For a nonempty set X. a subfamily µ ⊆ p(X) is said to be a generalized
topological space (briefly, GT) on X if Ø ∈ µ and unions of elements of µ belong to µ. A nonempty set X with a GT µ on X is said to be a generalized
topological space (briefly, GTS) and is denoted by (X, µ). Let (X, µ) be a GTS
and S ⊆ X. then S is said to be µ-dense if cµ(S) = X. (X, µ) is said to be
submaximal if each µ-dense subset of (X, µ) is a µ-open set.
A GTS (X, µ) is called µ-extremally disconnected or simply, extremally disconnected if the µ-closure of every µ-open set is µ-open.
A GTS(X, µ) is extremally disconnected iff for µ-open sets U and V with
U ∩V = Ø, we have cµ (U) ∩ cµ (V) = Ø.
A GTS (X,µ) is extremally disconnected iff for every µ-open set G and µ-closed set F with G⊆F, there exist a µ- open set G₁ and a µ- closed set F₁ such that G⊆F₁⊆G₁⊆F. A GTS (X,µ) is said to be σ-connected( resp, π-connected, α-connected, β-connected) if (X ,σ(µ)) (resp,(X,π(µ)), (X,α(µ)), (X,β(µ))) is connected. we show that if X is extremally disconnected then the three abave concepts are equivalen.

In this thesis we study two class of topological spaces which is called strongly compact and strongly lindelof .
The topological space (X,T) is called strongly compact if every preopen cover of X has a finite subcover. And also it is called strongly lindelof if every preopen of X has a countable subcover.
In the following, we charactrize these spaces and after study the relationship between this two class of spaces, we charactrize the maximal elements of these spaces. We also show that the topological space (X,T) is semi compact if and only if is S-closed and satisfy in the (C) condition. And if (X,T) is hausdorff and strongly lindelof space then X is a maximal strongly lindelof space if and only if X is P-space.

We obtain some properties of the KC-spaces and using them generalize theorems concerning T1-complement of hausdorff spaces to KC-spaces. We also introduce minimal KC-spaces and Katetov KC-spaces and provide the necessary and sufficient condition that a countable KC-space is a Katetov-KC. We will also show that every minimal KC-topology on a countable set is compact