L-pre-and L-semi-P-compact Spaces

: The purpose of this paper is to study a new types of compactness in the dual bitopological spaces. We shall introduce the concepts of L-pre-compactness and L- semi-P-compactness .


Introduction:
The concepts of bitopological space was initiated by Kelly [1].A set X equipped with two topologies and  2 is called a bitopological space denoted by . Navalagi [ , 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) [2].The smallest pre-closed subset of X containing A is called "pre-closure of A" and is denoted by pre-cl(A) [3]., 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-pclosed" set, the family of all semi-pclosed subsets of X is denoted by SPC(X).The smallest semi-p-closed set containing A is called semi-pclosure of A denoted by semi-pcl(A) [4].[3]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.[4]

Definition (2.1):
Let be a bitopological space and let G be a subset of X. then G is said to be:

Definition (2.2):
Let be a bitopological space and let A be a subset of X .The opposite direction of remark (2.5) is not true in general as the following example show:

Definition (2.7):
A bitopological space is said to be : 1-"L-pre-compact space " if and only if every L-pre-open cover of X has a finite sub cover.
2-"L-semi-p-compact space " if and only if every L-semi-p-open cover of X has a finite sub cover.

Remark (2.9):
The opposite direction of each case in proposition (2.8) is not true in general.As the following two examples show: 1-Let X be an infinite set with two topologies Let X=N with two topologies

Remark (2.11):
The opposite direction of each case in proposition (2.10) is not true in general.
As the following example show: Proposition (2.12):

Definition (2.15):
A bitopological space is said to be : T -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 G x  and H y  . [5]  2. "L-pre-T 2 -space" 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 T 2 -space" 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 U x  and V y  .

Remark (2.16):
An L-pre-compact subset of an Lpre -T 2 -space need not be L-preclosed.For example: , clear that A is an L-pre-compact subset of X, but it is not L-pre-closed.

Remark (2.17):
An L-semi -p-compact subset of an L-semi-p-T 2 -space need not be Lsemi-p-closed.For example: , clear that A is an L-semi-p-compact subset of X, but it is not L -semi-p-closed.

Proposition(2.20):
The L-continuous image of an Lcompact space is an L-compact.

Proof:
Suppose that is an L-

Proposition (2.22) :
The L-pre-irresolute image of an Lsemi-p-compact space is an L-precompact.

Theorem (2.23):
Let be abitopological space and let A be a subset of X.A point x in X is an L-pre-closure (Lsemi-p-closure) point of A if and only if every L-pre-neighbourhood (L-semip-neighbourhood) of x intersects A. Proof: Suppose that there exists an L-preneighbourhood (L-semi-pneighbourhood) M of x such that , that is, there exists an x  .Therefor  x  which is a contradiction hence every L-preneighbourhood (L-semi-pneighbourhood) of x must intersects A.
Conversely Assume that every L-preneighbourhood (L-semi-pneighbourhood) of x intersects A, and suppose that x is notL -pre-closure (Lsemi-p-closure) point of A,then  x  ,that is, there exists an L-pre-closed (Lsemi-p -closed) subset F of X with ,that is,there exists an element

Definition (2.25):
Let be abitopological space and let    , , , A X f be a net in X.Then f is said to be: 1-"L-pre-convergent" to a point ,then a point p of X is an L-pre-cluster(L-semi-p-cluster) point of f if and only if Assum that p is an L-pre-(L-semip-) cluster point of f and let M be an L-pre-(L-semi-p-) nhd.of p , then for each A a  , there exists an element for each A a  ,and suppose, if possible, pis not an L-pre-(L-semi-p-) cluster point of f, then there an L-pre-(L-semi-p-) nhd.Mof p and an element for every for this a which is a contradiction.Hence p must be an an L-pre-(L-semi-p-) cluster point of the net f.

Definition (2.28):
Let be a bitopological space and let Ғ be a filter on X .Apoint x inX is called 1-An" L-pre-cluster" point of Ғ if and only if each L-pre-nhd.Of x intersects every member of Ғ.
2-An" L-semi-p-cluster" point of Ғ if and only if each L-semi-p-nhd.Of x intersects every member of Ғ.

Theorem (2.29):
Let be a bitopological space and let Ғ be a filter on X .A point p inX is an L-pre-(L-semi-p-) cluster point of Ғ if and only if , for each  F Ғ.

Proof:
Suppose that p is an L-pre-(Lsemi-p-) cluster point of Ғ, then each L-pre-(L-semi-p-) nhd.M of p , F Ғ, then by theorem (2.23) every L-pre-(L-semi-p-) nhd.of p intersects F for each  F Ғ.Hence p is an L-pre-(L-semi-p-) cluster point of F.

Theorem (2.30):[6]
Let A be anon empty collection of subsets of a set X such that A has the FIP.Then there exists an ultrafilter Ғ containing A .

Theorem (2.32):
Let be a bitopological space.Then the following statements are equivalent 1-X an L-pre-(L-semi-p-) compact space , 2-Every collection of an L-pre-(Lsemi-p-) closed subsets of Xwith FIP has a non empty intersection ,and 3-Every filter on X has an L-pre-(Lsemi-p-) cluster point subsets of X also has the FIP, so by ( 2) there exists at least one point which is contradicts our supposition that  has no finite sub cover, thus A must have the FIP, it follows by theorem (2.30)that there exists an ultrafilter Ғ on X containing A .by (3) Ғ has an L-pre-(L-semi-p-) cluster point X x  , then by theorem (3.39) But X-G is an L-pre- (L-semi-p-) closed subset of X for each  G , therefore by propostion (2.24) Vol.8(4)2011 for each A a  ,so by theorem (3.37) p is an L-pre-(L-semi-p-) cluster point of f.
2] introduced the concepts of pre-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 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 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-popen subsets of X need not be a semip-open set.L-open set was studied by Al-Talkhany [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 of all Lopen subset 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 Lopen 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 *Department of Mathematics-Ibn-Al-Haitham College of Education -University of Baghdad need not be L-open set[5].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.[6]2-L-pre -and L-semi-p -compact spaces In this section we shall introduce a new typ of compactness namely L-pr -(L-semi-p-) compactness.We start with definition of L-pre-(L-semi-p-) open set.
1-" L-pre-open" set if and only if there exists a -pre-open set U such that 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 of all L-semi-P-open sub sets of X is denoted by .

1 .
By an "L-open cover of A" we mean a subcollection of the family L-O(X) which covers A . 2. By an "L -pre-open cover of A" we mean a subcollection of the family L-PO(X) which covers A.3.By an "L -semi-p-open cover of A" we mean a subcollection of the family L-SPO(X) which covers A. Remark (2.3): 1-Every L-open cover is an L-preopen.2-Every L-pre-open cover is an Lsemi-P-open.3-Every L-open cover is an L-semi-P-open.The converse of each case of remark (2.3) is not true in general as the following example shows: B is an L -semi-p-open cover, but it is neither Lpre-open nor L-open, and C is an Lpre-open cover,but it is not L-open cover.Remark (2.5): Every -pre-open( -semi-popen) cover of a sub set of a bitopological space is an L -pre-open "L-semi-p-open" respectively.
is an L -pre-open and L -semi-p-open cover, but it is neither -pre-open nor semi-p-open cover.
any function, then f is said to be: 1. "L-continuous" function if and only if the inverse image of any Lopen subset of Y is an L-open subset of X.[5] 2. "L-pre-irresolute" function if and only if the inverse image of an L-preopen subset of Y is an L-pre-open subset of X .3. "L-semi-p-irresolute" function if and only if the inverse image of an Lsemi-p-open subset of Y is an L-semip-open subset of X .
L-pre-irresolute (L-semi-pirresolute ) image of an L-pre-compact (L-semi-p-compact) space is an L-precompact (L-semi-p-compact) respectively.Proof: Suppose that is an L-pre-(L-semi-p-) irresolut and onto function and is an L-pre-(Lsemi-p-) compact space.Let be an L-pre-(L-semi-p-) open cover of ,it follows that is an L-pre-(Lsemi-p-) open cover of which is Lpre-(L-semi-p-) compact.So there are finitely many elements such that .Therefore Hence is an Lpre-(L-semi-p-) compact.
continuous and onto functionand 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.

FF
which is an L-preopen(L-semi-p -open) set.Now there is an L-pre-neighbourhood (L-semi-pneighbourhood) x must be an L-pre-(L-semi-p-) closure point of A Theorem (2.24): Let be abitopological space.Asubset A of X is an L-pre-(Lsemi-p-) closed if and only if o x in X if and only if for each L-prenhd.M of o x there exists an element L-semi-p-convergent" to a point o x in X if and only if for each Lsemi-p-nhd.M of in X.A point o x in X is called: 1-"L-pre-cluster point" of f if and only if for each A a  and for each L- 0101 pre-nhd.M of o x there exists an element a b  in A such that M f b  .2-"L-semi-p-cluster point" of f if and only if for each A a  and for each L- semi-p-nhd.M of

F
forms an L-pre-(L-semi-p-) open cover for X which is an L-pre-(L-semi-p-) compact space, then there exist fintiely many elements n a filter on X, then by remark(2.31)Ғ has the FIP, it follows that the collection 29)  x is an L-pre-(L-semi-p-) cluster point of Ғ .thus every filter on X has an L-pre-(Lsemi-p-) cluster point.3→1 Assume that every filter on X has an L-pre-(L-semi-p-) -cluster point.To show that X is an L-pre-(L-semi-p-) compact space and let  be an L-pre-(L-semi-p-) open cover of X and suppose ,if possible,  has no finite sub cover the collection A = fact that  is an L-pre-(L-semi-p-) open cover of X ,hence  must have a finite sub cover and consequently X is an L-pre-(Lsemi-p-) compact space.Theorem (2.33):Let be a bitopological space, if X is an L-pre-(L-semi-p-) compact space, then every net in X has an L-pre-(L-semi-p-) cluster point Proof: FIP , it follows by theorem(2.32)