ROUGH TOPOLOGY ON APPROXIMATION SPACES

: Though there are many studies on the topological properties of rough set approximations, only a few works have been done on the concept of rough topology. In this paper, a new definition of rough topology on an approximation space is proposed using the rough subsets of the extended approximation space. The basic concepts of a topological space are extended to the proposed rough topological space and the properties are investigated.


I. INTRODUCTION
The theory of rough sets was formulated by Z. Pawlak in 1982 [18].The mathematical framework of this potential theory has been enriched by the contribution of many researchers in various dimensions.An interesting and natural research area is the study of the interconnections between rough set theory and topology theory.The topological structure of rough sets forms an important base for information analysis and knowledge processing [6,16,20,28].
There are mainly two streams of study connecting rough set theory and topology theory.The first one is regarding the topology induced by rough set approximations.In his seminal paper itself, Z. Pawlak [18] pointed out that the set of all equivalence classes in an approximation space formed a base for a topology on the set U and the set of all equivalence classes of the rough equality relation formed a base for a topology on the power set of U.Both the topologies were found to be quasi discrete.Further studies on the topological properties of rough sets can be found in [3,10,11,13,15,20,21,27,29].The second stream consists of the studies on the different types of approximation spaces induced by different topological spaces [1,6,12,23,24,25].Some papers include results on both the streams [2,14].Regarding information systems, T. Herawan [4,5] worked on the topology on an information system.
Though there are many studies on the relations between rough set approximations and topological spaces, only a few works are there on the concept of rough topology.Q. Wu et al. [28] defined rough topology on a rough set by using a metric and then extended it to general topological space.M. L. Thivagar et al. [26] introduced the concept of rough topology which consists of the null set, the universal set, the lower approximation, upper approximation and the boundary region of a subset of U. B. P. Mathew and S. J. John [17] introduced the concept of rough topology on a rough set as a pair of topologies of exact subsets of the lower and upper approximations of the rough set under consideration.A follow up work was done by M. Ravindran and A. J. Divya [22] who studied the properties of compactness and connectedness and the separation axioms in rough topological spaces.But none of them regard rough topology as a rough subset of the power set.
In this paper, a new definition for rough topology on an approximation space is proposed using the rough subsets of the extended approximation space.The basic concepts of topology are extended to the proposed rough topology and the properties are investigated.This paper is organized as follows.In section 2, some preliminary concepts of rough set theory are recalled.In section 3, the concept of rough topology on an approximation space is introduced.Section 4 deals with some of the basic concepts of rough topology and the conclusion is given in section 5.

II. PRELIMINARIES
In this section, some basic notions of rough set theory are recalled.Further details of rough set theory can be found in [19].The basic concepts of topology theory are described in [9].

A. Rough Set Theory
The theory of rough sets was introduced by Z. Pawlak [18] in 1982.Consider the approximation space , , where U is a non-empty set of objects and R is an equivalence relation on U. The equivalence classes of R are called elementary sets and sets which can be expressed as union of some equivalence classes are called composed sets.
The lower and upper approximations [19] of , with respect to R are defined respectively as (1) (2) where, [x] R is the equivalence class of R containing x.In other words, is the union of equivalence classes that are contained in A and is the union of equivalence classes that have non-empty intersection with A. Thus, both lower and upper approximations are composed sets on , .
The set is called the positive region, is called the negative region and the set is called the boundary region.The properties of the rough set approximations are as follows: 1) 2) 3) for all

B. Operations on Rough Sets
The term 'rough set' has been used in two different viewpoints by Z. Pawlak.According to his original proposal, the equivalence classes of the rough equality relation were termed as rough sets [18].Later, in [19], a subset with or was called rough set.Rough sets may also be described using rough membership functions [19].There is yet another approach to rough set theory proposed by T. B. Iwinski [7].Throughout this paper, the word rough set refers to a pair , , where and are subsets of U such that and , for some [8].For convenience, a rough set may be denoted by , , where .The rough inclusion, rough union, rough intersection and rough complement operations of rough sets are defined in [8] as , , ,

B. Topology and Rough Sets
Let , be an approximation space.Then, the lower approximation operator satisfies the properties of an interior operator and hence it induces a topology on [1].A subset if and only if .Hence, consists of all composed sets on U. By properties (10) and (11), a set A will be closed iff . Hence all open subsets are closed and is the quasi discrete topology.The family of all equivalence classes form a basis for [1].
Let , be a rough subset of the approximation space (U, R).Let and be any two topologies which contain only exact subsets of and respectively.Then the pair , is said to be a rough topology on U [17].

III. ROUGH TOPOLOGY ON APPROXIMATION SPACES
Let (U, R) be an approximation space, where U is a nonempty set of objects and R is an equivalence relation on U. Consider the rough equality relation on , given by , (7) Then, is an equivalence relation on and the pair , is called the extended approximation space corresponding to (U,R) [1].Hence, we can extend the definition of rough set approximations to any subfamily .

Definition 3.1:
Let , be an approximation space and be the rough equality relation.Then, the -lower and -upper approximations of a sub family are given by (8) (9) respectively.

Corollary 3.1:
, where is the topology on U induced by R. Proof: Being the topology on U induced by R, consists of all composed subsets of U. Hence by theorem 3.2, .

Definition 3.2:
Let be a subfamily of .The pair , is called an -rough topology on U, if both and are topologies on U. The set U together with the -rough topology is called an -rough topological space.

Definition 3.3:
Let be a subfamily of .The pair , is called a lower -rough topology on U, if is a topology on U.

Definition 3.4:
Let be a subfamily of .The pair , is called an upper -rough topology on , if is a topology on .

Let
, be an -rough topology on .

Definition 3.8:
Any -rough topology on U which is equivalent to , is called a discrete -rough topology.

Theorem 3.6:
The pair , is an -rough topology on U, where is the discrete topology on U. Proof: The discrete topology on U consists of all subsets of U. Hence, .Since, , we get, .Therefore, both and are topologies on U. Thus, , is an rough topology on U.

Theorem 3.7:
The pair , is a discrete -rough topology on U. Proof: space such as rough open set, rough closed set, rough interior and rough closure were extended to the proposed rough topological space and the properties were investigated.