IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 On Pairwise Semi-p-separation Axioms in Bitopological Spaces R. N.Majeed Department of Mathematics,College of Education Ibn Al- Haitham, University of Baghdad Received in : 10 October 2010 Accepted in : 13 March 2011 Abstract In this paper, we define a new type of pairwise separation axioms called pairwise semi-p- separation axioms in bitopological spaces, also we study some properties of these spaces and relationships of each one with the ordinary separation axioms in the bitopological spaces. Keywords: Bitopological space, pairwise semi-p- - space, pairwise semi-p- - space, pairwise semi-p- - space, pairwise semi-p- space, pairwise semi-p- normal space. 1-Introduction The theory of bitopological spaces started with the paper of Kelly in [1]. A set equipped with two topologies is called a bitopological space. Since then several authors continued investigating such spaces. Furthermore, Kelly extended some of the standard results of separation axioms in a topological space to a bitopological space, such extensions are pairwise regular, pairwise Hausdorff and pairwise normal, concepts of pairwise and pairwise were introduced by Murdeshwar and Naimpally in [2]. The purpose of this paper is to introduce and investigate the notion of pairwise semi- p- separation axioms in bitopological spaces and study some properties of these spaces and relationships of each one with the ordinary separation axioms in the bitopological spaces. 2- Preliminaries In this section, we introduce some definitions and propositions, which is necessary for the paper. Definition 2.1[3]: A subset A of a topological space is called a pre-open set if . The complement of pre-open set is called pre-closed set. The family of all pre-open subsets of X is denoted by PO(X). The family of all p re-closed subsets of X is denoted by PC(X). Proposition 2.2 [4]: Let be a topological space, then: 1-Every open set is a pre-open set. 2-Every closed set is a pre-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 But the converse of (1) and (2) is not true in general. Proposition 2.3 [4]: The union of any family of pre-open sets is a pre-open set. Definition 2.4[3]: The union of all per-open sets contained in A is called the pre-interior of A, denoted by pre-int A. The intersection of all pre-closed sets containing A is called the per-closure of A, and is denoted by pre-cl A. Proposition 2.5 [4]: Let be a topological space and A, B be any two subsets of X, then: pre-cl A Definition 2.6 [4]: A subset A of a topological space is said to be semi-p-open set if and only if there exists a pre-open set in X, say U, such that The family of all semi-p-open sets of X is denoted by S-P(X). The complement of semi-p-open set is called semi-p-closed set. The family of all semi-p-closed sets of X is denoted by S-P-C(X). Proposition 2.7 [4]: 1- Every open (closed) set is semi-p-open (closed) set respectively. 2- Every pre-open (pre-closed) set is semi-p-open (semi-p-closed) set respectively. Also, the converse of (1) and (2) is not true in general. Proposition 2.8: The union of any family of semi-p-open sets is semi-p-open set. Proof: Let be any family of semi-p-open sets in X, we must prove is a semi-p-open set, since is semi-p-open set, for all , which implies there exists a pre- open set such that . Thus and from (Proposition 2.3 and 2.5) we have a pre-open set such that Hence is a semi- p-open set. ■ Definition 2.9 [4]: Let be a topological space and let A be any subset of X, then: 1- The union of all semi-p-open 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. Definition 2.10 [4]: Let be a topological space and let . A subset N of X is said to be semi-p- neighborhood of x if and only if there exists a semi-p-open set G, such that We shall use the symbol nbd. instead of the word neighborhood. If N is semi-p-open subset of X, then N is a semi-p-open nbd of x. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Proposition 2.11: Let be a topological space, then every semi-p-nbd is a semi-p-open set. Proof: Let N be any semi-p-nbds for each of its points, that is means for each , there exists a semi-p- open set G such that now we must prove N is a semi-p-open set, since and since N is a semi-p- nbd for all . Thus , and from (Proposition 2.8) we have N is a semi-p-open set. ■ Definition 2.12 [1]: Let X be a non-empty set, let be any two topologies on X, then is called a bitopological space. Note 2.13: In the space , we shall denote to the set of all semi-p- open sets in ) by S-P(X, (S-P(X, )) respectively. Definition 2.14 [2]: A bitopological space is said to be: 1- Pairwise if for every pair of points x and y in X such that there exists a -open set containing x but not y or y but not x or a -open set containing y but not x or x but not y. 2- Pairwise if for every pair of points x and y in X such that there exists a -open set U and a -open set V such that Definition 2.15[1]: A bitopological space is said to be: 1- Pairwise if every two distinct points in X can be separated by disjoint - open set and -open sets. 2- Pairwise regular space, if for each point and each -closed set F not containing x, there exists a -open set U and -open 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 -open, V is -open, 3-Pairwise semi-p-separation axioms We begin with the definition of pairwise semi-p- - spaces. Definition 3.1: A space is called pairwise semi-p- - space if for any pair of distinct points x and y in X, there exists a -semi-p-open set or -semi-p-open set which contains one of them but not the other. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Proposition 3.2: If a space is pairwise - space, then is pairwise semi-p- - space. Proof: For any such that , we must prove there exists a semi-p-open in which contains one of them but not the other. Now, let in X, since is pairwise - space, then there exists open set U in such that But from (Proposition 2.7 part (1)) there exists semi-p- open set U such that Thus is pairwise semi-p- - space. ■ Remark 3.3: The converse of (Proposition 3.2 ) is not true in general, as the following example shows: Example 1: Let X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { . Then, clearly the space is pairwise semi-p- - space, but not pairwise - space, since 2 in X but there is no open set U or U such that 2 Theorem 3.4 : For a space , the following are equivalent : (1) is pairwise semi-p - - space . (2) For every (3) For every the intersection of all and all is {x}. Proof: Suppose x≠ y in X, there exists a -semi-p-open set U containing x but not y or a - semi-p-open set V containing y but not x .That means mean either or Hence for a point x, y .Thus {x} . Suppose there exists y ≠ x such that y belongs to the intersection of all and all .Hence is not pairwise semi-p- - space, implies - semi - pcl {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 containing x or a not containing x .Therefore is pairwise semi-p - - space.■ IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Theorem 3.5: The product of an arbitrary family of pairwise semi -p- - spaces is pairwise semi - p - - space. Proof: Let be the product of an arbitrary family of pairwise semi -p- - spaces, where and are the product topologies on X generated by respectively and X = . Let and be two distinct points of X. Hence for some . But is pairwise semi -p - - space, therefore, there exists either a -semi-p-open set containing but not or a -semi-p-open set containing but not . Define and Then U is a - semi-p-open set and V is - semi-p-open set, also, U contains x but not y. Hence is pairwise semi - p - - space. ■ Definition 3.6: A space is called pairwise semi-p- - space, if for any pair of distinct points x and y in X, there exists a -semi-p-open set U and -semi-p-open set V such that and Proposition 3.7: If a space is pairwise - - space, then is pairwise semi-p- - space. Proof: For any in X, since is pairwise - - space, then there exists -open set U and -open set V such that and And since every open set is semi-p-open set ( by Proposition 2.7 part (1)), which implies U is semi-p-open set in containing x but not y and V is semi-p-open set in containing y but not x. Hence is pairwise semi-p- - space. ■ Remark 3.8: The converse of (Proposition 3.7) is not true in general as the following example shows: Consider Example 1, where: X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { . Then, clearly that the space is pairwise semi-p- - space, but not pairwise - - space, since in X, but there is no -open set containing 2 but not containing 3 and there is no -open set containing 3 but not 2. Theorem 3.9: The product of an arbitrary family of pairwise semi -p- - spaces is pairwise semi - p - - space. Proof: Similar to the proof of ( Theorem 3.5). ■ IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 Definition 3.10: A space is called pairwise semi-p- - space, if for any pair of distinct points x and y in X, there exists a -semi-p-open set U and -semi-p-open set V such that and . Proposition 3.11: If a space is pairwise - - space, then is pairwise semi-p- - space. Proof: similar of the proof of (Proposition 3.7). ■ Remark 3.12: The converse of (Proposition 3.11) is not true in general; consider example 1: X={1, 2, 3}, , PO(X, = S-P(X, = { , PO(X, = S-P(X, = { , clearly is pairwise semi-p- - space, but not pairwise - - space, since in X, but there is no two disjoint open sets in and , which contain 2 and 3 respectively. Theorem 3.13: For a space , the following are equivalent: 1- is pairwise semi-p- - space. 2- For each and for each such that , there exists a -semi-p-open set U containing x such that -semi-pclU. 3- For each , -semi-pclU: and U is -semi-p-open set}. 4- The diagonal is a semi-p-closed subset of Proof: Let such that , since is pairwise semi-p- - space, there exists -semi-p-open set U and -semi-p-open set V such that and . Hence -semi-pclU, since we have a semi-p-open set V such that , but . Suppose that there exists in X, such that -semi-pclU; and U is - semi-p-open set}; implies -semi-pclU; for all -semi-p-open set U, which is a contradiction, thus for each , -semi-pclU: and U is -semi-p-open set}. To prove is a semi-p-closed subset of , that is mean we must prove is semi-p-open subset of Let , which implies that In view of (3), there exists a -semi-p- open set U containing x and -semi-pclU. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 We know that -semi-pclU) = . Also, we have -semi-pclU). So -semi-pcl U) . But -semi-pclU) is a - semi-p- open set, so is a -semi-p-nbd of each of its points. 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 points. Therefore, there exists a -semi-p-open set containing and contained in then U is -semi-p-open set and V is -semi-p- open set, also and , since , . Thus is pairwise semi-p- - space. ■ Definition 3.14: A space is said to be pairwise semi-p-regular- space, if for each -closed set F and for each point , there exist - semi-p-open set U and - semi-p-open set V such that and , where i, j=1, 2 , . Proposition 3.15: Every pairwise regular space is pairwise semi-p-regular- space. Proof: Let F be any -closed set and let , such that , since is pairwise regular space, there exist - open set U and - open set V such that and . And from (Proposition 2.5 part (1)), we have - semi-p-open set U and - semi-p-open set V such that and . Hence is pairwise semi-p-regular- space. ■ Remark 3.16: The converse of (Proposition 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 pairwise semi-p-regular- space, but not pairwise regular space since {3} is closed set in and {3}, but for any - open set containing 1 and for any -open set containing {3}, its intersection is not empty. Theorem 3.17: A space is pairwise semi-p-regular- space if and only if for each point x in X and every - closed set F not containing x there is a - semi-p-open set U such that and ( Proof: Suppose is pairwise semi-p-regular- space, let and F is any - closed set such that , implies is -open set containing x and since is pairwise IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (3) 2011 semi-p-regular- space, there is a - semi-p-open set U such that Hence ( Conversely, let F be any - closed set and then there exists a - semi-p-open set U such that and ( Let V= ( then V is -semi-p-open set such that and , thus is pairwise semi-p-regular- space. ■ Definition 3.18: A space 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-open set U and - semi-p-open set V such that and , where i, j=1, 2 , . Proposition 3.19: Every pairwise normal space is pairwise semi-p-normal- space. Proof: Let A, B be two closed disjoint sets in (respectively), since X is pairwise normal space, there exist - open set U and - open set V such that and but from (Proposition 2.4 part (1)) U, V semi-p-open sets which contains A and B respectively. Thus is pairwise semi-p-normal- space. ■ Remark 3.20: The converse of Proposition 3.19 is not true in general, as the following example shows: Consider example 2, where: X={1, 2, 3}, , S-P(X, = { , S-P(X, = { . Then is pairwise semi-p-normal- space, but not pairwise normal space, since {3} and {2} are closed disjoint sets in respectively but for any open set in which containing {3} and any open set in which containing {2}, its intersection is not empty. References 1. Kelly, J. C. (1963), Bitopological Spaces, Proc. London Math. Soc. 13: 71-89. 2. Murdeshwar, N. G. and Naimpally, S. A. (1966), Quasi-uniform Compact Spaces, P. Noordhoff, Groningen. 3. Nour, T. M. (1995), A note on Five Separation Axioms in Bitopological Spaces, Indian J. pure appl. Math., 26(7): 669-674. 4. Al-Kazragi, R. B. (2004), On semi-p-open sets, M. Sc. Thesis, University of Baghdad, College of Education Ibn-Al- Haitham. 2011) 3( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد 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 ،