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In this paper, we have studied g-semi-open sets in generalized topological space. We have obtained some significant properties of g-semi-open sets and constructed various examples.

To study g-semi-open sets in generalized topological space and obtain some significant properties of g-semi-open sets and construct some examples.

First we recall the definition of generalized topological space, g-open sets and g-closed sets. Definition 2.1 [1] Let X be a non empty set and letτ_g be a family of subsets of X. Then τ_g is said to be a generalized topology on X, if following two conditions are satisfied viz.: ϕ∈τ_g; Arbitrary union of members of τ_g is a member of τ_g. The generalized topology τ_g is said to be strongif X∈τ_g, and the pair (X,τ_g) is called a generalized topological space. The members offamily τ_g are called g-open sets and their complements are called g-closed sets. From the above Definition 2.1 we observe that every topological space is a generalized topological space but the converse is not true. We have following Example. Example 2.1:Let X={a,b,c} and let τ_g={∅,X,{a,b},{b,c} }. Then τ_g is a generalized topology but is not a topology on X. Proposition 2.2[4]: Let (X,τ_g ) be a generalized topological space. Then ϕ and X are g-closed sets in X. Arbitrary intersection of g-closed sets is a g-closed set. Proposition 2.3: Let (X,τ_g ) be a generalized topological space and let A⊆X. Then A is g-open set in X iff for each point x∈A there exists a g-open set U in X such that x∈U⊆A . Corollary 2.4: Let (X,τ_g ) be a generalized topological space and let A⊆X. Then A is g-closed set in X iff for each point x∉A there exists a g-open set U in X such thatx∈U andU A =ϕ. Definition 2.5 [1]: LetXbe a generalized topological space and letA⊆X. Then g-interior of A is denoted by i_g(A) and is defined to be the union of all g-open sets contained in A. Theg-closure of A is denoted by c_g (A) and is defined to be the Intersection of all g-closed sets containing A. Remark:Since arbitrary union of g-open sets is a g-open set and arbitrary intersection of g-closed sets is a g-closed set, it follows that i_g (A)is a g-open set and c_g (A) is a g-closed set. Thus〖 i〗_g (A)isthelargestg-opensetcontained in A and c_g (A) is the smallest g-closed set containing A. Proposition 2.6 [4]: Let (X,τ_g )be a generalized topological space andletA ⊆X. Then Aisg-opensetiffi_g (A)=A. Aisg-closedsetiffc_g (A)=A. Theorem 2.7 [4]: Let(X,τ_g )be a generalized topological space and let A, B be subsets of X. Then following properties holds: i_g (ϕ)=ϕ,i_g (X)=X. If A⊆B then i_g (A)⊆i_g (B). i_g (A)∪i_g (B)⊆i_g (A∪B). i_g (A∩B)⊆i_g (A)∩i_g (B) . Theorem2.8[4] : Let(X,τ_g ) be a generalized topological space and let A, B be subsets of X. Then following properties holds: c_g (ϕ)=ϕ, c_g (X)=X. If A⊆B then c_g (A)⊆c_g (B). c_g (A)∪c_g (B)⊆c_g (A∪B). c_g (A∩B)⊆c_g (A)∩c_g (B). c_g (c_g (A) )=c_g (A). Theorem 2.9 [4]: Let (X,τ_g ) be a generalized topological space and {A_∝ }_(∝∈Λ) be a family of subsets of X. Then i_g (i_g (A) )=i_g (A). ⋃_(∝∈Λ)▒i_g 〖(A〗_∝)⊆i_g (⋃_(∝∈Λ)▒A_∝ ). i_g (⋂_(∝∈Λ)▒A_∝ )⊆ ⋂_(∝∈Λ)▒〖i_(g ) (A_∝)〗. Theorem 2.10 [4]:Let (X,τ_g ) be a generalized topological space and {A_∝ }_(∝∈Λ)be a family of subsets of X. Then (i) ⋃_(∝∈Λ)▒c_g 〖(A〗_∝)⊆c_g (⋃_(∝∈Λ)▒A_∝ ). (ii) c_g (⋂_(∝∈Λ)▒A_∝ )⊆ ⋂_(∝∈Λ)▒〖c_(g ) (A_∝)〗. Theorem 2.11 [4]:Let (X,τ_g ) be a generalized topological space and A⊆X. Then i_g (X-A)= X-c_g (A). c_g (X-A)= X-i_g (A).

g-Semi Open Sets
In this section we have obtained significant properties of g-semi open sets. Further we have constructed some useful examples.
Definition 3.1[2] : Let X be a generalized topological space and let A .Then the set A is said to be g-semi-open set, if .
Remark: The empty set  and whole set X are always g-semi-open set in any generalized topological space .
Proposition 3.2: Let X be a generalized topological space. If A is ag-open set in X then A is g-semi-open set.
Proof: Let X be a generalized  topological space and A  X. Suppose A is ag-open set in X. Then . Since , we have, . Hence A is g-semi-open set in X
However  the converse of above Proposition 3.2 is not necessarily true. In the following example we see that A is a g-semi-open set but A is not g-open set in X
Example 3.3: Let  and let consider generalized topology  on X. Suppose . Then we see that A is a g-semi-open set in X but not g-open set  in X.
Remark:Ina generalized topological space  if A is non empty g-semi-open subset of X, then is also a non empty subset of X.
Proposition 3.4:Let be a generalized  topological spaceandA . Then A is g-semi-open set iff  .
Proof: Let A be a g-semi-open set in X . Then we have This implies ) =  Since we have  Hence we find that  .
Conversely, suppose that . Since  we have  Thus A is g-semi-open set in X.Theorem 3.5 :  Let X be a generalized topological space and .Then A is g-semi open set if and only if there exist a g-open set U in X such that .
Proof. Let A be a g- semi open set in X .Then we have Suppose Then  is a g-open set in X and .Since ,we have  .Hence we deduce that .
Conversely suppose there exist a g-open set U in X such that .This implies ,and therefore .Then by  and ,we find that . Hence A is g-semi open set in X.Theorem 3.6:Let X be a generalized topological space and let  be a collection of g-semi-open sets in X. Then  is a g-semi-open set in X.
Proof: Let X be a generalized topological space and let  be a collection of g-semi-open sets in X. Then, , for all .  Put  . Wehave
⊇  Thus we conclude that . Hence A is a g-semi-open set in X.
In the following Example we see that intersection of two g-semi-open sets may not  a g-semi-open set
Example 3.7:  Let  and let us consider generalized topology   on X. Suppose  and . Then we see that A and B are g-semi-open sets in X but their intersection is {b}, which is not a  g-semi-open set in X.
Remark:In the above Theorem 3.6 it has been proved that arbitrary union of  g-semi-open sets is a g-semi-open set. Further in above Example 3.7 it is shown that intersection of two g-semi-open sets may not a g-semi-open set .Thus the collection of g-open sets in a generalized topological space form a generalized topology on X and this collection is finer than .
Theorem 3.8 : Let X be a generalized topological space and .Then A is g-semi open set if and only if  for each x there exists a g-semi open set U in X such that  x
Proof. Let A be a g-semi open set in X.Then clearly for any point x   there exist a g-semi open set viz.,A itself satisfying the desired condition.
Conversely suppose  having the property that for each x   there exists a g-semi open set U_x in X such that  Clearly we have . From Theorem 3.6we note that arbitary union of g-semi open sets is g-semi open, therefore A is a g-semi open set in X. 

Generalized topological spaces,g-interior,g-closure, g-semi open set.

In topology g-semi-open sets in generalized topological space were studied and some significant properties of g-semi-open sets were obtained. Also some examples were constucted.

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