Bitopological rough approximations with medical applications

1 Introduction

Rough set theory, proposed by Pawlak in the early 1980s (Pawlak, 1981), is an extension of set theory for the study of intelligent systems characterized by inexact, uncertain or insufficient information. Moreover, the theory may serve as a new mathematical tool to soft computing besides fuzzy set theory (Pawlak and Peters, 2007; Pawlak and Skowron, 2007a,b,c; Pawlak, 1981a, 1991, 2004; Peters et al., 2006a,b, 2007a,b; Peters and Henry, 2009; Peters and Ramanna, 2007, 2009; Peters, 2007a,b,c, 2008a,b, 2009), and has been successfully applied in machine learning, pattern recognition, expert systems, data analysis, and so on. Recently, lots of researchers are interested in the theory (Polkowski and Skowron, 1997; Polkowski, 2002; Puzio and Walczak, 2008; Randen and Husoy, 1999; Slowinski and Vanderpooten, 2000; Wasilewska, 1997; Yao, 1998a,b; Zadeh, 1965; Zakowski, 1983).

In Pawlak’s original rough set theory, partition or equivalence (indiscernibility) relation is an important and primitive concept. But, partition or equivalence relation is still restrictive for many applications. To address this issue, several interesting and meaningful extensions to equivalence relation have been proposed in the past, such as tolerance relations (Orłowska, 1985, 1998), similarity relations (Orłowska, 1998), and others (Abd El-Monsef et al., 2007; Gupta and Patnaik, 2008; Hassanien et al., 2009; Henry and Peters, 2008, 2009; Hurtut et al., 2008; Meghdadi et al., 2009). Particularly, Peters has used coverings of an universe for establishing the generalized rough set (Peters and Ramanna, 2007). And an extensive body of research works has been developed (Peters et al., 2007b, 2008; Peters, 2007a,b; Salama and Abu-Donia, 2006; Salama, 2008a,b,c, 2010). In 1997, Wasilewska defined the topological rough algebras. Furthermore, Pawlak (2004) in his long paper have characterized a measure of roughness making use of the concept of rough fuzzy sets in 1995. He also suggested some possible applications of the measure in pattern recognition and image analysis problems. Some results about rough sets and fuzzy sets are obtained by Pawlak and Skowron (2007b).

In this paper, we investigate some important and basic issues of generalized rough sets induced by topological structures. The plan of this paper is as follows.

In Section 2, we recall the basic concepts and properties of the Pawlak’s rough set theory. In Section 3, some new concepts and main results are considered in generalized rough sets induced by two topological structures. In Section 4, we define a measure of roughness based on generalized rough sets with the new approximations, and prove some properties of the measure. Finally, we give an example in order to indicate the use of the measure in Section 5.

2 Basic concepts and properties of the Pawlak’s rough set theory

This section presents a review of some fundamental notions of rough sets. We refer to Hassanien et al. (2009), Orłowska (1998), Pawlak and Skowron (2007a,b,c), Pawlak (1981a,b, 1991, 2004) for details.

Motivation for rough set theory has come from the need to represent subsets of an universe in terms of equivalence classes of a partition of that universe. The partition characterizes a topological space, called approximation space A = (U, R), where U is a set called the universe and R is an equivalence relation (Orłowska, 1985; Pawlak and Skowron, 2007c). The equivalence classes of R are also known as the granules, elementary sets or blocks; we will use [x]_R ⊆ U to denote the equivalence class containing x ∈ U. In the approximation space, we consider two operators $\overline{R}(X)=\{x\in U:[x]{\hspace{0.17em}}_R\cap X\ne \varphi \}$ and R(X) = {x ∈ U: [x]_R ⊆ X}, called the lower approximation and upper approximation of X ⊆ U, respectively. Also let ${\mathit{POS}}_R(X)=\underset{\_}{R}(X)$ denote the positive region of X, ${\mathit{NEG}}_R(X)=U-\overline{R}(X)$ denote the negative region of X and ${\mathit{BN}}_R(X)=\overline{R}(X)-\underset{\_}{R}(X)$ denote the borderline region of X.

The degree of completeness can also be characterized by the accuracy measure, in which ∣X∣ represents the cardinality of set X as follows: $${\alpha }_R(X)=\frac{|\underset{\_}{R}(X)|}{|\overline{R}(X)|}\text{,}\mathrm{where}X\ne \varphi$$ Accuracy measures try to express the degree of completeness of knowledge. ${\alpha }_R(X)$ is able to capture 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 its 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:

• If R(X) ≠ ϕ and $\overline{R}(X)\ne U$ , then X is roughly R-definable.

• If R(X) = ϕ and $\overline{R}(X)\ne U$ , then X is internally R-undefinable.

• If R(X) ≠ ϕ and $\overline{R}(X)=U$ , then X is externally R-undefinable.

• If R(X) = ϕ and $\overline{R}(X)=U$ , then X is totally R-undefinable.

We denote the set of all roughly R-definable (resp. internally R-undefinable, externally R-undefinable and totally R-undefinable) sets by RD(U) (resp. REUD(U), RUD(U) and RTUD(U)).

With ${\alpha }_R(X)$ and classifications above we can characterize rough sets by the size of the boundary region and structure. Rough sets are treated as a special case of relative sets and integrated with the notion of Belnap’s logic (Orłowska, 1998).

A topological space (Hurtut et al., 2008) is a pair (U, τ) consisting of a set U and family τ of subset of U satisfying the following conditions:

• ϕ, U ∈ τ.

• τ is closed under arbitrary union.

• τ is closed under finite intersection.

The pair (U, τ) is called a topological space, the elements of U are called points of the space, the subsets of U belonging to τ are called open sets in the space, and the complement of the subsets of U belonging to τ are called closed sets in the space; the family τ of open subsets of U is also called a topology for U.

${\overline{X}}_{\tau }=\bigcap \{F\subseteq U:X\subseteq F\text{,}F\in {\tau }^c\}$ is called τ-closure of a subset X ⊂ U. Evidently, ${\overline{X}}_{\tau }$ is the smallest closed subset of U which contains X. Note that X is closed iff $X={\overline{X}}_{\tau }$ .

${\underset{\_}{X}}_{\tau }=\bigcup \{G\subseteq U:G\subseteq X\text{,}G\in \tau \}$ is called the τ-interior of a subset X ⊆ U. Evidently, X_τ is the union of all open subsets of U which containing in X. Note that X is open iff X = X_τ. And $X_b={\overline{X}}_{\tau }-{\underset{\_}{X}}_{\tau }$ is called the τ-boundary of a subset X ⊆ U.

Let X be a subset of a topological spaces (U, τ). Let ${\overline{X}}_{\tau }$ , X_τ and X_b be closure, interior, and boundary of X, respectively. X is exact if X_b = ϕ, otherwise X is rough. It is clear X is exact iff ${\overline{X}}_{\tau }={\underset{\_}{X}}_{\tau }$ . In Pawlak space a subset X ⊆ U has two possibilities rough or exact.

3 Generalized rough sets induced by two topological structures

In this section, we introduce and investigate the concept of ulu₁₂ (ulu₂₁)-approximation space. Also, we introduce the concepts of ulu₁₂ (ulu₂₁)-lower approximation and ulu₁₂ (ulu₂₁ and 12-21-ulu)-upper approximation and study their properties.

We see from Table 1 that the degree of exactness of these subsets by using 12-21-ulu-accuracy measure is equal to 100% of the chosen subsets. Consequently 12-21-ulu-accuracy measure is refinement of the last measures.

The universe U can be divided into 24 regions with respect to any X ⊆ U as follows:

• The internal edge of X, Edg(X) = X − R(X).

• The τ-internal edge of X, ${\underset{\_}{\mathit{Edg}}}_{\tau }(X)=X-{\underset{\_}{R}}_{\tau }(X)$ .

• The 12-21-ulu-internal edge of X, ${\underset{\_}{\mathit{Edg}}}_{12\text{-}21\text{-}\mathit{ulu}}(X)=X-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• The external edge of X, $\overline{\mathit{Edg}}(X)=\overline{R}(X)-X$ .

• The τ-external edge of X, ${\overline{\mathit{Edg}}}_{\tau }(X)={\overline{R}}_{\tau }(X)-X$ .

• The 12-21-ulu-external edge of X, ${\overline{\mathit{Edg}}}_{12\text{-}21\text{-}\mathit{ulu}}(X)={\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-X$ .

• The boundary of X, $b(X)=\overline{R}(X)-\underset{\_}{R}(X)$ .

• The τ-boundary of X, $X_b={\overline{R}}_{\tau }(X)-{\underset{\_}{R}}_{\tau }(X)$ .

• The 12-21-ulu-boundary of X, $X_{12\text{-}21\text{-}\mathit{ulu}}={\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• The exterior of X, $X^{\mathit{ex}}=X-\overline{R}(X)$ .

• The τ-exterior of X, $X_{\tau }^{\mathit{ex}}=X-{\overline{R}}_{\tau }(X)$ .

• The 12-21-ulu-exterior of X, $X_{12\text{-}21\text{-}\mathit{ulu}}^{\mathit{ex}}=X-{\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• $\overline{R}(X)-{\underset{\_}{R}}_{\tau }(X)$ .

• $\overline{R}(X)-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• ${\overline{R}}_{\tau }(X)-\underset{\_}{R}(X)$ .

• ${\overline{R}}_{\tau }(X)-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• ${\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-\underset{\_}{R}(X)$ .

• ${\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-{\underset{\_}{R}}_{\tau }(X)$ .

• ${\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-{\underset{\_}{R}}_{\tau }(X)$ .

• ${\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)-\underset{\_}{R}(X)$ .

• ${\underset{\_}{R}}_{\tau }(X)-\underset{\_}{R}(X)$ .

• $\overline{R}(X)-{\overline{R}}_{\tau }(X)$ .

• $\overline{R}(X)-{\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

• ${\overline{R}}_{\tau }(X)-{\overline{R}}_{12\text{-}21\text{-}\mathit{ulu}}(X)$ .

4 Properties of 12-21-ulu-approximations

In this section, we introduce a generalization for some of the concepts of rough set theory by using the 12-21-ulu-lower and the 12-21-ulu-upper approximations.

The converse of Remark 4.2 may not be true in general as seen in the following examples.

We investigate 12-21-ulu-rough equality and 12-21-ulu-rough inclusion based on rough equality and inclusion.

The converse of all parts of Proposition 4.1, may not be true in general as seen in the following example.

The following example shows the above remark.

5 Medical applications

In this section, we briefly describe the rheumatic fever data sets used in this study as a topological application of data reduction (Abd El-Monsef et al., 2007; Salama and Abu-Donia, 2006, 2008; Salama, 2008a,b,c, 2010). No doubt that rheumatic fever is a very common disease and it has many symptoms that differ from one patient to another though the diagnosis is the same. So, we obtained the following example on four rheumatic fever patients. All patients were between 9 and 12 years old with a history of Arthurian which began from age 3 to 5 years. This disease has many symptoms and it usually starts at young age and persists with the patient all through his life.

Table 3 contains information on seven patients characterized by eight symptoms (attributes) which were used to decide the diagnosis for each patient (decision attribute), where the attributes are shown in Table 2.

Here we will give the main conventions that we will apply in this section. These conventions will be indicated by examples.

The structure $\mathit{GMIS}=(U\text{,}\mathit{At}\text{,}\{V_a:a\in \mathit{At}\}\text{,}{f}_a\text{,}\{{\eta }_B:B\subseteq \mathit{At}\})$ is called generalized multi-valued information system, where U is a non-empty finite set of objects (persons, planets, cars, digits, etc.) called the universe. Any set X ⊆ U is called a category in U. V_a is a collection of value sets corresponding to the attribute a ∈ At. f_a: U → V_a is a total information function such that f_a(x) ∈ V_a. ${\eta }_B$ is a binary relation defined on U, which is not necessary to be an equivalence relation. Here, we consider ${\eta }_a$ as an example of non-equivalence relation on U which is defined by: for $B\in \mathit{At}\text{,}{\eta }_B=\{(x\text{,}y):[f_a(x)]{\hspace{0.17em}}^c\subseteq f_a(y)\}\forall a\in B\text{,}B\subseteq \mathit{At}$ . Clearly, ${\eta }_B$ is not reflexive, not transitive, but it is symmetric. For a ∈ At, the class $A_{{\eta }_a}$ , is defined by: $A_{{\eta }_a}=\{{\eta }_{\mathit{ax}}:x\in U\}$ , where ${\eta }_{\mathit{ax}}=\{y:x{\eta }_{a}y\}$ .

If D is the decision attribute, then the generalized multi-valued information system will take the form $\mathit{GMIS}=(U\text{,}\mathit{At}\cup D\text{,}\{V_a:a\in \mathit{At}\}\text{,}{f}_a\text{,}\{{\eta }_B:B\subseteq \mathit{At}\})$ . In this case, we suggest the following non-equivalence relation for the decision attribute: $${\eta }_D=\{(x\text{,}y):f_D(x)\subseteq f_D(y)\}\text{.}$$

The concept of this relation is defined as η_Dx = {y: xη_Dy}. The set of all concepts is defined by $A_{{\eta }_a}=\{{\eta }_{\mathit{ax}}:x\in U\}$ . Also, if D is the decision attribute and for a ∈ At, we have ${\mathit{POS}}_a(D)={\cup }_{Y\in A_{{\eta }_D}}a-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(Y)$ , where $$a-{\underset{\_}{R}}_{12\text{-}21\text{-}\mathit{ulu}}(Y)=a-{\underset{\_}{R}}_{{\mathit{ulu}}_{12}}(Y)\cup a-{\underset{\_}{R}}_{{\mathit{ulu}}_{21}}(Y)$$ where $a-{\underset{\_}{R}}_{{\mathit{ulu}}_{12}}(Y)$ and $a-{\underset{\_}{R}}_{{\mathit{ulu}}_{21}}(Y)$ are the lower approximations defined in Definition 3.3, by using the attribute a ∈ At.

Let us take $\{A_{{\eta }_a}:a\in \mathit{At}\}$ as a subbase of a topological space τ_a (the set of all finite intersections and arbitrary unions of members of $A_{{\eta }_a})$ and $\{A_{{\eta }_B}:B\subseteq \mathit{At}\}$ as a subbase of a topological space τ_B. The decision makes the topology τ_D which has $\{A_{{\eta }_D}\}$ as a subbase. Hence, we can say that the set of attributes B ⊆ At is a reduct if τ_B < τ_D and B is a minimal, where τ_B < τ_D iff $\forall G\in {\tau }_B\text{,}\exists G^{\prime }\in {\tau }_D$ s.t. G ⊆ G′, G, G′ ≠ U.

A set of attribute B depends totally on a set of attributes A denoted by A ⇒ B, if all values sets of attributes from B are uniquely determined by values sets of attributes from A. Let A and B be subsets of At, we say that B depends on A in a degree K (0 ⩽ K ⩽ 1), denoted by: $A{\Rightarrow }_{K}B\mathrm{if}K=\gamma (A\text{,}B)=\frac{\left|{\mathit{POS}}_A(B)\right|}{\left|U\right|}$ .

If K = 1, B depends totally on A. If K < 1, B depends partially (in a degree K) on A.

If we take A = At and B = D in the above two issues, where At is the set of condition attributes and D is the decision attribute, then we say that, D depends totally on At, denoted by At ⇒ D, if all values of attributes from D are uniquely determined by values sets of attributes from At. Otherwise, we say that D depends on At in a degree K, denoted by At ⇒_K D.

Table 4 shows the coding of the data, which is described as follows: Sex (S) = {M, F} = {0, 1}, Pharyngitis (F) = {yes, no} = {1, 0}, Arthritis A = {a₀, a₁, a₂} = {0, 1, 2}, Carditis R = {affected, not affected} = {1, 0}, Chorea K = {yes, no} = {1, 0}, ESR E = normal, high = {0, 1}, Abdominal Pain P = {absent, present} = {0, 1} and Headache H = {yes, no} = {1, 0}. The decision attribute is Diagnosis D = {rheumatic arthritis, rheumatic carditis, rheumatic arthritis and carditis} = {d1, d2, d3}.

Then we constrain the MIS as shown in Table 5.

From the relation R_a = {(x, y): f_a(x) ⊆ f_a(y)}, where a is an element of the power set of the set of condition attributes {α, β, δ}. The two subbases of two topologies for each element of the power set of {α, β, δ} are defined as: ${\xi }_1^a=\{{\mathit{xR}}_a:x\in U\}$ , where xR_a = {y: xR_ay} and ${\xi }_2^a=\{R_{a}x:x\in U\}$ , where R_ax = {y: yR_ax}. Then according to Table 5 we have the following couples of topologies: $$\begin{array}{cc}\hfill & {\tau }_1^{\alpha }=\{U\text{,}\varphi \text{,}\{p_2\}\text{,}\{p_3\}\text{,}\{p_2\text{,}{p}_3\}\text{,}\{p_1\text{,}{p}_2\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\}\text{,}\hfill \\ \hfill & \{p_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_2\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\}\text{,}\hfill \\ \hfill & {\tau }_2^{\alpha }=\{U\text{,}\varphi \text{,}\{p_1\}\text{,}\{p_3\}\text{,}\{p_1\text{,}{p}_3\}\text{,}\{p_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_3\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\hfill \\ \hfill & \{p_1\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\}\text{,}\hfill \\ \hfill & {\tau }_1^{\beta }=\{U\text{,}\varphi \text{,}\{p_5\}\text{,}\{p_7\}\text{,}\{p_3\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_4\}\text{,}\{p_5\text{,}{p}_7\}\text{,}\{p_3\text{,}{p}_5\text{,}{p}_7\}\text{,}\hfill \\ \hfill & \{p_1\text{,}{p}_4\text{,}{p}_5\}\text{,}\{p_1\text{,}{p}_4\text{,}{p}_5\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_4\text{,}{p}_6\}\}\text{,}\hfill \\ \hfill & {\tau }_2^{\beta }=\{U\text{,}\varphi \text{,}\{p_3\text{,}{p}_5\text{,}{p}_7\}\text{,}\{p_2\text{,}{p}_3\text{,}{p}_6\}\text{,}\{p_2\text{,}{p}_3\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_4\text{,}{p}_6\}\text{,}\hfill \\ \hfill & \{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\}\text{,}\{p_2\text{,}{p}_3\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\}\text{,}\hfill \\ \hfill & \{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\text{,}{p}_7\}\}\text{,}\hfill \\ \hfill & {\tau }_1^{\delta }=\{U\text{,}\varphi \text{,}\{p_4\}\text{,}\{p_5\}\text{,}\{p_2\text{,}{p}_7\}\text{,}\{p_4\text{,}{p}_5\}\text{,}\{p_2\text{,}{p}_4\text{,}{p}_7\}\text{,}\{p_2\text{,}{p}_5\text{,}{p}_7\}\text{,}\hfill \\ \hfill & \{p_2\text{,}{p}_4\text{,}{p}_5\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\}\text{,}\hfill \\ \hfill & {\tau }_2^{\delta }=\{U\text{,}\varphi \text{,}\{p_4\}\text{,}\{p_1\text{,}{p}_3\text{,}{p}_6\}\text{,}\{p_1\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\}\text{,}\{p_1\text{,}{p}_3\text{,}{p}_5\text{,}{p}_6\}\text{,}\hfill \\ \hfill & \{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_3\text{,}{p}_4\text{,}{p}_5\text{,}{p}_6\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\text{,}{p}_7\}\}\hfill \\ \hfill & {\tau }_1^{\alpha \beta }={\tau }_1^{\alpha }\cap {\tau }_1^{\beta }=\{U\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_2^{\alpha \beta }={\tau }_2^{\alpha }\cap {\tau }_2^{\beta }=\{U\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_1^{\alpha \delta }={\tau }_1^{\alpha }\cap {\tau }_1^{\delta }=\{U\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_2^{\alpha \delta }={\tau }_2^{\alpha }\cap {\tau }_2^{\delta }=\{U\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_1^{\beta \delta }={\tau }_1^{\beta }\cap {\tau }_1^{\delta }=\{U\text{,}\varphi \text{,}\{p_5\}\}\text{,}\hfill \\ \hfill & {\tau }_2^{\beta \delta }={\tau }_2^{\beta }\cap {\tau }_2^{\delta }=\{U\text{,}\varphi \text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\text{,}{p}_7\}\}\text{,}\hfill \\ \hfill & {\tau }_1^{\alpha \beta \delta }={\tau }_1^{\alpha }\cap {\tau }_1^{\beta }\cap {\tau }_1^{\delta }=\{U\text{,}\varphi \}\text{,}\hfill \\ \hfill & {\tau }_2^{\alpha \beta \delta }={\tau }_2^{\alpha }\cap {\tau }_2^{\beta }\cap {\tau }_2^{\delta }=\{U\text{,}\varphi \}\text{.}\hfill \end{array}$$

Now we will deal with the decision attribute D applying the relation: η_D = {(x, y): D(x) ⊆ D(y)}, then the subbase of the decision topology is ${\xi }_1^D=\{{\mathit{xR}}_D:x\in U\}=\{\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_4\}\text{,}\{p_5\}\}$ . Then the decision topology is given by: τ_D = {U, ϕ, {p₁, p₂, p₃, p₆, p₇}, {p₄}, {p₅}, {p₄, p₅}, {p₁,p₂,p₃,p₄, p₆, p₇}, {p₁, p₂, p₃, p₅, p₆, p₇}}, the complement decision topology is $${\tau }_D^c=\{U\text{,}\varphi \text{,}\{p_4\text{,}{p}_5\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_5\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_4\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_1\text{,}{p}_2\text{,}{p}_3\text{,}{p}_6\text{,}{p}_7\}\text{,}\{p_5\}\text{,}\{p_4\}\}\text{.}$$

We can observe that ${\tau }_1^{\beta \delta }⩽{\tau }_D$ and ${\tau }_2^{\beta \delta }⩽{\tau }_D^c$ , which lead to {β, δ} = {F, A, E, R, P, H} which is the reduct and the core of our system. This means that we can remove the attributes {S, K} without losing any information.

6 Conclusion

It is well known that rough set theory has been regarded as a generalization of classical set theory in one way. Furthermore, this is an important mathematical tool to deal with uncertainty. As a natural need, it is a fruitful way to extend classical rough sets to generalized rough sets induced by topological spaces. In this paper, new lower and upper approximations are proposed in generalized rough set induced by a topological structure, and some important properties are obtained. Also, we define the concept of a rough membership function in generalized topological approximation spaces. It is a generalization of classical rough membership function of Pawlak rough sets. The rough membership function can be used to analyze which decision should be made according to a conditional attribute in decision table.