Generalized Fuzzy q-open sets

In this paper, we introduce the concepts of μ-fuzzy q-open sets which is generalization of simply open sets defined by Neubrunnove [9]. We also introduce and investigate, with the help of this new concept, the concepts of qiμ-Fuzzy open sets and qcμ-Fuzzy closed sets. The relations between these concepts are investigated and several examples are presented.


1.Introduction and Preliminaries
The potential of the notion of fuzzy set studied by L. A. Zadeh [12] was realized by the researchers and has successfully been applied for new investigations in all the branches of science and technology for more than last five decades. Since Chang [3] defined the concept of a fuzzy topology, then many authors investigated different properties of fuzzy open sets which are weaker than the property of openness of a fuzzy set in a fuzzy topological space. For example, ( [1], [10], [11], [13]) have considered such kind of properties of fuzzy sets and most of the collection forms a fuzzy supra topology therein. A significant contribution to the theory of generalized open sets has been reported by A. Csaszar ([6], [7], [8]) and extended by G. P. Chetty [5] in the context of fuzzy set theory with the name of generalized fuzzy topological space. Our aim is to study the parallel concept of topology in a given fuzzy space with an incomparable nature.
In the present paper, we introduce the concept of μ-fuzzy q-open sets and study some of their properties. Finally, we discuss about some fundamental properties of such structure and some related notions. In particular, we have shown that μ-fuzzy q-open sets is a weaker form of − − sets introduce by G. Palani Chettry [5] in 2008. Lastly, we define -Fuzzy open set which is a weaker than μ-fuzzy q-open set.
We , now, state a few definitions and results that are required in our work.

μ-Fuzzy q-interior and μ-Fuzzy q-closure.
Definition 3.1. Let λ by a fuzzy set of a GFTS ( , ) and defined the following sets: We call the μ-fuzzy q-interior of λ and , the μ-fuzzy q-closure of λ. ≤ It is similar to the proof of (c) . (e). Let be a fuzzy point of X such that ∈ . Then there exists a μ-fuzzy q-open set η such that ≤ with ∈ . Since ≤ , then ≤ , this implies that ≤ . Therefore ∈ . Hence ≤ .

4.
-Fuzzy set and -Fuzzy. λ ( resp. λ) is -fuzzy set (resp. -fuzzy set). Proof . Follows from Theorem 3.1. and the above definition.          is μ-fuzzy semiopen set ,then λ is μ-fuzzy semiopen , therefore λ is μ-fuzzy q-open . The proof of the other case is similar. Theorem 4.9. Let λ be a -fuzzy set (resp. -fuzzy set), then. ≤ (resp. ≤ ). Proof . Since λ is -fuzzy set ,then ≤ , then ≤ ≤ ⇒ ≤ . Thus ≤ ≤ . The proof of the other case is similar.  In the end of this paper, the following diagram give summary of the last results.