Fuzzy m-Structures, m-Open Multifunctions and Bitopological Spaces

In this paper we study different weak forms of open multifunctions from a fuzzy topological space into a fuzzy m-space. Further we study the same from a fuzzy topological space into a fuzzy bitopological space.


Introduction and preliminaries
The notion of fuzzy set was introduced by L.A. Zadeh in 1965.Since than the importance of the introduced notion was realised by researchers in various fields of science and has successfully been for investigations.The notion has been applied for introducing different types of fuzzy topological spaces and investigate their properties by Alimohammady and Roohi [1], Tripathy and Debnath [21][22], Tripathy and Ray [23][24] and many others.
Noiri and Popa [16] introduced the notions of minimal structures, m-spaces and m-continuity.Alimohammady and Roohi [1] introduced the concept of fuzzy minimal structure, fuzzy m-continuity and fuzzy minimal vector spaces.Using these concepts, several authors introduced and studied various types Let (X, τ ) be a fuzzy topological space and A be a fuzzy subset of X.The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively.
For the above definition, one may refer to Azad [2], Bin Shahna [3] and Mashhour, Ghanim and Fathalla [12].Definition 1.2.A fuzzy point x α is said to be quasi-coincident with A, denoted by x α qA, if and only if α + A(x) > 1 or α > A c (x). Definition 1.3.A fuzzy set A is said to be quasi-coincident with B and is denoted by AqB, if and only if there exists a x ∈ X such that A(x) + B(x) > 1.
It is clear that A and B are quasi-coincident at x both A(x) and B(x) are not zero at x and hence A and B intersect at x. Definition 1.4.A fuzzy set A in a fts (X, τ ) is called a quasi-neighborhood of x λ if and only if A 1 such that A 1 ⊆ A and x λ qA 1 .The family of all Qneighborhoods of x λ is called the system of Q-neighborhood of x λ .Intersection of two quasi-neighborhoods of x λ is a quasi-neighborhood.
In a fuzzy topological space (X, τ ) a fuzzy point x p is called a fuzzy θ-cluster point of a fuzzy set A if cl(V )qA holds for every open Q-neighbourhood V of x p (one may refer to Mukharjee and Sinha [13]).The union of all fuzzy θ-cluster points of A is called a fuzzy θ-closure of A, written as Cl θ (A) and A is called fuzzy θ-closed if A = Cl θ (A).The complements of fuzzy θ-closed sets are called fuzzy θ-open (one may refer to Mukharjee and Sinha [13]).The fuzzy θ-interior of a fuzzy set A in X, written as Int θ (A), is defined to be the fuzzy set (Cl θ (A c )) c (see for instance Ghosh [5]).
In a fuzzy topological spaces (X, τ ) a fuzzy point x p is called a fuzzy δ-cluster point of a fuzzy set A if every fuzzy regular open Q-neighbourhood of x p is quasicoincident with A (one may refer to Sinha [18]).The union of all fuzzy δ-cluster points of A is called a fuzzy δ-closure of A, written as Cl δ (A); and A is called fuzzy δ-closed if A = Cl δ (A).The complements of fuzzy δ-closed sets are called fuzzy δ-open (please refer to Sinha [17]).The fuzzy δ-interior of a fuzzy set A in X, written as Int δ (A), is defined to be the fuzzy set (Cl δ (A c )) c (one may refer to Ghosh [5]).
Throughout the paper, (X, τ ) and (Y, σ) (briefly X and Y ) denote fuzzy topological spaces and F : X → Y (respectively f : X → Y ) presents a multivalued (respectively single valued) function.For a fuzzy set A ≤ X, F + (A) and F − (A) are defined by see for instance Mukharjee and Malakar [14]).A subfamily m X of I X (where I X is the collections of all fuzzy sets from X into I = [0,1]) of a non empty set X is called fuzzy minimal structure (or briefly, fuzzy m-structure on X if α1 X ∈ m X for any α ∈ I = [0, 1] (one may refer to Alimohammady and Roohi [1]) By (X, m X ) (or briefly (X, m)), we denote a non-empty set X with a fuzzy minimal structure m X on X and call it a fuzzy m-space.Each member of m X is said to be fuzzy m X -open (or briefly m-open) and the complement of an m X -open set is said to be m X -closed (or briefly m-closed).
We procure the following two definitions, some lemmas and Remark due to Alimohammady and Roohi [1], those will be use in this article.Definition 2.2.Let X be a non empty set and m X an fuzzy m-structure on X.For a fuzzy subset A of X, the m X -closure of A and the m X -interior of A are defined as follows: (1) Lemma 2.1.Let (X, m X ) be an fuzzy m-space.For fuzzy subsets A and B of X, the following properties hold: Definition 2.3.A fuzzy minimal structure m X on a non-empty set X is said to have property B if the union of any family of fuzzy subsets belonging to m X belongs to m X .Lemma 2.2.Let (X, m X ) be an fuzzy m-space and m X satisfy the property B. Then for a fuzzy subset A of X, the following properties hold: Remark 2.1.Let (X, τ ) be a fuzzy topological space and m X = F SO(X) (respectively F P O(X), F αO(X), F βO(X), F BO(X)), then m X satisfies property B.
Note 3.1.The characterization theorems for the above multifunctions can also be established as in the earlier cases.
Competing interests: The authors declare that the article does not have competing interest.
of modifications of open functions and open multifunction in m-spaces.In this paper we define fuzzy K-open function based on K-open function in the sense of Kuratowski [10] and 120 B. C. Tripathy and S. Debnath introduce the notion of fuzzy K − m-open multifunction from a fuzzy topological space to fuzzy m-space, also between fuzzy bitopological spaces.

Koratowski [ 9 ]
defined K-open function in topological spaces.Based on this definition, we define K-open function in fuzzy topological spaces as follows.Definition 2.4.A function f : (X, τ ) → (Y, σ) is said to be fuzzy K-open if for each fuzzy point y p of Y and for each fuzzy open set U of X such that y p ∈ f (U ), there exists a fuzzy open set V of Y such that y p ∈ V ≤ f (U ).

Remark 2 . 2 .
A function f : (X, τ ) → (Y, σ) is fuzzy K-open if and only if f (U ) is fuzzy open in Y for each fuzzy open set U of X.Definition 2.5.Let (Y, m Y ) be a fuzzy m-space.A multifunction F : (X, τ ) → (Y, m Y ) is said to be fuzzy K − m-open if for each fuzzy point y p of Y and for each fuzzy open set U of X such that y p ∈ F (U ), there exists V ∈ m Y such that y p ∈ V ≤ F (U ).

Remark 2 . 3 .
(a) Let m Y have the property B.Then, it follows from Lemma 2.2 and Theorem 2.1 that F is fuzzy K − m-open if and only if F (U ) is fuzzy m Y -open for each open set U of X.(b) If F : (X, τ ) → (Y, σ) is a multifunction and m Y = σ (respectively F SO(Y ), F P O(Y ), FαO(Y), F βO(Y )), then by (a) we get definition 1.3.(c) If f : (X, τ ) → (Y, σ) is a function and m Y = σ (respectively F SO(Y ), F P O(Y ), F αO(Y ), F βO(Y )), then by (a) we get definition 1.2.