SLIGHTLY *g-CONTINUOUS FUNCTIONS

In this paper we introduce slightly *gα-contoinuos function and investigated the properties of slightly *gα-continous functions. By utilizing *gα-open sets, we derived the theorem deals with covering properties and axioms.

The aim of this paper is to introduce slightly *gα-continuous functions and investigate the properties of slightly *gα-continuous functions. By utilizing *gα-open sets, we derive the theorems which deals with covering properties and separation axioms.
1. gα-closed (Maki et al., 1993) if αcl(A)⊆U whenever A⊆U and U is α-open and the complement of gαclosed set is gα-open.
2. *gα-closed  if cl(A)⊆U whenever A⊆U and U is gα-open and the complement of *gα-closed set is *gα-open.

Definition
A function f: (X,τ)→(Y,σ) is called Throughout the present paper, X and Y are always topological spaces. Let A be a subspace of X. We denote the interior and closure of a set A by int(A) and cl(A), respectively. , 1983). The complement of α-open set is closed.

Definition
The intersection of all α-closed sets of X containing A is called the α-closure of A and is denoted by αcl(A).

Definition
A subset A of a space X is called 1. Slighly *gα-continuous at a point x X if for each clopen subset V in Y containing f(x),there exists a *gα-open subset U in X containing x such that f(U)⊂V.
2. Slightly *gα-continuous if it has this property at each point of X.
Proof. Let V be a clopen subset of Y. We have (f\A) -1 (V) =f -1 (V) ∩ A. Since f -1 (V) is *gα-open and A is open , then (f\A) -1 (V) is *gα-open in the relative topology of A. Thus f\A is slightly *gα-continuous.

Theorem
Let f: X→Y be a function and let g: X→XxY be the graph function of f, defined by g(x)=(x,f(x)) for every x X. Then g is slightly *gα-continuous if and only if f is slightly *gα-continuous.

Definition
A function f: X→Y is called

Theorem
Let f: X→Y and g: Y→Z be functions. Then, the following properties hold: 1. If f is *gα-irresolute and g is slightly *gαcontinuous, then g∘f: X→Z is slightly *gα-continuous.
3. If f is *gα-irresolute and g is slightly g-continuous, then g∘f: X→Z is slightly *gα-continuous.
(3): Let V be a clopen set in Z. Since g is continuous, then g -1 (V) is open in Y. Implies g -1 (V) is *gα-open in Y. Since f is *gα-irresolute then f -1 (g -1 (V)) is *gα-open in X. Therefore g∘f is slightly *gα-continuous.

Theorem
Let f: X→Y and g: Y→Z be functions. If f is *gα-open and surjective and g∘f: X→Z is slightly *gαcontinuous.
Combine the above two theorems, we get the following theorem.

Theorem.
Let f: X→Y be surjective, *gα-irresolute and *gα-open and g: Y→Z be a function. Then g∘f: X→Z is slightly *gα-continuous if and only if g is slightly *gαcontinuous.

Definition
1. A filter base Λ is said to be *gα-convergent to a point x in X if for any U∈*gαO(X) containing x, there exists a B∈ Λ such that B⊂U.

2.
A filter base Λ is said to be co-convergent to a point x in X if for any U∈CO(X) containing x, there exists a B∈ Λ such that B⊂U.

Theorem
If a function f: X→Y is slightly *gαcontinuous then for each point x∈ X and each filter base Λ in X *gα-converging to x, the filter base f(Λ) is co-convergent to f(x).
Proof. Let x∈ X and Λ be any filter base in Λ in X *gαconverging to x. Since f is slightly *gα-continuous, then for any V∈CO(Y) containing f(x), there exixts a U *gαO(X) containing x such that f(U)⊂V. Since Λ is *gα-converging to x, there exists a B∈ Λ such that B⊂U. This means that f(B)⊂Vand therefore that filter base f(Λ) is co-convergent to f(x).