Some Weaker Forms of Fuzzy Faintly Open Mappings

This paper is devoted to introduce and investigate some weak forms of fuzzy open mappings, namely fuzzy faintly semi open (fuzzy faintly semi closed), fuzzy faintly preopen (fuzzy faintly preclosed), fuzzy faintly α-open (fuzzy faintly α-closed), fuzzy faintly semi preopen (fuzzy faintly semi preclosed) and fuzzy faintly sp-open (fuzzy faintly sp-closed) mappings and their fundamental properties are obtained. Moreover, their relationship with other types of fuzzy open (closed) mappings are discussed.


Introduction
It is well accepted that in general topology, different types of mappings play significant roles in the process of formulating different topological concept.Thus with all round development of fuzzy topological concepts especially in the way of generalizations and extensions of the corresponding set topological notions, it has naturally been quit necessary to introduce different types of mappings between fuzzy topological spaces.We find a large community of mathematicians who has already contributed to a large extent in this direction.For example, Chang [3] has introduced fuzzy openness (fuzzy closedness), Azad [1] introduced fuzzy semi openness (fuzzy semi closedness), Bin Shahna [2] introduced fuzzy pre openness (fuzzy pre closedness) and fuzzy α -openness (fuzzy α -closedness), Thakur and Singh [6] introduced fuzzy semi pre openness (fuzzy semi pre closedness) and Hakeem A. Othman [4]

Preliminaries and notations
Throughout this paper by (X, τ) or simply by X we mean a fuzzy topological space ( f ts, shorty) and f : X → Y means a mapping f from a fuzzy topological space X to a fuzzy topological space Y .If A is a fuzzy set and p is a fuzzy singleton in X then N(p), Intλ , clλ , λ c denote respectively, the neighborhood system of p, the interior of λ , the closure of λ and complement of λ .Now, we mention the following definitions and results which are used in this paper concerning fuzzy topology.Definition 2.1.A fuzzy set λ in a fuzzy topological space (X, τ) is called: • Fuzzy semi open (Fuzzy semi closed) set [1] • Fuzzy preopen (Fuzzy preclosed) set [2] • Fuzzy semi preopen (Fuzzy semi preclosed) set [6] if there exists a fuzzy preopen (fuzzy preclosed • Fuzzy semi preopen (Fuzzy semi preclosed) [6] if f (λ ) is a fuzzy semi preopen (fuzzy semi preclosed) set in Y for each fuzzy open (fuzzy closed) set λ in X.

• Fuzzy sp-open (Fuzzy sp-closed) [4] if f (λ ) is a fuzzy sp-open (fuzzy sp-closed) set in Y for each fuzzy open (fuzzy closed
) set λ in X.

Main section
In this section, I define the following weak forms of faintly open mappings, some concepts related to them and their fundamental properties are obtained.Proposition 3.1.For a fuzzy topological space (X, τ) the following is valid.

is fuzzy faintly pre open [resp. fuzzy faintly semi open, fuzzy faintly αopen, fuzzy faintly semi preopen];
• Proof.We will prove the theorem only for fuzzy faintly pre open mappings.
By Theorem 3.5 there is a fuzzy preclosed set η ≥ µ such that θ − cl Now, I will introduce the Definition of fuzzy θ -irresolute and fuzzy β -irresolute mapping in order to discuss some results about the computation mappings.
Definition 3.2.A mapping f : (X, τ) → (Y, σ ) from a fuzzy topological space (X, τ) to another fuzzy topological space (Y, σ ) is said to be: The following examples illustrated new fuzzy mappings, fundamental properties of new fuzzy mappings and the reverse of the relations in Diagram (1).
has introduced fuzzy sp-openness (fuzzy sp-closedness) and they establish their various characteristic properties.In this paper, I introduce and study ten weak forms of faintly open mappings, called fuzzy faintly semi open (fuzzy faintly semi closed), fuzzy faintly preopen (fuzzy faintly preclosed), fuzzy faintly α-open (fuzzy faintly α-closed), fuzzy faintly semi preopen (fuzzy faintly semi preclosed) and fuzzy faintly sp-open (fuzzy faintly sp-closed) mappings, I study their basic properties and their relationship with other types of fuzzy open (closed) mappings.

Definition 3 . 1 .
A mapping f : (X, τ) → (Y, σ ) from a fuzzy topological space (X, τ) to another fuzzy topological space (Y, σ ) is said to be:•Fuzzy faintly pre open (Fuzzy faintly pre closed) if f (λ ) is a fuzzy pre open (fuzzy pre closed) set in Y for each fuzzy θ -open (fuzzy θ -closed) set λ in X.

•
Fuzzy faintly semi open (Fuzzy faintly semi closed) if f (λ ) is a fuzzy semi open (fuzzy semi closed) set in Y for each fuzzy θ -open (fuzzy θ -closed) set λ in X.