School of Science,Beijing University of Posts and Telecommunications,Beijing 100876,People’s Republic of China

?E-mail:maliwenmath@sina.com

Abstract:The authors study the covering rough sets by topological methods.They combine the covering rough sets and topological spaces by means of defining some new types of spaces called covering rough topological(CRT)spaces based on neighbourhoods or complementary neighbourhoods.As the separation axioms play a fundamental role in general topology,they introduce all these axioms into covering rough set theories and thoroughly study the equivalent conditions for every separation axiom in several CRT spaces.They also investigate the relationships between the separation axioms in these special spaces and reveal these relationships through diagrams in different CRT spaces.

1 Introduction

Rough set theoretical foundation was first set up by Z.Pawlak in 1982.It is an extension of set theory and intended to deal with incomplete information systems.In this theory,the equivalence relation is originally used.To solve various complicated problems in practice,the equivalence relations is relaxed to non-equivalent relations and coverings [1–7],even fuzzy relations and fuzzy coverings [8–11]etc.

Topology is closely related to rough set theory,because their common study objects are sets.Topology provides many valid ideas and methods for the investigation of rough set theory.Much valuable content of rough set theory was studied by researchers from topological point of view.For example,the relationships between topology and rough sets were investigated in[12,13].Methods for generating topologies through binary relations were provided in [14,15].Several important topological properties of binary relation-based rough sets were studied in[15–19].Discussions on topological properties of approximation operators can be found in[2,20–22].

Separation axioms are not only the most fundamental and important content in the general topology but also have valuable applications in information sciences.For example,it was used to investigate the rules of information generation and recovery in incomplete information systems [18],and was applied to restore the rough topological diagram for some incomplete diagrams [23].In this paper,we introduce all these separation axioms into covering rough set theory.We combine the covering rough sets and topological spaces by means of defining some new types of spaces called covering rough topological (CRT) spaces.Separation axioms in these CRT spaces are thoroughly investigated through studying the equivalent conditions for every separation axioms in each CRT space.We also discuss the relationships between the separation axioms in these special spaces and reveal the relationships through diagrams in different CRT spaces.

The remaining part of this paper is organised as follows.In Section 2,we recall some basic concepts and properties in topology,and in Section 3,we study some properties of several covering approximation operators.In Section 4,we define CRT spaces and thoroughly investigate separation axioms in these spaces,and we give the conclusion in the last section.

2 Basic concepts and properties in topology

Firstly,we recall basic topological knowledge which will be needed in the following discussion.The definitions and propositions listed here are cited from [24–26].

Definition 1:Let U be a set and T be a collection of subsets of U.If T satisfies the following conditions:

then T is called a topology on U.Every element AT is called an open subset of U,and U ?A is called a closed subset of U.The pair(U,T) is called a topological space.

If the rules (I1)?(I4) hold,we call int an interior operator on U.

Let U be a set.We can easily exam that TC={XU:cl(?X)=?X}and TI={XU:int(X)=X}are both topologies on U.We call TCthe topology generated by the closure operator cl and TIthe topology generated by the interior operator int.

For any subset X of a topological space,the closure operator cl and the interior operator int satisfy the formulas

We also have the following two properties for the closure operator cl in topological spaces.

Proposition 1:For each subset AU,we have cl(A)=where F is closed subset of U.

This proposition implies that cl(A)is the smallest closed subset of U containing A.The following proposition is a deduction of (C3) in Definition 2.

Proposition 2:Let U be a topological space,and A1,A2,...,AkU.Then we have

Definition 3:A topological space(U,T)is called a discrete space,if every subset of U is open.It is called a quasi-discrete space,if every open subset of U is also closed.

The separation axioms play a very important part in general topology.They lead to the investigation of various spaces,ranging from fairly general ones to those more and more special,and thus provide the theoretical foundations of many subjects.Through the study of these axioms,we can obtain deep understanding of the topological properties of common spaces.We list the axioms as follows:

Definition 4:A topological space U is called a T0-space,if for every pair of distinct points x,yU,there exists an open set containing exactly one of these point.

A topological space U is called a T1-space,if for every pair of distinct points x,yU,there exists an open subset XU such that xX and yX.

A topological space U is called a T2-space,or a Hausdorff space,if for every pair of distinct points x,yU,there exist two open subsets X,YU such that xX,yY and XY=

A topological space U is called a regular space,if for every xU and every closed subset FU satisfying xF,there exist open subsets X,YU,such that xX,FY and XY=.If a topological space U is both T1-space and regular space,we call it T3-space.

A topological space U is called a normal space,if for every pair of disjoint closed subsets A,BU,there exist open subsets X,YU such that AX,BY and XY=.If a topological space U is both T1-space and normal space,we call it T4-space.

The definitions of Ti-space (i=1,2,3,4),regular space and normal space are all called the separation axioms.In a general topological space,the relationship between the separation axioms can be shown through Fig.1,in which TiTjmeans that a Ti-space is also a Tj-space.

3 Some covering rough sets and their properties

In this section,the concepts of neighbourhood and complementary neighbourhood are recalled,based on which we give definitions to some special covering rough approximation spaces and show their properties.

3.1 Neighbourhood and complementary neighbourhood

The concepts of lower and upper approximations are key concepts in the rough set theory.Pawlak’s classical rough sets are based on equivalence relations on a universe U,i.e.the lower approximationand the upper approximationof XU are defined as follows:

where R is an equivalence relation on U,and U/R denotes the family of all equivalence classes(partitions)induced by R.Ifthen X is called a rough set.This concept was extended to covering-based rough sets by extending the partitions to coverings.In the following,we review some basic concepts and results in covering rough set theory.

Definition 5:[22]Let U be a universe set and C be a family of non-empty subsets of U.Ifthen C is called a covering of U,and the ordered pair (U,C) is called a covering approximation space.

Following the sense of Pawlak,a set XU is called a covering rough set,if its covering-induced lower approximation and upper approximation are not equal.

The neighbourhoods or complementary neighbourhoods of points are essential in defining the lower and upper approximations of sets in covering approximation spaces.

Definition 6:[7,22,27,28]Let(U,C)be a covering approximation space.We define the neighbourhood N(x) and complementary neighbourhood M(x) of an element xU as

where ?X denote the subset U ?X for XU,and if xC for each CC,we set M(x)=U.

We can also define the neighbourhoods and complementary neighbourhoods of sets [27].

Definition 7:[28]Let(U,C)be a covering approximation space.For each AU.We call N(A)={N(x):xA} the neighbourhood of set A and M(A)={M(x):xA} the complementary neighbourhood of set A.

Thus,N(M(x)) and M(N(x)) are both descriptions of xU,but M(N(x)) and N(M(x)) are not always equal [28].The neighbourhood and complementary neighbourhood have the following propositions:

Proposition 3:[27,28]Let (U,C) be a covering approximation space.For every x,yU,then

Proposition 4:[27,28]Let (U,C) be a covering approximation space.Then for each CC,we have

3.2 Four pairs of dual approximation operators

In this subsection,we recall four pairs of lower and upper approximations on the basis of the neighbourhoods and complementary neighbourhoods.We useto denote the lower and upper approximations of X,respectively.

Definition 8:[5,12,27–29]For every subset X in a covering approximation space (U,C),we define the lower approximations and upper approximations of X as follows:

The fact was proved in paper [28]that definition forms of the pairsandin Definition8 are equivalent tothe formsandin[29]respectively:

Fig.1 Relationship of separation axioms in general topological spaces

We can see that their definitions in Definition 8 are simpler than their equivalence forms in[29].This certainly facilitates our further study on corresponding covering approximation spaces.

Taking note of the fact that N(x) and M(x) are both subsets of M(N(x)) and N(M(x)),we can obtain the following relationships among

Proposition 5:For each X in (U,C),the relationships ofand Pi(X) (i=1,2,3,4) are shown through Fig.2,where the left elements are contained in the right elements as subsets when they are connected by one broken line.

3.3 Topological properties of the operators and

It follows from the discussions in [27]that the operatorsare interior operators,andare closure operators.However,the operatorsare not interior operators,andare not closure operators generally.We show this by the following example:

Example 1:Let U={x1,x2,x3,x4,x5} and C={C1,C2,C3},where C1={x1,x2},C2={x2,x3,x4},C3={x4,x5}.Considering the covering approximation space (U,C) and X={x2},Y={x1}.

We can easily obtain that

In the following,we discuss some conditions under whichandare interior and closure operators respectively.

Proposition 6:For the covering approximation space (U,C),the following statements are equivalent:

Proof:(i)?(ii) follows from the duality betweenandNext we show(i)?(iii).

Similarly,we can prove the following proposition.

Proposition 7:Let U be a finite set and C be a covering of U,then the following statements are equivalent

4 Separation axioms in CRT spaces

In this section,we first give the definition of CRT space,and then discuss the separation axioms in different CRT spaces.

Definition 9:Let U be a finite set and C be a covering of U.If T is a topology on U generated by one of the interior operatorsor closure operatorswe call(U,T) a CRT space.In the cases of i=3,4,we assume that the conditions (iii) in Propositions 7 and 8 are satisfied respectively.

We use (U,Ti) to denote the CRT space whose topology Tiis generated by

Since (U,T2) is similar to (U,T1),and (U,T4) is similar to(U,T3),in what follows,we only discuss the separation axioms in the CRT spaces (U,T1) and (U,T3).

4.1 Separation axioms in the CRT space (U,T1)

According to Propositions 4 and Definitions 2,6,we have the following property.

Proposition 8:In the CRT space (U,T1),every set CC is an open subset,and ?C is a closed subset.For any xU,M(x) is the minimal closed subset containing x,i.e.M(x) is the closure of{x};And N(x) is the minimal open subset containing x.We call N(x) the open expansion of {x}.Moreover,for any AU,M(A)is the closure of A,and N(A) is the open expansion of A.

Now we can discuss the separation axioms in the CRT space(U,T1).

Theorem 1:Let (U,T1) be a CRT space,then the following statements are equivalent:

(i) (U,T1) is a T0-space,

Proof:(i)?(ii).Suppose that (U,T1) is a T0-space.By contradiction,if there exists a y=x such that yM(x)N(x)for some xU,then we have yM(x) and yN(x).Thus xN(y) and yN(x) hold,and hence U is not T0-space.Therefore,M(x)N(x)={x} for each xU.

(ii)?(iii).We only need to show that M(x)=M(y)leads to x=y.Suppose that xy,then xM(x)=M(y),which implies yN(x).Therefore,we have yM(y)N(x)=M(x)N(x),and this contradicts to (ii).

(iii)?(iv).It is sufficient to show that N(x)=N(y) implies x=y.Suppose xy,then M(x)M(y) follows from (ii).Without loss of generality,we assume that M(x)?M(y).Then there is xM(y).Otherwise,according to Proposition 4,M(x)M(y) holds,which is a contradiction.Therefore,we have yN(x) contradicting to N(x)=N(y).

Theorem 2:For a CRT space(U,T1),the following statements are equivalent:

(i) (U,T1) is a T1-space,

(ii) (U,T1) is a discrete space,

(iii) M(x)=N(x)={x} holds for each xU,

Proof:(iii)?(iv)and(ii)?(i)are obvious.Since N(x)is an open subset and M(x)is a closed subset,(iii)?(ii)is obvious.So we only need to show(i)?(iii) and (iv)?(iii).

(i)?(iii).By contradiction,suppose M(x)=N(x)={x}not holds,then we have N(x){x}or M(x){x}.If N(x){x}holds,then there exists yN(x) such that yx.Due to the points x,y,U is not a T1-space,and this is a contradiction.If M(x){x} holds,then there is zM(x) such that zx.Therefore,xN(z),which also implies that U is not a T1-space.

We can easily deduce the following corollaries form Theorem 2.

Corollary 1:A CRT space(U,T1)is a T1-space,if and only if it is a Ti-space (i=2,3,4).

Corollary 2:If the CRT space(U,T1)is a T1-space,then it is also regular and normal.

Theorem 3:For CRT space (U,T1),the following statements are equivalent:

(i) (U,T1) is a regular space,

(iii) M(x)=N(x) holds for every xU,

(iv) (U,T1) is a quasi-discrete space,

(v) M(N(x))=N(x) holds for every xU,

(vi) N(M(x))=M(x) holds for every xU,

Proof:(ii)?(iii)?(iv)?(i) and (iii)?(vii)?(viii) are obvious.Next we show that (i)?(ii),(iv)?(v),(iv)?(vi) and(viii)?(iii).

(i)?(ii).Suppose that{N(x):xU} is not a partition of U.Then there exist x,yU such that N(x)N(y)and N(x)N(y).Without loss of generality,we assume that N(x)?N(y).Then we have xN(y).Take a zN(x)N(y),we also have zM(x).Otherwise,xN(z) holds,which,combined with zN(y),leads to N(x)N(y) contradicting to N(x)?N(y).Now we consider z and the closed subset M(x).It follows from zN(x) that zN(M(x)).Since N(M(x)) is the minimal open subset containing M(x),U is not a regular space,and this contradicts to (i).

(iv)?(v).Since every open subset in a quasi-discrete space is equal to its closure,we have M(N(x))=N(x),and (iv)?(v) is proved.Moreover,if M(N(x))=N(x) for each xU,then for every open subset X in U,noticing that M(X) is the closure of X,and considering Proposition 2,we have

Thus X is also a closed subset of U,and hence (U,T1) is a quasi-discrete space.So (v)?(iv) is correct.

(iv)?(vi).Implication (iv)?(vi) follows from the fact that every closed subset in quasi-discrete space is equal to its open expansion.Conversely,if N(M(x))=M(x) for each xU,then for every closed subset XU,we have

Namely,X is also an open subset of U.Thus (U,T1) is a quasi-discrete space,and (vi)?(iv) is proved.

This theorem indicates that all the approximations(i,j=1,2,3,4) in U are Pawlak’s original ones,if and only if the CRT space (U,T1) is a regular space.

The following corollaries are obvious.

Corollary 3:A regular CRT space (U,T1) is also normal.

Corollary 4:If (U,T1) is a regular CRT space,then M(N(x))=N(M(x)) for each xU.

The converse proposition of Corollary 4 is generally not correct.This can be seen from the following example.

Example 2:Let U={x1,x2,x3,x4} and C={C1,C2,C3},where C1={x1,x2,x3,x4},C2={x2,x4},and C3={x3,x4}.Consider the covering approximation space (U,C).

We can easily verify that

However,(U,T1) is not a regular space,since x2and the closed subset {x1,x3} cannot separated by disjoint open subsets.

Theorem 4:For a CRT space(U,T1),the following statements are equivalent:

(i) (U,T1) is a normal space,

Fig.2 Relationship of (X) and (X)(i=1,2,3,4)

Fig.3 Relationship of separation axioms in the CRT space (U,T1)

(ii)?(i).For each closed subset X(U,T1),there is

Thus for any disjoint closed subsets X,Y [U, we haveAccording to (ii),Since N(M(X)) and N(M(Y)) are open subsets of(U,T1) containing X and Y,respectively,(U,T1) is normal.□

According to the above discussion,the relationship of separation axioms in the CRT space (U,T1) can be shown through Fig.3.One can see the difference between this diagram and Fig.2.

The implications that not marked in Fig.3 are generally not hold in (U,T1).This can be seen from the following examples.

Example 3:Suppose that U={a,b,c} and C={C1,C2},where C1={a,b,c},C2={c}.

The CRT space(U,T1)is a normal space,because it has no disjoint closed subsets.Consider the points a and b,we can easily deduce that (U,T1) is not a T0-space.If consider the point c and the closed subset {a,b},we can also see that(U,T1) is not regular.

Example 3 shows that a normal CRT space may not be a T0-space.The following example shows the vice versa.

Example 4:Set U={a,b,c} and C={C1,C2},where C1={a,c},C2={b,c}.Consider the CRT space (U,T1).

It follows from Theorem 1 that(U,T1)is a T0-space.Considering the closed subsets{a},,we can see that(U,T1)is not a normal space.

A CRT T0-space may also not regular.Otherwise,by Corollary 3,it would be a normal space,and this is a contradiction.

Example 5:Suppose U={a,b,c}and C={C1,C2},where C1={a,b},C2={c}.The CRT space(U,T1)is a quasi-discrete space,i.e.a regular space.When considering the points a,b,we see that(U,T1) is not a T0-space.

From this example,we can see that a regular CRT space(U,T1)is generally not a T0-space.

4.2 Separation axioms in the CRT space (U,T3)

In this subsection,we assume that the condition(iii)in Proposition 6 is satisfied.

Theorem 5:For a CRT space (U,T3),the follows are equivalent:

(i) (U,T3) is a T0-space,

(iii) (U,T3) is a discrete space.

From this theorem,we have the relationships of T0,T1,T2,T3and T4in (U,T3).

Corollary 5:(U,T3) is a T0space,if and only if it is Ti(i=1,2,3,4) space.

Corollary 6:A T0space (U,T3) is regular and normal.

Definition 10:Let (U,C) be a covering approximation space and x,yU.If there are x1,x2,...,xmU such that xN(M(x1)),yN(M(xm)),and N(M(xi))N(M(xi+1))=(1=1,2,...,m ?1),we call that x and y are NM-connected,and we denote it by xy.

The next proposition follows from Proposition 4.

Proposition 9:Let (U,C) be a covering approximation space.Then for any x,y,z(U,C),we have

This proposition implies that the relationis an equivalent relation on U.

Definition 11:For covering approximation space (U,C),we call every equivalent class of U/an NM-connect branch of (U,C).

Example 6:Suppose U={xi:i=1,2,...,10}and C={Ci:i=1,2,...,7},where C1={x1,x2},C2={x2,x3,x6},C3={x4,x5,x6},C4={x5},C5={x7,x8},C6={x9,x10} and C7={x7,x9}.All the NM-connect branches of (U,C) are{xi:i=1,2,...,6},{x7,x8} and {x9,x10}.

Proposition 10:Let (U,C) be a covering approximation space.If AU is an NM-connect branch of (U,C),then A is an open subset of the CRT space (U,T3),and any proper subset of A is not an open subset.

Proof:Suppose that A is an NM-connect branch of (U,C).For any xA and yN(M(x)),we have xy and yA.Thus N(M(x)A and A{x:N(M(x))A}.It is obvious that A{x:N(M(x))A},so we have A={x:N(M(x))A}.According to the definition of (U,T3),A is an open subset.

Suppose that B is a proper subset of A,and B is an open subset.According the definition of NM-connect branch,there exist yA ?B,xB such yx.Then there are x1,x2,···,xnU such that yN(M(x1)),x1N(M(x2),···,xn?1N(M(xn))and xnN(M(x).Since B is an open subset,we have B={x:N(M(x))B}.Then xnB,xn?1B,···,x1B,yB followed successively.This contradicts to yA ?B,and the proof is completed.

Fig.4 Relationship of separation axioms in the CRT space (U,T3)

The following proposition and theorem follows from Definition 11 and Proposition 10.

Proposition 11:Let (U,C) be a covering approximation space,then all the NM-connect branches in the CRT space (U,T3)are also closed subsets.Therefore,(U,T3) is a quasi-discrete space.

Theorem 6:The CRT space (U,T3) is regular and normal.

The relationship of separation axioms in the CRT space(U,T3)are shown through Fig.4.

The following example shows that the implications not marked in Fig.4 are generally not hold in the CRT space (U,T3).

Example 7:Suppose (U,C) is the space in Example 5.The CRT space (U,T3) is a quasi-discrete space,regular space and normal space,but not a T0-space.

5 Conclusion

The separation axioms are not only important in topology,but also have valuable applications in information sciences.In this paper,we study the separation axioms in several CRT spaces whose topologies were generated by the lower or upper approximation operators of some neighbourhood-related covering rough sets.In each of the considered CRT space,we showed the equivalent conditions of every separation axiom,and revealed the relationship between different separation axioms.Our results are helpful to the research of data analysis and information recovery in different information systems.Based on this,we may also perform further theoretical and practical studies on such spaces.