# Fuzzy Approximating Spaces

Authors: Bin Qin
Journal: Journal of Applied Mathematics
 (2014)

---

## 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.


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## Body

## 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 Section2, 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 allA
∈
F
(
U
), we denote
(1)
R
A
=
{
(
x
,
y
)
∈
U
×
U
:
A
(
x
)
>
A
(
y
)
}
.Obviously,R
A
=
∅
⇔
A
=
a
¯ for some λ
∈
I.A fuzzy set is called a fuzzy point inU, 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,(2)
x
1
(
y
)
=
{
1
,
y
=
x
,
0
,
y
≠
x
.Remark 1.
Consider(3)
A
=
⋃
x
∈
U
‍
(
A
(
x
)
x
1
)
(
A
∈
F
(
U
)
)
.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
∈
J
A
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 denoteτ
c
=
{
A
∈
F
(
U
)
:
A
c
∈
τ
}.
It should be pointed out that if (i) in Definition2 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 topologyτ is called Alexandrov [27] if (ii) in Definition 2 is replaced by(
ii
)
′
{
A
j
:
j
∈
J
}
⊆
τ
⟹
⋂
j
∈
J
A
j
∈
τ.Definition 3 (see [28]).
LetR be a relation on U. ∀
x
∈
U, denote
(4)
R
p
(
x
)
=
{
y
∈
U
:
(
y
,
x
)
∈
R
}
,
R
s
(
x
)
=
{
y
∈
U
:
(
x
,
y
)
∈
R
}
.
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 thatR is called a fuzzy relation on U if R
∈
F
(
U
×
U
).Definition 4 (see [14, 29]).
LetR be a fuzzy relation on U. Then R is called(1)
reflexive, ifR
(
x
,
x
)
=
1 for any x
∈
U,
(2)
symmetric, ifR
(
x
,
y
)
=
R
(
y
,
x
) for any x
,
y
∈
U,
(3)
transitive, ifR
(
x
,
z
)
≥
R
(
x
,
y
)
∧
R
(
y
,
z
) for any x
,
y
,
z
∈
U.
LetR 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]).
LetR 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:
(5)
R
_
(
A
)
(
x
)
=
⋀
y
∈
U
(
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
)
(
x
∈
U
)
,
R
¯
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
(
x
,
y
)
)
(
x
∈
U
)
.
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
¯
(
x
1
)
(
y
)
=
R
(
y
,
x
) and R
_
(
(
x
1
)
c
)
(
y
)
=
1
-
R
(
y
,
x
)
(
x
,
y
∈
U
).Proposition 7 (see [30]).
LetR 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
∈
J
A
j
)
=
⋂
j
∈
J
(
R
_
(
A
j
)
), R
¯
(
⋃
j
∈
J
A
j
)
=
⋃
j
∈
J
(
R
¯
(
A
j
)
),
(5)
R
¯
(
λ
A
)
=
λ
R
¯
(
A
).Theorem 8 (see [14, 29, 30]).
LetR be a fuzzy relation on U. Then consider the following.(
1
)  R is reflexive ⇔
(
I
L
R
)
∀
A
∈
F
(
U
), R
_
(
A
)
⊆
A.
⇔
(
I
U
R
)
∀
A
∈
F
(
U
), A
⊆
R
¯
(
A
).
(
2
)  R is symmetric ⇔
(
I
L
S
)
∀
(
x
,
y
)
∈
U
×
U, R
_
(
(
x
1
)
c
)
(
y
)
=
R
_
(
(
y
1
)
c
)
(
x
).
⇔
(
I
U
S
)
∀
(
x
,
y
)
∈
U
×
U, R
¯
(
x
1
)
(
y
)
=
R
¯
(
y
1
)
(
x
).
(
3
)  R is transitive ⇔
(
I
L
T
)
∀
A
∈
F
(
U
), R
_
(
A
)
⊆
R
_
(
R
_
(
A
)
).
⇔
(
I
U
T
)
∀
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.
LetR 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 c
l
τ
(
z
1
)
(
y
)
=
λ. Note that R
_ is the interior operator of τ. Then
(6)
R
(
x
,
x
)
=
R
¯
(
x
1
)
(
x
)
=
c
l
τ
(
x
1
)
(
x
)
≥
x
1
(
x
)
=
1
.
Thus R is reflexive. By Remark 1, Remark 6, and Proposition 7(5),
(7)
R
(
x
,
y
)
∧
R
(
y
,
z
)
=
R
¯
(
y
1
)
(
x
)
∧
R
¯
(
z
1
)
(
y
)
=
R
¯
(
y
1
)
(
x
)
∧
c
l
τ
(
z
1
)
(
y
)
=
R
¯
(
y
1
)
(
x
)
∧
λ
=
λ
R
¯
(
y
1
)
(
x
)
=
R
¯
(
λ
y
1
)
(
x
)
=
c
l
τ
(
λ
y
1
)
(
x
)
=
c
l
τ
(
c
l
τ
(
z
1
)
(
y
)
y
1
)
(
x
)
≤
c
l
τ
(
⋃
t
∈
U
‍
(
c
l
τ
(
z
1
)
(
t
)
t
1
)
)
(
x
)
=
c
l
τ
(
c
l
τ
(
z
1
)
)
(
x
)
=
c
l
τ
(
z
1
)
(
x
)
=
R
(
x
,
z
)
.
Then R is transitive. Hence R is preorder.
(
2
) The proof is similar to (
1
).Proposition 11.
LetR be a fuzzy relation on U. Then, for each A
∈
F
(
U
) with R
A
≠
∅, consider the following.(1)
(a)
R
_
(
A
)
⊇
A
⇔
(
F
L
O
)
∀
(
x
,
y
)
∈
R
A
,
1
-
R
(
x
,
y
)
≥
A
(
x
)
∨
A
(
y
)
.
(b)
R
¯
(
A
)
⊆
A
⇔
(
F
U
O
)
∀
(
x
,
y
)
∈
R
A
,
R
(
y
,
x
)
≤
A
(
x
)
∧
A
(
y
)
.
(2)
If
R
is reflexive, then
(a)
R
_
(
A
)
=
A
⇔
(
F
L
R
)
∀
(
x
,
y
)
∈
R
A
,
1
-
R
(
x
,
y
)
≥
A
(
x
)
∨
A
(
y
)
.
(b)
R
¯
(
A
)
=
A
⇔
(
F
U
R
)
∀
(
x
,
y
)
∈
R
A
,
R
(
y
,
x
)
≤
A
(
x
)
∧
A
(
y
)
.Proof.
(
1
) (a) Necessity. Suppose that R
_
(
A
)
⊇
A. Note that ∀
x
∈
U,
(8)
⋀
y
∈
U
(
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
)
=
(
R
_
(
A
)
)
(
y
)
≥
A
(
x
)
.
Then ∀
x
,
y
∈
U, A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
≥
A
(
x
). Since ∀
(
x
,
y
)
∈
R
A, A
(
x
)
>
A
(
y
), we have
(9)
1
-
R
(
x
,
y
)
≥
A
(
x
)
=
A
(
x
)
∨
A
(
y
)
(
(
x
,
y
)
∈
R
A
)
.
Sufficiency. Suppose that(
F
L
O
) holds. Let x
∈
U. Consider the following.(i)
Ify
∈
(
R
A
)
s
(
x
), then
(10)
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
≥
A
(
y
)
∨
(
A
(
x
)
∨
A
(
y
)
)
≥
A
(
x
)
.
(ii)
Ify
∉
(
R
A
)
s
(
x
), then A
(
y
)
≥
A
(
x
) and so
(11)
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
≥
A
(
y
)
≥
A
(
x
)
.
HenceR
_
(
A
)
(
x
)
=
⋀
y
∈
U
(
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
)
≥
A
(
x
).
ThusR
_
(
A
)
⊇
A.
(b) Necessity. Suppose thatR
¯
(
A
)
⊆
A. Note that ∀
y
∈
U,
(12)
⋁
x
∈
U
(
A
(
x
)
∧
R
(
y
,
x
)
)
=
R
¯
(
A
)
(
y
)
≤
A
(
y
)
.
Then ∀
x
,
y
∈
U, A
(
x
)
∧
R
(
y
,
x
)
≤
A
(
y
). Since ∀
(
x
,
y
)
∈
R
A, A
(
x
)
>
A
(
y
), we have
(13)
R
(
y
,
x
)
≤
A
(
y
)
=
A
(
x
)
∧
A
(
y
)
(
(
x
,
y
)
∈
R
A
)
.
Sufficiency. Suppose that(
F
L
O
) holds. Let y
∈
U. Consider the following.(i)
Ifx
∈
(
R
A
)
p
(
y
), then (
x
,
y
)
∈
R
A and so
(14)
A
(
x
)
∨
R
(
y
,
x
)
≤
A
(
x
)
∧
(
A
(
x
)
∧
A
(
y
)
)
≤
A
(
y
)
.
(ii)
Ifx
∉
(
R
A
)
p
(
y
), then A
(
x
)
≤
A
(
y
) and so
(15)
A
(
x
)
∧
R
(
y
,
x
)
≤
A
(
x
)
≤
A
(
y
)
.
Hence(
R
¯
(
A
)
)
(
y
)
=
⋁
x
∈
U
(
A
(
x
)
∧
R
(
y
,
x
)
)
≤
A
(
y
).
ThusR
¯
(
A
)
⊆
A.
(2) holds by (1), the reflexivity ofR, and Theorem 8(1).


## 4. Relationships between Fuzzy Relations and Fuzzy Topologies

### 4.1.  Fuzzy Topology Induced by Fuzzy Relations

LetR be a fuzzy relation on U. Denote
(16)
σ
R
=
{
A
∈
F
(
U
)
:
A
⊆
R
_
(
A
)
}
,
τ
R
=
{
A
∈
F
(
U
)
:
A
=
R
_
(
A
)
}
,
θ
R
=
{
R
_
(
A
)
:
A
∈
F
(
U
)
}
;
s
R
=
⋀
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
,
t
R
=
⋁
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
.Remark 12.
LetR be a fuzzy relation on U. Then consider the following.(1)
τ
R
⊆
σ
R, τ
R
⊆
θ
R.
(2)
IfR is transitive, then τ
R
⊆
θ
R
⊆
σ
R.
(3)
IfR is reflexive, then τ
R
=
σ
R.
(4)
IfR is preorder, then σ
R
=
τ
R
=
θ
R.Theorem 13 (see [30]).
LetR 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.
LetR be a fuzzy relation on U. Then (1)
σ
R is an Alexandrov fuzzy topology on U,
(2)
ifR is reflexive, then ∀
A
∈
F
(
U
),
(17)
int
σ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
σ
R
(
A
)
.
(3)
A
∈
(
σ
R
)
c
⇔
R
¯
(
A
)
⊆
A.
(4)
∀
a
∈
I, a
¯
∈
(
σ
R
)
c.Proof.
(1) By Remark9(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),
(18)
⋂
j
∈
J
‍
A
j
⊆
⋂
j
∈
J
‍
R
_
(
A
j
)
=
R
_
(
⋂
j
∈
J
‍
A
j
)
,
⋃
j
∈
J
‍
A
j
⊆
⋃
j
∈
J
‍
R
_
(
A
j
)
⊆
R
_
(
⋃
j
∈
J
‍
A
j
)
.
Hence⋂
j
∈
J
A
j
,
⋃
j
∈
J
A
j
∈
σ
R. So σ
R is Alexandrov.
(2)∀
A
∈
F
(
U
), by Proposition 7(2),
(19)
in
t
σ
R
(
A
)
=
⋃
‍
{
B
∈
σ
R
:
B
⊆
A
}
⊆
⋃
‍
{
B
∈
σ
R
:
R
_
(
B
)
⊆
R
_
(
A
)
}
=
⋃
‍
{
B
∈
F
(
U
)
:
B
⊆
R
_
(
B
)
⊆
R
_
(
A
)
}
⊆
R
_
(
A
)
.
By Proposition 7(3),
(20)
c
l
σ
R
(
A
)
=
(
in
t
σ
R
(
A
c
)
)
c
⊇
(
R
_
(
A
c
)
)
c
=
R
¯
(
A
)
.
By the reflexivity of R and Theorem 8(1),
(21)
in
t
σ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
σ
R
(
A
)
.
(3) holds by Proposition7(3).
(4) holds by (3) and Remark9(1).Definition 15.
LetR be a fuzzy relation on U. σ
R is called the fuzzy topology induced by R on U.Definition 16.
LetR be a fuzzy relation on U. R is called pseudoconstant, if there exists a
∈
I such that, for any x
,
y
∈
U,
(22)
R
(
x
,
y
)
=
{
1
,
if
x
=
y
,
a
,
if
x
≠
y
.
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)σ
0
*
=
F
(
U
), σ
1
*
=
{
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
=
σ
1
*
∪
{
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 Proposition11(1) and Remark 17(3).
(2) holds by Remark17(1).


### 4.2.  Fuzzy Relations Induced by Fuzzy Topologies

Definition 19.
Letσ be a fuzzy topology on U. Define
(23)
R
σ
(
x
,
y
)
=
c
l
σ
(
y
1
)
(
x
)
(
x
,
y
∈
U
)
.
Then R
σ is called the fuzzy relation induced by σ on U.Theorem 20.
Letσ 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)
Ifa
¯
∈
σ
c whenever a
∈
I, then ∀
A
∈
F
(
U
),
(24)
R
σ
_
(
A
)
⊆
int
σ
(
A
)
⊆
A
⊆
c
l
σ
(
A
)
⊆
R
σ
¯
(
A
)
.Proof.
(1)∀
x
∈
U,
(25)
R
σ
(
x
,
x
)
=
c
l
σ
(
x
1
)
(
x
)
≥
(
x
1
)
(
x
)
=
1
¯
.
Then R
σ is reflexive.
(2)∀
A
∈
F
(
U
), by Remark 1 and Proposition 7(2),
(26)
c
l
σ
(
A
)
=
c
l
σ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
∩
y
1
)
⊆
⋃
y
∈
U
‍
(
c
l
σ
(
A
(
y
)
)
∩
c
l
σ
(
y
1
)
)
=
⋃
y
∈
U
‍
(
A
(
y
)
∩
c
l
σ
(
y
1
)
)
.
Then ∀
x
∈
U,
(27)
c
l
σ
(
A
)
(
x
)
≤
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
c
l
σ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
σ
(
x
,
y
)
)
=
R
σ
¯
(
A
)
(
x
)
.
Hence c
l
σ
(
A
)
⊆
R
σ
¯
(
A
).
By Proposition7(3),
(28)
in
t
σ
(
A
)
=
(
c
l
σ
(
A
c
)
)
c
⊇
(
R
σ
¯
(
A
c
)
)
c
=
R
σ
_
(
A
)
.
So
(29)
R
σ
_
(
A
)
⊆
in
t
σ
(
A
)
⊆
A
⊆
c
l
σ
(
A
)
⊆
R
σ
¯
(
A
)
.Theorem 21.
LetR be a preorder fuzzy relation, let σ
R be the fuzzy topology by R on U, and let R
σ
R be the fuzzy relation induced by σ
R on U. Then R
σ
R
=
R.Proof.
By Remark6, Remark 12(4), and Theorem 13(3),
(30)
R
(
x
,
y
)
=
R
¯
(
y
1
)
(
x
)
=
c
l
θ
R
(
y
1
)
(
x
)
=
c
l
σ
R
(
y
1
)
(
x
)
=
R
σ
R
(
x
,
y
)
n
n
n
n
n
n
n
n
n
n
n
n
n
(
x
,
y
∈
U
)
.
Then R
σ
R
=
R.


### 4.3. (CC) Axiom

The following condition for a fuzzy topologyσ on U is called (CC) axiom in [31],(CC) axiom: for anyλ
∈
I and A
∈
F
(
U
),
(31)
c
l
σ
(
λ
A
)
=
λ
c
l
σ
(
A
)
.Proposition 22.
Letσ 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,
(32)
c
l
σ
(
A
)
=
c
l
σ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
(
A
(
y
)
c
l
σ
(
y
1
)
)
.
Then ∀
x
∈
U,
(33)
c
l
σ
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
c
l
σ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
σ
(
x
,
y
)
)
=
R
σ
¯
(
A
)
(
x
)
.
Thus R
σ
¯
(
A
)
=
c
l
σ
(
A
). So R
σ
¯ is the closure operator of σ.
(2) holds by (1) and Theorem10.
(3)∀
a
∈
I, by (2), Proposition 7(3), and Remark 9(2),
(34)
in
t
σ
(
a
¯
)
=
(
c
l
σ
(
a
¯
c
)
)
c
=
(
R
σ
¯
(
a
¯
c
)
)
c
=
R
σ
_
(
a
¯
)
=
a
¯
.
Then a
¯
∈
σ.
(4) By (1) and Proposition7(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),
(35)
⋂
j
∈
J
‍
A
j
=
⋂
j
∈
J
‍
in
t
σ
(
A
j
)
=
⋂
j
∈
J
‍
R
σ
_
(
A
j
)
=
R
σ
_
(
⋂
j
∈
J
‍
A
j
)
=
in
t
σ
(
⋂
j
∈
J
‍
A
j
)
.
So ⋂
j
∈
J
A
j
∈
σ. Hence σ is Alexandrov.Proposition 23.
LetR be a preorder fuzzy relation on U. Then σ
R satisfies (CC) axiom.Proof.
For anyλ
∈
I and A
∈
F
(
U
), by Proposition 7(6) and Proposition 22,
(36)
c
l
σ
R
(
λ
A
)
=
R
¯
(
λ
A
)
=
λ
R
¯
(
A
)
=
λ
c
l
σ
R
(
A
)
.Theorem 24.
Letσ be a fuzzy topology on U, let R
σ be the fuzzy relation induced by σ on U, and let σ
R
σ be the fuzzy topology induced by R
σ on U. If σ satisfies (CC) axiom, then σ
R
σ
=
σ.Proof.
By Theorem20(1), R
σ is reflexive. ∀
x
,
y
,
z
∈
U, put c
l
σ
(
z
1
)
(
y
)
=
λ. By (CC) axiom and Remark 1,
(37)
λ
c
l
σ
(
y
1
)
=
c
l
σ
(
λ
y
1
)
=
c
l
σ
(
c
l
σ
(
z
1
)
(
y
)
y
1
)
⊆
c
l
σ
(
⋃
t
∈
U
‍
(
c
l
σ
(
z
1
)
(
t
)
t
1
)
)
=
c
l
σ
(
c
l
σ
(
z
1
)
)
=
c
l
σ
(
z
1
)
.
Then
(38)
R
σ
(
x
,
y
)
∧
R
σ
(
y
,
z
)
=
c
l
σ
(
y
1
)
(
x
)
∧
c
l
σ
(
z
1
)
(
y
)
=
c
l
σ
(
y
1
)
(
x
)
∧
λ
=
λ
∧
c
l
σ
(
y
1
)
(
x
)
=
(
λ
c
l
σ
(
y
1
)
)
(
x
)
≤
c
l
σ
(
z
1
)
(
x
)
=
R
σ
(
x
,
z
)
.
Thus R
σ is transitive and so R
σ is preorder.
∀
A
∈
F
(
U
), by Remark 12 and Theorem 13(3),
(39)
c
l
σ
R
σ
(
A
)
=
c
l
θ
R
σ
(
A
)
=
R
σ
¯
(
A
)
.
By (CC) axiom and Proposition 22, R
σ
¯
(
A
)
=
c
l
σ
(
A
). So c
l
σ
R
σ
(
A
)
=
c
l
σ
(
A
). Thus
(40)
in
t
σ
R
σ
(
A
)
=
(
c
l
σ
R
σ
(
A
c
)
)
c
=
(
c
l
σ
(
A
c
)
)
c
=
in
t
σ
(
A
)
.
Hence σ
R
σ
=
σ.Theorem 25.
Letτ be a fuzzy topology on U. Then the following are equivalent.(1)
τ satisfies (CC) axiom.
(2)
There exists a preorder fuzzy relationρ on U such that ρ
¯ is the closure operator of τ.
(3)
There exists a preorder fuzzy relationρ 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),
(41)
ρ
_
(
A
)
=
(
ρ
¯
(
A
c
)
)
c
=
(
c
l
τ
(
A
c
)
)
c
=
in
t
τ
(
A
)
.
Thusρ
_ 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 Remark6,
(42)
ρ
(
x
,
y
)
=
1
-
ρ
_
(
(
y
1
)
c
)
(
x
)
=
1
-
in
t
τ
(
(
y
1
)
c
)
(
x
)
=
c
l
τ
(
y
1
)
(
x
)
=
R
τ
(
x
,
y
)
(
x
,
y
∈
U
)
.
Thenρ
=
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),
(43)
c
l
τ
(
λ
A
)
=
R
τ
¯
(
λ
A
)
=
λ
R
τ
¯
(
A
)
=
λ
c
l
τ
(
A
)
.
Thus τ satisfies (CC) axiom.Theorem 26.
Let(44)
Σ
=
{
R
:
R
i
s
a
p
r
e
o
r
d
e
r
f
u
z
z
y
r
e
l
a
t
i
o
n
o
n
U
}
,
Γ
=
{
σ
:
σ
i
s
a
f
u
z
z
y
t
o
p
o
l
o
g
y
o
n
U
s
a
t
i
s
f
y
i
n
g
(
C
C
)
a
x
i
o
m
}
.
Then there exists a one-to-one correspondence between Σ and Γ.Proof.
f
:
Σ
→
Γ and g
:
Γ
→
Σ are defined as follows:
(45)
f
(
R
)
=
σ
R
(
R
∈
Σ
)
,
g
(
σ
)
=
R
σ
(
σ
∈
Γ
)
.
By Theorem21,
(46)
g
∘
f
=
i
Σ
,
where g
∘
f is the composition of f and g and i
Σ is the identity mapping on Γ.
By Theorem24,
(47)
f
∘
g
=
i
Γ
,
where f
∘
g is the composition of g and f and i
Γ is the identity mapping on Σ.
Hencef and g are two one-to-one correspondences. This proves that there exists a one-to-one correspondence between Σ and Γ.


## 4.1.  Fuzzy Topology Induced by Fuzzy Relations

LetR be a fuzzy relation on U. Denote
(16)
σ
R
=
{
A
∈
F
(
U
)
:
A
⊆
R
_
(
A
)
}
,
τ
R
=
{
A
∈
F
(
U
)
:
A
=
R
_
(
A
)
}
,
θ
R
=
{
R
_
(
A
)
:
A
∈
F
(
U
)
}
;
s
R
=
⋀
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
,
t
R
=
⋁
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
.Remark 12.
LetR be a fuzzy relation on U. Then consider the following.(1)
τ
R
⊆
σ
R, τ
R
⊆
θ
R.
(2)
IfR is transitive, then τ
R
⊆
θ
R
⊆
σ
R.
(3)
IfR is reflexive, then τ
R
=
σ
R.
(4)
IfR is preorder, then σ
R
=
τ
R
=
θ
R.Theorem 13 (see [30]).
LetR 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.
LetR be a fuzzy relation on U. Then (1)
σ
R is an Alexandrov fuzzy topology on U,
(2)
ifR is reflexive, then ∀
A
∈
F
(
U
),
(17)
int
σ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
σ
R
(
A
)
.
(3)
A
∈
(
σ
R
)
c
⇔
R
¯
(
A
)
⊆
A.
(4)
∀
a
∈
I, a
¯
∈
(
σ
R
)
c.Proof.
(1) By Remark9(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),
(18)
⋂
j
∈
J
‍
A
j
⊆
⋂
j
∈
J
‍
R
_
(
A
j
)
=
R
_
(
⋂
j
∈
J
‍
A
j
)
,
⋃
j
∈
J
‍
A
j
⊆
⋃
j
∈
J
‍
R
_
(
A
j
)
⊆
R
_
(
⋃
j
∈
J
‍
A
j
)
.
Hence⋂
j
∈
J
A
j
,
⋃
j
∈
J
A
j
∈
σ
R. So σ
R is Alexandrov.
(2)∀
A
∈
F
(
U
), by Proposition 7(2),
(19)
in
t
σ
R
(
A
)
=
⋃
‍
{
B
∈
σ
R
:
B
⊆
A
}
⊆
⋃
‍
{
B
∈
σ
R
:
R
_
(
B
)
⊆
R
_
(
A
)
}
=
⋃
‍
{
B
∈
F
(
U
)
:
B
⊆
R
_
(
B
)
⊆
R
_
(
A
)
}
⊆
R
_
(
A
)
.
By Proposition 7(3),
(20)
c
l
σ
R
(
A
)
=
(
in
t
σ
R
(
A
c
)
)
c
⊇
(
R
_
(
A
c
)
)
c
=
R
¯
(
A
)
.
By the reflexivity of R and Theorem 8(1),
(21)
in
t
σ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
σ
R
(
A
)
.
(3) holds by Proposition7(3).
(4) holds by (3) and Remark9(1).Definition 15.
LetR be a fuzzy relation on U. σ
R is called the fuzzy topology induced by R on U.Definition 16.
LetR be a fuzzy relation on U. R is called pseudoconstant, if there exists a
∈
I such that, for any x
,
y
∈
U,
(22)
R
(
x
,
y
)
=
{
1
,
if
x
=
y
,
a
,
if
x
≠
y
.
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)σ
0
*
=
F
(
U
), σ
1
*
=
{
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
=
σ
1
*
∪
{
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 Proposition11(1) and Remark 17(3).
(2) holds by Remark17(1).


## 4.2.  Fuzzy Relations Induced by Fuzzy Topologies

Definition 19.
Letσ be a fuzzy topology on U. Define
(23)
R
σ
(
x
,
y
)
=
c
l
σ
(
y
1
)
(
x
)
(
x
,
y
∈
U
)
.
Then R
σ is called the fuzzy relation induced by σ on U.Theorem 20.
Letσ 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)
Ifa
¯
∈
σ
c whenever a
∈
I, then ∀
A
∈
F
(
U
),
(24)
R
σ
_
(
A
)
⊆
int
σ
(
A
)
⊆
A
⊆
c
l
σ
(
A
)
⊆
R
σ
¯
(
A
)
.Proof.
(1)∀
x
∈
U,
(25)
R
σ
(
x
,
x
)
=
c
l
σ
(
x
1
)
(
x
)
≥
(
x
1
)
(
x
)
=
1
¯
.
Then R
σ is reflexive.
(2)∀
A
∈
F
(
U
), by Remark 1 and Proposition 7(2),
(26)
c
l
σ
(
A
)
=
c
l
σ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
∩
y
1
)
⊆
⋃
y
∈
U
‍
(
c
l
σ
(
A
(
y
)
)
∩
c
l
σ
(
y
1
)
)
=
⋃
y
∈
U
‍
(
A
(
y
)
∩
c
l
σ
(
y
1
)
)
.
Then ∀
x
∈
U,
(27)
c
l
σ
(
A
)
(
x
)
≤
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
c
l
σ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
σ
(
x
,
y
)
)
=
R
σ
¯
(
A
)
(
x
)
.
Hence c
l
σ
(
A
)
⊆
R
σ
¯
(
A
).
By Proposition7(3),
(28)
in
t
σ
(
A
)
=
(
c
l
σ
(
A
c
)
)
c
⊇
(
R
σ
¯
(
A
c
)
)
c
=
R
σ
_
(
A
)
.
So
(29)
R
σ
_
(
A
)
⊆
in
t
σ
(
A
)
⊆
A
⊆
c
l
σ
(
A
)
⊆
R
σ
¯
(
A
)
.Theorem 21.
LetR be a preorder fuzzy relation, let σ
R be the fuzzy topology by R on U, and let R
σ
R be the fuzzy relation induced by σ
R on U. Then R
σ
R
=
R.Proof.
By Remark6, Remark 12(4), and Theorem 13(3),
(30)
R
(
x
,
y
)
=
R
¯
(
y
1
)
(
x
)
=
c
l
θ
R
(
y
1
)
(
x
)
=
c
l
σ
R
(
y
1
)
(
x
)
=
R
σ
R
(
x
,
y
)
n
n
n
n
n
n
n
n
n
n
n
n
n
(
x
,
y
∈
U
)
.
Then R
σ
R
=
R.


## 4.3. (CC) Axiom

The following condition for a fuzzy topologyσ on U is called (CC) axiom in [31],(CC) axiom: for anyλ
∈
I and A
∈
F
(
U
),
(31)
c
l
σ
(
λ
A
)
=
λ
c
l
σ
(
A
)
.Proposition 22.
Letσ 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,
(32)
c
l
σ
(
A
)
=
c
l
σ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
c
l
σ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
(
A
(
y
)
c
l
σ
(
y
1
)
)
.
Then ∀
x
∈
U,
(33)
c
l
σ
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
c
l
σ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
σ
(
x
,
y
)
)
=
R
σ
¯
(
A
)
(
x
)
.
Thus R
σ
¯
(
A
)
=
c
l
σ
(
A
). So R
σ
¯ is the closure operator of σ.
(2) holds by (1) and Theorem10.
(3)∀
a
∈
I, by (2), Proposition 7(3), and Remark 9(2),
(34)
in
t
σ
(
a
¯
)
=
(
c
l
σ
(
a
¯
c
)
)
c
=
(
R
σ
¯
(
a
¯
c
)
)
c
=
R
σ
_
(
a
¯
)
=
a
¯
.
Then a
¯
∈
σ.
(4) By (1) and Proposition7(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),
(35)
⋂
j
∈
J
‍
A
j
=
⋂
j
∈
J
‍
in
t
σ
(
A
j
)
=
⋂
j
∈
J
‍
R
σ
_
(
A
j
)
=
R
σ
_
(
⋂
j
∈
J
‍
A
j
)
=
in
t
σ
(
⋂
j
∈
J
‍
A
j
)
.
So ⋂
j
∈
J
A
j
∈
σ. Hence σ is Alexandrov.Proposition 23.
LetR be a preorder fuzzy relation on U. Then σ
R satisfies (CC) axiom.Proof.
For anyλ
∈
I and A
∈
F
(
U
), by Proposition 7(6) and Proposition 22,
(36)
c
l
σ
R
(
λ
A
)
=
R
¯
(
λ
A
)
=
λ
R
¯
(
A
)
=
λ
c
l
σ
R
(
A
)
.Theorem 24.
Letσ be a fuzzy topology on U, let R
σ be the fuzzy relation induced by σ on U, and let σ
R
σ be the fuzzy topology induced by R
σ on U. If σ satisfies (CC) axiom, then σ
R
σ
=
σ.Proof.
By Theorem20(1), R
σ is reflexive. ∀
x
,
y
,
z
∈
U, put c
l
σ
(
z
1
)
(
y
)
=
λ. By (CC) axiom and Remark 1,
(37)
λ
c
l
σ
(
y
1
)
=
c
l
σ
(
λ
y
1
)
=
c
l
σ
(
c
l
σ
(
z
1
)
(
y
)
y
1
)
⊆
c
l
σ
(
⋃
t
∈
U
‍
(
c
l
σ
(
z
1
)
(
t
)
t
1
)
)
=
c
l
σ
(
c
l
σ
(
z
1
)
)
=
c
l
σ
(
z
1
)
.
Then
(38)
R
σ
(
x
,
y
)
∧
R
σ
(
y
,
z
)
=
c
l
σ
(
y
1
)
(
x
)
∧
c
l
σ
(
z
1
)
(
y
)
=
c
l
σ
(
y
1
)
(
x
)
∧
λ
=
λ
∧
c
l
σ
(
y
1
)
(
x
)
=
(
λ
c
l
σ
(
y
1
)
)
(
x
)
≤
c
l
σ
(
z
1
)
(
x
)
=
R
σ
(
x
,
z
)
.
Thus R
σ is transitive and so R
σ is preorder.
∀
A
∈
F
(
U
), by Remark 12 and Theorem 13(3),
(39)
c
l
σ
R
σ
(
A
)
=
c
l
θ
R
σ
(
A
)
=
R
σ
¯
(
A
)
.
By (CC) axiom and Proposition 22, R
σ
¯
(
A
)
=
c
l
σ
(
A
). So c
l
σ
R
σ
(
A
)
=
c
l
σ
(
A
). Thus
(40)
in
t
σ
R
σ
(
A
)
=
(
c
l
σ
R
σ
(
A
c
)
)
c
=
(
c
l
σ
(
A
c
)
)
c
=
in
t
σ
(
A
)
.
Hence σ
R
σ
=
σ.Theorem 25.
Letτ be a fuzzy topology on U. Then the following are equivalent.(1)
τ satisfies (CC) axiom.
(2)
There exists a preorder fuzzy relationρ on U such that ρ
¯ is the closure operator of τ.
(3)
There exists a preorder fuzzy relationρ 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),
(41)
ρ
_
(
A
)
=
(
ρ
¯
(
A
c
)
)
c
=
(
c
l
τ
(
A
c
)
)
c
=
in
t
τ
(
A
)
.
Thusρ
_ 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 Remark6,
(42)
ρ
(
x
,
y
)
=
1
-
ρ
_
(
(
y
1
)
c
)
(
x
)
=
1
-
in
t
τ
(
(
y
1
)
c
)
(
x
)
=
c
l
τ
(
y
1
)
(
x
)
=
R
τ
(
x
,
y
)
(
x
,
y
∈
U
)
.
Thenρ
=
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),
(43)
c
l
τ
(
λ
A
)
=
R
τ
¯
(
λ
A
)
=
λ
R
τ
¯
(
A
)
=
λ
c
l
τ
(
A
)
.
Thus τ satisfies (CC) axiom.Theorem 26.
Let(44)
Σ
=
{
R
:
R
i
s
a
p
r
e
o
r
d
e
r
f
u
z
z
y
r
e
l
a
t
i
o
n
o
n
U
}
,
Γ
=
{
σ
:
σ
i
s
a
f
u
z
z
y
t
o
p
o
l
o
g
y
o
n
U
s
a
t
i
s
f
y
i
n
g
(
C
C
)
a
x
i
o
m
}
.
Then there exists a one-to-one correspondence between Σ and Γ.Proof.
f
:
Σ
→
Γ and g
:
Γ
→
Σ are defined as follows:
(45)
f
(
R
)
=
σ
R
(
R
∈
Σ
)
,
g
(
σ
)
=
R
σ
(
σ
∈
Γ
)
.
By Theorem21,
(46)
g
∘
f
=
i
Σ
,
where g
∘
f is the composition of f and g and i
Σ is the identity mapping on Γ.
By Theorem24,
(47)
f
∘
g
=
i
Γ
,
where f
∘
g is the composition of g and f and i
Γ is the identity mapping on Σ.
Hencef 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 Section4, 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 relationρ on U such that ρ
¯ is the closure operator of τ.
(3)
There exists a preorder fuzzy relationρ 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 Theorems24 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
(48)
σ
=
σ
1
*
∪
{
A
∈
F
(
U
)
:
∀
(
x
,
y
)
∈
R
A
,
n
n
n
n
n
n
n
A
(
x
)
∨
A
(
y
)
≤
1
-
R
(
x
,
y
)
}
.Proof.
This holds by Theorem18(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.



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*Source: 405802-2014-06-15.xml*