Fuzzy Regularly Closed Sets in Michálek’s Fuzzy Topological Space

In this paper the Fuzzy Regularly closed subsets have been discussed with the help of the Fuzzy Topological Space defined by Michálek. Also we have discussed some characterstics of Fuzzy Regularly Closed sets.


Introduction
The commonly used concept of set can be expressed in a more general form by the Fuzzy sets introduced by Zadeh [5] in 1965. To introduce fuzzy sets Zadeh used the function form of an ordinary set that is, any subset A of X is characterized by its characteristic function χ A : X → {0,1}.
Using the fuzzy sets introduced by Zadeh, C L Chang [2] defined Fuzzy Topological space in 1968 for the first time. According to Chang, any family of fuzzy sets in X which include empty set and the complete set and which are closed under arbitrary union and finite intersection forms a fuzzy topology. But this definition is just a natural translation of the ordinary definition of topological space to fuzzy sets. In 1975 J Michálek [4] in his paper defined fuzzy topological space in a quite different manner. To define Fuzzy Topological Space Michálek first defined ordinary topological space in terms of functions and then defined Fuzzy Topological Space as an extension of these functions. From the introduction of Chang's Fuzzy Topology in 1968 this area caught the attention of researchers and many papers appeared there after in this area. But not much work has been carried out in the area of Fuzzy Topology by Michálek. In 2001, Francisco Galligo Lupranez in his paper [3] studied few characteristics of the Fuzzy Topological Space as given by Michálek . In 1981 K K Azad[1] came up with an idea of Fuzzy Regularly Closed subsets in Changs Fuzzy Topological Space. This work presents the concept of Fuzzy Regularly Closed Sets in Michálek sense and study some of its properties.
In the next section of this paper we give the necessary preliminary results required for the development of the main concepts.

Fuzzy Regularly Closed Sets and Its Properties
While in Chang's fuzzy topology, the interior of a fuzzy set is defined, Michálek discussed the fuzzy interior of a subset of . Here we introduce the concepts of fuzzy closure of the subset of and fuzzy regularly closed subset of in Michálek sense.
Therefore is a lower bound of .
ie, ≤ . Therefore, is the greatest lower bound.

Now put
Therefore is the least upper bound.
Therefore, FRC (X) is a complete lattice.

Note 3.
Similarly we can prove that FRO (X) is a complete lattice.

Conclusion
As FRC (X)) and FRO (X) are complete lattices, they can be used as the basic structure for the construction of topological spaces such as fuzzy extensions and fuzzy absolutes.