# On Metrization of the Topologies Induced by Fuzzy Metrics

Authors: Jianrong Wu; Hao Yang
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we focus on finding the metric functions in a fuzzy metric space. After introducing the concept of a stratified function in a fuzzy metric space, we prove that the topology generated by the family of stratified functions coincides with the topology generated by the fuzzy metric. Moreover, we get the concrete form of metric function under some special conditions.


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## Body

## 1. Introduction

Since Zadeh [1] first proposed fuzzy set theory in 1965, many researchers have defined concepts of fuzzy metric spaces and studied their properties in different ways [2–4]. Inspired by the notion of probabilistic metric spaces, Kramosil and Michalek [5] in 1975 introduced the notion of fuzzy metric, a fuzzy set in the Cartesian product X×X×R satisfying certain conditions. Later, George and Veeramani [6] used the concept of continuous t-norms to modify this definition of fuzzy metric space and showed that every fuzzy metric space generates a Hausdorff first-countable topology. So far the GV-fuzzy metric theory has been developed by many researchers. Many important topics about the classical metric spaces were developed to fuzzy metric spaces. In this process, it was found that the theory of fuzzy metric was very different from the classical theory of metric. For example, Gregori and Romaguera [7] proved that there exists a GV-fuzzy metric space which is not completable. Gregori and Romaguera [8] characterized the class of completable strong fuzzy metric spaces. Recently, some good related results about fixed point in fuzzy metric spaces were introduced. For example, one can see the works [9–11]. Meanwhile, the fuzzy metrics have been applied to domain theory, color image processing, and analysis of algorithms (see [12–17]).In 2000, Gregori and Romaguera [18] obtained a somewhat surprising stronger result. They proved that every GV-fuzzy metric generates a metrizable topology. This important result connects the GV-fuzzy metric and the classical metric. However, the form of the metric function has not been explored in existing literature. It is just the main goal of the present paper.In this paper, we first introduce the concept of a stratified function in a fuzzy metric space which is slightly different from the GV-fuzzy metric space and show that the topology induced by the family of stratified functions is compatible with the metrizable topology. Then, under some special conditions, we give the concrete metric function whose topology coincides with the metrizable topology.The structure of the paper is as follows. In the next section, we give the preliminary notions on fuzzy metrics, with which we deal. In Section3, we show our main results. Finally, we give our concluding remarks in Section 4.


## 2. Preliminaries

In this section, we first introduce some basic concepts and properties of fuzzy metric spaces.Definition 1.
(see [19]). A binary operation ∗: 0,1×0,1⟶0,1 is a continuous t-norm if it satisfies the following conditions:(1)
∗ is associative and commutative(2)
∗ is continuous(3)
a∗1=a for each a∈0,1(4)
a∗b≤c∗d whenever a≤c and b≤d with a,b,c,d∈0,1
The following continuoust-norms are used in this paper:(1)aΔ1b=maxa+b−1,0,aΔ2b=ab,aΔ3b=mina,b.
Also, we say thet-norm Δ′ is stronger than the t-norm Δ″ if aΔ′b≥aΔ″b for all a,b∈0,1. In such case, we denote it as Δ′≥Δ″. It is easy to see that Δ3≥Δ2≥Δ1.
In the sense of Gregori and Veeramani [6], a GV-fuzzy metric is defined by the follows.Definition 2.
LetX be a nonempty set and ∗ be a continuous t-norm. A fuzzy metric M on the set X is a mapping M: X2×0,∞⟶0,1 satisfying the following conditions: for all x,y,z∈X, s,t>0:(2)GV1Mx,y,t>0,GV2Mx,y,t=1if and only ifx=y,GV3Mx,y,t=My,x,t,GV4Mx,y,t∗My,z,s≤Mx,z,t+s,GV5Mx,y,⋅:0,∞⟶0,1is continuous.
IfM is a GV-fuzzy metric on X, then the 3-tuple X,M,∗ is said to be a GV-fuzzy metric space. In that case, if confusion is not possible, we call X a GV-fuzzy metric space for short. The following is a well-known result.Lemma 1.
(see [20]). Mx,y,⋅ is increasing for all x,y∈X.
Gregori and Veeramani proved in [7] that every GV-fuzzy metric M on X generates a topology τM which has as a base(3)BMx,r,t:x∈X,r∈0,1,t>0,where(4)BMx,r,t=y∈X:Mx,y,t>1−r,for all x∈X, r∈0,1, and t>0. They proved that for each x∈X, the family BMx,1/n,1/n:n∈N is a local base at x. A sequence xn in X,τM converges to x∈X if and only if limn⟶∞Mxn,x,t=1 for all t>0. Also, by using Kelley metrization lemma [21], they also proved that τM is a metrizable topology.


## 3. Main Results

First, we introduce the concept of a stratified function in a GV-fuzzy metric space.Definition 3.
X,M,∗ is a GV-fuzzy metric space. Let r∈0,1 and x,y∈X; set(5)drx,y=inft>0:Mx,y,t>1−r.
Then,dr is called a r-stratified function with respect to X,M,∗, dr:0<r<1, the family of stratified functions.
To avoid the occurrence of the empty set, by a fuzzy metric in the rest of this paper, we mean a GV-fuzzy metric satisfying(6)GV6limt⟶∞Mx,y,t=1,∀x,y∈X.Lemma 2.
LetX,M,∗ be a fuzzy metric space, r∈0,1, t>0, x,y∈X. Then,(1)
For anyλ>drx,y, Mx,y,λ>1−r(2)
The functiondr is decreasing with respect to r∈0,1(3)
BMx,r,t=Nrx,t, where(7)Nrx,t=y∈X:drx,y<t.(4)
The functionMx,y,⋅ is strictly increasing for the fixed points x,y∈X, if and only if for any r∈0,1,(8)drx,y=inft>0:Mx,y,t≥1−r.Proof.
(1)
Letλ>drx,y. From Definition 3, there exists 0<t<λ such that Mx,y,t>1−r. So, Mx,y,λ>1−r.(2)
It follows from Lemma1 directly.(3)
Lety∈BMx,r,t, that is, Mx,y,t>1−r. Since Mx,y,⋅ is continuous and increasing, there exists 0<t1<t such that Mx,y,t1>1−r. From (5), we know drx,y≤t1<t. So, y∈Nrx,t. From the arbitrariness of y, we know that(9)BMx,r,t⊆Nrx,t.Letz∈Nrx,t; then, drx,z<t. From (5), there exists 0<t2<t such that Mx,z,t2>1−r. Therefore, Mx,z,t≥Mx,z,t2>1−r. So, z∈BMx,r,t. From the arbitrariness of z, we know that Nrx,t⊆BMx,r,t.(4)
Suppose that (8) holds; however, Mx,y,⋅ is not strictly increasing. Then, there exist t1,t2∈t>0:0<Mx,y,t<1 such that t1<t2 and Mx,y,⋅≡1−r0 on t1,t2. Thus,(10)supt>0:Mx,y,t≤1−r0≥t2>t1≥inft>0:Mx,y,t≥1−r0.
It is easy to see that(11)supt>0:Mx,y,t≤1−r0=inft>0:Mx,y,t>1−r0.
Therefore,dr0x,y>inft>0:Mx,y,t≥1−r0, which conflicts with (8).
Conversely, supposeMx,y,⋅ is strictly increasing. Let(12)t0=drx,y=inft>0:Mx,y,t>1−r.
Obviously,inft>0:Mx,y,t≥1−r≤t0. If inft>0:Mx,y,t≥1−r<t0, then there is 0<t2<t0 such that Mx,y,t2≥1−r, so Mx,y,t0>1−r. Since Mx,y,⋅ is left continuous at t0, there is δ>0 such that Mx,y,t0−δ>1−r, which conflicts with the definition of t0. Thus, inft>0:Mx,y,t≥1−r≥t0. So,(13)drx,y=t0=inft>0:Mx,y,t>1−r.
Now, from Lemma2 (3), it is easy to see that the topology τM can be induced by the family of stratified functions. That is, we obtain the following theorem.Theorem 1.
LetD=dr:0<r<1 be the family of stratified functions with respect to a fuzzy metric space X,M,∗, Nrx,t be defined by (8), and(14)Bx=Nrx,t:r∈0,1,t>0.
Then,(1)
Bx is a base of neighborhoods at x∈X.(2)
The topologyτD generated by Bx:x∈X coincides with the topology τM.
Generally, a stratified function is not a pseudometric. In fact, we have the following result.Theorem 2.
LetX,M,∗ be a fuzzy metric space. A stratified function dr (r∈0,1) is a pseudometric on X if and only if M satisfies the following condition: for any x,y,z∈X, t1,t2>0, if Mx,z,t1>1−r, Mz,y,t2>1−r, then(15)Mx,y,t1+t2>1−r.Proof.
For anyr∈0,1, it is obvious that drx,y≥0, drx,y=dry,x, and drx,y=0 when x=y. Thus, to complete the proof, we only have to prove that drx,y≤drx,z+drz,y if and only if M satisfies condition (15).Sufficiency.
For anyε>0, from Lemma 2 (1), we obtain(16)Mx,z,drx,z+ε2>1−r,Mz,y,drz,y+ε2>1−r.
From (15), we have(17)Mx,y,drx,z+drz,y+ε>1−r.
Therefore,(18)drx,y≤drx,z+drz,y+ε.
From the arbitrariness ofε>0, we know drx,y≤drx,z+drz,y.Necessity.
Suppose thatMx,z,t1>1−r, Mz,y,t2>1−r. By (GV5), there exists δ>0 such that(19)Mx,z,t1−δ>1−r,Mz,y,t2−δ>1−r.
So,drx,z≤t1−δ and drz,y≤t2−δ. Since drx,y≤drx,z+drz,y, we have drx,y≤t1+t2−2δ<t1+t2. By the definition of drx,y, there exists t0<t1+t2 such that Mx,y,t0>1−r. Thus, Mx,y,t1+t2>1−r.Remark 1.
M satisfies (15) if ∗ = Δ3.
Now, we explore the metric which induces the topologyτM.Definition 4.
LetR+ = 0,∞. We call function f: R+⟶R+ satisfies(i)
The conditionC1, if f0=0, ft≡0, and f is increasing and continuous at 0.(ii)
The conditionC2, if f0=0, ft>0 as t>0, limt⟶+∞ft=+∞, ft1+t2≤ft1+ft2 for any t1,t2∈R+, and f is left continuous and increasing.Theorem 3.
X,M,∗ is fuzzy metric space; the functions K and G satisfy the conditions C1 and C2, respectively. Define a function d on X×X as(20)dx,y=infs>0:ifGt>s,thenMx,y,t+Ks≥1,∀x,y∈X.
If one of the following conditions is satisfied:(I)M satisfies condition (15)(II)∗≥Δ1, ∀r1,r2∈0,∞(21)Kr1+Kr2≤Kr1+r2,then d is a metric on X.Proof.
First, we prove the following fact: if(22)Gt>r>dx,y,then(23)Mx,y,t≥1−Kr.
In fact, ifGt>r>dx,y, from the definition of d, we obtain there exists 0<s<r such that if Gt>s, then Mx,y,t+Ks≥1. Therefore, if Gt>r, then Gt>s, and hence, Mx,y,t+Kr≥Mx,y,t+Ks≥1. That is, Mx,y,t≥1−Kr.
Next, we proved is a metric on X, that is, d satisfies the following properties: for any x,y,z∈X,(M1) dx,y≥0, dx,y=dy,x(M2) dx,y = 0 if and only if x=y(M3) dx,y≤dx,z+dz,y
The conclusion (M1) is obvious.
For the conclusion (M2), it is easy to see that dx,y = 0 if x=y. Now, we suppose dx,y = 0; however, x≠y. Then, there exists t0>0 such that Mx,y,t0≠1, that is, Mx,y,t0<1. Since K is continuous at 0, there exists 0<r0<Gt0 such that Kr0<1−Mx,y,t0. This is in direct contradiction to (23). Thus, dx,y = 0 implies that x=y.
To prove (M3), we take r1>dx,z and r2>dz,y arbitrarily. From (23), we know(i)
IfGt>r1, then(24)Mx,z,t≥1−Kr1≥1−Kr1+r2.(ii)
IfGt>r2, then(25)Mz,y,t≥1−Kr2≥1−Kr1+r2.
Now, suppose thatGt>r1+r2. Let(26)t0=infs:0<s≤t,Gs>r1.
Obviously,t0≤t. It is easy to prove that Gt0≤r1. In fact, if Gt0>r1, from the left continuity of G, there exists η>0 such that Gt0−η>r1. By the definition of t0, we conclude that t0≤t0−η, a contradiction. The fact Gt0≤r1 implies that Gt−t0≥Gt−Gt0>r2. By the left continuity of G again, we know there exists ε>0 such that Gt−t0+ε=Gt−t0−ε>r2. By the definition of t0, there exists 0<s1≤t such that Gs1>r1 and s1≤t0+ε. Noting that G is increasing, we obtain that Gt0+ε≥Gs1>r1. From (i) and (ii), we have(27)Mx,z,t0+ε≥1−Kr1≥1−Kr1+r2Mz,y,t−t0+ε≥1−Kr2≥1−Kr1+r2.
Combining conditions (I) and (II), we getMx,y,t≥1−Kr1+r2.
By the definition ofd, we know dx,y≤r1+r2. Since r1>dx,z and r2>dz,y, we obtain(28)dx,y≤dx,z+dz,y,directly.Theorem 4.
X,M,∗ is a fuzzy metric space. M satisfies condition (15) or ∗≥Δ1. Let(29)ρx,y=infa>0:Mx,y,a+a≥1,∀x,y∈X.
Then,ρ is a metric on X and the topology τρ induced by ρ coincides with the topology τM.Proof.
LetGt=t,Kt=t, ∀t≥0. Then, K and G satisfy the conditions C1 and C2, respectively. Besides, K satisfies condition (21). From Theorem 3, we know that(30)dx,y=infs>0:ift>s,thenMx,y,t+s≥1,∀x,y∈X,is a metric on X. Thus, to show that ρ is a metric, we only need to show dx,y = ρx,y, ∀x,y∈X. In fact, if a>0, Mx,y,a+a≥1 and t>a, then Mx,y,t+a≥Mx,y,a+a≥1. Therefore,(31)a>0:Mx,y,a+a≥1⊆s>0:ift>s,thenMx,y,t+s≥1.
From (29) and (30), we get ρx,y≥dx,y. On the other hand, from (30), for any ε>0, there exists s>0 such that s<dx,y+ε and Mx,y,t+s≥1 when t>s. Thus,(32)Mx,y,dx,y+ε+s≥1,and hence(33)Mx,y,dx,y+ε+dx,y+ε>1.
From (29), we get ρx,y≤dx,y+ε. From the arbitrariness of ε>0, we have ρx,y≤dx,y. Thus, dx,y = ρx,y.
Letr∈0,1, x∈X. We put Ux,r=y∈X:ρx,y<r. It is easy to show that(34)Ux,r⊆BMx,r,r⊆Ux,2r.
In fact, for anyy∈Ux,r, infa>0:Mx,y,a+a≥1<r. Thus, there is a>0 such that a<r and Mx,y,a+a≥1. So, Mx,y,r+r>1, that is, y∈BMx,r,r. Hence,(35)Ux,r⊆BMx,r,r.
On the other hand, for anyz∈BMx,r,r, Mx,z,r+r>1. From (29), we get ρx,z≤r<2r. So, z∈Ux,2r. Hence,(36)BMx,r,r⊆Ux,2r.
And (34) holds, which implies that τρ=τM directly.Lemma 3.
X,M,∗ is fuzzy metric space; the functions K and G satisfy the conditions C1 and C2, respectively. Then, the function d defined by (20) can be represented as follows,∀x,y∈X:(37)dx,y=sups>0:ifGt>s,thenMx,y,t+Ks<1.Proof.
In fact, we only need to show thatinfA=supB, where(38)A=a>0:ifGt>a,thenMx,y,t+Ka≥1,B=b>0:ifGt>b,thenMx,y,t+Kb<1.
Takea∈A, b∈B arbitrarily. Suppose b>a. If Gt>b, then Gt>a. From the definitions of A and B, we obtain(39)Mx,y,t+Ka≥1>Mx,y,t+Kb.
This is in direct contradiction to the condition thatK is increasing. So, a≥b, and hence, infA≥supB.
Now, letinfA=α,supB=β. For any δ>0, we know β+δ∉B, that is, β+δ∈A, which implies that infA≤β+δ. By the arbitrariness of δ, we know that infA≤β=supB. This completes the proof.Theorem 5.
X,M,∗ is a fuzzy metric space. If M satisfies condition (15) or ∗≥Δ1, then ∀x,y∈X:(40)ρx,y=supt>0:Mx,y,t+t<1,where ρ is defined by (29).Proof.
LetGt=t,Kt=t, ∀t≥0. Then, (40) follows from Lemma 3 directly.


## 4. Conclusions

In the present paper, we investigate some metric structures in a fuzzy metric space. Especially, we give the concrete form of metric function with respect to the metrizable topology for a fuzzy metric under two special cases. Based on the results in the paper, interesting future research studies about the related topics may be prospective. Moreover, the technique used in this paper is suggestive to discuss the related problem in the general case.



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