Q) be a bitopological space. (i) A set H is said to be p1-open if H is (P ∪Q)-open in X . (ii) A set M is said to be p1-closed if M is (P ∪Q)-closed in X . Definition 3.2. A bitopological space (X ,P ,Q) is said to be p1-regular if for each point x ∈ X , there is a p1-neighbourhood base of p1-closed sets, or, as is easily seen to be equivalent, if, for each point x ∈ X and each p1-closed set P such that x /∈ P, there are p1-open sets U and V such that x ∈ U , P ⊆ V, and U ∩ V = ;. Note that, below we shall use the notation p1-cl (U), which means that, the closure of U with respect to P ∪Q, or, in other words (P ∪Q)-cl (U). A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 329 Theorem 3.1. A bitopological space (X ,P ,Q) is p1-regular if and only if for each point x ∈ X and p1-open set H containing x, there exists a p1-open set U such that x ∈ U ⊆ p1- cl (U)⊆ H. Proof. (⇒) : Suppose (X ,P ,Q) is p1-regular. Let x ∈ X and H is a p1-open set containing x . Then G = X�H is a p1-closed set which x /∈ G. Since (X ,P ,Q) is p1-regular, then there are p1-open sets U and V such that x ∈ U , G ⊆ V , and U ∩ V = ;. Since U ⊆ X�V , then p1-cl (U) ⊆ p1-cl (X�V ) = X�V ⊆ X�G = H. Thus, x ∈ U ⊆ p1-cl (U)⊆ H as desired. (⇐) : Suppose the condition holds. Let x ∈ X and P is a p1-closed set such that x /∈ P. Then x ∈ X�P, and by hypothesis there exists a p1-open set U such that x ∈ U ⊆ p1-cl (U) ⊆ X�P. It follows that x ∈ U , P ⊆ X�p1-cl (U) and U ∩ � X�p1- cl (U) � = ;. This completes the proof. Definition 3.3. A bitopological space (X ,P ,Q) is said to be p2-normal if given p1-closed sets A and B with A∩ B = ;, there exist p1-open sets U and V such that A⊆ U, B ⊆ V , and U ∩ V = ;. Theorem 3.2. A space (X ,P ,Q) is p2-normal if and only if given a p1-closed set C and a p1-open set D such that C ⊆ D, there are a p1-open set G and a p1-closed set F such that C ⊆ G ⊆ F ⊆ D. Proof. (⇒) : Suppose (X ,P ,Q) is p2-normal. Let C be a p1-closed set and D a p1-open set such that C ⊆ D. Then K = X�D is a p1-closed set with K ∩ C = ;. Since (X ,P ,Q) is p2-normal, there exists p1-open sets U and G such that K ⊆ U , C ⊆ G, and U ∩ G = ;. Hence G ⊆ X�U ⊆ X�K = D. Thus C ⊆ G ⊆ X�U ⊆ D and the A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 330 result follows by taking X�U = F . (⇐) : Suppose the condition holds. Let A and B are p1-closed sets with A∩ B = ;. Then D = X�A is a p1-open set with B ⊆ D. By hypothesis, there are a p1-open set G and a p1-closed set F such that B ⊆ G ⊆ F ⊆ D. It follows that A= X�D ⊆ X�F , B ⊆ G and (X�F) ∩ G = ; where X�F and G are p1-open sets. This completes the proof. Example 3.1. Consider X = {a, b, c} with topologies P = {;, {a} , {c} , {a, c} , X} and Q = {;, {a} , {b} , {a, b} , {b, c} , X} defined on X . Then P ∪Q = {;, {a} , {b} , {c} , {a, b} , {a, c} , {b, c} , X} . Observe that P ∪Q is a discrete topology and p1-closed subsets of X are ;, {a} , {b} , {c} , {a, b} , {a, c} , {b, c} and X . It follows that (X ,P ,Q) does satisfy the condition in definition of p1-regular and p2-normal. Hence (X ,P ,Q) is p1-regular and p2-normal space. Example 3.2. Consider X = {a, b, c, d} with topologies P = {;, {a, b} , X} and Q = {;, {a} , {b, c, d} , X} defined on X . Then P ∪Q = {;, {a} , {a, b} , {b, c, d} , X} . Observe that p1-closed subsets of X are ;, {a} , {b, c, d} , {c, d} and X . Hence (X ,P ,Q) is p2-normal as we can checks. However (X ,P ,Q) is not p1-regular since the p1-closed set P = {c, d} satisfy b /∈ P, but do not exist the p1-open sets U and V such that b ∈ U, P ⊆ V and U ∩ V = ;. Note that P ∪Q is not a topology of X since {a, b} , {b, c, d} ∈ P ∪Q, but {a, b} ∩ {b, c, d} = {b} /∈ P ∪Q. A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 331 4. On p2-Lindelöf Spaces In this section, we shall introduce a new concept of pairwise compact spaces and pairwise Lindelöf spaces as the following. Definition 4.1. A bitopological space (X ,P ,Q) is said to be p2-compact if every p1-open cover of X has a finite subcover. Definition 4.2. A bitopological space (X ,P ,Q) is said to be p2-Lindelöf if every p1-open cover of X has a countable subcover. It is very clear that, every p2-compact space is p2-Lindelöf but not the converse by the following counter-example. Example 4.1. LetB be the collection of open-closed intervals in the real line R B = {(a, b] : a, b ∈ R, a < b} . Hence B is a base for the upper limit topology P on R. Similarly, the collection of closed-open intervals, B∗ = {[c, d) : c, d ∈ R, c < d} is a base for the lower limit topologyQ on R. Observe thatB ∪B∗ is the set of the form (a, b] , [a, b) , (a, d) , [c, b] and (a, b] ∪ [c, d), i.e., B ∪B∗ is a base for the P ∪Q. Thus (R,P ,Q) is a p2-Lindelöf space. But (R,P ,Q) is not p2-compact since for example {(n, n+ 1] : n ∈ Z} is a p1-open cover of R contains no finite subcover. Lemma 4.1. Every p1-closed subset of a p2-Lindelöf bitopological space is p2-Lindelöf. Proof. Let (X ,P ,Q) be a p2-Lindelöf bitopological space and let F is a p1-closed subset of X . If � Uα : α ∈∆ is a p1-open cover of F , then X = � ⋃ α∈∆ Uα � ⋃ (X�F). Hence the collection � Uα : α ∈∆ and X�F forms a p1-open cover of X . Since (X ,P ,Q) is p2-Lindelöf, there will be a countable subcover ¦ X�F, Uα1 , Uα2 , . . . © . But A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 332 F and X�F are disjoint; hence the subcollection of p1-open set ¦ Uαi : i ∈ N © also cover F , and so � Uα : α ∈∆ has a countable subcover. This completes the proof. Theorem 4.1. Every p1-regular and p2-Lindelöf bitopological space (X ,P ,Q) is p2- normal. Proof. Let A and B are p1-closed sets in X with A∩ B = ;. Since (X ,P ,Q) is p1-regular, then by Theorem 3.1, for each x in B and p1-open set X�A containing x , there is a p1-open set Px such that x ∈ Px ⊆ p1- cl(Px)⊆ X�A, i.e., p1-cl � Px � ∩ A= ;. The collection {Px : x ∈ B} forms a p1-open cover of B. Since (X ,P ,Q) is p2-Lindelöf, then B is also p2-Lindelöf by Lemma 4.1. Hence we obtain a countable p1-open cover of B, which we denote by � Pi : i ∈ N . Similarly, for each y in A and p1-open set X�B containing y, there is a p1-open set Q y such that y ∈ Q y ⊆ p1- cl � Q y � ⊆ X�B, i.e., p1-cl � Q y � ∩B = ;. The collection ¦ Q y : y ∈ A © forms a p1-open cover of A. Since (X ,P ,Q) is p2-Lindelöf, then A is also p2-Lindelöf by Lemma 4.1. Hence we obtain a countable p1-open cover of A, which we denote by � Q i : i ∈ N . Let Un =Qn� ⋃ � p1- cl � Vi � : i ≤ n and Vn = Pn� ⋃ � p1- cl � Ui � : i ≤ n . Since Un ∩ p1-cl � Vm � = ; for m ≤ n, it follows that Un ∩ Vm = ; for m≤ n. Similarly, Vm ∩ p1-cl � Un � = ; for n ≤ m, it follows that Vm ∩ Un = ; for n ≤ m. Thus Un ∩ Vm = ; for all m and n, and consequently U = ⋃ � Un : n ∈ N is disjoint A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 333 from V = ⋃ � Vn : n ∈ N . Finally, p1-cl � Vi � ∩ A and p1-cl � Ui � ∩ B are empty set for all i and hence the set U contains A and is p1-open, whilst the set V contains B and is p1-open. The proof is complete. 5. On p2-continuous Functions Now we shall give another concept of pairwise continuous functions and pairwise homeomorphism in the sense of A. Tallafha [5] and study its properties. Definition 5.1. A function f : (X ,P ,Q) → (Y,R ,T ) is said to be p2-continuous if the inverse image f −1 (U) ∈ P ∪Q for every U ∈ R ∪T , or equivalently, f −1 (U) is p1-open in X for every U is p1-open in Y . Definition 5.2. A function f : (X ,P ,Q)→ (Y,R ,T ) is said to be p2-homeomorphism if f is bijection, p2-continuous and f −1 : (Y,R ,T )→ (X ,P ,Q) is p2-continuous. The bitopological spaces (X ,P ,Q) and (Y,R ,T ) are then called p2-homeomorphic. A. Tallafha et. al. [5] called p2-continuous function in Definition 5.1 as p- continuous function and p2-homeomorphism in Definition 5.2 as p-homeomorphism. Theorem 5.1. If (X ,P ,Q) and (Y,R ,T ) are bitopological spaces and f : (X ,P ,Q)→ (Y,R ,T ) be a function, then the following statements are equivalent: (i) f is p2-continuous, (ii) for each x ∈ X and each p1-open set V in Y containing f (x), there exists a p1-open set U in X containing x such that f (U)⊆ V . (iii) f −1 (V ) is p1-closed in X for every p1-closed set V in Y , (iv) for every A⊆ X , f � p1- cl (A) � ⊆ p1-cl � f (A) � , A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 334 (v) for every B ⊆ Y , p1-cl � f −1 (B) � ⊆ f −1 � p1- cl (B) � . Proof. (i) ⇔ (ii) : Let x ∈ X and V is a p1-open set in Y containing f (x). By (i), f −1 (V ) is a p1-open set in X containing x . Take U = f −1 (V ), and then f (U) = f � f −1 (V ) � ⊆ V . Conversely, let U be a p1-open set in Y and let x ∈ f −1 (U). Then f (x) ∈ U and by (ii), there exists a p1-open set V in X containing x such that f (V ) ⊆ U . Hence x ∈ V ⊆ f −1 (U) and f −1 (U) = ⋃ x∈ f −1(U) V . This shows that f −1 (U) is p1-open set in X . Thus f is p2-continuous. (i)⇔ (iii) : Let V is p1-closed set in Y . Then Y�V is p1-open set in Y . Hence f −1 (Y�V ) is p1-open set in X by (i). Since f −1 (V ) = X� f −1 (Y�V ), it follows that f −1 (V ) is p1-closed in X . The converse can be proved similarly. (iii) ⇒ (iv) : Let V be any p1-closed set in Y containing f (A). Then f −1 (V ) is a p1-closed set in X containing A by (iii). Hence, p1-cl (A) ⊆ f −1 (V ), and it follows that f � p1- cl (A) � ⊆ f � f −1 (V ) � ⊆ V . Since this is true for any p1-closed set V in Y containing f (A), we have f � p1- cl (A) � ⊆ p1-cl � f (A) � . (iv)⇒ (v) : Let B ⊆ Y and A= f −1 (B). Then f (A) ⊆ B, and by (iv), f � p1- cl (A) � ⊆ p1-cl � f (A) � ⊆ p1-cl (B). This means p1-cl (A)⊆ f −1 � f � p1- cl (A) �� ⊆ f −1 � p1- cl (B) � . Thus we have p1 − cl � f −1 (B) � ⊆ f −1 �p1- cl (B) � by substituting A= f −1 (B). A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 335 (v) ⇒ (iii) : Let V be a p1-closed set in Y . Then V = p1-cl (V ). By (v), p1- cl � f −1 (V ) � ⊆ f −1 � p1- cl (V ) � = f −1 (V ). Since f −1 (V ) ⊆ p1-cl � f −1 (V ) � , we have f −1 (V ) = p1-cl � f −1 (V ) � . Therefore f −1 (V ) is p1-closed in X . Definition 5.3. A function f : (X ,P ,Q)→ (Y,R ,T ) is said to be p2-open if f (U) is p1-open in Y for every U is p1-open in X , and p2-closed if f (V ) is p1-closed in Y for every V is p1-closed in X . Theorem 5.2. Let f : (X ,P ,Q) → (Y,R ,T ) be a p2-continuous, surjective and p2- open function. If (X ,P ,Q) is p2-Lindelöf, then (Y,R ,T ) is p2-Lindelöf. Proof. Let (X ,P ,Q) is a p2-Lindelöf space. Suppose � Gi : i ∈∆ is a p1-open cover of Y , i.e., Y ⊆ ⋃ i∈∆ Gi with Gi ∈ R ∪T . Since f : (X ,P ,Q) → (Y,R ,T ) is p2-continuous and surjective, then f −1 � Gi � ∈ P ∪Q and X = f −1 (Y )⊆ f −1 ⋃ i∈∆ Gi ! = ⋃ i∈∆ f −1 �Gi � . Hence � f −1 � Gi � : i ∈∆ is a p1-open cover of X . But (X ,P ,Q) is p2-Lindelöf, so there exists a countable subcover of X , say ¦ f −1 � Gin � : n ∈ N © such that X ⊆ ⋃ n∈N f −1 � Gin � . Accordingly, Y = f (X ) ⊆ f ⋃ n∈N f −1 � Gin � ! = ⋃ n∈N f � f −1 � Gin �� ⊆ ⋃ n∈N Gin . Thus we obtain ¦ Gin : n ∈ N © is a countable subcover of Y since f : (X ,P ,Q) → (Y,R ,T ) is a p2-open function. This shows that (Y,R ,T ) is p2-Lindelöf. Example 5.1. Consider X = {a, b, c, d} with P is discrete topology and topology Q = {;, {a} , {a, b} , {a, b, c} , X} on X , and Y = � x , y, z, w with topologies R = � ;, {x} , � y , � x , y , � y, z, w , Y and T = � ;, {x} , � y, z, w , Y on Y . Observe that P ∪Q is a discrete topology and R ∪T = � ;, {x} , � y , � x , y , � y, z, w , Y A. Kılıçman and Z. Salleh / Eur. J. Pure Appl. Math, 2 (2009), (325-337) 336 on Y . Define a function f : (X ,P ,Q) → (Y,R ,T ) by f (a) = y, f (b) = f (d) = z and f (c) = w. Thus the function f : (X ,P ,Q)→ (Y,R ,T ) is p2-continuous since the inverse image of each member of R ∪T on Y is a member of P ∪Q on X . Example 5.2. Consider X = {a, b, c, d} with P = {;, {a} , X} and topology Q = {;, {a} , {a, b} , {a, b, c} , X} on X , and Y = � x , y, z, w with topologies R = � ;, {x} , � y , � x , y , � y, z, w , Y and T = � ;, {x} , � y, z, w , Y on Y . Observe that P ∪Q = {;, {a} , {a, b} , {a, b, c} , X} and R ∪T = � ;, {x} , � y , � x , y , � y, z, w , Y on Y . Define a function g : (X ,P ,Q)→ (Y,R ,T ) by g (a) = g (b) = x , g (c) = z and g (d) = w. Thus the function g : (X ,P ,Q) → (Y,R ,T ) is not p2-continuous since � y, z, w ∈ R ∪T but its inverse image g−1 �� y, z, w � = {c, d} /∈ P ∪Q. Example 5.3. Consider a function f : (X ,P ,Q) → (Y,R ,T ) as in Example 5.1. Ob- serve that the function f : (X ,P ,Q)→ (Y,R ,T ) is not p2-open since {b} ∈ P ∪Q but f ({b}) = {z} /∈ R ∪T . Example 5.4. Consider a function g : (X ,P ,Q) → (Y,R ,T ) as in Example 5.2. Ob- serve that the function g : (X ,P ,Q)→ (Y,R ,T ) is not p2-open since {a, b, c} ∈ P ∪Q but g ({a, b, c}) = {x , z} /∈ R ∪T . Example 5.5. The function f : (X ,P ,Q) → (Y,R ,T ) in Example 5.1 is not p2- homeomorphism since f −1 : (Y,R ,T )→ (X ,P ,Q) is not p2-continuous, and the func- tion g : (X ,P ,Q)→ (Y,R ,T ) in Example 5.2 is not p2-homeomorphism since it is not p2-continuous. Recall that, a property P of sets is called topological property if whenever a topo- logical space (X ,τ) has property P, then every space homeomorphic to (X ,τ) also has property P. In the case of bitopological space (X ,P ,Q), there are three types of topological properties since now we have three types of pairwise homeomorphism. REFERENCES 337 The first two types, the reader is suggested to refer [3] for the detail. Now if p2- homeomorphism considered, we shall call such property as p2-topological property. It is very clear that, Theorem 5.2 yields the following corollary. Corollary 5.1. A p2-Lindelöf property is p2-topological property. References [1] J. C. Kelly, Bitopological Spaces, Proc. London Math. Soc., (3) 13 (1963), 71-89. [2] J. L. Kelly, General Topology, Springer-Verlag, New York, 1955. [3] A. Kılıçman and Z. Salleh, On pairwise Lindelöf Bitopological spaces, Topology & Its Appl., 154 (8) (2007), 1600-1607. [4] A. Kılıçman and Z. Salleh, Mappings and Pairwise Continuity on Pairwise Lindelöf Bitopo- logical Spaces, Albanian J. Math., 1(2) (2007), 115–120. [5] A. Kılıçman and Z. Salleh, Pairwise almost Lindelöf bitopological spaces, Journal of Malaysian Mathematical Sciences, 1(2)(2007), 227–238. [6] A. Kılıçman and Z. Salleh, Pairwise Weakly Lindelöf Bitopological Spaces, Abstract and Applied Analysis, Volume 2008, Article ID 184243, 13 pages doi:10.1155/2008/184243. [7] W. J. Pervin, Foundations of General Topology, Academic Press, Inc., London, 1964. [8] A. Tallafha, A. Al-Bsoul and A. Fora, Countable dense homogeneous bitopological spaces, Tr. J. Math., 23 (1999), 233-242 c© TÜBİTAK. [9] S. Willard, General Topology, Addison-Wesley, Canada, 1970.