IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 On Pairwise Semi-p-separation Axioms in Bitopological Spaces R. N.Majeed Departme nt of Mathematics,College of Education Ibn Al- Haitham, Unive rsity of Baghdad Received in : 10 October 2010 Accepte d in : 13 March 2011 Abstract In this p ap er, we define a new type of p airwise sep aration axioms called p airwise semi-p - sep aration axioms in bitop ological sp aces, also we st udy some prop erties of these sp aces and relationship s of each one with t he ordinary sep aration axioms in the bitop ological sp aces. Keywords: Bitop ological sp ace, pairwise semi-p - - sp ace, pairwise semi-p - - sp ace, p airwise semi-p - - sp ace, pairwise semi-p - sp ace, pairwise semi-p - normal sp ace. 1-Introduction The theory of bitop ological sp aces st arted with the p ap er of Kelly in [1]. A set equipp ed with two top ologies is called a bitop ological sp ace. Since then several authors continued invest igatin g such sp aces. Furt hermore, Kelly extended some of the st andard results of sep aration axioms in a top ological sp ace to a bitop ological sp ace, such extensions are p airwise regu lar, p airwise Hausdorff and p airwise nor mal, concep ts of p airwise and p airwise were introduced by M urdeshwar and Naimpally in [2]. The p urp ose of this p ap er is to introduce and invest igate the notion of p airwise semi- p - sep aration axioms in bitop ological sp aces and st udy some p rop erties of these sp aces and relationship s of each one with t he ordinary sep aration axioms in the bitop ological sp aces. 2- Preliminaries In this section, we introduce some definitions and p rop ositions, which is necessary for the p ap er. De finition 2.1[3]: A subset A of a top ological sp ace is called a pre-open set if . The comp lement of p re-open set is called pre-closed set. The family of all p re-open subsets of X is denoted by PO(X). The family of all p re-closed subsets of X is denoted by PC(X). Proposi tion 2.2 [4]: Let be a top ological sp ace, then: 1-Every op en set is a pre-op en set. 2-Every closed set is a pre-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 But the converse of (1) and (2) is not true in general. Proposi tion 2.3 [4]: The union of any family of p re-op en sets is a p re-op en set. De finition 2.4[3]: The union of all p er-open sets contained in A is called the pre-interior of A, d enoted by p re-int A. The intersection of all p re-closed sets containin g A is called the per-closure o f A, and is denoted by p re-cl A. Proposi tion 2.5 [4]: Let be a top ological sp ace and A, B be any two subsets of X, t hen: p re-cl A De finition 2.6 [4]: A subset A of a top ological sp ace is said to be semi-p-open set if and on ly if there exist s a p re-op en set in X, say U, such that The family of all semi-p -op en sets of X is denoted by S-P(X). The complement of semi-p -op en set is called semi-p-closed set. The family of all semi-p -closed sets of X is denoted by S-P-C(X). Proposi tion 2.7 [4]: 1- Every op en (closed) set is semi-p-op en (closed) set resp ectively. 2- Every p re-op en (p re-closed) set is semi-p-op en (semi-p-closed) set resp ectively. Also, t he converse of (1) and (2) is not true in gener al. Proposi tion 2.8: The union of any family of semi-p -op en sets is semi-p -op en set. Proof: Let be any family of semi-p -op en sets in X, we must p rove is a semi-p -op en set, since is semi-p -op en set, for all , which implies there exists a p re- op en set such that . Thus and from (Prop osition 2.3 and 2.5) we have a pre-open set such that Hence is a semi- p-op en set. ■ De finition 2.9 [4]: Let be a top ological sp ace and let A be any subset of X, then: 1- The union of all semi-p -op en sets contained in A is called the semi-p-interior of A, denoted by semi-p -int A. 2- The intersection of all semi-p -closed sets containing A is called the semi-p-closure of A, and denoted by semi-p -cl A. De finition 2.10 [4]: Let be a top ological sp ace and let . A subset N of X is said to be semi-p- neighborhood of x if and only if there exist s a semi-p -op en set G, such that We shall use the sy mbol nbd. instead of the word neighborhood. If N is semi-p -op en subset of X, then N is a semi-p-op en nbd of x. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Proposi tion 2.11: Let be a top ological sp ace, then every semi-p -nbd is a semi-p-op en set. Proof: Let N be any semi-p -nbds for each of its p oints, t hat is means for each , there exist s a semi-p - op en set G such that now we must prove N is a semi-p -op en set, since and since N is a semi-p - nbd for all . Thus , and fro m (Prop osition 2.8) we have N is a semi-p-op en set. ■ De finition 2.12 [1]: Let X be a non- emp ty set, let be any two top ologies on X, then is called a bitop ological sp ace. Note 2.13: In the sp ace , we shall d enote to the set of all semi-p- op en sets in ) by S-P(X, (S-P(X, )) resp ectively. De finition 2.14 [2]: A bitop ological sp ace is said to be: 1- Pairwise if for every p air of p oints x and y in X such that there exist s a -op en set containing x but not y or y but not x or a -op en set containing y but not x or x but not y . 2- Pairwise if for every p air of p oints x and y in X such that there exist s a -op en set U and a -op en set V such that De finition 2.15[1]: A bitop ological sp ace is said to be: 1- Pairwise if every two dist inct p oints in X can be sep arated by disjoint - op en set and -op en sets. 2- Pairwise regul ar spa ce, if for each p oint and each -closed set F not containin g x, there exists a -op en set U and -op en set V such that where 3- Pairwise normal space, if for each -closed set A and -closed set B such that there exist sets U and V such that U is -op en, V is -op en, 3-Pairwise semi-p-separation axioms We begin with t he definition of pairwise semi-p - - sp aces. De finition 3.1: A sp ace is called pairwise semi-p- - space if for any p air of distinct p oints x and y in X, there exists a -semi-p -op en set or -semi-p -op en set which contains one of them but not the other. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 Proposi tion 3.2: If a sp ace is p airwise - sp ace, then is p airwise semi-p - - sp ace. Proof: For any such that , we must p rove there exists a semi-p -op en in which contains one of them but not the other. Now, let in X, since is p airwise - sp ace, then there exist s op en set U in such that But from (Prop osition 2.7 p art (1)) there exists semi-p- op en set U such that Thus is p airwise semi-p - - sp ace. ■ Remark 3.3: The converse of (Prop osition 3.2 ) is not true in general, as the following example shows: Exam ple 1: Let X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { . Then, clear ly the sp ace is p airwise semi-p - - sp ace, but not p airwise - sp ace, since 2 in X but there is no op en set U or U such that 2 The orem 3.4 : For a sp ace , the following are equivalent : (1) is p airwise semi- p - - sp ace . (2) For every (3) For every the intersection of all and all is {x} . Proof: Sup p ose x≠ y in X, there exists a - semi- p- op en set U containing x but not y or a - semi- p- op en set V containin g y but not x .That means mean either or Hence for a p oint x, y .Thus {x} . Sup p ose there exist s y ≠ x such that y belongs to the intersection of all and all .Hence is not p airwise semi- p- - sp ace, implies - semi - p cl {x} which is a contradiction, thus the intersection of all and all Let x ≠ y in X, since {x} = the intersection of all and Hence, there exists either on not containin g x or a not containing x .Therefore is p airwise semi- p - - sp ace.■ IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 The orem 3.5: The product of an arbitrary family of p airwise semi - p- - sp aces is p airwise semi - p - - sp ace. Proof: Let be the p roduct of an arbitrary family of p airwise semi - p- - sp aces, where and are the product top ologies on X generated by resp ectively and X = . Let and be two distinct p oints of X. Hence for some . But is p airwise semi - p - - sp ace, therefore, there exists either a -semi-p -op en set containing but not or a -semi-p -op en set containing but not . Define and Then U is a - semi-p -op en set and V is - semi-p -op en set, also, U contains x but not y . Hence is p airwise semi - p - - sp ace. ■ De finition 3.6: A sp ace is called pa irwise semi-p- - space, if for any p air of d istinct p oints x and y in X, there exists a -semi-p -op en set U and -semi-p -op en set V such that and Proposi tion 3.7: If a sp ace is p airwise - - sp ace, then is p airwise semi-p - - sp ace. Proof: For any in X, since is p airwise - - sp ace, then there exists -op en set U and -op en set V such that and And since every op en set is semi-p -op en set ( by Prop osition 2.7 p art (1)), which imp lies U is se mi-p -op en set in containin g x but not y and V is semi-p -op en set in containing y but not x. Hence is p airwise semi-p - - sp ace. ■ Remark 3.8: The converse of (Prop osition 3.7) is not true in general as the following example shows: Consider Examp le 1, where: X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { . Then, clearly that the sp ace is p airwise semi-p - - sp ace, but not pairwise - - sp ace, since in X, but there is no -op en set containing 2 but not containin g 3 and there is no -op en set containin g 3 but not 2. The orem 3.9: The product of an arbitrary family of p airwise semi - p- - sp aces is p airwise semi - p - - sp ace. Proof: Similar to t he proof of ( Theorem 3.5). ■ IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 De finition 3.10: A sp ace is called pa irwise semi-p- - space, if for any p air of d istinct p oints x and y in X, there exists a -semi-p -op en set U and -semi-p -op en set V such that and . Proposi tion 3.11: If a sp ace is p airwise - - sp ace, then is p airwise semi-p - - sp ace. Proof: similar of the p roof of (Prop osition 3.7). ■ Remark 3.12: The converse of (Prop osition 3.11) is not true in general; consider examp le 1: X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { , clearly is p airwise semi-p - - sp ace, but not p airwise - - sp ace, since in X, but there is no two disjoint op en sets in and , which contain 2 and 3 resp ectively. The orem 3.13: For a sp ace , the following are equivalent: 1- is p airwise semi-p - - sp ace. 2- For each and for each such that , there exists a -semi-p -op en set U containin g x such that -semi-p clU. 3- For each , -semi-p clU: and U is -semi-p -op en set}. 4- The diagon al is a semi-p-closed subset of Proof: Let such that , since is p airwise semi-p - - sp ace, there exists -semi-p -op en set U and -semi-p -op en set V such that and . Hence -semi-p clU, since we h ave a se mi-p -op en set V such that , but . Sup p ose that there exists in X, such that -semi-p clU; and U is - semi-p -op en set}; imp lies -semi-p clU; for all -semi-p -op en set U, which is a contradiction, thus for each , -semi-p clU: and U is -semi-p -op en set}. To p rove is a semi-p -closed subset of , that is mean we must p rove is semi-p -op en subset of Let , which imp lies that In view of (3), there exists a -semi-p - op en set U containing x and -semi-p clU. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 We know that -semi-p clU) = . Also, we have -semi-p clU). So -semi-p cl U) . But -semi-p clU) is a - semi-p - op en set, so is a -semi-p -nbd of each of its p oints. Thus is -semi-p - closed set. Let in X, hence . Since is -semi-p -closed set, is a semi-p -nbd of each of it is p oints. Therefore, there exists a -semi-p -op en set containin g and contained in then U is -semi-p -op en set and V is -semi-p - op en set, also and , since , . Thus is p airwise semi-p - - sp ace. ■ De finition 3.14: A space is said to be pairwise semi-p-regular- space, if for each -closed set F and for each p oint , there exist - semi-p -op en set U and - semi-p -op en set V such that and , where i, j=1, 2 , . Proposi tion 3.15: Every p airwise regular sp ace is p airwise semi-p -regular- sp ace. Proof: Let F be any -closed set and let , such that , since is p airwise regu lar sp ace, there exist - op en set U and - op en set V such that and . And from (Prop osition 2.5 part (1)), we have - semi-p -op en set U and - semi-p -op en set V such that and . Hence is p airwise semi-p -regular- sp ace. ■ Remark 3.16: The converse of (Prop osition 3.15) is not true in general, as the following example shows: Let X={1, 2, 3}, , then S-P(X, = { , S-P(X, = { . Then X is p airwise semi-p-regular- sp ace, but not p airwise regular sp ace since {3} is closed set in and {3}, but for any - op en set containing 1 and for any -op en set containing {3}, its intersection is not empty. The orem 3.17: A sp ace is p airwise semi-p -regular- sp ace if and only if for each p oint x in X and every - closed set F not containing x there is a - semi-p -op en set U such that and ( Proof: Sup p ose is p airwise semi-p -regular- sp ace, let and F is any - closed set such that , imp lies is -op en set containing x and since is p airwise IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 ( 3) 2011 semi-p -regular- sp ace, there is a - semi-p -op en set U such that Hence ( Conversely , let F be any - closed set and then there exists a - semi-p -op en set U such that and ( Let V= ( then V is -semi-p -op en set such that and , thus is p airwise semi-p -regular- sp ace. ■ De finition 3.18: A sp ace is said to be pairwise semi-p-normal- space, if for each -closed set A and - closed set B disjoint from A, there exist - semi-p -op en set U and - semi-p -op en set V such that and , where i, j=1, 2 , . Proposi tion 3.19: Every p airwise normal sp ace is p airwise semi-p -normal- sp ace. Proof: Let A, B be two closed disjoint sets in (resp ectively), since X is p airwise normal sp ace, there exist - op en set U and - op en set V such that and but from (Prop osition 2.4 part (1)) U, V semi-p-op en sets which contains A and B resp ectively. Thus is p airwise semi-p -normal- sp ace. ■ Remark 3.20: The converse of Prop osition 3.19 is not true in general, as the following example shows: Consider examp le 2, where: X={1, 2, 3}, , S-P(X, = { , S-P(X, = { . Then is p airwise semi-p -normal- sp ace, but not p airwise normal sp ace, since {3} and {2} are closed disjoint sets in resp ectively but for any op en set in which containin g {3} and any op en set in which containing {2}, its intersection is not empty. Re ferences 1. Kelly , J. C. (1963), Bitop ological Sp aces, Proc. London M ath. Soc. 13: 71-89. 2. M urdeshwar, N. G. and Naimp ally, S. A. (1966), Quasi-uniform Compact Sp aces, P. Noordhoff, Groningen. 3. Nour, T. M . (1995), A note on Five Separation Axioms in Bitop ological Sp aces, Indian J. p ure app l. M ath., 26(7): 669-674. 4. Al-Kazragi, R. B. (2004), On semi-p-op en sets, M . Sc. Thesis, University of Baghdad, College of Education Ibn-Al- Haitham. رفة والتطبیقیة المجلد 2011) 3( 24مجلة ابن الهیثم للعلوم الص على الفضاءات التبولوجیة – P –حول بدیھیات الفصل شبھ الثنائیة رشا ناصر مجید جامعة بغداد ،ابن الھیثم –قسم الریاضیا ت، كلیة التربیة 2010تشرین االول 10: استلم البحث في 2011 اذار 13 :في قبل البحث خالصةال في هذا البحث قمنا بتعریف نوع جدید من بدیهیات الفصل على الفضاءات التبولوجیة الثنائیة التي اسمیناها بدیهیات كل نوع مع بدیهیات الفصل االعتیادیة في سنا بعض خواص هذه الفضاءات وعالقاتكذلك در ، P –الفصل شبه . ات التبولوجیة الثنائیةالفضاء - p–الفضاء شبه ، - p–الفضاء شبه ، - p–الفضاء التبولوجي الثنائي، الفضاء شبه :یةفتاحالكلمات الم .االعتیادي – p–الفضاء شبه القیاسي، – p–الفضاء شبه ،