Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1909 
https://internationalpubls.com 

 Continuous and Irresolute Maps Via 𝜹-open Sets in Fermatean Fuzzy 

Topological Spaces And Application of MCDM Techniques 

 

  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, av12582@annamalaiuniversity.ac.in 

  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. 

pre9433@gmail.com 

Corresponding Author: S. Priya and V. Sagunthaladevi 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: In this paper, we develop the concept of Fermatean fuzzy (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 

𝛿𝛼 & 𝛿𝛽 or 𝑒∗)-continuity in Fermatean fuzzy topological spaces and specialize some 

of their basic properties with examples. Also, we discuss about properties and 

characterization of Fermatean fuzzy irresolute maps and application of Multiple Criteria 

Decision Making (MCDM) techniques to the real-world problem using a proposed 

entropy measure in Fermatean fuzzy topological spaces.  

Keywords: 𝔉ℱ𝛿𝑜𝑠, 𝔉ℱ𝛿𝐶𝑡𝑠, 𝔉ℱ𝛿𝐼𝑟𝑟 and Fermatean fuzzy entropy measure. 

AMS (2000) subject classification: 03E72, 54A40, 94D05 

 

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 

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 Pythagorean fuzzy set. Olgun et al., [9] defined a Pythagorean 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 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 10s (2025) 

  

1910 
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and weaker forms of Fermatean fuzzy continuous and irresolute maps in Fermatean fuzzy topological 

spaces and also specialized some of their basic properties with examples. In section 5, entropy measure 

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

version of entropy measure. Here we introduce entropy measure for Fermatean fuzzy sets and give an 

example for the decision making in real life problem. The paper is concluded 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, 

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, 



Communications on Applied Nonlinear Analysis 

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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}.  

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:   

    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 



Communications on Applied Nonlinear Analysis 

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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 Pythagorean fuzzy subset of a set can be 

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

Pythagorean 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 Pythagorean fuzzy topological space. Even an Fermatean fuzzy open set maybe 

neither an Intuitionistic fuzzy set nor Pythagorean fuzzy set.  

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

{1𝔉, 0𝔉, 𝐹1, 𝐹2} where 

 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 Pythagorean 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:  



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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), 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: 



Communications on Applied Nonlinear Analysis 

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1914 
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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 𝜹 (resp. 𝜹 pre, 𝜹 semi, 𝜹𝜶 and 𝜹𝜷)-continuous mappings 

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

mappings 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.  

Lemma 3.1   Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a function. Then the following statements hold.   

    1.  If 𝑆 and 𝑇 are 𝔉𝑓𝑠’s of 𝑋1 such that 𝑆 ⊆ 𝑇, then ℎ𝔉(𝑆) ⊆ ℎ𝔉(𝑇).  

    2.  If 𝑆 and 𝑇 are 𝔉𝑓𝑠’s of 𝑋2 such that 𝑆 ⊆ 𝑇, then ℎ𝔉
−1(𝑆) ⊆ ℎ𝔉

−1(𝑇).  

Lemma 3.2   Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a function. If 𝑆 is a 𝔉ℱ𝑠 of 𝑋1 and 𝑇 is a 𝔉ℱ𝑠 of 

𝑋2. Then   

    1.  ℎ𝔉(ℎ𝔉
−1(𝑆)) ⊆ 𝑆  

    2.  ℎ𝔉(ℎ𝔉
−1(𝑆)) = 𝑆 ⇔ ℎ𝔉 is surjective.  

    3.  ℎ𝔉
−1(ℎ𝔉(𝑆)) ⊇ 𝑆  

    4.  ℎ𝔉
−1(ℎ𝔉(𝑆)) = 𝑆 whenever ℎ𝔉 is injective.  

Theorem 3.1   Let (𝑋1, 𝜏1) and (𝑋2, 𝜏2) be two 𝔉ℱ𝑡𝑠’s and let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2), then   

    (i)  Every 𝔉ℱ𝛿𝐶𝑡𝑠 is a 𝔉ℱ𝐶𝑡𝑠.  

    (ii)  Every 𝔉ℱ𝛿𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝒫𝐶𝑡𝑠.  

    (iii)  Every 𝔉ℱ𝛿𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝒮𝐶𝑡𝑠.  

    (iv)  Every 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠.  

    (v)  Every 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠.  

    (vi)  Every 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝒫𝐶𝑡𝑠.  

    (vii)  Every 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 is a 𝔉ℱ𝛿𝒮𝐶𝑡𝑠.  

 But not converse.  

  Proof. (i) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝐶𝑡𝑠. Let 𝑆 be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) 

is 𝔉ℱ𝛿𝑜 set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝑜 set is 𝔉ℱ𝑜𝑠, ℎ𝔉
−1(𝑆) is 𝔉ℱ𝑜 set in (𝑋1, 𝜏1). Hence ℎ𝔉 

is 𝔉ℱ𝐶𝑡𝑠 function. 



Communications on Applied Nonlinear Analysis 

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(ii) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝐶𝑡𝑠 . Let 𝑆  be a 𝔉ℱ𝑜  set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆)  is 

𝔉ℱ𝛿𝑜 set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝑜 set is 𝔉ℱ𝛿𝒫𝑜𝑠, ℎ𝔉
−1(𝑆) is 𝔉ℱ𝛿𝒫𝑜 set in (𝑋1, 𝜏1). Hence 

ℎ𝔉 is 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 function. 

(iii) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝐶𝑡𝑠 . Let 𝑆  be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) is 

𝔉ℱ𝛿𝑜 set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝑜 set is 𝔉ℱ𝛿𝒮𝑜𝑠, ℎ𝔉
−1(𝑆) is 𝔉ℱ𝛿𝒮𝑜 set in (𝑋1, 𝜏1). Hence 

ℎ𝔉 is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 function. 

(iv) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝒮𝐶𝑡𝑠. Let 𝑆 be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) is 

𝔉ℱ𝛿𝒮𝑜  set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝒮𝑜  set is 𝔉ℱ𝛿𝛽𝑜𝑠 , ℎ𝔉
−1(𝑆)  is 𝔉ℱ𝛿𝛽𝑜  set in (𝑋1, 𝜏1). 

Hence ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 function. 

(v) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝒫𝐶𝑡𝑠. Let 𝑆 be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) is 

𝔉ℱ𝛿𝒫𝑜 set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝒫𝑜  set is 𝔉ℱ𝛿𝛽𝑜𝑠, ℎ𝔉
−1(𝑆) is 𝔉ℱ𝛿𝛽𝑜  set in (𝑋1, 𝜏1). 

Hence ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 function. 

(vi) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝛼𝐶𝑡𝑠. Let 𝑆 be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) is 

𝔉ℱ𝛿𝛼𝑜 set in (𝑋1, 𝜏1). Since every 𝔉ℱ𝛿𝛼𝑜  set is 𝔉ℱ𝛿𝒫𝑜𝑠, ℎ𝔉
−1(𝑆) is 𝔉ℱ𝛿𝒫𝑜  set in (𝑋1, 𝜏1). 

Hence ℎ𝔉 is 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 function. 

(vii) Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a 𝔉ℱ𝛿𝛼𝐶𝑡𝑠. Let 𝑆 be a 𝔉ℱ𝑜 set in (𝑋2, 𝜏2). Then ℎ𝔉
−1(𝑆) is 

𝔉ℱ𝛿𝛼𝑜  set in (𝑋1, 𝜏1) . Since every 𝔉ℱ𝛿𝛼𝑜  set is 𝔉ℱ𝛿𝒮𝑜𝑠 , ℎ𝔉
−1(𝑆) is 𝔉ℱ𝛿𝒮𝑜  set in (𝑋1, 𝜏1) . 

Hence ℎ𝔉 is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 function.              

Remark 3.1  The following Figure shows the relations among the different types of Fermatean fuzzy 

δ continuous mappings that were studied in this section.  

 

Figure : 𝕱𝓕𝜹𝑪𝒕𝒔 mappings in 𝕱𝓕𝒕𝒔 

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

𝛼𝐴1
(𝑎) = 0.4, 𝛽𝐴1

(𝑎) = 0.1, 𝛼𝐴1
(𝑏) = 0.6, 𝛽𝐴1

(𝑏) = 0.3; 

𝛼𝐴2
(𝑎) = 0.9, 𝛽𝐴2

(𝑎) = 0.2, 𝛼𝐴2
(𝑏) = 0.6, 𝛽𝐴2

(𝑏) = 0.3; 

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

function, Then ℎ𝔉 is 𝔉ℱ𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝛽𝐶𝑡𝑠) but not 𝔉ℱ𝛿𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝒮𝐶𝑡𝑠). Since, 𝐴2 is a 

𝔉ℱ𝑜 set in 𝑋2 but ℎ𝔉
−1(𝐴2) = 𝐴2 is not 𝔉ℱ𝛿𝑜 (resp. 𝔉ℱ𝛿𝒮𝑜 ) set in 𝑋1. 



Communications on Applied Nonlinear Analysis 

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1916 
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Example 3.2  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2 and 𝐴3 are defined as 

𝛼𝐴1
(𝑎) = 0.2, 𝛽𝐴1

(𝑎) = 0.8, 𝛼𝐴1
(𝑏) = 0.3, 𝛽𝐴1

(𝑏) = 0.7; 

𝛼𝐴2
(𝑎) = 0.1, 𝛽𝐴2

(𝑎) = 0.9, 𝛼𝐴2
(𝑏) = 0.1, 𝛽𝐴2

(𝑏) = 0.9; 

𝛼𝐴3
(𝑎) = 0.2, 𝛽𝐴3

(𝑎) = 0.8, 𝛼𝐴3
(𝑏) = 0.4, 𝛽𝐴3

(𝑏) = 0.6; 

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

identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝒫𝐶𝑡𝑠) but not 𝔉ℱ𝛿𝐶𝑡𝑠. Since, 𝐴1 is a 𝔉ℱ𝑜 set 

in 𝑋2 but ℎ𝔉
−1(𝐴1) = 𝐴1 is not 𝔉ℱ𝛿𝑜 set in 𝑋1. 

Example 3.3  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2 and 𝐴3 are defined as 

𝛼𝐴1
(𝑎) = 0.2, 𝛽𝐴1

(𝑎) = 0.6, 𝛼𝐴1
(𝑏) = 0.3, 𝛽𝐴1

(𝑏) = 0.5; 

𝛼𝐴2
(𝑎) = 0.6, 𝛽𝐴2

(𝑎) = 0.2, 𝛼𝐴2
(𝑏) = 0.5, 𝛽𝐴2

(𝑏) = 0.3; 

𝛼𝐴3
(𝑎) = 0.6, 𝛽𝐴3

(𝑎) = 0.6, 𝛼𝐴3
(𝑏) = 0.5, 𝛽𝐴3

(𝑏) = 0.5; 

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

identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝒫𝐶𝑡𝑠) but not 𝔉ℱ𝛼𝐶𝑡𝑠. Since, 𝐴3 is a 𝔉ℱ𝑜 set 

in 𝑋2 but ℎ𝔉
−1(𝐴3) = 𝐴3 is not 𝔉ℱ𝛼𝑜 set in 𝑋1. 

Example 3.4  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐵1, 𝐵2, 𝐵3 and 𝐵4 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; 

𝛼𝐵1
(𝑎) = 0.2, 𝛽𝐵1

(𝑎) = 0.8, 𝛼𝐵1
(𝑏) = 0.4, 𝛽𝐵1

(𝑏) = 0.6; 

𝛼𝐵2
(𝑎) = 0.1, 𝛽𝐵2

(𝑎) = 0.9, 𝛼𝐵2
(𝑏) = 0.3, 𝛽𝐵2

(𝑏) = 0.7; 

𝛼𝐵3
(𝑎) = 0.9, 𝛽𝐵3

(𝑎) = 0.1, 𝛼𝐵3
(𝑏) = 0.7, 𝛽𝐵3

(𝑏) = 0.3; 

𝛼𝐵4
(𝑎) = 0.2, 𝛽𝐵4

(𝑎) = 0.8, 𝛼𝐵4
(𝑏) = 0.3, 𝛽𝐵4

(𝑏) = 0.7; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4}  and 𝜏2 = {0𝔉, 1𝔉, 𝐵1, 𝐵2, 𝐵3, 𝐵4}  are 𝔉ℱ𝑡𝑠 ’s on 𝑋  and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 but not 𝔉ℱ𝛿𝒫𝐶𝑡𝑠. Since, 𝐵4 

is a 𝔉ℱ𝑜 set in 𝑋2 but ℎ𝔉
−1(𝐵4) = 𝐵4 is not 𝔉ℱ𝛿𝒫𝑜 set in 𝑋1. 

Theorem 3.2  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. A mapping ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) satisfies the 

following conditions are equivalent.   

    1.  ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠.  

    2.  The inverse ℎ𝔉
−1(𝐾) of all 𝔉ℱ𝑐𝑠 set 𝐾 in 𝑋2 is 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1917 
https://internationalpubls.com 

   Proof. The proof is directly, from ℎ𝔉
−1(𝐾) = ℎ𝔉

−1(𝐾) for all 𝔉ℱ𝑐𝑠 𝐾 of 𝑋2.              

Theorem 3.3  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. A mapping ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) satisfies the 

following conditions are hold.   

    (i)  ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(𝐿)) ⊆ 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿)), for all 𝔉ℱ𝑐𝑠 𝐿 in 𝑋1.  

    (ii)  𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) ⊆ ℎ𝔉

−1(𝔉ℱ𝛿𝑐𝑙(𝐾)), for all 𝔉ℱ𝑐𝑠 𝐾 in 𝑋2.  

   Proof. (i) Since 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿))  is a 𝔉ℱ𝛿𝑐𝑠  in 𝑋2  and ℎ𝔉  is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 , then ℎ𝔉
−1(𝔉ℱ𝛿𝑐𝑙( 

ℎ𝔉(𝐿)))  is 𝔉ℱ𝛿𝛽𝑐𝑠  in 𝑋1 . Now, since 𝐿 ⊆ ℎ𝔉
−1(𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿))) , 𝔉ℱ𝛿𝛽𝑐𝑙(𝐿) ⊆

ℎ𝔉
−1(𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿))). Therefore, ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(𝐿)) ⊆ 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿)). 

(ii) By replacing 𝐿  with 𝐾  in (i), we obtain ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾))) ⊆ 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(  ℎ𝔉

−1(𝐾))) ⊆

𝔉ℱ𝛿𝑐𝑙(𝐾). Hence, 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) ⊆ ℎ𝔉

−1(𝔉ℱ𝛿𝑐𝑙(𝐾)).              

Remark 3.2  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a mapping. If ℎ𝔉 

is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠, then   

    1.  ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(𝐿)) is not necessarily equal to 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐿)) where 𝐿 ∈ 𝑋1.  

    2.  𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) is not necessarily equal to ℎ𝔉

−1(𝔉ℱ𝛿𝑐𝑙(𝐾)) where 𝐾 ∈ 𝑋2.  

Example 3.5   Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐵1, 𝐵2, 𝐵3 and 𝐵4 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; 

𝛼𝐵1
(𝑎) = 0.2, 𝛽𝐵1

(𝑎) = 0.8, 𝛼𝐵1
(𝑏) = 0.4, 𝛽𝐵1

(𝑏) = 0.6; 

𝛼𝐵2
(𝑎) = 0.1, 𝛽𝐵2

(𝑎) = 0.9, 𝛼𝐵2
(𝑏) = 0.3, 𝛽𝐵2

(𝑏) = 0.7; 

𝛼𝐵3
(𝑎) = 0.9, 𝛽𝐵3

(𝑎) = 0.1, 𝛼𝐵3
(𝑏) = 0.7, 𝛽𝐵3

(𝑏) = 0.3; 

𝛼𝐵4
(𝑎) = 0.2, 𝛽𝐵4

(𝑎) = 0.8, 𝛼𝐵4
(𝑏) = 0.3, 𝛽𝐵4

(𝑏) = 0.7; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐵1, 𝐵2, 𝐵3, 𝐵4} and 𝜏2 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4} are 𝔉ℱ𝑡𝑠’s on 𝑋 and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠, [(i)] 

    1.  ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(𝐴1)) = 𝐴1 . But 𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐴1)) = 𝐴1
𝑐 . Thus ℎ𝔉(𝔉ℱ𝛿𝛽𝑐𝑙(𝐴1)) ≠

𝔉ℱ𝛿𝑐𝑙(ℎ𝔉(𝐴1)).  

    2.  𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐴1)) = 𝐴1 . But ℎ𝔉

−1(𝔉ℱ𝛿𝑐𝑙(𝐴1)) = 𝐴1
𝑐 . Thus 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉

−1(𝐴1))  ≠

ℎ𝔉
−1(𝔉ℱ𝛿𝑐𝑙(𝐴1)).  

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

1918 
https://internationalpubls.com 

Theorem 3.4  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a mapping. If ℎ𝔉 

is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠, then ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)) ⊆ 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉

−1(𝐿)), for all 𝔉ℱ𝑠 𝐿 in 𝑋2.  

  Proof. If ℎ𝔉  is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠  and 𝐿 ⊆ 𝑋2 . 𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)  is 𝔉ℱ𝛿𝑜𝑠  in 𝑋2  and hence, ℎ𝔉
−1(𝔉ℱ𝛿 

𝑖𝑛𝑡(𝐿))  is 𝔉ℱ𝛿𝛽𝑜𝑠  in 𝑋1 . Therefore 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿))) = ℎ𝔉

−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)) . Also, 

𝔉ℱ𝛿𝑖𝑛𝑡(𝐿) ⊆ 𝐿 , implies that ℎ𝔉
−1(𝔉ℱ  𝛿𝑖𝑛𝑡(𝐿)) ⊆ ℎ𝔉

−1(𝐿) . Therefore 

𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿))) ⊆ 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉

−1(𝐿)). That is ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)) ⊆ 𝔉ℱ𝛿𝛽𝑖𝑛𝑡 (ℎ𝔉

−1(𝐿)). 

Conversely, let ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)) ⊆ 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉

−1(𝐿)) for all subset 𝐿 of 𝑋2. If 𝐿 is 𝔉ℱ𝛿𝑜 in 𝑋2, 

then 𝔉ℱ𝛿𝑖𝑛𝑡(𝐿) = 𝐿 . By assumption, ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐿)) ⊆ 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉

−1(𝐿)) . Thus ℎ𝔉
−1(𝐿) ⊆

𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝐿)) . But 𝔉ℱ𝛿𝛽𝑖𝑛𝑡  (ℎ𝔉

−1(𝐿)) ⊆ ℎ𝔉
−1(𝐿) . Therefore 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉

−1(𝐿)) = ℎ𝔉
−1(𝐿) . 

That is, ℎ𝔉
−1(𝐿) is 𝔉ℱ𝛿𝛽𝑜 in 𝑋1, for all 𝔉ℱ𝛿𝑜𝑠 𝐿 in 𝑋2. Therefore ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 in 𝑋1.              

Remark 3.3  Let (𝑋1, 𝜏1) & (𝑋2, 𝜏2) be a 𝔉ℱ𝑡𝑠’s. Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be a mapping. If ℎ𝔉 

is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠, then 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝐾)) is not necessarily equal to h𝔉

−1(𝔉ℱδint(K)) where K ∈ X2.  

Example 3.6  In Example 3.5, ℎ𝔉  is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 .Then 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝐴1)) = 𝐴1 . But 

ℎ𝔉
−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐴1)) = 𝐴1

𝑐. Thus 𝔉ℱ𝛿𝛽𝑖𝑛𝑡(ℎ𝔉
−1(𝐴1)) ≠ ℎ𝔉

−1(𝔉ℱ𝛿𝑖𝑛𝑡(𝐴1)).  

Remark 3.4  Theorems 3.2, 3.3, 3.4 and Remarks 3.2, 3.3 are true for 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠 and 

𝔉ℱ𝛿𝛼𝑜𝑠.  

4  Fermatean fuzzy 𝜹 (resp. 𝜹 pre, 𝜹 semi, 𝜹𝜶 and 𝜹𝜷)-irresolute maps 

 In this section, we introduce the concept of Fermatean fuzzy irresoluteness called Fermatean fuzzy 

(resp. 𝛿 , 𝛿𝒫, 𝛿𝒮, 𝛿𝛼  and 𝛿𝛽) -irresolute maps by using 𝔉ℱ𝒮𝑜𝑠  (resp. 𝔉ℱ𝛿𝑜𝑠 , 𝔉ℱ𝛿𝒫𝑜𝑠,

𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠)’s and study some of their basic properties. This definition enables 

us to obtain conditions under which maps and inverse maps preserve respective open sets. 

Definition 4.1  A map ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) is said to be Fermatean fuzzy (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝛿𝛼 

and 𝛿𝛽) -irresolute (in short, 𝔉ℱ𝐼𝑟𝑟  (resp. 𝔉ℱ𝛿𝐼𝑟𝑟 , 𝔉ℱ𝛿𝒫𝐼𝑟𝑟, 𝔉ℱ𝛿𝒮𝐼𝑟𝑟, 𝔉ℱ𝛿𝛼𝐼𝑟𝑟  and 

𝔉ℱ𝛿𝛽𝐼𝑟𝑟) ) map if ℎ𝔉
−1(𝐾)  is a 𝔉ℱ𝒮𝑜𝑠  (resp. 𝔉ℱ𝛿𝑜𝑠 , 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠  and 

𝔉ℱ𝛿𝛽𝑜𝑠) in (𝑋1, 𝜏1) for each 𝔉ℱ𝒮𝑜𝑠 (resp. 𝔉ℱ𝛿𝑜𝑠, 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠) 

𝐾 of (𝑋2, 𝜏2).  

Theorem 4.1  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 𝔉ℱ𝐼𝑟𝑟 map is a 𝔉ℱ𝒮𝐶𝑡𝑠.  

    (ii)  Every 𝔉ℱ𝛿𝒮𝐼𝑟𝑟 map is a 𝔉ℱ𝛿𝒮𝐶𝑡𝑠.  

    (iii)  Every 𝔉ℱ𝛿𝒫𝐼𝑟𝑟 map is a 𝔉ℱ𝛿𝒫𝐶𝑡𝑠.  

    (iv)  Every 𝔉ℱ𝛿𝛼𝐼𝑟𝑟 map is a 𝔉ℱ𝛿𝛼𝐶𝑡𝑠.  

    (v)  Every 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 map is a 𝔉ℱ𝛿𝛽𝐶𝑡𝑠.  

   Proof. (i) Consider a 𝔉ℱ𝐼𝑟𝑟 map ℎ𝔉 and a 𝔉ℱ𝑜𝑠 𝐾 in 𝑋2. As each 𝔉ℱ𝑜𝑠 is a 𝔉ℱ𝒮𝑜𝑠, 𝐾 is a 



Communications on Applied Nonlinear Analysis 

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𝔉ℱ𝒮𝑜𝑠 in 𝑋2. By presumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝒮𝑜𝑠 in 𝑋1. Thus ℎ𝔉 is a 𝔉ℱ𝒮𝐶𝑡𝑠 map. 

(ii) Consider a 𝔉ℱ𝛿𝒮𝐼𝑟𝑟 map ℎ𝔉 and a 𝔉ℱ𝛿𝑜𝑠 𝐾 in 𝑋2. As each 𝔉ℱ𝛿𝑜𝑠 is a 𝔉ℱ𝑜𝑠 and 𝔉ℱ𝛿𝒮𝑜𝑠, 

𝐾 is a 𝔉ℱ𝛿𝑜𝑠 and 𝔉ℱ𝛿𝒮𝑜𝑠 in 𝑋2 . By presumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝒮𝑜𝑠 in 𝑋1 . Thus ℎ𝔉  is a 

𝔉ℱ𝛿𝒮𝐶𝑡𝑠 map. 

(iii) Consider a 𝔉ℱ𝛿𝒫𝐼𝑟𝑟  map ℎ𝔉  and a 𝔉ℱ𝛿𝑜𝑠  𝐾  in 𝑋2 . As each 𝔉ℱ𝛿𝑜𝑠  is a 𝔉ℱ𝑜𝑠  and 

𝔉ℱ𝛿𝒫𝑜𝑠, 𝐾 is a 𝔉ℱ𝛿𝑜𝑠 and 𝔉ℱ𝛿𝒫𝑜𝑠 in 𝑋2. By presumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝒫𝑜𝑠 in 𝑋1. Thus 

ℎ𝔉 is a 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 map. 

(iv) Consider a 𝔉ℱ𝛿𝛼𝐼𝑟𝑟  map ℎ𝔉  and a 𝔉ℱ𝛿𝑜𝑠  𝐾  in 𝑋2 . As each 𝔉ℱ𝛿𝑜𝑠  is a 𝔉ℱ𝑜𝑠  and 

𝔉ℱ𝛿𝛼𝑜𝑠, 𝐾 is a 𝔉ℱ𝛿𝑜𝑠 and 𝔉ℱ𝛿𝛼𝑜𝑠 in 𝑋2. By presumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛼𝑜𝑠 in 𝑋1. Thus 

ℎ𝔉 is a 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 map. 

(v) Consider a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 map ℎ𝔉 and a 𝔉ℱ𝛿𝑜𝑠 𝐾 in 𝑋2. As each 𝔉ℱ𝛿𝑜𝑠 is a 𝔉ℱ𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠, 

𝐾 is a 𝔉ℱ𝛿𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠 in 𝑋2. By presumption, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛽𝑜𝑠 in 𝑋1. Thus ℎ𝔉  is a 

𝔉ℱ𝛿𝛽𝐶𝑡𝑠 map.             

Example 4.1  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 𝔉ℱ𝒮𝐶𝑡𝑠 (resp. 𝔉ℱ𝛿𝒫𝐶𝑡𝑠) but not 𝔉ℱ𝐼𝑟𝑟 (resp. 𝔉ℱ𝛿𝒫𝐼𝑟𝑟). 

Since, 𝐴2
𝑐  is a 𝔉ℱ𝒮𝑜 (resp. 𝔉ℱ𝛿𝒫𝑜) set in 𝑋2 but ℎ𝔉

−1(𝐴2
𝑐) = 𝐴2

𝑐  is not 𝔉ℱ𝒮𝑜 (resp. 𝔉ℱ𝛿𝒫𝑜) set 

in 𝑋1. 

Example 4.2  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐵1, 𝐵2, 𝐵3 and 𝐵4 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; 

𝛼𝐵1
(𝑎) = 0.2, 𝛽𝐵1

(𝑎) = 0.8, 𝛼𝐵1
(𝑏) = 0.4, 𝛽𝐵1

(𝑏) = 0.6; 

𝛼𝐵2
(𝑎) = 0.1, 𝛽𝐵2

(𝑎) = 0.9, 𝛼𝐵2
(𝑏) = 0.3, 𝛽𝐵2

(𝑏) = 0.7; 

𝛼𝐵3
(𝑎) = 0.9, 𝛽𝐵3

(𝑎) = 0.1, 𝛼𝐵3
(𝑏) = 0.7, 𝛽𝐵3

(𝑏) = 0.3; 

𝛼𝐵4
(𝑎) = 0.2, 𝛽𝐵4

(𝑎) = 0.8, 𝛼𝐵4
(𝑏) = 0.3, 𝛽𝐵4

(𝑏) = 0.7; 



Communications on Applied Nonlinear Analysis 

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Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4}  and 𝜏2 = {0𝔉, 1𝔉, 𝐵1, 𝐵2, 𝐵3, 𝐵4}  are 𝔉ℱ𝑡𝑠 ’s on 𝑋  and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 but not 𝔉ℱ𝛿𝒮𝐼𝑟𝑟 . Since, 𝐴4 

is a 𝔉ℱ𝛿𝒮𝑜 set in 𝑋2 but ℎ𝔉
−1(𝐴4) = 𝐴4 is not 𝔉ℱ𝛿𝒮𝑜 set in 𝑋1. 

Example 4.3  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4, and 𝐴5 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) = 0.3; 

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; 

𝛼𝐴5
(𝑎) = 0.6, 𝛽𝐴5

(𝑎) = 0.4, 𝛼𝐴5
(𝑏) = 0.6, 𝛽𝐴5

(𝑏) = 0.4; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4}  and 𝜏2 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3}  are 𝔉ℱ𝑡𝑠 ’s on 𝑋  and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 but not 𝔉ℱ𝛿𝛼𝐼𝑟𝑟. Since, 𝐴5 

is a 𝔉ℱ𝛿𝛼𝑜 set in 𝑋2 but ℎ𝔉
−1(𝐴5) = 𝐴5 is not 𝔉ℱ𝛿𝛼𝑜 set in 𝑋1. 

Example 4.4  Let 𝑋1 = 𝑋2 = 𝑋 = {𝑎, 𝑏} and the 𝔉ℱ𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4, 𝐵1, 𝐵2, 𝐵3 and 𝐵4 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; 

𝛼𝐵1
(𝑎) = 0.4, 𝛽𝐵1

(𝑎) = 0.6, 𝛼𝐵1
(𝑏) = 0.5, 𝛽𝐵1

(𝑏) = 0.5; 

𝛼𝐵2
(𝑎) = 0.6, 𝛽𝐵2

(𝑎) = 0.4, 𝛼𝐵2
(𝑏) = 0.6, 𝛽𝐵2

(𝑏) = 0.4; 

𝛼𝐵3
(𝑎) = 0.7, 𝛽𝐵3

(𝑎) = 0.3, 𝛼𝐵3
(𝑏) = 0.6, 𝛽𝐵3

(𝑏) = 0.4; 

𝛼𝐵4
(𝑎) = 0.4, 𝛽𝐵4

(𝑎) = 0.6, 𝛼𝐵4
(𝑏) = 0.4, 𝛽𝐵4

(𝑏) = 0.6; 

Let 𝜏1 = {0𝔉, 1𝔉, 𝐴1, 𝐴2, 𝐴3, 𝐴4}  and 𝜏2 = {0𝔉, 1𝔉, 𝐵1, 𝐵2, 𝐵3, 𝐵4}  are 𝔉ℱ𝑡𝑠 ’s on 𝑋  and let 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) be an identity function, Then ℎ𝔉 is 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 but not 𝔉ℱ𝛿𝛽𝐼𝑟𝑟. Since, 𝐴4
𝑐  

is a 𝔉ℱ𝛿𝛽𝑜 set in 𝑋2 but ℎ𝔉
−1(𝐴4

𝑐) = 𝐴4
𝑐  is not 𝔉ℱ𝛿𝛽𝑜 set in 𝑋1. 

Definition 4.2  A 𝔉ℱ𝑡𝑠 (𝑋1, 𝜏1) is known as a Fermatean fuzzy 𝛿𝒮𝑈1

2

 (resp. 𝛿𝒫𝑈1

2

, 𝛿𝛼𝑈1

2

 and 

𝛿𝛽𝑈1

2

) (in short, 𝔉ℱ𝛿𝒮𝑈1

2

 (resp. 𝔉ℱ𝛿𝒫𝑈1

2

, 𝔉ℱ𝛿𝛼𝑈1

2

 and 𝔉ℱ𝛿𝛽𝑈1

2

))-space, if each 𝔉ℱ𝛿𝒮𝑜𝑠 (resp. 

𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝛼𝑜𝑠 and 𝔉ℱ𝛿𝛽𝑜𝑠) in 𝑋 is 𝔉ℱ𝑜𝑠 in 𝑋1.  

Theorem 4.2  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  be a 𝔉ℱ𝛿𝒮𝐼𝑟𝑟  (resp. 𝔉ℱ𝛿𝒫𝐼𝑟𝑟 , 𝔉ℱ𝛿𝛼𝐼𝑟𝑟  and 

𝔉ℱ𝛿𝛽𝐼𝑟𝑟  ) map. Then ℎ𝔉  is a 𝔉ℱ𝐶𝑡𝑠  map if 𝑋  is a 𝔉ℱ𝛿𝒮𝑈1

2

 (resp. 𝔉ℱ𝛿𝒫𝑈1

2

, 𝔉ℱ𝛿𝛼𝑈1

2

 and 



Communications on Applied Nonlinear Analysis 

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1921 
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𝔉ℱ𝛿𝛽𝑈1

2

)-space.  

   Proof. 

(i) Consider a 𝔉ℱ𝑜𝑠 𝐾 in 𝑋2. Then 𝐾 is a 𝔉ℱ𝛿𝒮𝑜𝑠 in 𝑋2. Therefore ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝒮𝑜𝑠 in 𝑋1. 

Since 𝑋1 is a 𝔉ℱ𝛿𝒮𝑈1

2

-space, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝑜𝑠 in 𝑋1. Hence ℎ𝔉 is a 𝔉ℱ𝐶𝑡𝑠 map. 

(ii) Consider a 𝔉ℱ𝑜𝑠 𝐾 in 𝑋2. Then 𝐾 is a 𝔉ℱ𝛿𝒫𝑜𝑠 in 𝑋2. Therefore ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝒫𝑜𝑠 in 

𝑋1. Since 𝑋 is a 𝔉ℱ𝛿𝒫𝑈1

2

-space, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝑜𝑠 in 𝑋1. Hence ℎ𝔉 is a 𝔉ℱ𝐶𝑡𝑠 map. 

(iii) Consider a 𝔉ℱ𝑜𝑠 𝐾 in 𝑋2. Then 𝐾 is a 𝔉ℱ𝛿𝛼𝑜𝑠 in 𝑋2. Therefore ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛼𝑜𝑠 in 

𝑋1. Since 𝑋1 is a 𝔉ℱ𝛿𝛼𝑈1

2

-space, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝑜𝑠 in 𝑋1. Hence ℎ𝔉 is a 𝔉ℱ𝐶𝑡𝑠 map. 

(iv) Consider a 𝔉ℱ𝑜𝑠 𝐾 in 𝑋2. Then 𝐾 is a 𝔉ℱ𝛿𝛽𝑜𝑠 in 𝑋2. Therefore ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛽𝑜𝑠 in 

𝑋1. Since 𝑋1 is a 𝔉ℱ𝛿𝛽𝑈1

2

-space, ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝑜𝑠 in 𝑋1. Hence ℎ𝔉 is a 𝔉ℱ𝐶𝑡𝑠 map.             

Theorem 4.3  Let ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  and 𝑔𝔉: (𝑋2, 𝜏2) → (𝑋3, 𝜏3)  be 𝔉ℱ𝛿𝐼𝑟𝑟  (resp. 

𝔉ℱ𝛿𝒮𝐼𝑟𝑟 , 𝔉ℱ𝛿𝒫𝐼𝑟𝑟 , 𝔉ℱ𝛿𝛼𝐼𝑟𝑟  and 𝔉ℱ𝛿𝛽𝐼𝑟𝑟  ) maps, then 𝑔𝔉 ∘ ℎ𝔉: (𝑋1, 𝜏1) → (𝑋3, 𝜏3)  is a 

𝔉ℱ𝛿𝐼𝑟𝑟 (resp. 𝔉ℱ𝛿𝒮𝐼𝑟𝑟, 𝔉ℱ𝛿𝒫𝐼𝑟𝑟, 𝔉ℱ𝛿𝛼𝐼𝑟𝑟 and 𝔉ℱ𝛿𝛽𝐼𝑟𝑟) map.  

   Proof. Consider a 𝔉ℱ𝛿𝑜𝑠 𝐾 in 𝑋3. So 𝑔𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝑜𝑠 in 𝑋2. As ℎ𝔉 is a 𝔉ℱ𝛿𝐼𝑟𝑟 map, 

ℎ𝔉
−1(𝑔𝔉

−1(𝐾)) is a 𝔉ℱ𝛿𝑜𝑠 in 𝑋1. Thus 𝑔𝔉 ∘ ℎ𝔉 is a 𝔉ℱ𝛿𝐼𝑟𝑟 map. The other cases are similar.              

Theorem 4.4  Consider a 𝔉ℱ𝛿𝐼𝑟𝑟  (resp. 𝔉ℱ𝛿𝒮𝐼𝑟𝑟 , 𝔉ℱ𝛿𝒫𝐼𝑟𝑟 , 𝔉ℱ𝛿𝛼𝐼𝑟𝑟  and 𝔉ℱ𝛿𝛽𝐼𝑟𝑟)  map 

ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2)  and a 𝔉ℱ𝛿𝐶𝑡𝑠  (resp. 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 , 𝔉ℱ𝛿𝛼𝐶𝑡𝑠  and 𝔉ℱ𝛿𝛽𝐶𝑡𝑠  ) 

map 𝑔𝔉: (𝑋2, 𝜏2) → (𝑋3, 𝜏3) . Then 𝑔𝔉 ∘ ℎ𝔉: (𝑋1, 𝜏1) → (𝑋3, 𝜏3)  is a 𝔉ℱ𝛿𝐶𝑡𝑠  (resp. 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 , 

𝔉ℱ𝛿𝒫𝐶𝑡𝑠, 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 and 𝔉ℱ𝛿𝛽𝐶𝑡𝑠 ) map.  

   Proof. Consider a 𝔉ℱ𝑜𝑠 𝐾  in 𝑋3 . So 𝑔𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝑜𝑠 in 𝑋2 . As ℎ𝔉  is a 𝔉ℱ𝛿𝐼𝑟𝑟 map, 

ℎ𝔉
−1(𝑔𝔉

−1(𝑈)) is a 𝔉ℱ𝛿𝑜𝑠 in 𝑋1. Thus 𝑔𝔉 ∘ ℎ𝔉 is a 𝔉ℱ𝛿𝐶𝑡𝑠 map. The other cases are similar.              

Theorem 4.5  Consider a map ℎ𝔉: (𝑋1, 𝜏1) → (𝑋2, 𝜏2) from a 𝔉ℱ𝑡𝑠  𝑋1  into a 𝔉ℱ𝑡𝑠  𝑋2 . The 

following are equivalent if 𝑋1 and 𝑋2 are 𝔉ℱ𝛿𝛽𝑈1

2

-spaces.   

    (i)  ℎ𝔉 is a 𝔉ℱ𝛿𝛽𝐼𝑟𝑟 map.  

    (ii)  ℎ𝔉
−1(𝐾) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1 for every 𝔉ℱ𝛿𝛽𝑐𝑠 𝐾 in 𝑋2.  

    (iii) 𝔉ℱ𝑐𝑙(ℎ𝔉
−1(𝐾)) ⊆ ℎ𝔉

−1(𝔉ℱ𝑐𝑙(𝐾)) for every 𝔉ℱ𝑠 𝐾 of 𝑋2.  

    Proof. (i) → (ii): Consider a 𝔉ℱ𝛿𝛽𝑐𝑠 𝐾 in 𝑋2. It follows 𝐾𝑐 is a 𝔉ℱ𝛿𝛽𝑜𝑠 in 𝑋2. As ℎ𝔉  is 

𝔉ℱ𝛿𝛽𝐼𝑟𝑟, ℎ𝔉
−1((𝐾)𝑐) is a 𝔉ℱ𝛿𝛽𝑜𝑠 in 𝑋1. We know that ℎ𝔉

−1((𝐾)𝑐) = (ℎ𝔉
−1(𝐾))

𝑐
. Hence ℎ𝔉

−1(𝐾) 

is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1. 

(ii) → (iii): Consider a 𝔉ℱ𝑠  𝐾  in 𝑋2  and 𝐾 ⊆ 𝔉ℱ𝛿𝛽𝑐𝑙(𝐾). Then ℎ𝔉
−1(𝐾) ⊆ ℎ𝔉

−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)). 

Since 𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋2. By presumption, ℎ𝔉
−1((𝔉ℱ𝛿𝛽𝑐𝑙(𝐾))) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1. 



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Also, as 𝑋1 is 𝔉ℱ𝛿𝛽𝑈1

2

-space, ℎ𝔉
−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1. 

(iii) →  (i): Consider a 𝔉ℱ𝛿𝛽𝑐𝑠  𝐾  in 𝑋2 . As 𝑋2  is 𝔉ℱ𝛿𝛽𝑈1

2

-space, 𝐾  is 𝔉ℱ𝑐𝑠  in 𝑋2  and 

𝔉ℱ𝑐𝑙(𝐾) = 𝐾 . Thus ℎ𝔉
−1(𝐾) = ℎ𝔉

−1(𝔉ℱ𝛿𝛽𝑐𝑙(𝐾)) ⊇ 𝔉ℱ𝛿𝛽𝑐𝑙(ℎ𝔉
−1(𝐾)) = 𝔉ℱ𝑐𝑙(ℎ𝔉

−1(𝐾)) . But 

clearly ℎ𝔉
−1(𝐾) ⊆ 𝔉ℱ𝑐𝑙(ℎ𝔉

−1(𝐾)) . Therefore 𝔉ℱ𝑐𝑙(ℎ𝔉
−1(𝐾)) = ℎ𝔉

−1(𝐾).  It follows ℎ𝔉
−1(𝐾)  is a 

𝔉ℱ𝑐𝑠 and so it is a 𝔉ℱ𝛿𝛽𝑐𝑠 in 𝑋1. Hence ℎ𝔉 is 𝔉ℱ𝛿𝛽𝑖𝑟𝑟 map. The proof is similar for other cases 

of 𝔉ℱ𝛿𝑜𝑠, 𝔉ℱ𝛿𝒫𝑜𝑠, 𝔉ℱ𝛿𝒮𝑜𝑠 and 𝔉ℱ𝛿𝛼𝑜𝑠.         

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, 𝛼𝐴(𝑥), 𝛽𝐴(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 (𝛼𝐴 − 𝛽𝐴)2;  𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖 ∈ 𝐴, where 𝜏𝔉𝑓𝑠(𝑈) denote the family of all 𝔉𝑓𝑠’s on 𝑈.  

Example 5.1  Consider an example of a shopping experience with different items. The pandemic 

situation of COVID-19 has broadened the doorstep of our shopping experience. Nowadays we depend 

on different methods of shopping like online (purchasing through internet, often through the websites 

or Apps), In-store (Visiting physically), Mobile (Using mobile to browse and purchase) shopping. 

Based on the reviews and ratings, we will find out the most reliable method of shopping for a specific 

item using the Fermatean fuzzy entropy measure. 

Table 1. Ratings of products based on the different shopping ways. 

 Gadgets (a) 
Gold jewellery 

(b) 

Food 

Products(c) 
Cloths (d) 

Online (1)  < 1, 𝑎; 0.6,0.3 >  < 1, 𝑏; 0.8,0.7 > < 1, 𝑐; 0.7,0.3 >  < 1, 𝑑; 0.5,0.3 >  

In-store (2)  < 2, 𝑎; 0.9,0.4 >  < 2, 𝑏; 0.4,0.7 >  < 2, 𝑐; 0.7,0.8 >  < 2, 𝑑; 0.2,0.8 >  

Mobile (3)  < 3, 𝑎; 0.6,0.8 >  < 3, 𝑏; 0.2,0.1 >  < 3, 𝑐; 0.9,0.2 >  < 3, 𝑑; 0.1,0.5 >  

 

 Clearly, all values in the Table 1 are 𝔉ℱ𝑠’s. Now we calculate the 휀𝔉ℱ𝑠 of each value. Table 2. 

Entropy measure of each Shopping for the different purchase. 

 Gadgets (a) Gold jewellery 

(b) 

Food 

Products(c) 

Cloths (d) 

Online (1) 0.91 0.99 0.84 0.96 

In-store (2) 0.75 0.91 0.99 0.64 

Mobile (3) 0.96 0.99 0.51 0.84 

  



Communications on Applied Nonlinear Analysis 

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1923 
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From Table 2, it is clear that 휀𝔉ℱ𝑠(1, 𝑐) < 휀𝔉ℱ𝑠(1, 𝑎) < 휀𝔉ℱ𝑠(1, 𝑑) ≤ 휀𝔉ℱ𝑠(1, 𝑏) 

Similarly 

휀𝔉ℱ𝑠(2, 𝑐) < 휀𝔉ℱ𝑠(2, 𝑏) ≤< 휀𝔉ℱ𝑠(2, 𝑎) < 휀𝔉ℱ𝑠(2, 𝑑) 

휀𝔉ℱ𝑠(3, 𝑏) < 휀𝔉ℱ𝑠(3, 𝑎) ≤ 휀𝔉ℱ𝑠(3, 𝑑) ≤ 휀𝔉ℱ𝑠(3, 𝑐) 

It is clear that People are most like to shop the online and Mobile shopping for food products and 

In-store shopping was buy a cloths.  

6  Conclusion 

In this paper, 𝔉ℱ𝛿𝐶𝑡𝑠 , 𝔉ℱ𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒮𝐶𝑡𝑠 , 𝔉ℱ𝛿𝒫𝐶𝑡𝑠 , 𝔉ℱ𝛿𝛼𝐶𝑡𝑠 , and 𝔉ℱ𝛿𝛽𝐶𝑡𝑠  respective 

irresolute map is defined using 𝔉ℱ𝛿𝑜, 𝔉ℱ𝛿𝒮𝑜, 𝔉ℱ𝛿𝒫𝑜, 𝔉ℱ𝛿𝛼𝑜 and 𝔉ℱ𝛿𝛽𝑜 set and its properties 

are analyzed with the examples. Then Fermatean fuzzy continuous maps are compared with other 

generalized Fermatean fuzzy continuous maps. Also we extended the concept of Fermatean fuzzy 

irresolute maps in Fermatean fuzzy topological spaces using 𝔉ℱ𝑜 sets. Some examples and basic 

relationships between the mappings were also discussed. In future, these can be extended to Fermatean 

fuzzy open, closed, homeomorphism and contra maps. Application for MCDM to the real world 

problem was solved with the proposed entropy measure. In future, MCDM to the real world problem 

can be developed to the 𝔉ℱ𝑡𝑠. 

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