On topologies defined by neighbourhood operators of approximation spaces

Abstract

We consider properties of an approximation space with a neighbourhood operator. Two types of topologies τ and σ are introduced on the approximation space using a neighbourhood operator and we show that they are the same Alexandrov topology, $\tau =\sigma$. Also, for a reflexive and transitive neighbourhood operator, the topological space $(U,\sigma )$ is a Hausdorff space if and only if σ is the discrete topology $\mathcal{P}(U)$, where U is a non-empty set which is not need to be finite. In addition, we prove an algebraic property that a pair $(n,\underset{\_}{apr})$ of operators defined by a neighbourhood operator n forms a Galois connection.

Introduction

Many methods have been proposed for finding useful information and rules from a data set with incomplete informations or contradictory items. Among them, the theory of rough sets proposed by Pawlak [8] is actively researched. The central concept is an approximation space. Two operators called an upper approximation and a lower approximation define specific subsets in the data set and we consider such subsets as representing knowledge. Therefore, it is important to determine a particular subset representing knowledge of the data set U. In this paper we take U as a set which is not need to be finite. The following methods have been proposed to specify subsets:

• A method using a binary relation R on U;

• A method using a covering which is a generalization of a partition;

• A method using neighbourhood operators.

  ⋯

With respect to (I), given $A\subseteq U$ and $R\subseteq U\times U$, Pawlak adopted an equivalence relation R as a binary relation and defined an upper approximation operator $R_{+}$ and a lower approximation operator $R_{-}$ by

$$R_{+}(A)=\{x\in U|R(x)\cap A\ne \varnothing \},R_{-}(A)=\{x\in U|R(x)\subseteq A\},\mathrm{where}R(x)=\{y\in U|xRy\}.$$

Then a pair of subsets $(R_{-}(A),R_{+}(A))$ was considered as knowledge about the data set A. After that, a generalized approximation space $(U,R)$, in which the binary relation R is not necessarily an equivalence relation, has been proposed and is studied [3], [4], [5], [12]. For example, for a reflexive relation R on U, the set $\mathcal{O}=\{A\subseteq U|R_{-}(A)=A\}$ of all fixed points of $R_{-}$ operator forms a topology on U in [3]. This is a generalization of the well known fact above. In addition, it is also proved in [5] that, for a reflexive relation R, its transitive closure $R^{⁎}$ defines a topology ${\mathcal{O}}^{\prime }=\{A\subseteq U|R_{-}^{⁎}(A)=A\}$ and $\mathcal{O}={\mathcal{O}}^{\prime }$. Moreover, for any binary relation S, its reflexive closure $S_{\omega }=S\cup \{\omega \}$ also defines a topology on U in [4], where $\omega =\{(x,x)|x\in U\}$.

Also, for the second method (II), instead of the partition defined by an equivalence relation, we use a covering $\mathcal{C}=\{A_i\subseteq U|A_i\ne \varnothing ,{\cup }_i{A}_i=U\}$. As is well known, an equivalence relation R gives the family $\mathcal{P}=\{R(x)|x\in U\}$ of subsets such that xRy if and only if $R(x)=R(y)$, in other words,

$$xRy\iff R(x)\cap R(y)\ne \varnothing .$$

On the other hand, coverings $\mathcal{C}$ permit the existence of subsets $A,B\in \mathcal{C}$ such that $A\ne B$ and $A\cap B\ne \varnothing$. Thus, the following most important property of partitions that, for $A,B\in \mathcal{P}$,

$$A\cap B\ne \varnothing \iff A=B$$

does not hold in general. For a given covering $\mathcal{C}$, we define an operator ν by

$$\nu (a)=\{K\in \mathcal{C}|a\in K\}\text{and}\nu (A)=\bigcup _{a\in A}\nu (a)\text{for}A\subseteq U.$$

In this case, the subset $\nu (A)$ is considered as representing knowledge [1], [7], [9], [11].

For the case of (III) which we treat here, a neighbourhood operator $n:U\to \mathcal{P}(U)$ directly gives a specific subset $n(x)\subseteq U$ [7], [9], [13]. For an element $x\in U$, $n(x)$ represents a neighbourhood of x. We note that $n(x)$ may not contain x. As to neighbourhood operators in rough sets, there are many papers treated them. For example, in [10], $n(x)\ne \varnothing$ is assumed and properties of serial (reflexive, symmetric, transitive, and so on) are investigated on a 1-neighbourhood system which is defined that each element $x\in U$ has exactly one neighbourhood $n(x)$. Also, in [2], 24 neighbourhood operators for a given covering are introduced and some properties of these operators, such as equalities and partial order relations between them, are proved. In [12], it is considered covering-based rough sets from the topological view and obtained the topological properties of this type of rough sets. Moreover, it is proved the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation. At last, there is also another paper from a different view. In [6], at least two methods for the development of the theory of rough sets, the constructive and the axiomatic approaches, are considered. In it, a new matrix view of the theory of rough sets is proposed and redefined a pair of lower and upper approximation operators using the matrix representation. It is proved that axioms of upper approximation operations guaranteed the existence of certain types of binary relations (or matrices) producing the same operators and the upper approximation of the Pawlak rough sets, rough fuzzy sets and rough sets of vectors over an arbitrary fuzzy lattice are characterized by the same independent axiomatic system.

As described above, there are various methods for giving specific subsets. In each case, if a topology is introduced on the (generalized) approximation space, then open sets (or closed sets) in the topological space can be considered as representing knowledge.

In this paper, we consider the case of (III), that is, using a lower approximation operator defined by a given neighbourhood operator, we introduce a topology and study properties of the topological space. It is easy to show that there is a one to one correspondence between binary relations and neighbourhood operators. Indeed, for a binary relation R on U, a neighbourhood operator $n_R$ can be defined by

$$n_R(x)=\{y\in U|(x,y)\in R\},$$

conversely, for a neighbourhood operator n, a binary relation $R_n$ is defined by

$$(x,y)\in R_n\iff y\in n(x),$$

and $R_{(n_R)}=R$, $n_{R_n}=n$.

However, it is easily understandable to use neighbourhood operators to specify subsets representing knowledge. Moreover, as proved later, although the topology σ defined by a neighbourhood operator n is the same as the topology τ defined by the lower approximation operator $\underset{\_}{apr}$, two operators n and $\underset{\_}{apr}$ are different. Therefore, it is important to consider topological properties of approximation spaces with neighbourhood operators.

We show that

• For any neighbourhood operator n on U, a pair $(n,\underset{\_}{apr})$ of the two operators n and $\underset{\_}{apr}$ forms a Galois connection, that, is, for all $A,B\subseteq U$,

  $$n(A)\subseteq B\iff A\subseteq \underset{\_}{apr}(B).$$

• For any neighbourhood operator n, two families $\sigma =\{A\subseteq U|n(A)\subseteq A\}$ and $\tau =\{A\subseteq U|A\subseteq \underset{\_}{apr}(A)\}$ of subsets are the same Alexandrov topology, although two operators n and $\underset{\_}{apr}$ are different.

• Moreover, if a neighbourhood operator n is reflexive and transitive, for the topological space $(U,\sigma )$, it is a Hausdorff space if and only if σ is a discrete topology.