A Grill between Weak Forms of Faint Continuity

The idea of grill topological space aimed at generalize the concept of topological space since it generates newtopology τG which helps to measure the things that was di cult to measure. Choquet [1] in 1947 was the first author introduced the idea of grill. It has been explored that there is some of similarity between Choquet;s concept and that ideals, nets and lters. In 2007, Roy and Mukherjee introduced the definition of τG which is related to two operators Φ and Ψ. They have determined the relation between τ and τG. A number of theories and characterization has been handeld in previous studies [2-4] whether respect to sets or functions. Hatir and Jafari [5] have the ability determine the defi nition of new categories of sets in a space which carries grill and topology and accordingly could obtain a new infection of continuity in terms of grill. Both Karthikeyan and Rajesh [6] have recently fixed the definition of a new category of functions which is called faintly G-precontinuous in a grill topological space. This paper aims at defining and studying new classes of functions called faintly G-γ continuous (resp. faintly G-α-continuous, faintly G-semi-continuous) between grill topological spaces. Some new descriptions and basic advantages of these classes of functions along with their relationships with certain types of function are investigated.


Introduction
The idea of grill topological space aimed at generalize the concept of topological space since it generates newtopology τ G which helps to measure the things that was di cult to measure. Choquet [1] in 1947 was the first author introduced the idea of grill. It has been explored that there is some of similarity between Choquet;s concept and that ideals, nets and lters. In 2007, Roy and Mukherjee introduced the definition of τ G which is related to two operators Φ and Ψ. They have determined the relation between τ and τ G. A number of theories and characterization has been handeld in previous studies [2][3][4] whether respect to sets or functions. Hatir and Jafari [5] have the ability determine the defi nition of new categories of sets in a space which carries grill and topology and accordingly could obtain a new infection of continuity in terms of grill. Both Karthikeyan and Rajesh [6] have recently fixed the definition of a new category of functions which is called faintly G-precontinuous in a grill topological space. This paper aims at defining and studying new classes of functions called faintly G-γ continuous (resp. faintly G-α-continuous, faintly G-semi-continuous) between grill topological spaces. Some new descriptions and basic advantages of these classes of functions along with their relationships with certain types of function are investigated.

Preliminaries
"In what follows, by a space Y we shall mean space Y which carries topology τ. For AY, we shall adopt the usual notations Cl(A) and Int(A) to A, respectively denote the closure and interior of A in Y. Also, the power set of Y will be written as P(Y). A nonempty sub collecting G of a space Y is called a grill [2] if

Remark 2.1
(1) "The minimal grill is G={Y} in any space Y which carries topology τ. (2) The maximal grill is G=P(Y)\{ф} in any topological space (Y; τ)." Since the grill depends on the two mappings Φand Ψ which is generated a unique grill topological space finer than on space Y denoted byτ G on Y have been discussion [6]. Definition 2.2: "Let Y be a space which carries topology and G be a grill on Y. A subset A in Y is called [3]: (2) G-preopen if A ⊆ Int((ΨA)); The family of all G-semi-open (resp. G-α-open, G-γ-open, G-preopen) sets of (Y,τ,G) containing a point y∈Y is denoted by GSO(Y; y) (resp. G α O(Y; y), G γ O(Y; y), GPO(Y; y))".

Proof:
The proof is similar to the proof of Theorem 3.2.

Theorem 3.5
Every G-γ-continuous function is faintly G-continuous.
Proof: It is clear from Definitions 3.1 and Theorem 3.2.
The converse of Theorem 3.5. is not true in general as it can be seen in the following example.     If h has this property at each point of Y, then it is said to be faintly G-precontinuous".

Proof:
The proof is similar to the proof of Theorem 3.2.

Definition 3.16:
A space (Y,τ,G) is named G-γ-connected if Y cannot be written as a union of two nonempty disjoint G-γ-open sets.
(2) G-γ-T 2 (resp. θ-T 2 [11]) if for each pair of different points y and z of Y, there exists disjoint G-γ-open (resp. θ-open) sets A and B in Y such that y ∈ A and z ∈ B.
Proof: Let Z be a θ-T 1 -space. For any two different points y and z in Y, then there exist A;W ∈ σ θ such that h(y)∈ A, h(z) ∉ A, h(y) ∉ W and h(z) ∈W for Z is a θ-T 1 -space. Because of h is faintly G-γ-continuous, h -1 (A) and h -1 (W) are G-γ-open subsets of (Y,τ,G) such that h -1 (A) and h -1 (W) containing y and z, respectively z ∉h -1 (A) and y ∉ h -1 (W). Then Y is G-γ-T 1.
Proof: For any two several points y and z of Y and since Z is a θ-T 2space, then there exists h(y)∈ A and h(z)∈ B for disjoint θ-open sets A and B in Z. h -1 (A) and h -1 (B) are G-γ-open subset of (Y,τ,G) such that h -1 (A) and h -1 (B) containing y and z, respectively by h is faintly G-γcontinuous injection. Therefore, h -1 (A)∩h -1 (B)=ф for A∩B=ф. Then Y is G-γ-T 2 .
Let h: (Y,τ,G) → (Z; σ) be a function. A function g: Y →Y ×Z, defined by g(y)=(y; h(y)) for every y ∈ Y, is called the graph function of h.
Proof: Since (y; z) ∈ (Y × Z) \ g(h), then h(y) ≠ z. Because of Z is θ-T 2 , there exist θ-open sets A and B in Z such that h(y) ∈ A; z ∈ B and B ∩ A=ф. Since h is faintly G-γ-continuous, h -1 (A) ∈ G γ O(Y, y). Let U=h -1 (A), we have h(U) ⊆ A. Therefore, we obtain h(A) ∩ B=ф. This shows that g(h) is θ-G-closed.

Conclusion
The study of faintly grill topological spaces is very important. It is generalization of faintly topological spaces. So we introduced neoteric classes of functions called faintly G-γ-continuous, faint G-αcontinuous and faint G-semi-continuous in grill topological spaces that helps us in many applications such as computer and information systems. Furthermore relationships between different classes are introduced. Also, some of their basic properties of different types of functions between grill topological spaces are obtained.