2011) 1( 24المجلد بیقیةالھیثم للعلوم الصرفة والتط بنا مجلة -L فضاءات الرص من النوع سعاد جدعان جامعة بغداد ،ابن الهیثم -كلیة التربیة ،قسم الریاضیات 2010 ایار 25 في البحث استلم 2010 ایلول 27 في البحث قبل الخالصة ـة أنــواع جدیــدة مــن التــراص فــي الفضــاءات التبلوجیــه الثنائیــة ســنقدم التــراص مـــن أذ ،الغــرض مــن هــذا البحــث دراسـ ال -النوع IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 L- compact Spaces S. Gedaan Department of Mathematics, Ibn-Al-Haitham College of Education, University of Baghdad Received in May,25,2010 Accepted in S ept,27,2010 Abstract The purpose of this paper is to study a new types of compactness in bitopological spaces. We shall introduce the concepts of L- compactness. Introduction The concept of bitopological space was initiated by Kelly[1].A set X equipped with two Topologies and 2 is called a bitopological space denoted by . By a directed set we mean a pair (A, ) consisting of a non-empty set A and a binary relation  defined on A and satisfies the following conditions: (1) a  a for each a  A. (2) If a  b and b  c, then a  c for each a, b, and c in A. (3) For each two members a and b of A, there exists a member c  A such that c  a and c b. If (A, ) is a directed set and f is a function of A into a non-empty set X, then f is called a "net" in X and is denoted by (f, X, A, ). The image of a  A under f is denoted by fa and a net in X will be sometimes denoted by {fa: a  A}.[2] A "filter" on a non-empty set X is a non-empty family F of subsets of X with the following properties: (1)   F. (2) If F  F and F  H, then H  F. (3) If F  F and H  F, then F  H  F. A filter on a non-empty set is said to be an ultrafilter if and only if it is not properly contained in any other filter on this set.[2] IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 L-open set was studied by Al-swid[2], 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[3].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[2].Al-Talkahny [3],introduced two new concepts “L- 2T -spaces” and “L-continuous functions ”. A bitopological space is said to be “L- 2T -space” if and only if for each pair of distinct points x and y in X,there exist two disjoint L-open subset 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-compactness Definition(2.1) Let be a bitopological space and let A be a subset of X. By an “L-open cover of A” we mean a subcollection of the family L-O(X) which covers A. Remark(2.2): Every -open cover in a bitopological space is an L-open cover. The converse of remark (2.2) is not true in general as the following example shows: Example (2.3)             4,3,2,,F 1,, 3,2,1,3,2,1,, 4,3,2,1 2 2 1      X X X X                         4,3,2,3,1,4,3,1,2,1,4,1,4,2,1,3,2,1,3,2,1,,XXOL  L et     4,3,2,1C , note that C is an L-open cover of X, but it is not -open cover. Definition(2.4) A bitopological space is said to be “L-compact space” if and only if every L- open cover of X has a finit subcover. Proposition (2.5) IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 If a bitopological space is an L-compact space, then   1 ,X is a compact space. Proof: follows from remark (2.2). Remark (2.6) The opposite direction of proposition (2.5) is not true in general, as the following example shows: Let X and let ox   ox,,1   I2 =The indiscrete topology     UorUxUXOL o; Note that  1,  is compact but  21 ,,  is not L-compact. Proposition (2.7) An L-closed subset of an L-compact space is L-compact. Proof: Let A be an L-closed subset of an L-compact space and let    :G be an L-open cover of A .Then   cAG   : forms an L-open cover of X which is L- compact space. So there are finitely many elements n ,,, 21  such that c n i AGX i  1   ,it follows that  n i i GA 1   .Hence A is an L-compact. Corollary (2.8) An L-closed subset of an L-compact space is -compact. Proof: Follows from proposition (2.7) and (2.5). Corollary (2.9) A -closed subset of an L-compact space is L-compact. Proof: Since every -closed set is an L-closed set and by proposition (2.7). Corollary (2.10) A -closed subset of an L-compact space is -compact. IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Proof: Follows from corollary(2.9) and proposition (2.5). Proposition(2.11) The L-continuous image of an L-compact space is an L-compact. Proof: Suppose that is an L-continuous and onto function and is an L-compact space. Let be an L-open cover of , it follows that is an L-open cover of which is L-compact.So there are finitely many elements such that .Therefore ,hence is an L-compact. Corollary (2.12) Let be an L-continuous function, then is a compact subset of for each L-compact subset of . Proof: Follows from propositions (2.11) and (2.5). It is known that every compact subset of any 2T -space is closed. If we change the concepts of compact, 2T and closed by the concepts L-compact- 2T and L-closed, then this fact being invalid in general, as the following example shows: Example (2.13)          I X X       2 1 2,1,2,1,, 3,2,1             3,2,3,1,2,1,2,1,,XXOL              1,2,3,3,1,3,2,,XXCL  . Clear that X is an L- 2T -space. If  2,1A ,then A is an L-compact subset of X, but it is not L-closed. Definition (2.14): [3] Let be a bitopological space and let A be a subset of X, Xx .Then A is called an IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 L-neighborhood of x if and only if there is an L-open set G in X such that AGx  . Definition (2.15) [3] Let be a bitopological space and let A be a subset of X.The intersection of all L-closed set containing A is called “L-closure of A”denoted by L-cl(A). Theorem (2.16) [4] L et be a bitopological space and let A be a subset of X. A point x in X is an L-closure point of A if and only if every L-open neighborhood of x intersects A. Definition (2.17) [4] Let be a bitopological space and let  ,,, AXf be a net in X ,then f is said to be “L-convergent ”to a point ox in X if and only if for each L-open neighborhood N of ox ,there exists an element Aao such that Nfa  for each oaa  . Definition (2.18) [4] Let be a bitopological space and let  ,,, AXf be a net in X. A point ox in X is called an “L-cluster point of f” if and only if for each Aa and for each L-open neighborhood N of ox ,there exists an element ab  in A such that Nfb . Theorem (2.19) [4] Let be a bitopological space and let  ,,, AXf be a net in X. For each Aa let   AinaxxfM  : ,then a point p of X is an L-cluster point of f if and only if  MclLp  for each Aa . Definition (2.20) Let be a bitopological space and let Ғ be a filter on X . A point x in X is called an “L-cluster point of Ғ” if and only if each L-open neighborhood of x intersects every member of Ғ. Theorem (2.21) Let be a bitopological space and let Ғ be a filter on X . A point p in X is an L- cluster point of Ғ if and only if  FclLp  for each F Ғ. Proof: the “first direction” Suppose that p is an L-cluster point of Ғ. then for each L-open neighborhood G of p , FG for each F Ғ, it follows by theorem (2.16) that  FclLp  for each F Ғ. IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 The “second direction” Assume that  FclLp  for each F Ғ, then by theorem (2.16) every L-open neighborhood of p intersects F for each F Ғ. Hence p is an L-cluster point of Ғ Definition (2.22) [2] A collection of sets is said to have the finite intersection property (FIP) if and only if the intersection of each finite subcollection of it is non empty. Remark (2.23) [2] Every filter in a non- empty set X has the FIP. Theorem (2.24) [3] Let A be a non empty collection of subsets of a set X such that A has the FIP. Then there exists an ultra filter Ғ containing A . Proposition (2.25) [4] Let A be a subset of a bitopological space . Then A is an L-closed set if and only if  AclLA  . Theorem (2.26) Let be a bitopological space. Then the following statements are equivalent: 1- X is an L-compact space, 2- Every collection of L-closed subsets of X with the FIP has anon empty intersection, and 3- Every filter on X has an L-cluster point. Proof: 1→2 Let    :F be a collection of L-closed subset of X with the FIP. suppose that      F , it follows by De-Morgan Laws that XF c      therefore         :cF forms an L-open cover for X which is an L-compact space, then there exists finitely many elements n ,,, 21  such that XF n i c i    1  . Again by De-Morgan Laws we have that     n i i F 1 which is a contradiction since    :F has the FIP. Hence      F IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 2→3 Let Ғ be a filter on X, then by remark (2.23) Ғ has the FIP, it follows that the collection    FFclL : Ғ }of L-closed subsets of X also has the FIP, so by (2) there exists at least one point   :  FFclLx  Ғ } then by theorem (2.21) x is an L-cluster point of Ғ . Thus every filter on X has an L-cluster point. 3→1 Assume that every filter on X has an L-cluster point and let  be an L-open cover of X. suppose ,if possible, has no finite sub cover the collection   :  GGX has the FIP, for if there is a finite sub collection   1: niGX i  of such that    1: niGX i this implies that   XniGi  1: which contradicts our supposition that has no finite sub cover, thus must have the FIP, it follows by theorem (2.24)that there exists an ultra filter Ғ on X containing .by (3) Ғ has an L-cluster point Xx , then by theorem (2.21 )  FclLx  for each F Ғ, in particular  GXclLx  for each G . But X-G is an L-closed subset of X for each G , therefore by propostion (2.25)   GXGXclL  for every G . This implies   GGXx : }, so by De-Morgen Laws   GGXx : },that is,   GGx : },which is a contradiction with the fact that  is an L-open cover of X ,hence  must have a finite sub cover and consequently X is an L-compact space. Proposition (2.27): Let be a bitopological space. If X is an L-compact space, then every net in X has an L-cluster point. Proof: let  ,,, AXf be a net in X . for each Aa let  AinaxK f xa  : . Since A is directed by  ,so the collection  AaKa : has the FIP. Hence   AaKclL a  : also has the FIP , it follows by theorem (2.26)      Aa aKclL let   Aa aKclLp   , then  aKclLp  for each Aa ,so by theorem (2.19) p is an L-cluster point of f. Refrences 1. Kelly, J. C. (1963)” Bitopological spaces”, Proc.London Math.Soc.13, p.p.71-89. 2. Sharma, L.J.N. (2000)”Topology”,Krishna Prakashan Media (P) Ltd, India, Twenty Fifth Edition. 3. 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.,. 4. AL-Khafaji, A.H. (2005) “On L-Proper Actions”, M.Sc. Thesis, University of AL- Mustansiriyah .