A Novel Constructing Continuous and Topology Approach to Fuzzy β-covering

Abstract

Fuzzy β-covering(Fβ-C) plays a key role in processing real-valued data sets and covering plays an important role in the topological spaces. Thus they have attracted much attention. But the relationship between Fβ-C and topology has not been studied. This inspires the research of Fβ-C from the perspective of topology. In this paper, we construct Fβ-C rough continuous and homeomorphism mappings by using Fβ-C operator. We not only obtain some equivalent descriptions of the mappings but also profoundly reveal the relationship of two Fβ-C approximation spaces. We give the classification method of Fβ-C approximation spaces with the help of homeomorphism mapping, propose a new method to construct topology induced by Fβ-C operator and investigate the properties in the topological spaces further. Finally, we obtain the necessary and sufficient conditions for Fβ-C operators to be topological closure operators.

Keywords

Fuzzy β-covering Fuzzy β-covering mapping Fuzzy β-covering operator Topology

1 Introduction

1.1 Research background and related works

The concept of covering rough set were first presented by Zakowski in 1983 [1]. It can not only solve the limitations of classical rough sets, but also reveal potential rules in many kinds information systems. Therefore, many scholars have carried out a lot of research work in this field. For instance, Zhu [2] studied covering rough set theory from a topological point. Ma [3, 4] first proposed the complementary neighborhoods and has investigated the properties and topological importance of covering rough set. However, it is very poor in processing real-valued data sets. Fuzzy set theory came into being by Zadeh [5, 6] which is useful to overcome the limitations and has been used in uncertainties, such as data mining, feature selection and fraud detection. Subsequently, the research on dealing with the hybridization by using rough set and fuzzy set have proposed in [7, 8]. Dubois et al. [9, 10] first defined fuzzy rough sets based on these theories. Some models were constructed based on fuzzy rough set [11, 12], which can be viewed as bridge between covering rough set theory and fuzzy rough set theory. Zhang [13] obtained the invariance of covering rough sets by using compatible mappings. Li [14, 15] studied covering generalized approximation spaces by using topological methods. Wang et al. [16, 17] obtained the if and only if conditions for the approximation operator to become a topological closure operator and investigated membership functions. Zhang et al. [18 –22] studied the rough set of general relations. Atef et al [39] constructed three new coverings based variable precision fuzzy rough sets using the concept of Ma et al.

Nevertheless, fuzzy covering is difficult to elaborate the granularity information of fuzzy covering families in practical applications. For the sake of this problem, Ma [23] defined Fβ-C and investigated two types of Fβ-C rough set. It is more general model because when β = 1, a Fβ-C reduces to a fuzzy covering. Therefore, it quickly attracted the attention of a large number of scholars. Yang et al. [24, 25] defined some new Fβ-C rough set model according to fuzzy complementary β-neighborhood operators. Huang et al. [26] depicted the discernibility ability of fuzzy coverings based on a novel uncertainty measure w.r.t fuzzy β-neighborhood. Dai et al. [27] defined new Fβ-C and studied their applications. Zhang et al. [28] explored properties of Fβ-C approximation spaces. Zhang et al. [34] proposed four types of fuzzy soft β-coverings and described the algorithm decision making process of the new method in detail based on the fuzzy rough set of fuzzy soft β-coverings. Xu et al. [35] given an updated attribute algorithm in the approximate space of motion Fβ-C. These models mainly focused on model generalization, but rarely involved the relationship between Fβ-C and topology, and the classification of Fβ-C spaces.

Topological structure is one of the most important structures of mathematics, which provide mathematic tool for dealing with information systems and rough sets [29]. For example, Lashin et al. [36] used topology to obtain distinguish ability matrices and discernible functions to reduce knowledge and make decisions about information systems. Qin et al. [37] worked to discuss the relationship between fuzzy rough set models and fuzzy topology on a finite universe. Salama et al. [38] defined new pre-topological approximations and pre-topological measurements and used them to determine the reduction of incomplete information systems. Yang et al. [30] defined topology by lower approximation operator. Zhao [31] proposed topology induced by the covering. Yu et al. [32] obtained topology by lower and upper approximation operators. What are the characteristics of the topology induced by Fβ-C?

1.2 Motivation and contributions

The motivations of this study are elaborated as follows.

With the increase of Fβ-C spaces, Fβ-C operators also appear in large numbers. There is one-to-one correspondence between operators and space properties. In order to classify Fβ-C spaces, it is natural to ask how to distinguish these approximation operators effectively. In this paper, a new method from topological point of view is proposed to deal with this question.

Continuous mapping plays a key role in general topology and other fields. It is a bridge connecting various spaces. Therefore, we define Fβ-C rough continuous mapping and homeomorphism mapping and get some equivalent descriptions in order to improve classification efficiency and save resources. Meanwhile, we propose a novel method to construct topology based on Fβ-C. With the help of this method, we can form a unified situation about using Fβ-C operator to construct topology. From Fβ-C relation and fuzzy β-neighborhood, we explore the if and only if conditions for the upper approximation operator to be a topological closure operator from different perspective and obtain some properties in topological spaces. From the view of topology, we get the essence of Fβ-C from another perspective.

1.3 Structure and organization

The structure of this article is arranged as follows. In Section 1, we introduce the background and motivation. In Section 2, several preliminary concepts in rough approximation space and topology space are briefly recalled. In Section 3, we propose some new concepts Fβ-C rough continuous mapping, Fβ-C rough homeomorphism mapping, obtain some equivalent descriptions, and provide the classification method of Fβ-C spaces. In Section 4, we apply Fβ-C rough set to topological space, generalize the method of topology induced by rough sets and propose a new topology induced by Fβ-C, prove the if and only if conditions to be a topological closure operator and obtain some interesting properties of the topology space. Section 5 concludes the paper.

2 Preliminaries

2.1 Fuzzy set theory and Fβ-C

Let U be a non-empty set and * an arbitrary relation. For any X ⊆ U, the operators of X are defined: $\begin{matrix} \underline{*} (X) = {x \in U : * (x) \subseteq X} \\ \bar{*} (X) = {x \in U : * (x) \cap X \neq \emptyset} \end{matrix}$

* is a relation ∀ x, y, z ∈ U, if (x, y)∈ * and (x, z)∈ * implies (y, z)∈ *, then * is called Euclidean;

Definition 2.1. [5] Let U be a non-empty finite set, I means the unit interval [0, 1]. A fuzzy set K in U is defined as K : U ⟶ I and K (x) is referred to as the membership degree of u relative to K. I^U indicates the set of all fuzzy sets on U. Given K ∈ I^U, then K is denoted as $K = \frac{K (x_{1})}{x_{1}} + \frac{K (x_{2})}{x_{2}} + . . . + \frac{K (x_{n})}{x_{n}} .$

Definition 2.2. [23] Let U be a universe and $C = {K_{1}, K_{2}, . . ., K_{m}}$ be a subset of the fuzzy power set of U, β ∈ (0, 1]. If $(⋃_{i = 1}^{m} K_{i}) (x) \geq β$ for all x ∈ U, then $C$ is called a fuzzy β-covering(Fβ-C).

Definition 2.3. [23] Let Δ be a family of Fβ-Cs with each $C_{i} \in Δ$ being finite. The tuple (U, Δ) is called a fuzzy β-covering approximation space(Fβ-CAS).

Definition 2.4. [23] Let (U, Δ) be a Fβ-CAS, where $C_{i} = {K_{1}, K_{2}, . . ., K_{m}} \in Δ$ . For each x ∈ U, then the fuzzy β-neighborhood of x is defined by

$[x]_{Δ}^{β} = ⋂ {K_{j} : K_{j} \in C_{i} \in Δ, K_{j} (x) \geq β} .$ (2.1)

Definition 2.5. [23] Let (U, Δ) be a Fβ-CAS. Fβ-C relation R_Δ on U is defined as: $R_{Δ} = [x]_{Δ}^{β} (y) x, y \in U .$ (2.2)

Definition 2.6. [23] Let (U, Δ) be a Fβ-CAS. For each x ∈ U, β-neighborhood $N_{x}^{β}$ of x is defined as: $N_{x}^{β} = {y \in U : R_{Δ} \geq β} .$ (2.3)

Definition 2.7. [23] Let (U, Δ) be a Fβ-CAS. ∀X ⊆ U, the operators are defined as:

$\underline{R_{Δ}} (X) = {x : N_{x}^{β} \subseteq X}$ (2.4) $\bar{R_{Δ}} (X) = {x : X \cap N_{x}^{β} \neq \emptyset}$ (2.5)

2.2 Basic concepts in topology

Next, we introduce some basic definitions and lemma of topology.

Definition 2.8. [33] The family $O$ of subsets of X is called topology if it satisfies the following conditions:

(A₁) $\emptyset \in O$ and $X \in O$ ;

(A₂) If $U_{1} \in O$ and $U_{2} \in O$ , then $U_{1} \cap U_{2} \in O$ ;

(A₃) $A \subseteq O$ , then $\cup A \in O$ .

The elements of X are called points of the space, and every element of $O$ is called open set.

Definition 2.9. [33] A set F ⊆ X is called closed in topological space $(X, O)$ if its complement X ∖ F is an open set. If ∀x ∈ X, there exists open set U ⊆ X such that x ∈ U, then U is called a neighborhood of x.

Definition 2.10. [33] A family $B \subseteq O$ is called a base for a topological space $(X, O)$ if $B$ has the following properties:

(B₁) For any $U_{1}, U_{2} \in B$ and every point x ∈ U₁ ∩ U₂, there exists a $U \in B$ such that x ∈ U ⊆ U₁ ∩ U₂;

(B₂) For every x ∈ X, there exists a $U \in B$ such that x ∈ U.

Definition 2.11. [33] The collection ${B (x) : x \in X}$ is called a neighbourhood system, if for any neighbourhood V of x, there exists a $U \in B (x)$ such that x ∈ U ⊆ V.

Definition 2.12. [33] For any X, Y ⊆ U, an operator H: $P (U) \to P (U)$ is called a topological closure operator on U if it satisfied:

(C₁) H (X ∪ Y) = H (X) ∪ H (Y);

(C₂) X ⊆ H (X);

(C₃) H (∅) = ∅;

(C₄) H (H (X)) = H (X).

Lemma 2.13. Let X and Y be two sets and f : X → Y a mapping, then:

(1) ∀A ⊆ X, we have A ⊆ f^-1 (f (A));

(2) ∀B ⊆ Y, we have f (f^-1 (B)) ⊆ B, if f is surjection, then f (f^-1 (B)) = B;

(3) ∀A ⊆ X and B ⊆ Y, there is f (A) ⊆ B if and only if A ⊆ f^-1 (B).

3 Fβ-C rough continuous and homeomorphism mappings

Fβ-CAS is the most important generation of fuzzy covering approximation space, many scholars have defined many fuzzy covering approximation operators. It is very important to generalize these operators. In this section, Fβ-C rough continuous and homeomorphism mappings are proposed by using the upper operator $\bar{R_{Δ}}$ of a Fβ-CAS and obtain some properties of the Fβ-C rough continuous mapping and the Fβ-C rough homeomorphism mapping. Fβ-C spaces are classified by means of Fβ-C rough homeomorphism mapping.

Lemma 3.1.Let (U, Δ) be a Fβ-CAS. For any X ⊆ U, X_i ⊆ U (i ∈ I), then have:

(1) $\bar{R_{Δ}} (X^{c}) = (\underline{R_{Δ}} (X))^{c}$ , where X^c represents the complement of X in U;

(2) $\underline{R_{Δ}} (⋂_{i \in I} X_{i}) = ⋂_{i \in I} \underline{R_{Δ}} (X_{i})$ ;

(3) $\bar{R_{Δ}} (⋃_{i \in I} X_{i}) = ⋃_{i \in I} \bar{R_{Δ}} (X_{i})$ ;

(4) $\bar{R_{Δ}} (X) = ⋃_{x \in X} \bar{R_{Δ}} ({x})$ .

Proof. (1) For $\forall x \in \bar{R_{Δ}} (X^{c})$ , we have $N_{x}^{β} ⋂ (X^{c}) \neq \emptyset$ by Definition 2.7, then $N_{x}^{β} ⊈ X$ . Hence $x \notin \underline{R_{Δ}} (X)$ , and then $x \in (\underline{R_{Δ}} (X))^{c}$ , which implies $\bar{R_{Δ}} (X^{c}) \subseteq (\bar{R_{Δ}} (X))^{c}$ .

We prove the converse. For $\forall y \in (\underline{R_{Δ}} (X))^{c}$ , we have $y \notin \underline{R_{Δ}} (X)$ . By Definition 2.7, we can obtain $N_{y}^{β} ⊈ X$ , and claim that $N_{y}^{β} ⋂ (X^{c}) \neq \emptyset$ . If $N_{y}^{β} ⋂ (X^{c}) = \emptyset$ , then $N_{y}^{β} \subseteq X$ . It is contradiction to $N_{y}^{β} ⊈ X$ . Therefore $y \in \bar{R_{Δ}} (X^{c})$ . It follows that $\bar{R_{Δ}} (X^{c}) = (\underline{R_{Δ}} (X))^{c}$ .

(2) Since ⋂_i∈IX_i ⊆ X_i for ∀ i ∈ I, then $\underline{R_{Δ}} (⋂_{i \in I} X_{i})$ $\subseteq \underline{R_{Δ}} (X_{i})$ , thus $\underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq ⋂_{i \in I} \underline{R_{Δ}} (X_{i})$ , we shall prove the converse.

$\forall x \in ⋂_{i \in I} \underline{R_{Δ}} (X_{i})$ , we have $x \in \underline{R_{Δ}} (X_{i})$ for any i ∈ I. According to Definition 2.7, we can obtain $N_{x}^{β} \subseteq X_{i}$ . By the arbitrariness of i, we have $N_{x}^{β} \subseteq ⋂_{i \in I} X_{i}$ . Thus we have $x \in \underline{R_{Δ}} (⋂_{i \in I} X_{i})$ . From the above, we know that $\underline{R_{Δ}} (⋂_{i \in I} X_{i}) = ⋂_{i \in I} \underline{R_{Δ}} (X_{i})$ .

(3) The proof is similar to (2).

(4) It is obvious $⋃_{x \in X} \bar{R_{Δ}} ({x}) \subseteq \bar{R_{Δ}} (X)$ by (3), we will prove that $\bar{R_{Δ}} (X) \subseteq ⋃_{x \in X} \bar{R_{Δ}} ({x})$ . For $\forall y \in \bar{R_{Δ}} (X)$ , according to Definition 2.7, we obtain $N_{y}^{β} ⋂ X \neq \emptyset$ . Thus $\exists x_{0} \in N_{y}^{β} ⋂ X$ such that $N_{y}^{β} ⋂ {x_{0}} \neq \emptyset$ . Therefore, $y \in \bar{R_{Δ}} ({x_{0}})$ . By the arbitrariness of y, we can obtain $\bar{R_{Δ}} (X) \subseteq ⋃_{x \in X} \bar{R_{Δ}} ({x})$ . From the above, it follows that $\bar{R_{Δ}} (X) = ⋃_{x \in X} \bar{R_{Δ}} ({x})$ .

Definition 3.2. Let (U₁, Δ₁) and (U₂, Δ₂) be two Fβ-CASs and f : U₁ → U₂ a mapping. If $f (\bar{R_{Δ}} (X)) \subseteq \bar{R_{Δ}} (f (X))$ for any X ⊆ U₁, then we call f is a Fβ-C rough continuous mapping from (U₁, Δ₁) to (U₂, Δ₂); If f is a bijective, f and f^-1 are Fβ-C rough continuous mappings, then f is called a Fβ-C rough homeomorphism mapping from (U₁, Δ₁) to (U₂, Δ₂). Where f (X) = {f (x) : x ∈ X}.

From Lemma 2.13, we can obtain the following results:

Proposition 3.3. Let (U₁, Δ₁) and (U₂, Δ₂) be two Fβ-CASs and f : U₁ → U₂ is a mapping. Then the followings are equivalent:

(1) f is a Fβ-C rough continuous mapping from (U₁, Δ₁) to (U₂, Δ₂);

(2) For any Y ⊆ U₂, $\bar{R_{Δ}} (f^{- 1} (Y)) \subseteq f^{- 1} (\bar{R_{Δ}} (Y))$ ;

(3) For any Y ⊆ U₂, $f^{- 1} (\underline{R_{Δ}} (Y)) \subseteq \underline{R_{Δ}} (f^{- 1} (Y))$ ;

(4) For any x ∈ U₁, $f (\bar{R_{Δ}} {x}) \subseteq \bar{R_{Δ}} (f (x))$ .

Proof. (1) ⇒ (2). For ∀ Y ⊆ U₂, by Definition 3.2 and Lemma 2.13 (2), it is easy to check $f (\bar{R_{Δ}} (f^{- 1} (Y))) \subseteq \bar{R_{Δ}} (f (f^{- 1} (Y))) \subseteq \bar{R_{Δ}} (Y)$ . We have $\bar{R_{Δ}} (f^{- 1} (Y)) \subseteq f^{- 1} (\bar{R_{Δ}} (Y))$ according to Lemma 2.13 (3).

(2) ⇒ (1). For ∀ X ⊆ U₁, we have

$f^{- 1} (\bar{R_{Δ}} (f (X))) \supseteq \bar{R_{Δ}} (f^{- 1} (f (X))) \supseteq \bar{R_{Δ}} (X)$ by (2). So $f (\bar{R_{Δ}} (X)) \subseteq \bar{R_{Δ}} (f (X))$ by Lemma 2.13 (3). Therefore f is a Fβ-C rough continuous mapping from (U₁, Δ₁) to (U₂, Δ₂).

(2) ⇒ (3). For ∀ Y ⊆ U₂, $f^{- 1} (\bar{R_{Δ}} (Y^{c})) \supseteq \bar{R_{Δ}} (f^{- 1}) (Y^{c}))$ by (2). From Lemma 3.1 (1), we have $f^{- 1} (\bar{R_{Δ}} (Y^{c})) = f^{- 1} ((\underline{R_{Δ}} (Y))^{c}) = (f^{- 1} (\underline{R_{Δ}} (Y)))^{c}$ , it implies $f^{- 1} (\underline{R_{Δ}} (Y)) \subseteq \underline{R_{Δ}} (f^{- 1} (Y))$ for any Y ⊆ U₂.

(3) ⇒ (2). It is easy to prove, so we omit the proof.

(1) ⇒ (4). It is obvious by Definition 3.2.

(4) ⇒ (1). For ∀ X ⊆ U₁, X = ∪ _x∈X {x}. We have $f (\bar{R_{Δ}} {x}) \subseteq \bar{R_{Δ}} (f (x))$ by (4). From Lemma 3.1 (4) and f preserves the union operations, thus $f (\bar{R_{Δ}} (X)) = f (\bar{R_{Δ}} (U_{x \in X} {x})) = U_{x \in X} f (\bar{R_{Δ}} ({x})) \subseteq U_{x \in X} \bar{R_{Δ}} (f (x)) = \bar{R_{Δ}} (f (X))$ . therefore, (1) ⇔ (4).

Theorem 3.4. Let (U₁, Δ₁), (U₂, Δ₂) and (U₃, Δ₃) be Fβ-CASs. If f : (U₁, Δ₁) → (U₂, Δ₂) and g : (U₂, Δ₂) → (U₃, Δ₃) are both Fβ-C rough continuous mappings, then g ∘ f : (U₁, Δ₁) → (U₃, Δ₃) is also a Fβ-C rough continuous mapping.

Proof. For ∀ X ⊆ U₁, f is a Fβ-C rough continuous mapping, from Definition 3.2, we have $f (\bar{R_{Δ}} (X)) \subseteq \bar{R_{Δ}} (f (X))$ . Since f (X) ⊆ U₂ and g is a Fβ-C rough continuous mapping. It is easy to obtain $g (\bar{R_{Δ}} (f (X))) \subseteq \bar{R_{Δ}} (g (f (X)))$ . Therefore, $g (f (\bar{R_{Δ}} (X))) \subseteq \bar{R_{Δ}} (g (f (X)))$ , thus $g \circ f (\bar{R_{Δ}} (X)) \subseteq \bar{R_{Δ}} (g \circ f (X))$ , and then g ∘ f is a Fβ-C rough continuous mapping.

By Definition 3.2, Proposition 3.3 and Theorem 3.4, it is easy to obtain theorems of Fβ-C rough homeomorphism mapping.

Theorem 3.5. Let (U₁, Δ₁) and (U₂, Δ₂) be two Fβ-CASs and f a bijective mapping. Then the following conditions are equivalent:

(1) If f is a Fβ-C rough homeomorphism mapping from (U₁, Δ₁) to (U₂, Δ₂), then f^-1 is a Fβ-C rough homeomorphism mapping from (U₂, Δ₂) to (U₁, Δ₁);

(2) ∀X ⊆ U₁, then $f (\bar{R_{Δ}} (X)) = \bar{R_{Δ}} (f (X))$ ;

(3) ∀Y ⊆ U₂, then $f^{- 1} (\underline{R_{Δ}} (Y)) = \underline{R_{Δ}} (f^{- 1} (Y))$ ;

(4) ∀Y ⊆ U₂, then $f^{- 1} (\bar{R_{Δ}} (Y)) = \bar{R_{Δ}} (f^{- 1} (Y))$ .

Theorem 3.6. Let (U₁, Δ₁),(U₂, Δ₂) and (U₃, Δ₃) be Fβ-CASs, we have:

(1) Identity mapping i : (U₁, Δ₁) → (U₁, Δ₁) is a Fβ-C rough homeomorphism mapping.

(2) If f : (U₁, Δ₁) → (U₂, Δ₂) is a Fβ-C rough homeomorphic mapping, then f^-1 : (U₂, Δ₂) → (U₁, Δ₁) is also a Fβ-C rough homeomorphism mapping.

(3) If f : (U₁, Δ₁) → (U₂, Δ₂) and g : (U₂, Δ₂) → (U₃, Δ₃) are Fβ-C rough homeomorphism mappings, then g ∘ f : (U₁, Δ₁) → (U₃, Δ₃) is also a Fβ-C rough homeomorphism mapping.

With the use of the mappings, we can use Fβ-C rough homeomorphism mapping to classify Fβ-CASs, that is to say, two Fβ-CASs belong to the same class, if they can establish Fβ-C rough homeomorphism mapping. It also will helpfully solve some other applicational problems in Fβ-CASs, such as attributes reduction and describing the similarity of objects.

4 A new method of constructing topology by Fβ-C operators

Zhao defined topology by coverings in [31] and only used in covering approximation space. L. Yang et al. defined open set by $\underline{R} (X) = A^{0}$ and construct topology. Inspiring by them, we construct a new topology induced by Fβ-C operators.

Theorem 4.1. Let (U, Δ) be a Fβ-CAS and R_Δ a Fβ-C relation on U. We construct topology τ by R_Δ as follows: V ⊆ U is called open set in U if ∀x ∈ V, ∃ a finite family ${X_{i} : i \in I} \subseteq P (U)$ such that $x \in \underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq V$ .

Proof. We prove that τ satisfies Definition 2.8.

(1) It is obvious ∅ and U in τ by the definition of open set. So they satisfy (O₁) of Definition 2.8.

(2) For ∀ V₁, V₂ ∈ τ, we show that V₁ ⋂ V₂ ∈ τ. For ∀x ∈ V₁ ⋂ V₂, there exist finite family {X_{i
_k} : k ∈ K} and {X_{j
_m} : m ∈ M} such that $x \in \underline{R_{Δ}} (⋂_{k \in I} X_{i_{k}}) \subseteq V_{1}$ and $x \in \underline{R_{Δ}} (⋂_{m \in L} X_{j_{m}}) \subseteq V_{2}$ . Thus $x \in (\underline{R_{Δ}} (⋂_{k \in I} X_{i_{k}})) ⋂ (\underline{R_{Δ}} (⋂_{m \in L} X_{j_{m}}))$ = $\underline{R_{Δ}} ((⋂_{m \in L} X_{j_{m}}) ⋂ (⋂_{k \in I} X_{i_{k}})) \subseteq V_{1} ⋂ V_{2}$ . This prove that any finite intersections of elements of τ are still in τ. It satisfies (O₂) of Definition 2.8.

(3) Let {V_α : α ∈ J} be a family of elements of τ. We need to prove that V = ⋃ _α∈JV_α ∈ τ. For ∀x ∈ V, it must exist indexed α₀ ∈ J such that x ∈ V_α₀. Since V_α₀ is open and R_Δ a fuzzy β covering relation on U, we can find a finite family ${X_{i} : i \in I} \subseteq P (U)$ such that $x \in \underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq V_{α_{0}} \subseteq V$ , so V is open, which satisfies (O₃) of Definition 2.8. Therefore the desired result is proved.

Example 4.2. Let (U, Δ) be a Fβ - CAS, where $U = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}, Δ = {C_{1}, C_{2}}, C_{1} = {K_{11}, K_{12}, K_{13}}, C_{2} = {K_{21}, K_{22}, K_{23}}$ . β = 0.5.

$C_{1}$	x ₁	x ₂	x ₃	x ₄	x ₅
K ₁₁	0.7	0.6	0.3	0.1	0.4
K ₁₂	0.4	0.2	0.5	0.6	0.7
K ₁₃	0.3	0.7	0.6	0.8	0.3

Then $[x_{1}]_{C_{1}}^{β} = K_{11}$ , $[x_{2}]_{C_{1}}^{β} = K_{11} ⋂ K_{13}$ , $[x_{3}]_{C_{1}}^{β} = K_{12} ⋂ K_{13}$ , $[x_{4}]_{C_{1}}^{β} = K_{11} ⋂ K_{13}$ , $[x_{5}]_{C_{1}}^{β} = K_{12}$ .

$C_{2}$	x ₁	x ₂	x ₃	x ₄	x ₅
K ₂₁	0.8	0.7	0.4	0.4	0.6
K ₂₂	0.9	0.8	0.5	0.5	0.5
K ₂₃	0.2	0.7	0.6	0.7	0.4

Thus $[x_{1}]_{C_{2}}^{β} = K_{21} ⋂ K_{22}$ , $[x_{2}]_{C_{2}}^{β} = K_{21} ⋂ K_{22} ⋂ K_{23}$ , $[x_{3}]_{C_{2}}^{β} = K_{22} ⋂ K_{23}$ , $[x_{4}]_{C_{2}}^{β} = K_{22} ⋂ K_{23}$ , $[x_{5}]_{C_{2}}^{β} = K_{21} ⋂ K_{22}$ .

According to Definition 2.4, the $[x]_{Δ}^{β}$ are as follows.

Δ	x ₁	x ₂	x ₃	x ₄	x ₅
$[x_{1}]_{Δ}^{β}$	0.7	0.6	0.3	0.1	0.4
$[x_{2}]_{Δ}^{β}$	0.2	0.6	0.3	0.1	0.3
$[x_{3}]_{Δ}^{β}$	0.2	0.2	0.5	0.5	0.3
$[x_{4}]_{Δ}^{β}$	0.2	0.2	0.5	0.5	0.3
$[x_{5}]_{Δ}^{β}$	0.4	0.2	0.4	0.4	0.5

Based on Definition 2.5 and Definition 2.6, we got that $N_{x_{1}}^{β} = {x_{1}, x_{2}}$ , $N_{x_{2}}^{β} = {x_{2}}$ , $N_{x_{3}}^{β} = {x_{3}, x_{4}}$ , $N_{x_{4}}^{β} = {x_{3}, x_{4}}$ , $N_{x_{5}}^{β} = {x_{5}}$ .

Take V = {x₃, x₄} ⊆ U, then for ∀x ∈ V, ∃ a finite family $X = {{x_{1}, x_{3}, x_{4}, x_{5}}, {x_{1}, x_{3}, x_{4}}, {x_{1}, x_{2}, x_{3}, x_{4}}}$ such that $x \in \underline{R_{Δ}} (⋂ X) = {x_{3}, x_{4}} \subseteq V$ .

So τ = {∅ , {x₃} , {x₄} , {x₃, x₄}} is a topology.

Operators establish a bridge between general topology and rough sets,which provides a method for exchange information system[29]. We discuss the if and only if conditions for $\bar{R_{Δ}}$ to be a topological closure operator from two different perspective.

From the idea of fuzzy β neighborhood, we have the following results:

Definition 4.3. For any $N_{x}^{β}, N_{y}^{β}$ , when $N_{x}^{β} \cap N_{y}^{β} \neq \emptyset$ , if exists z such that $N_{z}^{β} \subseteq N_{x}^{β} \cap N_{y}^{β}$ , then $N_{z}^{β}$ is called a minimum neighborhood.

Theorem 4.4. $\bar{R_{Δ}}$ is a topological closure operator if and only if for any x ∈ A exist at least one minimum neighborhood.

Proof. (⇒) For ∀ x ∈ A we have $N_{x}^{β} \cap A \neq \emptyset$ by the definition of $\bar{R_{Δ}}$ . Take $y \in N_{x}^{β} \cap A$ , then we obtain $N_{x}^{β} \cap N_{y}^{β} \cap A \neq \emptyset$ , thus $N_{x}^{β} \cap N_{y}^{β} \neq \emptyset$ and there must be a $N_{z}^{β} \subseteq N_{x}^{β} \cap N_{y}^{β}$ such that $N_{x}^{β} \cap A \neq \emptyset$ .

(←) we shall prove $\bar{R_{Δ}}$ satisfies Definition 2.12.

(1) For $\forall x \in \bar{R_{Δ}} (A ⋃ B)$ , we have $N_{x}^{β} \cap (A ⋃ B) \neq \emptyset$ by Definition 2.7, thus $(N_{x}^{β} \cap A) ⋃ (N_{x}^{β} \cap B) \neq \emptyset$ , then $N_{x}^{β} \cap A \neq \emptyset$ or $N_{x}^{β} \cap B \neq \emptyset$ , hence $x \in \bar{R_{Δ}} (A)$ or $x \in \bar{R_{Δ}} (B)$ . Therefore $\bar{R_{Δ}} (A ⋃ B) \subseteq \bar{R_{Δ}} (A) ⋃ \bar{R_{Δ}} (B)$ . We shall prove the converse.

Monotonic increment from upper approximation, we easy prove the converse. Therefore

$\bar{R_{Δ}} (A) ⋃ \bar{R_{Δ}} (B) = \bar{R_{Δ}} (A ⋃ B)$ .

(2) For ∀ x ∈ A, then $N_{x}^{β} \cap A \neq \emptyset$ , thus $x \in \bar{R_{Δ}} (A)$ by Definition 2.7. Therefore $A \subseteq \bar{R_{Δ}} (A)$ .

(3) $\bar{R_{Δ}} (\emptyset) = \emptyset$ is obvious.

(4) From (2), we obtain $\bar{R_{Δ}} (A) \subseteq \bar{R_{Δ}} (\bar{R_{Δ}} (A))$ and shall prove $\bar{R_{Δ}} (\bar{R_{Δ}} (A)) \subseteq \bar{R_{Δ}} (A)$ . For $\forall y \in \bar{R_{Δ}} (\bar{R_{Δ}} (A))$ , we have $N_{y}^{β} \cap \bar{R_{Δ}} (A) \neq \emptyset$ by Definition 2.7. Then $\exists z \in N_{y}^{β} \cap \bar{R_{Δ}} (A)$ such that $N_{z}^{β} \cap N_{y}^{β} \neq \emptyset$ and $N_{z}^{β} \cap A \neq \emptyset$ . According to Definition 4.3, ∃ m such that $N_{m}^{β} \subseteq N_{z}^{β} \cap N_{y}^{β}$ . From $N_{z}^{β} \cap A \neq \emptyset$ , we obtain $N_{m}^{β} \cap A \neq \emptyset$ , then $N_{y}^{β} \cap A \neq \emptyset$ , thus $y \in \bar{R_{Δ}} (A)$ . Hence $\bar{R_{Δ}} (\bar{R_{Δ}} (A)) = \bar{R_{Δ}} (A)$ is proved.

From Fβ-C relation, we have results as follow:

Lemma 4.5. Let (U, Δ) be a Fβ-CAS. R_Δ is transitive if and only if ∀X ⊆ U, $\bar{R_{Δ}} (\bar{R_{Δ}} (X)) \subseteq \bar{R_{Δ}} (X)$ .

Proof. (⇒) For $\forall x \in \bar{R_{Δ}} (\bar{R_{Δ}} (X))$ , we have $N_{x}^{β} \cap \bar{R_{Δ}} (X) \neq \emptyset$ by Definition 2.7, then $\exists y \in N_{x}^{β} \cap \bar{R_{Δ}} (X)$ such that $N_{y}^{β} \cap X \neq \emptyset$ , and then $\exists m \in N_{y}^{β} \cap X$ such that $N_{m}^{β} \cap X \neq \emptyset$ . From $y \in N_{x}^{β}, m \in N_{y}^{β}$ , we have (x, y) ∈ R_▵ ≥ β, (y, m) ∈ R_▵ ≥ β by Definition 2.6. Since R_Δ is transitive, we obtain that (x, m) ∈ R_▵ ≥ β, then $m \in N_{x}^{β}$ . Thus we have $N_{x}^{β} \cap X \neq \emptyset$ . Since x is arbitrary, then we have $x \in \bar{R_{Δ}} (X)$ .

(←) For $\forall x \in \bar{R_{Δ}} (\bar{R_{Δ}} (X))$ , then $x \in \bar{R_{Δ}} (X)$ which means for $N_{x}^{β} \cap \bar{R_{Δ}} (X) \neq \emptyset$ , thus $N_{x}^{β} \cap X \neq \emptyset$ . From $N_{x}^{β} \cap \bar{R_{Δ}} (X) \neq \emptyset$ , then $\exists y \in N_{x}^{β} \cap \bar{R_{Δ}} (X)$ such that $N_{y}^{β} \cap X \neq \emptyset$ , thus $N_{y}^{β} \subseteq N_{x}^{β}$ . For $y \in N_{x}^{β}$ , we obtain (x, y) ∈ R_▵ ≥ β by Definition 2.6. If we have $m \in N_{y}^{β} \subseteq N_{x}^{β}$ , then (y, m) ∈ R_▵ ≥ β and (x, m) ∈ R_▵ ≥ β. Hence R_Δ is transitive.

Theorem 4.6. $\bar{R_{Δ}}$ is a topological closure operator if and only if R_Δ is transitive.

Proof. (⇒)If $\bar{R_{Δ}}$ is a topological closure operator, then $\bar{R_{Δ}}$ satisfied the conditions of Definition 2.12, according to Definition 2.12 (C₄) and Lemma 4.5, R_Δ is transitive.

(←) We shall prove $\bar{R_{Δ}}$ satisfies Definition 2.12.

(1) For $\forall x \in \bar{R_{Δ}} (A ⋃ B)$ , then $N_{x}^{β} \cap (A ⋃ B) \neq \emptyset$ by Definition 2.7, thus $(N_{x}^{β} \cap A) ⋃ (N_{x}^{β} \cap B) \neq \emptyset$ , then $N_{x}^{β} \cap A \neq \emptyset$ or $N_{x}^{β} \cap B \neq \emptyset$ , hence $x \in \bar{R_{Δ}} (A)$ or $x \in \bar{R_{Δ}} (B)$ . Therefore $\bar{R_{Δ}} (A ⋃ B) \subseteq \bar{R_{Δ}} (A) ⋃ \bar{R_{Δ}} (B)$ . We shall prove the converse.

Since $\bar{R_{Δ}} (A) \subseteq \bar{R_{Δ}} (A ⋃ B)$ , $\bar{R_{Δ}} (B) \subseteq \bar{R_{Δ}} (A) ⋃ B)$ is obvious, then $\bar{R_{Δ}} (A) ⋃ \bar{R_{Δ}} (B) \subseteq \bar{R_{Δ}} (A) ⋃ B)$ . Therefore $\bar{R_{Δ}} (A) ⋃ \bar{R_{Δ}} (B) = \bar{R_{Δ}} (A ⋃ B)$ .

(2) For ∀ x ∈ A, then $N_{x}^{β} \cap A \neq \emptyset$ , thus $x \in \bar{R_{Δ}} (A)$ by Definition 2.7. Therefore $A \subseteq \bar{R_{Δ}} (A)$ .

(3) $\bar{R_{Δ}} (\emptyset) = \emptyset$ ia obvious.

(4) Since R_Δ is transitive, then $\bar{R_{Δ}} (\bar{R_{Δ}} (A)) \subseteq \bar{R_{Δ}} (A)$ is obviously by Lemma 4.5. From (2) we have $\bar{R_{Δ}} (A) \subseteq \bar{R_{Δ}} (\bar{R_{Δ}} (A))$ . Hence $\bar{R_{Δ}} (A) = \bar{R_{Δ}} (\bar{R_{Δ}} (A))$ .

Remark 4.7. (1) ∀X ⊆ U, if x ∈ X and R_Δ is a Fβ-C relation on U, We have $\underline{R_{Δ}} (X) \in τ$ by Theorem 4.1.

(2) If R_Δ is a Fβ-C relation and $\underline{R_{Δ}} (X) = X^{0}$ , then $\underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq \underline{R_{Δ}} (X) = X^{0}$ by Theorem 4.1 and X⁰ ∈ τ.

Theorem 4.8. Let (U, Δ) be a Fβ-CAS, R_Δ a Fβ-C relation on U and τ a topology induced by R_Δ. Then the following properties holds:

(1) For each x ∈ U, ${X_{i} : i \in I} \subseteq P (U)$ is all the subsets of U which contains x and R_Δ is a Fβ-C relation on U. If V ∈ τ is a subset of U and x ∈ V, then $\underline{R_{Δ}} (⋂_{i \in I} X_{i})$ is the smallest subset of U and $x \in \underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq V$ . Denoted by $C (x) = ⋂ {\underline{R_{Δ}} (X_{i}) : x \in \underline{R_{Δ}} (X_{i}), i \in I}$ .

(2) If V ∈ τ is a subset of U, then V = ⋃_x∈VC (x).

(3) Let ${\underline{R_{Δ}} (X_{i}) : x \notin \underline{R_{Δ}} (X_{i}), i \in I}$ , then $\bar{{x}}$ = $U \ ⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i})$ ;

(4) Let R_Δ be a Fβ-C relation on U. ∀X ⊆ U, we have $int (\underline{R_{Δ}} (X)) = \underline{R_{Δ}} (X)$ . Where $int (\underline{R_{Δ}} (X))$ represents the interior of $\underline{R_{Δ}} (X)$ .

(5) Let ${\underline{R_{Δ}} (X_{i}) : x \in \underline{R_{Δ}} (X_{i}), i \in I}$ be a family subsets of U, then ${\underline{R_{Δ}} (X_{i}) : x \in \underline{R_{Δ}} (X_{i}), i \in I}$ is a base for (U, τ) at the point x.

Proof. (1) Let V ∈ τ be a subset of U and x ∈ V. There is a finite subset family ${X_{k} : k \in K} \subseteq P (U)$ such that $x \in \underline{R_{Δ}} (⋂_{i \in I} X_{i}) \subseteq V$ by Theorem 4.1. ${X_{i} : i \in I} \subseteq P (U)$ is all the subsets of U which contains x, then C (x) is the smallest subset of U containing x and $C (x) \subseteq \underline{R_{Δ}} (⋂_{k \in K} X_{k})$ , thus $x \in C (x) \subseteq \underline{R_{Δ}} (⋂_{k \in K} X_{k}) \subseteq V$ .

(2) Let V ∈ τ be a subset of U and x ∈ V, we can obtain C (x) ⊆ V by (1). Since {C (x) : x ∈ V} is a Fβ-C of V, then V ⊆ ⋃ _x∈VC (x), thus we have V = ⋃_x∈VC (x).

(3) Pick $M (x) = ⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i})$ . $\underline{R_{Δ}} (X_{i})$ is an open subset of U for very i ∈ I by Remark 4.7 (1), the union $⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i})$ is open by Definition 2.8 (A₃). It follows that the set

$U \ ⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i})$ is closed and

$x \in (U \ ⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i}))$ . $\bar{{x}}$ is the smallest closed set containing x. Obviously,

$\bar{{x}} \subseteq U \ ⋃_{x \notin \underline{R_{Δ}} (X_{i}), i \in I} \underline{R_{Δ}} (X_{i})$ . We prove the converse.

For $\forall y \in U \ \bar{{x}}$ . Since $U \ \bar{{x}}$ is an open set containing y, then $C (y) \subseteq U \ \bar{{x}}$ by (1), therefore $C (y) \cap \bar{{x}} = \emptyset$ and x ∉ C (y). There exists a $\underline{R_{Δ}} (X_{i_{0}})$ of ${\underline{R_{Δ}} (X_{i}) : x \notin \underline{R_{Δ}} (X_{i}), i \in I}$ such that $y \in \underline{R_{Δ}} (X_{i_{0}}) \land x \notin \underline{R_{Δ}} (X_{i_{0}})$ . Thus y ∈ M (x) and $U \ \bar{{x}} \subseteq M (x)$ , therefore $U \ M (x) \subseteq \bar{{x}}$ .

(4) Let R_Δ be a Fβ-C relation on U, ∀X ⊆ U, from Remark 4.7 (2) it follows that $\underline{R_{Δ}} (X)$ is an open set, then we obtain $\underline{R_{Δ}} (X) = int (\underline{R_{Δ}} (X))$ directly by definition of open set.

(5) Let $B (x)$ = ${\underline{R_{Δ}} (X_{i}) : x \in \underline{R_{Δ}} (X_{i}), i \in I}$ . For $\forall V_{1}, V_{2} \in B (x)$ and every x ∈ V₁ ∩ V₂, there exist finite collections {X_{i
_k} : k ∈ K} and {X_{j
_m} : m ∈ M} such that $x \in \underline{R_{Δ}} (⋂_{k \in K} X_{i_{k}}) \subseteq V_{1}$ and $x \in \underline{R_{Δ}} (⋂_{m \in M} X_{j_{m}}) \subseteq V_{2}$ . Thus

$x \in (\underline{R_{Δ}} (⋂_{k \in K} X_{i_{k}})) ⋂ (\underline{R_{Δ}} (⋂_{m \in L} X_{j_{m}}))$ = $\underline{R_{Δ}} ((⋂_{m \in M} X_{j_{m}}) ⋂ (⋂_{k \in I} X_{i_{k}})) \subseteq V_{1} ⋂ V_{2}$ . Pick X₀ = (⋂ _m∈LX_{j
_m}) ⋂ (⋂ _k∈IX_{i
_k}), then $\underline{R_{Δ}} (X_{0}) \in B (x)$ which satisfies (B₁) of Definition 2.10. $B (x)$ is obvious that it satisfies (B₂) of Definition 2.10. ${\underline{R_{Δ}} (X_{i}) : x \in \underline{R_{Δ}} (X_{i}), i \in I}$ is a base for (U, τ) at the point x.

Lemma 4.9. Let X be a topological space. ∀A ⊆ X, then the point x belong to $\bar{A}$ if and only if for every neighborhood U of x, we have U⋂ A ≠ ∅.

Theorem 4.10. Let (U, Δ) be a Fβ-CAS, R_Δ a Fβ-C relation on U and τ a topology induced by R_Δ, then we have:

(1) ∀A ⊆ U and x ∈ U, $x \in \bar{A}$ if and only if C (x)∩ A ≠ ∅ if and only if there exists a base $B (x)$ at x such that for every $V \in B (x)$ , V∩ A ≠ ∅.

(2) ∀x, y ∈ U and x ≠ y, then C (x) = C (y) if and only if $\bar{{x}} = \bar{{y}}$ .

(3) Let $P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}}$ = ${y \in U : \forall i \in I, \underline{R_{Δ}} (X_{i}) \in {\underline{R_{Δ}} (X_{i}) : i \in I} (x \in \underline{R_{Δ}} (X_{i}) \leftarrow \to y \in \underline{R_{Δ}} (X_{i}))}$ . For any x ∈ U, then $P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}}$ = $C (x) \cap \bar{{x}}$ .

(4) Let F ⊆ U be a closed set, then $F = ⋃_{x \in F} \bar{{x}}$ = $⋃_{x \in F} P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}}$ .

Proof. (1) For $\forall x \in \bar{A}$ , from Lemma 4.9, $x \in \bar{A}$ ⇔ C (x)∩ A ≠ ∅. $x \in \bar{A}$ ⇔ ∃ a base $B (x)$ at x such that for every $V \in B (x)$ , V∩ A ≠ ∅ can be proved in a similar way.

If C (x)∩ A ≠ ∅, then V∩ A ≠ ∅ is obvious. It remains to show that if there exists a base $B (x)$ at x such that for every $V \in B (x)$ , V∩ A ≠ ∅, then $x \in \bar{A}$ . Suppose that $x \in \bar{A}$ dose not hold, i.e., $x \notin \bar{A}$ . There exists a closed set F such that x ∉ F. For the open set V = U ∖ F we have x ∈ V and V∩ A = ∅. For every base $B (x)$ at x, there exists a such that and from V∩ A = ∅, it follows that , i,e, for every $V \in B (x)$ , V∩ A ≠ ∅ dose not hold.

(2) If C (x) = C (y), we have $x \in \bar{{y}}$ and $y \in \bar{{x}}$ from (1). It is well known that $\bar{{x}}$ is the smallest closed set containing x, thus $\bar{{x}} \subseteq \bar{{y}}$ . Similarly, $\bar{{y}} \subseteq \bar{{x}}$ , therefore, we have $\bar{{x}} = \bar{{y}}$ . If $\bar{{x}} = \bar{{y}}$ , we shall prove C (x) = C (y). Since $x \in \bar{{y}}$ , we have {y}∩ C (x) ≠ ∅, thus y ∈ C (x), therefore C (y) ⊆ C (x). Similarly, we have C (x) ⊆ C (y), thus C (x) = C (y).

(3) It is not difficult to prove that $P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}} = {y \in U : \forall i \in I, \underline{R_{Δ}} (X_{i}) \in {\underline{*} (X_{i}) : i \in I} (x \in \underline{R_{Δ}} (X_{i}) \leftarrow \to y \in \underline{R_{Δ}} (X_{i}))}$ = {y ∈ U : C (x) = C (y)}. Suppose y is an element of $P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}}$ , then y ∈ C (x), thus $y \in \bar{{x}}$ , and then $y \in C (x) \cap \bar{{x}}$ , therefore $P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}} \subseteq C (x) \cap \bar{{x}}$ . We shall prove the converse. For any $y \in C (x) \cap \bar{{x}}$ . We have y ∈ C (x) and x ∈ C (y) by (1); therefore, C (x) = C (y), thus we have $y \in P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}}$ .

(4) For any x ∈ F, we have $\bar{{x}} \subseteq F$ , and then $⋃_{x \in F} \bar{{x}} \subseteq F$ . ${\bar{{x}} : x \in F}$ is an closed covering of F, thus $F \subseteq ⋃_{x \in F} \bar{{x}}$ , therefore, $F = ⋃_{x \in F} \bar{{x}}$ . $F \subseteq ⋃_{x \in F} P_{x}^{{\underline{R_{Δ}} (X_{i}) : i \in I}} \subseteq ⋃_{x \in F} \bar{{x}} = F$ by (3).

5 Conclusions

In this paper, we have proposed new concepts and investigated relationship among Fβ-CAS, and obtained some interesting properties of Fβ-CAS by Fβ-C rough continuous mapping and homeomorphism mapping. We construct topology by Fβ-C relation and explore the if and only if conditions of upper operator to be a topological closure. In the future, we will use the constructed topology and proven correlation properties to construct new uncertainty measures and try to apply them to anomaly detection and feature selection in the information system.

Footnotes

Acknowledgments

This work is supported by Natural Science Foundation of China(NO.12261096), Guangxi Natural Science Foundation (NO. 2020GXNSFAA159155), Guangxi One Thousand Young and Middle-aged College and University Backbone Teachers Cultivation Program (No. [2019] 5).

References

Zakowski

, Approximations in the space (U, p), Demonstratio Mathematica 16(3) (1983), 761–770.

Zhu

, Topological approaches to covering rough sets, Information Sciences 177(6) (2007), 1499–1508.

, On some types of neighborhood-related covering rough sets, International Journal of Approximate Reasoning 53 (2012), 901–911.

, Some twin approximation operators on covering approximation spaces, International Journal of Approximate Reasoning 56 (2015), 59–70.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

Zadeh

L.A.

, Fuzzy logic computing with words, IEEE Transactions in fuzzy system 4(1) (1996), 103–111.

Lin

T.Y.

Topological and fuzzy rough sets, in: R. Sowi²a nski (Ed.), Intelligent Decision Support-Handbook of Applications and Advances of Rough Sets Theory, Kluwer Academic Publishers, Boston, (1992), pp. 287–304.

Nanda

and Majumdar

, Fuzzy rough sets, Fuzzy Sets System 45 (1992), 157–160.

Dubois

, Prade

Putting rough sets and fuzzy sets together, in: R. Sowia²a nski (Ed.), Intelligent Decision Support: Handbook of Applications Advances of the Sets Theory, Kluwer, Dordrecht, (1992), pp. 203–232.

10.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990), 191–209.

11.

Feng

, Zhang

S.P.

and Mi

J.S.

, The reduction and fusion of fuzzy covering systems based on the evidence theory, International Journal of Approximate Reasoning 53 (2012), 87–103.

12.

Inuiguchi

Classification and approximation oriented fuzzy rough sets, in: B. Bouchon-Meunier, et al. (Eds.), Proceedings of Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2004, cd-rom.

13.

T.J.

, Leung

and Zhang

W.X.

, Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning 48 (2008), 836–856.

14.

M.F.

, Han

H.H.

, Si

Y.F.

Properties and axiomatization of fuzzy rough sets based on fuzzy coverings, in: Proceedings of the 2012 Interna-tional Conference on Machine Learning and Cybernetics, (2012), pp 184–189.

15.

Zhang

Y.L.

and Luo

M.K.

, Relationships beween coveringbasedrough sets and relation-based rough sets, Information Sciences 225 (2013), 55–71.

16.

Chen

J.K.

, Lin

Y.J.

, Lin

G.P.

, Li

J.J.

and Ma

Z.M.

, Intuitionistic Fuzzy Rough Set-Based Granular Structures and Attribute Subset Selection, Information Sciences 325 (2015), 87–97.

17.

, Topological Method in Covering Generalized Rough Set Theory, Pattern Recognition and Artificial Intelligence 17(1) (2004), 7–10.

18.

Wang

and Li

Q.G.

, The rough membership function based on C10 and its applications, Journal of Intelligent and Fuzzy Systems 32 (2017), 279–289.

19.

Wang

, Wu

Q.J.

, He

J.L.

and Shang

, Approximation Operator Based on Neighborhoods Systems, Symmetry 10(11) (2018), 539.

20.

Zhang

, Wu

, Liang

, Li

Rough Set Theory and Method, Beijing Science Press, 2001.

21.

Zhu

, Topological approaches to covering rough sets, Information Sciences 177(6) (2007), 1499–1508.

22.

and Li

, Definable subset in covering approximation spaces, International Journal of Computational and Mathematical Sciences 5(1) (2011), 31–34.

23.

L.W.

, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Sets System (294) (2016), 1–17.

24.

Yang

and Hu

B.Q.

, On some types of fuzzy covering-based rough sets, Fuzzy Sets System 312 (2017), 36–65.

25.

Yang

and Hu

B.Q.

, Fuzzy neighborhood operators and derived fuzzy coverings, Fuzzy Sets System 370 (2019), 1–33.

26.

Huang

Z.H.

, Li

J.J.

Discernibility measures for fuzzy β-covering and their application, IEEE Transactions on Cybernetics (2021).

27.

Dai

J.H.

, Zou

X.T.

, Wu

W.Z.

Novel fuzzy β-covering rough set models and their applications, Information Sciences (2022).

28.

Zhang

X.H.

and Wang

J.Q.

, Fuzzy β-covering approximation spaces, International Journal of Approximate Reasoning 126 (2020), 27–47.

29.

, Wang

and Li

, Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method, Symmetry 11(2) (2019), 199.

30.

Yang

and Xu

, Topological properties of generalized approximation spaces, Information Sciences 181 (2011), 3570–3580.

31.

Zhao

, On some type of covering rough sets from topological points of view, International Journal of Approximation Reasoning 68 (2016), 1–14.

32.

and Zhan

, On the topological properties of generalized rough sets, Information Sciences 263 (2014), 141–152.

33.

Engelking

General Topology, Polish Scientific Publishers, Warsaw, 1977.

34.

Zhang

, Zhan

J.M.

, Wu

W.Z.

and Alcantud

J.C.R.

, Fuzzy β-covering based ()-fuzzy rough set models and applications to multi-attribute decision-making, Computers and Industrial Engineering 128 (2019), 605–621.

35.

, Li

and Yu

, Updating reducts in fuzzy β-covering via matrix approaches while coarsening and refining a covering element, Journal of Applied Mathematics and Computing 63(1-2) (2020), 717–737.

36.

Lashin

E.F.

and Medhat

, Topological reduction of information systems, Chaos, Solitons and Fractals 25 (2005), 277–286.

37.

Qin

and Pei

, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005), 601–613.

38.

Salama

A.S.

, Topological solution of missing attribute values problem in incomplete information tables, Information Sciences 180 (2010), 631–639.

39.

Atef

and Azzam

A.A.

, Covering fuzzy rough sets via variable precision, Journal of Mathematics 2021 (2021), 1–10.