Study on Fuzzy 𝜷 - Supercontinuous Function in Fuzzy Topological Spaces

The main aim of this research is introducing and characterizing new concepts of fuzzy continuity using the definition of fuzzy β-sets and fuzzy βs -sets including fuzzy 𝛽 - supercontinuous. Some of their basic properties are examined. After that some theorem by using these properties have been proved.


Introduction
The concept of fuzzy sets was introduced by Zadeh in 1965 [4] and it has an important role in the study of fuzzy topology which was explained by Chang in 1968 [1]. Chang also investigated fuzzy continuity that was proved to be of fundamental importance in the field of fuzzy topology. Mapping is a significant tool for the research of topological notions and for building a new topological space, Chang (1968) extended the concepts of continuous, open and closed mappings in fuzzy topological spaces. Therefore, various concepts in general topology have been extended to fuzzy topological spaces. The class of b -open sets in general topology was defined by Andrijevic in 1996 [2]. In fuzzy topological space the notion of fuzzy semi open sets and fuzzy semi closed sets were first explained in 1981 by Azad [5] and also he defined the concept of semi continuous and semi open functions in fuzzy topological spaces. In this study, we introduce definition of fuzzy -supercontinuous function and illustrate a deep study on their properties and its related with other types of fuzzy continuous functions.

Preliminaries
In this section, we illustrate some basic definition and results.  1) Let F be fuzzy set in a fuzzy topological space (µ, ). Then its fuzzy b-closure is defined by Fbcl(F) = ˄ {L ≥F: L is a fb-closed set of (µ, ). 2) Let F be fuzzy set in fuzzy topological space (µ, ). Then its fuzzy b-interior ifdefined by Fbint(F)=˅{G≤F: G is afb-open set of (µ, )}. Now we will recall some details of fuzzy β-sets and fuzzy βs -sets.
Definition 2.17 [7] A subset A of a fuzzy topological space (µ, ) is said to be fuzzy β-set if A is fuzzy b-open and fuzzy locally closed set. The family of all fuzzy β-sets is denoted by Fβs(µ). It is clear that F βs(µ)=FLC(µ, )˄FBO(µ, ).
Proposition 2.19. [7] If µ is a sub maximal space, then the union of any family of fuzzy β-sets is fuzzy β-set. The class of all F β -set in(µ, )will bedenoted by Fβs. We will denote to the complement of F βs-set by 1-F βs-set. 2) If topological space μ is a sub maximal externally disconnected, then Fβs = F βs(µ) = .

Definition 2.26 [9]
Let be any non-empty sets and : → is a function, then for Let be a fuzzy set in . any fuzzy set in then the inverse image of under , we can write as −1 ( ) and it is fuzzy set in and defined by Since −1 ( ) is fuzzy β-set subset of µ for each ∈ . Hence, is fuzzy β-set. Therefore, we have, is fuzzy β-supercontinuous.  Since −1 ( ) is fuzzy βset in µ such that, ( ) ∈ ,therefore ∈ −1 ( ).
Proof: If follows directly from theorem (3.5).    Proof: Let be a fuzzy β-set in . Since is extremally disconnected space, then is fuzzy open set.

Conclusion.
In this research, we continue the study of continuity, and we defined new concepts of fuzzy continuous that is fuzzy -supercontinuous. Moreover, we examined the related properties and proved some theorems.