GENERALIZED TOPOLOGICAL APPROXIMATION SPACES AND THEIR MEDICAL APPLICATIONS

We generalize Pawlak's approximation space to topological approximation spaces using some topological near open sets, such as regular open sets, semi-open sets, pre-open sets,  -open sets,  -open sets and  -open sets and others. The properties of the topological approximations space will be studied. Finally, We applied our results on medical data.


Introduction
Rough set theory, proposed by Pawlak in [1], is an extension of set theory for the study of intelligent systems characterized by inexact, uncertain or insufficient information.Many suggestions have been made for generalizing and interpreting rough sets [2][3][4].Moreover, this theory may serve as a new mathematical tool to soft computing and has been successfully applied in machine learning, information sciences, expert systems, data reduction, and so on.Recently, lots of researchers are interested to generalize this theory in many fields of applications [5][6][7][8][9].Several interesting and meaningful generalizations to equivalence relation have been proposed in the past, such as topological bases and subbases [10,11,12].Particularly, some researchers have used coverings of the universe of discourse for establishing the generalized rough sets by coverings [13,14].
Rough set theory is a recent approach for reasoning about data.This theory depends basically on a certain topological structure and has achieved a great success in many fields of real life applications.The concept of a topological rough set given by Wiweger [15] in 1989 is one of the most important topological generalization of rough sets.In 1983 Abd El-Monsef in [16] introduced the concept of  -open sets.This paper is organized as follows: In Section 2 we give topological basic concepts.Also, Section 3 discussed the fundamentals of rough sets and investigated the concept of topological approximation space.Section 4 is devoted to introduce medical application example.The paper , s conclusion is given in Section 5.

Topological Basic Concepts
More details about the following topological near open sets found in [17][18][19][20].Definition 2. 1 [17,18] A subset A of a topological space ) , is the intersection of all semi-closed (resp. -closed, semi-pre-closed) sets that contains A and is denoted by ).The union of all semi-open subsets of U is called the semi-interior of A and is denoted by (3) generalized semi-closed set (briefly − gs closed ) if . The equivalence classes of R are also known as the granules, elementary sets or blocks; we will use . In the approximation space, we consider two operators called the lower approximation and upper approximation of The degree of completeness can also be characterized by the accuracy measure, in which X represents the cardinality of set X as follows:

Topological approximation spaces
this section, we introduce and investigate the concept of topological approximation space.Also, we introduce the concepts of topological lower approximation and topological upper approximation and study their properties.Definition 3.1 Let  = ( , ) be an space with general relation  and   is the topology associated with  .Then the triple   = (, ,   ) is called a topological approximation space.
Let (, ,   ) be a topological approximation space.The Universe U can be divided into many regions with respect to any  ⊆  and with respect to any m{semi, pre, α, β, regular, semi-regular, δ, g, sg, gs, αg, gα, gα^(**), gsp, δg, Q} as follows: Let (, ,   ) be a topological approximation space.For any m{semi, pre, α, β, regular, semi-regular, δ, g, sg, gs, αg, gα, gα^(**), gsp, δg, Q}and for any subset XU  we define the following memberships: x belongs strong to X if ,x belongs weak to X if topology associated with this relation is τR= {U, φ, {a}, {c, d}, {a, c, d}} .So (, ,   )is a topological approximation space.Let X = {b, c, d}, we have b is m-strong belongs to X but b is not strong belongs to X .Also, let Y = {c} be another subset of U. Then we have d is weak belongs to  but d is not m-weak belongs to Y.
The degree of topological completeness can also be characterized by the topological accuracy measure (maccuracy), in which represents the cardinality of set as follows: , where According to Example 3.1, Table 1 showing the differences among the degree of Pawlak's accuracy measure α(X) and βaccuracy measure αβ (X) for some subsets of U if we take m=β.The β-accuracy measure of the class of all β-open sets is accurate than the others measures and the following example and its followed diagram illustrate this fact.

.
The complement of a − sg closed set is called a − sg open set.
rules: 1: If patient  =  and  =  and ℎ =  Then  = , 2: If patient  =  ℎ and  =  and ℎ =  Then  = , 3: If patient  = ℎ and  =  and ℎ =  Then  = .According to  -rough set approach ) is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ).) is the set of all  −  sets in   = (, ,   ).(6) Semi-regular lower approximation of ,where () and () are the set of all  −  sets and  −  sets in   = (, ,   ) is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ).
Rprewhere () is the set of all  −  sets in   = (, ,   ).) is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ).(5) Regular-upper approximation of , where () is the set of all  −  sets in   = (, ,   ).(6) Semi-regular upper approximation of , where () and () is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  sets in   = (, ,   ). is the set of all  −  in   = (, ,   ).Motivation for topological rough set theory has come from the need to represent subsets of a universe in terms of topological classes of the topological base generated by the general binary relation defined on the universe.That base characterizes a topological space, called topological approximation space   = (, ,   ).The topological classes of  are also known as the topological granules, topological elementary sets or topological blocks; we will use ,  ∈ {, , , , ,  − , , , , , , ,  * * , , , } to denote the

Table 2
is shown the patients information system and respective symptoms, and the data are of the discrete type.

Table 3 :
Reduced of Table 1 using Pawlak's rough approach According to Table 2 above we have the following Decision Table 2 is reduced to Table 4 below.