Strongly C-Lindelof Spaces

In this paper, we define another type of Lindelof which is called strongly cLindelof, and we introduce some properties about this type of Lindelof and the relationships with Lindelof , cLindelof and strongly Lindelof spaces.


Introduction:
A topological space ) , (  X is said to be Lindelof space if and only if every open cover of X has a countable subcover [1]

The intersection of all preclosed sets in X
which contain A is called the preclosure of A and denoted by A cl -pre .3. The prederived set of A is the set of all elements x of X satisfies the condition, that for every  X is called strongly Lindelof space if and only if every preopen cover of X has a countable subcover [2].In this paper, we introduce the concept of strongly c- y regularity of X ,there are two open sets  X is called strongly c-Lindelof space if and only if for every preclosed set study some properties of this kind of Lindelof space.We also study the relationships among Lindelof spaces, c-Lindelof spaces, strongly Lindelof spaces and strongly c-Lindelof spaces.

Remark
Every strongly Lindelof space is Lindelof space .Proof: Let ) , (  X be a strongly Lindelof space and let   X which is strongly Lindelof space.Therefore,there exists a countable number of Hence every open cover of X has a countable subcover , therefore X is Lindelof space.Remark Every Lindelof space is c-Lindelof space .
Proof: Let ) , (  X be a Lindelof space and let Therefore,there exists a countable number of

Definition [1]
A topological space X is a regular space if and only if whenever A is closed in X and A x  ,then there are disjoint open sets U and V with U x  and V A  .A space X is said to be a for X which has no countable subcover .Since X is c-Lindelof , then there is a countable subfamily

Proposition
Every strongly c-Lindelof and 3 T -space is strongly Lindelof space .Proof: Let ) , (  X be a 3 T strongly c-Lindelof space.Assume X is not strongly Lindelof space , then there is a preopen cover       : U for X which has no countable subcover.Since X is strongly c-Lindelof , then there is a countable subfamily .Then X is strongly Lindelof.Note : From remark (1.1) and remark (1.4) we have every strongly Lindelof space is a c-Lindelof space.

Theorem [2]
If the set of accumulation points of the space X is finite ,then X is strongly Lindelof ,whenever it is Lindelof space.

Theorem
Every c-Lindelof and 3 T -space is strongly Lindelof space, whenever the set of accumulation points of T -c-Lindelof space such that the set of accumulation points of X is finite.Remark (1.3)  gives X is Lindelof and theorem (1.5) gives X is strongly Lindelof.

Strongly C-Lindelof Spaces:
In this section, we give the definition of strongly c-Lindelof, and we also study the relationships among Lindelof spaces, c-Lindelof spaces, strongly Lindelof spaces and strongly c-Lindelof spaces.

Definition
A topological space ) , (  X is called strongly c-Lindelof space if and only if for every preclosed set Every strongly Lindelof space is strongly c-Lindelof space .
Proof: Let ) , (  X be a strongly Lindelof space and let X A  be any preclosed subset of X .Let is a preopen cover of X which is strongly Lindelof space .Therefore,there exists a countable number of which is a contradiction.Therefore X is c- Lindelof space.

Proposition
In a 3 T -space X ,if the set of accumulation points of X is finite, then the concepts of c- Lindelof and strongly c-Lindelof are coincident.

Proof:
If X is strongly c-Lindelof space then by proposition (2.6) it is c-Lindelof.Conversely, if T -c-Lindelof space, then by remark 3),it is Lindelof,and since the set of accumulation points of X is finite, then by proposition (2.5) it is strongly c-Lindelof space. .

Corollary
Every strongly c-Lindelof and 3 T -space is Lindelof space .Proof: T -strongly c-Lindelof space, then by proposition (2.3), X is strongly Lindelof,and by remark (1.4), X is Lindelof.Proposition If the set of accumulation points of the space X is finite ,then X is strongly c-Lindelof space whenever it is a Lindelof space.

Proof:
Let X be a Lindelof space such that the set of accumulation points of X is finite, then by theorem (1.5), X is strongly Lindelof,and by remark (2.2),it is strongly c-Lindelof space.

Proposition
Every strongly c-Lindelof space is c-Lindelof space .Proof: Let X be a strongly c-Lindelof space, to prove it is c-Lindelof.If not, then there is a closed set

 3 T
space if and only if it is regular and  1 T space.Remark Every c-Lindelof and 3 T -space is Lindelof space .Proof: Let ) , (  X be a 3T -c-Lindelof space.Assume X is not Lindelof space , then there is an open cover