Fuzzy Bicontinuous Maps in Fuzzy Biclosure Spaces

The purpose of this paper is to introduce the notion of fuzzy bicontinuous maps and fuzzy biclosed (fuzzy biopen) maps in fuzzy biclosure spaces and study some of their properties.


Introduction
The concept of fuzzy set was first introduced by Zadeh [1] in his classical paper in 1965. Zadeh's introduction of the notion of a fuzzy set in a universe has inspired many mathematicians to generalize the main concepts and structures of present day mathematics into the framework of fuzzy sets. The theory of general topology is based on the set operation of union, intersection and complementation. Fuzzy sets do have the same kind of operations. Inspired by these observations Chang [2] extended the concepts of point set topology to fuzzy sets and laid the foundation of the fuzzy topology.
In recent years, fuzzy topology has been found to be very useful in solving many practical problems. Du.et. al. [3] fuzzified the very successful 9-intersection Egenhofer model [4] for depicting topological relations in Geographic Information Systems (GIS) query. Today fuzzy topology has firmly established as one of the basic discipline of fuzzy mathematics, and have a fundamental role to play in pure and applied sciences.
Closure spaces which is a generalization of topological spaces were introduced by E. Čech [5]. Today, the theory of closure space is one of the most popular theory of mathematics which finds many interesting applications in the areas of fuzzy sets, combinatorics, genetics or quantum mechanics. In 1985 fuzzy closure spaces were first studied by A.S. Mashhour and M.H. Ghanim [6]. Boonpok [7] studied the notion of biclosure spaces. Such spaces are equipped with two arbitrary closure operators. He extended some of the standard results of separation axioms from closure spaces to biclosure spaces. Thereafter a large number of papers have been written to generalize the concept of closure space to biclosure space.
Recently, Tapi and Navalakhe [8] have introduced the notion of fuzzy biclosure spaces and generalized the concept of topological spaces to fuzzy closure space and fuzzy biclosure space.
In this paper we have fuzzified the paper of Boonpok [9] in which he had studied the concept of Bicontinuous maps in biclosure spaces. Tapi and Navalakhe [10] had already worked on the concept of Pairwise Fuzzy Bicontinuous maps in Fuzzy biclosure spaces.
The focus of this paper is to introduce and study the concept of fuzzy bicontinuous maps and fuzzy biclosed (fuzzy biopen) maps in fuzzy biclosure spaces. We have also investigated some of the important characterizations and properties of fuzzy bicontinuous maps and fuzzy biclosed (fuzzy biopen) maps in fuzzy biclosure spaces

Preliminaries
In order to make this paper self contained, we briefly recall certain definitions and results. Let X be an arbitrary set, [0,1] I = and X I be a family of all fuzzy sets of X . For a fuzzy set µ of X , cl( µ ), int( µ ) and 1 µ − will denote the closure of µ , the interior of µ and the complement of µ respectively whereas the constant fuzzy sets taking on the values 0 and 1 on X are denoted by 0 X and 1 X respectively.
is a fuzzy closed subset of ( ) 1 2 , , X u u for every fuzzy closed subset µ of ( ) Clearly, it is easy to prove that a map ( ) ( ) is a fuzzy open subset of ( ) 1 2 , , X u u for every fuzzy open subset u of ( ) Definition 2.5 [9]. Let ( ) 1 2 , , X u u and ( ) , , the product of the family of fuzzy closure spaces { } , :

IJPMS Volume 17
be a family of fuzzy biclosure spaces. Then for each J β ∈ , the projection map ( ) ( ) be a family of fuzzy biclosure spaces and let J β ∈ .
is a fuzzy closed subset of ( )

Fuzzy bicontinuous maps
In this section, we introduce the concept of fuzzy bicontinuous maps in fuzzy biclosure spaces and investigate some of the properties of fuzzy bicontinuous maps in fuzzy biclosure spaces.
Hence, the map f is fuzzy bicontinuous .
Consequently, the map h f  is fuzzy bicontinuous. Proof. Let f be fuzzy bicontinuous. Since α π is fuzzy bicontinuous for each J α ∈ , it follows that f α π  is fuzzy bicontinuous for each J α ∈ . Conversely, let the map f α π  be fuzzy bicontinuous for each J α ∈ . Suppose that the map f is not fuzzy bicontinuous. Then there exists a fuzzy subset η of X such that This contradicts the fuzzy bicontinuity of the map f β π  . Consequently, the map f is fuzzy bicontinuous. , , , , Then the map f is fuzzy bicontinuous if and only if f α is fuzzy bicontinuous for each J α ∈ .

Proof.
Let the map f be fuzzy bicontinuous, let J β ∈ and X β µ ≤ . Then Therefore, the map f is fuzzy bicontinuous.

Fuzzy biclosed maps
This section of the paper is aimed to introduce the notion of fuzzy biclosed (fuzzy biopen) maps in fuzzy biclosure spaces and to investigate some of the important characterization of fuzzy biclosed (fuzzy biopen) maps in fuzzy biclosure spaces.  Y v . Therefore, the map f is fuzzy biclosed.
The following statement is evident.