R-Generalized Fuzzy Closed Sets with Respect to an Ideal in Fuzzy Topological Spaces

The work in this paper is a generalization The concept of r-generalized fuzzy closed sets in fuzzy topological spaces was introduced by Kim. In this paper, we introduce and study the concept of r-generalized fuzzy closed sets with respect to an ideal in an ideal fuzzy topological space in Sostak sense.


Definition 1.1
A mapping τ: I X →I is called a fuzzy topology on X if it satisfies the following conditions [17]: The pair ( ) X τ is called a fuzzy topological space (for short, fts).
A fuzzy set λ is called r-generalized fuzzy closed (for short, r-gfc) if C γ ( λ;γ)whenever λ≤μ and ( ) Let (X,τ,I) be a fits.The simplest fuzzy ideal on X are I 0 ,I 1 : I X →I where ( ) ( ) If we take I=I 0 , for each A∊I X we have A * r =C τ (A,r).

Definition 1.4
Let (X,τ,I) be a fuzzy ideal topological space [16].Let µ, λ∊I X , the r-fuzzy open local function µ * r of µ is the union of all fuzzy points x t such that if ρ∊Q(x t ,γ) and I(λ) ≥ r then there is at least one y∊X for which ρ(y)+ µ(y)−1>λ(y).

Theorem 1.1
Let (X,τ) be a fts.Then for each r∈I 0 , λ ∈ I X we define an operator C τ : I X ×I 0 → I X as follows: For λ, µ∊I X and r, s∊I 0 , the operator C τ satisfies the following conditions: Let (X,τ) be a fts.Then for each r ∈ I 0 , λ ∈ I X we define an operator Iτ : I X ×I 0 → I X as follows [18]: For λ,µ ∈ I X and r,s ∈ I 0 , the operator I τ satisfies the following conditions:

Lemma 2.2
Every r-gfc set is r-gfIc.

Example
The converse Lemma 2.2 is not true.Let X={a,b} be a set.We define fuzzy topology and fuzzy ideal τ, I : I X → I as follows ( ) ( ) Then µ is r-gfIc set because, ( ) Let (X,τ,I) be an fuzzy ideal topological space, µ, λ ∈ I X and r ∈ I 0 .If µ and λ are r-gfIc sets, then µ∨λ is r-gfIc.

Remark
The intersection of two r-gfIc sets need not be an r-gfIc set as shown by the following example.

Definition 2.2
Let (X,τ,I) be fuzzy ideal topological space, µ∊I X and γ∊I 0 .A fuzzy set µ is called r-fuzzy generalized open with respect to an ideal I (briefly, r-gfIo) if 1 µ − is r-gfIc set.

Proof
Suppose that µ is r-gfIo sets.Suppose λ ≤ µ and ( ) By assumption, and µ ≤ λ, and µ r-gfIo relative to λ and λ is r-gfIo relative to X, then µ r-gfIo relative to X.

Proof
Obvious.