On IVF Approximating Spaces

Abstract

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

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 [𝑎_𝑗,𝑏_𝑗]∈[𝐼] (𝑗=1,2), we define 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,𝑏₁or𝑎₁,𝑏₁^𝑐=1−𝑏₁,1−𝑎₁. (1)

Obviously,  ([𝑎,𝑏]^𝑐)^𝑐=[𝑎,𝑏] for each [𝑎,𝑏]∈[𝐼].

Definition 1 (see [6, 7]). For each {[𝑎_𝑗,𝑏_𝑗]∶𝑗∈𝐽}⊆[𝐼], one define _𝑗∈𝐽𝑎_𝑗,𝑏_𝑗=_𝑗∈𝐽𝑎_𝑗,_𝑗∈𝐽𝑏_𝑗,_𝑗∈𝐽𝑎_𝑗,𝑏_𝑗=_𝑗∈𝐽𝑎_𝑗,_𝑗∈𝐽𝑏_𝑗, (2) where ⋁_𝑗∈𝐽𝑎_𝑗=sup{𝑎_𝑗∶𝑗∈𝐽}and⋀_𝑗∈𝐽𝑎_𝑗=inf{𝑎_𝑗∶𝑗∈𝐽}.

Definition 2 (see [6, 7]). An IVF set 𝐴 in 𝑈 is defined by a mapping 𝐴∶𝑈→[𝐼].
Denote 𝐴(𝑥)=𝐴⁻(𝑥),𝐴⁺(𝑥)(𝑥∈𝑈). (3) 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 in 𝑈 is denoted by 𝐹^(𝑖)(𝑈).

Let 𝑎,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 on 𝐹^(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=𝐵^𝑐⟺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, _𝑗∈𝐽𝐴(𝑥)=_𝑗∈𝐽𝐴(𝑥),_𝑗∈𝐽𝐴(𝑥)=_𝑗∈𝐽𝐴(𝑥), (4) where {𝐴_𝑗∶𝑗∈𝐽}⊆𝐹^(𝑖)(𝑈).(6)([𝑎,𝑏]𝐴)(𝑥)=[𝑎,𝑏]∧[𝐴⁻(𝑥),𝐴⁺(𝑥)] for each 𝑥∈𝑈.

Obviously, 𝐴=𝐵⟺𝐴⁻=𝐵⁻,𝐴⁺=𝐵⁺,[𝑎,𝑏]^𝑐=[𝑎,𝑏]^𝑐([𝑎,𝑏]∈[𝐼]). (5)

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 𝐴𝑦=[𝑎,𝑏],𝑦=𝑥,0,𝑦≠𝑥. (6) We denote 𝐴 by 𝑥_[𝑎,𝑏].

If [𝑎,𝑏]=1, then 𝑥₁𝑦=1,𝑦=𝑥,0,𝑦≠𝑥. (7)

Remark 4. 𝐴=⋃_𝑥∈𝑈(𝐴(𝑥)𝑥₁)(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){𝐴_𝑗∶𝑗∈𝐽}⊆𝜏⇒⋃_𝑗∈𝐽𝐴_𝑗∈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 topology 𝜏 is called Alexandrov, if (ii) in Definition 5 is replaced by(ii)′{𝐴_𝑗∶𝑗∈𝐽}⊆𝜏⇒⋂_𝑗∈𝐽𝐴_𝑗∈𝜏.
We denote  𝜏^𝑐={𝐴∶𝐴^𝑐∈𝜏}.
The interior and closure of 𝐴∈𝐹^(𝑖)(𝑈) denoted, respectively, by int(𝐴) and cl(𝐴), are defined as follows: int(𝐴)orint_𝜏(𝐴)={},(𝐴)𝐵∈𝜏∶𝐵⊆𝐴clorcl_𝜏(𝐴)=𝐵∈𝜏^𝑐∶𝐵⊇𝐴. (8)

Proposition 6 (see [8]). Let 𝜏 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(𝐴^𝑐)=(cl(𝐴))^𝑐, cl(𝐴^𝑐)=(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 that 𝑅 is called an IVF relation on 𝑈 if 𝑅∈𝐹^(𝑖)(𝑈×𝑈).

Definition 7 (see [9]). Let 𝑅 be an IVF relation on 𝑈. Then, 𝑅 is called(1)reflexive, if 𝑅(𝑥,𝑥)=1 for each 𝑥∈𝑈,(2)symmetric, if 𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥) for any 𝑥,𝑦∈𝑈,(3)transitive, if 𝑅(𝑥,𝑧)≥𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧) for any 𝑥,𝑦,𝑧∈𝑈.

Let 𝑅 be an IVF relation on 𝑈. 𝑅 is called preorder if 𝑅 is reflexive and transitive (see [11]).

Definition 8 (see [9]). Let 𝑅 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: 𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦∨1−𝑅𝑥,𝑦(𝑥∈𝑈),𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅𝑥,𝑦(𝑥∈𝑈). (9)

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 each 𝑥,𝑦∈𝑈, 𝑅𝑥₁𝑦=𝑅𝑦,𝑥,𝑅𝑥₁^𝑐𝑦=1−𝑅𝑦,𝑥; (10)(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), 𝑅(𝐴)⁻=𝑅⁺(𝐴⁻),𝑅(𝐴)⁺=𝑅⁻𝐴⁺,𝑅(𝐴)⁻=𝑅⁻(𝐴⁻),𝑅(𝐴)⁺=𝑅⁺𝐴⁺. (11)

Proposition 11. Let (𝑈,𝑅) be an IVF approximation space. Then, for any 𝐴,𝐵∈𝐹^(𝑖)(𝑈), {𝐴_𝑗∶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(𝐴^𝑐)=(𝑅(𝐴))^𝑐, 𝑅(𝐴^𝑐)=(𝑅(𝐴))^𝑐,(4)𝑅(⋂_𝑗∈𝐽𝐴_𝑗)=⋂_𝑗∈𝐽𝑅(𝐴_𝑗), 𝑅(⋃_𝑗∈𝐽𝐴_𝑗)=⋃_𝑗∈𝐽𝑅(𝐴_𝑗),(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 each 𝑥∈𝑈,  by Proposition 10, 𝑅𝐴^𝑐(𝑥)=_𝑦∈𝑈1−𝐴⁺𝑦∨1−𝑅⁺𝑥,𝑦_𝑦∈𝑈,1−𝐴⁻𝑦∨1−𝑅⁻=𝑥,𝑦_𝑦∈𝑈1−𝐴⁺𝑦∧𝑅⁺𝑥,𝑦_𝑦∈𝑈,1−𝐴⁻𝑦∧𝑅⁻=𝑥,𝑦1−_𝑦∈𝑈𝐴⁺𝑦∧𝑅⁺𝑥,𝑦,1−_𝑦∈𝑈𝐴⁻𝑦∧𝑅⁻=𝑥,𝑦1−𝑅(𝐴)⁺(𝑥),1−𝑅(𝐴)⁻(𝑥=)1−𝑅(𝐴)⁻(𝑥),𝑅(𝐴)⁺(𝑥=)1−𝑅(𝐴)(𝑥)=𝑅(𝐴)^𝑐(𝑥). (12) Then, 𝑅(𝐴^𝑐)=(𝑅(𝐴))^𝑐.
Pick 𝐴=𝐵^𝑐. Since 𝑅(𝐵^𝑐)=(𝑅(𝐵))^𝑐, 𝑅𝐴^𝑐=𝑅(𝐵)=𝑅(𝐵)^𝑐𝑐=𝑅𝐵^𝑐𝑐=𝑅(𝐴)^𝑐. (13)
(4) For each 𝑥∈𝑈, by 𝑅_𝑗∈𝐽𝐴_𝑗(𝑥)=_{𝑦∈𝑈𝑗∈𝐽}𝐴_𝑗𝑦∨=1−𝑅𝑥,𝑦_{𝑦∈𝑈𝑗∈𝐽}𝐴_𝑗𝑦∨=1−𝑅𝑥,𝑦_{𝑦∈𝑈𝑗∈𝐽}𝐴_𝑗𝑦∨=1−𝑅𝑥,𝑦_{𝑗∈𝐽𝑦∈𝑈}𝐴_𝑗𝑦∨=1−𝑅𝑥,𝑦_𝑗∈𝐽𝑅(𝐴)(𝑥)=_𝑗∈𝐽𝑅𝐴_𝑗(𝑥), (14) we have 𝑅(⋂_𝑗∈𝐽𝐴_𝑗)=⋂_𝑗∈𝐽𝑅(𝐴_𝑗).
By (3) and 𝑅(⋂_𝑗∈𝐽(𝐴_𝑗)^𝑐)=⋂_𝑗∈𝐽𝑅((𝐴_𝑗)^𝑐), we have 𝑅_𝑗∈𝐽𝐴_𝑗^𝑐=𝑅_𝑗∈𝐽𝐴_𝑗^𝑐=𝑅_𝑗∈𝐽𝐴^𝑐_𝑗=_𝑗∈𝐽𝑅𝐴^𝑐_𝑗=_𝑗∈𝐽𝑅𝐴_𝑗^𝑐=_𝑗∈𝐽𝑅𝐴_𝑗^𝑐. (15) Then, 𝑅(⋃_𝑗∈𝐽𝐴_𝑗)=⋃_𝑗∈𝐽𝑅(𝐴_𝑗).
(5) For each 𝑥∈𝑈,  by Proposition 10, 𝑅]𝐴𝑥)=([𝑎,𝑏)(_𝑦∈𝑈]𝐴)𝑦=([𝑎,𝑏∧𝑅𝑥,𝑦_𝑦∈𝑈[]𝑦=𝑎,𝑏∧𝐴∧𝑅𝑥,𝑦_𝑦∈𝑈[]∧𝐴𝑦=[]∧𝑎,𝑏∧𝑅𝑥,𝑦𝑎,𝑏_𝑦∈𝑈𝐴𝑦=[]∧∧𝑅𝑥,𝑦𝑎,𝑏𝑅(𝐴)(𝑥)=[𝑎,𝑏]𝑅(𝐴)(𝑥). (16) Then, 𝑅([𝑎,𝑏]𝐴)=[𝑎,𝑏]𝑅(𝐴).
Similarly, we can prove that 𝑅([𝑎,𝑏]∪𝐴)=[𝑎,𝑏]∪𝑅(𝐴).

Theorem 12. Let 𝑅 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 Propositions 6(4) and 11(3), (1) and (2) are equivalent. We only need to prove that (2) implies the reflexivity and transitivity of 𝑅.
By Remark 9(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, 𝑅(𝑥,𝑥)=𝑅𝑥₁(𝑥)=cl_𝜏𝑥₁(𝑥)≥𝑥₁(𝑥)=1. (17) Thus, 𝑅 is reflexive.
For any 𝑥,𝑦,𝑧∈𝑈, denote cl_𝜏(𝑧₁)(𝑦)=[𝑎,𝑏], and by Remark 4, Remark 9(1) and, Proposition 11(5), 𝑅𝑥,𝑦∧𝑅𝑦,𝑧=𝑅𝑦₁(𝑥)∧𝑅𝑧₁𝑦=𝑅𝑦₁(𝑥)∧cl_𝜏𝑧₁𝑦=𝑅𝑦₁(𝑥)∧[]=[]𝑎,𝑏𝑎,𝑏𝑅𝑦₁(𝑥)=𝑅[𝑎,𝑏]𝑦₁(𝑥)=cl_𝜏[𝑎,𝑏]𝑦₁(𝑥)=cl_𝜏cl_𝜏𝑧₁𝑦𝑦₁(𝑥)≤𝑐𝑙_{𝜏𝑡∈𝑈}cl_𝜏𝑧₁(𝑡)𝑡₁(𝑥)=cl_𝜏cl_𝜏𝑧₁(𝑥)=cl_𝜏𝑧₁(𝑥)=𝑅(𝑥,𝑧). (18) Then, 𝑅 is transitive.

Theorem 13. Let (𝑈,𝑅) be an IVF approximation space. Then, (1)𝑅𝑖𝑠𝑟𝑒𝑓𝑙𝑒𝑥𝑖𝑣𝑒⟺(𝐼𝐿𝑅)∀𝐴∈𝐹^(𝑖)(𝑈),𝑅(𝐴)⟺()⊆𝐴𝐼𝑈𝑅∀𝐴∈𝐹^(𝑖)(𝑈),𝐴⊆𝑅(𝐴),(2)()∀𝑅𝑅𝑖𝑠𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐⟺𝐼𝐿𝑆𝑥,𝑦∈𝑈×𝑈,𝑥₁^𝑐𝑦=𝑅𝑦₁^𝑐(𝑥)⟺(𝐼𝑈𝑆)∀𝑥,𝑦∈𝑈×𝑈,𝑅𝑥₁𝑦=𝑅𝑦₁𝑥),(3)()(𝑅𝑖𝑠𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑣𝑒⟺𝐼𝐿𝑇∀𝐴∈𝐹^(𝑖)(𝑈),𝑅(𝐴)⊆𝑅𝑅(𝐴)⟺(𝐼𝑈𝑇)∀𝐴∈𝐹^(𝑖)(𝑈),𝑅𝑅(𝐴)⊆𝑅(𝐴). (19)(20)(21)

Proof. (1) By Proposition 11(3), (ILR) and (IUR) are equivalent. We only need to prove that the reflexivity of 𝑅 is equivalent to (IUR).
Assume that 𝑅 is reflexive. For any 𝐴∈𝐹^(𝑖)(𝑈) and 𝑥∈𝑈, by the reflexivity of 𝑅, 𝑅(𝑥,𝑥)=1. Then, 𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦(𝑥)()(𝑥).∧𝑅𝑥,𝑦≥𝐴∧𝑅𝑥,𝑥=𝐴 (22) Thus, 𝐴⊆𝑅(𝐴).
Conversely, assume that (IUR) holds. For each 𝑥∈𝑈, pick 𝐴=𝑥₁. By (IUR), we have 𝑥₁⊆𝑅(𝑥₁). By Remark 9(1), 1=𝑥₁(𝑥)≤𝑅𝑥₁(𝑥)=𝑅(𝑥,𝑥)≤1. (23) Then, 𝑅(𝑥,𝑥)=1. Thus, 𝑅 is reflexive.
(2) By Proposition 11(3), (ILS) and (IUS) are equivalent. We only need to prove that the symmetry of R is equivalent to (IUS).
For any 𝑥,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 Proposition 11(3), (ILT) and (IUT) are equivalent. We only need to prove that the transitivity of R is equivalent to (IUT).
Assume that 𝑅 is transitive. Then, 𝑅(𝑥,𝑧)≥⋁_𝑦∈𝑈(𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧))  for any 𝑥,𝑦,𝑧∈𝑈. Denote 𝑎_𝑥𝑧=⋁_𝑦∈𝑈(𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧)). Then, for any 𝐴∈𝐹^(𝑖)(𝑈) and 𝑥∈𝑈, 𝑅𝑅(𝐴)(𝑥)=_𝑦∈𝑈𝑅(𝐴)𝑦=∧𝑅𝑥,𝑦_{𝑦∈𝑈𝑧∈𝑈}𝐴(𝑧)=∧𝑅𝑦,𝑧∧𝑅𝑥,𝑦_{𝑦∈𝑈𝑧∈𝑈}𝐴(𝑧)=∧𝑅𝑦,𝑧∧𝑅𝑥,𝑦_{𝑦∈𝑈𝑧∈𝑈}𝐴(𝑧)∧𝑅≤𝑦,𝑧∧𝑅𝑥,𝑦_{𝑦∈𝑈𝑧∈𝑈}𝐴(𝑧)∧𝑎_𝑥𝑧=_𝑧∈𝑈𝐴(𝑧)∧𝑎_𝑥𝑧≤_𝑧∈𝑈(𝐴(𝑧)∧𝑅(𝑥,𝑧))=𝑅(𝐴)(𝑥). (24) So, 𝑅(𝑅(𝐴))⊆𝑅(𝐴).
Conversely, assume that (IUT) holds. For any 𝑥,𝑦,𝑧∈𝑈, by (IUT), 𝑅𝑅𝑧₁⊆𝑅𝑧₁. (25) By Remark 9(1), 𝑅𝑥,𝑦∧𝑅𝑦,𝑧≤_𝑡∈𝑈(𝑅()(=𝑥,𝑡∧𝑅𝑡,𝑧))_𝑡∈𝑈𝑅(𝑥,𝑡)∧𝑅𝑧₁(𝑡)=𝑅𝑅𝑧₁(𝑥)≤𝑅𝑧₁(𝑥)=𝑅(𝑥,𝑧). (26)
Hence, 𝑅 is transitive.

Corollary 14. Let (𝑈,𝑅) be an IVF approximation space. If 𝑅 is preorder, then 𝑅𝑅(𝐴)=𝑅(𝐴),𝑅𝑅(𝐴)=𝑅(𝐴),𝐴∈𝐹^(𝑖)(𝑈). (27)

Proof. This holds by Theorem 13.

4. Relationships between IVF Relations and IVF Topologies

Let 𝑅 be an IVF relation on 𝑈. We denote 𝜏_𝑅=𝐴∈𝐹^(𝑖)(𝑈)∶𝑅(𝐴),𝜃=𝐴_𝑅=𝑅(𝐴)∶𝐴∈𝐹^(𝑖)(𝑈). (28)

4.1. IVF Topologies Induced by IVF Relations

Theorem 15. Let 𝑅 be an IVF relation on 𝑈. If 𝑅 is reflexive, then 𝜏_𝑅 is an IVF topology on 𝑈.

Proof. (i) By Proposition 11(1), 𝑅(1)=1. Then, 1∈𝜏_𝑅.
By Theorem 13(1), 𝑅(0)⊆0. Then, 𝑅(0)=0. So, 0∈𝜏_𝑅.
(ii) Let 𝐴,𝐵∈𝜏_𝑅. By Proposition 11(4), 𝑅(𝐴∩𝐵)=𝑅(𝐴)∩𝑅(𝐵). (29) Then, 𝑅(𝐴∩𝐵)=𝐴∩𝐵. Thus, 𝐴∩𝐵∈𝜏_𝑅.
(iii) Let {𝐴_𝑗∶𝑗∈𝐽}⊆𝜏_𝑅. Then, 𝑅(𝐴_𝑗)=𝐴_𝑗 for each 𝑗∈𝐽. By Proposition 11(2), 𝑅_𝑗∈𝐽𝐴_𝑗⊇_𝑗∈𝐽𝑅𝐴_𝑗=_𝑗∈𝐽𝐴_𝑗. (30) By Theorem 13(1), 𝑅(⋃_𝑗∈𝐽𝐴_𝑗)⊆⋃_𝑗∈𝐽𝐴_𝑗.
Then, 𝑅(⋃_𝑗∈𝐽𝐴_𝑗)=⋃_𝑗∈𝐽𝐴_𝑗, and so ⋃_𝑗∈𝐽𝐴_𝑗∈𝜏_𝑅.
Thus, 𝜏_𝑅 is an IVF topology on 𝑈.

Definition 16. Let 𝑅 be an IVF relation on 𝑈. If 𝑅 is reflexive, then 𝜏_𝑅 is called the IVF topology induced by 𝑅 on 𝑈.

Theorem 17. Let 𝑅 be a reflexive IVF relation on 𝑈, and let 𝜏_𝑅 be the IVF topology induced by 𝑅 on 𝑈. Then, the following properties hold:(1)𝜏_𝑅⊆𝜃_𝑅,(2)for each 𝐴∈𝐹^(𝑖)(𝑈), int_𝜏𝑅(𝐴)⊆𝑅(𝐴)⊆𝐴⊆𝑅(𝐴)⊆cl_𝜏𝑅(𝐴), (31)(3)for each [𝑎,𝑏]∈[𝐼], [𝑎,𝑏]∈𝜏_𝑅∩𝜏^𝑐_𝑅.

Proof. (1) This is obvious.
(2) For each 𝐴∈𝐹^(𝑖)(𝑈), by Proposition 11(2), we have int_𝜏𝑅(𝐴)=𝐵∶𝐵∈𝜏_𝑅=𝑅,𝐵⊆𝐴(𝐵)∶𝐵∈𝜏_𝑅,𝐵⊆𝐴⊆𝑅(𝐴). (32)
By Propositions 6(4) and 11(3), cl_𝜏𝑅(𝐴)=int_𝜏𝑅𝐴^𝑐𝑐⊇𝑅𝐴^𝑐𝑐=𝑅(𝐴). (33)
By Theorem 13(1), int_𝜏𝑅(𝐴)⊆𝑅(𝐴)⊆𝐴⊆𝑅(𝐴)⊆cl_𝜏𝑅(𝐴). (34)
(3) For each [𝑎,𝑏]∈[𝐼], by Remark 9(2) and Theorem 13(1), 𝑅([𝑎,𝑏])=[𝑎,𝑏]. Then, [𝑎,𝑏]∈𝜏_𝑅. By Proposition 6(4), cl_𝜏𝑅[𝑎,𝑏]=int_𝜏𝑅[𝑎,𝑏]^𝑐𝑐=int_𝜏𝑅[𝑎,𝑏]^𝑐𝑐=[𝑎,𝑏]^𝑐𝑐=[𝑎,𝑏]. (35) So, [𝑎,𝑏]∈𝜏^𝑐_𝑅.

Theorem 18. Let 𝑅 be a reflexive IVF relation on 𝑈, and let 𝜏_𝑅 be the IVF topology induced by 𝑅 on 𝑈. If 𝑅 is transitive, then(1)𝜏_𝑅=𝜃_𝑅,(2)𝑅 is the interior operator of 𝜏_𝑅,(3)𝑅 is the closure operator of 𝜏_𝑅.

Proof. (1) Obviously, 𝜏_𝑅⊆𝑅(𝐴)∶𝐴∈𝐹^(𝑖)(𝑈). (36) By Corollary 14, 𝜏_𝑅⊇{𝑅(𝐴)∶𝐴∈𝐹^(𝑖)(𝑈)}. Then, 𝜏_𝑅={𝑅(𝐴)∶𝐴∈𝐹^(𝑖)(𝑈)}=𝜃_𝑅.
(2) It suffices to show that for each 𝐴∈𝐹^(𝑖)(𝑈), 𝑅(𝐴)=int_𝜏𝑅(𝐴), (37) where int_𝜏𝑅(𝐴)=⋃{𝐵∈𝜏_𝑅∶𝐵⊆𝐴}.
Since 𝑅(𝐴)∈𝜃_𝑅, by (1), 𝑅(𝐴)∈𝜏_𝑅.
By Theorem 13(1), 𝑅(𝐴)⊆𝐴. Then, 𝑅(𝐴)⊆int_𝜏𝑅(𝐴).
By (1), int_𝜏𝑅(𝐴)⊆𝑅(𝐴). Then, 𝑅(𝐴)=int_𝜏𝑅(𝐴).
(3) This holds by (2), Proposition 6(4), and Proposition 11(3).

Example 19. Let 𝑈={𝑥,𝑦,𝑧}, and let 𝑅 be a reflexive IVF relation on 𝑈. 𝑅 is defined as follows: 𝑅𝑥,𝑦=𝑅(𝑥,𝑧)=𝑅(𝑧,𝑥)=𝑅=[],𝑅=0,𝑦,𝑥0.2,0.7𝑦,𝑧1,𝑅𝑧,𝑦=[0.3,0.8]. (38) Pick 𝐴=0𝑥+[0.4,0.5]𝑦+1𝑧,𝐵=1𝑥+[0.5,0.6]𝑦+0𝑧. (39)
(1) We have 𝑅𝑧,𝑦∧𝑅𝑦,𝑥=[0.2,0.7]≰0=𝑅(𝑧,𝑥). (40)
Then, 𝑅 is not transitive.
(2) Since 𝑅(𝐴)=0𝑥+[0.3,0.5]𝑦+[0.4,0.7]𝑧,𝑅𝑅(𝐴)=0𝑥+[0.3,0.5]𝑦+[0.3,0.7]𝑧, (41) we have 𝑅(𝑅(𝐴))≠𝑅(𝐴). Then, 𝑅(𝐴)∉𝜏_𝑅. Thus, 𝜏_𝑅≠𝑅(𝐴)∶𝐴∈𝐹^(𝑖)(𝑈),int_𝜏𝑅(𝐴)≠𝑅(𝐴). (42)
Obviously, 𝐵^𝑐=𝐴. By Proposition 11(3), 𝑅(𝐵)^𝑐=𝑅𝐵^𝑐=𝑅(𝐴)∉𝜏_𝑅. (43) Then, 𝑅(𝐵)∉𝜏^𝑐_𝑅. Thus, cl_𝜏𝑅(𝐵)≠𝑅(𝐵).

4.2. IVF Relations Induced by IVF Topologies

Definition 20. Let 𝜏 be an IVF topology on 𝑈. Define an IVF relation 𝑅_𝜏 on 𝑈 by 𝑅_𝜏𝑥,𝑦=cl_𝜏𝑦₁(𝑥) (44) for each (𝑥,𝑦)∈𝑈×𝑈. Then, 𝑅_𝜏 is called the IVF relation induced by 𝜏 on 𝑈.
An IVF topology 𝜏 on 𝑈 is said to satisfy the following: (C₁)𝑎𝑥𝑖𝑜𝑚:cl_𝜏([𝑎,𝑏]𝐴)=[𝑎,𝑏]cl_𝜏(𝐴)𝑓𝑜𝑟𝑎𝑛𝑦[𝑎,𝑏]∈[𝐼]𝑎𝑛𝑑𝐴∈𝐹^(𝑖)(𝑈), (C₂)𝑎𝑥𝑖𝑜𝑚:cl_𝜏(⋃_𝑗∈𝐽𝐴_𝑗)=⋃_𝑗∈𝐽cl_𝜏(𝐴_𝑗)𝑓𝑜𝑟𝑎𝑛𝑦{𝐴_𝑗∶𝑗∈𝐽}⊆𝐹^(𝑖)(𝑈).

Theorem 21. Let 𝜏 be an IVF topology on 𝑈, and let 𝑅_𝜏 be the IVF relation induced by 𝜏 on 𝑈. Then, the following properties hold.(1)𝑅_𝜏 is reflexive.(2)If 𝜏 satisfies (𝐶₂) axiom and {[𝑎,𝑏]∶[𝑎,𝑏]∈[𝐼]}⊆𝜏, then 𝑅_𝜏(𝐴)⊆int_𝜏(𝐴)⊆𝐴⊆cl_𝜏(𝐴)⊆𝑅_𝜏(𝐴)𝐴∈𝐹^(𝑖)(𝑈). (45)

Proof. (1) For each 𝑥∈𝑈, 𝑅_𝜏(𝑥,𝑥)=cl_𝜏𝑥₁(𝑥)≥𝑥₁(𝑥)=1. (46) Then, 𝑅_𝜏 is reflexive.
(2) Since {[𝑎,𝑏]∶[𝑎,𝑏]∈[𝐼]}⊆𝜏, we have {[𝑎,𝑏]∶[𝑎,𝑏]∈[𝐼]}⊆𝜏^𝑐. For each 𝐴∈𝐹^(𝑖)(𝑈), by Remark 4, (𝐶₂) axiom, and Proposition 11, cl_𝜏(𝐴)=cl_{𝜏𝑦∈𝑈}𝐴𝑦𝑦₁=_𝑦∈𝑈cl_𝜏𝐴𝑦𝑦₁=_𝑦∈𝑈cl_𝜏𝐴𝑦∩𝑦₁⊆_𝑦∈𝑈cl_𝜏𝐴𝑦∩cl_𝜏𝑦₁=_𝑦∈𝑈𝐴𝑦∩cl_𝜏𝑦₁. (47) Then, for each 𝑥∈𝑈, cl_𝜏(𝐴)(𝑥)≤_𝑦∈𝑈𝐴𝑦(𝑥)∧cl_𝜏𝑦₁(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅_𝜏𝑥,𝑦=𝑅_𝜏(𝐴)(𝑥). (48) Hence, cl_𝜏(𝐴)⊆𝑅_𝜏(𝐴).
By Propositions 6(4) and 11(3), int_𝜏(𝐴)=cl_𝜏𝐴^𝑐𝑐⊇𝑅_𝜏𝐴^𝑐𝑐=𝑅_𝜏(𝐴), (49) so 𝑅_𝜏(𝐴)⊆int_𝜏(𝐴)⊆𝐴⊆cl_𝜏(𝐴)⊆𝑅_𝜏(𝐴). (50)

Proposition 22. Let 𝜏 be an IVF topology on 𝑈. If 𝜏 satisfies (𝐶₁) and (𝐶₂) axioms, then(1)𝑅_𝜏 is the closure operator of  𝜏,(2)𝑅_𝜏 is the interior operator of   𝜏,(3)for each [𝑎,𝑏]∈[𝐼],  [𝑎,𝑏]∈𝜏,(4)𝜏 is Alexandrov.

Proof. (1) For each 𝐴∈𝐹^(𝑖)(𝑈), by Remark 4, (𝐶₁) axiom, and (𝐶₂) axiom, cl_𝜏(𝐴)=cl_{𝜏𝑦∈𝑈}𝐴𝑦𝑦₁=_𝑦∈𝑈cl_𝜏𝐴𝑦𝑦₁=_𝑦∈𝑈𝐴𝑦cl_𝜏𝑦₁. (51) Then, for each 𝑥∈𝑈, cl_𝜏(𝐴)(𝑥)=_𝑦∈𝑈𝐴𝑦(𝑥)∧cl_𝜏𝑦₁(𝑥)=_𝑦∈𝑈𝐴𝑦∧𝑅_𝜏𝑥,𝑦=𝑅_𝜏(𝐴)(𝑥). (52) Hence, 𝑅_𝜏(𝐴)=cl_𝜏(𝐴). Thus, 𝑅_𝜏 is the closure operator of 𝜏.
(2) This holds by (1), Proposition 6(4) and, Proposition 11(3).
(3) For each [𝑎,𝑏]∈[𝐼], by (2), Remark 9(2), and Proposition 6(2), [𝑎,𝑏]⊇int_𝜏[𝑎,𝑏]=𝑅[𝑎,𝑏]⊇[𝑎,𝑏]. (53) Then  int_𝜏([𝑎,𝑏])=[𝑎,𝑏], and so [𝑎,𝑏]∈𝜏.
(4) Let {𝐴_𝑗∶𝑗∈𝐽}⊆𝜏. By (2), for each 𝑗∈𝐽, 𝐴_𝑗=int_𝜏𝐴_𝑗=𝑅𝐴_𝑗. (54) By (2) and Proposition 11(4), _𝑗∈𝐽𝐴_𝑗=_𝑗∈𝐽𝑅𝐴_𝑗=𝑅_𝑗∈𝐽𝐴_𝑗=int_{𝜏𝑗∈𝐽}𝐴_𝑗. (55) So ⋂_𝑗∈𝐽𝐴_𝑗∈𝜏.
Hence, 𝜏 is Alexandrov.

Proposition 23. Let 𝑅 be a preorder IVF relation on 𝑈. Then, 𝜏_𝑅 satisfies (𝐶₁) and (𝐶₂) axioms.

Proof. For any [𝑎,𝑏]∈[𝐼] and 𝐴∈𝐹^(𝑖)(𝑈), by Theorem 18(3) and Proposition 11(5), cl_𝜏𝑅([𝑎,𝑏]𝐴)=𝑅([𝑎,𝑏]𝐴)=[𝑎,𝑏]𝑅(𝐴)=[𝑎,𝑏]cl_𝜏𝑅(𝐴). (56) Thus,  𝜏_𝑅 satisfies (𝐶₁) axiom.
For any {𝐴_𝑗∶𝑗∈𝐽}⊆𝐹^(𝑖)(𝑈), by Proposition 11(4) and Theorem 18, cl_{𝜏𝑅𝑗∈𝐽}𝐴_𝑗=cl_{𝜃𝑅𝑗∈𝐽}𝐴_𝑗=𝑅_𝑗∈𝐽𝐴_𝑗=_𝑗∈𝐽𝑅𝐴_𝑗=_𝑗∈𝐽cl_𝜃𝑅𝐴_𝑗=_𝑗∈𝐽cl_𝜏𝑅𝐴_𝑗. (57) Thus, 𝜏_𝑅 satisfies (𝐶₂) axiom.

5. IVF Approximating Spaces

As can be seen from Section 4, 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 𝜏_𝑅=𝜏, then (𝑈,𝜏) is called an IVF approximating space.

Theorem 25. Let 𝜏 be an IVF topology on 𝑈. Let 𝑅_𝜏 be the IVF relation induced by (𝑈,𝜏), and let 𝜏_𝑅𝜏 be the IVF topology induced by 𝑅_𝜏 on 𝑈. If 𝜏 satisfies (𝐶₁) and (𝐶₂) axioms, then 𝜏_𝑅𝜏=𝜏.

Proof. By Theorem 21(1), 𝑅_𝜏 is reflexive. For any 𝑥,𝑦,𝑧∈𝑈, put cl(𝑧₁)(𝑦)=[𝑎,𝑏]. By Remark 4 and Proposition 11(2), [𝑎,𝑏]cl_𝜏𝑦₁=cl_𝜏[𝑎,𝑏]𝑦₁=cl_𝜏𝑐𝑙_𝜏𝑧₁𝑦𝑦₁⊆cl_{𝜏𝑡∈𝑈}𝑐𝑙_𝜏𝑧₁(𝑡)𝑡₁=cl_𝜏𝑐𝑙_𝜏𝑧₁=cl_𝜏𝑧₁. (58) Then, 𝑅_𝜏𝑥,𝑦∧𝑅_𝜏𝑦,𝑧=cl_𝜏𝑦₁(𝑥)∧cl_𝜏𝑧₁𝑦=cl_𝜏𝑦₁𝑥)∧[]=[](𝑎,𝑏𝑎,𝑏∧cl_𝜏𝑦₁(𝑥)=[𝑎,𝑏]cl_𝜏𝑦₁(𝑥)≤cl_𝜏𝑧₁(𝑥)=𝑅_𝜏(𝑥,𝑧). (59) So, 𝑅 is transitive.
So, 𝑅_𝜏 is preorder. For each 𝐴∈𝐹^(𝑖)(𝑈), by Theorem 18, cl_𝜏𝑅𝜏(𝐴)=cl_𝜃𝑅𝜏(𝐴)=𝑅_𝜏(𝐴). (60) Since 𝜏 satisfies (𝐶₁) and (𝐶₂) axioms, by Proposition 22(1), 𝑅_𝜏(𝐴)=cl_𝜏(𝐴). So, cl_𝜏𝑅𝜏(𝐴)=cl_𝜏(𝐴).
Thus, 𝜏_𝑅𝜏=𝜏.

Theorem 26. Let 𝜏 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 {𝐴_𝑗∶𝑗∈𝐽}⊆𝐹^(𝑖)(𝑈), int_𝜏[𝑎,𝑏]∪𝐴=[𝑎,𝑏]∪int_𝜏(𝐴),int_{𝜏𝑗∈𝐽}𝐴_𝑗=_𝑗∈𝐽int_𝜏𝐴_𝑗. (61)(3)There exists a preorder IVF relation 𝜌 on 𝑈 such that 𝜌 is the closure operator of 𝜏.(4)There exists a preorder IVF relation 𝜌 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) is obvious.
(1) ⟹ (3). Suppose that 𝜏 satisfies (𝐶₁) and (𝐶₂) axioms. Pick 𝜌=𝑅_𝜏. 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), 𝜌(𝐴)=𝜌𝐴^𝑐𝑐=cl_𝜏𝐴^𝑐𝑐=int_𝜏(𝐴). (62) 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), 𝜌𝑥,𝑦=1−𝜌𝑦₁^𝑐(𝑥)=1−int_𝜏𝑦₁^𝑐(𝑥)=cl_𝜏𝑦₁(𝑥)=𝑅_𝜏𝑥,𝑦. (63) Then, 𝜌=𝑅_𝜏. Note that 𝜌 is the interior operator of 𝜏. Then, 𝑅_𝜏 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), cl_𝜏([𝑎,𝑏]𝐴)=𝑅_𝜏([𝑎,𝑏]𝐴)=[𝑎,𝑏]𝑅_𝜏(𝐴)=[𝑎,𝑏]cl_𝜏(𝐴). (64) Thus, 𝜏 satisfies (𝐶₁) axiom.
For any {𝐴_𝑗∶𝑗∈𝐽}⊆𝐹^(𝑖)(𝑈), by Proposition 11(4), cl_{𝜏𝑗∈𝐽}𝐴_𝑗=𝑅_{𝜏𝑗∈𝐽}𝐴_𝑗=_𝑗∈𝐽𝑅_𝜏𝐴_𝑗=_𝑗∈𝐽cl_𝜏𝐴_𝑗. (65) 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), int[𝑎,𝑏]∪𝐴=[𝑎,𝑏]∪int(𝐴). (66)(3)There exists a preorder IVF relation 𝑅 on 𝑈 such that 𝑅 is the closure operator of 𝜏.(4)There exists a preorder IVF relation 𝑅 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 Theorems 25 and 26.

Example 28. {[𝑎,𝑏]∶[𝑎,𝑏]∈[𝐼]} is an IVF approximating space.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11061004), the Natural Science Foundation of Guangxi (no. 2013GXNSFBA019016), and Guangxi University Science and Technology Research Project (no. 2013ZD020).