ABSTRACT

The main aim of this work is to expand and study some types of topological spaces  by -open sets .

In this work,we extend these concepts by using -open sets to new definitions for -connected space, -compact space,countably ωb-compact, -cluser Point, -lindelof space,then we study the relations between the above mentioned with other concepts like   – , – , -regular, -normal,During the work,some important and new concepts have been illustrated including nearly -compact,nearly -lindelof in addition studing the behavior of  these qualities under the in-fluence of certain types of functions we also dealt with the concepts of -closed, -open functions, -continuou

the properties of these functions .

the following are among our main results:

1- Let  be a bijective function  .

i-  If f is b-open and X is -space then Y is -space.

ii- If f is b-continuous and Y is -space then X is -space .

2- The door space is –   if and only if it is – .

3- The door space is –  if and only if it is –

4- Let X be topological space, then the following statements are.equivalent:

i- X is -compact.

ii- Every maximal filterbase -converges to some points of  X.

iii- Every filterbase -accumulates at some points of .X.

5- A topological space X is ωb-compact if and only if each net in X, has  at least one ωb-cluster point .

6- Let  be an.almost contra-ωb-continuous, onto the following  statement

 are equivalent

i- if X is ωb-compact, then Y is S-closed.

ii- if X is ωb-compact, then Y is S-Lindelof  .

iii- if X is countably ωb-compact,thenY is countably S-closed.

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