# On IVF Approximating Spaces

Authors: Dingwei Zheng; Rongchen Cui; Zhaowen Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce the concept of IVF approximating spaces and obtain decision conditions that every IVF topological space is an IVF approximating space.


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## Body

## 1. Introduction

Rough set theory was proposed by Pawlak [1] as a mathematical tool to handle imprecision and uncertainty in data analysis. It 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]).As a generalization of Zadeh’s fuzzy set, interval-valued fuzzy (IVF, for short) sets were introduced by Gorzałczany [6] and Türksen [7]. Mondal and Samanta [8] defined topology of IVF sets and studied their properties.By replacing crisp relations with IVF relations, Sun et al. [9] introduced IVF rough sets based on an IVF approximation space, defined IVF information systems, and discussed their attribute reduction. Gong et al. [10] presented IVF rough sets based on approximation spaces and studied the knowledge discovery in IVF information systems.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.The purpose of this paper is to investigate IVF approximating space, that is, a particular type of IVF topological spaces where the given IVF topology coincides with the IVF topology induced by some reflexive IVF relation.


## 2. Preliminaries

Throughout this paper, “interval-valued fuzzy” is denoted briefly by “IVF.”U denotes a nonempty set called the universe. I denotes [0,1], and [I] denotes {[a,b]:a,b∈Ianda≤b}. F(i)(U) denotes the family of all IVF sets in U. a- denotes [a,a] for each a∈[0,1].For any[aj,bj]∈[I] (j=1,2), we define
(1)[a1,b1]=[a2,b2]⟺a1=a2,b1=b2,[a1,b1]≤[a2,b2]⟺a1≤a2,b1≤b2,[a1,b1]<[a2,b2]⟺[a1,b1]≤[a2,b2],[a1,b1]≠[a2,b2],1--[a1,b1]or[a1,b1]c=[1-b1,1-a1].Obviously,([a,b]c)c=[a,b] for each [a,b]∈[I].Definition 1 (see [6, 7]).
For each{[aj,bj]:j∈J}⊆[I], one define
(2)⋁j∈J[aj,bj]=[⋁j∈Jaj,⋁j∈Jbj],⋀j∈J[aj,bj]=[⋀j∈Jaj,⋀j∈Jbj],
where ⋁j∈Jaj=sup{aj:j∈J}and⋀j∈Jaj=inf{aj:j∈J}.Definition 2 (see [6, 7]).
An IVF setA in U is defined by a mapping A:U→[I].
Denote(3)A(x)=[A-(x),A+(x)](x∈U).
Then A-(x) (resp., A+(x)) is called the lower (resp., upper) degree at which x belongs to A. A- (resp., A+ ) is called the lower (resp., upper) IVF set of A.The set of all IVF sets inU is denoted by F(i)(U).Leta,b∈I. [a,b]~ represents the IVF set which satisfies [a,b]~(x)=[a,b] for each x∈U. We denoted [a,a]~ by a~.We recall some basic operations onF(i)(U) as follows [6, 7]: for any A,B∈F(i)(U) and [a,b]∈[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=Bc⟺A(x)=B(x)c 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,(4)(⋃j∈J‍A)(x)=⋁j∈JA(x),(⋂j∈J‍A)(x)=⋀j∈JA(x),
where {Aj:j∈J}⊆F(i)(U).(6)
([a,b]A)(x)=[a,b]∧[A-(x),A+(x)] for each x∈U.Obviously,(5)A=B⟺A-=B-,A+=B+,([a,b]~)c=[a,b]c~([a,b]∈[I]).Definition 3 (see [8]).
A
∈
F
(
i
)
(
U
) is called an IVF point in U, if there exist [a,b]∈[I]-{0-} and x∈U such that
(6)A(y)={[a,b],y=x,0-,y≠x.
We denote A by x[a,b].If[a,b]=1-, then
(7)x1-(y)={1-,y=x,0-,y≠x.Remark 4.
A
=
⋃
x
∈
U
(
A
(
x
)
x
1
-
)
(
A
∈
F
(
i
)
(
U
)
).Definition 5 (see [8]).
τ
⊆
F
(
i
)
(
U
) is called an IVF topology on U, if(i)
0~,1~∈τ,(ii)
A,B∈τ⇒A∩B∈τ,(iii)
{Aj:j∈J}⊆τ⇒⋃j∈JAj∈τ.
The pair(U,τ) is called an IVF topological space. Every member of τ is called an IVF open set in U. Its complement is called an IVF closed set in U.
An IVF topologyτ is called Alexandrov, if (ii) in Definition 5 is replaced by(ii)′
{Aj:j∈J}⊆τ⇒⋂j∈JAj∈τ.
We denoteτc={A:Ac∈τ}.
The interior and closure ofA∈F(i)(U) denoted, respectively, by int(A) and cl(A), are defined as follows:
(8)int(A)orintτ(A)=⋃‍{B∈τ:B⊆A},cl(A)orclτ(A)=⋂‍{B∈τc:B⊇A}.Proposition 6 (see [8]).
Letτ be an IVF topology on U. Then, for any A,B∈F(i)(U),(1)
int(1~)=1~, cl(0~)=0~,(2)
int(A)⊆A⊆cl(A),(3)
A⊆B⇒int(A)⊆int(B), cl(A)⊆cl(B),(4)
int(Ac)=(cl(A))c, cl(Ac)=(int(A))c,(5)
int(A∩B)=int(A)∩int(B), cl(A∪B)=cl(A)∪cl(B),(6)
int(int(A))=int(A), cl(cl(A))=cl(A).


## 3. IVF Approximation Spaces and IVF Rough Sets

Recall thatR is called an IVF relation on U if R∈F(i)(U×U).Definition 7 (see [9]).
LetR be an IVF relation on U. Then, R is called(1)
reflexive, ifR(x,x)=1- for each 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 an IVF relation on U. R is called preorder if R is reflexive and transitive (see [11]).Definition 8 (see [9]).
LetR be an IVF relation on U. The pair (U,R) is called an IVF approximation space. For each A∈F(i)(U), the IVF lower and the IVF upper approximations of A with respect to (U,R), denoted by R_(A) and R¯(A), are two IVF sets and are, respectively, defined as follows:
(9)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 IVF rough set of A with respect to (U,R).Remark 9.
Let(U,R) be an IVF approximation space. Then,(1)
for eachx,y∈U,
(10)R¯(x1-)(y)=R(y,x),R_((x1-)c)(y)=1--R(y,x);(2)
for each[a,b]∈[I], R_([a,b]~)⊇[a,b]~⊇R¯([a,b]~).Proposition 10 (see [9]).
Let(U,R) be an IVF approximation space. Then, for each A∈F(i)(U),
(11)(R_(A))-=R+_(A-),(R_(A))+=R-_(A+),(R¯(A))-=R-¯(A-),(R¯(A))+=R+¯(A+).Proposition 11.
Let(U,R) be an IVF approximation space. Then, for any A,B∈F(i)(U), {Aj:j∈J}⊆F(i)(U), and [a,b]∈[I],(1)
R_(1~)=1~,R¯(0~)=0~,(2)
A⊆B⇒R_(A)⊆R_(B),R¯(A)⊆R¯(B),(3)
R_(Ac)=(R¯(A))c,R¯(Ac)=(R_(A))c,(4)
R_(⋂j∈JAj)=⋂j∈JR_(Aj),R¯(⋃j∈JAj)=⋃j∈JR¯(Aj),(5)
R_([a,b]~∪A)=[a,b]~∪R_(A),R¯([a,b]A)=[a,b]R¯(A).Proof.
(1) and (2) are obvious.
(3) For eachx∈U,  by Proposition 10,
(12)R_(Ac)(x)=[⋀y∈U((1-A+(y))∨(1-R+(x,y))),⋀y∈U((1-A-(y))∨(1-R-(x,y)))]=[⋀y∈U(1-A+(y)∧R+(x,y)),⋀y∈U(1-A-(y)∧R-(x,y))]=[1-⋁y∈U(A+(y)∧R+(x,y)),1-⋁y∈U(A-(y)∧R-(x,y))]=[1-(R¯(A))+(x),1-(R¯(A))-(x)]=1--[(R¯(A))-(x),(R¯(A))+(x)]=1--R¯(A)(x)=(R¯(A))c(x).
Then, R_(Ac)=(R¯(A))c.
PickA=Bc. Since R_(Bc)=(R¯(B))c,
(13)R¯(Ac)=R¯(B)=((R¯(B))c)c=(R_(Bc))c=(R_(A))c.
(4) For eachx∈U, by
(14)R_(⋂j∈J‍Aj)(x)=⋀y∈U((⋂j∈J‍Aj)(y)∨(1--R(x,y)))=⋀y∈U((⋀j∈JAj(y))∨(1--R(x,y)))=⋀y∈U(⋀j∈J(Aj(y)∨(1--R(x,y))))=⋀j∈J(⋀y∈U(Aj(y)∨(1--R(x,y))))=⋀j∈JR_(A)(x)=(⋂j∈J‍R_(Aj))(x),
we have R_(⋂j∈JAj)=⋂j∈JR_(Aj).
By (3) andR_(⋂j∈J(Aj)c)=⋂j∈JR_((Aj)c), we have
(15)(R¯(⋃j∈J‍Aj))c=R_((⋃j∈J‍Aj)c)=R_(⋂j∈J‍Ajc)=⋂j∈J‍R_(Ajc)=⋂j∈J‍(R¯(Aj))c=(⋃j∈J‍R¯(Aj))c.
Then, R¯(⋃j∈JAj)=⋃j∈JR¯(Aj).
(5) For eachx∈U,  by Proposition 10,
(16)R¯([a,b]A)(x)=⋁y∈U(([a,b]A)(y)∧R(x,y))=⋁y∈U(([a,b]∧A(y))∧R(x,y))=⋁y∈U([a,b]∧(A(y)∧R(x,y)))=[a,b]∧(⋁y∈U(A(y)∧R(x,y)))=[a,b]∧R¯(A)(x)=([a,b]R¯(A))(x).
Then, R¯([a,b]A)=[a,b]R¯(A).
Similarly, we can prove thatR_([a,b]~∪A)=[a,b]~∪R_(A).Theorem 12.
LetR be an IVF relation on U, and let τ be an IVF topology on U. If one of the following conditions is satisfied, then R is preorder.(1)
R_ is the interior operator of τ.(2)
R¯ is the closure operator of τ.Proof.
By Propositions6(4) and 11(3), (1) and (2) are equivalent. We only need to prove that (2) implies the reflexivity and transitivity of R.
By Remark9(1),  R¯(x1-)(y)=R(y,x) for any x,y∈U. Note that R¯ is the closure operator of τ. Then, for each x∈U,
(17)R(x,x)=R¯(x1-)(x)=clτ(x1-)(x)≥x1-(x)=1-.
Thus, R is reflexive.
For anyx,y,z∈U, denote clτ(z1-)(y)=[a,b], and by Remark 4, Remark 9(1) and, Proposition 11(5),
(18)R(x,y)∧R(y,z)=R¯(y1-)(x)∧R¯(z1-)(y)=R¯(y1-)(x)∧clτ(z1-)(y)=R¯(y1-)(x)∧[a,b]=[a,b]R¯(y1-)(x)=R¯([a,b]y1-)(x)=clτ([a,b]y1-)(x)=clτ(clτ(z1-)(y)y1-)(x)≤clτ(⋃t∈U‍(clτ(z1-)(t)t1-))(x)=clτ(clτ(z1-))(x)=clτ(z1-)(x)=R(x,z).
Then, R is transitive.Theorem 13.
Let(U,R) be an IVF approximation space. Then,
(19)(1)Risreflexive⟺(ILR)∀A∈F(i)(U),kkkR_(A)⊆A⟺(IUR)∀A∈F(i)(U),kkkA⊆R¯(A),(20)(2)Rissymmetric⟺(ILS)∀(x,y)∈U×U,kkkR_((x1-)c)(y)=R_((y1-)c)(x)⟺(IUS)∀(x,y)∈U×U,kkkR¯(x1-)(y)=R¯(y1-)(x),(21)(3)Ristransitive⟺(ILT)∀A∈F(i)(U),kkkR_(A)⊆R_(R_(A))⟺(IUT)∀A∈F(i)(U),kkkR¯(R¯(A))⊆R¯(A).Proof.
(1) By Proposition11(3), (ILR) and (IUR) are equivalent. We only need to prove that the reflexivity of R is equivalent to (IUR).
Assume thatR is reflexive. For any A∈F(i)(U) and x∈U, by the reflexivity of R, R(x,x)=1-. Then,
(22)(R¯(A))(x)=⋁y∈U(A(y)∧R(x,y))≥A(x)∧R(x,x)=A(x).
Thus, A⊆R¯(A).
Conversely, assume that (IUR) holds. For eachx∈U, pick A=x1-. By (IUR), we have x1-⊆R¯(x1-). By Remark 9(1),
(23)1-=x1-(x)≤R¯(x1-)(x)=R(x,x)≤1-.
Then, R(x,x)=1-. Thus, R is reflexive.
(2) By Proposition11(3), (ILS) and (IUS) are equivalent. We only need to prove that the symmetry of R is equivalent to (IUS).
For anyx,y∈U, by Remark 9(1), R¯(y1-)(x)=R(x,y) and R¯(x1-)(y)=R(y,x). So, the symmetry of R is equivalent to (IUS).
(3) By Proposition11(3), (ILT) and (IUT) are equivalent. We only need to prove that the transitivity of R is equivalent to (IUT).
Assume thatR is transitive. Then, R(x,z)≥⋁y∈U(R(x,y)∧R(y,z))  for any x,y,z∈U. Denote axz=⋁y∈U(R(x,y)∧R(y,z)). Then, for any A∈F(i)(U) and x∈U,
(24)R¯(R¯(A))(x)=⋁y∈U(R¯(A)(y)∧R(x,y))=⋁y∈U((⋁z∈U(A(z)∧R(y,z)))∧R(x,y))=⋁y∈U(⋁z∈U((A(z)∧R(y,z))∧R(x,y)))=⋁y∈U(⋁z∈U(A(z)∧(R(y,z)∧R(x,y))))≤⋁y∈U(⋁z∈U(A(z)∧axz))=⋁z∈U(A(z)∧axz)≤⋁z∈U(A(z)∧R(x,z))=R¯(A)(x).
So, R¯(R¯(A))⊆R¯(A).
Conversely, assume that (IUT) holds. For anyx,y,z∈U, by (IUT),
(25)R¯(R¯(z1-))⊆R¯(z1-).
By Remark 9(1),
(26)R(x,y)∧R(y,z)≤⋁t∈U(R(x,t)∧R(t,z))=⋁t∈U(R(x,t)∧R¯(z1-)(t))=R¯(R¯(z1-))(x)≤R¯(z1-)(x)=R(x,z).
Hence,R is transitive.Corollary 14.
Let(U,R) be an IVF approximation space. If R is preorder, then
(27)R_(R_(A))=R_(A),R¯(R¯(A))=R¯(A),(A∈F(i)(U)).Proof.
This holds by Theorem13.


## 4. Relationships between IVF Relations and IVF Topologies

LetR be an IVF relation on U. We denote
(28)τR={A∈F(i)(U):R_(A)=A},θR={R_(A):A∈F(i)(U)}.


### 4.1. IVF Topologies Induced by IVF Relations

Theorem 15.
LetR be an IVF relation on U. If R is reflexive, then τR is an IVF topology on U.Proof.
(i) By Proposition11(1), R_(1~)=1~. Then, 1~∈τR.
By Theorem13(1), R_(0~)⊆0~. Then, R_(0~)=0~. So, 0~∈τR.
(ii) LetA,B∈τR. By Proposition 11(4),
(29)R_(A∩B)=R_(A)∩R_(B).
Then, R_(A∩B)=A∩B. Thus, A∩B∈τR.
(iii) Let{Aj:j∈J}⊆τR. Then, R_(Aj)=Aj for each j∈J. By Proposition 11(2),
(30)R_(⋃j∈J‍Aj)⊇⋃j∈J‍R_(Aj)=⋃j∈J‍Aj.
By Theorem 13(1), R_(⋃j∈JAj)⊆⋃j∈JAj.
Then,R_(⋃j∈JAj)=⋃j∈JAj, and so ⋃j∈JAj∈τR.
Thus,τR is an IVF topology on U.Definition 16.
LetR be an IVF relation on U. If R is reflexive, then τR is called the IVF topology induced by R on U.Theorem 17.
LetR be a reflexive IVF relation on U, and let τR be the IVF topology induced by R on U. Then, the following properties hold:(1)
τR⊆θR,(2)
for eachA∈F(i)(U),
(31)intτR(A)⊆R_(A)⊆A⊆R¯(A)⊆clτR(A),(3)
for each[a,b]∈[I], [a,b]~∈τR∩τRc.Proof.
(1) This is obvious.
(2) For eachA∈F(i)(U), by Proposition 11(2), we have
(32)intτR(A)=⋃‍{B:B∈τR,B⊆A}=⋃‍{R_(B):B∈τR,B⊆A}⊆R_(A).
By Propositions6(4) and 11(3),
(33)clτR(A)=(intτR(Ac))c⊇(R_(Ac))c=R¯(A).
By Theorem13(1),
(34)intτR(A)⊆R_(A)⊆A⊆R¯(A)⊆clτR(A).
(3) For each[a,b]∈[I], by Remark 9(2) and Theorem 13(1), R_([a,b]~)=[a,b]~. Then, [a,b]~∈τR. By Proposition 6(4),
(35)clτR([a,b]~)=(intτR(([a,b]~)c))c=(intτR([a,b]c~))c=([a,b]~c)c=[a,b]~.
So, [a,b]~∈τRc.Theorem 18.
LetR be a reflexive IVF relation on U, and let τR be the IVF topology induced by R on U. If R is transitive, then(1)
τR=θR,(2)
R_ is the interior operator of τR,(3)
R¯ is the closure operator of τR.Proof.
(1) Obviously,(36)τR⊆{R_(A):A∈F(i)(U)}.
By Corollary 14, τR⊇{R_(A):A∈F(i)(U)}. Then, τR={R_(A):A∈F(i)(U)}=θR.
(2) It suffices to show that for eachA∈F(i)(U),
(37)R_(A)=intτR(A),
where intτR(A)=⋃‍{B∈τR:B⊆A}.
SinceR_(A)∈θR, by (1), R_(A)∈τR.
By Theorem13(1), R_(A)⊆A. Then, R_(A)⊆intτR(A).
By (1),intτR(A)⊆R_(A). Then, R_(A)=intτR(A).
(3) This holds by (2), Proposition6(4), and Proposition 11(3).Example 19.
LetU={x,y,z}, and let R be a reflexive IVF relation on U. R is defined as follows:
(38)R(x,y)=R(x,z)=R(z,x)=0-,R(y,x)=[0.2,0.7],R(y,z)=1-,R(z,y)=[0.3,0.8].
Pick
(39)A=0¯x+[0.4,0.5]y+1-z,B=1-x+[0.5,0.6]y+0¯z.
(
1
) We have
(40)R(z,y)∧R(y,x)=[0.2,0.7]≰0-=R(z,x).
Then,R is not transitive.
(
2
) Since
(41)R_(A)=0-x+[0.3,0.5]y+[0.4,0.7]z,R_(R_(A))=0-x+[0.3,0.5]y+[0.3,0.7]z,
we have R_(R_(A))≠R_(A). Then, R_(A)∉τR. Thus,
(42)τR≠{R_(A):A∈F(i)(U)},intτR(A)≠R_(A).
Obviously,Bc=A. By Proposition 11(3),
(43)(R¯(B))c=R_(Bc)=R_(A)∉τR.
Then, R¯(B)∉τRc. Thus, clτR(B)≠R¯(B).


### 4.2. IVF Relations Induced by IVF Topologies

Definition 20.
Letτ be an IVF topology on U. Define an IVF relation Rτ on U by
(44)Rτ(x,y)=clτ(y1-)(x)
for each (x,y)∈U×U. Then, Rτ is called the IVF relation induced by τ on U.
An IVF topologyτ on U is said to satisfy the following: (C1)
axiom:clτ([a,b]A)=[a,b]clτ(A)forany[a,b]∈[I]andA∈F(i)(U),(C2)
axiom:clτ(⋃j∈JAj)=⋃j∈Jclτ(Aj)forany{Aj:j∈J}⊆F(i)(U).Theorem 21.
Letτ be an IVF topology on U, and let Rτ be the IVF relation induced by τ on U. Then, the following properties hold.(1)
Rτ is reflexive.(2)
Ifτ satisfies (C2) axiom and {[a,b]~:[a,b]∈[I]}⊆τ, then
(45)Rτ_(A)⊆intτ(A)⊆A⊆clτ(A)⊆Rτ¯(A)(A∈F(i)(U)).Proof.
(1) For eachx∈U,
(46)Rτ(x,x)=clτ(x1-)(x)≥(x1-)(x)=1-.
Then, Rτ is reflexive.
(2) Since{[a,b]~:[a,b]∈[I]}⊆τ, we have {[a,b]~:[a,b]∈[I]}⊆τc. For each A∈F(i)(U), by Remark 4, (C2) axiom, and Proposition 11,
(47)clτ(A)=clτ(⋃y∈U‍(A(y)y1-))=⋃y∈U‍clτ(A(y)y1-)=⋃y∈U‍clτ(A(y)~∩y1-)⊆⋃y∈U‍(clτ(A(y)~)∩clτ(y1-))=⋃y∈U‍(A(y)~∩clτ(y1-)).
Then, for each x∈U,
(48)clτ(A)(x)≤⋁y∈U(A(y)~(x)∧clτ(y1-)(x))=⋁y∈U(A(y)∧Rτ(x,y))=Rτ¯(A)(x).
Hence, clτ(A)⊆Rτ¯(A).
By Propositions6(4) and 11(3),
(49)intτ(A)=(clτ(Ac))c⊇(Rτ¯(Ac))c=Rτ_(A),
so
(50)Rτ_(A)⊆intτ(A)⊆A⊆clτ(A)⊆Rτ¯(A).Proposition 22.
Letτ be an IVF topology on U. If τ satisfies (C1) and (C2) axioms, then(1)
Rτ¯is the closure operator ofτ,(2)
Rτ_is the interior operator ofτ,(3)
for each[a,b]∈[I],[a,b]~∈τ,(4)
τis Alexandrov.Proof.
(1) For eachA∈F(i)(U), by Remark 4, (C1) axiom, and (C2) axiom,
(51)clτ(A)=clτ(⋃y∈U‍(A(y)y1-))=⋃y∈U‍clτ(A(y)y1-)=⋃y∈U‍(A(y)clτ(y1-)).
Then, for each x∈U,
(52)clτ(A)(x)=⋁y∈U(A(y)(x)∧clτ(y1-)(x))=⋁y∈U(A(y)∧Rτ(x,y))=Rτ¯(A)(x).
Hence, Rτ¯(A)=clτ(A). Thus, Rτ¯ is the closure operator of τ.
(2) This holds by (1), Proposition6(4) and, Proposition 11(3).
(3) For each[a,b]∈[I], by (2), Remark 9(2), and Proposition 6(2),
(53)[a,b]~⊇intτ([a,b]~)=R_([a,b]~)⊇[a,b]~.
Then  intτ([a,b]~)=[a,b]~, and so [a,b]~∈τ.
(4) Let{Aj:j∈J}⊆τ. By (2), for each j∈J,
(54)Aj=intτ(Aj)=R_(Aj).
By (2) and Proposition 11(4),
(55)⋂j∈J‍Aj=⋂j∈J‍R_(Aj)=R_(⋂j∈J‍Aj)=intτ(⋂j∈J‍Aj).
So ⋂j∈JAj∈τ.
Hence,τ is Alexandrov.Proposition 23.
LetR be a preorder IVF relation on U. Then, τR satisfies (C1) and (C2) axioms.Proof.
For any[a,b]∈[I] and A∈F(i)(U), by Theorem 18(3) and Proposition 11(5),
(56)clτR([a,b]A)=R¯([a,b]A)=[a,b]R¯(A)=[a,b]clτR(A).
Thus,  τR satisfies (C1) axiom.
For any{Aj:j∈J}⊆F(i)(U), by Proposition 11(4) and Theorem 18,
(57)clτR(⋃j∈J‍Aj)=clθR(⋃j∈J‍Aj)=R¯(⋃j∈J‍Aj)=⋃j∈J‍R¯(Aj)=⋃j∈J‍clθR(Aj)=⋃j∈J‍clτR(Aj).
Thus, τR satisfies (C2) axiom.


## 4.1. IVF Topologies Induced by IVF Relations

Theorem 15.
LetR be an IVF relation on U. If R is reflexive, then τR is an IVF topology on U.Proof.
(i) By Proposition11(1), R_(1~)=1~. Then, 1~∈τR.
By Theorem13(1), R_(0~)⊆0~. Then, R_(0~)=0~. So, 0~∈τR.
(ii) LetA,B∈τR. By Proposition 11(4),
(29)R_(A∩B)=R_(A)∩R_(B).
Then, R_(A∩B)=A∩B. Thus, A∩B∈τR.
(iii) Let{Aj:j∈J}⊆τR. Then, R_(Aj)=Aj for each j∈J. By Proposition 11(2),
(30)R_(⋃j∈J‍Aj)⊇⋃j∈J‍R_(Aj)=⋃j∈J‍Aj.
By Theorem 13(1), R_(⋃j∈JAj)⊆⋃j∈JAj.
Then,R_(⋃j∈JAj)=⋃j∈JAj, and so ⋃j∈JAj∈τR.
Thus,τR is an IVF topology on U.Definition 16.
LetR be an IVF relation on U. If R is reflexive, then τR is called the IVF topology induced by R on U.Theorem 17.
LetR be a reflexive IVF relation on U, and let τR be the IVF topology induced by R on U. Then, the following properties hold:(1)
τR⊆θR,(2)
for eachA∈F(i)(U),
(31)intτR(A)⊆R_(A)⊆A⊆R¯(A)⊆clτR(A),(3)
for each[a,b]∈[I], [a,b]~∈τR∩τRc.Proof.
(1) This is obvious.
(2) For eachA∈F(i)(U), by Proposition 11(2), we have
(32)intτR(A)=⋃‍{B:B∈τR,B⊆A}=⋃‍{R_(B):B∈τR,B⊆A}⊆R_(A).
By Propositions6(4) and 11(3),
(33)clτR(A)=(intτR(Ac))c⊇(R_(Ac))c=R¯(A).
By Theorem13(1),
(34)intτR(A)⊆R_(A)⊆A⊆R¯(A)⊆clτR(A).
(3) For each[a,b]∈[I], by Remark 9(2) and Theorem 13(1), R_([a,b]~)=[a,b]~. Then, [a,b]~∈τR. By Proposition 6(4),
(35)clτR([a,b]~)=(intτR(([a,b]~)c))c=(intτR([a,b]c~))c=([a,b]~c)c=[a,b]~.
So, [a,b]~∈τRc.Theorem 18.
LetR be a reflexive IVF relation on U, and let τR be the IVF topology induced by R on U. If R is transitive, then(1)
τR=θR,(2)
R_ is the interior operator of τR,(3)
R¯ is the closure operator of τR.Proof.
(1) Obviously,(36)τR⊆{R_(A):A∈F(i)(U)}.
By Corollary 14, τR⊇{R_(A):A∈F(i)(U)}. Then, τR={R_(A):A∈F(i)(U)}=θR.
(2) It suffices to show that for eachA∈F(i)(U),
(37)R_(A)=intτR(A),
where intτR(A)=⋃‍{B∈τR:B⊆A}.
SinceR_(A)∈θR, by (1), R_(A)∈τR.
By Theorem13(1), R_(A)⊆A. Then, R_(A)⊆intτR(A).
By (1),intτR(A)⊆R_(A). Then, R_(A)=intτR(A).
(3) This holds by (2), Proposition6(4), and Proposition 11(3).Example 19.
LetU={x,y,z}, and let R be a reflexive IVF relation on U. R is defined as follows:
(38)R(x,y)=R(x,z)=R(z,x)=0-,R(y,x)=[0.2,0.7],R(y,z)=1-,R(z,y)=[0.3,0.8].
Pick
(39)A=0¯x+[0.4,0.5]y+1-z,B=1-x+[0.5,0.6]y+0¯z.
(
1
) We have
(40)R(z,y)∧R(y,x)=[0.2,0.7]≰0-=R(z,x).
Then,R is not transitive.
(
2
) Since
(41)R_(A)=0-x+[0.3,0.5]y+[0.4,0.7]z,R_(R_(A))=0-x+[0.3,0.5]y+[0.3,0.7]z,
we have R_(R_(A))≠R_(A). Then, R_(A)∉τR. Thus,
(42)τR≠{R_(A):A∈F(i)(U)},intτR(A)≠R_(A).
Obviously,Bc=A. By Proposition 11(3),
(43)(R¯(B))c=R_(Bc)=R_(A)∉τR.
Then, R¯(B)∉τRc. Thus, clτR(B)≠R¯(B).


## 4.2. IVF Relations Induced by IVF Topologies

Definition 20.
Letτ be an IVF topology on U. Define an IVF relation Rτ on U by
(44)Rτ(x,y)=clτ(y1-)(x)
for each (x,y)∈U×U. Then, Rτ is called the IVF relation induced by τ on U.
An IVF topologyτ on U is said to satisfy the following: (C1)
axiom:clτ([a,b]A)=[a,b]clτ(A)forany[a,b]∈[I]andA∈F(i)(U),(C2)
axiom:clτ(⋃j∈JAj)=⋃j∈Jclτ(Aj)forany{Aj:j∈J}⊆F(i)(U).Theorem 21.
Letτ be an IVF topology on U, and let Rτ be the IVF relation induced by τ on U. Then, the following properties hold.(1)
Rτ is reflexive.(2)
Ifτ satisfies (C2) axiom and {[a,b]~:[a,b]∈[I]}⊆τ, then
(45)Rτ_(A)⊆intτ(A)⊆A⊆clτ(A)⊆Rτ¯(A)(A∈F(i)(U)).Proof.
(1) For eachx∈U,
(46)Rτ(x,x)=clτ(x1-)(x)≥(x1-)(x)=1-.
Then, Rτ is reflexive.
(2) Since{[a,b]~:[a,b]∈[I]}⊆τ, we have {[a,b]~:[a,b]∈[I]}⊆τc. For each A∈F(i)(U), by Remark 4, (C2) axiom, and Proposition 11,
(47)clτ(A)=clτ(⋃y∈U‍(A(y)y1-))=⋃y∈U‍clτ(A(y)y1-)=⋃y∈U‍clτ(A(y)~∩y1-)⊆⋃y∈U‍(clτ(A(y)~)∩clτ(y1-))=⋃y∈U‍(A(y)~∩clτ(y1-)).
Then, for each x∈U,
(48)clτ(A)(x)≤⋁y∈U(A(y)~(x)∧clτ(y1-)(x))=⋁y∈U(A(y)∧Rτ(x,y))=Rτ¯(A)(x).
Hence, clτ(A)⊆Rτ¯(A).
By Propositions6(4) and 11(3),
(49)intτ(A)=(clτ(Ac))c⊇(Rτ¯(Ac))c=Rτ_(A),
so
(50)Rτ_(A)⊆intτ(A)⊆A⊆clτ(A)⊆Rτ¯(A).Proposition 22.
Letτ be an IVF topology on U. If τ satisfies (C1) and (C2) axioms, then(1)
Rτ¯is the closure operator ofτ,(2)
Rτ_is the interior operator ofτ,(3)
for each[a,b]∈[I],[a,b]~∈τ,(4)
τis Alexandrov.Proof.
(1) For eachA∈F(i)(U), by Remark 4, (C1) axiom, and (C2) axiom,
(51)clτ(A)=clτ(⋃y∈U‍(A(y)y1-))=⋃y∈U‍clτ(A(y)y1-)=⋃y∈U‍(A(y)clτ(y1-)).
Then, for each x∈U,
(52)clτ(A)(x)=⋁y∈U(A(y)(x)∧clτ(y1-)(x))=⋁y∈U(A(y)∧Rτ(x,y))=Rτ¯(A)(x).
Hence, Rτ¯(A)=clτ(A). Thus, Rτ¯ is the closure operator of τ.
(2) This holds by (1), Proposition6(4) and, Proposition 11(3).
(3) For each[a,b]∈[I], by (2), Remark 9(2), and Proposition 6(2),
(53)[a,b]~⊇intτ([a,b]~)=R_([a,b]~)⊇[a,b]~.
Then  intτ([a,b]~)=[a,b]~, and so [a,b]~∈τ.
(4) Let{Aj:j∈J}⊆τ. By (2), for each j∈J,
(54)Aj=intτ(Aj)=R_(Aj).
By (2) and Proposition 11(4),
(55)⋂j∈J‍Aj=⋂j∈J‍R_(Aj)=R_(⋂j∈J‍Aj)=intτ(⋂j∈J‍Aj).
So ⋂j∈JAj∈τ.
Hence,τ is Alexandrov.Proposition 23.
LetR be a preorder IVF relation on U. Then, τR satisfies (C1) and (C2) axioms.Proof.
For any[a,b]∈[I] and A∈F(i)(U), by Theorem 18(3) and Proposition 11(5),
(56)clτR([a,b]A)=R¯([a,b]A)=[a,b]R¯(A)=[a,b]clτR(A).
Thus,  τR satisfies (C1) axiom.
For any{Aj:j∈J}⊆F(i)(U), by Proposition 11(4) and Theorem 18,
(57)clτR(⋃j∈J‍Aj)=clθR(⋃j∈J‍Aj)=R¯(⋃j∈J‍Aj)=⋃j∈J‍R¯(Aj)=⋃j∈J‍clθR(Aj)=⋃j∈J‍clτR(Aj).
Thus, τR satisfies (C2) axiom.


## 5. IVF Approximating Spaces

As can be seen from Section4, a reflexive IVF relation yields an IVF topology. In this section, we consider the reverse problem; that is, under which conditions can an IVF topology be associated with an IVF relation which produces the given IVF topology?Definition 24.
Let(U,τ) be an IVF topological space. If there exists a reflexive IVF relation on U such that τR=τ, then (U,τ) is called an IVF approximating space.Theorem 25.
Letτ be an IVF topology on U. Let Rτ be the IVF relation induced by (U,τ), and let τRτ be the IVF topology induced by Rτ on U. If τ satisfies (C1) and (C2) axioms, then τRτ=τ.Proof.
By Theorem21(1), Rτ is reflexive. For any x,y,z∈U, put cl(z1-)(y)=[a,b]. By Remark 4 and Proposition 11(2),
(58)[a,b]clτ(y1-)=clτ([a,b]y1-)=clτ(clτ(z1-)(y)y1-)⊆clτ(⋃t∈U‍(clτ(z1-)(t)t1-))=clτ(clτ(z1-))=clτ(z1-).
Then,
(59)Rτ(x,y)∧Rτ(y,z)=clτ(y1-)(x)∧clτ(z1-)(y)=clτ(y1-)(x)∧[a,b]=[a,b]∧clτ(y1-)(x)=([a,b]clτ(y1-))(x)≤clτ(z1-)(x)=Rτ(x,z).
So, R is transitive.
So,Rτ is preorder. For each A∈F(i)(U), by Theorem 18,
(60)clτRτ(A)=clθRτ(A)=Rτ¯(A).
Since τ satisfies (C1) and (C2) axioms, by Proposition 22(1), Rτ¯(A)=clτ(A). So, clτRτ(A)=clτ(A).
Thus,τRτ=τ.Theorem 26.
Letτ be an IVF topology on U. Then, the following are equivalent.(1)
τ satisfies (C1) and (C2) axioms.(2)
For any[a,b]∈[I], A∈F(i)(U) and {Aj:j∈J}⊆F(i)(U),
(61)intτ([a,b]~∪A)=[a,b]~∪intτ(A),intτ(⋂j∈J‍Aj)=⋂j∈J‍intτ(Aj).(3)
There exists a preorder IVF relationρ on U such that ρ¯ is the closure operator of τ.(4)
There exists a preorder IVF relationρ on U such that ρ_ is the interior operator of τ.(5)
Rτ¯ is the closure operator of τ.(6)
Rτ_ is the interior operator of τ.Proof.
(1)⟺ (2) is obvious.
(1)⟹ (3). Suppose that τ satisfies (C1) and (C2) axioms. Pick ρ=Rτ. By Proposition 22(1), ρ¯ is the closure operator of τ. By Theorem 12(2), ρ is preorder.
(3)⟹ (4). Let ρ¯ be the closure operator of τ for some preorder IVF relation ρ on U. For each A∈F(i)(U), by Propositions 6(4) and 11(3),
(62)ρ_(A)=(ρ¯(Ac))c=(clτ(Ac))c=intτ(A).
Thus, ρ_ is the interior operator of τ.
(4)⟹ (6). Let ρ_ be the interior operator of τ for some preorder IVF relation ρ on U. For each (x,y)∈U×U, by Remark 9(1),
(63)ρ(x,y)=1--ρ_((y1-)c)(x)=1--intτ((y1-)c)(x)=clτ(y1-)(x)=Rτ(x,y).
Then, ρ=Rτ. Note that ρ_ is the interior operator of τ. Then, Rτ_ is the interior operator of τ.
(6)⟺ (5). This holds by Propositions 6(4) and 11(3).
(5)⟹ (1). For any [a,b]∈[I] and A∈F(i)(U), by Proposition 11(5),
(64)clτ([a,b]A)=Rτ¯([a,b]A)=[a,b]Rτ¯(A)=[a,b]clτ(A).
Thus, τ satisfies (C1) axiom.
For any{Aj:j∈J}⊆F(i)(U), by Proposition 11(4),
(65)clτ(⋃j∈J‍Aj)=Rτ¯(⋃j∈J‍Aj)=⋃j∈J‍Rτ¯(Aj)=⋃j∈J‍clτ(Aj).
Thus, τ satisfies (C2) axiom.Theorem 27.
Let(U,τ) be an IVF topological space. If one of the following conditions is satisfied, then (U,τ) is an IVF approximating space.(1)
τ satisfies (C1) and (C2) axioms.(2)
For any[a,b]∈[I] and A∈F(i)(U),
(66)int([a,b]~∪A)=[a,b]~∪int(A).(3)
There exists a preorder IVF relationR on U such that R¯ is the closure operator of τ.(4)
There exists a preorder IVF relationR on U such that R_ is the interior operator of τ.(5)
Rτ¯ is the closure operator of τ.(6)
Rτ_ is the interior operator of τ.Proof.
These hold by Theorems25 and 26.Example 28.
{
[
a
,
b
]
~
:
[
a
,
b
]
∈
[
I
]
} is an IVF approximating space.



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*Source: 494857-2013-07-31.xml*