Communications on Applied Nonlinear Analysis 

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Homeomorphism via 𝜹𝜷-open Sets in Fermatean Fuzzy Topological 

Spaces and Application in Entropy Measure 

 

A. Vadivel1 ,  V. Sagunthaladevi2  and   S. Priya3 

 1PG and Research Department of Mathematics, Arignar Anna Government Arts College, Namakkal - 637 002, India. 

avmaths@gmail.com 

  3Department of Mathematics, M.Kumarasamy College of Engineering, Karur - 639 113, India. 

sagunthala98v@gmail.com  

  1,2,3Department of Mathematics, Annamalai University, Annamalai Nagar - 608 002, India. 

Corresponding Author: S. Priya 

pre9433@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: Classical set theory failed to cover non-probabilistic uncertain situations in to 

the set format, but fuzzy set theory can do this job perfectly with the fuzzy number of 

vagueness of non-probabilistic uncertainty. Topologist adopt this fuzzy set and fit in to 

the topological concepts to extent and apply their innovations for the needs and growth of 

the humans. Even though the fuzzy concept convert every situation, it hire the concept of 

intuitionistic fuzzy set to evident the importance of non-membership of the situation. 

Pythagorean fuzzy set is one of by membership and non-membership, more forceful to 

seize indeterminacy to cover the uncertain situations which are unable to covered by 

intuitionistic fuzzy sets. Then any Intuitionistic fuzzy subset or Phythagorean fuzzy 

subset of a set can be considered as Fermatean fuzzy subset, we observe that any 

Intuitionstic fuzzy topological space or Phythagorean fuzzy topological space is a 

Fermatean fuzzy topological space as well. In this paper, we contribute our concept of 𝛿 

(resp. 𝛿𝑃, 𝛿𝑆, 𝛿𝛼, 𝛿𝛽  )-homeomorphism, 𝐶 -homeomorphism in Fermatean fuzzy 

topological spaces and the properties for the Fermatean field of fuzzy topological spaces. 

To register importance of Fermatean fuzzy sets we applied a proposed entropy measure 

for multiple criteria decision making problem. 

Keywords: Fermatean fuzzy homeomorphism, Fermatean fuzzy 𝐶 - homeomorphism, 

entropy measure. 

AMS (2000) subject classification: 06F35, 03G10, 03B52. 

 

1  Introduction 

 Fuzzy sets were introduced Zadeh [15] in 1965. The fuzzy set concept was the basis of mathematical 

testing of the fuzzy concept that exists in our real world and the formation of new branches in 

mathematics. The fuzzy set concept corresponding to unexplained physical situations gives useful 

applications on many topics such as statistics, data processing and linguistics. A lot of research has 

been done on this subject since 1965. In 1968, Chang [6] defined the concept of fuzzy topological 

space and generalized some basic notions of topology such as open set, closed set, continuity and 

compactness to fuzzy topological spaces. The idea of intuitionistic fuzzy set was first published by 



Communications on Applied Nonlinear Analysis 

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Atanassov [1] and many works by the same author and his colleagues appeared in the literature [2, 5]. 

Coker [7] initiated a study of intuitionistic fuzzy topological spaces. Later Yager [13] launched a non 

standard fuzzy set referred to as Phythagorean fuzzy set. Olgun et al., [9] defined a Phythagorean fuzzy 

topological spaces. Fermatean fuzzy sets proposed by Senapati and Yager in 2020 [10], can handle 

uncertain information more easily in the process of decision making. They defined basic operations 

over the Fermatean fuzzy sets. Hariwan Z. Ibrahim defined a Fermatean fuzzy topological spaces and 

the continuity of a function defind among Fermatean fuzzy topological spaces. 

The aim of this paper is as follows. In Section 2, some basic definitions of 𝑓𝑠’s, 𝑖𝑓𝑠’s, 𝑝𝑓𝑠’s and 

Fermatean fuzzy sets are briefly reviewed. In section 3 and 4, we develop the concept of some stronger 

and weaker forms of Fermatean fuzzy 𝛿𝛽 homeomorphism and 𝐶-homeomorphism in Fermatean 

fuzzy topological spaces and also specialized some of their basic properties with examples. Entropy 

measure was introduced by Zadeh [16] for classical fuzzy sets. Many authors developed and created 

for their version of entropy measure. Here in section 5, we introduce entropy measure for Fermatean 

fuzzy sets and give an example for the decision making in real life problem. Finally we conclude in 

section 6. 

2  Preliminaries 

 We recall some basic notions of fuzzy sets, 𝐼𝐹𝑆’s, 𝑃𝐹𝑆’s and 𝔉ℱ𝑠’s. 

Definition 2.1  [15] Let 𝑋 be a nonempty set. A fuzzy set 𝐴 in 𝑋 is characterized by a membership 

function 𝜇𝐴: 𝑋 → [0,1]. That is:  

 𝜇𝐴(𝑥) = {
1,    if        𝑥 ∈ 𝑋
0,    if        𝑥 ∉ 𝑋
(0,1)    if 𝑥 ispartlyin 𝑋.

 

Alternatively, a fuzzy set 𝐴 in 𝑋 is an object having the form 𝐴 = {< 𝑥, 𝜇𝐴(𝑥) > |𝑥 ∈ 𝑋} or 𝐴 =

{⟨
𝜇𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋},  where the function 𝜇𝐴(𝑥): 𝑋 → [0,1]  defines the degree of membership of the 

element, 𝑥 ∈ 𝑋. 

The closer the membership value 𝜇𝐴(𝑥) to 1, the more 𝑥 belongs to 𝐴, where the grades 1 and 0 

represent full membership and full nonmembership. Fuzzy set is a collection of objects with graded 

membership, that is, having degree of membership. Fuzzy set is an extension of the classical notion of 

set. In classical set theory, the membership of elements in a set is assessed in a binary terms according 

to a bivalent condition; an element either belongs or does not belong to the set. Classical bivalent sets 

are in fuzzy set theory called crisp sets. Fuzzy sets are generalized classical sets, since the indicator 

function of classical sets is special cases of the membership functions of fuzzy sets, if the latter only 

take values 0 or 1. Fuzzy sets theory permits the gradual assessment of the membership of element in 

a set; this is described with the aid of a membership function valued in the real unit interval [0,1]. 

Let us consider two examples: 

(i) all employees of 𝑋𝑌𝑍 who are over 1.8𝑚 in height; (ii) all employees of 𝑋𝑌𝑍 who are tall. The 

first example is a classical set with a universe (all 𝑋𝑌𝑍 employees) and a membership rule that divides 

the universe into members (those over 1.8𝑚) and nonmembers. The second example is a fuzzy set, 



Communications on Applied Nonlinear Analysis 

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because some employees are definitely in the set and some are definitely not in the set, but some are 

borderline. 

This distinction between the ins, the outs, and the borderline is made more exact by the membership 

function, 𝜇. If we return to our second example and let 𝐴 represent the fuzzy set of all tall employees 

and 𝑥 represent a member of the universe 𝑋 (i.e. all employees), then 𝜇𝐴(𝑥) would be 𝜇𝐴(𝑥) = 1 if 

𝑥 is definitely tall or 𝜇𝐴(𝑥) = 0 if 𝑥 is definitely not tall or 0 < 𝜇𝐴(𝑥) < 1 for borderline cases.  

Definition 2.2  [1] The intuitionistic fuzzy sets are defined on a non-empty sets 𝑋 as objects having 

the form 𝐼 = {〈𝑥, 𝛼𝐼(𝑥), 𝛽𝐼(𝑥)〉: 𝑥 ∈ 𝑋} , where 𝛼𝐼(𝑥): 𝑋 → [0,1] and 𝛽𝐼(𝑥): 𝑋 → [0,1]  denote the 

degree of memebership and the degree of non-memebership of each element 𝑥 ∈ 𝑋 to the set 𝐼, 

respectively, and 0 ≤ 𝛼𝐼(𝑥) + 𝛽𝐼(𝑥) ≤ 1, for all 𝑥 ∈ 𝑋.  

Definition 2.3  [1, 2, 3, 4] Let a nonempty set 𝑋 be fixed. An 𝐼𝐹𝑆 𝐴 in 𝑋 is an object having the 

form: 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜆𝐴(𝑥) > |𝑥 ∈ 𝑋}  or 𝐴 = {⟨
𝜇𝐴(𝑥),𝜆𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋},  where the functions 

𝜇𝐴(𝑥): 𝑋 → [0,1]  and 𝜆𝐴(𝑥): 𝑋 → [0,1]  define the degree of membership and the degree of 

nonmembership, respectively, of the element 𝑥 ∈ 𝑋 to 𝐴, which is a subset of 𝑋, and for every 𝑥 ∈

𝑋: 0 ≤ 𝜇𝐴(𝑥) + 𝜆𝐴(𝑥) ≤ 1. For each 𝐴 in 𝑋: 𝜋𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) − 𝜆𝐴(𝑥) is the intuitionistic fuzzy 

set index or hesitation margin of 𝑥  in 𝑋.  The hesitation margin 𝜋𝐴(𝑥)  is the degree of 

nondeterminacy of 𝑥 ∈ 𝑋 to the set 𝐴 and 𝜋𝐴(𝑥) ∈ [0,1]. The hesitation margin is the function that 

expresses lack of knowledge of whether 𝑥 ∈ 𝑋 or 𝑥 ∉ 𝑋. Thus: 𝜇𝐴(𝑥) + 𝜆𝐴(𝑥) + 𝜋𝐴(𝑥) = 1.  

Example 2.1  Let 𝑋 = {𝑥, 𝑦, 𝑧} be a fixed universe of discourse and 𝐴 = {⟨
0.6,0.1

𝑥
⟩ , ⟨

0.8,0.1

𝑦
⟩ , ⟨

0.5,0.3

𝑧
⟩}, 

be the intuitionistic fuzzy set in 𝑋. The hesitation margins of the elements 𝑥, 𝑦, 𝑧 to 𝐴 are as follows: 

𝜋𝐴(𝑥) = 0.3, 𝜋𝐴(𝑦) = 0.1 and 𝜋𝐴(𝑧) = 0.2.  

Definition 2.4  [12, 13, 14] Let 𝑋 be a universal set. Then, a Pythagorean fuzzy set 𝐴, which is a set 

of ordered pairs over 𝑋 , is defined by the following: 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜆𝐴(𝑥)|𝑥 ∈ 𝑋}  or 𝐴 =

{⟨
𝜇𝐴(𝑥),𝜆𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋}, where the functions 𝜇𝐴(𝑥): 𝑋 → [0,1] and 𝜆𝐴(𝑥): 𝑋 → [0,1] define the degree 

of membership and the degree of nonmembership, respectively, of the element 𝑥 ∈ 𝑋 to 𝐴, which is a 

subset of 𝑋 , and for every 𝑥 ∈ 𝑋 , 0 ≤ (𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2 ≤ 1 . Supposing (𝜇𝐴(𝑥))2 +

(𝜆𝐴(𝑥))2 ≤ 1 , then there is a degree of indeterminacy of 𝑥 ∈ 𝑋  to 𝐴  defined by 𝜋𝐴(𝑥) =

√1 − [(𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2]  and 𝜋𝐴(𝑥) ∈ [0,1] . In what follows, (𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2 +

(𝜋𝐴(𝑥))2 = 1. Otherwise, 𝜋𝐴(𝑥) = 0 whenever (𝜇𝐴(𝑥))2 + (𝜆𝐴(𝑥))2 = 1. We denote the set of all 

𝑃𝐹𝑆’s over 𝑋 by 𝑝𝑓𝑠(𝑋).  

Definition 2.5  [10] Let 𝑋 be a universe of discourse. A Fermatean fuzzy set (𝔉ℱ𝑠) 𝐹 in 𝑋 is an 

object having the form 𝐹 = {< 𝑥, 𝛼𝐹(𝑥), 𝛽𝐹(𝑥) >: 𝑥 ∈ 𝑋} where 𝛼𝐹(𝑥): 𝑋 → [0,1] and 𝛽𝐹(𝑥): 𝑋 →

[0,1], including the condition 0 ≤ (𝛼𝐹(𝑥))3 + (𝛽𝐹(𝑥))3 ≤ 1, for all 𝑥 ∈ 𝑋. The numbers 𝛼𝐹(𝑥) and 

𝛽𝐹(𝑥) denote, respectively, the degree of memebership and the degree of non-memebership of the 

element 𝑥  in the set 𝐹 . For any 𝔉ℱ𝑠  𝐹  and 𝑥 ∈ 𝑋 , 𝜋𝐹(𝑥) = √1 − [(𝛼𝐹(𝑥))3 − (𝛽𝐹(𝑥))3]
3

 is 

identified as the degree of interminancy of 𝑥 to 𝐹. In the interest of simplicity, we shall mention the 

symbol 𝐹 = (𝛼𝐹 , 𝛽𝐹) for the 𝔉ℱ𝑠 𝐹 = {< 𝑥, 𝛼𝐹(𝑥), 𝛽𝐹(𝑥): 𝑥 ∈ 𝑋}.  



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Definition 2.6  [10] Let 𝐹 = (𝛼𝐹, 𝛽𝐹), 𝐹1 = (𝛼𝐹1
, 𝛽𝐹1

) and 𝐹2 = (𝛼𝐹2
, 𝛽𝐹2

), be three Fermatean 

fuzzy sets (𝔉ℱ𝑠’s), then their operations are defined as follows: [(i)]  

    1.  𝐹1 ∩ 𝐹2 = (𝑚𝑖𝑛{𝛼𝐹1
, 𝛼𝐹2

}, 𝑚𝑎𝑥{𝛽𝐹1
, 𝛽𝐹2

}).  

    2.  𝐹1 ∪ 𝐹2 = (𝑚𝑎𝑥{𝛼𝐹1
, 𝛼𝐹2

}, 𝑚𝑖𝑛{𝛽𝐹1
, 𝛽𝐹2

}).  

    3.  𝐹𝑐 = (𝛽𝐹 , 𝛼𝐹).  

Remark 2.1 If 𝛼𝐹1
= 𝛼𝐹2

 and 𝛽𝐹1
= 𝛽𝐹2

, then 𝐹1 = 𝐹2  

 Note that, for understanding the Fermatean fuzzy set better, we give an instance to illuminate the 

understandability of the Fermatean fuzzy set. The point when someone needs will plan as much 

craving for the level for an alternative 𝑠𝑖 on a criterion 𝐶𝑗, he might provide for the degree on which 

that alternative 𝑠𝑖 fulfils those criteria 𝐶𝑗 likewise 0.85, what is more correspondingly the elective 𝑠𝑖 

dissatisfies the criterion 𝐶𝑗  similarly as 0.65. We can definitely get 0.85 + 0.65 = 1.5 > 1, and, 

therefore, it does not follow the condition of intuitionistic fuzzy sets. Also, we can get (0.85)2 +

(0.65)2 = 0.7225 + 0.4225 = 1.145 > 1 , which does not obey the constraint condition of 

Pythagorean fuzzy set. However, we can get (0.85)3 + (0.65)3 = 0.614125 + 0.274625 =

0.88875 ≤ 1, which is good enough to apply the Fermatean fuzzy set to control it [10]. Throughout 

this paper, we use the notation 1𝔉 for the Fermatean fuzzy subset (1,0) and we use the notation 0𝔉 

for the Fermatean fuzzy subset (0,1), that is, 𝛼1𝔉
= 1, 𝛽1𝔉

= 0, 𝛼0𝔉
= 0, 𝛽0𝔉

= 1. A Fermatean 

fuzzy subset 𝔉 of a non-empty set 𝑋 is a pair (𝛼𝔉, 𝛽𝔉) of a membership function (𝛼𝔉(𝑥): 𝑋 → [0,1] 

and a non-membership function (𝛽𝔉(𝑥): 𝑋 → [0,1] with (𝛼𝔉(𝑥))3 + (𝛽𝔉(𝑥))3 = (𝛾𝔉(𝑥))3 for any 

𝑥 ∈ 𝑋 where 𝛾𝔉(𝑥): 𝑋 → [0,1] is a function which is called the strength of commitment at point 𝑥. 

Definition 2.7  [8] Let 𝑋 be a non empty set and 𝜏 be a family of Fermatean fuzzy subsets of 𝑋. If  

    1.  1𝔉, 0𝔉 ∈ 𝜏  

    2.  for any 𝐹1, 𝐹2 ∈ 𝜏, we have 𝐹1 ∩ 𝐹2 ∈ 𝜏,  

    3.  for any {𝐹𝑖}𝑖∈𝐼 ⊂ 𝜏, we have ⋃𝑖∈𝐼 𝐹𝑖 ∈ 𝜏 where 𝐼 is an arbitrary index set then 𝜏 is called a 

Fermatean fuzzy topology on 𝑋.  

 The pair (𝑋, 𝜏) is said to be a Fermatean fuzzy topological space. Each member of 𝜏 is called an 

Fermatean fuzzy oprn set. The complement of an Fermatean fuzzy open set is called a Fermatean fuzzy 

closed set.  

 Remark 2.2  [8] As any Intuitionistic fuzzy subset or Phythagorean fuzzy subset of a set can be 

considered as Fermatean fuzzy subset, we observe that any Intuitionstic fuzzy topological space or 

Phythagorean fuzzy topological space is a Fermatean fuzzy topological space as well. On the other 

hand, it is obvious that a Fermatean fuzzy topological space need not be Intuitionistic fuzzy 

topological space and Phythagorean fuzzy topological space. Even an Fermatean fuzzy open set maybe 

neither an Intuitionistic fuzzy set nor Phythagorean fuzzy set.  

Example 2.2  [8] Let 𝑋 = {𝑐1, 𝑐2}. Consider the following family Fermatean fuzzy subsets 𝜏 =

{1𝔉, 0𝔉, 𝐹1, 𝐹2} where 



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 𝐹1 = {〈𝑐1, 𝛼𝐹1
(𝑐1) = 0.4, 𝛽𝐹1

(𝑐1) = 0.6〉, 〈𝑐2, 𝛼𝐹1
(𝑐2) = 0.1, 𝛽𝐹1

(𝑐2) = 0.3〉}  

and 𝐹2 = {〈𝑐1, 𝛼𝐹2
(𝑐1) = 0.9, 𝛽𝐹2

(𝑐1) = 0.6〉, 〈𝑐2, 𝛼𝐹2
(𝑐2) = 0.2, 𝛽𝐹2

(𝑐2) = 0.3〉} . Observe that 

(𝑋, 𝜏) is a Fermatean fuzzy topological space but (𝑋, 𝜏) is neither Intuitionistic fuzzy topological 

space nor Phythagorean fuzzy topological space. 

Definition 2.8  [8] Let (𝑋, 𝜏) be an 𝔉ℱ𝑡𝑠 and 𝐴 = {< 𝑎, 𝛼𝐴(𝑎), 𝛽𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝔉ℱ𝑠 in 

𝑋. Then the Fermatean fuzzy interior and the Fermatean fuzzy closure of 𝐴 are denoted by 𝔉ℱ𝑖𝑛𝑡(𝐴) 

and 𝔉ℱ𝑐𝑙(𝐴)  and are defined as follows: 𝔉ℱ𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 𝑖𝑠𝑎 𝔉ℱ𝑜𝑠 𝑎𝑛𝑑 𝐺 ⊆ 𝐴}  and 

𝔉ℱ𝑐𝑙(𝐴) =∩ {𝐾|𝐾 𝑖𝑠𝑎 𝔉ℱ𝑐𝑠 𝑎𝑛𝑑 𝐴 ⊆ 𝐾}. Also, it can be established that 𝔉ℱ𝑐𝑙(𝐴) is an 𝔉ℱ𝑐𝑠 and 

𝔉ℱ𝑖𝑛𝑡(𝐴) is an 𝔉ℱ𝑜𝑠, 𝐴 is an 𝔉ℱ𝑐𝑠 if and only if 𝔉ℱ𝑐𝑙(𝐴) = 𝐴 and 𝐴 is an 𝔉ℱ𝑜𝑠 if and only if 

𝔉ℱ𝑖𝑛𝑡(𝐴) = 𝐴. We say that 𝐴 is 𝔉ℱ-dense if 𝔉ℱ𝑐𝑙(𝐴) = 1𝔉.  

Lemma 2.1  [8]  For any Fermatean fuzzy set 𝐴 in (𝑋, 𝜏), we have 1𝔉 − 𝔉ℱ𝑖𝑛𝑡(𝐴) = 𝔉ℱ𝑐𝑙(1𝔉 −

𝐴) and 1𝔉 − 𝔉ℱ𝑐𝑙(𝐴) = 𝔉ℱ𝑖𝑛𝑡(1𝔉 − 𝐴).  

Definition 2.9  [11] Let (𝑋, 𝜏) be an 𝔉ℱ𝑡𝑠 and 𝐴 be an 𝔉ℱ𝑠. Then 𝐴 is said to be an Fermatean 

fuzzy (i) regular open set (𝔉ℱ𝑟𝑜𝑠 in short) if 𝐴 = 𝔉ℱ𝑖𝑛𝑡(𝔉ℱ𝑐𝑙(𝐴)). (ii) regular closed set (𝔉ℱ𝑟𝑐𝑠 

in short) if 𝐴 = 𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(𝐴)). By Lemma 2.1, it follows that 𝐴 is an 𝔉ℱ𝑟𝑜𝑠 iff 𝐴̅ is an 𝔉ℱ𝑟𝑐𝑠.  

Definition 2.10  [11]  Let (𝑋, 𝜏) be an 𝔉ℱ𝑡𝑠 and 𝐴 = {< 𝑎, 𝛼𝐴(𝑎), 𝛽𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝔉ℱ𝑠 

in 𝑋. Then the 𝛿-interior and the 𝛿-closure of 𝐴 are denoted by 𝔉ℱ𝛿𝑖𝑛𝑡(𝐴) and 𝔉ℱ𝛿𝑐𝑙(𝐴) and 

are defined as follows. 𝔉ℱ𝛿𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 is an 𝔉ℱ𝑟𝑜𝑠 and 𝐺 ⊆ 𝐴}, 𝔉ℱ𝛿𝑐𝑙(𝐴) =∩ {𝐾|𝐾 is an 

𝔉ℱ𝑟𝑐𝑠 and 𝐴 ⊆ 𝐾}.  

Definition 2.11  [11] Let (𝑋, 𝜏) be an 𝔉ℱ𝑡𝑠 and 𝐴 = {< 𝑎, 𝛼𝐴(𝑎), 𝛽𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝔉ℱ𝑠 in 

𝑋. A set 𝐴 is said to be 𝔉ℱ  

    1.  𝛿-open set (briefly, 𝔉ℱ𝛿𝑜𝑠) if 𝐴 = 𝔉ℱ𝛿𝑖𝑛𝑡(𝐴),  

    2.  𝛿-pre open set (briefly, 𝔉ℱ𝛿𝒫𝑜𝑠) if 𝐴 ⊆ 𝔉ℱ𝑖𝑛𝑡(𝔉ℱ𝛿𝑐𝑙(𝐴)).  

    3.  𝛿-semi open set (briefly, 𝔉ℱ𝛿𝒮𝑜𝑠) if 𝐴 ⊆ 𝔉ℱ𝑐𝑙(𝔉ℱ𝛿𝑖𝑛𝑡(𝐴)).  

    4.  𝛿 - 𝛼  open set or 𝑎 -open set (briefly, 𝔉ℱ𝛿𝛼𝑜𝑠  or 𝔉ℱ𝑎𝑜𝑠 ) if 𝐴 ⊆

𝔉ℱ𝑖𝑛𝑡(𝔉ℱ𝑐𝑙(𝔉ℱ𝛿𝑖𝑛𝑡(𝐴))).  

    5.  𝛿 - 𝛽  open set or 𝑒∗ -open set (briefly, 𝔉ℱ𝛿𝛽𝑜𝑠  or 𝔉ℱ𝑒∗𝑜𝑠 ) if 𝐴 ⊆

𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(𝔉ℱ𝛿𝑐𝑙(𝐴))).  

    6.  𝛿  (resp. 𝛿 -pre, 𝛿 -semi, 𝛿 - 𝛼  and 𝛿 - 𝛽 ) dense if 𝔉ℱ𝛿𝑐𝑙(𝐴)  (resp. 

𝔉ℱ𝛿𝑝𝑐𝑙(𝐴), 𝔉ℱ𝛿𝒮𝑐𝑙(𝐴), 𝔉ℱ𝛿𝛼𝑐𝑙(𝐴) and 𝔉ℱ𝛿𝛽𝑐𝑙(𝐴)) = 1𝔉. 

The complement of an 𝔉ℱ𝛿𝑜𝑠 (resp. 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠) is called an 𝔉ℱ𝛿 

(resp. 𝔉ℱ𝛿𝒫, 𝔉ℱ𝛿𝒮,  𝔉ℱ𝛿𝛼  and 𝔉ℱ𝛿𝛽)  closed set (briefly, 𝔉ℱ𝛿𝑐𝑠  (resp. 

𝔉ℱ𝛿𝒫𝑐𝑠, 𝔉ℱ𝛿𝒮𝑐𝑠, 𝔉ℱ𝛿𝛼𝑐𝑠 and 𝔉ℱ𝛿𝛽𝑐𝑠)) in 𝑋. 

The family of all 𝔉ℱ𝛿𝑜𝑠  (resp. 𝔉ℱ𝛿𝑐𝑠, 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒫𝑐𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝒮𝑐𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠,

𝔉ℱ𝛿𝛼𝑐𝑠, 𝔉ℱ𝛿𝛽𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑐𝑠) of 𝑋 is denoted by 𝔉ℱ𝛿𝑂𝑆(𝑋) (resp. 𝔉ℱ𝛿𝐶𝑆(𝑋), 𝔉ℱ𝛿𝒫𝑂𝑆(𝑋),



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1900 
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𝔉ℱ𝛿𝒫𝐶𝑆(𝑋), 𝔉ℱ𝛿𝒮𝑂𝑆(𝑋), 𝔉ℱ𝛿𝒮𝐶𝑆(𝑋), 𝔉ℱ𝛿𝛼𝑂𝑆(𝑋), 𝔉ℱ𝛿𝛼𝐶𝑆(𝑋),  𝔉ℱ𝛿𝛽𝑂𝑆(𝑋)  and 

𝔉ℱ𝛿𝛽𝐶𝑆(𝑋)).  

Definition 2.12  [11] Let (𝑋, 𝜏) be an 𝔉ℱ𝑡𝑠 and 𝐴 = {< 𝑎, 𝛼𝐴(𝑎), 𝛽𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝔉ℱ𝑠 in 

𝑋 . Then the 𝔉ℱ𝛿 -pre (resp. 𝔉ℱ𝛿 -semi, 𝔉ℱ𝛿𝛼  and 𝔉ℱ𝛿𝛽) -interior and the 𝔉ℱ𝛿 -pre (resp. 

𝔉ℱ𝛿 -semi, 𝔉ℱ𝛿𝛼  and 𝔉ℱ𝛿𝛽) -closure of 𝐴  are denoted by 𝔉ℱ𝛿𝒫𝑖𝑛𝑡(𝐴)  (resp. 𝔉ℱ𝛿𝒮𝑖𝑛𝑡(𝐴) , 

𝔉ℱ𝛿𝛼𝑖𝑛𝑡(𝐴)  and 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(𝐴))  and the 𝔉ℱ𝛿𝒫𝑐𝑙(𝐴)  (resp. 𝔉ℱ𝛿𝒮𝑐𝑙(𝐴), 𝔉ℱ𝛿𝛼𝑐𝑙(𝐴)  and 

𝔉ℱ𝛿𝛽𝑐𝑙(𝐴)) and are defined as follows: 

𝔉ℱ𝛿𝒫𝑖𝑛𝑡(𝐴) (resp. 𝔉ℱ𝛿𝒮𝑖𝑛𝑡(𝐴), 𝔉ℱ𝛿𝛼𝑖𝑛𝑡(𝐴) and 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(𝐴)) =∪ {𝐺|𝐺  in a 𝔉ℱ𝛿𝒫𝑜𝑠 (resp. 

𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠  and 𝔉ℱ𝛿𝛽𝑜𝑠 ) and 𝐺 ⊆ 𝐴}  and 𝔉ℱ𝛿𝒫𝑐𝑙(𝐴)  (resp. 𝔉ℱ𝛿𝒮𝑐𝑙(𝐴), 𝔉ℱ𝛿𝛼𝑐𝑙(𝐴) 

and 𝔉ℱ𝛿𝛽𝑐𝑙(𝐴)) =∩ {𝐾|𝐾 is an 𝔉ℱ𝛿𝒫𝑐𝑠 (resp. 𝔉ℱ𝛿𝒮𝑐𝑠, 𝔉ℱ𝛿𝛼𝑐𝑠,  𝔉ℱ𝛿𝛽𝑐𝑠) and 𝐴 ⊆ 𝐾}.  

3  Fermatean fuzzy 𝜹𝜷-homeomorphism 

 In this section, we introduce Fermatean fuzzy 𝛿  (resp. 𝛿  pre, 𝛿  semi, 𝛿𝛼  and 

𝛿𝛽)-homeomorphism and discuss some of their properties. 

Definition 3.1   Let (𝑋1, 𝜏1) and (𝑋2, 𝜏2) be two 𝔉ℱ𝑡𝑠’s. Then a function ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) 

is said to be a Fermatean fuzzy 𝛿 (resp. 𝛿 pre, 𝛿 semi, 𝛿𝛼 and 𝛿𝛽) continuous (briefly, 𝔉ℱ𝛿𝐶𝑡𝑠 

(resp. 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 , 𝔉ℱ𝛿𝛼𝐶𝑡𝑠  and 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 )) function if ℎ𝔉
−1(𝐺)  is 𝔉ℱ𝛿𝑜  (resp. 

𝔉ℱ𝛿𝒫𝑜, 𝔉ℱ𝛿𝒮𝑜, 𝔉ℱ𝛿𝛼𝑜 & 𝔉ℱ𝛿𝛽𝑜) set in 𝑋1 for all 𝔉ℱ𝑜 set 𝐺 in 𝑋2.  

Definition 3.2  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) is said to 

be a Fermatean fuzzy (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝛿𝛼 & 𝛿𝛽 or 𝑒∗)- open map (briefly, 𝔉ℱ𝑂 (resp. 𝔉ℱ𝛿𝑂, 

𝔉ℱ𝛿𝒫𝑂, 𝔉ℱ𝛿𝒮𝑂, 𝔉ℱ𝛿𝛼𝑂, & 𝔉ℱ𝛿𝛽𝑂 or 𝔉ℱ𝑒∗𝑂)) if the image of every 𝔉ℱ𝑜𝑠 in 𝑋1 is a 𝔉ℱ𝑜𝑠 

(resp. 𝔉ℱ𝛿𝑜𝑠, 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠 & 𝔉ℱ𝛿𝛽𝑜𝑠 or 𝔉ℱ𝑒∗𝑜𝑠) in 𝑋2.  

Definition 3.3  A bijection ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) is called a Fermatean fuzzy (resp. 𝛿, 𝛿𝛼, 𝛿𝒮, 

𝛿𝒫  &  𝛿𝛽  or 𝑒∗ )-homeomorphism (briefly, 𝔉ℱ𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝐻𝑜𝑚 , 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 , 

𝔉ℱ𝛿𝒫𝐻𝑜𝑚 & 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 or 𝔉ℱ𝑒∗𝐻𝑜𝑚)) if ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝐶𝑡𝑠, 𝔉ℱ𝛿𝛼𝐶𝑡𝑠, 

𝔉ℱ𝛿𝒮𝐶𝑡𝑠, 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 & 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 or 𝔉ℱ𝑒∗𝐶𝑡𝑠) mappings.  

Theorem 3.1  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a mapping. Then 

the following statements are hold for 𝔉ℱ𝑡𝑠, but not conversely.   

    (i)  Every 𝔉ℱ𝛿𝐻𝑜𝑚 is a 𝔉ℱ𝐻𝑜𝑚.  

    (ii)  Every 𝔉ℱ𝛿𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝒮𝐻𝑜𝑚.  

    (iii)  Every 𝔉ℱ𝛿𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝒫𝐻𝑜𝑚.  

    (iv)  Every 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚.  

    (v)  Every 𝔉ℱ𝛿𝒫𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚.  

    (vi)  Every 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝒮𝐻𝑜𝑚.  

    (vii)  Every 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 is a 𝔉ℱ𝛿𝒫𝐻𝑜𝑚.  

  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1901 
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  Proof. (i) Let ℎ𝔉 be 𝔉ℱ𝛿𝐻𝑜𝑚, then ℎ𝔉  and ℎ𝔉
−1 are 𝔉ℱ𝛿𝐶𝑡𝑠. But every 𝔉ℱ𝛿𝐶𝑡𝑠 function is 

𝔉ℱ𝐶𝑡𝑠. Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝐻𝑜𝑚. 

(ii) Let ℎ𝔉 be 𝔉ℱ𝛿𝐻𝑜𝑚, then ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝐶𝑡𝑠. But every 𝔉ℱ𝛿𝐶𝑡𝑠 function is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠. 

Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝒮𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝒮𝐻𝑜𝑚. 

(iii) Let ℎ𝔉 be 𝔉ℱ𝛿𝐻𝑜𝑚, then ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝐶𝑡𝑠. But every 𝔉ℱ𝛿𝐶𝑡𝑠 function is 𝔉ℱ𝛿𝒫𝐶𝑡𝑠. 

Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝒫𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝒫𝐻𝑜𝑚. 

(iv) Let ℎ𝔉  be 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 , then ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 . But every 𝔉ℱ𝛿𝒮𝐶𝑡𝑠  function is 

𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚. 

(v) Let ℎ𝔉  be 𝔉ℱ𝛿𝒫𝐻𝑜𝑚 , then ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 . But every 𝔉ℱ𝛿𝒫𝐶𝑡𝑠  function is 

𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚. 

(vi) Let ℎ𝔉  be 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 , then ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 . But every 𝔉ℱ𝛿𝛼𝐶𝑡𝑠  function is 

𝔉ℱ𝛿𝒮𝐶𝑡𝑠. Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝒮𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝒮𝐻𝑜𝑚. 

(vii) Let ℎ𝔉  be 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 , then ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 . But every 𝔉ℱ𝛿𝛼𝐶𝑡𝑠  function is 

𝔉ℱ𝛿𝒫𝐶𝑡𝑠. Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝒫𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝒫𝐻𝑜𝑚.              

Example 3.1  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3 and 𝐴4 are defined as 

𝛼𝐴1
(𝑎) = 0.2, 𝛽𝐴1

(𝑎) = 0.8, 𝛼𝐴1
(𝑏) = 0.4, 𝛽𝐴1

(𝑏) = 0.6; 

𝛼𝐴2
(𝑎) = 0.1, 𝛽𝐴2

(𝑎) = 0.9, 𝛼𝐴2
(𝑏) = 0.3, 𝛽𝐴2

(𝑏) = 0.7; 

𝛼𝐴3
(𝑎) = 0.9, 𝛽𝐴3

(𝑎) = 0.1, 𝛼𝐴3
(𝑏) = 0.7, 𝛽𝐴3

(𝑏) = 0.7; 

𝛼𝐴4
(𝑎) = 0.2, 𝛽𝐴4

(𝑎) = 0.8, 𝛼𝐴4
(𝑏) = 0.3, 𝛽𝐴4

(𝑏) = 0.7; 

 Let 𝜏1 = 𝜏2 = 𝜏 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4} be a 𝔉ℱ𝑡𝑠 on 𝑋 and let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an 

identity function, Then ℎ𝔉  is 𝔉ℱ𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒫𝐻𝑜𝑚 and 𝔉ℱ𝛿𝒫𝐻𝑜𝑚) but not 

𝔉ℱ𝛿𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 , 𝔉ℱ𝛿𝐻𝑜𝑚  and 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 ). Since, 𝐴4  is a 𝔉ℱ𝑜  set in 𝑋2  but 

ℎ𝔉
−1(𝐴4) = 𝐴4 is not 𝔉ℱ𝛿𝑜 (resp. 𝔉ℱ𝛿𝒮𝑜, 𝔉ℱ𝛿𝑜 and 𝔉ℱ𝛿𝛼𝑜 ) set in 𝑋1. 

Example 3.2  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3 and 𝐴4 are defined as 

𝛼𝐴1
(𝑎) = 0.4, 𝛽𝐴1

(𝑎) = 0.6, 𝛼𝐴1
(𝑏) = 0.5, 𝛽𝐴1

(𝑏) = 0.5; 

𝛼𝐴2
(𝑎) = 0.6, 𝛽𝐴2

(𝑎) = 0.4, 𝛼𝐴2
(𝑏) = 0.6, 𝛽𝐴2

(𝑏) = 0.4; 

𝛼𝐴3
(𝑎) = 0.7, 𝛽𝐴3

(𝑎) = 0.3, 𝛼𝐴3
(𝑏) = 0.6, 𝛽𝐴3

(𝑏) = 0.4; 

𝛼𝐴4
(𝑎) = 0.4, 𝛽𝐴4

(𝑎) = 0.6, 𝛼𝐴4
(𝑏) = 0.4, 𝛽𝐴4

(𝑏) = 0.6; 

Let 𝜏1 = 𝜏2 = 𝜏 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4}  be a 𝔉ℱ𝑡𝑠  on 𝑋  and let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an 

identity function, Then ℎ𝔉  is 𝔉ℱ𝛿𝒮𝐻𝑜𝑚  but not 𝔉ℱ𝛿𝐻𝑜𝑚 . Since, 𝐴3  is a 𝔉ℱ𝑜  set in 𝑋2  but 

ℎ𝔉
−1(𝐴3) = 𝐴3 is not 𝔉ℱ𝛿𝑜 set in 𝑋1. 



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Example 3.3  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐵1 and 𝐵2 are defined as 

𝛼𝐴1
(𝑎) = 0.2, 𝛽𝐴1

(𝑎) = 0.7, 𝛼𝐴1
(𝑏) = 0.1, 𝛽𝐴1

(𝑏) = 0.8; 

𝛼𝐴2
(𝑎) = 0.3, 𝛽𝐴2

(𝑎) = 0.6, 𝛼𝐴2
(𝑏) = 0.4, 𝛽𝐴2

(𝑏) = 0.5; 

𝛼𝐵1
(𝑎) = 0.1, 𝛽𝐵1

(𝑎) = 0.9, 𝛼𝐵1
(𝑏) = 0.2, 𝛽𝐵1

(𝑏) = 0.9; 

𝛼𝐵2
(𝑎) = 0.2, 𝛽𝐵2

(𝑎) = 0.3, 𝛼𝐵2
(𝑏) = 0.4, 𝛽𝐵2

(𝑏) = 0.7; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2} and 𝜏2 = {0𝔉, 1𝔉, 𝐵1, 𝐵2} are 𝔉ℱ𝑡𝑠’s on 𝑋 and let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) 

be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 but not 𝔉ℱ𝛿𝒫𝐻𝑜𝑚. Since, 𝐴2 is a 𝔉ℱ𝑜 set in 𝑋2 

but ℎ𝔉
−1(𝐴2) = 𝐴2 is not 𝔉ℱ𝛿𝒫𝑜 set in 𝑋1.  

Theorem 3.2  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a bijective mapping. If ℎ𝔉 is 𝔉ℱ𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝐶𝑡𝑠, 

𝔉ℱ𝛿𝛼𝐶𝑡𝑠, 𝔉ℱ𝛿𝒮𝐶𝑡𝑠, 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 & 𝔉ℱ𝛿𝛽𝐶𝑡𝑠), then the followings statements are equivalent:  

    (i)  ℎ𝔉 is a 𝔉ℱ𝐶 (resp. 𝔉ℱ𝛿𝐶, 𝔉ℱ𝛿𝛼𝐶, 𝔉ℱ𝛿𝒮𝐶, 𝔉ℱ𝛿𝒫𝐶 & 𝔉ℱ𝛿𝛽𝐶 or 𝔉ℱ𝑒∗𝐶) mapping.  

    (ii)  ℎ𝔉 is a 𝔉ℱ𝑂 (resp. 𝔉ℱ𝛿𝑂, 𝔉ℱ𝛿𝛼𝑂, 𝔉ℱ𝛿𝒮𝑂, 𝔉ℱ𝛿𝒫𝑂 & 𝔉ℱ𝛿𝛽𝑂 or 𝔉ℱ𝑒∗𝐶) mapping.  

    (iii)  ℎ𝔉
−1 is a 𝔉ℱ𝐻𝑜𝑚 (resp. 𝔉ℱ𝛿𝐻𝑜𝑚, 𝔉ℱ𝛿𝛼𝐻𝑜𝑚, 𝔉ℱ𝛿𝒮𝐻𝑜𝑚, 𝔉ℱ𝛿𝒫𝐻𝑜𝑚 & 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 

or 𝔉ℱ𝑒∗𝐻𝑜𝑚) .  

 Proof. (i) ⇒ (ii) : Assume that ℎ𝔉 is a bijective mapping and a 𝔉ℱ𝛿𝛽𝐶 mapping. Hence, ℎ𝔉
−1 is a 

𝔉ℱ𝛿𝛽𝐶𝑡𝑠 mapping. We know that each 𝔉ℱ𝑜𝑠 in (𝑋1, 𝜏1) is a 𝔉ℱ𝛿𝛽𝑜𝑠 in (𝑋2, 𝜏2). Hence, ℎ𝔉 is a 

𝔉ℱ𝛿𝛽𝑂 mapping. 

(ii) ⇒ (iii) : Let ℎ𝔉  be a bijective and 𝔉ℱ𝛿𝛽𝑂 mapping. Further, ℎ𝔉
−1 is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 mapping. 

Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Therefore, ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚. 

(iii) ⇒  (i): Let ℎ𝔉  be a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 . Then ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 . Since each 𝔉ℱ𝛿𝑐𝑠  in 

(𝑋1, 𝜏1) is 𝔉ℱ𝑐𝑠 in (𝑋1, 𝜏1) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2), ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐶 mapping. The proof of other 

cases are similar.              

Definition 3.4  A 𝔉ℱ𝑡𝑠 (𝑋, 𝜏) is said to be a Fermatean fuzzy 𝛼𝑇1

2

 (resp. 𝛿𝒮𝑇1

2

, 𝛿𝒫𝑇1

2

 and 𝛿𝛽𝑇1

2

) 

(briefly, 𝔉ℱ𝛿𝛼𝑇1

2

 (resp. 𝔉ℱ𝛿𝒮𝑇1

2

, 𝔉ℱ𝛿𝒫𝑇1

2

 and 𝔉ℱ𝛿𝛽𝑇1

2

))-space if every 𝔉ℱ𝛿𝛼𝑐𝑠  (resp. 

𝔉ℱ𝛿𝒮𝑐𝑠, 𝔉ℱ𝛿𝒫𝑐𝑠, and 𝔉ℱ𝛿𝛽𝑐𝑠) is 𝔉ℱ𝑐𝑠 in (𝑋, 𝜏).  

Theorem 3.3  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  be a 𝔉ℱ𝛿𝛼𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒫𝐻𝑜𝑚  & 

𝔉ℱ𝛿𝛽𝐻𝑜𝑚  or 𝔉ℱ𝑒∗𝐻𝑜𝑚) . Then ℎ𝔉  is a 𝔉ℱ𝐻𝑜𝑚  if (𝑋1, 𝜏1) and (𝑋2, 𝜏2) are 𝔉ℱ𝛿𝛼𝑇1

2

 (resp. 

𝔉ℱ𝛿𝒮𝑇1

2

, 𝔉ℱ𝛿𝒫𝑇1

2

 and 𝔉ℱ𝛿𝛽𝑇1

2

)-space.  

 Proof. Assume that 𝐾  is a 𝔉ℱ𝑐𝑠  in (𝑋2, 𝜏2) . Then ℎ𝔉
−1(𝐾)  is a 𝔉ℱ𝛿𝛽𝑐𝑠  in (𝑋1, 𝜏1) . Since 

(𝑋1, 𝜏1)  is a 𝔉ℱ𝛿𝛽𝑇1

2

-space, ℎ𝔉
−1(𝐾)  is a 𝔉ℱ𝑐𝑠  in (𝑋1, 𝜏1) . Therefore, ℎ𝔉  is 𝔉ℱ𝐶𝑡𝑠 . By 

hypothesis, ℎ𝔉
−1 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠. Let 𝐿 be a 𝔉ℱ𝑐𝑠 in (𝑋1, 𝜏1). Then, (ℎ𝔉

−1)−1(𝐿) = ℎ𝔉(𝐿) is a 𝔉ℱ𝑐𝑠 



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in (𝑋2, 𝜏2) , by presumption. Since (𝑋2, 𝜏2)  is a 𝔉ℱ𝛿𝛽𝑇1

2

-space, ℎ𝔉(𝐿)  is a 𝔉ℱ𝑐𝑠  in (𝑋2, 𝜏2) . 

Hence, ℎ𝔉
−1 is 𝔉ℱ𝐶𝑡𝑠. Hence, ℎ𝔉 is a 𝔉ℱ𝐻𝑜𝑚. Proof of other cases are similar.              

Theorem 3.4  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  be a mapping. Then the following are equivalent if 

(𝑋2, 𝜏2) is a 𝔉ℱ𝛿𝛼𝑇1

2

 (resp. 𝔉ℱ𝛿𝒮𝑇1

2

, 𝔉ℱ𝛿𝒫𝑇1

2

 and 𝔉ℱ𝛿𝛽𝑇1

2

)-space: 

    (i)  ℎ𝔉 is 𝔉ℱ𝛿𝛼𝐶 (resp. 𝔉ℱ𝛿𝒮𝐶, 𝔉ℱ𝛿𝒫𝐶 and 𝔉ℱ𝛿𝛽𝐶) mapping.  

    (ii)  If 𝐾  is a 𝔉ℱ𝑜𝑠  in (𝑋1, 𝜏1), then ℎ𝔉(𝑋1, 𝜏1) is 𝔉ℱ𝛿𝛼𝑜𝑠  (resp. 𝔉ℱ𝛿𝒮𝑜𝑠 , 𝔉ℱ𝛿𝒫𝑜𝑠  and 

𝔉ℱ𝛿𝛽𝑜𝑠) in (𝑋2, 𝜏2).  

    (iii)  ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾)) ⊆ 𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(ℎ𝔉(𝐾))) for every 𝔉ℱ𝑠 𝐾 in (𝑋1, 𝜏1).  

  Proof. (i) ⇒ (ii): Obvious. 

(ii) ⇒  (iii): Let 𝐾  be a 𝔉ℱ𝑠  in (𝑋1, 𝜏1) . Then, 𝔉ℱ𝑖𝑛𝑡(𝐾)  is a 𝔉ℱ𝑜𝑠  in (𝑋1, 𝜏1) . Then, 

ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾) is a 𝔉ℱ𝛿𝛽𝑜𝑠 in (𝑋2, 𝜏2). Since (𝑋2, 𝜏2) is a 𝔉ℱ𝛿𝛽𝑇1

2

-space, so ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾)) is 

a𝔉ℱ𝑜𝑠 in (𝑋2, 𝜏2).  

Therefore, ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾)) = 𝔉ℱ𝑖𝑛𝑡 (ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾))) ⊆ 𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(ℎ𝔉(𝐾))). 

(iii) ⇒ (i): Let 𝐾 be a 𝔉ℱ𝑐𝑠 in (𝑋1, 𝜏1). Then, 𝐾𝑐 is a 𝔉ℱ𝑜𝑠 in (𝑋1, 𝜏1). From, ℎ𝔉(𝔉ℱ𝑖𝑛𝑡(𝐾)𝑐) 

⊆ 𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(ℎ𝔉(𝐾)𝑐)), ℎ𝔉((𝐾)𝑐) ⊆ 𝔉ℱ𝑐𝑙(𝔉ℱ𝑖𝑛𝑡(ℎ𝔉(𝐾)𝑐)). Therefore, ℎ𝔉((𝐾)𝑐)  is 𝔉ℱ𝛿𝛽𝑜𝑠 

in (𝑋2, 𝜏2). Therefore, ℎ𝔉(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋1, 𝜏1). Hence, ℎ𝔉 is a 𝔉ℱ𝐶 mapping. The proof of 

other cases are similar.             

Theorem 3.5  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) and 𝑔𝔉: (𝑋2, 𝜏2) → (𝑋3, 𝜏3) be 𝔉ℱ𝛿𝛼𝐶  (resp. 𝔉ℱ𝛿𝒮𝐶 , 

𝔉ℱ𝛿𝒫𝐶 and 𝔉ℱ𝛿𝛽𝐶), where (𝑋1, 𝜏1) and (𝑋3, 𝜏3) are two 𝔉ℱ𝑡𝑠’s and (𝑋2, 𝜏2) a 𝔉ℱ𝛿𝑇1

2

 (resp. 

𝔉ℱ𝛿𝛼𝑇1

2

, 𝔉ℱ𝛿𝒮𝑇1

2

, 𝔉ℱ𝛿𝒫𝑇1

2

 and 𝔉ℱ𝛿𝛽𝑇1

2

)-space, then the composition 𝑔𝔉 ∘ ℎ𝔉 is 𝔉ℱ𝛿𝛼𝐶 (resp. 

𝔉ℱ𝛿𝒮𝐶, 𝔉ℱ𝛿𝒫𝐶 and 𝔉ℱ𝛿𝛽𝐶).  

 Proof. Let 𝐾 be a 𝔉ℱ𝑐𝑠 in (𝑋1, 𝜏1). Since ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶 and ℎ𝔉(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2), by 

assumption, ℎ𝔉(𝐾) is a 𝔉ℱ𝑐𝑠 in (𝑋2, 𝜏2). Since 𝑔𝔉  is 𝔉ℱ𝛿𝛽𝐶 , then 𝑔𝔉(ℎ𝔉(𝐾)) is 𝔉ℱ𝛿𝛽𝑐𝑠  in 

(𝑋3, 𝜏3) and 𝑔𝔉(ℎ𝔉(𝐾)) = (𝑔𝔉 ∘ ℎ𝔉)(𝐾). Therefore, 𝑔𝔉 ∘ ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶.              

Theorem 3.6  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  and 𝑔𝔉: (𝑋2, 𝜏2) → (𝑋3, 𝜏3)  be two 𝔉ℱ𝑡𝑠 ’s, then the 

following hold:  

    (i)  If 𝑔𝔉 ∘ ℎ𝔉  is 𝔉ℱ𝛿𝑂  (resp. 𝔉ℱ𝛿𝛼𝑂 , 𝔉ℱ𝛿𝒮𝑂 , 𝔉ℱ𝛿𝒫𝑂  and 𝔉ℱ𝛿𝛽𝑂 ) and ℎ𝔉  is 𝔉ℱ𝐶𝑡𝑠 , 

then 𝑔𝔉 is 𝔉ℱ𝛿𝑂 (resp. 𝔉ℱ𝛿𝛼𝑂, 𝔉ℱ𝛿𝒮𝑂, 𝔉ℱ𝛿𝒫𝑂 and 𝔉ℱ𝛿𝛽𝑂).  

    (ii)  If 𝑔𝔉 ∘ ℎ𝔉  is 𝔉ℱ𝑂  and 𝑔𝔉  is 𝔉ℱ𝛿𝐶𝑡𝑠  (resp. 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒫𝐶𝑡𝑠  and 

𝔉ℱ𝛿𝛽𝐶𝑡𝑠), then ℎ𝔉 is 𝔉ℱ𝛿𝑂 (resp. 𝔉ℱ𝛿𝛼𝑂, 𝔉ℱ𝛿𝒮𝑂, 𝔉ℱ𝛿𝒫𝑂 and 𝔉ℱ𝛿𝛽𝑂).  

  Proof. (i) Let 𝐾 be a 𝔉ℱ𝑜𝑠 in (𝑋2, 𝜏2). As ℎ𝔉 is 𝔉ℱ𝐶𝑡𝑠 mapping, ℎ𝔉
−1(𝐾) is 𝔉ℱ𝑜𝑠 in (𝑋1, 𝜏1). 

As 𝑔𝔉 ∘ ℎ𝔉  is 𝔉ℱ𝛿𝛽𝑂  mapping, (𝑔𝔉 ∘ ℎ𝔉)(ℎ𝔉
−1(𝐾)) = 𝑔𝔉(ℎ𝔉(ℎ𝔉

−1(𝐾))) = 𝑔𝔉(𝐾)  is 𝔉ℱ𝛿𝛽𝑜𝑠  in 



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1904 
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(𝑋3, 𝜏3). Thus ℎ𝔉 is 𝔉ℱ𝛿𝛽𝑂 mapping. 

The other case is similar.              

4  Fermatean fuzzy 𝜹𝜷-𝑪 homeomorphism 

Definition 4.1  A bijection ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) is called a Fermatean fuzzy (resp. 𝛿, 𝛿𝛼, 𝛿𝒮, 

𝛿𝒫  &  𝛿𝛽  or 𝑒∗ )- 𝐶  homeomorphism (briefly, 𝔉ℱ𝐶𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝛼𝐶𝐻𝑜𝑚 , 

𝔉ℱ𝛿𝒮𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒫𝐶𝐻𝑜𝑚  &  𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚  or 𝔉ℱ𝑒∗𝐶𝐻𝑜𝑚 )) if ℎ𝔉  and ℎ𝔉
−1  are 𝔉ℱ𝐼𝑟𝑟  (resp. 

𝔉ℱ𝛿𝐼𝑟𝑟, 𝔉ℱ𝛿𝛼𝐼𝑟𝑟, 𝔉ℱ𝛿𝒮𝐼𝑟𝑟, 𝔉ℱ𝛿𝒫𝐼𝑟𝑟 & 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 or 𝔉ℱ𝑒∗𝐼𝑟𝑟) mappings.  

Theorem 4.1  Each 𝔉ℱ𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝐻𝑜𝑚 , 𝔉ℱ𝛿𝛼𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒮𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒫𝐶𝐻𝑜𝑚  & 

𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚  or 𝔉ℱ𝑒∗𝐶𝐻𝑜𝑚 ) is a 𝔉ℱ𝐶𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝛼𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒮𝐻𝑜𝑚 , 

𝔉ℱ𝛿𝒫𝐻𝑜𝑚 & 𝔉ℱ𝛿𝛽𝐻𝑜𝑚 or 𝔉ℱ𝑒∗𝐻𝑜𝑚). But not conversely.  

 Proof. Let us assume that 𝐾 be a 𝔉ℱ𝛿𝑐𝑠 in (𝑋2, 𝜏2) is 𝔉ℱ𝑐𝑠. This shows that 𝐾 is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 

(𝑋2, 𝜏2). By assumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠  in (𝑋1, 𝜏1). Hence, ℎ𝔉  is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠  mapping. 

Hence, ℎ𝔉 and ℎ𝔉
−1 are 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 mappings. Hence ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐻𝑜𝑚. The proof of other cases 

are similar.              

Example 4.1  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3 and 𝐴4 are defined as 

𝛼𝐴1
(𝑎) = 0.2, 𝛽𝐴1

(𝑎) = 0.8, 𝛼𝐴1
(𝑏) = 0.4, 𝛽𝐴1

(𝑏) = 0.6; 

𝛼𝐴2
(𝑎) = 0.1, 𝛽𝐴2

(𝑎) = 0.9, 𝛼𝐴2
(𝑏) = 0.3, 𝛽𝐴2

(𝑏) = 0.7; 

𝛼𝐴3
(𝑎) = 0.9, 𝛽𝐴3

(𝑎) = 0.1, 𝛼𝐴3
(𝑏) = 0.7, 𝛽𝐴3

(𝑏) = 0.3; 

𝛼𝐴4
(𝑎) = 0.2, 𝛽𝐴4

(𝑎) = 0.8, 𝛼𝐴4
(𝑏) = 0.3, 𝛽𝐴4

(𝑏) = 0.7; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4} , 𝜏2 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3}  be a 𝔉ℱ𝑡𝑠  on 𝑋1  and 𝑋2 ; and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉  is 𝔉ℱ𝐶𝐻𝑜𝑚 (resp. 𝔉ℱ𝛿𝐶𝐻𝑜𝑚) but not 

𝔉ℱ𝐻𝑜𝑚 (resp. 𝔉ℱ𝛿𝐻𝑜𝑚). Since, 𝐴4 is a 𝔉ℱ𝑜 set in 𝑋1 but (ℎ𝔉
−1)−1(𝐴4) = 𝐴4 is not 𝔉ℱ𝑜 (resp. 

𝔉ℱ𝛿𝑜 ) set in 𝑋2. 

Theorem 4.2   If ℎ𝔉: (𝑋1, 𝜏1)  → (𝑋2, 𝜏2)  is a 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚 , then 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) ⊆

ℎ𝔉
−1(𝔉ℱ𝑐𝑙(𝐾)) for each 𝔉ℱ𝑠 𝐾 in (𝑋2, 𝜏2).  

 Proof. Let 𝐾 be a 𝔉ℱ𝑠 in (𝑋2, 𝜏2). Then, 𝔉ℱ𝑐𝑙(𝐾) is a 𝔉ℱ𝑐𝑠 in (𝑋2, 𝜏2), and every 𝔉ℱ𝑐𝑠 is a 

𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2). Assume ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 and ℎ𝔉
−1(𝔉ℱ𝑐𝑙(𝐾)) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋1, 𝜏1). Then, 

𝔉ℱ𝑐𝑙(ℎ𝔉
−1(𝔉ℱ𝑐𝑙(𝐾))) = ℎ𝔉

−1(𝔉ℱ𝑐𝑙(𝐾)).  

Here, 𝔉ℱ𝛿𝛽𝑐𝑙 (ℎ𝔉
−1(𝐾)) ⊆ 𝔉ℱ𝛿𝛽𝑐𝑙 (ℎ𝔉

−1(𝔉ℱ𝑐𝑙(𝐾))) = ℎ𝔉
−1(𝔉ℱ𝑐𝑙(𝐾)). 

Therefore, 𝔉ℱ𝛿𝛽𝑐𝑙 (ℎ𝔉
−1(𝐾)) ⊆  ℎ𝔉

−1(𝔉ℱ𝑐𝑙(𝐾)) for every 𝔉ℱ𝑠 𝐾 in (𝑋2, 𝜏2).              

Theorem 4.3   Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  be a 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚 . Then 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) =

ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) for each 𝔉ℱ𝑠 𝐾 in (𝑋2, 𝜏2).  



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 Proof. Since ℎ𝔉  is a 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚 , ℎ𝔉  is a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟  mapping. Let 𝐾  be a 𝔉ℱ𝑠  in (𝑋2, 𝜏2) . 

Clearly, 𝔉ℱ𝛿𝛽𝑐𝑙(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2). Then 𝔉ℱ𝛿𝛽𝑐𝑙(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2).  

Since ℎ𝔉
−1(𝐾) ⊆ ℎ𝔉

−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)),  then 𝔉ℱ𝛿𝛽𝑐𝑙 (ℎ𝔉
−1(𝐾)) ⊆ 𝔉ℱ𝛿𝛽𝑐𝑙 (ℎ𝔉

−1 (𝔉ℱ𝛿𝛽𝑐𝑙(𝐾))) =

ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)). Therefore, 𝔉ℱ 𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾)) ⊆ ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)). Let ℎ𝔉 be a 𝔉ℱ𝛿𝛽𝐶 𝐻𝑜𝑚. 

ℎ𝔉
−1  is a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟  mapping. Let us consider 𝔉ℱ𝑠  ℎ𝔉

−1(𝐾)  in (𝑋1, 𝜏1),  which implies 

𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾))  is a 𝔉ℱ𝛿𝛽𝑐𝑠  in (𝑋1, 𝜏1).  Hence, 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾))  is a 𝔉ℱ𝛿𝛽𝑐𝑠  in (𝑋1, 𝜏1) . 

This implies that (ℎ𝔉
−1)−1(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾))) = ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾))) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2). 

 This proves 𝐾 = (ℎ𝔉
−1)−1(ℎ𝔉

−1(𝐾)) ⊆ (ℎ𝔉
−1)−1(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾))) = ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾))) . 

Therefore, 𝔉ℱ𝛿𝛽𝑐𝑙(𝐾) ⊆ 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝑝(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)))) = ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾))). since ℎ𝔉
−1 is a 

𝔉ℱ𝛿𝛽𝐼𝑟𝑟 mapping. Hence, ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) ⊆ ℎ𝔉

−1(ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)))) = 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾)). 

That is, ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) ⊆ 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐾)). Hence, 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) = ℎ𝔉

−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)).              

Remark 4.1  Theorems 4.2 and 4.3 are also true if ℎ𝔉  is a 𝔉ℱ𝐶𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿𝐶𝐻𝑜𝑚 , 

𝔉ℱ𝛿𝛼𝐶𝐻𝑜𝑚, 𝔉ℱ𝛿𝒮𝐶𝐻𝑜𝑚 & 𝔉ℱ𝛿𝒫𝐶𝐻𝑜𝑚.)  

Theorem 4.4  If ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  and 𝑔𝔉: (𝑋2, 𝜏2) → (𝑋3, 𝜏3)  are 𝔉ℱ𝐶𝐻𝑜𝑚  (resp. 𝔉ℱ𝛿 

𝐶𝐻𝑜𝑚, 𝔉ℱ𝛿𝛼𝐶𝐻𝑜𝑚, 𝔉ℱ𝛿𝒮𝐶𝐻𝑜𝑚, 𝔉ℱ𝛿𝒫𝐶𝐻𝑜𝑚 & 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚 or 𝔉ℱ𝑒∗𝐶𝐻𝑜𝑚)’s, then 𝑔𝔉 ∘ ℎ𝔉 

is a 𝔉ℱ𝐶𝐻𝑜𝑚 (resp. 𝔉ℱ𝛿𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝛼𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒮𝐶𝐻𝑜𝑚 , 𝔉ℱ𝛿𝒫𝐶𝐻𝑜𝑚  &  𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚  or 

𝔉ℱ𝑒∗𝐶𝐻𝑜𝑚).  

 Proof. Let ℎ𝔉 and 𝑔𝔉 be two 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚’s. Assume 𝐾 is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋3, 𝜏3). Then, 𝑔𝔉
−1(𝐾) 

is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2). Then, by hypothesis, ℎ𝔉
−1(𝑔𝔉

−1(𝐾)) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋1, 𝜏1). Hence, 𝑔𝔉 ∘

ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 mapping. Now, let 𝐾 be a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋1, 𝜏1). Then, by presumption, ℎ𝔉(𝐾) is a 

𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋2, 𝜏2). Then, by hypothesis, 𝑔𝔉(ℎ𝔉(𝐾)) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in (𝑋3, 𝜏3). This implies that 

𝑔𝔉 ∘ ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 mapping. Hence, 𝑔𝔉 ∘ ℎ𝔉  is a 𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚. The proof of other cases are 

similar.              

5  Application 

 Entropy as a measure of fuzziness was first proposed by Zadeh [16]. Later many mathematicians 

defined several entropy measures. In this section, we focus on defining an entropy measure for 𝔉𝑓𝑠 

that connects the degree of membership and non-membership. As an example, we have applied the 

proposed entropy measure in the field of decision making. 

Definition 5.1  Let 𝐴 = {< 𝑥, 𝜇𝐴(𝑥), 𝜆𝐴(𝑥)|𝑥 ∈ 𝑋} be a 𝔉𝑓𝑠 in 𝑈. The new entropy measure for 𝐴 

denoted by 휀𝔉𝑓𝑠(𝐴),  is a function, 휀𝔉𝑓𝑠: 𝜏𝔉𝑓𝑠(𝑈) → [0,1]  and is defined as 휀𝔉𝑓𝑠(𝐴) = 1 −
1

𝑛
∑𝑛

𝑖=1 (𝜇𝐴 − 𝜆𝐴)2;  𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖 ∈ 𝐴, where 𝜏𝔉𝑓𝑠(𝑈) denote the family of all 𝔉𝑓𝑠’s on 𝑈.  

Example 5.1  Assume that a certain Institution wants to assign a permanent faculty member from the 

set of candidates {𝑃1, 𝑃2, 𝑃3, 𝑃4, 𝑃5}. For this, the Institution authorities consider the following four 

criteria 𝐶 = {𝐶𝑖: 𝑖 = 1,2,3,4} , where: •  𝐶1  represents the number of research publications, 

conferences and 𝐹𝐷𝑃 participated, • 𝐶2 represents the teaching experience, • 𝐶3 represents the 



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regularity and punctuality, • 𝐶4 represents the behavior with students through the class. 

After a deep discussion, a committee (forms by the Institution authorities) proposed a performance of 

these candidates environment as given in Table. Every ordered pair given in Table represents the 

membership and non-membership degrees of a candidate corresponding criteria (or attribute). Assume 

that the proposed approach for accessing the best candidate with appreciation to every criterion 

provided the committee. Then, we compute the entropy measure for each candidate to decide who is 

the optimal candidate(s). 

Table 1. Selection criteria for the candidates. 

 Person 1 (P1) Person 2 (P2) Person 3 (P3) Person 4 (P4) Person 5 (P5) 

(C1) < 𝐶1, 𝑃1; 0.3,0.7

> 

< 𝐶1, 𝑃2; 0.7,0.2

> 

< 𝐶1, 𝑃3; 0.2,0.3

> 

< 𝐶1, 𝑃4; 0.3,0.4

> 

< 𝐶1, 𝑃5; 0.1,0.3

> 

(C2) < 𝐶2, 𝑃1; 0.7,0.6

> 

< 𝐶2, 𝑃2; 0.7,0.1

> 

< 𝐶2, 𝑃3; 0.7,0.5

> 

< 𝐶2, 𝑃4; 0.3,0.4

> 

< 𝐶2, 𝑃5; 0.2,0.7

> 

(C3) < 𝐶3, 𝑃1; 0.3,0.2

> 

< 𝐶3, 𝑃2; 0.2,0.4

> 

< 𝐶3, 𝑃3; 0.9,0.2

> 

< 𝐶3, 𝑃4; 0.2,0.5

> 

< 𝐶3, 𝑃5; 0.6,0.2

> 

(C4) < 𝐶4, 𝑃1; 0.1,0.7

> 

< 𝐶4, 𝑃2; 0.3,0.3

> 

< 𝐶4, 𝑃3; 0.1,0.3

> 

< 𝐶4, 𝑃4; 0.5,0.1

> 

< 𝐶4, 𝑃5; 0.1,0.9

> 

 

 Clearly, all values in the Table 1 are 𝔉ℱ𝑠’s. Now we calculate the 휀𝔉ℱ𝑠 of each value.  

Table 2. Entropy measure of each candidate based on their criteria. 

  Person 1 (P1)  Person 2 (P2)  Person 3 (P3)  Person 4 (P4)  Person 5 (P5)  

(C1) 0.84 0.75 0.99 0.99 0.96 

(C2) 0.99 0.64 0.96 0.99 0.75 

(C3) 0.99 0.96 0.51 0.91 0.84 

(C4) 0.64 1.0 0.96 0.84 0.36 

 

 From Table 2, it is clear that, 

 휀𝔉ℱ𝑠(𝐶1, 𝑃2) < 휀𝔉ℱ𝑠(𝐶1, 𝑃1) < 휀𝔉ℱ𝑠(𝐶1, 𝑃5) < 휀𝔉ℱ𝑠(𝐶1, 𝑃3) ≤ 휀𝔉ℱ𝑠(𝐶1, 𝑃4) 

Similarly  

휀𝔉ℱ𝑠(𝐶2, 𝑃2) < 휀𝔉ℱ𝑠(𝐶2, 𝑃5) < 휀𝔉ℱ𝑠(𝐶2, 𝑃3) < 휀𝔉ℱ𝑠(𝐶2, 𝑃1) ≤ 휀𝔉ℱ𝑠(𝐶2, 𝑃4) 

휀𝔉ℱ𝑠(𝐶3, 𝑃3) < 휀𝔉ℱ𝑠(𝐶3, 𝑃5) < 휀𝔉ℱ𝑠(𝐶3, 𝑃4) < 휀𝔉ℱ𝑠(𝐶3, 𝑃2) ≤ 휀𝔉ℱ𝑠(𝐶3, 𝑃1) 

휀𝔉ℱ𝑠(𝐶4, 𝑃5) < 휀𝔉ℱ𝑠(𝐶4, 𝑃1) < 휀𝔉ℱ𝑠(𝐶4, 𝑃4) < 휀𝔉ℱ𝑠(𝐶4, 𝑃3) ≤ 휀𝔉ℱ𝑠(𝐶4, 𝑃2) 



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It is clear that based on the criteria 1 and 2, person 2 be the selection for the permanent post, criteria 3, 

person 3 be the selection for the permanent post, criteria 4, person 5 be the selection for the permanent 

post.  

6  Conclusion 

 Fuzzy Topological spaces are the classical Topological spaces which characterises the membership 

values alone. Intuitionistic Fuzzy topological spaces portates the membership as well as the 

non-membership values. Pythagorean fuzzy topological spaces extent its arm to cover the missed ones 

of the Intuitionistic fuzzy topological spaces. Fermatean fuzzy topological spaces shorten the 

Pythagorean fuzzy sets of any cardinality in to a tiny set which represents the same in nano 

approximation with boundary space. Our contribution to this area is the concepts of 𝔉ℱ𝛿𝛽𝐻𝑜𝑚, 

𝔉ℱ𝛿𝛽𝐶𝐻𝑜𝑚 and 𝔉ℱ𝛿𝛽𝑇1

2

-space and derived some of its related characteristics. Finally applied a 

entropy measure to the multiple criteria decision making with the help of Fermatean fuzzy sets. In 

future we will employee some entropy or any other measures for comparing in decision making to the 

field of medical diagnosis and teaching learning process. We also takeup this idea into the diverse 

fuzzy environment for real world application purpose. 

References 

[1]  K. T. Atanassov (1983),  Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia. 

[2]  K. T. Atanassov (1986),  Intuitionistic fuzzy sets, Fuzzy Sets Syst.  20, 87-96. 

[3]  K. T. Atanassov (1999),  Intuitionistic fuzzy sets: theory and applications, Physica, Heidelberg. 

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