Rough topological space using equivalence relation

To generalise the basic rough set definitions, the topology induced by equivalence relations is used. The proposed topological structure opens the way for the implementation of a broad range of topological facts and techniques in the granular computing process, including the introduction of the definition of topological membership functions that incorporates the concept of rough and fuzzy sets. There is an overlap between rough set theory and several other theories dealing with incomplete knowledge


Introduction
It is possible to consider rough set theory as a new mathematical method for imperfect data analysis. In many fields, such as decision support, engineering, the environment, finance, medicine and others, the theory has found applications. The theory provides an attempt to deal with confusion or vagueness. The Rough set theory has drawn the attention of many scientists and practitioners who have ultimately contributed to its development and applications. The Preliminaries: This section will briefly review basic concepts and outcomes of the rough sets dependent relationship and rough topological space with equivalence relationship and some important definitions. [1][2][3][4][5]

Definition
Let X is subset of U, let R be a relation of equivalence to U. Define the following, then,the Lower Approximation of X with regard to R is the set of all objects that can be identified with certainty as members of X with regard to R. It is defined by

Definition
The upper approximation of X with respect to R is the set of all objects that can be identified with certainty with respect to R as potential members of X. It is defined by ={x:R[x]∩X≠Ø}

Definition
The difference between the upper and lower approximations is the boundary region of the set.
Lower and upper approximation diagram ) ( X R Intuitively, the boundary region of the set consists of all elements that, by the use of available information, cannot be identified uniquely as a set or its complement. It is defined by BN B (X) = R * X -R * X A set, if it has a non-empty boundary area, is said to be a rough set. If the boundary region is empty, the Crisp or Exact Set set is the set.

Definition:
A topological space [3] is a pair (X, T) consisting of a collection of X and a family of subsets of X that satisfy the following requirements: i) ∅, X ∈ T ii) Under an arbitrary union, T is closed.

Observations:
 A relation of equivalence causes a partitioning of the universe.  The partitions can be used to create the universe 's new subsets.  The same value of the decision attribute is assigned to subsets that are most often of interest. Example:  All subsets of Z integers are declared open, so Z in the so-called discrete topology is a topological space.

Definition:
Topological equivalence here, Topologists study (topological) spaces up to homeomorphism, just as algebraists study groups up to isomorphism or matrices up to linear conjugacy. [6][7][8] Topological space Fundamental Study: A map f: X ¬ Y between topological spaces is a homeomorphism if it is continuous and bijective with the inverse continuous. If there is an X-Y homeomorphism, then it is said that X and Y are homeomorphic or often topologically identical. A property of a topological space that is the same is said to be a topological invariant for any two homeomorphic spaces. Obviously, the relationship of being homeomorphic is a relationship of equivalence (in the technical sense: reflexive, symmetric, and transitive). Topological spaces are thus divided into classes of equivalence, often referred to as homeomorphic classes. In this relation, the topologist is often represented as a person who (since these two objects are homeomorphic) can not distinguish a coffee cup from a doughnut. In other words, from the intrinsic point of view, two homeomorphic topological spaces are identical or indistinguishable in the same way as isomorphic groups are indistinguishable from the abstract group theory point of view, or two conjugate n x n matrices are indistinguishable as linear ndimensional vector space transformations without a fixed vector space.
Proof: Consider any subset A= (A L ,A U )of the Discrete Rough Topological Space(X,T) . Where X= (X L ,X U ) , T= (T L ,T U ). Being the lower approximation of A, A L is an exact subset of X L and hence A L ∈ T L therefore A is lower rough open. Also being the upper approximation of A, A U is exact subset of X U and hence A U ∈ T U therefore A is upper rough open. That is A= (A L ,A U ) is lower and upper rough open and therefore rough open subset of X. Since A is arbitrary, every subset of a Discrete Rough Topological Space is rough open.

Conclusion:
We found that every Rough topological space satisfies equivalence relation in this paper.