On Fuzzy soft Regularly nowhere dense sets

In this paper, several characterizations of fuzzy soft regularly nowhere dense sets, fuzzy soft regularly dense, fuzzy soft regularly residual, several examples are given to illustrate the concepts introduced in this paper.


Introduction
Modeling uncertain data has been an interesting subject in engineering, eco-nomics, environmental and social sciences. Crisp set theory helped a little for formal modeling, reasoning and computing uncertain data. However, there are many com-plex problems in various branches of science that involve data which is not always crisp. We can not successfully use classical methods because of various types of uncertainties present in these problems.In 1999 Molodtsov [8] introduced the Soft set theory (SST) as a new mathematical tool to deal with uncertain data which is free from such difficulties. This theory has proved to be useful in many different fields such as:decision making ,data analysis forecasting and simulation .Later on Maji et al. [7] introduced several operations of soft sets. Pei and Miao [10], M. I. Ali et al. [1] introduced and studied several soft set operations. The concept of FSS is introduced and studied [ 3-6, 9] a more generalized concept, which is a combination of fuzzy set and fuzzy soft set and studied its properties. The aim of this paper is to introduce the concepts of fuzzy soft regularly nowhere dense sets.

2.1[3].Definition
The fuzzy soft set ∈FS( , ) is said to be null fuzzy soft set and it is denoted by , if for all ∈ , ( ) is the null fuzzy soft set 0 ̅ of U, where 0 ̅ (x) = 0 for all x∈ .

2.3[3].Definition
A fuzzy soft set is said to be a fuzzy soft subset of a fuzzy soft set over a common universe if ⊆ and ( ) ⊆ ( ) for all ∈ , . ., if ( ) ≤ μ ( ) for all ∈ and for all ∈ and denoted by ⊆ ̆.

2.5[11].Definition
A FSS is a FSTS (U,E, ) is called a fuzzy soft dense if there exists no fuzzy soft closed set GB in (U,E, ), such that FA< GB< 1, that is cl fs (FA) =1.

2.6[11].Definition
Let (U,E, ) be a fuzzy soft topology. A Fuzzy soft set FA in (U,E, ) is called fuzzy soft first where FAᵢ 's are fuzzy soft nowhere dense sets in (U,E, ). Any other fuzzy soft set in (U,E, ) is said to be of fuzzy soft second category.

2.7[12].Definition
Let be the collection of fuzzy soft sets over . Then is called a fuzzy soft topology on if satisfies the following axioms: • , ̅ belong to . • The union of any number of fuzzy soft sets in belongs to .
• The intersection of any two fuzzy soft sets belongs to . The triplet (U, , ) is called a fuzzy soft topological space over U. The members of are called fuzzy soft open sets in U and complements of them are called fuzzy soft closed sets in U.

2.8[ 12].Definition
The union of all fuzzy soft open subsets of over ( , ) is called the interior of and is denoted by ( FA).

[12].Definition
Let ∈ FS ( , ) be a fuzzy soft set. Then the intersection of all closed sets, each containing A, is called the closure of and is denoted by ( ).

2.11.[12].Remarks
• For any fuzzy soft set in a fuzzy soft topological space ( , , ), it is easy to seethat ( ( )) c = ( ) and ( (FA )) c = ( ). • For any fuzzy soft A subset of a fuzzy soft topological space ( , , , ), we define the fuzzy soft subspace topology on by if KD = ∧ ̆G B For some GB . • For any fuzzy soft in fuzzy soft subspace of a fuzzy soft topological space, we denote to the interior and closure of in by ( ) and ( ), respectively.

2.17[2]: Theorem
Let (X , T, E ) be fuzzy soft topological space and , ∈FSS(X)E . Thenthe following properties are satisfied for the fuzzy semi interior operator, denoted by FSint.

3.6.Definition
Let FA be a FSRFCS in a FSTS (U,E, ), then 1−FA is called a fuzzy soft residual set in (U,E, ).

3.7.Proposition
If FA is a fuzzy soft residual set in (U,E, ), then FA is a FSR residual set in (U,E, ).

Proof:
Let

3.9.Proposition
If FA is a fuzzy soft semiclosed set with int fs (FA) ≠ 0 in a FSTS (U,E, ), then FA is not a fuzzy soft nowhere dense set in (U,E, ).

Proof:
Let FA be a fuzzy soft semi-closed set with int fs (FA) ≠ 0 in (U,E, ). By proposition 3.8, int fs [cl fs (FA)]≠0, implies that FA is not a fuzzy soft nowhere dense set in (U,E, ).

3.11.,Proposition
If FA is a fuzzy soft closed set with int fs (FA) ≠ 0 in a FSTS (U,E, ), then FA is not a fuzzy soft nowhere dense set in (U,E, ).

Proof:
Let FA be a fuzzy soft closed set with int fs (FA) ≠ 0 in (U,E, ). By proposition 3.10, int fs (FA) ≠ 0, implies that then FA is not a fuzzy soft nowhere dense set in (U,E, ).

3.15,Proposition
If FAis a fuzzy soft closed set with int fs (FA)= 0 in a FSTS (U,E, ), then FA is a FSR nowhere dense set in (U,E, ).

Proof:
Let FA is a fuzzy soft closed set with int fs (FA) ≠ 0 in (U,E, ). Since FA is a fuzzy soft closed in (U,E, ) , then cl fs (FA) = FA implies that int fs [cl fs (FA)] = int fs (FA)=0. Thus FA is a fuzzy soft nowhere dense set and hence FA is a fuzzy soft residual nowhere dense set in (U,E, ).