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
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.