# On Fuzzy Rough Sets and Their Topological Structures

Authors: Weidong Tang; Jinzhao Wu; Dingwei Zheng
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. 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 satisfying (CC) axiom is proved, the concept of fuzzy approximating spaces is introduced, and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained, which illustrates that we can research fuzzy relations or fuzzy approximation spaces by means of topology and vice versa. Moreover, fuzzy pseudoclosure operators are examined.


---

## 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 (see [2–4]). 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 [8–10], and so on.Various fuzzy generalizations of rough approximations have been proposed [11–16]. 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 [11] 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 [17], approximate reasoning [5], medical time series, and case generation [18].There are two approaches to the development of fuzzy rough set theory as in classical cases. One is the constructive approach in which lower and upper approximation operators are constructed from fuzzy relations. The other one is the axiomatic approach in which the lower and upper approximation operators satisfy a set of axioms that are the same as those constructed ones. Morsi and Yakout [19] provide the axioms for fuzzy rough set model. Wu et al. [16] presented a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. Yeung et al. [20] researched the generalization of fuzzy rough sets and established a connection between fuzzy preorders and saturated Kuratowski fuzzy closure operators satisfying an additional condition.Topological structure is an important base for knowledge extraction and processing. Therefore, an interesting and natural research topic in rough set theory is to study the relationship between rough sets and topologies. Many authors studied topological structures of rough sets [21–26]. 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 [21, 27].In the study of topological structures of fuzzy rough sets or fuzzy approximation spaces, Qin and Pei [28] investigated topological structures of fuzzy rough sets and pointed out that there exists a one-to-one correspondence between the set of all reflexive, transitive fuzzy relations and the set of all fuzzy topologies satisfying (TC) axiom; Hao and Li [29] discussed the relationship between L-fuzzy rough set and L-topology. Zhou et al. [30] and Wu and Zhou [31] studied intuitionistic fuzzy rough sets and intuitionistic fuzzy topologies.Topological structures of approximation spaces may have some applications in information science. In practical, it often happens that some attribute values for an object of an information system are missing. For such incomplete information systems, Kryszkiewicz [32, 33] and Salama [34] studied the rule generation and information recovery by rough set approach and topological method, respectively.The purpose of this paper is to investigate further topological structures of fuzzy approximation spaces.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 and fuzzy relations are researched by means of topology or lower and upper sets. In Section 4, topological structures of fuzzy approximation spaces are given. In Section 5, fuzzy topology satisfying (CC) axiom is investigated, concept of fuzzy approximating spaces is introduced, and the decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained. In Section 6, fuzzy pseudoclosure operators are considered. Some concluding remarks appear in Section 7.


## 2. Preliminaries

Throughout this paper,U denotes a nonempty finite set called the universe of discourse. I denotes [
0,1
]. 2
U denotes the family of all subsets of U. F
(
U
) denotes the set of all fuzzy sets in U. For a
∈
I, a
¯ denotes the constant fuzzy set in U.We recall some basic operations onF
(
U
) as follows [35]: for any A
,
B
∈
F
(
U
) and λ
∈
I,(1)
A
=
B
⇔
A
(
x
)
=
B
(
x
) for each x
∈
U,
(2)
A
⊆
B
⇔
A
(
x
)
≤
B
(
x
) for each x
∈
U,
(3)
A
=
B
c
⇔
A
(
x
)
=
1
-
B
(
x
) for each x
∈
U,
(4)
(
A
∩
B
)
(
x
)
=
A
(
x
)
∧
B
(
x
) for each x
∈
U,
(5)
(
A
∪
B
)
(
x
)
=
A
(
x
)
∨
B
(
x
) for each x
∈
U.Moreover,(1)
(
⋃
j
∈
J
‍
A
)
(
x
)
=
⋁
j
∈
J
A
(
x
)
,
(
⋂
j
∈
J
‍
A
)
(
x
)
=
⋀
j
∈
J
A
(
x
)
,where{
A
j
:
j
∈
J
}
⊆
F
(
U
).(
6
)
(
λ
A
)
(
x
)
=
λ
∧
A
(
x
) for each x
∈
U.For eachA
∈
F
(
U
), we denote
(2)
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 [36]). Denote
(3)
P
(
U
)
=
{
x
λ
:
x
∈
U
,
λ
∈
(
0,1
]
}
.Specially,(4)
x
1
(
y
)
=
{
1
,
y
=
x
,
0
,
y
∈
U
-
{
x
}
.For a fuzzy pointx
λ and A
∈
F
(
U
), we defined x
λ
∈
A by x
λ
⊆
A.Obviously,(5)
x
λ
∈
A
⇔
λ
≤
A
(
x
)
.Remark 1.
ConsiderA
=
⋃
x
∈
U
(
A
(
x
)
x
1
) and A
=
⋂
x
∈
U
(
A
(
x
)
¯
∪
x
1
)
(
A
∈
F
(
U
)
).Definition 2 (see [37]).
Considerτ
⊆
F
(
U
) is called a fuzzy topology on U, if(i)
for eacha
∈
I, a
¯
∈
τ;
(ii)
A
,
B
∈
τ implies A
∩
B
∈
τ;
(iii)
{
A
j
:
j
∈
J
}
⊆
τ implies ⋃
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
}.
The interior and closure ofA denoted, respectively, by int
⁡
(
A
) and cl
⁡
(
A
) for each A
∈
F
(
U
) are defined as follows:
(6)
int
⁡
(
A
)
or
int
⁡
τ
(
A
)
=
⋃
‍
{
B
∈
τ
:
B
⊆
A
}
,
cl
⁡
(
A
)
or
c
l
τ
(
A
)
=
⋂
‍
{
B
∈
τ
c
:
B
⊇
A
}
.It should be pointed out that if (i) in Definition2 is replaced by(
i
)
′
0
¯
,
1
¯
∈
τ,then τ is a fuzzy topology in the sense of Chang [38]. 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 [39] if (ii) in Definition 2 is replaced by(
ii
)
′
{
A
j
:
j
∈
J
}
⊆
τ
⇒
⋂
j
∈
J
A
j
∈
τ.Definition 3 (see [36]).
Let(
U
,
τ
) be a fuzzy topological space and let x
λ
∈
P
(
U
) and A
∈
F
(
U
). A is called a closed remote neighborhood of x
λ, if A
∈
τ
c and x
λ
∉
A.Denote(7)
η
-
(
x
λ
)
=
{
f
A
∈
F
(
U
)
:
A
is
closed
remote
neighborhood
of
x
λ
f
}
.Definition 4 (see [36]).
Let(
U
,
τ
) be a fuzzy topological space. Consider the following.(1)
(
U
,
τ
) is called T
-
1, if for any x
λ
,
x
μ
∈
P
(
U
) and μ
<
λ, there exists A
∈
η
-
(
x
λ
) such that x
μ
∈
A, or, there exists B
∈
η
-
(
x
μ
) such that x
λ
∈
B.
(2)
(
U
,
τ
) is sub-T
0, if for any x
,
y
∈
U and x
≠
y, there exist λ
∈
(
0,1
] and A
∈
η
-
(
x
λ
) such that y
λ
∈
A, or, there exist λ
∈
(
0,1
] and B
∈
η
-
(
y
λ
) such that x
λ
∈
B.Definition 5 (see [40]).
LetR be a relation on U. For each x
∈
U, denote
(8)
R
p
(
x
)
=
{
y
∈
U
:
(
y
,
x
)
∈
R
}
,
R
s
(
x
)
=
{
y
∈
U
:
(
x
,
y
)
∈
R
}
.
R
p
(
x
) and R
s
(
x
) are called the predecessor and successor neighborhood of x, respectively.


## 3. Fuzzy Rough Sets and Fuzzy Rough Approximation Operators

In this section, we review rough approximation operators and rough sets in fuzzy environments and further researched their properties.Recall thatR is a fuzzy relation on U if R
∈
F
(
U
×
U
). R may be represented by a matrix. That is, if U
=
{
t
1
,
t
2
,
…
,
t
n
}, then R may be represented by the following matrix:
(9)
(
R
(
t
1
,
t
1
)
R
(
t
1
,
t
2
)
⋯
R
(
t
1
,
t
n
)
R
(
t
2
,
t
1
)
R
(
t
2
,
t
2
)
⋯
R
(
t
2
,
t
n
)
⋯
⋯
⋯
⋯
R
(
t
n
,
t
1
)
R
(
t
n
,
t
2
)
⋯
R
(
t
n
,
t
n
)
)
.LetR be a fuzzy relation on U. R is called reflexive if R
(
x
,
x
)
=
1 for each x
∈
U. R is called symmetric if R
(
x
,
y
)
=
R
(
y
,
x
) for any x
,
y
∈
U. R is called transitive if R
(
x
,
z
)
≥
R
(
x
,
y
)
∧
R
(
y
,
z
) for any x
,
y
,
z
∈
U. R is called preorder if R is reflexive and transitive.LetR be a fuzzy relation on U. R
-
1 is called the inverse of R if R
-
1
(
x
,
y
)
=
R
(
y
,
x
) for any x
,
y
∈
U. For each z
∈
U, we define fuzzy sets [
z
]
R
:
U
→
I, [
z
]
R
(
x
)
=
R
(
z
,
x
), and [
z
]
R
:
U
→
I, [
z
]
R
(
x
)
=
R
(
x
,
z
).Definition 6 (see [28]).
LetR be a fuzzy relation on U. The pair (
U
,
R
) is called a fuzzy approximation space. The fuzzy lower and the fuzzy upper approximation of A
∈
F
(
U
) with respect to (
U
,
R
), denoted by R
_
(
A
) and R
¯
(
A
), are respectively, defined as follows:
(10)
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 7.
Let(
U
,
R
) be a fuzzy approximation space. Then
(11)
R
¯
(
x
1
)
(
y
)
=
R
(
y
,
x
)
,
R
_
(
(
x
1
)
c
)
(
y
)
=
1
-
R
(
y
,
x
)
(
x
,
y
∈
U
)
.
Let(
U
,
R
) be a fuzzy approximation space. (
U
,
R
) is call reflexive (resp., preorder) if R is reflexive (resp., preorder).Proposition 8 (see [15, 28]).
Let(
U
,
R
) be a fuzzy approximation space. 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
_
(
A
∩
B
)
=
R
_
(
A
)
∩
R
_
(
B
), R
¯
(
A
∪
B
)
=
R
¯
(
A
)
∪
R
¯
(
B
),
(5)
R
_
(
⋂
j
∈
J
A
j
)
=
⋂
j
∈
J
(
R
_
(
A
j
)
), R
¯
(
⋃
j
∈
J
A
j
)
=
⋃
j
∈
J
(
R
¯
(
A
j
)
).
(6)
R
_
(
λ
¯
∪
A
)
=
λ
¯
∪
R
_
(
A
), R
¯
(
λ
A
)
=
λ
R
¯
(
A
).Theorem 9 (see [16, 28, 41]).
Let(
U
,
R
) be a fuzzy approximation space. Then Consider the following
(12)
(
1
)
R
i
s
r
e
f
l
e
x
i
v
e
⇔
(
I
L
R
)
∀
A
∈
F
(
U
)
,
R
_
(
A
)
⊆
A
⇔
(
I
U
R
)
∀
A
∈
F
(
U
)
,
A
⊆
R
¯
(
A
)
.
(
2
)
R
i
s
s
y
m
m
e
t
r
i
c
⇔
(
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
i
s
t
r
a
n
s
i
t
i
v
e
⇔
(
I
L
T
)
∀
A
∈
F
(
U
)
,
R
_
(
A
)
⊆
R
_
(
R
_
(
A
)
)
⇔
(
I
U
T
)
∀
A
∈
F
(
U
)
,
R
¯
(
R
¯
(
A
)
)
⊆
R
¯
(
A
)
.Theorem 10 (see [16, 28]).
Let(
U
,
R
) be a preorder fuzzy approximation space. Then for each A
∈
F
(
U
),
(13)
R
_
(
R
_
(
A
)
)
=
R
_
(
A
)
,
R
¯
(
R
¯
(
A
)
)
=
R
¯
(
A
)
.Remark 11.
(1) For eacha
∈
I, R
¯
(
a
¯
)
⊆
a
¯
⊆
R
_
(
a
¯
).
(2) IfR is reflexive, then for each a
∈
I, R
_
(
a
¯
)
=
a
¯
=
R
¯
(
a
¯
).Definition 12.
Let(
U
,
R
) be a fuzzy approximation space.(1)
A
∈
F
(
U
) is called an upper set if A
(
x
)
⋀
R
(
x
,
y
)
≤
A
(
y
) for any x
,
y
∈
U.
(2)
A
∈
F
(
U
) is called a lower set if A
(
y
)
⋀
R
(
x
,
y
)
≤
A
(
x
) for any x
,
y
∈
U.Proposition 13.
Let
(
U
,
R
)
be a fuzzy approximation space. Then the following are equivalent:
(1)
R
¯
(
A
)
⊆
A;
(2)
A  is a lower set in
(
U
,
R
);
(3)
A  is an upper set in
(
U
,
R
-
1
).Proof.
(1)⇒(2). Suppose that R
¯
(
A
)
⊆
A. Since for each x
∈
U,
(14)
⋁
y
∈
U
(
A
(
y
)
∧
R
(
x
,
y
)
)
=
R
¯
(
A
)
(
x
)
≤
A
(
x
)
,
we have
(15)
A
(
y
)
∧
R
(
x
,
y
)
≤
A
(
x
)
(
x
,
y
∈
U
)
.
Thus A is a lower set in (
U
,
R
).
(2)⇒(3). This is obvious.
(3)⇒(1). Suppose that A is an upper set in (
X
,
R
-
1
). Then for any x
,
y
∈
U, A
(
x
)
∧
R
-
1
(
x
,
y
)
≤
A
(
y
). So A
(
x
)
∧
R
(
y
,
x
)
≤
A
(
y
). Thus
(16)
R
¯
(
A
)
(
y
)
=
⋁
x
∈
U
(
A
(
x
)
∧
R
(
y
,
x
)
)
≤
A
(
y
)
(
y
∈
U
)
.
Hence R
¯
(
A
)
⊆
A.Corollary 14.
Let(
U
,
R
) be a fuzzy approximation space. If R is reflexive, then the following are equivalent:(1)
R
¯
(
A
)
=
A;
(2)
A is a lower set in (
U
,
R
);
(3)
A is an upper set in(
U
,
R
-
1
).Proof.
This holds by Theorem9(1) and Proposition 13.Theorem 15.
Let(
U
,
R
) be a fuzzy approximation space. Then
(17)
(
1
)
R
i
s
r
e
f
l
e
x
i
v
e
⇔
(
I
L
S
′
)
∀
x
∈
U
,
[
x
]
R
(
x
)
=
1
⇔
(
I
U
R
′
)
∀
x
∈
U
,
[
x
]
R
(
x
)
=
1
.
(
2
)
R
i
s
s
y
m
m
e
t
r
i
c
⇔
(
I
L
S
′
)
∀
x
∈
U
,
[
x
]
R
=
[
x
]
R
⇒
∀
A
∈
F
(
U
)
,
A
i
s
a
l
o
w
e
r
s
e
t
i
f
a
n
d
o
n
l
y
i
f
A
i
s
a
n
u
p
p
e
r
s
e
t
.
(
3
)
R
i
s
t
r
a
n
s
i
t
i
v
e
⇔
(
I
L
T
′
)
∀
x
∈
U
,
[
x
]
R
i
s
a
l
o
w
e
r
s
e
t
⇔
(
I
U
T
′
)
∀
x
∈
U
,
[
x
]
R
i
s
a
n
u
p
p
e
r
s
e
t
⇔
(
I
U
T
′′
)
∀
A
∈
F
(
U
)
,
R
¯
(
A
)
i
s
a
l
o
w
e
r
s
e
t
.Proof.
(1) and (2) are obvious.
(3)(18)
(
I
U
T
′′
)
⇔
∀
A

∈

F
(
U
)
,
R
¯
(
R
¯
(
A
)
)

⊆

R
¯
(
A
)
.
(
Proposition
13
)
⇔
R
is
transitive
.
(
Theorem
9
(
3
)
)
⇔
∀
x
,
y
,
z
∈
U
,
R
(
x
,
y
)
∧
R
(
y
,
z
)
≤
R
(
x
,
z
)
⇔
∀
x
,
y
,
z
∈
U
,
R
(
x
,
y
)
∧
[
z
]
R
(
y
)
≤
[
z
]
R
(
x
)
⇔
(
I
L
T
′
)
∀
x
∈
U
,
[
x
]
R
is
a
lower
set
⇔
∀
x
,
y
,
z
∈
U
,
[
x
]
R
(
y
)
∧
R
(
y
,
z
)
≤
[
x
]
R
(
z
)
⇔
(
I
U
T
′
)
∀
x
∈
U
,
[
x
]
R
is
an
upper
set
.Theorem16 below illustrates that fuzzy relations can be researched by means of topology.Theorem 16.
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) By Remark7, R
¯
(
x
1
)
(
y
)
=
R
(
y
,
x
) for each x
∈
U. Note that R
_ is the interior operator of τ. Then for each x
∈
U,
(19)
R
(
x
,
x
)
=
R
¯
(
x
1
)
(
x
)
=
c
l
τ
(
x
1
)
(
x
)
≥
x
1
(
x
)
=
1
.
Thus R is reflexive.
For anyx
,
y
,
z
∈
U, denote c
l
τ
(
z
1
)
(
y
)
=
λ, and by Remarks 1 and 7 and Proposition 8(6),
(20)
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
)
.
ThenR is transitive. Thus R is preorder.
(2) The proof is similar to (1).Proposition 17.
Let(
U
,
R
) be a fuzzy approximation space. 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)
IfR 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 for each x
∈
U,
(21)
⋀
y
∈
U
(
A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
)
=
(
R
_
(
A
)
)
(
y
)
≥
A
(
x
)
.
Then A
(
y
)
∨
(
1
-
R
(
x
,
y
)
)
≥
A
(
x
) for any x
,
y
∈
U. Since A
(
x
)
>
A
(
y
) for each (
x
,
y
)
∈
R
A, we have
(22)
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
(23)
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
(24)
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 that R
¯
(
A
)
⊆
A. Note that for each y
∈
U,
(25)
⋁
x
∈
U
(
A
(
x
)
∧
R
(
y
,
x
)
)
=
R
¯
(
A
)
(
y
)
≤
A
(
y
)
.
Then A
(
x
)
∧
R
(
y
,
x
)
≤
A
(
y
)
(
x
,
y
∈
U
) for any x
,
y
∈
U. Since A
(
x
)
>
A
(
y
) for each (
x
,
y
)
∈
R
A, we have
(26)
R
(
y
,
x
)
≤
A
(
y
)
=
A
(
x
)
∧
A
(
y
)
(
(
x
,
y
)
∈
R
A
)
.
Sufficiency. Suppose that (
F
L
O
) holds. Let y
∈
U.
(i) Ifx
∈
(
R
A
)
p
(
y
), then (
x
,
y
)
∈
R
A and so
(27)
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
(28)
A
(
x
)
∧
R
(
y
,
x
)
≤
A
(
x
)
≤
A
(
y
)
.
Thus(
R
¯
(
A
)
)
(
y
)
=
⋁
x
∈
U
(
A
(
x
)
∧
R
(
y
,
x
)
)
≤
A
(
y
).
HenceR
¯
(
A
)
⊆
A.
(2) This holds by (1), the reflexivity ofR and Theorem 9(1).Proposition 18.
Let(
U
,
R
1
) and (
U
,
R
2
) be two fuzzy approximation spaces. Then for each A
∈
F
(
U
).(1)
(a)
R
1
⊆
R
2
⇒
R
1
_
(
A
)
⊇
R
2
_
(
A
), R
1
¯
(
A
)
⊆
R
2
¯
(
A
).
(b)
(
R
1
∪
R
2
_
)
(
A
)
=
R
1
_
(
A
)
∩
R
2
_
(
A
), (
R
1
∪
R
2
¯
)
(
A
)
=
R
1
¯
(
A
)
∪
R
2
¯
(
A
).
(2)
If
R
1
and
R
2
are reflexive, then
(a)
R
2
_
(
R
1
_
(
A
)
)
⊆
(
R
1
∪
R
2
_
)
(
A
), R
1
_
(
R
2
_
(
A
)
)
⊆
(
R
1
∪
R
2
_
)
(
A
).
(b)
R
2
¯
(
R
1
¯
(
A
)
)
⊇
(
R
1
∪
R
2
¯
)
(
A
), R
1
¯
(
R
2
¯
(
A
)
)
⊇
(
R
1
∪
R
2
¯
)
(
A
).
(3)
If
R
1
is reflexive,
R
2
is transitive and
R
1
⊆
R
2
, then
(29)
R
1
_
(
R
2
_
(
A
)
)
=
R
2
_
(
A
)
,
R
1
¯
(
R
2
¯
(
A
)
)
=
R
2
¯
(
A
)
.Proof.
(1) (a) This is obvious.
(b) For eachx
∈
U, since
(30)
(
R
1
∪
R
2
¯
)
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
∧
(
R
1
∪
R
2
)
(
x
,
y
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
(
R
1
(
x
,
y
)
∨
R
2
(
x
,
y
)
)
)
=
⋁
y
∈
U
(
(
A
(
y
)
∧
R
1
(
x
,
y
)
)
∨
(
A
(
y
)
∧
R
2
(
x
,
y
)
)
)
=
(
⋁
y
∈
U
(
A
(
y
)
∧
R
1
(
x
,
y
)
)
)
∨
(
⋁
y
∈
U
(
A
(
y
)
∨
R
2
(
x
,
y
)
)
)
=
R
1
¯
(
A
)
(
x
)
∨
R
2
¯
(
A
)
(
x
)
=
(
R
1
¯
(
A
)
∪
R
2
¯
(
A
)
)
(
x
)
,
we have
(31)
(
R
1
∪
R
2
¯
)
(
A
)
=
R
1
¯
(
A
)
∪
R
2
¯
(
A
)
.
By Proposition8(3),
(32)
R
1
_
(
A
)
∩
R
2
_
(
A
)
=
(
R
1
¯
(
A
c
)
)
c
∩
(
R
2
¯
(
A
c
)
)
c
=
(
R
1
¯
(
A
c
)
∪
R
2
¯
(
A
c
)
)
c
=
(
(
R
1
∪
R
2
¯
)
(
A
c
)
)
c
=
(
R
1
∪
R
2
_
)
(
A
)
.
(2) (a) By Theorem9(1), R
2
_
(
R
1
_
(
A
)
)
⊆
R
2
_
(
A
) and R
2
_
(
R
1
_
(
A
)
)
⊆
R
1
_
(
A
). By (1),
(33)
R
2
_
(
R
1
_
(
A
)
)
⊆
R
1
_
(
A
)
∩
R
2
_
(
A
)
=
(
R
1
∪
R
2
_
)
(
A
)
.
Similarly, we can prove thatR
1
_
(
R
2
_
(
A
)
)
⊆
(
R
1
∪
R
2
_
)
(
A
).
(b) The proof is similar to (a).
(3) By Theorem9(1), R
1
¯
(
R
2
¯
(
A
)
)
⊇
R
2
¯
(
A
).
For eachx
∈
U, by R
1
⊆
R
2 and the transitivity of R
2, we have
(34)
R
1
¯
(
R
2
¯
(
A
)
)
(
x
)
=
⋁
y
∈
U
(
R
2
¯
(
A
)
(
y
)
∧
R
1
(
x
,
y
)
)
=
⋁
y
∈
U
(
(
⋁
z
∈
U
(
A
(
z
)
∧
R
2
(
y
,
z
)
)
)
∧
R
1
(
x
,
y
)
)
=
⋁
y
∈
U
(
⋁
z
∈
U
(
(
A
(
z
)
∧
R
2
(
y
,
z
)
)
∧
R
1
(
x
,
y
)
)
)
=
⋁
y
∈
U
(
⋁
z
∈
U
(
A
(
z
)
∧
(
R
2
(
y
,
z
)
∧
R
1
(
x
,
y
)
)
)
)
≤
⋁
y
∈
U
(
⋁
z
∈
U
(
A
(
z
)
∧
(
R
2
(
y
,
z
)
∧
R
2
(
x
,
y
)
)
)
)
≤
⋁
y
∈
U
(
⋁
z
∈
U
(
A
(
z
)
∧
R
2
(
x
,
z
)
)
)
=
⋁
z
∈
U
(
A
(
z
)
∧
R
2
(
x
,
z
)
)
=
R
2
¯
(
A
)
(
x
)
.
Then R
1
¯
(
R
2
¯
(
A
)
)
⊆
R
2
¯
(
A
). So R
1
¯
(
R
2
¯
(
A
)
)
=
R
2
¯
(
A
).
By Proposition8(3),
(35)
R
1
_
(
R
2
_
(
A
)
)
=
R
1
_
(
(
R
2
¯
(
A
c
)
)
c
)
=
(
R
1
¯
(
R
2
¯
(
A
c
)
)
)
c
=
(
R
2
¯
(
A
c
)
)
c
=
R
2
_
(
A
)
.


## 4. Fuzzy Approximation Spaces versus Fuzzy Topologies

In this section we give topological structures of fuzzy approximation spaces (i.e., structures of the topology induced by fuzzy approximation spaces).For eachR
∈
F
(
U
×
U
), denote
(36)
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
)
.Proposition 19.
Let(
U
,
R
) be a fuzzy approximation space. If R is preorder, then
(37)
τ
R
=
θ
R
.Proof.
Obviously,τ
R
⊆
θ
R. By Theorem 10, τ
R
⊇
θ
R. Thus τ
R
=
θ
R.


### 4.1. The Fuzzy Topology Induced by a Reflexive Fuzzy Approximation Space

Theorem 20.
Let(
U
,
R
) be a reflexive fuzzy approximation space. Then Consider the following.(1)
τ
R is an Alexandrov fuzzy topology on U.
(2)
For eachA
∈
F
(
U
),
(38)
in
t
τ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
τ
R
(
A
)
.
(3)
A
∈
(
τ
R
)
c
⇔
A
=
R
¯
(
A
).
(4)
For eacha
∈
I, a
¯
∈
(
τ
R
)
c.
(5)
τ
R
=
{
A
c
:
A
i
s
a
l
o
w
e
r
s
e
t
i
n
(
U
,
R
)
}, τ
R
-
1
=
{
A
c
:
A
i
s
a
n
u
p
p
e
r
s
e
t
i
n
(
U
,
R
)
}.Proof.
(1) (i) For eacha
∈
I, by the reflexivity of R and Remark 11(2), R
_
(
a
¯
)
=
a
¯. Then a
¯
∈
τ
R.
(ii) Let{
A
j
:
j
∈
J
}
⊆
τ
R. Then R
_
(
A
j
)
=
A
j for each j
∈
J. By Proposition 8(5),
(39)
R
_
(
⋂
j
∈
J
‍
A
j
)
=
⋂
j
∈
J
‍
R
_
(
A
j
)
=
⋂
j
∈
J
‍
A
j
.
Hence⋂
j
∈
J
A
j
∈
τ
R.
(iii) Let{
A
j
:
j
∈
J
}
⊆
τ
R. Then R
_
(
A
j
)
=
A
j for each j
∈
J. By the reflexivity of R and Theorem 9(1), R
_
(
⋃
j
∈
J
A
j
)
⊆
⋃
j
∈
J
A
j. By Proposition 8(2),
(40)
R
_
(
⋃
j
∈
J
‍
A
j
)
⊇
⋃
j
∈
J
‍
R
_
(
A
j
)
=
⋃
j
∈
J
‍
A
j
.
Then R
_
(
⋃
j
∈
J
A
j
)
=
⋃
j
∈
J
A
j. So ⋃
j
∈
J
A
j
∈
τ
R.
Henceτ
R is Alexandrov.
(2) For eachA
∈
F
(
U
), by Proposition 8(2),
(41)
int
⁡
τ
R
(
A
)
=
⋃
‍
{
B
∈
τ
R
:
B
⊆
A
}
⊆
⋃
‍
{
B
∈
τ
R
:
R
_
(
B
)
⊆
R
_
(
A
)
}
=
⋃
‍
{
B
∈
F
(
U
)
:
B
=
R
_
(
B
)
⊆
R
_
(
A
)
}
⊆
R
_
(
A
)
.
By Proposition 8(3),
(42)
cl
⁡
τ
R
(
A
)
=
(
int
⁡
τ
R
(
A
c
)
)
c
⊇
(
R
_
(
A
c
)
)
c
=
R
¯
(
A
)
.
By the reflexivity ofR and Theorem 9(1),
(43)
int
⁡
τ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
cl
⁡
τ
R
(
A
)
.
(3) This holds by Proposition8(3).
(4) This holds by (3) and Remark11(2).
(5) This holds by Proposition8(3) and Corollary 14.Example 21.
LetU
=
{
x
,
y
,
z
,
w
} and let R be a fuzzy relation on U, where
(44)
R
=
(
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
)
.
Then R is not reflexive.
For anyA
∈
F
(
U
) and t
∈
U,
(45)
R
_
(
A
)
(
t
)
=
⋀
s
∈
U
(
A
(
s
)
∨
(
1
-
R
(
t
,
s
)
)
)
=
⋀
s
∈
U
-
{
t
}
A
(
s
)
.
Suppose thatA
(
x
)
≤
A
(
y
)
≤
A
(
z
)
≤
A
(
w
). Since R
_
(
A
)
=
A, we have
(46)
A
(
x
)
∧
A
(
y
)
∧
A
(
z
)
=
A
(
w
)
.
Then A
(
t
)
≥
A
(
w
) for each t
∈
{
x
,
y
,
z
}. So A
(
x
)
=
A
(
y
)
=
A
(
z
)
=
A
(
w
). Thus τ
R
=
{
a
¯
:
a
∈
I
}.
Obviously,τ
R is an Alexandrov fuzzy topology on U.Definition 22.
Let(
U
,
R
) be a reflexive fuzzy approximation space. τ
R is called the fuzzy topology induced by (
U
,
R
) or induced by R on U.Theorem 23.
Let(
U
,
R
) be a reflexive fuzzy approximation space and let τ
R be the fuzzy topology induced by (
U
,
R
). Then the following properties hold.(1)
(
U
,
τ
R
) is not connected.
(2)
(
U
,
τ
R
) is T
-
1.
(3)
(a)
If(
U
,
τ
R
) is sub-T
0, then for any x
,
y
∈
U with x
≠
y, R
(
x
,
y
)
∧
R
(
y
,
x
)
<
1.
(b)
Ift
R
<
1, then (
U
,
τ
R
) is sub-T
0.Proof.
(1) This holds by Theorem20.
(2) For anyx
λ
,
x
μ
∈
P
(
U
) with μ
<
λ, put u
∈
(
μ
,
λ
) and we have x
λ
∉
u
¯. By Theorem 20(4), u
¯
∈
η
-
(
x
λ
). Note that x
μ
∈
u
¯. Then (
U
,
τ
R
) is T
-
1.
(3) (a) For anyx
,
y
∈
U with x
≠
y, since (
U
,
τ
R
) is sub-T
0, then there exist λ
∈
(
0,1
] and A
∈
η
-
(
x
λ
) such that y
λ
∈
A, or there exist λ
∈
(
0,1
] and B
∈
η
-
(
y
λ
) such that x
λ
∈
B.(i)
If there existλ
∈
(
0,1
] and A
∈
η
-
(
x
λ
) such that y
λ
∈
A, then A
(
y
)
≥
λ
>
A
(
x
). By Theorem 20(3) and A
∈
(
τ
R
)
c, R
¯
(
A
)
=
A. By (
y
,
x
)
∈
R
A and Proposition 17(2),
(47)
R
(
x
,
y
)
≤
A
(
x
)
∧
A
(
y
)
=
A
(
x
)
<
λ
≤
1
.
Then R
(
x
,
y
)
<
1.
(ii)
If there existλ
∈
(
0,1
] and B
∈
η
-
(
y
λ
) such that x
λ
∈
B, then we can prove that R
(
y
,
x
)
<
1.
SoR
(
x
,
y
)
∧
R
(
y
,
x
)
<
1.
(b) Letx
,
y
∈
U with x
≠
y. By t
R
<
1, there exists λ
∈
(
t
R
,
1
]. Put
(48)
A
(
t
)
=
{
t
R
,
t
=
x
,
λ
,
t
≠
x
.
Then R
A
=
{
(
t
,
x
)
:
t
∈
U
-
{
x
}
}. For each (
t
,
x
)
∈
R
A,
(49)
R
(
x
,
t
)
≤
⋁
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
=
t
R
=
λ
∧
t
R
=
A
(
t
)
∧
A
(
x
)
.
By Proposition 17(2), R
¯
(
A
)
=
A. By Theorem 20(3), A
∈
(
τ
R
)
c. Note that A
(
x
)
=
t
R
<
λ. Then A
∈
η
-
(
x
λ
). Obviously, y
λ
∈
A. Hence (
U
,
τ
R
) is sub-T
0.Proposition 24.
LetR
1 and R
2 be two reflexive fuzzy relations on U. Then(1)
R
1
⊆
R
2
⇒
τ
R
2
⊆
τ
R
1.
(2)
(50)
τ
R
1
⊆
τ
R
2
⇒
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
(
x
∈
U
)
.
(3)
R
1
∪
R
2
is reflexive.
(4)
τ
R
1
∪
R
2
is an Alexandrov fuzzy topology on
U
.
(5)
τ
R
1
∪
R
2
=
τ
R
1
∩
τ
R
2
.Proof.
(1) For eachA
∈
τ
R
2, R
2
_
(
A
)
=
A. By R
1
⊆
R
2, Proposition 18(1), the reflexivity of R
1 and Theorem 9(1),
(51)
A
=
R
2
_
(
A
)
⊆
R
1
_
(
A
)
⊆
A
.
Hence A
∈
τ
R
1. Thus τ
R
2
⊆
τ
R
1.
(2) Letx
∈
U. Pick λ
=
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
).(i)
Ifλ
=
0, then
(52)
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
1
=
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
.
(ii)
Ifλ
>
0, put A
=
x
λ
∈
P
(
U
), then R
A
=
{
(
x
,
y
)
:
y
∈
U
-
{
x
}
}. For any (
x
,
y
)
∈
R
A, by
(53)
1
-
R
1
(
x
,
y
)
≥
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
=
λ
=
A
(
x
)
∨
A
(
y
)and Proposition 17(2), R
1
_
(
A
)
=
A. So A
∈
τ
1. Note that τ
R
1
⊆
τ
R
2. Then A
∈
τ
2. So R
2
_
(
A
)
=
A. For each (
x
,
y
)
∈
R
A, by Proposition 17(2),
(54)
1
-
R
2
(
x
,
y
)
≥
A
(
x
)
∨
A
(
y
)
=
λ
.
Hence(55)
1
-
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
=
⋀
y
∈
U
-
{
x
}
(
1
-
R
2
(
x
,
y
)
)
≥
λ
=
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
.
So⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
).
(3) This is obvious.
(4) This holds by (3) and Theorem20(1).
(5) By (1),τ
R
1
∪
R
2
⊆
τ
R
1
∩
τ
R
2.
For eachA
∈
τ
R
1
∩
τ
R
2, we have A
=
R
1
_
(
A
) and A
=
R
2
_
(
A
).
By Proposition18(1),
(56)
A
=
R
1
_
(
A
)
∩
R
2
_
(
A
)
=
(
R
1
∪
R
2
_
)
(
A
)
.
Then A
∈
τ
R
1
∪
R
2. Thus τ
R
1
∪
R
2
⊇
τ
R
1
∩
τ
R
2.
Henceτ
R
1
∪
R
2
=
τ
R
1
∩
τ
R
2.Definition 25.
LetR be a fuzzy relation on U. R is called pseudoconstant if there exists a
∈
I such that for any x
,
y
∈
U,
(57)
R
(
x
,
y
)
=
{
1
,
x
=
y
,
a
,
x
≠
y
.
We write R by a
* or a
U
×
U
*.That is,(58)
a
*
(
x
,
y
)
=
{
1
,
x
=
y
,
a
,
x
≠
y
.Obviously, every pseudoconstant fuzzy relation is an equivalence fuzzy relation.Remark 26.
(1) For anya
,
b
∈
I, a
≤
b implies a
*
⊆
b
*.
(2) For eacha
∈
I, θ
a
*
=
τ
a
*.
(3)τ
0
*
=
F
(
U
), τ
1
*
=
{
a
¯
:
a
∈
I
}.Remark 27.
LetR be a reflexive fuzzy relation on U. Then
(59)
s
R
*
⊆
R
⊆
t
R
*
.The following theorem gives the topological structure of reflexive fuzzy approximation spaces.Theorem 28.
Let(
U
,
R
) be a reflexive fuzzy approximation space.(1)
Considerτ
R
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
∀
(
x
,
y
)
∈
R
A
,
A
(
x
)
∨
A
(
y
)
≤
1
-
R
(
x
,
y
)
}.
(2)
Considerτ
t
R
*
⊆
τ
R
⊆
τ
s
R
*.
(3)
For eacha
∈
I, τ
a
*
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
R
A
≠
∅
,
⋁
x
∈
U
A
(
x
)
≤
1
-
a
}.Proof.
(1) This holds by Proposition17(2), Theorem 20(1), and Remark 26(3).
(2) This holds by Proposition24(1) and Remark 27.
(3) This holds by (1).


### 4.2. Fuzzy Topologies Induced by Preorder Fuzzy Approximation Spaces

Theorem 29 (see [28]).
Let(
U
,
R
) be a preorder fuzzy approximation space. 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.Example 30.
LetU
=
{
x
,
y
,
z
} and let R be a fuzzy relation on U where
(60)
R
=
(
1
0
0
0.7
1
1
0
0.8
1
)
.
Then R is reflexive. Note that
(61)
R
(
z
,
y
)
∧
R
(
y
,
x
)
=
0.7
≰
0
=
R
(
z
,
x
)
.
Then R is not transitive. So R is not preorder.
Pick(62)
A
=
0
x
+
0.4
y
+
1
z
,
B
=
1
x
+
0.6
y
+
0
z
.
(1)
Since(63)
R
_
(
A
)
=
0
x
+
0.3
y
+
0.4
z
,
(64)
R
_
(
R
_
(
A
)
)
=
0
x
+
0.3
y
+
0.3
z
,
we have R
_
(
R
_
(
A
)
)
≠
R
_
(
A
). Then R
_
(
A
)
∉
τ
R. Thus
(65)
θ
R
≠
τ
R
.
(2)
Obviously,B
c
=
A. Note that R
_
(
A
)
∉
τ
R and int
⁡
τ
R
(
B
)
∈
τ
R. Then
(66)
int
⁡
τ
R
(
A
)
≠
R
_
(
A
)
,
(67)
cl
⁡
τ
R
(
B
)
≠
R
¯
(
B
)
.
Thus, R
_ is not the interior operator of τ
R and R
¯ is not the closure operator of τ
R.Example 31.
LetU
=
{
x
,
y
,
z
,
w
} and let R be the fuzzy relation on U where
(68)
R
=
(
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
)
.
Then R is neither reflexive nor transitive.
For eachA
∈
F
(
U
), since
(69)
R
_
(
A
)
=
A
(
y
)
x
+
A
(
z
)
y
+
A
(
w
)
z
+
A
(
x
)
w
=
B
=
R
¯
(
A
)
,
R
_
(
B
)
=
A
(
z
)
x
+
A
(
w
)
y
+
A
(
x
)
z
+
A
(
y
)
w
=
C
=
R
¯
(
B
)
,
R
_
(
C
)
=
A
(
w
)
x
+
A
(
x
)
y
+
A
(
y
)
z
+
A
(
z
)
w
=
D
=
R
¯
(
C
)
,
R
_
(
D
)
=
A
(
x
)
x
+
A
(
y
)
y
+
A
(
z
)
z
+
A
(
w
)
w
=
A
=
R
¯
(
D
)
,
we have R
_
(
R
_
(
R
_
(
R
_
(
A
)
)
)
)
=
A. Then θ
R
=
F
(
U
).
Obviously,θ
R is a fuzzy topology on U.
PutW
=
x
0.3
∈
P
(
U
). Then
(70)
cl
⁡
θ
R
(
W
)
=
x
0.3
≠
w
0.3
=
R
¯
(
W
)
,
int
⁡
θ
R
(
W
)
=
x
0.3
≠
w
0.3
=
R
2
_
(
W
)
.
Thus,R
_ is not the interior operator of θ
R and R
¯ is not the closure operator of θ
R.The following theorem reveals fuzzy relations can be researched by means of topology.Theorem 32.
LetR
1 and R
2 be two preorder fuzzy relations on U. Then(1)
θ
R
2
=
θ
R
1
⇔
R
1
=
R
2.
(2)
θ
R
1
∩
θ
R
2
⊆
θ
R
1
∪
R
2.Proof
(1) Necessity. Suppose thatθ
R
2
=
θ
R
1. By Remark 7 and Theorem 29(3),
(71)
R
1
(
x
,
y
)
=
R
1
¯
(
y
1
)
(
x
)
=
cl
⁡
R
τ
1
(
y
1
)
(
x
)
=
cl
⁡
R
τ
2
(
y
1
)
(
x
)
=
R
2
(
x
,
y
)
for any x
,
y
∈
U. Then R
1
=
R
2.
Sufficiency. This is obvious.
(2) By Propositions19 and 24(5),
(72)
θ
R
1
∩
θ
R
2
=
τ
R
1
∩
τ
R
2
=
τ
R
1
∪
R
2
⊆
θ
R
1
∪
R
2
.Example 33.
LetU
=
{
x
,
y
,
z
,
w
} and let R
1 and R
2 be two fuzzy relations on U, where
(73)
R
1
=
(
1
0
1
0
0
1
0
1
0
1
1
0
1
0
0
1
)
,
R
2
=
(
1
0
0
1
1
1
0
0
0
1
1
0
0
0
1
1
)
.
Then R
1 and R
2 are reflexive. Note that
(74)
R
1
(
x
,
z
)
∧
R
1
(
z
,
y
)
=
1
≰
0
=
R
1
(
x
,
y
)
,
R
2
(
x
,
w
)
∧
R
2
(
w
,
z
)
=
1
≰
0
=
R
2
(
x
,
z
)
.
Then R
1 and R
2 are not transitive. So R
1 and R
2 are not preorder.
For anyA
,
B
∈
F
(
U
), since
(75)
R
1
_
(
A
)
=
A
(
x
)
∧
A
(
z
)
x
+
A
(
y
)
∧
A
(
w
)
y
+
A
(
y
)
∧
A
(
z
)
z
+
A
(
x
)
∧
A
(
w
)
w
=
A
,
R
2
_
(
B
)
=
B
(
x
)
∧
B
(
w
)
x
+
B
(
x
)
∧
B
(
y
)
y
+
B
(
y
)
∧
B
(
z
)
z
+
B
(
z
)
∧
B
(
w
)
w
=
B
,
we have
(76)
A
(
z
)
≥
A
(
x
)
≥
A
(
w
)
≥
A
(
y
)
≥
A
(
z
)
,
B
(
w
)
≥
B
(
x
)
≥
B
(
y
)
≥
B
(
z
)
≥
B
(
w
)
.
So A
(
x
)
=
A
(
y
)
=
A
(
z
)
=
A
(
w
) and B
(
x
)
=
B
(
y
)
=
B
(
z
)
=
B
(
w
).
Thusτ
R
1
=
{
a
¯
:
a
∈
I
}
=
τ
R
2. But R
1
≠
R
2.


## 4.1. The Fuzzy Topology Induced by a Reflexive Fuzzy Approximation Space

Theorem 20.
Let(
U
,
R
) be a reflexive fuzzy approximation space. Then Consider the following.(1)
τ
R is an Alexandrov fuzzy topology on U.
(2)
For eachA
∈
F
(
U
),
(38)
in
t
τ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
c
l
τ
R
(
A
)
.
(3)
A
∈
(
τ
R
)
c
⇔
A
=
R
¯
(
A
).
(4)
For eacha
∈
I, a
¯
∈
(
τ
R
)
c.
(5)
τ
R
=
{
A
c
:
A
i
s
a
l
o
w
e
r
s
e
t
i
n
(
U
,
R
)
}, τ
R
-
1
=
{
A
c
:
A
i
s
a
n
u
p
p
e
r
s
e
t
i
n
(
U
,
R
)
}.Proof.
(1) (i) For eacha
∈
I, by the reflexivity of R and Remark 11(2), R
_
(
a
¯
)
=
a
¯. Then a
¯
∈
τ
R.
(ii) Let{
A
j
:
j
∈
J
}
⊆
τ
R. Then R
_
(
A
j
)
=
A
j for each j
∈
J. By Proposition 8(5),
(39)
R
_
(
⋂
j
∈
J
‍
A
j
)
=
⋂
j
∈
J
‍
R
_
(
A
j
)
=
⋂
j
∈
J
‍
A
j
.
Hence⋂
j
∈
J
A
j
∈
τ
R.
(iii) Let{
A
j
:
j
∈
J
}
⊆
τ
R. Then R
_
(
A
j
)
=
A
j for each j
∈
J. By the reflexivity of R and Theorem 9(1), R
_
(
⋃
j
∈
J
A
j
)
⊆
⋃
j
∈
J
A
j. By Proposition 8(2),
(40)
R
_
(
⋃
j
∈
J
‍
A
j
)
⊇
⋃
j
∈
J
‍
R
_
(
A
j
)
=
⋃
j
∈
J
‍
A
j
.
Then R
_
(
⋃
j
∈
J
A
j
)
=
⋃
j
∈
J
A
j. So ⋃
j
∈
J
A
j
∈
τ
R.
Henceτ
R is Alexandrov.
(2) For eachA
∈
F
(
U
), by Proposition 8(2),
(41)
int
⁡
τ
R
(
A
)
=
⋃
‍
{
B
∈
τ
R
:
B
⊆
A
}
⊆
⋃
‍
{
B
∈
τ
R
:
R
_
(
B
)
⊆
R
_
(
A
)
}
=
⋃
‍
{
B
∈
F
(
U
)
:
B
=
R
_
(
B
)
⊆
R
_
(
A
)
}
⊆
R
_
(
A
)
.
By Proposition 8(3),
(42)
cl
⁡
τ
R
(
A
)
=
(
int
⁡
τ
R
(
A
c
)
)
c
⊇
(
R
_
(
A
c
)
)
c
=
R
¯
(
A
)
.
By the reflexivity ofR and Theorem 9(1),
(43)
int
⁡
τ
R
(
A
)
⊆
R
_
(
A
)
⊆
A
⊆
R
¯
(
A
)
⊆
cl
⁡
τ
R
(
A
)
.
(3) This holds by Proposition8(3).
(4) This holds by (3) and Remark11(2).
(5) This holds by Proposition8(3) and Corollary 14.Example 21.
LetU
=
{
x
,
y
,
z
,
w
} and let R be a fuzzy relation on U, where
(44)
R
=
(
0
1
1
1
1
0
1
1
1
1
0
1
1
1
1
0
)
.
Then R is not reflexive.
For anyA
∈
F
(
U
) and t
∈
U,
(45)
R
_
(
A
)
(
t
)
=
⋀
s
∈
U
(
A
(
s
)
∨
(
1
-
R
(
t
,
s
)
)
)
=
⋀
s
∈
U
-
{
t
}
A
(
s
)
.
Suppose thatA
(
x
)
≤
A
(
y
)
≤
A
(
z
)
≤
A
(
w
). Since R
_
(
A
)
=
A, we have
(46)
A
(
x
)
∧
A
(
y
)
∧
A
(
z
)
=
A
(
w
)
.
Then A
(
t
)
≥
A
(
w
) for each t
∈
{
x
,
y
,
z
}. So A
(
x
)
=
A
(
y
)
=
A
(
z
)
=
A
(
w
). Thus τ
R
=
{
a
¯
:
a
∈
I
}.
Obviously,τ
R is an Alexandrov fuzzy topology on U.Definition 22.
Let(
U
,
R
) be a reflexive fuzzy approximation space. τ
R is called the fuzzy topology induced by (
U
,
R
) or induced by R on U.Theorem 23.
Let(
U
,
R
) be a reflexive fuzzy approximation space and let τ
R be the fuzzy topology induced by (
U
,
R
). Then the following properties hold.(1)
(
U
,
τ
R
) is not connected.
(2)
(
U
,
τ
R
) is T
-
1.
(3)
(a)
If(
U
,
τ
R
) is sub-T
0, then for any x
,
y
∈
U with x
≠
y, R
(
x
,
y
)
∧
R
(
y
,
x
)
<
1.
(b)
Ift
R
<
1, then (
U
,
τ
R
) is sub-T
0.Proof.
(1) This holds by Theorem20.
(2) For anyx
λ
,
x
μ
∈
P
(
U
) with μ
<
λ, put u
∈
(
μ
,
λ
) and we have x
λ
∉
u
¯. By Theorem 20(4), u
¯
∈
η
-
(
x
λ
). Note that x
μ
∈
u
¯. Then (
U
,
τ
R
) is T
-
1.
(3) (a) For anyx
,
y
∈
U with x
≠
y, since (
U
,
τ
R
) is sub-T
0, then there exist λ
∈
(
0,1
] and A
∈
η
-
(
x
λ
) such that y
λ
∈
A, or there exist λ
∈
(
0,1
] and B
∈
η
-
(
y
λ
) such that x
λ
∈
B.(i)
If there existλ
∈
(
0,1
] and A
∈
η
-
(
x
λ
) such that y
λ
∈
A, then A
(
y
)
≥
λ
>
A
(
x
). By Theorem 20(3) and A
∈
(
τ
R
)
c, R
¯
(
A
)
=
A. By (
y
,
x
)
∈
R
A and Proposition 17(2),
(47)
R
(
x
,
y
)
≤
A
(
x
)
∧
A
(
y
)
=
A
(
x
)
<
λ
≤
1
.
Then R
(
x
,
y
)
<
1.
(ii)
If there existλ
∈
(
0,1
] and B
∈
η
-
(
y
λ
) such that x
λ
∈
B, then we can prove that R
(
y
,
x
)
<
1.
SoR
(
x
,
y
)
∧
R
(
y
,
x
)
<
1.
(b) Letx
,
y
∈
U with x
≠
y. By t
R
<
1, there exists λ
∈
(
t
R
,
1
]. Put
(48)
A
(
t
)
=
{
t
R
,
t
=
x
,
λ
,
t
≠
x
.
Then R
A
=
{
(
t
,
x
)
:
t
∈
U
-
{
x
}
}. For each (
t
,
x
)
∈
R
A,
(49)
R
(
x
,
t
)
≤
⋁
x
,
y
∈
U
,
x
≠
y
R
(
x
,
y
)
=
t
R
=
λ
∧
t
R
=
A
(
t
)
∧
A
(
x
)
.
By Proposition 17(2), R
¯
(
A
)
=
A. By Theorem 20(3), A
∈
(
τ
R
)
c. Note that A
(
x
)
=
t
R
<
λ. Then A
∈
η
-
(
x
λ
). Obviously, y
λ
∈
A. Hence (
U
,
τ
R
) is sub-T
0.Proposition 24.
LetR
1 and R
2 be two reflexive fuzzy relations on U. Then(1)
R
1
⊆
R
2
⇒
τ
R
2
⊆
τ
R
1.
(2)
(50)
τ
R
1
⊆
τ
R
2
⇒
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
(
x
∈
U
)
.
(3)
R
1
∪
R
2
is reflexive.
(4)
τ
R
1
∪
R
2
is an Alexandrov fuzzy topology on
U
.
(5)
τ
R
1
∪
R
2
=
τ
R
1
∩
τ
R
2
.Proof.
(1) For eachA
∈
τ
R
2, R
2
_
(
A
)
=
A. By R
1
⊆
R
2, Proposition 18(1), the reflexivity of R
1 and Theorem 9(1),
(51)
A
=
R
2
_
(
A
)
⊆
R
1
_
(
A
)
⊆
A
.
Hence A
∈
τ
R
1. Thus τ
R
2
⊆
τ
R
1.
(2) Letx
∈
U. Pick λ
=
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
).(i)
Ifλ
=
0, then
(52)
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
1
=
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
.
(ii)
Ifλ
>
0, put A
=
x
λ
∈
P
(
U
), then R
A
=
{
(
x
,
y
)
:
y
∈
U
-
{
x
}
}. For any (
x
,
y
)
∈
R
A, by
(53)
1
-
R
1
(
x
,
y
)
≥
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
=
λ
=
A
(
x
)
∨
A
(
y
)and Proposition 17(2), R
1
_
(
A
)
=
A. So A
∈
τ
1. Note that τ
R
1
⊆
τ
R
2. Then A
∈
τ
2. So R
2
_
(
A
)
=
A. For each (
x
,
y
)
∈
R
A, by Proposition 17(2),
(54)
1
-
R
2
(
x
,
y
)
≥
A
(
x
)
∨
A
(
y
)
=
λ
.
Hence(55)
1
-
⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
=
⋀
y
∈
U
-
{
x
}
(
1
-
R
2
(
x
,
y
)
)
≥
λ
=
1
-
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
)
.
So⋁
y
∈
U
-
{
x
}
R
2
(
x
,
y
)
≤
⋁
y
∈
U
-
{
x
}
R
1
(
x
,
y
).
(3) This is obvious.
(4) This holds by (3) and Theorem20(1).
(5) By (1),τ
R
1
∪
R
2
⊆
τ
R
1
∩
τ
R
2.
For eachA
∈
τ
R
1
∩
τ
R
2, we have A
=
R
1
_
(
A
) and A
=
R
2
_
(
A
).
By Proposition18(1),
(56)
A
=
R
1
_
(
A
)
∩
R
2
_
(
A
)
=
(
R
1
∪
R
2
_
)
(
A
)
.
Then A
∈
τ
R
1
∪
R
2. Thus τ
R
1
∪
R
2
⊇
τ
R
1
∩
τ
R
2.
Henceτ
R
1
∪
R
2
=
τ
R
1
∩
τ
R
2.Definition 25.
LetR be a fuzzy relation on U. R is called pseudoconstant if there exists a
∈
I such that for any x
,
y
∈
U,
(57)
R
(
x
,
y
)
=
{
1
,
x
=
y
,
a
,
x
≠
y
.
We write R by a
* or a
U
×
U
*.That is,(58)
a
*
(
x
,
y
)
=
{
1
,
x
=
y
,
a
,
x
≠
y
.Obviously, every pseudoconstant fuzzy relation is an equivalence fuzzy relation.Remark 26.
(1) For anya
,
b
∈
I, a
≤
b implies a
*
⊆
b
*.
(2) For eacha
∈
I, θ
a
*
=
τ
a
*.
(3)τ
0
*
=
F
(
U
), τ
1
*
=
{
a
¯
:
a
∈
I
}.Remark 27.
LetR be a reflexive fuzzy relation on U. Then
(59)
s
R
*
⊆
R
⊆
t
R
*
.The following theorem gives the topological structure of reflexive fuzzy approximation spaces.Theorem 28.
Let(
U
,
R
) be a reflexive fuzzy approximation space.(1)
Considerτ
R
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
∀
(
x
,
y
)
∈
R
A
,
A
(
x
)
∨
A
(
y
)
≤
1
-
R
(
x
,
y
)
}.
(2)
Considerτ
t
R
*
⊆
τ
R
⊆
τ
s
R
*.
(3)
For eacha
∈
I, τ
a
*
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
R
A
≠
∅
,
⋁
x
∈
U
A
(
x
)
≤
1
-
a
}.Proof.
(1) This holds by Proposition17(2), Theorem 20(1), and Remark 26(3).
(2) This holds by Proposition24(1) and Remark 27.
(3) This holds by (1).


## 4.2. Fuzzy Topologies Induced by Preorder Fuzzy Approximation Spaces

Theorem 29 (see [28]).
Let(
U
,
R
) be a preorder fuzzy approximation space. 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.Example 30.
LetU
=
{
x
,
y
,
z
} and let R be a fuzzy relation on U where
(60)
R
=
(
1
0
0
0.7
1
1
0
0.8
1
)
.
Then R is reflexive. Note that
(61)
R
(
z
,
y
)
∧
R
(
y
,
x
)
=
0.7
≰
0
=
R
(
z
,
x
)
.
Then R is not transitive. So R is not preorder.
Pick(62)
A
=
0
x
+
0.4
y
+
1
z
,
B
=
1
x
+
0.6
y
+
0
z
.
(1)
Since(63)
R
_
(
A
)
=
0
x
+
0.3
y
+
0.4
z
,
(64)
R
_
(
R
_
(
A
)
)
=
0
x
+
0.3
y
+
0.3
z
,
we have R
_
(
R
_
(
A
)
)
≠
R
_
(
A
). Then R
_
(
A
)
∉
τ
R. Thus
(65)
θ
R
≠
τ
R
.
(2)
Obviously,B
c
=
A. Note that R
_
(
A
)
∉
τ
R and int
⁡
τ
R
(
B
)
∈
τ
R. Then
(66)
int
⁡
τ
R
(
A
)
≠
R
_
(
A
)
,
(67)
cl
⁡
τ
R
(
B
)
≠
R
¯
(
B
)
.
Thus, R
_ is not the interior operator of τ
R and R
¯ is not the closure operator of τ
R.Example 31.
LetU
=
{
x
,
y
,
z
,
w
} and let R be the fuzzy relation on U where
(68)
R
=
(
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
)
.
Then R is neither reflexive nor transitive.
For eachA
∈
F
(
U
), since
(69)
R
_
(
A
)
=
A
(
y
)
x
+
A
(
z
)
y
+
A
(
w
)
z
+
A
(
x
)
w
=
B
=
R
¯
(
A
)
,
R
_
(
B
)
=
A
(
z
)
x
+
A
(
w
)
y
+
A
(
x
)
z
+
A
(
y
)
w
=
C
=
R
¯
(
B
)
,
R
_
(
C
)
=
A
(
w
)
x
+
A
(
x
)
y
+
A
(
y
)
z
+
A
(
z
)
w
=
D
=
R
¯
(
C
)
,
R
_
(
D
)
=
A
(
x
)
x
+
A
(
y
)
y
+
A
(
z
)
z
+
A
(
w
)
w
=
A
=
R
¯
(
D
)
,
we have R
_
(
R
_
(
R
_
(
R
_
(
A
)
)
)
)
=
A. Then θ
R
=
F
(
U
).
Obviously,θ
R is a fuzzy topology on U.
PutW
=
x
0.3
∈
P
(
U
). Then
(70)
cl
⁡
θ
R
(
W
)
=
x
0.3
≠
w
0.3
=
R
¯
(
W
)
,
int
⁡
θ
R
(
W
)
=
x
0.3
≠
w
0.3
=
R
2
_
(
W
)
.
Thus,R
_ is not the interior operator of θ
R and R
¯ is not the closure operator of θ
R.The following theorem reveals fuzzy relations can be researched by means of topology.Theorem 32.
LetR
1 and R
2 be two preorder fuzzy relations on U. Then(1)
θ
R
2
=
θ
R
1
⇔
R
1
=
R
2.
(2)
θ
R
1
∩
θ
R
2
⊆
θ
R
1
∪
R
2.Proof
(1) Necessity. Suppose thatθ
R
2
=
θ
R
1. By Remark 7 and Theorem 29(3),
(71)
R
1
(
x
,
y
)
=
R
1
¯
(
y
1
)
(
x
)
=
cl
⁡
R
τ
1
(
y
1
)
(
x
)
=
cl
⁡
R
τ
2
(
y
1
)
(
x
)
=
R
2
(
x
,
y
)
for any x
,
y
∈
U. Then R
1
=
R
2.
Sufficiency. This is obvious.
(2) By Propositions19 and 24(5),
(72)
θ
R
1
∩
θ
R
2
=
τ
R
1
∩
τ
R
2
=
τ
R
1
∪
R
2
⊆
θ
R
1
∪
R
2
.Example 33.
LetU
=
{
x
,
y
,
z
,
w
} and let R
1 and R
2 be two fuzzy relations on U, where
(73)
R
1
=
(
1
0
1
0
0
1
0
1
0
1
1
0
1
0
0
1
)
,
R
2
=
(
1
0
0
1
1
1
0
0
0
1
1
0
0
0
1
1
)
.
Then R
1 and R
2 are reflexive. Note that
(74)
R
1
(
x
,
z
)
∧
R
1
(
z
,
y
)
=
1
≰
0
=
R
1
(
x
,
y
)
,
R
2
(
x
,
w
)
∧
R
2
(
w
,
z
)
=
1
≰
0
=
R
2
(
x
,
z
)
.
Then R
1 and R
2 are not transitive. So R
1 and R
2 are not preorder.
For anyA
,
B
∈
F
(
U
), since
(75)
R
1
_
(
A
)
=
A
(
x
)
∧
A
(
z
)
x
+
A
(
y
)
∧
A
(
w
)
y
+
A
(
y
)
∧
A
(
z
)
z
+
A
(
x
)
∧
A
(
w
)
w
=
A
,
R
2
_
(
B
)
=
B
(
x
)
∧
B
(
w
)
x
+
B
(
x
)
∧
B
(
y
)
y
+
B
(
y
)
∧
B
(
z
)
z
+
B
(
z
)
∧
B
(
w
)
w
=
B
,
we have
(76)
A
(
z
)
≥
A
(
x
)
≥
A
(
w
)
≥
A
(
y
)
≥
A
(
z
)
,
B
(
w
)
≥
B
(
x
)
≥
B
(
y
)
≥
B
(
z
)
≥
B
(
w
)
.
So A
(
x
)
=
A
(
y
)
=
A
(
z
)
=
A
(
w
) and B
(
x
)
=
B
(
y
)
=
B
(
z
)
=
B
(
w
).
Thusτ
R
1
=
{
a
¯
:
a
∈
I
}
=
τ
R
2. But R
1
≠
R
2.


## 5. Fuzzy Topologies versus Fuzzy Approximation Spaces

As can be seen from Section4, a reflexive fuzzy approximation space yields a fuzzy topology. In this section, we consider the reverse problem; that is, under which conditions can a fuzzy topology be associated with a fuzzy approximation space which produces the given fuzzy topology?In what follows, we investigate fuzzy approximation spaces or fuzzy relations induced by fuzzy topologies, introduce the concept of fuzzy approximating spaces, and consider fuzzy topologies which satisfy (CC) axiom.


### 5.1.  Fuzzy Approximation Spaces Induced by Fuzzy Topologies

Definition 34.
Letτ be a fuzzy topology on U. Define a fuzzy relation R
τ on U by
(77)
R
τ
(
x
,
y
)
=
cl
⁡
τ
(
y
1
)
(
x
)
(
(
x
,
y
)
∈
U
×
U
)
.
Then R
τ is called the fuzzy relation induced by τ on U and (
U
,
R
τ
) is called the fuzzy approximation space induced by τ on U.Theorem 35.
Letτ be a fuzzy topology on U and let R
τ be the fuzzy relation induced by τ on U. Then consider the following.(1)
R
τ is reflexive.
(2)
(78)
R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
)
⊆
A
⊆
cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
)
(
A
∈
F
(
U
)
)
.Proof.
(1) For eachx
∈
U,
(79)
R
τ
(
x
,
x
)
=
cl
⁡
τ
(
x
1
)
(
x
)
≥
(
x
1
)
(
x
)
=
1
.
Then R
τ is reflexive.
(2) For eachA
∈
F
(
U
), by Remark 1 and Proposition 8(2),
(80)
cl
⁡
τ
(
A
)
=
cl
⁡
τ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
¯
∩
y
1
)
⊆
⋃
y
∈
U
‍
(
cl
⁡
τ
(
A
(
y
)
¯
)
∩
cl
⁡
τ
(
y
1
)
)
=
⋃
y
∈
U
‍
(
A
(
y
)
¯
∩
cl
⁡
τ
(
y
1
)
)
.
Then for each x
∈
U,
(81)
cl
⁡
τ
(
A
)
(
x
)
≤
⋁
y
∈
U
(
A
(
y
)
¯
(
x
)
∧
cl
⁡
τ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
τ
(
x
,
y
)
)
=
R
τ
¯
(
A
)
(
x
)
.
Thus cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
).
By Proposition8(3),
(82)
int
⁡
τ
(
A
)
=
(
cl
⁡
τ
(
A
c
)
)
c
⊇
(
R
τ
¯
(
A
c
)
)
c
=
R
τ
_
(
A
)
.
Hence
(83)
R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
)
⊆
A
⊆
cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
)
.Theorem 36.
LetR be a reflexive fuzzy relation, let τ
R be the fuzzy topology by R on U, and let R
τ
R be the fuzzy relation induced by τ
R on U. If R is transitive, then R
τ
R
=
R.Proof.
For each(
x
,
y
)
∈
U
×
U, by Remark 7, Proposition 19, and Theorem 29(3),
(84)
R
(
x
,
y
)
=
R
¯
(
y
1
)
(
x
)
=
cl
⁡
θ
R
(
y
1
)
(
x
)
=
cl
⁡
τ
R
(
y
1
)
(
x
)
.
Note that R
τ
R
(
x
,
y
)
=
cl
⁡
τ
R
(
y
1
)
(
x
). Then R
τ
R
(
x
,
y
)
=
R
(
x
,
y
). Thus
(85)
R
τ
R
=
R
.


### 5.2. (CC) Axiom

The following condition for a fuzzy topologyτ on U is called (CC) axiom:(CC)
axiom: for anyλ
∈
I and A
∈
F
(
U
),
(86)
cl
⁡
τ
(
λ
A
)
=
λ
cl
⁡
τ
(
A
)
.Theorem 37.
Letτ be a fuzzy topology on U. If τ satisfies (CC) axiom, then(1)
R
τ
¯ is the closure operator of τ.
(2)
τ is Alexandrov.Proof.
(1) For eachA
∈
F
(
U
), by Remark 1 and (CC) axiom,
(87)
cl
⁡
τ
(
A
)
=
cl
⁡
τ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
(
A
(
y
)
cl
⁡
τ
(
y
1
)
)
.
Then for each x
∈
U,
(88)
cl
⁡
τ
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
cl
⁡
τ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
τ
(
x
,
y
)
)
=
R
τ
¯
(
A
)
(
x
)
.
Thus R
τ
¯
(
A
)
=
cl
⁡
τ
(
A
). So R
τ
¯ is the closure operator of τ.
(2) For eachA
∈
F
(
U
), by (CC) axiom and Theorem 37(1), R
¯
(
A
c
)
=
c
l
τ
(
A
c
). Then R
_
(
A
)
=
int
⁡
τ
(
A
).
Let{
A
j
:
j
∈
J
}
⊆
τ. Then int
⁡
(
A
j
)
=
A
j for each j
∈
J. Note that R
_
(
A
j
)
=
int
⁡
τ
(
A
j
). Then A
j
=
R
_
(
A
j
). By Proposition 8(5),
(89)
⋂
j
∈
J
‍
A
j
=
⋂
j
∈
J
‍
R
_
(
A
j
)
=
R
_
(
⋂
j
∈
J
‍
A
j
)
=
int
⁡
τ
(
⋂
j
∈
J
‍
A
j
)
.
So ⋂
j
∈
J
A
j
∈
τ. Hence τ is Alexandrov.For any𝒜
,
ℬ
⊆
F
(
U
), define
(5.2)
𝒜
∧
ℬ
=
{
A
∩
B
:
A
∈
𝒜
,
B
∈
ℬ
}
,
𝒜
∨
ℬ
=
{
A
∪
B
:
A
∈
𝒜
,
B
∈
ℬ
}
,
∩
𝒜
=
⋂
A
∈
𝒜
‍
A
,
∪
𝒜
=
⋃
A
∈
𝒜
‍
A
.Example 38.
LetU
=
{
x
,
y
,
z
}. Put τ
=
⋃
i
=
1
5
τ
i, where
(91)
τ
1
=
{
a
¯
:
a
∈
I
}
,
τ
2
=
{
A
a
b
:
a
∈
[
0.5,0.7
]
,
b
∈
I
}
0000
with
A
a
b
=
a
x
+
0.7
y
+
b
z
,
τ
3
=
{
B
a
b
:
a
∈
[
0.5,0.7
]
,
b
∈
I
}
with
B
a
b
=
0.5
x
+
a
y
+
b
z
,
000
τ
4
=
{
C
a
b
:
a
,
b
∈
I
}
with
C
a
b
=
a
x
+
a
y
+
b
z
,
τ
5
=
{
D
a
b
c
:
0.5
≤
a
≤
b
≤
0.7
,
c
∈
I
}
00000000000
with
D
a
b
c
=
a
x
+
b
y
+
c
z
.
(1) For any𝒞
i
⊆
τ
i,
(
1
*
)
⋂
‍
𝒞
i
⊆
τ
i
,
⋃
‍
𝒞
i
⊆
τ
i
(
i
=
1,2
,
3,4
,
5
)
.
For anyd
∈
I, A
a
b
∈
τ
2 and B
a
b
∈
τ
3,
(92)
d
¯
∩
A
a
b
=
{
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
d
∈
[
0,0.5
)
,
a
x
+
d
y
+
d
∧
b
z
∈
τ
5
,
a
<
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
a
≥
d
,
d
∈
[
0.5,0.7
)
,
a
x
+
0.7
y
+
d
∧
b
z
∈
τ
2
,
d
∈
[
0.7,1
]
,
d
¯
∪
A
a
b
=
{
d
∨
a
x
+
0.7
y
+
d
∨
b
z
∈
τ
2
,
d
∈
[
0,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
d
∈
[
0.7,1
]
,
d
¯
∩
B
a
b
=
{
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
d
∈
[
0,0.5
)
,
0.5
x
+
d
∧
a
y
+
d
∧
b
z
∈
τ
3
,
d
∈
[
0.5,1
]
,
d
¯
∪
B
a
b
=
{
0.5
x
+
a
y
+
d
∨
b
z
∈
τ
3
,
d
∈
[
0,0.5
)
,
d
x
+
a
y
+
d
∨
b
z
∈
τ
5
,
a
>
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
a
≤
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
d
∈
[
0.7,1
]
.
Then
(
2
*
)
τ
1
∧
τ
2
⊆
τ
4
∪
τ
5
∪
τ
2
⊆
τ
,
τ
1
∨
τ
2
⊆
τ
2
∪
τ
4
⊆
τ
,
τ
1
∧
τ
3
⊆
τ
4
∪
τ
3
⊆
τ
,
τ
1
∨
τ
3
⊆
τ
3
∪
τ
5
∪
τ
4
⊆
τ
.
Similarly, we can prove that(
3
*
)
τ
1
∧
τ
4
⊆
τ
4
⊆
τ
,
τ
1
∨
τ
4
⊆
τ
4
⊆
τ
;
τ
1
∧
τ
5
⊆
τ
4
∪
τ
5
⊆
τ
,
τ
1
∨
τ
5
⊆
τ
5
∪
τ
4
⊆
τ
;
τ
2
∧
τ
3
⊆
τ
3
⊆
τ
,
τ
2
∨
τ
3
⊆
τ
2
⊆
τ
;
τ
2
∧
τ
4
⊆
τ
4
∪
τ
5
∪
τ
2
⊆
τ
,
τ
2
∨
τ
4
⊆
τ
2
∪
τ
4
⊆
τ
;
τ
2
∧
τ
5
⊆
τ
5
⊆
τ
,
τ
2
∨
τ
5
⊆
τ
2
⊆
τ
;
τ
3
∧
τ
4
⊆
τ
4
∪
τ
3
⊆
τ
,
τ
3
∨
τ
4
⊆
τ
3
∪
τ
5
∪
τ
4
⊆
τ
;
τ
3
∧
τ
5
⊆
τ
3
⊆
τ
,
τ
3
∨
τ
5
⊆
τ
5
⊆
τ
;
τ
4
∧
τ
5
⊆
τ
4
∪
τ
5
⊆
τ
,
τ
4
∨
τ
5
⊆
τ
5
∪
τ
4
⊆
τ
.
(2)(
i
) For each a
∈
I, a
¯
∈
τ
1
⊆
τ.
(
ii
) Let 𝒜
⊆
τ and 𝒜
≠
∅. Then there exist 𝒜
1, 𝒜
2, 𝒜
3, 𝒜
4, and 𝒜
5 such that 𝒜
=
⋃
i
=
1
5
𝒜
i and
(93)
∅
≠
𝒜
i
⊆
τ
i
or
𝒜
i
=
∅
(
i
=
1,2
,
3,4
,
5
)
.
If 𝒜
i
⊆
τ
i, then by (
1
*
),
(94)
⋂
A
∈
𝒜
i
‍
A
=
⋂
‍
𝒜
i
∈
τ
i
,
⋃
A
∈
𝒜
i
‍
A
=
⋃
‍
𝒜
i
∈
τ
i
.
By(
2
*
) and (
3
*
),
(95)
⋂
A
∈
𝒜
‍
A
=
⋂
i
=
1
5
‍
(
⋂
A
∈
𝒜
i
‍
A
)
∈
τ
,
⋃
A
∈
𝒜
‍
A
=
⋃
i
=
1
5
‍
(
⋃
A
∈
𝒜
i
‍
A
)
∈
τ
.
Thus,τ is an Alexandrov fuzzy topology on U.
(3) Putλ
=
0.4 and A
=
x
1
∈
P
(
U
). Then λ
A
=
x
0.4. Note that(i)
⋂
‍
{
B
∈
τ
1
c
:
B
⊇
x
0.4
}
=
0.4
¯, ⋂
‍
{
B
∈
τ
1
c
:
B
⊇
A
}
=
1
¯;
(ii)
⋂
‍
{
B
∈
τ
2
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
2
c
:
B
⊇
A
}
=
∅;
(iii)
⋂
‍
{
B
∈
τ
3
c
:
B
⊇
x
0.4
}
=
(
0.5
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
3
c
:
B
⊇
A
}
=
∅;
(iv)
⋂
‍
{
B
∈
τ
4
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.4
/
y
)
+
(
0
/
z
), ⋂
‍
{
B
∈
τ
4
c
:
B
⊇
A
}
=
(
1
/
x
)
+
(
1
/
y
)
+
(
0
/
z
);
(v)
⋂
‍
{
B
∈
τ
5
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
5
c
:
B
⊇
A
}
=
∅.
Then(96)
cl
⁡
τ
(
λ
A
)
=
cl
⁡
τ
(
x
0.4
)
=
0.4
x
+
0.3
y
+
0
z
,
cl
⁡
τ
(
A
)
=
1
x
+
1
y
+
0
z
.
Then(97)
cl
⁡
τ
(
λ
A
)
=
0.4
x
+
0.3
y
+
0
z
≠
0.4
x
+
0.4
y
+
0
z
=
λ
cl
⁡
τ
(
A
)
.
Thus,τ does not satisfy (CC) axiom.Theorem 39.
LetR be a preorder fuzzy relation on U. Then τ
R satisfies (CC) axiom.Proof.
For anyλ
∈
I and A
∈
F
(
U
), by Theorem 29(3) and Proposition 8(6),
(98)
cl
⁡
τ
R
(
λ
A
)
=
R
¯
(
λ
A
)
=
λ
R
¯
(
A
)
=
λ
cl
⁡
τ
R
(
A
)
.
Thus τ
R satisfies (CC) axiom.Theorem 40.
Letτ be a fuzzy topology on U, let R
τ be the fuzzy relation induced by τ on U, and let τ
R
τ be the fuzzy topology induced by R
τ on U. Then
(99)
τ
R
τ
=
τ
i
f
f
τ
s
a
t
i
s
f
i
e
s
(
CC
)
a
x
i
o
m
.Proof
Necessity. For each A
∈
F
(
U
), by Theorem 35(2), R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
). By Theorems 35(1) and 20(2),
(100)
int
⁡
τ
(
A
)
=
int
⁡
τ
R
τ
(
A
)
⊆
R
τ
_
(
A
)
.
Then
(101)
int
⁡
τ
(
A
)
=
R
τ
_
(
A
)
(
A
∈
U
)
.
Thus R
_ is the closure operator of τ.
Sufficiency. By Theorem 35(1), R
τ is reflexive. For any x
,
y
,
z
∈
U, put cl
⁡
τ
(
z
1
)
(
y
)
=
λ. By Remark 1 and Proposition 8(2),
(102)
λ
cl
⁡
τ
(
y
1
)
=
cl
⁡
τ
(
λ
y
1
)
=
cl
⁡
τ
(
cl
⁡
τ
(
z
1
)
(
y
)
y
1
)
⊆
cl
⁡
τ
(
⋃
t
∈
U
‍
(
cl
⁡
τ
(
z
1
)
(
t
)
t
1
)
)
=
cl
⁡
τ
(
cl
⁡
τ
(
z
1
)
)
=
cl
⁡
τ
(
z
1
)
.
Then
(103)
R
τ
(
x
,
y
)
∧
R
τ
(
y
,
z
)
=
cl
⁡
τ
(
y
1
)
(
x
)
∧
cl
⁡
τ
(
z
1
)
(
y
)
=
cl
⁡
τ
(
y
1
)
(
x
)
∧
λ
=
λ
∧
cl
⁡
τ
(
y
1
)
(
x
)
=
(
λ
cl
⁡
τ
(
y
1
)
)
(
x
)
≤
cl
⁡
τ
(
z
1
)
(
x
)
=
R
τ
(
x
,
z
)
.
Then R
τ is transitive.
SoR
τ is preorder. For each A
∈
F
(
U
), by Proposition 19 and Theorem 29(3)
(104)
cl
⁡
τ
R
τ
(
A
)
=
cl
⁡
θ
R
τ
(
A
)
=
R
τ
¯
(
A
)
.
By (CC) axiom and Theorem 37(1), R
τ
¯
(
A
)
=
cl
⁡
τ
(
A
). So cl
⁡
τ
R
τ
(
A
)
=
cl
⁡
τ
(
A
). Thus
(105)
τ
R
τ
=
τ
.Theorem 41.
Letτ be a fuzzy topology on U. Then the following are equivalent.(1)
τ satisfies (CC) axiom.
(2)
For anyλ
∈
I and A
∈
F
(
U
),
(106)
int
⁡
τ
(
λ
¯
∪
A
)
=
λ
¯
∪
int
⁡
τ
(
A
)
.
(3)
There exists a preorder fuzzy relationρ on U such that ρ
¯ is the closure operator of τ.
(4)
There exists a preorder fuzzy relationρ on U such that ρ
_ is the interior operator of τ.
(5)
R
τ
¯ is the closure operator of τ.
(6)
R
τ
_ is the interior operator of τ.Proof.
(1)⇔ (2). This is obvious.
(1)⇒ (3). Suppose that τ satisfies (CC) axiom. Pick ρ
=
R
τ. By Theorem 37(1), ρ
¯ is the closure operator of τ. By Theorem 16(1), ρ
¯ is preorder.
(3)⇒ (4). Let ρ
¯ be the closure operator of τ for some preorder fuzzy relation ρ on U. For each A
∈
F
(
U
), by Proposition 8(3),
(107)
ρ
_
(
A
)
=
(
ρ
¯
(
A
c
)
)
c
=
(
cl
⁡
τ
(
A
c
)
)
c
=
int
⁡
τ
(
A
)
.
Thus, ρ
_ is the interior operator of τ.
(4)⇒ (6). Let ρ
_ be the interior operator of τ for some preorder fuzzy relation ρ on U.
For(
x
,
y
)
∈
U
×
U, by Remark 7,
(108)
ρ
(
x
,
y
)
=
1
-
ρ
_
(
(
y
1
)
c
)
(
x
)
=
1
-
int
⁡
τ
(
(
y
1
)
c
)
(
x
)
=
cl
⁡
τ
(
y
1
)
(
x
)
=
R
τ
(
x
,
y
)
.
Then ρ
=
R
τ. Note that ρ
_ is the interior operator of τ. Then R
τ
_ is the interior operator of τ.
(6)⇒ (5) holds by Proposition 8(3).
(5)⇒ (1). For any a
∈
I and A
∈
F
(
U
), by Proposition 8(6),
(109)
cl
⁡
τ
(
λ
A
)
=
R
τ
¯
(
λ
A
)
=
λ
R
τ
¯
(
A
)
=
λ
cl
⁡
τ
(
A
)
.
Thus τ satisfies (CC) axiom.The following theorem reveals that we can research fuzzy relations or fuzzy approximation spaces by means topology and vice versa.Theorem 42.
Let(110)
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
=
{
τ
:
τ
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.
Two mappingsf
:
Σ
→
Γ and g
:
Γ
→
Σ are defined as follows:
(111)
f
(
R
)
=
τ
R
(
R
∈
Σ
)
,
g
(
τ
)
=
R
τ
(
τ
∈
Γ
)
.
By Theorem36,
(112)
g
∘
f
=
i
Σ
,
where g
∘
f is the composition of f and g and i
Σ is the identity mapping on Γ.
By Theorem40,
(113)
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 prove that there exists a one-to-one correspondence between Σ and Γ.


### 5.3. Fuzzy Approximating Spaces

Below, we introduce the concept of fuzzy approximating spaces, which enable us to study fuzzy relations or fuzzy approximation spaces by means of topology.Definition 43.
Let(
U
,
σ
) be a fuzzy topological space. If there exists a reflexive fuzzy relation R on U such that τ
R
=
σ, then (
U
,
σ
) is called a fuzzy approximating space.Theorem 44.
Let(
U
,
τ
) be a fuzzy topological space. If one of the following conditions satisfies, then (
U
,
τ
) is a fuzzy approximating space.(1)
τ satisfies (CC) axiom.
(2)
For anyλ
∈
I and A
∈
F
(
U
),
(114)
int
⁡
τ
(
λ
¯
∪
A
)
=
λ
¯
∪
int
⁡
τ
(
A
)
.
(3)
There exists a preorder fuzzy relationR on U such that R
¯ is the closure operator of τ.
(4)
There exists a preorder fuzzy relationR on U such that R
_ is the interior operator of τ.
(5)
R
τ
¯ is the closure operator of τ.
(6)
R
τ
_ is the interior operator of τ.Proof.
These hold by Theorems40 and 41.By Remark26, Theorems 39 and 44, we have the following example.Example 45.
{
a
¯
:
a
∈
I
} is a fuzzy approximating space.Theorem 46.
Let(
U
,
τ
) be a fuzzy topological space. Then (
U
,
τ
) is a fuzzy approximating space if and only if there exist a reflexive fuzzy relation R such that
(115)
τ
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
∀
(
x
,
y
)
∈
R
A
,
A
c
(
x
)
∧
A
c
(
y
)
≥
R
(
x
,
y
)
}
.Proof.
This holds by Proposition17(2), Theorems 20(1) and 28.


## 5.1.  Fuzzy Approximation Spaces Induced by Fuzzy Topologies

Definition 34.
Letτ be a fuzzy topology on U. Define a fuzzy relation R
τ on U by
(77)
R
τ
(
x
,
y
)
=
cl
⁡
τ
(
y
1
)
(
x
)
(
(
x
,
y
)
∈
U
×
U
)
.
Then R
τ is called the fuzzy relation induced by τ on U and (
U
,
R
τ
) is called the fuzzy approximation space induced by τ on U.Theorem 35.
Letτ be a fuzzy topology on U and let R
τ be the fuzzy relation induced by τ on U. Then consider the following.(1)
R
τ is reflexive.
(2)
(78)
R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
)
⊆
A
⊆
cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
)
(
A
∈
F
(
U
)
)
.Proof.
(1) For eachx
∈
U,
(79)
R
τ
(
x
,
x
)
=
cl
⁡
τ
(
x
1
)
(
x
)
≥
(
x
1
)
(
x
)
=
1
.
Then R
τ is reflexive.
(2) For eachA
∈
F
(
U
), by Remark 1 and Proposition 8(2),
(80)
cl
⁡
τ
(
A
)
=
cl
⁡
τ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
¯
∩
y
1
)
⊆
⋃
y
∈
U
‍
(
cl
⁡
τ
(
A
(
y
)
¯
)
∩
cl
⁡
τ
(
y
1
)
)
=
⋃
y
∈
U
‍
(
A
(
y
)
¯
∩
cl
⁡
τ
(
y
1
)
)
.
Then for each x
∈
U,
(81)
cl
⁡
τ
(
A
)
(
x
)
≤
⋁
y
∈
U
(
A
(
y
)
¯
(
x
)
∧
cl
⁡
τ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
τ
(
x
,
y
)
)
=
R
τ
¯
(
A
)
(
x
)
.
Thus cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
).
By Proposition8(3),
(82)
int
⁡
τ
(
A
)
=
(
cl
⁡
τ
(
A
c
)
)
c
⊇
(
R
τ
¯
(
A
c
)
)
c
=
R
τ
_
(
A
)
.
Hence
(83)
R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
)
⊆
A
⊆
cl
⁡
τ
(
A
)
⊆
R
τ
¯
(
A
)
.Theorem 36.
LetR be a reflexive fuzzy relation, let τ
R be the fuzzy topology by R on U, and let R
τ
R be the fuzzy relation induced by τ
R on U. If R is transitive, then R
τ
R
=
R.Proof.
For each(
x
,
y
)
∈
U
×
U, by Remark 7, Proposition 19, and Theorem 29(3),
(84)
R
(
x
,
y
)
=
R
¯
(
y
1
)
(
x
)
=
cl
⁡
θ
R
(
y
1
)
(
x
)
=
cl
⁡
τ
R
(
y
1
)
(
x
)
.
Note that R
τ
R
(
x
,
y
)
=
cl
⁡
τ
R
(
y
1
)
(
x
). Then R
τ
R
(
x
,
y
)
=
R
(
x
,
y
). Thus
(85)
R
τ
R
=
R
.


## 5.2. (CC) Axiom

The following condition for a fuzzy topologyτ on U is called (CC) axiom:(CC)
axiom: for anyλ
∈
I and A
∈
F
(
U
),
(86)
cl
⁡
τ
(
λ
A
)
=
λ
cl
⁡
τ
(
A
)
.Theorem 37.
Letτ be a fuzzy topology on U. If τ satisfies (CC) axiom, then(1)
R
τ
¯ is the closure operator of τ.
(2)
τ is Alexandrov.Proof.
(1) For eachA
∈
F
(
U
), by Remark 1 and (CC) axiom,
(87)
cl
⁡
τ
(
A
)
=
cl
⁡
τ
(
⋃
y
∈
U
‍
(
A
(
y
)
y
1
)
)
=
⋃
y
∈
U
‍
cl
⁡
τ
(
A
(
y
)
y
1
)
=
⋃
y
∈
U
‍
(
A
(
y
)
cl
⁡
τ
(
y
1
)
)
.
Then for each x
∈
U,
(88)
cl
⁡
τ
(
A
)
(
x
)
=
⋁
y
∈
U
(
A
(
y
)
(
x
)
∧
cl
⁡
τ
(
y
1
)
(
x
)
)
=
⋁
y
∈
U
(
A
(
y
)
∧
R
τ
(
x
,
y
)
)
=
R
τ
¯
(
A
)
(
x
)
.
Thus R
τ
¯
(
A
)
=
cl
⁡
τ
(
A
). So R
τ
¯ is the closure operator of τ.
(2) For eachA
∈
F
(
U
), by (CC) axiom and Theorem 37(1), R
¯
(
A
c
)
=
c
l
τ
(
A
c
). Then R
_
(
A
)
=
int
⁡
τ
(
A
).
Let{
A
j
:
j
∈
J
}
⊆
τ. Then int
⁡
(
A
j
)
=
A
j for each j
∈
J. Note that R
_
(
A
j
)
=
int
⁡
τ
(
A
j
). Then A
j
=
R
_
(
A
j
). By Proposition 8(5),
(89)
⋂
j
∈
J
‍
A
j
=
⋂
j
∈
J
‍
R
_
(
A
j
)
=
R
_
(
⋂
j
∈
J
‍
A
j
)
=
int
⁡
τ
(
⋂
j
∈
J
‍
A
j
)
.
So ⋂
j
∈
J
A
j
∈
τ. Hence τ is Alexandrov.For any𝒜
,
ℬ
⊆
F
(
U
), define
(5.2)
𝒜
∧
ℬ
=
{
A
∩
B
:
A
∈
𝒜
,
B
∈
ℬ
}
,
𝒜
∨
ℬ
=
{
A
∪
B
:
A
∈
𝒜
,
B
∈
ℬ
}
,
∩
𝒜
=
⋂
A
∈
𝒜
‍
A
,
∪
𝒜
=
⋃
A
∈
𝒜
‍
A
.Example 38.
LetU
=
{
x
,
y
,
z
}. Put τ
=
⋃
i
=
1
5
τ
i, where
(91)
τ
1
=
{
a
¯
:
a
∈
I
}
,
τ
2
=
{
A
a
b
:
a
∈
[
0.5,0.7
]
,
b
∈
I
}
0000
with
A
a
b
=
a
x
+
0.7
y
+
b
z
,
τ
3
=
{
B
a
b
:
a
∈
[
0.5,0.7
]
,
b
∈
I
}
with
B
a
b
=
0.5
x
+
a
y
+
b
z
,
000
τ
4
=
{
C
a
b
:
a
,
b
∈
I
}
with
C
a
b
=
a
x
+
a
y
+
b
z
,
τ
5
=
{
D
a
b
c
:
0.5
≤
a
≤
b
≤
0.7
,
c
∈
I
}
00000000000
with
D
a
b
c
=
a
x
+
b
y
+
c
z
.
(1) For any𝒞
i
⊆
τ
i,
(
1
*
)
⋂
‍
𝒞
i
⊆
τ
i
,
⋃
‍
𝒞
i
⊆
τ
i
(
i
=
1,2
,
3,4
,
5
)
.
For anyd
∈
I, A
a
b
∈
τ
2 and B
a
b
∈
τ
3,
(92)
d
¯
∩
A
a
b
=
{
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
d
∈
[
0,0.5
)
,
a
x
+
d
y
+
d
∧
b
z
∈
τ
5
,
a
<
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
a
≥
d
,
d
∈
[
0.5,0.7
)
,
a
x
+
0.7
y
+
d
∧
b
z
∈
τ
2
,
d
∈
[
0.7,1
]
,
d
¯
∪
A
a
b
=
{
d
∨
a
x
+
0.7
y
+
d
∨
b
z
∈
τ
2
,
d
∈
[
0,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
d
∈
[
0.7,1
]
,
d
¯
∩
B
a
b
=
{
d
x
+
d
y
+
d
∧
b
z
∈
τ
4
,
d
∈
[
0,0.5
)
,
0.5
x
+
d
∧
a
y
+
d
∧
b
z
∈
τ
3
,
d
∈
[
0.5,1
]
,
d
¯
∪
B
a
b
=
{
0.5
x
+
a
y
+
d
∨
b
z
∈
τ
3
,
d
∈
[
0,0.5
)
,
d
x
+
a
y
+
d
∨
b
z
∈
τ
5
,
a
>
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
a
≤
d
,
d
∈
[
0.5,0.7
)
,
d
x
+
d
y
+
d
∨
b
z
∈
τ
4
,
d
∈
[
0.7,1
]
.
Then
(
2
*
)
τ
1
∧
τ
2
⊆
τ
4
∪
τ
5
∪
τ
2
⊆
τ
,
τ
1
∨
τ
2
⊆
τ
2
∪
τ
4
⊆
τ
,
τ
1
∧
τ
3
⊆
τ
4
∪
τ
3
⊆
τ
,
τ
1
∨
τ
3
⊆
τ
3
∪
τ
5
∪
τ
4
⊆
τ
.
Similarly, we can prove that(
3
*
)
τ
1
∧
τ
4
⊆
τ
4
⊆
τ
,
τ
1
∨
τ
4
⊆
τ
4
⊆
τ
;
τ
1
∧
τ
5
⊆
τ
4
∪
τ
5
⊆
τ
,
τ
1
∨
τ
5
⊆
τ
5
∪
τ
4
⊆
τ
;
τ
2
∧
τ
3
⊆
τ
3
⊆
τ
,
τ
2
∨
τ
3
⊆
τ
2
⊆
τ
;
τ
2
∧
τ
4
⊆
τ
4
∪
τ
5
∪
τ
2
⊆
τ
,
τ
2
∨
τ
4
⊆
τ
2
∪
τ
4
⊆
τ
;
τ
2
∧
τ
5
⊆
τ
5
⊆
τ
,
τ
2
∨
τ
5
⊆
τ
2
⊆
τ
;
τ
3
∧
τ
4
⊆
τ
4
∪
τ
3
⊆
τ
,
τ
3
∨
τ
4
⊆
τ
3
∪
τ
5
∪
τ
4
⊆
τ
;
τ
3
∧
τ
5
⊆
τ
3
⊆
τ
,
τ
3
∨
τ
5
⊆
τ
5
⊆
τ
;
τ
4
∧
τ
5
⊆
τ
4
∪
τ
5
⊆
τ
,
τ
4
∨
τ
5
⊆
τ
5
∪
τ
4
⊆
τ
.
(2)(
i
) For each a
∈
I, a
¯
∈
τ
1
⊆
τ.
(
ii
) Let 𝒜
⊆
τ and 𝒜
≠
∅. Then there exist 𝒜
1, 𝒜
2, 𝒜
3, 𝒜
4, and 𝒜
5 such that 𝒜
=
⋃
i
=
1
5
𝒜
i and
(93)
∅
≠
𝒜
i
⊆
τ
i
or
𝒜
i
=
∅
(
i
=
1,2
,
3,4
,
5
)
.
If 𝒜
i
⊆
τ
i, then by (
1
*
),
(94)
⋂
A
∈
𝒜
i
‍
A
=
⋂
‍
𝒜
i
∈
τ
i
,
⋃
A
∈
𝒜
i
‍
A
=
⋃
‍
𝒜
i
∈
τ
i
.
By(
2
*
) and (
3
*
),
(95)
⋂
A
∈
𝒜
‍
A
=
⋂
i
=
1
5
‍
(
⋂
A
∈
𝒜
i
‍
A
)
∈
τ
,
⋃
A
∈
𝒜
‍
A
=
⋃
i
=
1
5
‍
(
⋃
A
∈
𝒜
i
‍
A
)
∈
τ
.
Thus,τ is an Alexandrov fuzzy topology on U.
(3) Putλ
=
0.4 and A
=
x
1
∈
P
(
U
). Then λ
A
=
x
0.4. Note that(i)
⋂
‍
{
B
∈
τ
1
c
:
B
⊇
x
0.4
}
=
0.4
¯, ⋂
‍
{
B
∈
τ
1
c
:
B
⊇
A
}
=
1
¯;
(ii)
⋂
‍
{
B
∈
τ
2
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
2
c
:
B
⊇
A
}
=
∅;
(iii)
⋂
‍
{
B
∈
τ
3
c
:
B
⊇
x
0.4
}
=
(
0.5
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
3
c
:
B
⊇
A
}
=
∅;
(iv)
⋂
‍
{
B
∈
τ
4
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.4
/
y
)
+
(
0
/
z
), ⋂
‍
{
B
∈
τ
4
c
:
B
⊇
A
}
=
(
1
/
x
)
+
(
1
/
y
)
+
(
0
/
z
);
(v)
⋂
‍
{
B
∈
τ
5
c
:
B
⊇
x
0.4
}
=
(
0.4
/
x
)
+
(
0.3
/
y
)
+
(
0
/
z
), {
B
∈
τ
5
c
:
B
⊇
A
}
=
∅.
Then(96)
cl
⁡
τ
(
λ
A
)
=
cl
⁡
τ
(
x
0.4
)
=
0.4
x
+
0.3
y
+
0
z
,
cl
⁡
τ
(
A
)
=
1
x
+
1
y
+
0
z
.
Then(97)
cl
⁡
τ
(
λ
A
)
=
0.4
x
+
0.3
y
+
0
z
≠
0.4
x
+
0.4
y
+
0
z
=
λ
cl
⁡
τ
(
A
)
.
Thus,τ does not satisfy (CC) axiom.Theorem 39.
LetR be a preorder fuzzy relation on U. Then τ
R satisfies (CC) axiom.Proof.
For anyλ
∈
I and A
∈
F
(
U
), by Theorem 29(3) and Proposition 8(6),
(98)
cl
⁡
τ
R
(
λ
A
)
=
R
¯
(
λ
A
)
=
λ
R
¯
(
A
)
=
λ
cl
⁡
τ
R
(
A
)
.
Thus τ
R satisfies (CC) axiom.Theorem 40.
Letτ be a fuzzy topology on U, let R
τ be the fuzzy relation induced by τ on U, and let τ
R
τ be the fuzzy topology induced by R
τ on U. Then
(99)
τ
R
τ
=
τ
i
f
f
τ
s
a
t
i
s
f
i
e
s
(
CC
)
a
x
i
o
m
.Proof
Necessity. For each A
∈
F
(
U
), by Theorem 35(2), R
τ
_
(
A
)
⊆
int
⁡
τ
(
A
). By Theorems 35(1) and 20(2),
(100)
int
⁡
τ
(
A
)
=
int
⁡
τ
R
τ
(
A
)
⊆
R
τ
_
(
A
)
.
Then
(101)
int
⁡
τ
(
A
)
=
R
τ
_
(
A
)
(
A
∈
U
)
.
Thus R
_ is the closure operator of τ.
Sufficiency. By Theorem 35(1), R
τ is reflexive. For any x
,
y
,
z
∈
U, put cl
⁡
τ
(
z
1
)
(
y
)
=
λ. By Remark 1 and Proposition 8(2),
(102)
λ
cl
⁡
τ
(
y
1
)
=
cl
⁡
τ
(
λ
y
1
)
=
cl
⁡
τ
(
cl
⁡
τ
(
z
1
)
(
y
)
y
1
)
⊆
cl
⁡
τ
(
⋃
t
∈
U
‍
(
cl
⁡
τ
(
z
1
)
(
t
)
t
1
)
)
=
cl
⁡
τ
(
cl
⁡
τ
(
z
1
)
)
=
cl
⁡
τ
(
z
1
)
.
Then
(103)
R
τ
(
x
,
y
)
∧
R
τ
(
y
,
z
)
=
cl
⁡
τ
(
y
1
)
(
x
)
∧
cl
⁡
τ
(
z
1
)
(
y
)
=
cl
⁡
τ
(
y
1
)
(
x
)
∧
λ
=
λ
∧
cl
⁡
τ
(
y
1
)
(
x
)
=
(
λ
cl
⁡
τ
(
y
1
)
)
(
x
)
≤
cl
⁡
τ
(
z
1
)
(
x
)
=
R
τ
(
x
,
z
)
.
Then R
τ is transitive.
SoR
τ is preorder. For each A
∈
F
(
U
), by Proposition 19 and Theorem 29(3)
(104)
cl
⁡
τ
R
τ
(
A
)
=
cl
⁡
θ
R
τ
(
A
)
=
R
τ
¯
(
A
)
.
By (CC) axiom and Theorem 37(1), R
τ
¯
(
A
)
=
cl
⁡
τ
(
A
). So cl
⁡
τ
R
τ
(
A
)
=
cl
⁡
τ
(
A
). Thus
(105)
τ
R
τ
=
τ
.Theorem 41.
Letτ be a fuzzy topology on U. Then the following are equivalent.(1)
τ satisfies (CC) axiom.
(2)
For anyλ
∈
I and A
∈
F
(
U
),
(106)
int
⁡
τ
(
λ
¯
∪
A
)
=
λ
¯
∪
int
⁡
τ
(
A
)
.
(3)
There exists a preorder fuzzy relationρ on U such that ρ
¯ is the closure operator of τ.
(4)
There exists a preorder fuzzy relationρ on U such that ρ
_ is the interior operator of τ.
(5)
R
τ
¯ is the closure operator of τ.
(6)
R
τ
_ is the interior operator of τ.Proof.
(1)⇔ (2). This is obvious.
(1)⇒ (3). Suppose that τ satisfies (CC) axiom. Pick ρ
=
R
τ. By Theorem 37(1), ρ
¯ is the closure operator of τ. By Theorem 16(1), ρ
¯ is preorder.
(3)⇒ (4). Let ρ
¯ be the closure operator of τ for some preorder fuzzy relation ρ on U. For each A
∈
F
(
U
), by Proposition 8(3),
(107)
ρ
_
(
A
)
=
(
ρ
¯
(
A
c
)
)
c
=
(
cl
⁡
τ
(
A
c
)
)
c
=
int
⁡
τ
(
A
)
.
Thus, ρ
_ is the interior operator of τ.
(4)⇒ (6). Let ρ
_ be the interior operator of τ for some preorder fuzzy relation ρ on U.
For(
x
,
y
)
∈
U
×
U, by Remark 7,
(108)
ρ
(
x
,
y
)
=
1
-
ρ
_
(
(
y
1
)
c
)
(
x
)
=
1
-
int
⁡
τ
(
(
y
1
)
c
)
(
x
)
=
cl
⁡
τ
(
y
1
)
(
x
)
=
R
τ
(
x
,
y
)
.
Then ρ
=
R
τ. Note that ρ
_ is the interior operator of τ. Then R
τ
_ is the interior operator of τ.
(6)⇒ (5) holds by Proposition 8(3).
(5)⇒ (1). For any a
∈
I and A
∈
F
(
U
), by Proposition 8(6),
(109)
cl
⁡
τ
(
λ
A
)
=
R
τ
¯
(
λ
A
)
=
λ
R
τ
¯
(
A
)
=
λ
cl
⁡
τ
(
A
)
.
Thus τ satisfies (CC) axiom.The following theorem reveals that we can research fuzzy relations or fuzzy approximation spaces by means topology and vice versa.Theorem 42.
Let(110)
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
=
{
τ
:
τ
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.
Two mappingsf
:
Σ
→
Γ and g
:
Γ
→
Σ are defined as follows:
(111)
f
(
R
)
=
τ
R
(
R
∈
Σ
)
,
g
(
τ
)
=
R
τ
(
τ
∈
Γ
)
.
By Theorem36,
(112)
g
∘
f
=
i
Σ
,
where g
∘
f is the composition of f and g and i
Σ is the identity mapping on Γ.
By Theorem40,
(113)
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 prove that there exists a one-to-one correspondence between Σ and Γ.


## 5.3. Fuzzy Approximating Spaces

Below, we introduce the concept of fuzzy approximating spaces, which enable us to study fuzzy relations or fuzzy approximation spaces by means of topology.Definition 43.
Let(
U
,
σ
) be a fuzzy topological space. If there exists a reflexive fuzzy relation R on U such that τ
R
=
σ, then (
U
,
σ
) is called a fuzzy approximating space.Theorem 44.
Let(
U
,
τ
) be a fuzzy topological space. If one of the following conditions satisfies, then (
U
,
τ
) is a fuzzy approximating space.(1)
τ satisfies (CC) axiom.
(2)
For anyλ
∈
I and A
∈
F
(
U
),
(114)
int
⁡
τ
(
λ
¯
∪
A
)
=
λ
¯
∪
int
⁡
τ
(
A
)
.
(3)
There exists a preorder fuzzy relationR on U such that R
¯ is the closure operator of τ.
(4)
There exists a preorder fuzzy relationR on U such that R
_ is the interior operator of τ.
(5)
R
τ
¯ is the closure operator of τ.
(6)
R
τ
_ is the interior operator of τ.Proof.
These hold by Theorems40 and 41.By Remark26, Theorems 39 and 44, we have the following example.Example 45.
{
a
¯
:
a
∈
I
} is a fuzzy approximating space.Theorem 46.
Let(
U
,
τ
) be a fuzzy topological space. Then (
U
,
τ
) is a fuzzy approximating space if and only if there exist a reflexive fuzzy relation R such that
(115)
τ
=
τ
1
*
∪
{
A
∈
F
(
U
)
:
∀
(
x
,
y
)
∈
R
A
,
A
c
(
x
)
∧
A
c
(
y
)
≥
R
(
x
,
y
)
}
.Proof.
This holds by Proposition17(2), Theorems 20(1) and 28.


## 6. Fuzzy Pseudoclosure Operators

In this section, we will define the concept of fuzzy pseudo-closure operators induced by a fuzzy topology and investigate their properties.For anyλ
∈
I
,
X
∈
2
U, we define
(116)
(
λ
X
)
(
x
)
=
{
λ
,
x
∈
X
,
0
,
x
∈
U
-
X
.Denote(117)
E
(
U
)
=
{
λ
X
:
λ
∈
I
,
X
∈
2
U
}
.Obviously,E
(
U
)
⊆
F
(
U
).Definition 47.
Letτ be a fuzzy topology on U. Define
(118)
S
τ
(
A
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
A
λ
)
(
A
∈
F
(
U
)
)
.
Then S
τ
:
F
(
U
)
→
F
(
U
) is called the fuzzy pseudo-closure operator induced by τ.Theorems48(5) and 49 below illustrate the meaning on fuzzy pseudoclosure operators.Theorem 48.
Letτ be a fuzzy topology on U and let S
τ be the fuzzy pseudo-closure operator induced τ. Then for any A
,
B
∈
F
(
U
), consider the following.(1)
ConsiderS
τ
(
0
¯
)
=
0
¯.
(2)
ConsiderA
⊆
S
τ
(
A
)
⊆
cl
⁡
τ
(
A
).
(3)
ConsiderS
τ
(
A
∪
B
)
⊇
S
τ
(
A
)
∪
S
τ
(
B
). Consider S
τ
(
A
∩
B
)
⊆
S
τ
(
A
)
∩
S
τ
(
B
).
(4)
ConsiderA
∈
τ
c
⇒
S
τ
(
A
)
=
A.
(5)
ConsiderS
τ coincides with cl
⁡
τ as operators from E
(
U
) to F
(
U
).Proof.
(1) For anyλ
∈
I and x
∈
U, since
(119)
(
λ
0
¯
λ
)
(
x
)
=
λ
∧
0
¯
λ
(
x
)
=
{
0
∧
1
=
0
,
λ
=
0
,
λ
∧
0
=
0
,
λ
∈
(
0,1
]
.
We have λ
0
¯
λ
=
0
¯. Thus
(120)
S
τ
(
0
¯
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
0
¯
λ
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
0
¯
)
=
0
¯
.
(2) ByA
=
⋃
λ
∈
I
λ
A
λ,
(121)
A
=
⋃
λ
∈
I
‍
λ
A
λ
⊆
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
A
λ
)
=
S
τ
(
A
)
,
S
τ
(
A
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
A
λ
)
⊆
cl
⁡
τ
(
⋃
λ
∈
I
‍
λ
A
λ
)
=
cl
⁡
τ
(
A
)
.
(3) For anyA
,
B
∈
F
(
U
), λ
∈
I, and x
∈
U, put
(122)
C
(
x
)
=
{
1
,
x
∈
A
λ
,
0
,
x
∈
U
-
A
λ
,
D
(
x
)
=
{
1
,
x
∈
B
λ
,
0
,
x
∈
U
-
B
λ
.
Obviously, λ
A
λ
=
λ
¯
∩
C
,
λ
B
λ
=
λ
¯
∩
D. Then
(123)
λ
(
A
λ
∪
B
λ
)
=
λ
¯
∩
(
C
∪
D
)
,
λ
(
A
λ
∩
B
λ
)
=
λ
¯
∩
(
C
∩
D
)
.
We can easily prove that
(124)
(
A
∪
B
)
λ
⊇
A
λ
∪
B
λ
,
(
A
∩
B
)
λ
=
A
λ
∩
B
λ
.
Thus
(125)
S
τ
(
A
∪
B
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
(
A
∪
B
)
λ
)
⊇
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
(
A
λ
∪
B
λ
)
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
(
C
∪
D
)
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
(
λ
¯
∩
C
)
∪
(
λ
¯
∩
D
)
)
=
⋃
λ
∈
I
‍
(
cl
⁡
τ
(
λ
¯
∩
C
)
∪
cl
⁡
τ
(
λ
¯
∩
D
)
)
=
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
C
)
)
∪
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
D
)
)
=
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
A
λ
)
)
∪
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
B
λ
)
)
=
S
τ
(
A
)
∪
S
τ
(
B
)
,
S
τ
(
A
∩
B
)
=
⋂
λ
∈
I
‍
cl
⁡
τ
(
λ
(
A
∩
B
)
λ
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
(
A
λ
∩
B
λ
)
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
(
C
∩
D
)
)
=
⋃
λ
∈
I
‍
cl
⁡
τ
(
(
λ
¯
∩
C
)
∩
(
λ
¯
∩
D
)
)
⊆
⋃
λ
∈
I
‍
(
cl
⁡
τ
(
λ
¯
∩
C
)
∩
cl
⁡
τ
(
λ
¯
∩
D
)
)
⊆
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
C
)
)
∩
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
¯
∩
D
)
)
=
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
A
λ
)
)
∩
(
⋃
λ
∈
I
‍
cl
⁡
τ
(
λ
B
λ
)
)
=
S
τ
(
A
)
∩
S
τ
(
B
)
.
(4) By (2),(126)
cl
⁡
τ
(
A
)
⊆
S
(
cl
⁡
τ
(
A
)
)
⊆
cl
⁡
τ
(
cl
⁡
τ
(
A
)
)
=
cl
⁡
τ
(
A
)
.
Then(127)
S
τ
(
A
)
=
S
τ
(
cl
⁡
τ
(
A
)
)
=
cl
⁡
τ
(
A
)
=
A
.
(5) LetA
∈
E
(
U
). Then there exist λ
∈
I and X
∈
2
U such that A
=
λ
X.
(i) Ifλ
≠
0, then for each x
∈
U,
(128)
A
λ
(
x
)
=
(
λ
X
)
λ
(
x
)
=
{
1
,
(
λ
X
)
(
x
)
≥
λ
0
,
(
λ
X
)
(
x
)
<
λ
=
{
1
,
x
∈
X
,
0
,
x
∈
U
-
X
.
Thus A
λ
=
X. So
(129)
S
τ
(
A
)
=
⋃
μ
∈
I
‍
cl
⁡
τ
(
μ
A
μ
)
⊇
cl
⁡
τ
(
λ
A
λ
)
=
cl
⁡
τ
(
λ
X
)
=
cl
⁡
τ
(
A
)
.
Note thatS
τ
(
A
)
⊆
cl
⁡
(
A
) by (2). Then S
τ
(
A
)
=
cl
⁡
τ
(
A
).
(ii) Ifλ
=
0, then A
=
0
¯. By (1), S
τ
(
0
¯
)
=
0
¯.
Thus(130)
S
τ
(
A
)
=
cl
⁡
τ
(
A
)
(
A
∈
E
(
U
)
)
.
Hence(131)
S
τ
coincides
with
cl
⁡
τ
as
operators
from
E
(
U
)
to
F
(
U
)
.Theorem 49.
Let(
U
,
R
) be a reflexive fuzzy approximation space. Then
(132)
S
τ
R
(
A
)
=
R
¯
(
A
)
(
A
∈
E
(
U
)
)
,
where τ
R is the fuzzy topology induced by (
U
,
R
).Proof.
For eachA
∈
E
(
U
), by Proposition 19 and Theorem 29(3),
(133)
cl
⁡
τ
R
(
A
)
=
cl
⁡
θ
R
(
A
)
=
R
¯
(
A
)
.
By Theorem 48(5), S
τ
R
(
A
)
=
cl
⁡
τ
R
(
A
). Thus
(134)
S
τ
R
(
A
)
=
R
¯
(
A
)
.


## 7. Conclusions

In this paper, relationships between fuzzy approximation spaces and fuzzy topology were discussed. We proved 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. We presented the concept of pseudoconstant fuzzy relations and gave topological structures of fuzzy approximation spaces by using them. We introduced the concept of fuzzy approximating spaces and obtained decision conditions that a fuzzy topological space is a fuzzy approximating space.The results of this paper may have at least the following functions. Fuzzy relations can be researched by means of topology or lower and upper sets (see Theorems15, 16, and 32). The concept of fuzzy approximating spaces and 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 satisfying (CC) axiom enable us to study fuzzy relations or fuzzy approximation spaces by means of topology (see Theorem 42 and Definition 43). Separation axioms can provide means to distinguish various fuzzy rough sets and give more local information or knowledge (see Theorem 23). We hope that one can find applications of topological structures of fuzzy approximation spaces for real life problems.



---

*Source: 546372-2014-01-29.xml*