Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                            Volume 41, 2021, pp. 137-145 

 

              
 

137 

 

 

 

A characterization of strong fuzzy 

diameter zero in intuitionistic fuzzy 

metric spaces 

 
 

S. Yahya Mohamed* 

E. Naargees Begum† 
  
 

 

Abstract  

The idea of intuitionistic fuzzy metric space introduced by Park 

(2004). In this paper, we introduce the notion of strong 

intuitionistic fuzzy diameter zero for a family of subsets based on 

the intuitionistic fuzzy diameter for a subset of 𝐴. Then we 
introduce nested sequence of subsets having strong intuitionistic 

fuzzy diameter zero using their intuitionistic fuzzy diameter. 

Keywords: strong fuzzy diameter; intuitionistic fuzzy metric 

space; strong completeness. 

2010 AMS subject classification: 05C72, 54E50, 03F55.‡ 
 

 
 

 
*PG and Research Department of Mathematics, Affiliated to Bharathidasan University, 

Government Arts College, Trichy, Tamilnadu, India. yahya_md@yahoo.com.  
† Department of Mathematics, Dr. R.K. Shanmugam College of Arts and Science, Indili, 

Tamilnadu, India; mathsnb@gmail.com. 
‡ Received on September 18, 2021. Accepted on December 1, 2021. Published on December 

31, 2021. doi: 10.23755/rm.v41i0.661. ISSN: 1592-7415. eISSN: 2282-8214. © The Authors. 

This paper is published under the CC-BY licence agreement. 



S. Yahya Mohamed and E. Naargees Begum 

 

138 

 

1 Introduction  

The theory of fuzzy sets was introduced by L.A. Zadeh [17] in 1965. 

Kramosil and Michalek [6] introduced the fuzzy metric spaces (FM-spaces) by 

generalizing the concept of probabilistic metric spaces to fuzzy situation. 

George and Veeramani [4] modified the concept of fuzzy metric space 

introduced by Kramosil and Michalek [6] with a view to obtain a Hausdorff 

topology on fuzzy metric spaces which have very important applications in 

quantum particle particularly in connection with both string and E-infinity 

theory. In 2004, Park [8] defined the concept of intuitionistic fuzzy metric 

space with the help of continuous t-norms and continuous t-conorms. 

 

Several researchers have shown interest in the intuitionistic fuzzy metric 

space successfully applied in many fields, it can be found in [5, 10, 11, 13, 14, 

15, 16]. Theory of fuzzy sets have been widely used and developed in different 

fields of sciences, including mathematical programing, theory of modeling, 

theory of optimal control, theory of neural network, engineering and medical 

sciences, coloured image processing, etc. 

 

In this paper, the concept of characterization of strong fuzzy diameter zero  

in intuitionistic fuzzy metric spaces are introduced and also discuss some 

properties of strong fuzzy diameter zero in intuitionistic fuzzy metric spaces. 

 

2  Preliminaries  

Definition 2.1[17] Let 𝑋 be a nonempty set. A fuzzy set 𝐴 in 𝑋 is 

characterized by its membership function 𝜇𝐴 ∶  𝑋 →  [0, 1] and 𝜇𝐴(𝑥) is 

interpreted as the degree of membership of element 𝑥 in fuzzy set 𝐴 for each 

𝑥 ∈  𝑋. It is clear that 𝐴 is completely determined by the set of tuples 𝐴 =

 {(𝑥, 𝜇𝐴(𝑥))|𝑥 ∈  𝑋}.         

 

Definition 2.2[4] The 3-tuple (𝐴, 𝑀,∗) is said to be a fuzzy metric space if 𝐴 

be a non-empty set and ∗ be a continuous t-norm. A fuzzy set 𝐴2 × (0, ∞) is 

called a fuzzy metric on 𝐴 if 𝑎, 𝑏, 𝑐 ∈  𝐴 and 𝑠, 𝑡 > 0, the following condition 

holds: 

1. 𝑀 (𝑎, 𝑏, 𝑡) = 0 

2. 𝑀 (𝑎, 𝑏, 𝑡) = 1 if and only if 𝑎 = 𝑏 



A characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric 

spaces 
 

139 

 

3. 𝑀 (𝑎, 𝑏, 𝑡)  =  𝑀(𝑏, 𝑎, 𝑡 ) 

4. 𝑀 (𝑎, 𝑏, 𝑡 +  𝑠)  ≥  𝑀(𝑎, 𝑏, 𝑡) ∗ 𝑀(𝑎, 𝑏, 𝑠) 

5. 𝑀 (𝑎, 𝑏, •): (0, ∞) → [0, 1] is left continuous 

The function 𝑀(𝑎, 𝑏, 𝑡) denote the degree of nearness between 𝑎 and 𝑏 with 

respect to t respectively. 

 

Definition 2.3[1, 2] Let a set 𝐸 be fixed. An intuitionistic fuzzy set 𝐴 in 𝐸 is 

an object of the following 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜐𝐴(𝑥)), 𝑥 ∈ 𝐸 } where the 

functions 𝜇𝐴(𝑥): 𝐸 → [0, 1] and 𝜐𝐴 (𝑥 ): 𝐸 → [0, 1] determine the degree of 
membership and the degree of non-membership of the element 𝑥 ∈ 𝐸, 
respectively, and for every 𝑥 ∈ 𝐸: 0 ≤ 𝜇𝐴(𝑥) + 𝜐𝐴(𝑥) ≤ 1, when          
𝜐𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) for all 𝑥 ∈ 𝐸 is an ordinary fuzzy set. In addition, for each 
IFS 𝐴 in 𝐸, if 𝜋𝐴(𝑥) = 1 −  𝜇𝐴(𝑥) − 𝜐𝐴(𝑥). Then 𝜇𝐴(𝑥) is called the degree of 
indeterminacy of 𝑥 to 𝐴 or called the degree of hesitancy of 𝑥 to 𝐴. It is 
obvious that  0 ≤ πA(x) ≤ 1, for each 𝑥 ∈ 𝐸. 
 

Definition 2.4 [7] A 5-tuple (𝐴, 𝑀, 𝑁,∗,∘) is said to be an intuitionistic fuzzy 
metric space if 𝐴 is an arbitrary set, ∗ is a continuous t-norm, ∘ is a continuous 
t- conorm and, 𝑀, 𝑁 are fuzzy sets on 𝐴2  ×  [0, ∞) satisfying the conditions: 

1. 𝑀(𝑎, 𝑏, 𝑡) +  𝑁(𝑎, 𝑏, 𝑡) ≤  1,  for all 𝑎, 𝑏 ∈  𝐴 and 𝑡 ˃ 0 
2. 𝑀(𝑎, 𝑏, 0) =  0,  for all  𝑎, 𝑏 ∈ 𝐴 
3. 𝑀(𝑎, 𝑏, 𝑡) =  1, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 ˃ 0 if and only if 𝑎 =  𝑏 
4. 𝑀(𝑎, 𝑏, 𝑡) =  𝑀(𝑏, 𝑎, 𝑡), for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 
5. 𝑀(𝑎, 𝑏, 𝑡) ∗ 𝑀(𝑏, 𝑐, 𝑠) ≤  𝑀(𝑎, 𝑐, 𝑡 + 𝑠), for all 𝑎, 𝑏, 𝑐 ∈ 𝐴 and 𝑠, 𝑡 ˃ 0 
6. 𝑀(𝑎, 𝑏,•): [0, ∞) → [0, ∞] is left continuous for all 𝑎, 𝑏 ∈ 𝐴 
7. 𝑙𝑖𝑚

𝑡→∞
𝑀(𝑎, 𝑏, 𝑡) = 1, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 

8. 𝑁(𝑎, 𝑏, 0) =  1, for all a, 𝑏 ∈ 𝐴  
9. 𝑁(𝑎, 𝑏, 𝑡) =  0, for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 if and only if 𝑎 = 𝑏 
10. 𝑁(𝑎, 𝑏, 𝑡) =  𝑁(𝑏, 𝑎, 𝑡), for all 𝑎, 𝑏 ∈ 𝐴 and 𝑡 > 0 
11. 𝑁(𝑎, 𝑏, 𝑡) ∘  𝑁(𝑏, 𝑐, 𝑠) ≥  𝑁(𝑎, 𝑐, 𝑡 + 𝑠), for all 𝑎, 𝑏, 𝑐 ∈ 𝐴 and 𝑠, 𝑡 > 0 
12. 𝑁(𝑎, 𝑏,•): [0, ∞) → [0,1] is right continuous for all 𝑎, 𝑏 ∈ 𝐴 
13.  𝑙𝑖𝑚

𝑡→∞
𝑁(𝑎, 𝑏, 𝑡) = 0, for all 𝑎, 𝑏 ∈ 𝐴. 

The functions 𝑀(𝑎, 𝑏, 𝑡) and 𝑁(𝑎, 𝑏, 𝑡) denote the degree of nearness and 
the degree of non-nearness between 𝑎 and 𝑏 w.r.t  𝑡 respectively. 

 

Definition 2.5 [9] The fuzzy diameter of a non-empty set 𝐵 of a fuzzy metric 

space 𝐴, with respect to t, is the function 𝜑𝐵 : (0, +∞) →  [0, 1] given by               

φB(t) = 𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈ 𝐵} for each 𝑡 ∈ 𝑅
+. 



S. Yahya Mohamed and E. Naargees Begum 

 

140 

 

 

Definition 2.6 [9] A collection of sets  {𝐵𝑖 }𝑖∈𝐼  is said to have fuzzy diameter 

zero if given 𝑟 ∈ (0, 1) and 𝑡 ∈  𝑅+ there exists  𝑖 ∈  𝐼 such that     

M(a, b, t)  >  1 −  r for all 𝑎, 𝑏 ∈ 𝐵𝑖. 

 

3 Strong fuzzy diameter zero in intuitionistic 

fuzzy metric spaces 

Definition 3.1 The fuzzy diameter of a non-empty set 𝐵 of a intuitionistic 
fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘), with respect to 𝑡, is the function           
φB: (0, +∞) →  [0, 1] given by 𝜑𝐵 (𝑡) =  𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈  𝐵}  and 
𝜓𝐵 : (0, +∞) →  [0, 1] given by  𝜓𝐵 (𝑡) =  𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈  𝐵} for each 
𝑡 ∈ 𝑅+ 

 

Definition 3.2 A collection of sets {𝐵𝑖}𝑖∈𝐼  of a intuitionistic fuzzy metric space 
(𝐴, 𝑀, 𝑁,∗,∘) is said to have fuzzy diameter zero if given 𝑟 ∈ (0, 1) and 𝑡 ∈ 𝑅+ 
there exists 𝑖 ∈ 𝐼 such that 𝑀(𝑎, 𝑏, 𝑡)  >  1 −  𝑟  𝑁(𝑎, 𝑏, 𝑡) <  𝑟 for all 𝑎, 𝑏 ∈
𝐵𝑖 . 

Theorem 3.3 Let {𝐵𝑛}𝑛∈ℕ be a nested sequence of sets of the intuitionistic 
fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘). Then the following statements are 
equivalent:  

(i) {𝐵𝑛}𝑛∈ℕ has fuzzy diameter zero.  

(ii)  𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡) = 1, 𝑙𝑖𝑚𝑛→∞
𝜓𝐵𝑛 (𝑡) = 0 for all 𝑡 ∈  𝑅

+. 

Proof:  

(i)→(ii):  
Let  𝑡 ∈ 𝑅+. Given 𝑟 ∈ (0, 1) exists 𝑛𝑟,𝑡 ∈  ℕ  such that                  

𝑀(𝑎, 𝑏, 𝑡)  >  1 –  𝑟, 𝑁(𝑎, 𝑏, 𝑡) < 𝑟 for each 𝑎, 𝑏 ∈ 𝐵𝑛  with 𝑛 ≥ 𝑛𝑟,𝑡  .    
Then,           𝜑𝐵𝑛 (𝑡) =  𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈  𝐵𝑛} ≥  1 –  𝑟 and  

                 𝜓𝐵𝑛 (𝑡) =  𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡): 𝑎, 𝑏 ∈  𝐵𝑛} ≤  1 –  𝑟  for all n ≥ nr,t.  

Hence, 

𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡) = 1 and  𝑙𝑖𝑚𝑛→∞
𝜓𝐵𝑛 (𝑡) = 0, since r is arbitrary in  (0,1). 

(ii)→(i): 
Suppose 𝑙𝑖𝑚

𝑛→∞
𝜑𝐵𝑛 (𝑡) = 1 and 𝑙𝑖𝑚𝑛→∞

𝜓𝐵𝑛 (𝑡) = 0, for all 𝑡 ∈  𝑅
+.  

Let 𝑡 ∈ 𝑅+and let 𝑟 ∈ (0, 1).  
We can find 𝑛𝑟,𝑡 ∈  ℕ  such that 𝜑𝐵𝑛 (𝑡) > 1 − 𝑟 and  𝜓𝐵𝑛 (𝑡)  < 𝑟 for all  ≥

𝑛𝑟,𝑡. Thus, 𝑀(𝑎, 𝑏, 𝑡)  >  1 −  𝑟 and 𝑁(𝑎, 𝑏, 𝑡)  < 𝑟 for each 𝑎, 𝑏 ∈ 𝐵𝑛 with 
𝑛 ≥ 𝑛𝑟,𝑡 i.e., {𝐵𝑛}𝑛∈ℕ has fuzzy diameter zero.  



A characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric 

spaces 
 

141 

 

Definition 3.4 A family of non-empty sets {𝐵𝑖}𝑖∈𝐼  of a intuitionistic fuzzy 

metric space (𝐴, 𝑀, 𝑁,∗,∘) has strong fuzzy diameter zero if for 𝑟 ∈ (0, 1) 

there exists 𝑖 ∈  𝐼 such that 𝑀(𝑎, 𝑏, 𝑡)  >  1 –  𝑟 and  𝑁(𝑎, 𝑏, 𝑡)  < 𝑟 for each 

𝑎, 𝑏 ∈ 𝐵𝑛 and all 𝑡 ∈ 𝑅
+. 

Theorem 3.5 Let (𝐴, 𝑀, 𝑁,∗,∘) be an intuitionistic fuzzy metric space and let 

{𝐵𝑛}𝑛∈ℕ be a nested sequence of sets of 𝐴. Then the following statements are 

equivalent. 

(i) {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero.  

(ii) 𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡𝑛) = 1, 𝑙𝑖𝑚𝑛→∞
𝜓𝐵𝑛 (𝑡𝑛) = 0  for every decreasing and 

increasing sequence of positive real numbers {𝑡𝑛}𝑛∈ℕ that converges 

and diverges respectively.  

Proof: 

(i) → (ii): Let {𝑡𝑛}𝑛∈ℕ be a decreasing increasing sequence of positive real 

numbers that converges and diverges respectively. Given 𝑟 ∈ (0, 1), we can 

find 𝑛𝑟 ∈ ℕ such that 𝑀(𝑎, 𝑏, 𝑡)  >  1 –  𝑟 and 𝑁(𝑎, 𝑏, 𝑡)  < 𝑟 for each a, b ∈

Bn with 𝑛 ≥ 𝑛𝑟  and all 𝑡 ∈ 𝑅
+. In particular, 𝑀(𝑎, 𝑏, 𝑡𝑛)  >  1 –  𝑟 and 

𝑁(𝑎, 𝑏, 𝑡𝑛 )  < 𝑟  for all a, b ∈ Bn with 𝑛 ≥ 𝑛𝑟 , i.e., 𝜑𝐵𝑛 (𝑡𝑛) ≥ 1 − 𝑟, 

𝜓𝐵𝑛 (𝑡𝑛)  ≤ 𝑟 for all 𝑛 ≥ 𝑛𝑟 , i.e., 𝑙𝑖𝑚𝑛→∞
𝜑𝐵𝑛 (𝑡𝑛) = 1,𝑙𝑖𝑚𝑛→∞

𝜓𝐵𝑛 (𝑡𝑛) = 0.  

(ii) → (i): Suppose that {𝐵𝑛}𝑛∈ℕ has not strong fuzzy diameter zero. Let r ∈
(0, 1) such that 𝐼 =  { 𝑛 ∈ ℕ: 𝑀(𝑎, 𝑏, 𝑡)  ≤  1 –  𝑟, 𝑁(𝑎, 𝑏, 𝑡) ≥ 𝑟 for some 
𝑎, 𝑏 ∈ 𝐵𝑛 and some 𝑡 ∈  𝑅

+}, is infinite. Take 𝑛1 =  𝑚𝑖𝑛 𝐼. Then, there exist 
𝑎𝑛1 , 𝑏𝑛1 ∈ 𝐵𝑛1  such that 𝑀(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 )  ≤  1 –  𝑟, 𝑁(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 )  ≥ 𝑟 with 

0 <  𝑡𝑛1 < 1. 

Take 𝑛2 > 𝑛1, with 𝑛2  ∈  ℕ, such that 𝑀(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 )  ≤

 1 –  𝑟, 𝑁(𝑎𝑛1 , 𝑏𝑛1 , 𝑡𝑛1 )  ≥ 𝑟 for some 𝑎𝑛2 , 𝑏𝑛2 ∈ 𝐵𝑛2  and 0 <  𝑡𝑛2 <

 𝑚𝑖𝑛{𝑡𝑛1  ,
1

2
 }. In this way, we construct, by induction, a sequence {𝑡𝑛𝑖 }𝑖∈ℕ such 

that 𝑀(𝑎𝑛𝑖 , 𝑏𝑛𝑖 , 𝑡𝑛𝑖 )  ≤  1 –  𝑟, 𝑁(𝑎𝑛𝑖 , 𝑏𝑛𝑖 , 𝑡𝑛𝑖 )  ≥ 𝑟 for some 𝑎𝑛𝑖 , 𝑏𝑛𝑖 ∈

𝐵𝑛𝑖 , 𝑛𝑖  ∈  ℕ with 𝑛𝑖 > 𝑛𝑖−1 and 0 <  𝑡𝑛𝑖 <  𝑚𝑖𝑛{𝑡𝑛𝑖−1  ,
1

𝑖
 }.  

Then, 

𝜑𝐵𝑛𝑖
(𝑡𝑛𝑖 ) =  𝑖𝑛𝑓{𝑀(𝑎, 𝑏, 𝑡𝑛𝑖 ): 𝑎, 𝑏 ∈ 𝐵𝑛𝑖  } ≤  1 –  𝑟, 

                     𝜓𝐵𝑛𝑖
(𝑡𝑛𝑖 ) =  𝑠𝑢𝑝{𝑁(𝑎, 𝑏, 𝑡𝑛𝑖 ): 𝑎, 𝑏 ∈ 𝐵𝑛𝑖  } ≥  𝑟 for all 𝑖 ∈ ℕ . 

Hence {𝜑𝐵𝑛𝑖
(𝑡𝑛𝑖 )}𝑖∈ℕ, {𝜓𝐵𝑛𝑖

(𝑡𝑛𝑖 )}𝑖∈ℕ does not converge and diverge 



S. Yahya Mohamed and E. Naargees Begum 

 

142 

 

respectively. Now, {𝑡𝑛𝑖 }𝑖∈ℕ is a subsequence of the decreasing and increasing 

sequence {tn}n∈ℕ that converges and diverges respectively, given by 

𝑡𝑛 = {
𝑡𝑛1                     𝑛 ≤ 𝑛1

𝑡𝑛 𝑖+1     𝑛𝑖 ≤ 𝑛 ≤ 𝑛𝑖+1
 

and the sequence {𝜑𝐵𝑛 (𝑡𝑛)}𝑛∈ℕ, {𝜓𝐵𝑛 (𝑡𝑛)}𝑛∈ℕ does not converge and diverge 

respectively. Thus, we get the contradiction. 

Theorem 3.6  Let {𝐵𝑛}𝑛∈ℕ be a nested sequence of sets with fuzzy diameter 

zero in a intuitionistic fuzzy metric space (𝐴, 𝑀, 𝑁,∗,∘). {𝐵𝑛}𝑛∈ℕ has strong 

fuzzy diameter zero if and only if {𝐵𝑛}  is a singleton set after a certain stage.  

Proof:  

Suppose {𝐵𝑛}𝑛∈ℕ is not eventually constant. Put 𝑝𝑛 =

𝑠𝑢𝑝{𝑑(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐵𝑛 }, 𝑞𝑛 = 𝑖𝑛𝑓{𝑑(𝑎, 𝑏): 𝑎, 𝑏 ∈ 𝐵𝑛 } and take 𝑡𝑛 =

 𝑝𝑛 and 𝑡𝑛 =  𝑞𝑛  for all 𝑛 ∈ ℕ. Then, {𝑡𝑛 }𝑛∈ℕ is a decreasing and increasing 

sequence of positive real numbers converges and diverges respectively.  

Then, 𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡) = 𝑙𝑖𝑚 𝑖𝑛𝑓{𝑀𝑑(𝑎, 𝑏, 𝑡𝑛): 𝑎, 𝑏 ∈ 𝐵𝑛 }
𝑛→∞

 

                                             = 𝑙𝑖𝑚
𝑛→∞

𝑡𝑛

𝑡𝑛+𝑑𝑖𝑎𝑚(𝐵𝑛)
 

                                             = 𝑙𝑖𝑚
𝑛→∞

𝑝𝑛

𝑝𝑛+𝑝𝑛
=

1

2
  

and 

𝑙𝑖𝑚
𝑛→∞

𝜓𝐵𝑛 (𝑡) = 𝑙𝑖𝑚 𝑠𝑢𝑝{𝑁𝑑(𝑎, 𝑏, 𝑡𝑛 ): 𝑎, 𝑏 ∈ 𝐵𝑛}
𝑛→∞

 

                                            = 𝑙𝑖𝑚
𝑛→∞

𝑡𝑛

𝑡𝑛+𝑑𝑖𝑎𝑚(𝐵𝑛)
 

                                            = 𝑙𝑖𝑚
𝑛→∞

𝑞𝑛

𝑞𝑛+𝑞𝑛
=

1

2
  

Hence {𝐵𝑛}𝑛∈ℕ has not strong fuzzy diameter zero. 

Theorem 3.7 Let (𝐴, 𝑀, 𝑁,∗,∘) be a intuitionistic fuzzy metric space. If 

{𝐵𝑛}𝑛∈ℕ is a nested sequence of sets of A which has strong fuzzy diameter zero 

then {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero.  

Proof:  

First, we prove that 𝜑�̅� (𝑡 ) = 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) = 𝜓𝐵 (𝑡 ) for every subset 

𝐵 of 𝐴 . Indeed, take 𝑎, 𝑏 ∈  𝐵. Then, we can find two sequences {𝑎𝑛}𝑛∈ℕ and 



A characterization of strong fuzzy diameter zero in intuitionistic fuzzy metric 

spaces 
 

143 

 

{𝑏𝑛}𝑛∈ℕ  in 𝐵 that converge to 𝑎 and 𝑏, respectively. Let 𝑡 ∈  𝑅
+ and an 

arbitrary ε ∈ (0, 1). We have that 

 𝑀(𝑎, 𝑏, 𝑡 +  2𝜀) ≥  𝑀(𝑎, 𝑏𝑛, 𝜀) ∗  𝑀(𝑎𝑛, 𝑏𝑛, 𝑡) ∗  𝑀(𝑏𝑛, 𝑏, 𝜀) 

                             ≥  𝑀(𝑎, 𝑎𝑛, 𝜀) ∗  𝜑𝐵 (𝑡 )  ∗  𝑀(𝑏𝑛, 𝑏, 𝜀),  

𝑁(𝑎, 𝑏, 𝑡 +  2𝜀) ≤  𝑁(𝑎, 𝑏𝑛, 𝜀) ∘  𝑁(𝑎𝑛, 𝑏𝑛, 𝑡) ∘  𝑁(𝑏𝑛, 𝑏, 𝜀) 

                               ≤  𝑁(𝑎, 𝑎𝑛, 𝜀)  ∘  𝜓𝐵 (𝑡 )  ∘  𝑁(𝑏𝑛, 𝑏, 𝜀)  

and taking limit on the inequality when n tends to ∞, we obtain 

𝑀(𝑎, 𝑏, 𝑡 +  2𝜀)  ≥  1 ∗ 𝜑𝐵 (𝑡 )  ∗  1 =  𝜑𝐵 (𝑡 ) , 

𝑁(𝑎, 𝑏, 𝑡 +  2𝜀) ≤  1 ∘  𝜑𝐵 (𝑡 ) ∘   1 =  𝜑𝐵 (𝑡 ) . 

Since ε is arbitrary, due to the continuity of 𝑀(𝑎, 𝑏, 𝑡), 𝑁(𝑎, 𝑏, 𝑡) we obtain 
𝑀(𝑎, 𝑏, 𝑡)  ≥  𝜑𝐵 (𝑡 ), 𝑁(𝑎, 𝑏, 𝑡)  ≤ 𝜓𝐵 (𝑡 ) and then  𝜑�̅� (𝑡 ) ≥ 𝜑𝐵 (𝑡 ), 

𝜓�̅� (𝑡 ) ≥ 𝜓𝐵 (𝑡 ). On the other hand, we have 𝜑�̅� (𝑡 ) ≤ 𝜑𝐵 (𝑡 ),             

𝜓�̅� (𝑡 ) ≤ 𝜓𝐵 (𝑡 ) and hence 𝜑�̅� (𝑡 ) = 𝜑𝐵 (𝑡 ), 𝜓�̅� (𝑡 ) = 𝜓𝐵 (𝑡 ). Let {𝑡𝑛}𝑛∈ℕ be 

a decreasing and increasing sequence of positive real numbers convergent and 

divergent respectively. By theorem 3.5, we have that                 𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡𝑛) =

1, 𝑙𝑖𝑚
𝑛→∞

𝜓𝐵𝑛 (𝑡𝑛) = 0.  

Then, by our last argument, we have that, 

𝑙𝑖𝑚
𝑛→∞

𝜑𝐵𝑛 (𝑡𝑛) = 𝑙𝑖𝑚𝑛→∞
 𝜑�̅�𝑛 (𝑡𝑛) = 1, 

𝑙𝑖𝑚
𝑛→∞

𝜓𝐵𝑛 (𝑡𝑛) = 𝑙𝑖𝑚𝑛→∞
 𝜓�̅�𝑛 (𝑡𝑛) = 0,  

and consequently, by theorem 3.5, {𝐵𝑛}𝑛∈ℕ has strong fuzzy diameter zero. 
 

4 Conclusion 

Intuitionistic fuzzy set theory plays a vital role in uncertain situations in 

all aspects. In this paper, the characterizations of strong fuzzy diameter zero in 

intuitionistic fuzzy metric spaces are discussed and proved that the nested 

sequences having the strong fuzzy diameter zero in intuitionistic fuzzy metric 

space. We have also provided that nested sequences of subsets has strong 

fuzzy diameter zero if and only if singleton set after a certain stage in a 

intuitionistic fuzzy metric spaces. 

 

 



S. Yahya Mohamed and E. Naargees Begum 

 

144 

 

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