Fuzzy Approximating Spaces

Abstract

Relationships between fuzzy relations and fuzzy topologies are deeply researched. The concept of fuzzy approximating spaces is introduced and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained.

1. Introduction

Rough set theory, proposed by Pawlak [1], is a new mathematical tool for data reasoning. It may be seen as an extension of classical set theory and has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields [2–5].

The basic structure of rough set theory is an approximation space. Based on it, lower and upper approximations can be induced. Using these approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules. A key notion in Pawlak rough set model is equivalence relations. The equivalence classes are the building blocks for the construction of these approximations. In the real world, the equivalence relation is, however, too restrictive for many practical applications. To address this issue, many interesting and meaningful extensions of Pawlak rough sets have been presented. Equivalence relations can be replaced by tolerance relations [6], similarity relations [7], binary relations [8, 9], and so on.

Various fuzzy generalizations of rough approximations have been proposed [10–14]. The most common fuzzy rough set is obtained by replacing the crisp relations with fuzzy relations on the universe and crisp subsets with fuzzy sets. Dubois and Prade [10] first proposed the concept of rough fuzzy sets and fuzzy rough sets and pointed out that a rough fuzzy set is a special case of a fuzzy rough set. Now, fuzzy rough sets have been used to solve practical problems such as data mining [15], approximate reasoning [5], and medical time series.

An interesting and natural research topic in rough set theory is to study the relationship between rough sets and topologies. Many authors studied topological properties of rough sets [16–21]. It is known that the pair of lower and upper approximation operators induced by a reflexive and transitive relation is exactly the pair of interior and closure operators of a topology [16, 22].

The purpose of this paper is to investigate further topological properties of fuzzy rough sets.

The remaining part of this paper is organized as follows. In Section 2, we recall some basic concepts about fuzzy sets and fuzzy topologies. In Section 3, fuzzy rough approximation operators are further investigated. In Section 4, relationships between fuzzy approximation spaces and fuzzy topologies are established. In Section 5, the concept of fuzzy approximating spaces is introduced and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained. Conclusions are in Section 6.

2. Preliminaries

Throughout this paper, 𝑈 denotes a nonempty finite set, 𝐼 denotes [0,1], and 𝐹(𝑈) denotes the set of all fuzzy sets in 𝑈. For 𝑎∈𝐼, 𝑎 denotes the constant fuzzy set in 𝑈.

For all 𝐴∈𝐹(𝑈), we denote 𝑅_𝐴=𝑥,𝑦∈𝑈×𝑈∶𝐴(𝑥)>𝐴𝑦. (1)

Obviously, 𝑅_𝐴=∅⇔𝐴=𝑎 for some 𝜆∈𝐼.

A fuzzy set is called a fuzzy point in 𝑈, if it takes the value 0 for each 𝑦∈𝑈 except one, say, 𝑥∈𝑈. If its value at 𝑥 is 𝜆(0<𝜆≤1), we denote this fuzzy point by 𝑥_𝜆, where the point 𝑥 is called its support and 𝜆 is called its height (see [23, 24]).

Specially, 𝑥₁𝑦=1,𝑦=𝑥,0,𝑦≠𝑥. (2)

Remark 1. Consider 𝐴=_𝑥∈𝑈𝐴(𝑥)𝑥₁(𝐴∈𝐹(𝑈)). (3)

Definition 2 (see [25]). 𝜏⊆𝐹(𝑈) is called a fuzzy topology on 𝑈, if (i) ∀𝑎∈𝐼,𝑎∈𝜏, (ii) 𝐴,𝐵∈𝜏⇒𝐴∩𝐵∈𝜏, (iii) {𝐴_𝑗∶𝑗∈𝐽}⊆𝜏⇒⋃_𝑗∈𝐽𝐴_𝑗∈𝜏.
The pair (𝑈,𝜏) is called a fuzzy topological space. Every member of 𝜏 is called a fuzzy open set in 𝑈. Its complement is called a fuzzy closed set in 𝑈.
We denote 𝜏^𝑐={𝐴∈𝐹(𝑈)∶𝐴^𝑐∈𝜏}.
It should be pointed out that if (i) in Definition 2 is replaced [26] by (i)^′0,1∈𝜏,then 𝜏 is a fuzzy topology in the sense of Chang [26]. We can see that a fuzzy topology in the sense of Lowen must be a fuzzy topology in the sense of Chang. In this paper, we always consider the fuzzy topology in the sense of Lowen.
A fuzzy topology 𝜏 is called Alexandrov [27] if (ii) in Definition 2 is replaced by  (ii)^′{𝐴_𝑗∶𝑗∈𝐽}⊆𝜏⟹⋂_𝑗∈𝐽𝐴_𝑗∈𝜏.

Definition 3 (see [28]). Let 𝑅 be a relation on 𝑈. ∀𝑥∈𝑈, denote 𝑅_𝑝(𝑥)=,𝑅𝑦∈𝑈∶𝑦,𝑥∈𝑅_𝑠(𝑥)=𝑦∈𝑈∶𝑥,𝑦∈𝑅. (4) Then 𝑅_𝑝(𝑥) and 𝑅_𝑠(𝑥) are called the predecessor and successor neighborhood of 𝑥, respectively.

3. Fuzzy Approximation Spaces and Fuzzy Rough Approximation Operators

Recall that 𝑅 is called a fuzzy relation on 𝑈 if 𝑅∈𝐹(𝑈×𝑈).

Definition 4 (see [14, 29]). Let 𝑅 be a fuzzy relation on 𝑈. Then 𝑅 is called(1)reflexive, if 𝑅(𝑥,𝑥)=1 for any 𝑥∈𝑈,(2)symmetric, if 𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥) for any 𝑥,𝑦∈𝑈,(3)transitive, if 𝑅(𝑥,𝑧)≥𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧) for any 𝑥,𝑦,𝑧∈𝑈.
Let 𝑅 be a fuzzy relation on 𝑈. 𝑅 is called preorder if 𝑅 is reflexive and transitive. 𝑅 is called equivalence if 𝑅 is reflexive, symmetric, and transitive.

Definition 5 (see [14, 29]). Let 𝑅 be a fuzzy relation on 𝑈. The pair (𝑈,𝑅) is called a fuzzy approximation space. Based on (𝑈,𝑅), the fuzzy lower and the fuzzy upper approximation of 𝐴∈𝐹(𝑈), denoted, respectively, by 𝑅(𝐴) and 𝑅(𝐴), are defined as follows: 𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦∨1−𝑅𝑥,𝑦(𝑥∈𝑈),𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅𝑥,𝑦(𝑥∈𝑈). (5)
The pair (𝑅(𝐴),𝑅(𝐴)) is called the fuzzy rough set of 𝐴 with respect to (𝑈,𝑅).
𝑅∶𝐹(𝑈)→𝐹(𝑈) and 𝑅∶𝐹(𝑈)→𝐹(𝑈) are called the fuzzy lower approximation operator and the fuzzy upper approximation operator, respectively. In general, we refer to 𝑅 and 𝑅 as the fuzzy rough approximation operators.

Remark 6. 𝑅(𝑥₁)(𝑦)=𝑅(𝑦,𝑥) and 𝑅((𝑥₁)^𝑐)(𝑦)=1−𝑅(𝑦,𝑥)(𝑥,𝑦∈𝑈).

Proposition 7 (see [30]). Let 𝑅 be a fuzzy relation on 𝑈. Then, for any 𝐴,𝐵∈𝐹(𝑈), {𝐴_𝑗∶𝑗∈𝐽}⊆𝐹(𝑈) and 𝜆∈𝐼,(1)𝑅(1)=1, 𝑅(0)=0,(2)𝐴⊆𝐵⇒𝑅(𝐴)⊆𝑅(𝐵), 𝑅(𝐴)⊆𝑅(𝐵),(3)𝑅(𝐴^𝑐)=(𝑅(𝐴))^𝑐, 𝑅(𝐴^𝑐)=(𝑅(𝐴))^𝑐,(4)𝑅(⋂_𝑗∈𝐽𝐴_𝑗)=⋂_𝑗∈𝐽(𝑅(𝐴_𝑗)), 𝑅(⋃_𝑗∈𝐽𝐴_𝑗)=⋃_𝑗∈𝐽(𝑅(𝐴_𝑗)),(5)𝑅(𝜆𝐴)=𝜆𝑅(𝐴).

Theorem 8 (see [14, 29, 30]). Let 𝑅 be a fuzzy relation on 𝑈. Then consider the following. (1)  𝑅 is reflexive ⇔(𝐼𝐿𝑅)∀𝐴∈𝐹(𝑈), 𝑅(𝐴)⊆𝐴. ⇔(𝐼𝑈𝑅)∀𝐴∈𝐹(𝑈), 𝐴⊆𝑅(𝐴). (2)  𝑅 is symmetric ⇔(𝐼𝐿𝑆)∀(𝑥,𝑦)∈𝑈×𝑈, 𝑅((𝑥₁)^𝑐)(𝑦)=𝑅((𝑦₁)^𝑐)(𝑥). ⇔(𝐼𝑈𝑆)∀(𝑥,𝑦)∈𝑈×𝑈, 𝑅(𝑥₁)(𝑦)=𝑅(𝑦₁)(𝑥). (3)  𝑅 is transitive ⇔(𝐼𝐿𝑇)∀𝐴∈𝐹(𝑈), 𝑅(𝐴)⊆𝑅(𝑅(𝐴)). ⇔(𝐼𝑈𝑇)∀𝐴∈𝐹(𝑈), 𝑅(𝑅(𝐴))⊆𝑅(𝐴).

Remark 9. (1)  ∀𝑎∈𝐼, 𝑅(𝑎)⊆𝑎⊆𝑅(𝑎);
(2) if 𝑅 is reflexive, then ∀𝑎∈𝐼, 𝑅(𝑎)=𝑎=𝑅(𝑎).

Theorem 10. Let 𝑅 be a fuzzy relation on 𝑈 and let 𝜏 be a fuzzy topology on 𝑈. If one of the following conditions is satisfied, then 𝑅 is preorder:(1)𝑅 is the closure operator of 𝜏,(2)𝑅 is the interior operator of 𝜏.

Proof. (1) Let 𝑥,𝑦,𝑧∈𝑈. Put 𝑐𝑙_𝜏(𝑧₁)(𝑦)=𝜆. Note that 𝑅 is the interior operator of 𝜏. Then 𝑅(𝑥,𝑥)=𝑅𝑥₁(𝑥)=𝑐𝑙_𝜏𝑥₁(𝑥)≥𝑥₁(𝑥)=1. (6) Thus 𝑅 is reflexive. By Remark 1, Remark 6, and Proposition 7(5), 𝑅=𝑥,𝑦∧𝑅𝑦,𝑧𝑅𝑦₁(𝑥)∧𝑅𝑧₁𝑦=𝑅𝑦₁(𝑥)∧𝑐𝑙_𝜏𝑧₁𝑦=𝑅𝑦₁(𝑥)∧𝜆=𝜆𝑅𝑦₁(𝑥)=𝑅𝜆𝑦₁(𝑥)=𝑐𝑙_𝜏𝜆𝑦₁(𝑥)=𝑐𝑙_𝜏𝑐𝑙_𝜏𝑧₁𝑦𝑦₁(𝑥)≤𝑐𝑙_{𝜏𝑡∈𝑈}𝑐𝑙_𝜏𝑧₁(𝑡)𝑡₁(𝑥)=𝑐𝑙_𝜏𝑐𝑙_𝜏𝑧₁(𝑥)=𝑐𝑙_𝜏𝑧₁(𝑥)=𝑅(𝑥,𝑧). (7) Then 𝑅 is transitive. Hence 𝑅 is preorder.
(2) The proof is similar to (1).

Proposition 11. Let 𝑅 be a fuzzy relation on 𝑈. Then, for each 𝐴∈𝐹(𝑈) with 𝑅_𝐴≠∅, consider the following. (1)(a)𝑅(𝐴)⊇𝐴⇔(𝐹𝐿𝑂)∀(𝑥,𝑦)∈𝑅_𝐴, 1−𝑅(𝑥,𝑦)≥𝐴(𝑥)∨𝐴(𝑦). (b)𝑅(𝐴)⊆𝐴⇔(𝐹𝑈𝑂)∀(𝑥,𝑦)∈𝑅_𝐴, 𝑅(𝑦,𝑥)≤𝐴(𝑥)∧𝐴(𝑦). (2) If 𝑅 is reflexive, then (a)𝑅(𝐴)=𝐴⇔(𝐹𝐿𝑅)∀(𝑥,𝑦)∈𝑅_𝐴, 1−𝑅(𝑥,𝑦)≥𝐴(𝑥)∨𝐴(𝑦). (b)𝑅(𝐴)=𝐴⇔(𝐹𝑈𝑅)∀(𝑥,𝑦)∈𝑅_𝐴, 𝑅(𝑦,𝑥)≤𝐴(𝑥)∧𝐴(𝑦).

Proof. (1) (a) Necessity. Suppose that 𝑅(𝐴)⊇𝐴. Note that ∀𝑥∈𝑈, _𝑦∈𝑈𝐴𝑦∨1−𝑅𝑥,𝑦=𝑅(𝐴)𝑦≥𝐴(𝑥). (8) Then ∀𝑥,𝑦∈𝑈, 𝐴(𝑦)∨(1−𝑅(𝑥,𝑦))≥𝐴(𝑥). Since ∀(𝑥,𝑦)∈𝑅_𝐴, 𝐴(𝑥)>𝐴(𝑦), we have 1−𝑅𝑥,𝑦≥𝐴(𝑥)=𝐴(𝑥)∨𝐴𝑦𝑥,𝑦∈𝑅_𝐴. (9)
Sufficiency. Suppose that (𝐹𝐿𝑂) holds. Let 𝑥∈𝑈. Consider the following.(i)If 𝑦∈(𝑅_𝐴)_𝑠(𝑥), then 𝐴𝑦∨1−𝑅𝑥,𝑦≥𝐴𝑦∨𝐴(𝑥)∨𝐴𝑦≥𝐴(𝑥). (10)(ii)If 𝑦∉(𝑅_𝐴)_𝑠(𝑥), then 𝐴(𝑦)≥𝐴(𝑥) and so 𝐴𝑦∨1−𝑅𝑥,𝑦≥𝐴𝑦≥𝐴(𝑥). (11)
Hence 𝑅(𝐴)(𝑥)=⋀_𝑦∈𝑈(𝐴(𝑦)∨(1−𝑅(𝑥,𝑦)))≥𝐴(𝑥).
Thus 𝑅(𝐴)⊇𝐴.
(b) Necessity. Suppose that 𝑅(𝐴)⊆𝐴. Note that ∀𝑦∈𝑈, _𝑥∈𝑈𝐴(𝑥)∧𝑅𝑦,𝑥=𝑅(𝐴)𝑦≤𝐴𝑦. (12) Then ∀𝑥,𝑦∈𝑈, 𝐴(𝑥)∧𝑅(𝑦,𝑥)≤𝐴(𝑦). Since ∀(𝑥,𝑦)∈𝑅_𝐴, 𝐴(𝑥)>𝐴(𝑦), we have 𝑅𝑦,𝑥≤𝐴𝑦=𝐴(𝑥)∧𝐴𝑦𝑥,𝑦∈𝑅_𝐴. (13)
Sufficiency. Suppose that (𝐹𝐿𝑂) holds. Let 𝑦∈𝑈. Consider the following.(i)If 𝑥∈(𝑅_𝐴)_𝑝(𝑦), then (𝑥,𝑦)∈𝑅_𝐴 and so 𝐴(𝑥)∨𝑅𝑦,𝑥≤𝐴(𝑥)∧𝐴(𝑥)∧𝐴𝑦≤𝐴𝑦. (14)(ii)If 𝑥∉(𝑅_𝐴)_𝑝(𝑦), then 𝐴(𝑥)≤𝐴(𝑦) and so 𝐴(𝑥)∧𝑅𝑦,𝑥≤𝐴(𝑥)≤𝐴𝑦. (15)
Hence (𝑅(𝐴))(𝑦)=⋁_𝑥∈𝑈(𝐴(𝑥)∧𝑅(𝑦,𝑥))≤𝐴(𝑦).
Thus 𝑅(𝐴)⊆𝐴.
(2) holds by (1), the reflexivity of 𝑅, and Theorem 8(1).

4. Relationships between Fuzzy Relations and Fuzzy Topologies

4.1. Fuzzy Topology Induced by Fuzzy Relations

Let 𝑅 be a fuzzy relation on 𝑈. Denote 𝜎_𝑅=𝐴∈𝐹(𝑈)∶𝐴⊆𝑅(𝐴,𝜏_𝑅)=𝐴∈𝐹(𝑈)∶𝐴=𝑅(𝐴),𝜃_𝑅=𝑅(𝐴)(𝑈;𝑠∶𝐴∈𝐹)_𝑅=_{𝑥,𝑦∈𝑈,𝑥≠𝑦}𝑅𝑥,𝑦,𝑡_𝑅=_{𝑥,𝑦∈𝑈,𝑥≠𝑦}𝑅𝑥,𝑦. (16)

Remark 12. Let 𝑅 be a fuzzy relation on 𝑈. Then consider the following.(1)𝜏_𝑅⊆𝜎_𝑅, 𝜏_𝑅⊆𝜃_𝑅.(2)If 𝑅 is transitive, then 𝜏_𝑅⊆𝜃_𝑅⊆𝜎_𝑅.(3)If 𝑅 is reflexive, then 𝜏_𝑅=𝜎_𝑅.(4)If 𝑅 is preorder, then 𝜎_𝑅=𝜏_𝑅=𝜃_𝑅.

Theorem 13 (see [30]). Let 𝑅 be a preorder fuzzy relation on 𝑈. Then consider the following.(1)𝜃_𝑅 is a fuzzy topology on 𝑈.(2)𝑅 is the interior operator of 𝜃_𝑅.(3)𝑅 is the closure operator of 𝜃_𝑅.

Theorem 14. Let 𝑅 be a fuzzy relation on 𝑈. Then (1)𝜎_𝑅 is an Alexandrov fuzzy topology on 𝑈,(2)if 𝑅 is reflexive, then ∀𝐴∈𝐹(𝑈), int_𝜎𝑅(𝐴)⊆𝑅(𝐴)⊆𝐴⊆𝑅(𝐴)⊆𝑐𝑙_𝜎𝑅(𝐴). (17)(3)𝐴∈(𝜎_𝑅)^𝑐⇔𝑅(𝐴)⊆𝐴.(4)∀𝑎∈𝐼, 𝑎∈(𝜎_𝑅)^𝑐.

Proof. (1) By Remark 9(1), 𝑎∈𝜎_𝑅(𝑎∈𝐼).
Let {𝐴_𝑗∶𝑗∈𝐽}⊆𝜎_𝑅. Then ∀𝑗∈𝐽, 𝐴_𝑗⊆𝑅(𝐴_𝑗). By Proposition 7(4), _𝑗∈𝐽𝐴_𝑗⊆_𝑗∈𝐽𝑅𝐴_𝑗=𝑅_𝑗∈𝐽𝐴_𝑗,_𝑗∈𝐽𝐴_𝑗⊆_𝑗∈𝐽𝑅𝐴_𝑗⊆𝑅_𝑗∈𝐽𝐴_𝑗. (18)
Hence ⋂_𝑗∈𝐽𝐴_𝑗,⋃_𝑗∈𝐽𝐴_𝑗∈𝜎_𝑅. So 𝜎_𝑅 is Alexandrov.
(2) ∀𝐴∈𝐹(𝑈), by Proposition 7(2), int_𝜎𝑅(𝐴)=𝐵∈𝜎_𝑅∶𝐵⊆𝐴⊆𝐵∈𝜎_𝑅∶𝑅(𝐵)⊆𝑅(𝐴=(𝑈))𝐵∈𝐹∶𝐵⊆𝑅(𝐵)⊆𝑅(𝐴)⊆𝑅(𝐴). (19) By Proposition 7(3), 𝑐𝑙_𝜎𝑅(𝐴)=int_𝜎𝑅𝐴^𝑐𝑐⊇𝑅𝐴^𝑐𝑐=𝑅(𝐴). (20) By the reflexivity of 𝑅 and Theorem 8(1), int_𝜎𝑅(𝐴)⊆𝑅(𝐴)⊆𝐴⊆𝑅(𝐴)⊆𝑐𝑙_𝜎𝑅(𝐴). (21)
(3) holds by Proposition 7(3).
(4) holds by (3) and Remark 9(1).

Definition 15. Let 𝑅 be a fuzzy relation on 𝑈. 𝜎_𝑅 is called the fuzzy topology induced by 𝑅 on 𝑈.

Definition 16. Let 𝑅 be a fuzzy relation on 𝑈. 𝑅 is called pseudoconstant, if there exists 𝑎∈𝐼 such that, for any 𝑥,𝑦∈𝑈, 𝑅𝑥,𝑦=1,if𝑥=𝑦,𝑎,if𝑥≠𝑦. (22) We write 𝑅 by 𝑎^∗ or 𝑎^∗_𝑈×𝑈.

Obviously, every pseudoconstant fuzzy relation is an equivalence fuzzy relation.

Remark 17. (1) ∀𝑎,𝑏∈𝐼, 𝑎≤𝑏 implies 𝑎^∗⊆𝑏^∗.
(2) ∀𝑎∈𝐼, 𝜎_𝑎∗=𝜏_𝑎∗=𝜃_𝑎∗.
(3) 𝜎₀^∗=𝐹(𝑈), 𝜎₁^∗={𝑎∶𝑎∈𝐼}.
(4) ∀𝑅∈𝐹(𝑈×𝑈), 𝑠^∗_𝑅⊆𝑅⊆𝑡^∗_𝑅.

The following theorem gives the topological structure of fuzzy approximation spaces.

Theorem 18. Let (𝑈,𝑅) be a fuzzy approximation space. Then (1)𝜎_𝑅=𝜎₁^∗∪{𝐴∈𝐹(𝑈)∶∀(𝑥,𝑦)∈𝑅_𝐴,𝐴(𝑥)∨𝐴(𝑦)≤1−𝑅(𝑥,𝑦)}, (2)𝜎_𝑡^∗_𝑅⊆𝜎_𝑅⊆𝜎_𝑠^∗_𝑅.

Proof. (1) holds by Proposition 11(1) and Remark 17(3).
(2) holds by Remark 17(1).

4.2. Fuzzy Relations Induced by Fuzzy Topologies

Definition 19. Let 𝜎 be a fuzzy topology on 𝑈. Define 𝑅_𝜎𝑥,𝑦=𝑐𝑙_𝜎𝑦₁(𝑥)𝑥,𝑦∈𝑈. (23) Then 𝑅_𝜎 is called the fuzzy relation induced by 𝜎 on 𝑈.

Theorem 20. Let 𝜎 be a fuzzy topology on 𝑈 and let 𝑅_𝜎 be the fuzzy relation induced by 𝜎 on 𝑈. Then(1)𝑅_𝜎 is reflexive,(2)If 𝑎∈𝜎^𝑐 whenever 𝑎∈𝐼, then ∀𝐴∈𝐹(𝑈), 𝑅_𝜎(𝐴)⊆int_𝜎(𝐴)⊆𝐴⊆𝑐𝑙_𝜎(𝐴)⊆𝑅_𝜎(𝐴). (24)

Proof. (1) ∀𝑥∈𝑈, 𝑅_𝜎(𝑥,𝑥)=𝑐𝑙_𝜎𝑥₁(𝑥)≥𝑥₁(𝑥)=1. (25) Then 𝑅_𝜎 is reflexive.
(2) ∀𝐴∈𝐹(𝑈), by Remark 1 and Proposition 7(2), 𝑐𝑙_𝜎(𝐴)=𝑐𝑙_{𝜎𝑦∈𝑈}𝐴𝑦𝑦₁=_𝑦∈𝑈𝑐𝑙_𝜎𝐴𝑦𝑦₁=_𝑦∈𝑈𝑐𝑙_𝜎𝐴𝑦∩𝑦₁⊆_𝑦∈𝑈𝑐𝑙_𝜎𝐴𝑦∩𝑐𝑙_𝜎𝑦₁=_𝑦∈𝑈𝐴𝑦∩𝑐𝑙_𝜎𝑦₁. (26) Then ∀𝑥∈𝑈, 𝑐𝑙_𝜎(𝐴)(𝑥)≤_𝑦∈𝑈𝐴𝑦(𝑥)∧𝑐𝑙_𝜎𝑦₁(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅_𝜎𝑥,𝑦=𝑅_𝜎(𝐴)(𝑥). (27) Hence 𝑐𝑙_𝜎(𝐴)⊆𝑅_𝜎(𝐴).
By Proposition 7(3), int_𝜎(𝐴)=𝑐𝑙_𝜎𝐴^𝑐𝑐⊇𝑅_𝜎𝐴^𝑐𝑐=𝑅_𝜎(𝐴). (28) So 𝑅_𝜎(𝐴)⊆int_𝜎(𝐴)⊆𝐴⊆𝑐𝑙_𝜎(𝐴)⊆𝑅_𝜎(𝐴). (29)

Theorem 21. Let 𝑅 be a preorder fuzzy relation, let 𝜎_𝑅 be the fuzzy topology by 𝑅 on 𝑈, and let 𝑅_𝜎𝑅 be the fuzzy relation induced by 𝜎_𝑅 on 𝑈. Then 𝑅_𝜎𝑅=𝑅.

Proof. By Remark 6, Remark 12(4), and Theorem 13(3), 𝑅𝑥,𝑦=𝑅𝑦₁(𝑥)=𝑐𝑙_𝜃𝑅𝑦₁(𝑥)=𝑐𝑙_𝜎𝑅𝑦₁(𝑥)=𝑅_𝜎𝑅.𝑥,𝑦𝑥,𝑦∈𝑈 (30) Then 𝑅_𝜎𝑅=𝑅.

4.3. (CC) Axiom

The following condition for a fuzzy topology 𝜎 on 𝑈 is called (CC) axiom in [31],

(CC) axiom: for any 𝜆∈𝐼 and 𝐴∈𝐹(𝑈), 𝑐𝑙_𝜎(𝜆𝐴)=𝜆𝑐𝑙_𝜎(𝐴). (31)

Proposition 22. Let 𝜎 be a fuzzy topology on 𝑈. If 𝜎 satisfied the (CC) axiom, then(1)𝑅_𝜎 is the closure operator of 𝜎,(2)𝑅_𝜎 is a preorder fuzzy relation on 𝑈,(3)∀𝑎∈𝐼, 𝑎∈𝜎,(4)𝜎 is Alexandrov.

Proof. (1) ∀𝐴∈𝐹(𝑈), by Remark 1 and (CC) axiom, 𝑐𝑙_𝜎(𝐴)=𝑐𝑙_{𝜎𝑦∈𝑈}𝐴𝑦𝑦₁=_𝑦∈𝑈𝑐𝑙_𝜎𝐴𝑦𝑦₁=_𝑦∈𝑈𝐴𝑦𝑐𝑙_𝜎𝑦₁. (32) Then ∀𝑥∈𝑈, 𝑐𝑙_𝜎(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦(𝑥)∧𝑐𝑙_𝜎𝑦₁(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅_𝜎𝑥,𝑦=𝑅_𝜎(𝐴)(𝑥). (33) Thus 𝑅_𝜎(𝐴)=𝑐𝑙_𝜎(𝐴). So 𝑅_𝜎 is the closure operator of 𝜎.
(2) holds by (1) and Theorem 10.
(3) ∀𝑎∈𝐼, by (2), Proposition 7(3), and Remark 9(2), int_𝜎(𝑎)=𝑐𝑙_𝜎𝑎^𝑐𝑐=𝑅_𝜎𝑎^𝑐𝑐=𝑅_𝜎(𝑎)=𝑎. (34) Then 𝑎∈𝜎.
(4) By (1) and Proposition 7(3), 𝑅_𝜎 is the interior operator of 𝜎.
Let {𝐴_𝑗∶𝑗∈𝐽}⊆𝜎. Then ∀𝑗∈𝐽, int(𝐴_𝑗)=𝐴_𝑗. By Proposition 7(4), _𝑗∈𝐽𝐴_𝑗=_𝑗∈𝐽int_𝜎𝐴_𝑗=_𝑗∈𝐽𝑅_𝜎𝐴_𝑗=𝑅_{𝜎𝑗∈𝐽}𝐴_𝑗=int_{𝜎𝑗∈𝐽}𝐴_𝑗. (35) So ⋂_𝑗∈𝐽𝐴_𝑗∈𝜎. Hence 𝜎 is Alexandrov.

Proposition 23. Let 𝑅 be a preorder fuzzy relation on 𝑈. Then 𝜎_𝑅 satisfies (CC) axiom.

Proof. For any 𝜆∈𝐼 and 𝐴∈𝐹(𝑈), by Proposition 7(6) and Proposition 22, 𝑐𝑙_𝜎𝑅(𝜆𝐴)=𝑅(𝜆𝐴)=𝜆𝑅(𝐴)=𝜆𝑐𝑙_𝜎𝑅(𝐴). (36)

Theorem 24. Let 𝜎 be a fuzzy topology on 𝑈, let 𝑅_𝜎 be the fuzzy relation induced by 𝜎 on 𝑈, and let 𝜎_𝑅𝜎 be the fuzzy topology induced by 𝑅_𝜎 on 𝑈. If 𝜎 satisfies (CC) axiom, then 𝜎_𝑅𝜎=𝜎.

Proof. By Theorem 20(1), 𝑅_𝜎 is reflexive. ∀𝑥,𝑦,𝑧∈𝑈, put 𝑐𝑙_𝜎(𝑧₁)(𝑦)=𝜆. By (CC) axiom and Remark 1, 𝜆𝑐𝑙_𝜎𝑦₁=𝑐𝑙_𝜎𝜆𝑦₁=𝑐𝑙_𝜎𝑐𝑙_𝜎𝑧₁𝑦𝑦₁⊆𝑐𝑙_{𝜎𝑡∈𝑈}𝑐𝑙_𝜎𝑧₁(𝑡)𝑡₁=𝑐𝑙_𝜎𝑐𝑙_𝜎𝑧₁=𝑐𝑙_𝜎𝑧₁. (37) Then 𝑅_𝜎𝑥,𝑦∧𝑅_𝜎𝑦,𝑧=𝑐𝑙_𝜎𝑦₁(𝑥)∧𝑐𝑙_𝜎𝑧₁𝑦=𝑐𝑙_𝜎𝑦₁(𝑥)∧𝜆=𝜆∧𝑐𝑙_𝜎𝑦₁(𝑥)=𝜆𝑐𝑙_𝜎𝑦₁(𝑥)≤𝑐𝑙_𝜎𝑧₁(𝑥)=𝑅_𝜎(𝑥,𝑧). (38) Thus 𝑅_𝜎 is transitive and so 𝑅_𝜎 is preorder.
∀𝐴∈𝐹(𝑈), by Remark 12 and Theorem 13(3), 𝑐𝑙_𝜎𝑅𝜎(𝐴)=𝑐𝑙_𝜃𝑅𝜎(𝐴)=𝑅_𝜎(𝐴). (39) By (CC) axiom and Proposition 22, 𝑅_𝜎(𝐴)=𝑐𝑙_𝜎(𝐴). So 𝑐𝑙_𝜎𝑅𝜎(𝐴)=𝑐𝑙_𝜎(𝐴). Thus int_𝜎𝑅𝜎(𝐴)=𝑐𝑙_𝜎𝑅𝜎𝐴^𝑐𝑐=𝑐𝑙_𝜎𝐴^𝑐𝑐=int_𝜎(𝐴). (40) Hence 𝜎_𝑅𝜎=𝜎.

Theorem 25. Let 𝜏 be a fuzzy topology on 𝑈. Then the following are equivalent.(1)𝜏 satisfies (CC) axiom.(2)There exists a preorder fuzzy relation 𝜌 on 𝑈 such that 𝜌 is the closure operator of 𝜏.(3)There exists a preorder fuzzy relation 𝜌 on 𝑈 such that 𝜌 is the interior operator of 𝜏.(4)𝑅_𝜏 is the closure operator of 𝜏.(5)𝑅_𝜏 is the interior operator of 𝜏.

Proof. (1) ⇒ (2). Suppose that 𝜏 satisfies (CC) axiom. Pick 𝜌=𝑅_𝜏. By Proposition 22, 𝜌 is the closure operator of 𝜏. By Theorem 10, 𝜌 is preorder.
(2) ⇒ (3). Let 𝜌 be the closure operator of 𝜏 for some preorder fuzzy relation 𝜌 on 𝑈. ∀𝐴∈𝐹(𝑈), by Proposition 7(3), 𝜌(𝐴)=𝜌𝐴^𝑐𝑐=𝑐𝑙_𝜏𝐴^𝑐𝑐=int_𝜏(𝐴). (41)
Thus 𝜌 is the interior operator of 𝜏.
(3) ⇒ (5). Let 𝜌 be the interior operator of 𝜏 for some preorder fuzzy relation 𝜌 on 𝑈.
By Remark 6, 𝜌𝑥,𝑦=1−𝜌𝑦₁^𝑐(𝑥)=1−int_𝜏𝑦₁^𝑐(𝑥)=𝑐𝑙_𝜏𝑦₁(𝑥)=𝑅_𝜏𝑥,𝑦𝑥,𝑦∈𝑈. (42)
Then 𝜌=𝑅_𝜏. Thus 𝑅_𝜏 is the interior operator of 𝜏.
(5) ⇒ (4) holds by Proposition 7(3).
(4) ⇒ (1). For any 𝜆∈𝐼 and 𝐴∈𝐹(𝑈), by Proposition 7(6), 𝑐𝑙_𝜏(𝜆𝐴)=𝑅_𝜏(𝜆𝐴)=𝜆𝑅_𝜏(𝐴)=𝜆𝑐𝑙_𝜏(𝐴). (43) Thus 𝜏 satisfies (CC) axiom.

Theorem 26. Let ,().Σ=𝑅∶𝑅𝑖𝑠𝑎𝑝𝑟𝑒𝑜𝑟𝑑𝑒𝑟𝑓𝑢𝑧𝑧𝑦𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑜𝑛𝑈Γ=𝜎∶𝜎𝑖𝑠𝑎𝑓𝑢𝑧𝑧𝑦𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦𝑜𝑛𝑈𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔𝐶𝐶𝑎𝑥𝑖𝑜𝑚 (44) Then there exists a one-to-one correspondence between Σ and Γ.

Proof. 𝑓∶Σ→Γ and 𝑔∶Γ→Σ are defined as follows: 𝑓(𝑅)=𝜎_𝑅(),𝑔(𝜎)𝑅∈Σ=𝑅_𝜎(𝜎∈Γ). (45)
By Theorem 21, 𝑔∘𝑓=𝑖_Σ, (46) where 𝑔∘𝑓 is the composition of 𝑓 and 𝑔 and 𝑖_Σ is the identity mapping on Γ.
By Theorem 24, 𝑓∘𝑔=𝑖_Γ, (47) where 𝑓∘𝑔 is the composition of 𝑔 and 𝑓 and 𝑖_Γ is the identity mapping on Σ.
Hence 𝑓 and 𝑔 are two one-to-one correspondences. This proves that there exists a one-to-one correspondence between Σ and Γ.

5. Fuzzy Approximating Spaces

As can be seen from Section 4, a fuzzy relation yields a fuzzy topology. In this section, we consider the reverse problem; that is, when can the given fuzzy topology coincide with the fuzzy topology induced by some fuzzy relation?

Definition 27. Let (𝑈,𝜎) be a fuzzy topological space. If there exists a fuzzy relation on 𝑈 such that 𝜎_𝑅=𝜎, then (𝑈,𝜎) is called a fuzzy approximating space.

Theorem 28. Let (𝑈,𝜎) be a fuzzy topological space. If one of the following conditions is satisfied, then (𝑈,𝜏) is a fuzzy approximating space.(1)𝜏 satisfies (CC) axiom.(2)There exists a preorder fuzzy relation 𝜌 on 𝑈 such that 𝜌 is the closure operator of 𝜏.(3)There exists a preorder fuzzy relation 𝜌 on 𝑈 such that 𝜌 is the interior operator of 𝜏.(4)𝑅_𝜏 is the closure operator of 𝜏.(5)𝑅_𝜏 is the interior operator of 𝜏.

Proof. These hold by Theorems 24 and 25.

Theorem 29. Let (𝑈,𝜎) be a fuzzy topological space. Then (𝑈,𝜎) is a fuzzy approximating space if and only if there exists a fuzzy relation 𝑅 such that 𝜎=𝜎₁^∗∪𝐴∈𝐹(𝑈)∶∀𝑥,𝑦∈𝑅_𝐴,𝐴(𝑥)∨𝐴𝑦≤1−𝑅𝑥,𝑦. (48)

Proof. This holds by Theorem 18(1).

Example 30. {𝑎∶𝑎∈𝐼} is a fuzzy approximating space.

6. Conclusions

In this paper, relationships between fuzzy relations and fuzzy topology were discussed. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies which satisfy (CC) axiom was proved. We introduced the concept of fuzzy approximating spaces and gave decision conditions that a fuzzy topological space is a fuzzy approximating space.

The results of this paper illustrate that fuzzy relations can be researched by means of topology. We hope that one can find applications of topological properties of fuzzy rough sets in information sciences. In future work, we will do similar exploration of fuzzy neighborhood spaces like this paper.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 11261005 and 11161029), the Natural Science Foundation of Guangxi (no. 2012GXNSFDA276040), Guangxi University Science and Technology Research Project (no. 2013ZD061), and the Natural Science Foundation for Young Scholar of Guangxi Province (2013GXNSFBA019020).