Estimation of fuzzy metric spaces from metric spaces


 

 

 

 

Ratio Mathematica                                             
 

64 
 

 

 

Estimation of fuzzy metric spaces from 

metric spaces 
 

 

Senthil Kumar Pichai* 

Thiruveni Packirisamy† 

 
                                  

 

Abstract 

In this paper we estimate the new way for analyzing the fuzzy metric spaces 

from metric spaces using fuzzy fixed point theorem and vice versa. We derive 

some definitions and theorems for analyzing the metric spaces with new 

structure of fuzzy metric space using fixed point theorems. Also, we have 

given new examples for fuzzy metric spaces using fixed point theorem. 

Keywords: metric space, fuzzy logic, fixed point theorem. 

2020 AMS subject classifications: 54E35, 03B52, 47H10 

 

 

 

 

 

 

 

 
  * Department of Mathematics, Rajah Serfoji Government College (Autonomous) (Affiliated to    

Bharathidasan University), Thanjavur, India;  pskmaths@rsgc.ac.in. 
  †Department of Mathematics, Rajah Serfoji Government College (Autonomous) (Affiliated to  

Bharathidasan University), Thanjavur, India;   smile.thiruveni@gmail.com. 
Received on October 10, 2021. Accepted on December 15, 2021. Published on December 31, 

2021. doi: 10.23755/rm.v41i0.670. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This 

paper is published under the CC-BY licence agreement. 

 
 

Volume 41, 2021, pp. 64-70

mailto:pskmaths@rsgc.ac.in
mailto:smile.thiruveni@gmail.com


 

Estimation of fuzzy metric spaces from metric spaces 

65 
 

1. Introduction 
      In earlier of 1906, Maurice Freechet provided the concept of metric spaces. 

In 1922, Banach developed a reliable result called Banach contraction principle 

on the basis of fixed point theory [15]. Later in 1965, Zadeh [10] defined the 

way of fuzzy set in metric spaces. This fuzzy concept used in various field of 

engineering, science and technology. This fuzzy concept is used to analyze the 

complex state as to easy by simple condition and conversion. Then several 

mathematicians gave their various concept towards the concept of fuzzy metric 

space Erceg [4] [5], Diamond and Kolden [7], George & Veeramani [6], Gregori 

and Romaguera [11]. The analysis of fuzzy metric spaces was introduced in 

various way and its topologies developed by many researchers.  On the line of 

this, we frame a new structure to analyze the fuzzy metric spaces from metric 

spaces using fuzzy fixed point theorem [13] [14]. In this paper, we discuss and 

give various definition and theorem to estimate the fuzzy metric space from 

metric space and converse also using  fixed point theorem[1][9][12][16]. 

2. Prefatory 
 In this part, we look back on some basic concepts and results in both 

metric and fuzzy based metric spaces. 

Definition 2.1: [2] A metric space is given by a set 𝑋 and a distance function d̅ ∶
 X ×  X → R defined on 𝑋 such that 𝑎,𝑏,𝑐 ∈ 𝑋 

(𝑖) d̅(𝑎,𝑏) ≥ 0,d̅(𝑎,𝑏) = 0 ↔ 𝑎 = 𝑏 

(𝑖𝑖) d̅(𝑎,𝑏) = d̅(𝑏,𝑎) 

(𝑖𝑖𝑖) d̅(𝑎,𝑐) ≤ d̅(𝑎,𝑏)+ d̅ (𝑏,𝑐) 

 

Definition 2.2: [10] A fuzzy set 𝐴 in 𝑋 is a function with domain 𝑋 and values 
in [0, 1]. 

Definition 2.3: [8] A binary operation ∗ ∶ [0,1]
2
→ [0,1] is called a continuous 

triangular norm (called t- norm) if it satisfies the following conditions: 

(i) * is associative and commutative, 

(ii) * is continuous, 

(iii) 𝑝∗1 = 𝑝 for all 𝑝,𝑞,𝑟 ∈ [0,1], 

(iv) 𝑝∗𝑞 ≤ 𝑟 ∗𝑡 whenever 𝑝 ≤ 𝑟 and 𝑞 ≤ 𝑝 for all 𝑝,𝑞,𝑟,𝑡 ∈ [0,1] 



 

Senthil Kumar Pichai and Thiruveni Packirisamy 

 

66 
 

Examples of t-norm are 𝑝∗𝑞 = 𝑝𝑞,𝑝 ∗𝑞 = min{𝑝,𝑞} and  𝑝∗𝑞 = max {𝑝,𝑞}. 

Definition 2.4: [3, 6] The 3-tuple (𝑋,𝐹�̅�,∗) is called a fuzzy metric space if 𝑋 is 
an arbitrary (non-empty) set, * is a continuous t-norm and 𝐹�̅� is a fuzzy set on 
𝑋2 ×[0,∞) satisfying the following conditions, for all 𝑎,𝑏,𝑐 ∈ 𝑋, each t and 
𝑢 > 0 
(i) 𝐹�̅�(𝑎,𝑏,𝑡) > 0 
(ii) 𝐹�̅�(𝑎,𝑏,𝑡) = 0 if and only if 𝑎 = 𝑏, 
(iii) 𝐹�̅�(𝑎,𝑏,𝑡) ∗𝐹�̅�(𝑏,𝑎,𝑡), 
(iv) 𝐹�̅�(𝑎,𝑏,𝑡) ∗𝐹�̅�(𝑏,𝑐,𝑢) ≤ 𝐹�̅�(𝑎,𝑐,𝑡 +𝑢), 
(v) 𝐹�̅�(𝑎,𝑏,°):(0,∞) → [0,1] is continuous. 
Then is 𝐹�̅� called a fuzzy metric on 𝑋. Then 𝐹�̅�(𝑎,𝑏,𝑡) denotes the degree of 
nearness between 𝑎 and 𝑏 with respect to 𝑡. 

Lemma 2.5: (𝑋,𝐹�̅�,∗) is non-decreasing for all 𝑎,𝑏 ∈ 𝑋. 
 

3.  Main Results 

  The main aim of this paper is to estimate the fuzzy metric spaces from any 

ordinary metric spaces and converse also and justify the Banach fixed point. 

Theorem 3.1: Let �̅� and 𝐹�̅� are metric and fuzzy metric respectively, so the 
following diagram   

 

                                           𝑋  ×   𝑋 ×   ℛ́+  
𝐹�̅�
⇒     𝐼 

     𝑑𝑝𝑟 ↓                                 ↑  𝛽 

𝑋   ×   𝑋                  
�̅�
⇒     ℛ́ + 

Figure (1): commutative diagram 

 

is commutative. Where, �̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some 

metric �̅�(𝑎,𝑏) = 𝑦 > 0 and 𝛽:(𝑡𝑦) → 1−sin(𝑡𝑦) =: �̂�𝜖𝐼. Further more 𝐹�̅� =
𝛽 °  �̅� ° �̅�𝑥𝑦. 

Proof: According to the below equation it is very easy to check that 𝛽 is 
continuous. 

sin−1(𝑡𝑦) in ℛ́+  is continuous ⟹ 𝛽 is continuous. 
Now we justify that �̅� ° �̅�𝑥𝑦 = 𝛽

−1 ° 𝐹�̅�. For (a,b,c) ∈ X×X×ℛ́
+, we have, 

 �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = �̅�(𝑡𝑎,𝑡𝑎𝑏) = 𝑡𝑦 ≔ 𝑞 > 0. 

On the other hand, 

𝛽−1 °𝐹�̅�(𝑎,𝑏,𝑡) = 𝛽
−1(�̂�) = 𝛽−1(1−sin𝑡𝑦)= sin−1{1−(1−sin𝑡𝑦)} = 𝑡𝑦 

Therefore, the above diagram (Figure 1) is commutative. 



 

Estimation of fuzzy metric spaces from metric spaces 

67 
 

Lemma 3.2: 

Let 𝑡1, 𝑡2 ∈ ℛ́
+, if 𝑡1 ≤ 𝑡2, then 𝛽(𝑡1) ≥ 𝛽(𝑡2) and 𝛽(𝑡1 +𝑡2) ≥

max { 𝛽(𝑡1),𝛽(𝑡2)}. 
Proof: 

If 𝑡1 ≤ 𝑡2, this     ⟹ sin𝑡1 ≤ sin𝑡2 ⟹ −sin𝑡1 ≥ −sin𝑡2 ⟹  1−
sin𝑡1 ≥ 1−sin𝑡2 

Hence, 𝛽(𝑡1) ≥ 𝛽(𝑡2). Now, the second declaration is clear. 
 

Theorem 3.3: Let (𝑋,  �̅�) be the metric space and (𝑋,𝐹�̅�,∗) is a fuzzy metric 
space with                𝑝∗𝑞 = max {𝑝,𝑞} for all 𝑝,𝑞 ∈ 𝐼. Then for all 𝑎,𝑏,𝑐 ∈
𝑋,𝑡,𝑞,𝑦 ∈ ℛ́+, we have (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗) is a fuzzy metric space. 

Proof: 

We verify the fuzzy metric space conditions (i), (ii), (iii) from the above 

definition (2.4), 

For (i) 𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = 𝛽 °  �̅�(𝑡𝑎,𝑡𝑏) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑏)} = �̂� > 0 .  

For (ii) 𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎,𝑎,𝑡) = 𝛽 °  �̅�(𝑡𝑎,𝑡𝑎) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑎)} = 𝛽(0) = 1  

For (iii) 𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = 𝛽 °  �̅�(𝑡𝑎,𝑡𝑏) = 𝛽 { �̅�(𝑡𝑎,𝑡𝑏)} = 𝛽(𝑡𝑦) = �̂�. 

Another side, 

𝛽 °  �̅� ° �̅�𝑥𝑦(𝑏,𝑎,𝑡) = 𝛽 °  �̅�(𝑡𝑏,𝑡𝑎) = 𝛽 { �̅�(𝑡𝑏,𝑡𝑎)} = 𝛽(𝑡𝑦) = �̂� > 0 

Therefore, symmetric condition satisfied. 

Next, we check the condition (iv) 

𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎,𝑐,𝑡 +𝑢) = 𝛽 °  �̅� {(𝑡+𝑢)𝑎,(𝑡+𝑢)𝑐) 

 =𝛽 {�̅�(𝑡+𝑢)𝑎,(𝑡+𝑢)𝑐) 
 =𝛽 { �̅�(𝑡+𝑢)𝑟)} 
 =𝛽 (𝑡𝑟 +𝑢𝑟) 

Using the above lemma,  𝛽(𝑡𝑟 +𝑢𝑟) ≥ max{𝛽(𝑡𝑟),𝛽(𝑢𝑟)} 
 =𝛽(𝑡𝑟)∗𝛽(𝑢𝑟) 
  =𝛽{�̅�(𝑡𝑎,𝑡𝑏)}∗𝛽{�̅�(𝑢𝑎,𝑢𝑐)} 
   =𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) ∗𝛽 °  �̅� ° �̅�𝑥𝑦(𝑏,𝑎,𝑢) 

For (v) is trivially true. Therefore, (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗) is a fuzzy metric space. 

Comments: 
On the other hand we reach the metric space from the fuzzy metric space 
basis on the above commutative diagram (Figure 1). So, if (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗

) is a fuzzy metric space, then the associative metric is 

(𝑋 ,𝛽 −1 ° 𝐹�̅� °  �̅�𝑥𝑦
−1
). 

Definition 3.4: Let (X, �̅�) be a metric space on X, and {𝑎𝑛} be a sequence in X 
the n is {𝑎𝑛} called converge Sequence to some fixed 𝑎 ∈ 𝑋 𝑖𝑓 𝜀 > 0,𝑁 𝜖 ℕ,  
�̅�(𝑎𝑛,𝑎) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 > 𝑁 
We represent 𝑎𝑛 → a if {𝑎𝑛} converge to 𝑎 ; and {𝑎𝑛} is called a Cauchy 



 

Senthil Kumar Pichai and Thiruveni Packirisamy 

 

68 
 

sequence.          �̅�(𝑎𝑛,𝑎𝑚) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛,𝑚 > 𝑁 
Definition 3.5: Let (X, d̅)and(𝑋,𝐹�̅�,∗) are metric and fuzzy metric space on X, 

respectively. And {𝑎𝑛} is a sequence in 𝑋 then the following is equivalent. 
(i) {𝑎𝑛} is convergent in the metric space (𝑋,�̅�) 

(ii)  �̅�(𝑎𝑛,𝑎) < 𝜖 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 > 𝑁 

(iii) {𝑎𝑛} is convergent in the fuzzy metric space (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗) 

(iv) For any 0 < 𝜀 < 1 and 𝑡 > 0 there exists 𝑛 > 𝑁 such that 
𝛽 °  �̅� ° �̅�𝑥𝑦(𝑎𝑛,𝑎, 𝑡) > 1−𝜀 

Definition 3.6: A metric space (𝑋,𝑑̅) is complete if every Cauchy sequence in 

𝑋 is convergent. 

Definition 3.7: A fuzzy metric space (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗) is complete if and only 

if (𝑋,𝑑̅) is complete. 

In the below mentioned theorem we prove that if any self map has fixed point 

theorems in the metric space, then we induced fuzzy metric space  for the same 

point and vice versa. We point to Mihet & Shen et al. [5] for fixed point 
theorems in the fuzzy metric spaces. 

Theorem 3.8: Let (𝑋,𝑑̅) be a complete metric space on 𝑋, suppose the 
mapping 𝑆:𝑋 → 𝑋 satisfy the contractive condition, thus �̅�(𝑆𝑎,𝑆𝑏) <
𝑘 �̅�(𝑎,𝑏), for all 𝑎,𝑏 ∈ 𝑋,𝑘 ∈ [0,1) is a constant. If 𝑇 has a unique fixed 
point in 𝑋 with respect to  the metric (𝑋,𝑑̅), then 𝑇 has a unique fixed point 

with respect to the induced fuzzy metric (𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗). 

Proof: Suppose that 𝑇 has a unique fixed point in 𝑋 with respect to the metric 
space (𝑋,𝑑̅). So, we have 𝑑̅(𝑆𝑎,𝑎) = 0 for some 𝑥. 

Therefore   𝐹�̅�(𝑆𝑎,𝑎,𝑡) =  𝛽 °  �̅� ° �̅�𝑥𝑦(𝑆𝑎,𝑎,𝑡) 

= 𝛽 ° {𝑠(𝑆𝑎),𝑠(𝑆𝑏)} 
     = 𝛽(𝑡𝑦) = 𝛽(0) = 1 

This represents that 𝑆𝑎 = 𝑎 with respect to the fuzzy metric space 
(𝑋,𝛽 °  �̅� ° �̅�𝑥𝑦,∗) 

If there is another fixed point 𝑏 ∈ 𝑋 then, 
   𝐹�̅�(𝑎,𝑏, 𝑡) =  𝛽 °  �̅� ° �̅�𝑥𝑦(𝑆𝑎,𝑆𝑏,𝑡) 

                                = 𝛽 ° {𝑠(𝑆𝑎),𝑠(𝑆𝑏)} 
= 𝛽(𝑡𝑦) = 𝛽(0) = 1 

and therefore, 𝑎 = 𝑏. 

Theorem 3.9: Let �̅� and 𝐹�̅� are metric and fuzzy metric respectively and 
commutative.  

Where, �̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some metric �̅�(𝑎,𝑏) =

𝑦 > 0 and 𝛽:(𝑡𝑦) → 1−cos(𝑡𝑦) =:�̂�𝜖𝐼. Further more 𝐹�̅� = 𝛽 °  �̅� ° �̅�𝑥𝑦.   



 

Estimation of fuzzy metric spaces from metric spaces 

69 
 

Proof for this theorem is similar to the proof of theorem 3.1. 

 

Example 3.10: Let �̅� and 𝐹�̅� are metric and fuzzy metric respectively.  

�̅�𝑥𝑦:(𝑎,𝑏,𝑡) → (𝑡𝑎,𝑡𝑏), �̅�(𝑡𝑎,𝑡𝑏) → 𝑡𝑦 for some metric �̅�(𝑎,𝑏) = 𝑦 > 0 and                       

𝛽:(𝑡𝑦) → 1−𝑠𝑖𝑛2(𝑡𝑦) =:�̂�𝜖𝐼. Furthermore 𝐹�̅� = 𝛽 °  �̅� ° �̅�𝑥𝑦. And we prove 

it is commutative. 

Solution: According to the below equation it is very easy to check that 𝛽 is 
continuous. 

 sin2(𝑡𝑦) in ℛ́+  is continuous ⟹ 𝛽 is continuous. 
Now we justify that  �̅� ° �̅�𝑥𝑦 = 𝛽

−1 ° 𝐹�̅�. For (a,b,c) ∈ X×X×ℛ́
+, we have,  

  �̅� ° �̅�𝑥𝑦(𝑎,𝑏,𝑡) = �̅�(𝑡𝑎,𝑡𝑎𝑏) = 𝑡𝑦 ≔ 𝑞 > 0. 

On the other hand, 

𝛽−1 °𝐹�̅�(𝑎,𝑏,𝑡) = 𝛽
−1(�̂�)

= 𝛽−1(1−𝑠𝑖𝑛2(𝑡𝑦))= sin−1{√1−(1−𝑠𝑖𝑛2(𝑡𝑦)) } = 𝑡𝑦 

Therefore, it is commutative. 

4. Conclusion 

     In this article we design a new structure to estimate the fuzzy metric space 

with the help of metric space using fuzzy fixed point theorems and converse 

also. We discussed the various definitions and provided the proof for the same 

definitions with new structure using fuzzy fixed point theorems. We have also 

given some examples which satisfies the condition of our new structure basis 

on fixed point theorem. This paper makes away to analysis the applications of 

fuzzy metric space in various field of engineering. 

 

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