مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 مجموعات مفتوحة من النوع L-pre and L-semi-p جاسم سعاد جدعان د، إبن الهیثم-كلیة التربیة ،قسم الریاضیات جامعة بغدا 2011 آب 8: قبل البحث في 2010 تشرین الثاني 24:استلم البحث في الخالصة . لمفتوحة في الفضاءات التبولوجیة الثنائیة الغرض من هذا البحث دراسة انواع جدیدة من المجموعات ا مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 L- pre-open and L-semi-p-open Sets S. G. Gasim Department of Mathematics, College of Education Ibn-Al-Haitham , University of Baghdad Received in: 24 November 2010 Accepted in : 8 February 2011 Abstract The purpose of this paper is to study new types of open sets in bitopological spaces. We shall introduce the concepts of L- pre-open and L-semi-p-open sets. Keywords : pre- open- set, semi- p- open - set, L-pre-open, L-semi-p-open 1-Introduction Navalagi [1] introduced the concepts of p re-open and semi-P-open sets. A subset A of a topological space  ,X is said to be “pre-open” set if and only if  AclA int , the family of all pre open subsets of X is denoted by PO(X).The complement of a pre-open set is called pre-closed set, the family of all pre- closed subsets of X is denoted by PC(X) [1].The smallest pre- closed subset of X containing A is called “pre-closure of A” and is denoted by pre-cl(A)[2]. Let  ,X be a topological space, a subset A of X is said to be “semi-P-open” set if and only if there exists a pre-open subset U of X such that  UclpreAU  , the family of all semi –p-open subsets of X is denoted by SPO(X).The complement of a semi-p-open set is called “semi-p-closed” set, the family of all semi-p-closed subsets of X is denoted by SPC(x). The smallest semi-p-closed set containing A is called semi-p-closure of A denoted by semi-p-cl(A)[3].[2]shows that every open set is a pre-open and the union of any family of pre- open subsets of X is a pre-open set, but the intersection of any two pre-open subsets of X need not be apre-open set.[3] shows that every pre-open set is a semi –p-open and consequentiy every open set is a semi-p-open. Also she shows that the union of any family of semi-p-open subsets of X is a semi-p-open set,but the intersection of any two semi-p-open subsets of X need not be a semi-p-open set. The concepts of bitopological space was initiated by Kelly[4].A set X equipped with two topologies and 2 is called a bitopological space denoted by . L-open set was studied by Al-swid[5], asubset G of a bitopological space is said to be “L –open” set if and only if there exists a -open set U such that  UclGU 2 ,the family of all L-open subsets of X is denoted by L-O(X).The complement of an L-open set is called “L-closed” set,the family of all L-closed subsets of X is denoted by L-C(X).In a bitopological space every -open set is an L-open set[5].The union of any family of L-open subsets of X is an L-open set, but the intersection of any two L-open subsets of X need not be L-open set[5].Al-Talkahny [6],introduces two new concepts “L- 2T -spaces” and “L-continuous functions ”. A bitopological space is مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 said to be “L- 2T -space” if and only if for each pair of distinct points x and y in X,there exists two disjoint L-open subsets G and H of X such that Gx and Hy .Let   21 ,,X ,         21 ,,Y be any bitopological spaces and let YXf : be any function, then f is said to be “L-continuous” function if and only if the inverse image of any L-open subset of Y is an L-open subset of X. 2- L-pre - open and L-semi-P- open Sets Definition 2.1 Let be a bitopological space and let G be a subset of X. then G is said to be: 1- “ L-pre-open” set if and only if there exists a -pre-open set U such that  UclGU 2 .the family of all L-pre-open sub sets of X is denoted by . 2- “ L-semi-P-open” set if and only if there exists a - semi-P-open setU such that  UclGU 2 .the family of all L- semi-P-open sub sets of X is denoted by . Remark(2.2): 1- The complement of an L-pre-open subset of a bitopological space X is called an L- pre-closed set. The family of all L- pre-closed sub sets of X is denoted by . 2- The complement of an L-semi-P-open sub set of a bitopological space X is called an L-semi-P-closed set. The family of all L- semi-P-closed sub sets of X is denoted by . Remark (2.3): In a bitopological space : 1- Every L-open set is an L- pre-open set. 2- Every L-pre-open set is an L-semi-P-open set. 3- Every L-open set is an L-semi-P-open set. The converse of each case of remark (2.3) is not true in general as the following example shows: Example (2.4): Let =the discrete topology Then                dcadbacabacbXOLXpoL ,,,,,,,,,,, مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Note that, is an L-pre-open set, but it is not L- open. And is an L-semi-P-open set but it is neither L-pre-open nor L-open. Remark(2.5) In a bitopological space : 1- Every -pre-open set is an L-pre-open set. 2- Every -semi- p-open set is an L-semi-p-open set. The opposite direction of each case in remark (2.5) is not true in general, as the following example shows: Example (2.6): Let =the indiscrete topology                  cbadcadbacabacbaXXpo ,,.,,,,,,,,,,,,,,1               dcbdcdbdaXPOXPOL ,,,,,,,,1      XPOLXSPOL  Note that,  da, is an L-pre-open set, but it is not -pre- open. And is an L-semi-P- open set but it is not –semi-P-open. Proposition (2.7): The union of any family of L -pre-open (L-semi-P-open) subsets of a bitopological space is an L -pre-open (L-semi-P-open) respectively. Proof: Let    :G be a family of L -pre-open (L-semi-P-open) subsets of X , then for each G there exists a -pre-open( -semi- p-open) set U in X such that  UGU cl   2  .So           UUGU clcl           22 .But U     is a -pre-open( -semi- p-open).Hence U     is an L -pre-open (L-semi-P-open) respectively. Remark (2.8): The intersection of any two L -pre-open (L-semi-P-open) sets need not be L -pre-open (L-semi-P-open) respectively. For example مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Let            2,1,, 4,1,4,1,, 4,3,2,1 2 1     X X X    Note that   4,3,3,1 are two L -pre-open (L-semi-P-open) sets, but      34,33,1  is neither L -pre-open nor L-semi-P-open. Definition (2.9): Let be abitopological space and let Xx ,a subset M of X is said to be : 1- An “L-pre-neighbourhood” of x if and only if there exists an L-pre-open set G such that MGx  2- An “L-semi-p- neighbourhood” of x if and only if there exists an L-semi-p-open set G such that MGx  Definition (2.10): Let be abitopological space and let A be a subset of X, then: 1- The intersection of all L-pre-closed subset of X containing A is called “L-pre- closure of A” and is dented by L-pcl(A). 2- The intersection of all L-semi-p-closed subset of X containing A is called “L-semi- p-closure of A” and is dented by L-spcl(A). Theorem (2.11): Let be abitopological space and let A be a subset of X.A point x in X is an L-pre-closure (L-semi-p-closure) point of A if and only if every L-pre- neighbourhood (L-semi-p- neighbourhood) of x intersects A. Proof: The “only if” part Assum that x is an L-pre-closure (L-semi-p-closure) of A , then x    closedpsemiLclosedpreLanisFandFAXF  : .Suppose that there exists an L-pre-neighbourhood (L-semi-p- neighbourhood) M of x such that AM  , that is, there exists an L-pre-open(L-semi-p -open) set G such that MGx  ,then such that GM cc A  ,but G c is an L-pre-closed (L-semi-p-closed) with G c x  . Therefore x  which is a contradiction hence every L-pre-neighborhood (L-semi- p- neighborhood) of x must intersect A. The “if” part Assume that every L-pre-neighborhood (L-semi-p- neighborhood) of x intersects A, and suppose that x is not L -pre-closure (L-semi-p-closure) point of A, then x  ,that is, there exists an L-pre-closed (L-semi-p -closed) subset F of X with FA  such that Fx  ,it مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 follows that Fx c  which is an L-pre-open(L-semi-p -open) set. Now there is an L-pre- neighborhood (L-semi-p- neighborhood) Fc of x with F c A  .that implies to contradiction with our assumption. Hence x must be an L-pre-(L-semi-p-) closure point of A Theorem (2.12): Let be a bi topological space. A subset A of X is an L-pre-(L-semi-p-) closed if and only if     ASPclLAPclLA  Proof: The “only if” part Suppose that     XSPCLXPCLA  and     ASPclLAPclLA  .Since     ASPclLAPclLA  , so      AASPclLAPclL  ,that is, there exists an element     ASPclLAPclLr  and Ar ,it follows that cAr which is an L-pre-(L- semi-p-) open set. Then by theorem (3.33) A c A  which is a contradiction with the fact cAA  .Hence     ASPclLAPclLA  The “if” part Assume that     ASPclLAPclLA  , but     ASPclLAPclL  is an L- pre-(L-semi-p-) closed subset of X by definition (3.32). So A is an L-pre-(L-semi-p-) closed set. Definition (2.13): A bi topological space is said to be : 1- if and only if for each pair of distinct points x and y,there are two disjoint L-pre-open subsets U and V of X such that Ux  and Vy  . 2- if and only if for each pair of distinct points x and y,there are two disjoint L-semi-p-open subsets U and V of X such that Ux  and Vy  . Proposition (2.14) 1- . 2- . 3- . Proof: follows from remark (2.3). Remark (2.15): The opposite direction of each case proposition (2.6) is not true in general. As the following two examples show: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012                             3,1,3,2,2,1 2,1,, 3,1,3,1,, 2,1,, 3,2,1 2 1 XOLXPOL XXOL X X X         Clear that is space, but it is not Clear that is space, but it is neither space nor . Definition (2.16): Let be any function, then f is said to be: 1- “L-pre-irresolute” function if and only if the inverse image of an L-pre-open subset of Y is an L-pre-open subset of X 2- “L-semi-p-irresolute” function if and only if the inverse image of an L-semi-p-open subset of Y is an L-semi-p-open subset of X. It is clear that their is no relation among the concepts of L-continuous, L-pre- irresolute and L-semi-p-irresolute function. See the following examples: Example(2.17):                                      4,3,1,4,2,3,2,4,1,3,1,4,3,2 4,2,1,3,2,1,2,1,2,1,, 2,1,2,1,, 2,1,2,1,, 4,3,2,1 2 1 XPOLXSPOL XXPOL XXOL D X X           مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 It is clear that is L-pre-(L-semi-p-)irresolute function but it is not L-continuous function. Eample(2.18): It is clear that is L-pre-(L-semi-p-)irresolute function and L-continuous function. Theorem(2.19): A bi topological space space if and only if for each pair of distinct points there exists L-pre-(L-semi-p-)irresolute function f from which is L-pre-(L-semi-p-) -space such that . Proof: "first direction" Suppos that space.If we take the identity function clear that is L-pre (L-semi-p)-irresolute function .Now let ,it follows that . "second direction" Let be an L-pre-(L-semi-p-) irresolute function and space and let ,then by ypothesis مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 . So there are L-pre(L-semi-p)-open sets Definition(2 .20): A function is called: 1. " L-pre-open" function if and only if 2. "L-semi-p-open" function if and only if 3. "L-pre-closed" function if and only if 4. "L-semi-p-closed" function if and only if . proposition(2 .21): Proof: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 References 1.Navalagi ,G.B.(2000)”Definition Bank in General Topology” which is available at Topology Atlas –Survey Articles Section. 2. Nasir ,A.I.( 2005)”some Kinds of strongly Compact and Pair- wise compact Spaces”M.SC.Thesis, University of Baghdad ,Iraq,. 3.Al-Khazaraji, R.B.(2004)” On Semi-p-open Sets”, M.Sc. Thesis, College of Education Ibn Al-Haitham, University of Baghdad,. 4. Kelly, J.C.(1963)” Bitopological spaces”, Proc.London Math.Soc.13:71-89. 5.Al-Swid, L.A.(1994)”On New Separation Axioms in Bitopological Spaces” to appear. 6.AL-Talkhany, Y.K.(2001)”Separation Axioms in Bitopological spaces “, Research submitted to college of Education Babylon University as apartial Fulfillment of the Requirement for Degree of master of science in Math.