## 1 Introduction

The concept of rough set which was first proposed by Pawlak [11] is an extension of set theory for the study of the intelligent systems characterized by insufficient and incomplete information in 1982. It is a useful and powerful tool in many fields, such as data analysis, granularity or vagueness. It has also been applied successfully in process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis, and so on. Many researchers have made some significant contributions to developing the rough theory [3,4,7-8,12,14-15,17-26]. However, a problem with Pawlak's rough set theory is that partition or equivalence relation is explicitly used in the definition of the lower and upper approximations. Such a partition or equivalence relation is too restrictive for many applications because it can only deal with complete information systems. To address this issue, generalizations of rough set theory were considered by scholars. One approach was to extend equivalence relation to tolerance [5-6,32] and others [20-21,33-34]. Another important approach was to relax the partition to a covering of the universe. In 1983, W.Zakowski generalized the classical rough set theory by using coverings of a universe instead of partitions [27]. The covering-based rough sets is one of the most important generalization of the classical Pawlak rough sets. Such generalization leads to various covering approximation operators that are both of theoretical and practical importance [28-30]. The relationships between properties of covering-based approximation and their corresponding coverings have attracted intensive research. Based on the mutual correspondence of the concepts of extension and intension, E.Bryniarski [36] and Z.Bonikowski et al. [37] gave the second type of covering-based rough sets. The third and the fourth type of covering-based rough sets were introduced in [38]. Subsequently, W.Zhu utilized the topological method to characterize covering rough sets [18]. W.Zhu and F. Wang discussed the relationship between properties of four types of covering-based upper approximation operators and their corresponding coverings [21-23]. T. Yang et al. researched attribute reduction of covering information systems [39]. G. Liu studied the two types of rough sets induced by coverings and obtained some interesting results. G. Cattaneo, D. Ciucci, and G. Liu obtained the algebraic structures of generalized rough set theory [1,2,40,44]. G. Liu also used axiomatic method to characterize covering-based rough sets [41-43]. X. Bian et al. gave characterizations of covering-based approximation spaces being closure operators [31]. Ge et al. proposed not only general but also topological characterizations of coverings for these operators being closure operators [25,29-30]. T.Lin et al. defined some approximation operators which are based on neighborhood systems and did some research on them [52]. In addition, Y. Zhang et al. discussed the relationships between generalized rough sets based on covering and reflexive neighborhood system [49] and proposed the operator *NS*(*U*) and *S*. We also give characterizations for

This paper is organized as follows. Section 2 recalls the main ideas of generalized rough set and covering approximations. Section 3 gives the properties of *NS*(*U*) and some examples. Section 4 studies the characterization of *NS*(*U*) for *S* and *S* for covering-based upper approximation operator

## 2 Background

In this section, we introduce the fundamental concepts that are used in this paper. *U* is the universe of discourse and *U*. If *U*, none of sets in *U*.

([46]). *A mapping* *n* : *U* → *P*(*U*) *is called a neighborhood operator If* *n*(*x*) ≠ ∅ *for all* *x* ∈ *U*, *n* *is called a serial neighborhood operator If* *x* ∈ *n*(*x*) *for all* *x* ∈ *U*, *n* *is called a reflexive neighborhood operator*

([50-51]). *A neighborhood system of an object* *x* ∈ *U*, *denoted by* *NS*(*x*), *is a non-empty family of neighborhoods of* *x*. *The set* {*NS*(*x*) : *x* ∈ *U*} *is called as a neighborhood system of* *U*, *and it is denoted by* *NS*(*U*). *Let* *NS*(*U*) *be a neighborhood system of* *U*.

*NS*(*U*) *is said to be serial if for any* *x* ∈ *U* *and* *n*(*x*) ∈ *NS*(*x*), *n*(*x*) *is non-empty* (*called* *Fré*(*V*) *Space in* [32]).

*NS*(*U*) *is said to be reflexive, if for any* *x* ∈ *U* *and* *n*(*x*) ∈ *NS*(*x*), *x* ∈ *n*(*x*).

*NS*(*U*) *is said to be symmetric, if for any* *x*, *y* ∈ *U*, *n*(*x*) ∈ *NS*(*x*) *and* *n*(*y*) ∈ *NS*(*y*), *x* ∈ *n*(*y*) ⇒ *y* ∈ *n*(*x*).

*NS*(*U*) *is said to be transitive, if for any* *x*, *y*, *z* ∈ *U*, *n*(*y*) ∈ *NS*(*y*) *and* *n*(*z*) ∈ *NS*(*z*), *x* ∈ *n*(*y*) *and* *y* ∈ *n*(*z*) ⇒ *x* ∈ *n*(*z*).

(Covering approximation space [23]). *If* *U* *is an universe and* *is a covering of* *U*, *then we call* *U* *together with covering* *a covering approximation space, denoted by*

([52]). *Let* *NS*(*U*) *be a neighborhood system of U. The lower and upper operators of* *X* *are defined as follows*:

([49]). *Let* *NS*(*U*) *be a neighborhood system of* *U*.

*NS*(*U*) *is referred to as weak-unary, if for any* *x* ∈ *U* *and* *n*_{1}(*x*), *n*_{2}(*x*) ∈ *NS*(*x*), *there exists an* *n*_{3}(*x*) ∈ *NS*(*x*) *such that* *n*_{3}(*x*)⊆ *n*_{1}(*x*)∩ *n*_{2}(*x*);

*NS*(*U*) *is referred to as weak-transitive, if for any* *x* ∈ *U* *and* *n*(*x*) ∈ *NS*(*x*), *there exists an* *n*_{1}(*x*) ∈ *NS*(*x*) *satisfying thatfor any* *y* ∈ *n*_{1}(*x*), *there exists an* *n*(*y*) ∈*NS*(*y*) *such that* *n*(*y*)⊆ *n*(*x*);

*NS*(*U*) *is referred to as a* *weak-S*_{4} *neighborhood system, if* *NS*(*x*) *is reflexive and weak-transitive*.

The following topological concepts and facts are elementary and can be found in[47,51]. We list them below for the purpose of this paper being self-contained.

- A topological space is a pair (
*U*, τ) consisting of a set*U*and a family τ of subsets of*U*satisfying the following conditions: (a) ∅ ∈ τ and*U*∈ τ (b) If*U*_{1},*U*_{2}∈ τ, then*U*_{1}∩*U*_{2}∈ τ; (c)If , then$\mathcal{A}\subseteq \tau $ is called a topology on$\cup \mathcal{A}\in \tau .\tau $ *U*and the members of τ are called open sets of (*U*, τ). The complementary set of an open set is called a closed set. - A set
*F*is called a clopen set, if*F*in (*U*, τ) is both an open set and a closed set. - A family
is called a base for ($\mathcal{B}\subseteq \tau $ *U*, τ) if for every non-empty open subset*O*of*U*and each*x*∈*O*, there exists a set such that$B\in \mathcal{B}$ *x*∈*B*⊆*O*. Equivalently, a family if every non-empty open subset$\mathcal{B}\subseteq \tau $ *O*of*U*can be represented as the union of a subfamily of .$\mathcal{B}$ - For any
*x*∈*U*, a family is called alocal base at$\mathcal{B}\subseteq \tau $ *x*for (*U*τ) if*x*∈*B*for each , and for every open subset$B\in \mathcal{B}$ *O*of*U*with*x*∈*O*, there exists a set such that$B\in \mathcal{B}$ *B*⊆*O*. - If
is a partition of$\mathcal{P}$ *U*, the topology τ = {*O*⊆*U**O*} is the union of some members of is called a pseudo-discrete topology in [13](also called a closed-open topology in [12]).$\mathcal{P}\cup \{\mathrm{\varnothing}\}$ - Let (
*U*, τ) be a topological space. If for each pair of points*x*,*y*∈*U*with*x*≠*y*, there exist open sets*O*,*O'*such that*x*∈*O*,*y*∈*O*′ and*O*∩*O*′ = ∅, then (*U*, τ) is called a*T*_{2}-space and τ is called a*T*_{2}-topology.

(Induced topology and subspace). *Let* (*U*, τ) *be a topological space and* *X* ⊆ *U*. *It is easy to check that* τ′ = {*O*∩ *X* : *O* ∈ τ} *is a topology on X*. τ′ *is called a topology induced by* *X*, *and the topology space* (*X*, τ′) *is called a subspace of* (*U*, τ).

(Closure operator). *An operator* *H*: *P*(*U*) → *P*(*U*)*ia* *called a closure operator on* *U* *if it satisfies the following conditions*: *for any* *X*, *Y* ⊆ *U*,

(Interior operator). *An operator* *I*: *P*(*U*) → *P*(*U*)*ia* *called* *a* *interior operator on* *U* *if it satisfies the following conditions*: *for any* *X*, *Y* ⊆ *U*,

(Dual operator). *Assume that* *H*, *I*: *P*(*U*) → *P*(*U*) *are two operators on U. If for any* *X* ⊆ *U*, *H*(*X*) = ∼ *I*(∼ *X*). *We say that* *H*, *I are dual operators or* *H* *is the dual operator of* *I*.

([49]). *Let* *NS*(*U*) *be a neighborhood system of U. Then the following are equivalent*:

;${\overline{apr}}_{NS}(X\cup Y)={\overline{apr}}_{NS}(X)\cup {\overline{apr}}_{NS}(Y)for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X,Y\subseteq U$ ;${\underset{\_}{apr}}_{NS}(X\cup Y)={\underset{\_}{apr}}_{NS}(X)\cup {\underset{\_}{apr}}_{NS}(Y)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X,Y\subseteq U$ *NS*(*U*)*is weak-unary*.

([49]). *Let* *NS*(*U*) *be a neighborhood system of U. Then the following are equivalent*:

${\overline{apr}}_{NS}({\overline{apr}}_{NS}(X))\subseteq {\overline{apr}}_{NS}(X)\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X\subseteq U$ ${\underset{\_}{apr}}_{NS}(X)\subseteq {\underset{\_}{apr}}_{NS}({\underset{\_}{apr}}_{NS}(Y))\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}X\subseteq U$ *NS*(*U*)*is weak-transitive*.

## 3 Some propositions of NS(U)

In this section, we will discuss some properties of *NS*(*U*).

*Let* *NS*(*U*) *be a neighborhood system of U*. *NS*(*U*) *is said to be Euclidean, if for any* *x*, *y* ∈ *U*, *and* *n*(*x*) ∈ *NS*(*x*), *y* ∈ *n*(*x*) ⇒ *n*(*x*) ⊆ *n*(*y*).

*Let* *NS*(*U*) *be a neighborhood system of U. If* *NS*(*U*) *is Euclidean, then* *and* *for any* *X* ⊆ *U*.

For any *x* ∈ *U*, if *n*(*x*)∈ *NS*(*U*), We have *n*(*y*) ∈ *NS*(*y*) such that *n*(*y*) ⊆ *X*. Since *NS*(*U*) is Euclidean, we have *n*(*x*) ⊆ *n*(*y*). It follows that *n*(*x*) ⊆ *X*. From the Definition 2.4, We have

We can get

The inverse of Proposition 2.10 does not hold. An example is given as follows:

*Let* *U* = {*a*, *b*, *c*}, *NS*(*a*) = {{*a*}}, *NS*(*b*) = {{*b*}, {*a*, *b*}} *and* *NS*(*c*) = {{*c*}}. *Then*:

*Hence* *for any* *X* ⊆ *U*. *Since* *a* ∈ *n*(*b*) = {*a*, *b*} ∈ *NS*(*b*) *and* *n*(*b*) ⊈ *n*(*a*). *We obtain that* *NS*(*U*) *is not Euclidean*.

*Let* *NS*(*U*) *be a neighborhood system of U. Then the following are equivalent*:

*NS*(*U*)*is transitive*;*For any**x*,*y*∈*U*,*if**x*∈*n*(*y*),*then**n*(*x*)⊆*n*(*y*).

(1) ⇒ (2) For any *x*, *y* ∈ *U* and *x* ∈ *n*(*y*), if *n*(*x*) ⊈ *n*(*y*), there exists *p* ∈ *n*(*x*) and *p* ∉ *n*(*y*). Since *x* ∈ *n*(*y*) and *NS*(*U*) is transitive, we have *p* ∈ *n*(*y*). It is contradictory to *p* ∉ *n*(*y*).

(2) ⇒ (1) for any *x*, *y*, *z* ∈ *U*, *n*(*y*) ∈ *NS*(*y*) and *n*(*z*) ∈ *NS*(*z*), *x* ∈ *n*(*y*) and *y* ∈ *n*(*z*). It is easy to prove *NS*(*U*) is transitive. □

*Let* *NS*(*U*) *be a neighborhood system of* *U* *and* *n* : *U* → *P*(*U*) *is a reflexive mapping. Then the following are equivalent*:

- {
*n*(*x*) :*x*∈*U*}*forms a partition of**U*; *NS*(*U*)*is reflexive, transitive and Euclidean*.

(1) ⇒ (2) For any *x*,*y* ∈ *U* and *y* ∈ *n*(*x*). Since {*n*(*x*) : *x* ∈ *U*} forms a partition of *U* and *n* : *U* → *P*(*U*) is a reflexive mapping, then *n*(*x*) = *n*(*y*). By the Definition 3.1 and Proposition 2.11, it is easy to prove *NS*(*U*) is reflexive, transitive and Euclidean.

(2) ⇒ (1) for any *x*, *y* ∈ *U*, if *n*(*x*) ≠ *n*(*y*), then *n*(*x*)>∩ *n*(*y*) = ∅. Otherwise, we take a *z* ∈ *n*(*x*)∩ *n*(*y*), since *NS*(*U*) is reflexive, transitive and Euclidean, then *n*(*z*) = *n*(*x*) = *n*(*y*). It is contradictory to *n*(*x*) ≠ *n*(*y*). □

*Let* *NS*(*U*) *be a reflexive neighborhood system of U*. *NS*(*U*) *is weak-unary if and only if there is a topology on* *U* *such that* *NS*(*x*) *is a local base for any* *x* ∈ *U*.

(1) ⇒ (2) For any *x* ∈ *U*, *n*_{1}(*x*), *n*_{2}(*x*) ∈ *NS*(*x*) and *x* ∈ *n*_{1}(*x*) ∩ *n*_{2}(*x*), since *NS*(*U*) is weak-unary, there exists *n*_{3}(*x*) such that *n*_{3}(*x*) ⊆ *n*_{1}(*x*) ∩ *n*_{2}(*x*). We have *x* ∈ *n*_{3}(*x*) because *NS*(*U*) is a reflexive neighborhood system of *U*. So *NS*(*x*) is alocal base for any *x* ∈ *U*.

(2) ⇒ (1) By the definition of local base, it is easy to prove *NS*(*U*) is weak-unary. □

## 4 Characterization of *NS*(*U*) for ${\overline{apr}}_{NS}$ being a closure operator

In Section 3, we discuss the properties of *NS*(*U*). One natural question thus arise: When is

(General characterization of of *NS*(*U*) for *Let* *NS*(*U*) *be a neigh*- *borhood system of U*. *is a closure operator if and only if* *NS*(*U*) *is reflexive, weak-unary and weak*- *transitive*.

(⊆) Assume that *X*, *Y* ⊆ *U*. According to Proposition 6 [49] and Proposition 7 [49], we obtain *NS*(*U*) is weak-unary and weak-transitive.

(⇐) Assume that *NS*(*U*) is weak-unary and weak-transitive. We prove that *H*_{i})(for *i* = 1, 2, 3, 4). By the Definition of *H*_{3}) is satisfied. We prove (*H*_{1}) and (*H*_{4}). Since *NS*(*U*) is weak-unary and weak-transitive, by the Proposition 6 [49] and Proposition 7 [49], we obtain *X*, *Y* ⊆ *U*. Since *NS*(*U*) is a reflexive neighborhood system of *U*, so *X* ⊆ *U*. Therefore *X* ⊆ *U*. It is obvious that (*H*_{2}) holds. Thus

(Topological characterization of of *NS*(*U*) for *Let* *NS*(*U*) *be a reflexive neighborhood system of U. Then* *is a closure operator if and only if* *is a base for some topology* τ *on* *U*.

(⇒) Assume that *U*. Since *NS*(*U*) is a reflexive neighborhood system of *U*. It is easy to prove *x* ∈ *n*(*x*) for each *n*(*x*) ∈ *NS*(*x*) and *x* ∈ *U*. Thus *U*. For any *x*, *y*, *z* ∈ *U*, *n*(*y*) ∈ *NS*(*y*), *n*(*z*) ∈ *NS*(*z*) and *x* ∈ *n*(*y*) ∩ *n*(*z*). Since *NS*(*U*) is weak-unary. Therefore, there exists a *n*_{0}(*x*) ∈ *NS*(*x*) such that *n*_{0} (*x*) ⊆ *n*(*y*) ∩ (*z*). By the definition of base, we have *U*.

(⇐) Assume that *U*. We prove that *H*_{i})(*i* = 1,2,3,4). By Definition of *H*_{3}) is satisfied. So to prove (*H*_{1}), we prove (*H*_{1}) holds. Since *U*, by Definition 2.5, for any *x* ∈ *U* and *n*_{1}(*x*), *n*_{2}(*x*) ∈ *NS*(*x*), there exists an *n*_{3}(*x*) ∈ *NS*(*x*) such that *n*_{3}(*x*) ⊆ *n*_{1}(*x*) ∩ *n*_{2}(*x*); so *NS*(*U*) is weak-unary. By Proposition 2.10, we have *X*, *Y* ⊆ *U*. Since *NS*(*U*) is a reflexive neighborhood system of *U*, *X* ⊆ *U*. So to prove (*H*_{1}), we only need to prove *A* ⊆ *U*. Let *n*(*x*) ∈ *NS*(*x*). Pick *p* ∈ *n*(*x*) and *n*(*p*) ∩ *A* ≠ ∅ for any *n*(*p*) ∈ *NS*(*p*). From *p* ∈ *n*(*x*), we obtain *n*(*p*) ⊆ *n*(*x*), therefore *n*(*x*) ∩ *A* ≠ ∅. By the arbitrariness of *n*, we have *A* ⊆ *U*. Since *NS*(*U*) is a reflexive neighborhood system of *U*, (*H*_{2}) is obvious satisfied. Therefore

*Let* *NS*(*U*) *be a reflexive neighborhood system of U. Then* *is a closure operator if and only* *if* *is a base for some topology* τ *on* *U* *and* *NS*(*U*) *is transitive*.

It is easy to prove by Proposition 3.4 and Theorem 4.2. □

## 5 Characterization of covering *S* for ${\overline{apr}}_{S}$ being a closure operator

Zhang, Li and Lin defined a special covering *S* and investigated twenty-three types of covering-based rough sets proposed in [49] which can be treated as the generalized rough sets based on neighborhood systems. In this section, we will discuss the properties of *S* for

([49]). *A family of subsets of universe* *U* *is called a closure system over* *U* *if it contains* *U* *and is closed under set intersection. Given a closure system* *one can define its dual system* *as follows*:

(Subsystem based definition [49]). *Suppose* *is a pair of subsystems of* *is a closure system and* *is the dual system of* *A pair of lower and upper approximation operators* *with respect to* *S* *is defined as*:

$(\overline{S},\phantom{\rule{thickmathspace}{0ex}}\subseteq )$ *is a complete lattice*.$\overline{S}$ *may not have element*∅.*Fig*.??*gives an intuitive illustration*(*2*)*of the Remark 5.3*:*If*$\overline{S}$ *has no less than two single sets, then* .$\mathrm{\varnothing}\in \overline{S}$ *The converse may not hold*:

By Definition 5.2, it is easy to obtain properties of lower and upper approximation operators as follows:

*Let* *S* = (*S*̅, *S*̲ be a pair *of subsystems of* *for any* *X*, *Y* ∈ *U*, *we have*:

;${\underset{\_}{apr}}_{S}(\mathrm{\varnothing})=\mathrm{\varnothing}$ ;${\overline{apr}}_{S}(U)=U$ ;$X\subseteq Y\Rightarrow {\overline{apr}}_{S}(X)\subseteq {\overline{apr}}_{S}(Y),{\underset{\_}{apr}}_{S}(X)\subseteq {\underset{\_}{apr}}_{S}(Y)$ ${\overline{apr}}_{S}({\overline{apr}}_{S}(X))={\overline{apr}}_{S}(X)$ *for any**X*⊆*U*; .${\overline{apr}}_{S}(X)=-{\underset{\_}{apr}}_{S}(-X),{\underset{\_}{apr}}_{S}(X)=-{\overline{apr}}_{S}(-X)$

However, the following properties may not hold:

(1)

*Let* *It is easy to see* *contains* *U* *and is closed under set intersection. But*

(2)

*Let* *Then*

(3) *X*, *Y* ⊆ *U*.

*Let* *Let* *X* = {*a*}, *Y* = {*b*}, *then* *Hence*

([53]). *Suppose* *R* *is an arbitrary relation on U. With respect to* *R*, *we can define the left neighborhoods of an element* *x* *in* *U* *as follows*:

([53]). *For an arbitrary relation* *R*, *by substituting equivalence class* [*x*]_{R} *with right neighborhood* *l*_{R}(*x*), *we define the operators* *and* *from* *P*(*U*) *to itselfas follows*:

([53]). *If* *S* *is another binary relation on* *U* *and* *for any* *X* ⊆ *U*, *then* *R* = *S*.

*Let* *be a pair of subsystems of* *and* *U* *is finite. If* *for any* *X*, *Y* ⊆ *U*, *then there exists a unique reflexive and transitive relation* *R* *on* *U* *such that* *for any* *X* ⊆ *U*.

Using *U*, for any *x* ∈ *U*, we choose all *x*} ⊆ *S* which forms the family *S*′, via left neighborhood of an element *x* ∈ *U*, we construct the binary *R* on *U* as follows:

*l*_{R}(*x*) = {*y* ∈ *U* : *y* ∈ ∩ *S* It is clear that *R* is a reflexive relation. Since *X*, *Y* ⊆ *U*, we have *X* ⊆ *U*, this means that *R* is transitive. Thus *R* is a reflexive and transitive relation. The unique *R* comes from Lemma 5.10. □

*Let* *S* = (*S*̅ *S*̲) be a pair *of subsystems* of *then the following are equivalent*:

$\overline{S}$ *has element*∅; ;${\overline{apr}}_{S}(\mathrm{\varnothing})=\mathrm{\varnothing}$ .${\underset{\_}{apr}}_{S}(U)=U$

(1) ⇒ (2) By the Definition 5.2, it is easy to prove.

(2) ⇔ (3). It can be obtained by the duality.

(2) ⇒ (1) If *S* ∈ *S*̅. Thus ∅ ⊆ ∩ *S*̅. Since *S*̅ ∈ *S*̅ and ∩*S*̅≠∅. From the Definition 5.2, we have

*For any* *X*, *Y* ⊆ *U*, *if and only if* *A* ∪ *B* ∈ *S*̅ *for any* *A*, *B* ∈ *S*̅.

(⇒) Suppose there exists *S*_{1}, *S*_{2} ∈ *S*̅, it is easy to prove *S*_{1} ∪ *S*_{2} ∈ *S*̅.

(⇐) For any *X*, *Y* ⊆ *U*, we have:

Thus, we have *X* ∪ *Y* ⊆ *S*_{1} ∪ *S*_{2} ∈ *S*̅. Hence

(General characterization of coverings *S* for *Let* *be a pair of subsystems of* *is a closure operator if and only if* *and* *for any*

(⇒) Assume *S*_{1} ∪ *S*_{2} ∈ *S*̅ for any *S*_{1}, *S*_{2} ∈ *S*̅.

(⇐) Assume ∅ ∈ *S*̅ and *S*_{1} ∪ *S*_{2} ∈ *S*̅ for any *S*_{1}, *S*_{2} ∈ *S*̅. We need to prove *H*_{3}) is satisfied. By the Lemma 5.13, we have *X*, *Y* ⊆ *U*. Thus, (*H*_{1}) is satisfied. By the Definition 5.2, it is easy to prove *X* ⊆ *U*.

Then we only need to prove (*H*_{4}) holds. For any *X* ⊆ *U*, *X* ⊆ *U*. By the Definition 5.2, we have *S*_{1} ∈ *S*̅. Therefore *X* ⊆ *U*. Hence

*Let* *S* = (*S*̅, *S*̲ *be a pair* *of subsystems* *of* *S*̅ *and* *A* ∪ *B* ∈ *S*̅ *for any* *A*, *B* ∈ *S*̅ *if and only if* *is a topology on* *U*.

(⇒) Assume *ŝ* is a topology on *U*. Since ∅, *U* ∈ *S*̅, so ∅, *U* ∈ ŝ.

For any *X*, *Y* ∈ ŝ, we prove *X* ∩ *Y* ∈ ŝ. There exists *S*_{1}, *S*_{1} ∈ *S*̅ such that *X* = *U*\ *S*_{1} and *Y* = *U*\ *S*_{2}. Thus *X* ∩ *Y* = (*U*\ *S*_{1}) ∩ (*U*\ *S*_{2}) = *U*\ (*S*_{1} ∪ *S*_{2}). By the condition, we obtain (*S*_{1} ∪ *S*_{2}) ∈ *S*̅. From the definition of ŝ, we have *U*\ (*S*_{1} ∪ *S*_{2}) ∈ ŝ.

Let {*A*_{i} : *i* ∈ *I*} ⊆ ŝ, there exists {*S*_{j} : *i* ∈ *I*} ⊆ *S*̅ such that *A*_{i} = *U*\ *S*_{j} for each *i* ∈ *I*, thus *ŝ* = {*U* *S* : *S* ∈ *S*̅} is a topology on *U*.

(⇐) Similarly, we can prove the converse. □

From the Lemma 5.15, we have the following conclusion:

(Topological characterization of coverings *S* for *Let* *S* = (*S*̅, *S*̲ *is a pair of subsystems of* *is a closure operator if and only if* *ŝ* = {*U*\ *S* : *S* ∈ *S*̅} *is a topology on* *U*.

## 6 Conclusions

In this paper, we not only obtained the properties of *NS*(*U*) and *S*, but also investigated two type approximation operators. We give general characterization of covering *S* for covering-based upper approximation operator *S* for

This work is supported by the Natural Science Foundation of China (No.11371130), the Natural Science Foundation of Guangxi (No. 2014GXNSFBA118015), Guangxi Universities Key Lab of Complex System optimization and Big Data Processing (No.2016CSOBDP0004) and Key research and development of Hunan province (No.2016JC2014).

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