Applications of Some Topological Near Open Sets to Knowledge Discovery

In this paper, we use some topological near open sets to introduce the rough set concepts such as near open lower and near open upper approximations. Also, we study the concept of near open, rough set and some of their basic properties. We will study the comparison among near open concepts and rough set concepts. We also studied the effect of this concept to motivate the knowledge discovery processing. Keywords—Topological spaces; Rough sets; Knowledge discovery; open sets; Accuracy measure


INTRODUCTION
Information technology is huge and need more accurate measures to discover their valuable knwoledge.An important field of this technology is the information discovery via available information.Rough set theory and their topological generalizations using some topological near open sets [1,2,4,5,7,12,13,15,17,18,19,20] are a recent, an accurate and applicable approach for reasoning about data.Many researchers generalized rough sets, but the concept of a topological rough set given by Wiweger [16] in 1989 is the basic starting point.This generalization is basically starting with a topological space and defined the approximation via the interior and the closure operators on topological spaces.The concept of β -open set was introduced in 1983 by M. E. Abd El-Monsef and others [3].
In this paper, we introduce and investigate the concept of near open approximation space.These spaces help to get a new classification for the universe.Also, we investigate the concept of near open lower and near open upper approximations.We study near open, rough sets, the comparison between this concept and rough sets is also studied.Also, we give some counter examples.

II. ROUGH SET BASIC CONCEPTS
Classical rough set theory has come from the need to represent subsets of a universe in terms of equivalence classes of a partition of that universe.The partition characterizes a topological space, called approximation space ) , ( = R X K , where X is a set called the universe and R is an equivalence relation [8,14].The equivalence classes of R are also known as the granules, elementary sets or blocks; we will use X R x ⊆ to denote the equivalence class containing X x ∈ .
In the approximation space, we consider two sets, namely x ⊆ ∈ that is called the lower and the upper approximation of The degree of completeness can also be characterized by the accuracy measure, in which Accuracy measures try to express the degree of completeness of knowledge.
) ( A R α is able to imprisonment, how large the boundary region of the data sets is; however, we cannot easily capture the structure of the knowledge.A fundamental advantage of rough set theory is the ability to handle a category that cannot be sharply defined given a knowledge base.Characteristics of the potential data sets can be measured through the rough sets framework.We can measure inexactness and express topological characterization of imprecision with: We denote the set of all, roughly R -definable (resp.Internally R -undefinable, externally R -undefinable and totally R -undefinable) sets by ) ( X RD A topological space [6] is a pair ) , ( τ X consisting of a set X and family τ of subsets of X satisfying the following conditions: (1) (2) τ is closed under arbitrary union.
The pair ) , ( τ X is called a topological space, the elements of X are called points of the space, the subsets of X belonging to are called open set in the space, and the complement of the subsets of X belonging, to τ are called closed set in the space; the family τ of open subsets of X is also called a topology for Let A be a subset of a topological space ) , ( τ is a near open approximation space.
, for any X A ⊆ .The Universe X can be divided into 12 regions with respect to any X A ⊆ as shown in Figure 1.
will be defined well in A , while this point was undefinable in Pawlak's approximation spaces.Also, the elements of the region [ ] do not belong to A , while these elements was not well defined in Pawlak's approximation spaces.
In our study, reduce the boundary of A , in Pawlak's approximation space by near open boundary of A .Also, we extend exterior of A which contains the elements did not belong to A by near open exterior of A .(1) be a near open approximation space and X A ⊆ .Then (1) The converse of Remark 3.4 may not be true in general as seen in the following example.
We can characterize the degree of completeness by a new tool named β -accuracy measure defined as follows: We see that the degree of exactness of the set } { = a A    given by the condition: .Then we say that above, we can characterize rough sets by their size of the boundary region.
NEAR OPEN ROUGH CLASSIFICATION In this section, we introduce and investigate the concept of near open approximation space.Also, we introduce the concepts of near open lower approximation and near open upper approximation and study their properties.Definition 3.1.Let X be a finite non-empty universe.The pair ) , ( β R X is called a near open approximation space where β R is a general relation used to get a subbase for a topology τ on X and a class of β -open sets ) ( X O β .Remark 3.1.In Definition 3.1, we use the symbol β R to avoid confusion with R which refers to an equivalence relation. of multi valued information system of seven patients (A, B and C are conditional attributes and D is the decision attribute) of the universe U , and a relation R defined on X by } open approximation space near open lower (resp.near open upper)approximation of any non-empty subset A of X is defined as:

Fig. 1 .
Fig. 1.Regions of the universe Remark 3.2.The study of near open approximation spaces is a generalization for the study of approximation spaces.Because of the elements of the regions [

Proposition 3 . 1 .
For any near open approximation space ) , ( β R X , the following are hold of any

,
the following are hold of any

.
The following are hold to any β ∈ , say, near open strong and near open weak memberships respectively which defined by:

3 .
According to Definition 3.4, near open lower and near open upper approximations of a set by using accuracy measure equal to 50% and by using near open accuracy measure equal to 100% .Consequently near open accuracy measure is the accurate measure in this case.We investigate near open, rough equality and near open rough inclusion based on rough equality and inclusion which introduced by Pawlak and Novotny in[10,11].say that A and B are: (i) Near open roughly bottom equal if

)
OF NEAR OPEN, ROUGH SETS In this section, we introduced a new concept of near open rough set.Definition 4.1.For any near open approximation space

( 1 )
Every exact set in X is near open exact.(2) Every near open rough set in X is rough.Proof.Obvious.The converse of all parts of Proposition 4.2 may not be true in general as seen in the following example.

TABLE I
According to Example 3.1, we can deduce Table 2 below that show the degree of accuracy measure

TABLE II .
ACCURACY MEASURES