Communications on Applied Nonlinear Analysis 

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Contra 𝑴-continuous Maps in Pythagorean Fuzzy Topological Spaces 
 

B. Vijayalakshmi 𝟏,   M. Ramalakshmi 𝟐, A. Vadivel  𝟑  and 

G. Saravanakumar4 

 1Department of Mathematics, Government Arts College, Chidambaram, Tamil Nadu-608 102; Mathematics Section 

(FEAT), Annamalai University, Annamalai Nagar - 608 002, TamilNadu. 

 2Department of Mathematics, Sri Meenakshi Government Arts College for Women (A), Madurai- 625 002  3 Arignar 

Anna Government Arts College, Namakkal - 637 002, India. 

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

 4Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology (Deemed 

to be University), Avadi, Chennai-600062, India 

 ∗Correspondence: mathvijaya2006au@gmail.com, avmaths@gmail.com 
1mathvijaya2006au@gmail.com 2ramalakshmikrishnan23@gmail.com 3 avmaths@gmail.com 

4saravananguru2612@gmail.com, 

 

Article History: 

Received: 20-10-2024 

Revised: 04-12-2024 

Accepted: 11-12-2024 

Abstract:   

In this paper, we introduce and investigate Pythagorean fuzzy contra 𝑀-continuous 

maps in Pythagorean fuzzy topological spaces and also discuss about some properties 

and characterization of Pythagorean fuzzy contra 𝑀-irresolute maps.  

Keywords: Pythagorean fuzzy 𝑀-closed sets, Pythagorean fuzzy contra 𝑀-continuous 

maps and Pythagorean fuzzy contra 𝑀-irresolute maps. 

AMS (2000) subject classification: 03E72, 54A10, 54A40, 54C05, 54C10. 

1. Introduction 

Considering the imprecision in decision-making, Zadeh [36] introduced the idea of fuzzy set which 

has a membership function, 𝜇 that assigns to each element of the universe of discourse, a number from 

the unit interval [0,1] to indicate the degree of belongingness to the set under consideration. The notion 

of fuzzy sets generalizes classical sets theory by allowing intermediate situations between the whole 

and nothing. In a fuzzy set, a membership function is defined to describe the degree of membership of 

an element to a class. The membership value ranges from 0 to 1, where 0 shows that the element does 

not belong to a class, 1 means belongs, and other values indicate the degree of membership to a class. 

For fuzzy sets, the membership function replaced the characteristic function in crisp sets. The concept 

of fuzzy set theory seems to be inconclusive because of the exclusion of nonmembership function and 

the disregard for the possibility of hesitation margin. 

Atanassov critically studied these shortcomings and proposed a concept called intuitionistic fuzzy sets 

(𝐼𝐹𝑆s) [1, 2, 4, 5]. The construct (that is, 𝐼𝐹𝑆’s) incorporates both membership function, 𝜇 and 

nonmembership function, 𝜈 with hesitation margin, 𝜋 (that is, neither membership nor non-

membership functions), such that 𝜇 + 𝜈 ≤ 1 and 𝜇 + 𝜈 + 𝜋 = 1. Atanassov [3] introduced 

intuitionistic fuzzy sets of second type (𝐼𝐹𝑆𝑆𝑇) with the property that the sum of the square of the 

membership and non-membership degrees is less than or equal to one. This concept generalizes 𝐼𝐹𝑆’s 

in a way. The notion of 𝐼𝐹𝑆’s provides a flexible framework to elaborate uncertainty and vagueness. 

The idea of 𝐼𝐹𝑆 seems to be resourceful in modelling many real-life situations like medical diagnosis 

[7, 8, 12, 28, 29], career determination [10], selection process [11], and multi-criteria decision-making 

[15, 16, 17], among others. 

mailto:avmaths@gmail.com
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There are situations where 𝜇 + 𝜈 ≥ 1 unlike the cases capture in 𝐼𝐹𝑆’s. This limitation in 𝐼𝐹𝑆 naturally 

led to a construct, called Pythagorean fuzzy sets (𝑝𝑓𝑠’s). Pythagorean fuzzy set (𝑝𝑓𝑠) proposed in [33, 

34, 35] is a new tool to deal with vagueness considering the membership grade, 𝜇 and non-membership 

grade, 𝜈 satisfying the conditions 𝜇 + 𝜈 ≤ 1 or 𝜇 + 𝜈 ≥ 1, and also, it follows that 𝜇2 + 𝜈2 + 𝜋2 = 1, 
where 𝜋 is the Pythagorean fuzzy set index. In fact, the origin of Pythagorean fuzzy sets emanated 

from 𝐼𝐹𝑆𝑆𝑇 earlier studied in the literature. As a generalized set, 𝑃𝐹𝑆 has close relationship with 𝐼𝐹𝑆. 

The construct of 𝑃𝐹𝑆’s can be used to characterize uncertain information more sufficiently and 

accurately than 𝐼𝐹𝑆. Garg [14] presented an improved score function for the ranking order of interval-

valued Pythagorean fuzzy sets (𝐼𝑉𝑃𝐹𝑆s). Based on it, a Pythagorean fuzzy technique for order of 

preference by similarity to ideal solution (𝑇𝑂𝑃𝑆𝐼𝑆) method by taking the preferences of the experts in 

the form of interval-valued Pythagorean fuzzy decision matrices was discussed. Other explorations of 

the theory of 𝑃𝐹𝑆’s can be found in [6, 9, 13, 18, 19, 25, 26]. Saha [27] defined 𝛿-open sets in 

topological spaces. Vadivel et al. [31] introduced 𝛿-open sets in a neutrosophic topological space. The 

notion of M-open sets in topological spaces were introduced by El-Maghrabi and Al-Juhani [23] in 

2011 and studied some of their properties. The class of sets namely, 𝑀-open sets are playing more 

important role in topological spaces, because of their applications in various fields of Mathematics and 

other real fields. Recently, Jeeva et al. [20, 21, 22] introduced neutrosophic soft 𝑀-open sets in 

neutrosophic topological spaces and developed the concepts of neutrosophic soft 𝑀-Continuity and 

𝑀-Irresolute maps. 

2. Preliminaries 

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

Definition 2.1  [36] 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. 



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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, 2, 4, 5] Let a nonempty set 𝑋 be fixed. An 𝐼𝐹𝑆 𝐴 in 𝑋 is an object having the 

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

𝑥
⟩ |𝑥 ∈ 𝑋}, where the functions 

𝜇𝐴(𝑥): 𝑋 → [0,1] and 𝜈𝐴(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) + 𝜈𝐴(𝑥) ≤ 1. For each 𝐴 in 𝑋: 𝜋𝐴(𝑥) = 1 − 𝜇𝐴(𝑥) − 𝜈𝐴(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) + 𝜈𝐴(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 2.3  [33, 34, 35] 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), 𝜈𝐴(𝑥)|𝑥 ∈ 𝑋} or 𝐴 =

{⟨
𝜇𝐴(𝑥),𝜈𝐴(𝑥)

𝑥
⟩ |𝑥 ∈ 𝑋}, where the functions 𝜇𝐴(𝑥): 𝑋 → [0,1] and 𝜈𝐴(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 + (𝜈𝐴(𝑥))2 ≤ 1. Supposing (𝜇𝐴(𝑥))2 + (𝜈𝐴(𝑥))2 ≤
1, then there is a degree of indeterminacy of 𝑥 ∈ 𝑋 to 𝐴 defined by 𝜋𝐴(𝑥) =

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

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

𝑃𝐹𝑆’s over 𝑋 by 𝑝𝑓𝑠(𝑋).  

Definition 2.4  [35] Let 𝐴 and 𝐵 be 𝑝𝑓𝑠’s of the forms 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋} and 𝐵 =
{< 𝑎, 𝜆𝐵(𝑎), 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}. Then  

    1.  𝐴 ⊆ 𝐵 if and only if 𝜆𝐴(𝑎) ≤ 𝜆𝐵(𝑎) and 𝜇𝐴(𝑎) ≥ 𝜇𝐵(𝑎) for all 𝑎 ∈ 𝑋.  

    2.  𝐴 = 𝐵 if and only if 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴.  

    3.  �̅� = {< 𝑎, 𝜇𝐴(𝑎), 𝜆𝐴(𝑎) > |𝑎 ∈ 𝑋}.  

    4.  𝐴 ∩ 𝐵 = {< 𝑎, 𝜆𝐴(𝑎) ∧ 𝜆𝐵(𝑎), 𝜇𝐴(𝑎) ∨ 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}.  

    5.  𝐴 ∪ 𝐵 = {< 𝑎, 𝜆𝐴(𝑎) ∨ 𝜆𝐵(𝑎), 𝜇𝐴(𝑎) ∧ 𝜇𝐵(𝑎) > |𝑎 ∈ 𝑋}.  

    6.  0𝑋 = {< 𝑎, 0,1 > |𝑎 ∈ 𝑋} and 1𝑋 = {< 𝑎, 1,0 > |𝑎 ∈ 𝑋}.  

    7.  1̅ = 0 and 0̅ = 1.  

 Definition 2.5  [24] An Pythagorean fuzzy topology by subsets of a non-empty set 𝑋 is a family 𝜏 of 

𝑝𝑓𝑠’s satisfying the following axioms.   

    1.  𝜙, 𝑋 ∈ 𝜏.  

    2.  𝐺1 ∩ 𝐺2 ∈ 𝜏 for every 𝐺1, 𝐺2 ∈ 𝜏 and  



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    3.  ⋃ 𝐺𝑖 ∈ 𝜏 for any arbitrary family {𝐺𝑖|𝑖 ∈ 𝑗} ⊆ 𝜏. The pair (𝑋, 𝜏) is called an Pythagorean 

fuzzy topological space (𝑝𝑓𝑡𝑠 in short) and any 𝑝𝑓𝑠 𝐺 in 𝜏 is called an Pythagorean fuzzy open set 

(𝑝𝑓𝑜𝑠 in short) in 𝑋. The complement �̅� of an Pythagorean fuzzy open set 𝐴 in an 𝑝𝑓𝑡𝑠(𝑋, 𝜏) is 

called an Pythagorean fuzzy closed set (𝑝𝑓𝑐𝑠 in short).  

Definition 2.6  [24] Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋} be an 𝑝𝑓𝑠 in 𝑋. 
Then the interior and the closure of 𝐴 are denoted by 𝑝𝑓𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝑐𝑙(𝐴) and are defined as 

follows: 𝑝𝑓𝑐𝑙(𝐴) =∩ {𝐾|𝐾 𝑖𝑠𝑎𝑛 𝑝𝑓𝑐𝑠 𝑎𝑛𝑑 𝐴 ⊆ 𝐾} and 𝑝𝑓𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 𝑖𝑠𝑎𝑛 𝑝𝑓𝑜𝑠 𝑎𝑛𝑑 𝐺 ⊆ 𝐴}. 

Also, it can be established that 𝑝𝑓𝑐𝑙(𝐴) is an 𝑝𝑓𝑐𝑠 and 𝑝𝑓𝑖𝑛𝑡(𝐴) is an 𝑝𝑓𝑜𝑠, 𝐴 is an 𝑝𝑓𝑐𝑠 if and 

only if 𝑝𝑓𝑐𝑙(𝐴) = 𝐴 and 𝐴 is an 𝑝𝑓𝑜𝑠 if and only if 𝑝𝑓𝑖𝑛𝑡(𝐴) = 𝐴. We say that 𝐴 is 𝑝𝑓-dense if 

𝑝𝑓𝑐𝑙(𝐴) = 𝑋.  

Lemma 2.1  [30]  For any Pythagorean fuzzy set 𝐴 in (𝑋, 𝜏), we have 𝑋 − 𝑝𝑓𝑖𝑛𝑡(𝐴) = 𝑝𝑓𝑐𝑙(𝑋 − 𝐴) 

and 𝑋 − 𝑝𝑓𝑐𝑙(𝐴) = 𝑝𝑓𝑖𝑛𝑡(𝑋 − 𝐴).  

Definition 2.7  [30] Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 be an 𝑝𝑓𝑠. Then 𝐴 is said to be an Pythagorean 

fuzzy (i) regular open set (𝑝𝑓𝑟𝑜𝑠 in short) if 𝐴 = 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝑐𝑙(𝐴)). (ii) regular closed set (𝑝𝑓𝑟𝑐𝑠 in 

short) if 𝐴 = 𝑝𝑓𝑐𝑙(𝑝𝑓𝑖𝑛𝑡(𝐴)). By Lemma 2.1, it follows that 𝐴 is an 𝑝𝑓𝑟𝑜𝑠 iff �̅� is an 𝑝𝑓𝑟𝑐𝑠.  

Definition 2.8  [32] Let (𝑋1, 𝛤𝑃) (or 𝑋1) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 

𝑝𝑓𝑠 in 𝑋1. Then the (i) 𝑝𝑓𝛿-interior of 𝐴 are denoted by 𝑝𝑓𝛿𝑖𝑛𝑡(𝐴) and are defined as follows. 

𝑝𝑓𝛿𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 is an 𝑝𝑓𝑟𝑜𝑠 and 𝐺 ⊆ 𝐴}. (ii) 𝑝𝑓𝛿-closure of 𝐴 are denoted by 𝑝𝑓𝛿𝑐𝑙(𝐴) and 

are defined as follows. 𝑝𝑓𝛿𝑐𝑙(𝐴) =∩ {𝐾|𝐾 is an 𝑝𝑓𝑟𝑐𝑠 and 𝐴 ⊆ 𝐾}.  

Definition 2.9  [32]  Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 

𝑋1. A set 𝐴 is said to be 𝑝𝑓 [(i)]  

    1.  𝛿-open set (briefly, 𝑝𝑓𝛿𝑜𝑠) if 𝐴 = 𝑝𝑓𝛿𝑖𝑛𝑡(𝐴),  

    2.  𝛿-pre open set (briefly, 𝑝𝑓𝛿𝒫𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)),  

    3.  𝛿-semi open set (briefly, 𝑝𝑓𝛿𝒮𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝛿𝑖𝑛𝑡(𝐴)),  

    4.  𝑒 open set (briefly, 𝑝𝑓𝑒𝑜𝑠 ) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝛿𝑖𝑛𝑡(𝐴)) ∪ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)),  

    5.  𝛿 (resp. 𝛿-pre, 𝛿-semi and 𝑒) dense if 𝑝𝑓𝛿𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒫𝑐𝑙(𝐴), 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 

𝑝𝑓𝑒𝑐𝑙(𝐴)) = 𝑋1. 

The complement of an 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠 and 𝑝𝑓𝑒𝑜𝑠) is called an 𝑝𝑓𝛿 (resp. 𝑝𝑓𝛿𝒫, 𝑝𝑓𝛿𝒮 

and 𝑝𝑓𝑒) closed set (briefly, 𝑝𝑓𝛿𝑐𝑠 (resp. 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠 and 𝑝𝑓𝑒𝑐𝑠)) in 𝑋1. 

The family of all 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝑐𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑒𝑜𝑠 and 𝑝𝑓𝑒𝑐𝑠) of 𝑋1 

is denoted by 𝑝𝑓𝛿𝑂𝑆(𝑋1), (resp. 

𝑝𝑓𝛿𝐶𝑆(𝑋1), 𝑝𝑓𝛿𝒫𝑂𝑆(𝑋1), 𝑝𝑓𝛿𝒫𝐶𝑆(𝑋1), 𝑝𝑓𝛿𝒮𝑂𝑆(𝑋1), 𝑝𝑓𝛿𝒮𝐶𝑆(𝑋1), 𝑝𝑓𝑒𝑂𝑆(𝑋1) and 𝑝𝑓𝑒𝐶𝑆(𝑋1)). 

Definition 2.10  [32] Let (𝑋, 𝜏) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 𝑋1. 

Then the (i) 𝑝𝑓𝛿-pre (resp. 𝑝𝑓𝛿-semi and 𝑝𝑓𝑒)-interior of 𝐴 are denoted by 𝑝𝑓𝛿𝒫𝑖𝑛𝑡(𝐴) (resp. 

𝑝𝑓𝛿𝒮𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝑒𝑖𝑛𝑡(𝐴)) and are defined as follows: 𝑝𝑓𝛿𝒫𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑖𝑛𝑡(𝐴) and 

𝑝𝑓𝑒𝑖𝑛𝑡(𝐴) =∪ {𝐺|𝐺 in a 𝑝𝑓𝛿𝒫𝑜𝑠 (resp. 𝑝𝑓𝛿𝒮𝑜𝑠 and 𝑝𝑓𝑒𝑜𝑠) and 𝐺 ⊆ 𝐴}, (ii) 𝑝𝑓𝛿-pre (resp. 𝑝𝑓𝛿-

semi and 𝑝𝑓𝑒)-closure of 𝐴 are denoted by 𝑝𝑓𝛿𝒫𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 𝑝𝑓𝑒𝑐𝑙(𝐴)) and are 

defined as follows: 𝑝𝑓𝛿𝒫𝑐𝑙(𝐴) (resp. 𝑝𝑓𝛿𝒮𝑐𝑙(𝐴) and 𝑝𝑓𝑒𝑐𝑙(𝐴)) =∩ {𝐾|𝐾 is an 𝑝𝑓𝛿𝒫𝑐𝑠 (resp. 

𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑒𝑐𝑠) and 𝐴 ⊆ 𝐾}.  

Definition 2.11  [32]  Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 

𝑋1. A set 𝐴 is said to be 𝑝𝑓  



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1.  𝜃-interior of 𝐴 (briefly, 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)) is defined by 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴) =∪ {𝑝𝑓𝑖𝑛𝑡(𝐵): 𝐵 ⊆
𝐴 & 𝐵 isa 𝑝𝑓𝑐𝑠 in 𝑋1}.  

2.  𝜃-open set (briefly, 𝑝𝑓𝜃𝑜𝑠) if 𝐴 = 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴).  

3.  𝜃 -semi open set (briefly, 𝑝𝑓𝜃𝒮𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)).  

4.  𝑀-open set (briefly, 𝑝𝑓𝑀𝑜𝑠) if 𝐴 ⊆ 𝑝𝑓𝑐𝑙(𝑝𝑓𝜃𝑖𝑛𝑡(𝐴)) ∪ 𝑝𝑓𝑖𝑛𝑡(𝑝𝑓𝛿𝑐𝑙(𝐴)).  

  The complement of a 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝜃𝑜𝑠 & 𝑝𝑓𝜃𝒮𝑜𝑠) is called an 𝑝𝑓𝑀 (resp. 𝑝𝑓𝜃 & 𝑝𝑓𝜃𝒮) closed 

set (briefly, 𝑝𝑓𝑀𝑐𝑠 (resp. 𝑝𝑓𝜃𝑐𝑠 & 𝑝𝑓𝜃𝒮𝑐𝑠)) in 𝑋1. 

The family of all 𝑝𝑓𝜃𝑜𝑠 (resp. 𝑝𝑓𝜃𝑐𝑠, 𝑝𝑓𝜃𝒮𝑜𝑠, 𝑝𝑓𝜃𝒮𝑐𝑠, 𝑝𝑓𝑀𝑜𝑠 and 𝑝𝑓𝑀𝑐𝑠) of 𝑋1 is denoted by 

𝑝𝑓𝜃𝑂𝑆(𝑋1), (resp. 𝑝𝑓𝜃𝐶𝑆(𝑋1), 𝑝𝑓𝜃𝒮𝑂𝑆(𝑋1), 𝑝𝑓𝜃𝒮𝐶𝑆(𝑋1), 𝑝𝑓𝑀𝑂𝑆(𝑋1) and 𝑝𝑓𝑀𝐶𝑆(𝑋1)). 

Definition 2.12  [32] Let (𝑋1, 𝛤𝑃) be an 𝑝𝑓𝑡𝑠 and 𝐴 = {< 𝑎, 𝜆𝐴(𝑎), 𝜇𝐴(𝑎) > |𝑎 ∈ 𝑋1} be an 𝑝𝑓𝑠 in 

𝑋1. Then the 𝑝𝑓  

1.  𝑀 (resp. 𝑝𝑓𝜃-semi )-interior of 𝐴 (briefly,  𝑝𝑓𝑀𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝜃𝒮𝑖𝑛𝑡(𝐴)) is defined by 

𝑝𝑓𝑀𝑖𝑛𝑡(𝐴) (resp. 𝑝𝑓𝜃𝑖𝑛𝑡(𝐴) and 𝑝𝑓𝜃𝒮𝑖𝑛𝑡(𝐴)) =∪ {𝐵: 𝐵 ⊆ 𝐴 and 𝐵 is a 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝜃𝒮𝑜𝑠) in 

𝑋1}.  

2.  𝑀 (resp. 𝜃-semi )-closure of 𝐴 (briefly, 𝑝𝑓𝑀𝑐𝑙(𝐴) (resp. 𝑝𝑓𝜃𝒮𝑐𝑙(𝐴)) is defined by 𝑝𝑓𝑀𝑐𝑙(𝐴) 

(resp. 𝑝𝑓𝜃𝒮𝑐𝑙(𝐴)) =∩ {𝐵: 𝐴 ⊆ 𝐵 and 𝐴 is a 𝑝𝑓𝑀𝑐𝑠 (resp. 𝑝𝑓𝜃𝒮𝑐𝑠) in 𝑋1}.  

Definition 2.13  Let (𝑋1, 𝛤𝑃) and (𝑋2, 𝛹𝑃) be any two 𝑝𝑓𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 

said to be a Pythagorean fuzzy (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝑒, 𝜃, 𝜃𝒮 and 𝑀 )-continuous (briefly, 𝑝𝑓𝐶𝑡𝑠 (resp. 

𝑝𝑓𝛿𝐶𝑡𝑠, 𝑝𝑓𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑒𝐶𝑡𝑠, 𝑝𝑓𝜃𝐶𝑡𝑠, 𝑝𝑓𝜃𝒮𝐶𝑡𝑠 and 𝑝𝑓𝑀𝐶𝑡𝑠)) if the inverse image of 

every 𝑝𝑓𝑜𝑠 in (𝑋2, 𝛹𝑃) is a 𝑝𝑓𝑜𝑠 (resp. 𝑝𝑓𝛿𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝑒𝑜𝑠, 𝑝𝑓𝜃𝑜𝑠, 𝑝𝑓𝜃𝒮𝑜𝑠 and 

𝑝𝑓𝑀𝑜𝑠) in (𝑋1, 𝛤𝑃).  

3  Pythagorean fuzzy contra 𝑴-continuous maps 

Definition 3.1  Let (𝑋1, 𝛤𝑃) and (𝑋2, 𝛹𝑃) be any two 𝑝𝑓𝑡𝑠’s. A mapping ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 

said to be a Pythagorean fuzzy contra (resp. 𝛿, 𝛿𝒫, 𝛿𝒮, 𝑒, 𝜃, 𝜃𝒮 and 𝑀 )-continuous (briefly, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠)) if the inverse image of every 𝑝𝑓𝑜𝑠 in 

(𝑋2, 𝛹𝑃) is a 𝑝𝑓𝑐𝑠 (resp. 𝑝𝑓𝛿𝑐𝑠, 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝑒𝑐𝑠, 𝑝𝑓𝜃𝑐𝑠, 𝑝𝑓𝜃𝒮𝑐𝑠 and 𝑝𝑓𝑀𝑐𝑠) in (𝑋1, 𝛤𝑃).  

Proposition 3.1  Let (𝑋1, 𝛤𝑃) & (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑡𝑠’s. Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a mapping. 

Then the following statements are hold for 𝑝𝑓𝑡𝑠, but not conversely. 

    1.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠.  

    2.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠.  

    3.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠.  

    4.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠.  

    5.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠.  

    6.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠.  

    7.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠.  

    8.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠.  

    9.  Every 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠.  



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  Proof.  

1.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝜃𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝜃𝑐𝑠 is a 𝑝𝑓𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠. 

2.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝜃𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝜃𝑐𝑠 is a 𝑝𝑓𝜃𝒮𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠. 

3.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝜃𝒮𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝜃𝒮𝑐𝑠 is a 𝑝𝑓𝑀𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠. 

4.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝛿𝑐𝑠 is a 𝑝𝑓𝛿𝒮𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝛿𝒮𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠. 

5.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝛿𝑐𝑠 is a 𝑝𝑓𝛿𝒫𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝛿𝒫𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠. 

6.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝛿𝒮𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝛿𝒮𝑐𝑠 is a 𝑝𝑓𝑒𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑒𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠. 

7.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝛿𝒫𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝛿𝒫𝑐𝑠 is a 𝑝𝑓𝑀𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠. 

8.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝑀𝑐𝑠 is a 𝑝𝑓𝑒𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑒𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠.  

9.  Let 𝐵 be a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, ℎ𝑃
−1(𝐵) is 𝑝𝑓𝛿𝑐𝑠 in (𝑋1, Γ𝑃). Since 

every 𝑝𝑓𝛿𝑐𝑠 is a 𝑝𝑓𝑐𝑠, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠.              

 Remark 3.1  We obtain the following diagram from the results are discussed above.  

       

Note: 𝐴 → 𝐵 denotes 𝐴 implies 𝐵, but not conversely. 

Example 3.1   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3  & 𝐴4 in 𝑋1 and 𝐵1, 𝐵2, 𝐵3  & 𝐵4 in 𝑋2 are 

defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 



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 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.40 >} 

 𝐵2 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐵3 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐵4 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.70,0.30 >}. 

 Then, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1, 𝐵2, 𝐵3, 𝐵4}. Let ℎ𝑃: (𝑋1, Γ𝑃) →
(𝑋2, Ψ𝑃) be an identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, because the set 𝐵1 

is 𝑝𝑓𝑜𝑠 in 𝑋2 but ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝜃𝑐𝑠 in 𝑋1.  

Example 3.2   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋1  & 𝐵1 in 𝑋2 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 Here, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠) but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠 (resp. 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠), because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋2 but ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝜃𝑐𝑠 (resp. 𝑝𝑓𝛿𝑐𝑠) in 𝑋1.  

Example 3.3   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋1  & 𝐵1 in 𝑋2 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.60,0.40 >}. 

 Here, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 

𝑋2 but ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝜃𝒮𝑐𝑠 in 𝑋1.  

Example 3.4   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋1  & 𝐵1 in 𝑋2 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.20,0.40 >, < 𝑥2, 0.40,0.40 >} 



Communications on Applied Nonlinear Analysis 

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 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 

𝑋2 but ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝑀𝑐𝑠 in 𝑋1.  

Example 3.5   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋1  & 𝐵1 in 𝑋2 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.80,0.20 >, < 𝑥2, 0.70,0.30 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 ) but not 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠), because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 𝑋2 but 

ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝛿𝑐𝑠 (resp. 𝑝𝑓𝛿𝒮𝑐𝑠 and 𝑝𝑓𝛿𝑐𝑠) in 𝑋1.  

Example 3.6   Let 𝑋1 = 𝑋2 = {𝑥1, 𝑥2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4 in 𝑋1  & 𝐵1 in 𝑋2 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.60 >} 

 Now, we have Γ𝑃 = {0𝑋 , 1𝑋 , 𝐴1, 𝐴2, 𝐴3, 𝐴4} and Ψ𝑃 = {0𝑋 , 1𝑋 , 𝐵1}. Let ℎ𝑃: (𝑋1, Γ𝑃) → (𝑋2, Ψ𝑃) be an 

identity mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, because the set 𝐵1 is 𝑝𝑓𝑜𝑠 in 

𝑋2 but ℎ𝑃
−1(𝐵1) = 𝐵1 is not 𝑝𝑓𝛿𝒫𝑐𝑠 in 𝑋1.  

 Theorem 3.1  A map ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 ) iff the 

inverse image of each 𝑝𝑓𝑐𝑠 in (𝑋2, 𝛹𝑃) is 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝑜𝑠, 𝑝𝑓𝛿𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝜃𝑜𝑠, 

𝑝𝑓𝑒𝑜𝑠 and 𝑝𝑓𝜃𝒮𝑜𝑠 ) in (𝑋1, 𝛤𝑃).  

  Proof. Let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋2, Ψ𝑃). This implies 𝐵𝑐 is 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠, 

ℎ𝑃
−1(𝐵𝑐) is 𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Since, ℎ𝑃

−1(𝐵𝑐) = (ℎ𝑃
−1(𝐵))𝑐, ℎ𝑃

−1(𝐵) is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋1, Γ𝑃). 

Conversely, let 𝐵 be a 𝑝𝑓𝑐𝑠 in (𝑋2, Ψ𝑃). Then, 𝐵𝑐 is a 𝑝𝑓𝑜𝑠 in (𝑋2, Ψ𝑃). By hypothesis ℎ𝑃
−1(𝐵𝑐) is 

𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Since, ℎ𝑃
−1(𝐵𝑐) = (ℎ𝑃

−1(𝐵))𝑐, (ℎ𝑃
−1(𝐵))𝑐 is a 𝑝𝑓𝑀𝑐𝑠 in (𝑋1, Γ𝑃). Therefore, ℎ𝑃

−1(𝐵) 

is a 𝑝𝑓𝑀𝑜𝑠 in (𝑋1, Γ𝑃). Hence, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠. The proof of other cases are similar  

Definition 3.2  A 𝑝𝑓𝑡𝑠 (𝑋1, 𝛤𝑃) is said to be a Pythagorean fuzzy 𝑀𝑈1/2 (resp.𝑝𝑓𝛿𝒮𝑈1/2, 𝑝𝑓𝛿𝒫𝑈1/2, 

𝑝𝑓𝜃𝑈1/2, 𝑝𝑓𝑒𝑈1/2 and 𝑝𝑓𝜃𝒮𝑈1/2 )-space, if every 𝑝𝑓𝑀𝑜𝑠 (resp. 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝜃𝑜𝑠, 𝑝𝑓𝑒𝑜𝑠 

and 𝑝𝑓𝜃𝒮𝑜𝑠 ) in 𝑋1 is a 𝑝𝑓𝑜𝑠 in 𝑋1.  

Theorem 3.2  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 ), then ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 

if 𝑋1 is a 𝑝𝑓𝑀𝑈1/2) (resp. 𝑝𝑓𝛿𝒮𝑈1/2, 𝑝𝑓𝛿𝒫𝑈1/2, 𝑝𝑓𝜃𝑈1/2, 𝑝𝑓𝑒𝑈1/2 and 𝑝𝑓𝜃𝒮𝑈1/2 )-space.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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  Proof. Let 𝐵 be a 𝑃𝑓𝑜𝑠 in 𝑋2. Then, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1, by hypothesis. Since 𝑋1 is a 𝑝𝑓𝑀𝑈1/2-

space, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠. The proof of other cases are similar.              

Theorem 3.3   Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) be 

a 𝑝𝑓𝐶𝑡𝑠, then 𝑔𝑃 ∘ ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠.  

  Proof. Let 𝐾 be a 𝑝𝑓𝑜𝑠 in 𝑋3. Then, 𝑔𝑃
−1(𝐾) is a 𝑝𝑓𝑜𝑠 in 𝑋2, by hypothesis. Since ℎ𝑃 is a 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map, ℎ𝑃
−1(𝑔𝑃

−1(𝐾)) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1. Hence 𝑔𝑃 ∘ 𝑓𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map. The 

proof of other cases are similar.              

Theorem 3.4   Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map. Then, the following conditions 

are hold.  

    1.  ℎ𝑃(𝑝𝑓𝑀𝑐𝑙(𝐴)) ≥ 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃(𝐴)), for all 𝑝𝑓𝑐𝑠 𝐴 in 𝑋1.  

    2.  𝑝𝑓𝑀𝑐𝑙(ℎ𝑃
−1(𝐵)) ≥ ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵)), for all 𝑝𝑓𝑐𝑠 𝐵 in 𝑋2.  

  Proof.  

    1.  Since 𝑝𝑓𝑀𝑐𝑙(ℎ𝑃(𝐴)) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋2 and ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠, then ℎ𝑃
−1(𝑝𝑓𝑀𝑐𝑙(ℎ𝑃(𝐴))) is 

𝑝𝑓𝑀𝑜𝑠 in 𝑋1. Now, since 𝐴 ≥ ℎ𝑃
−1(𝑝𝑓𝑖𝑛𝑡(ℎ𝑃(𝐴))), 𝑝𝑓𝑀𝑐𝑙(𝐴) ≥ ℎ𝑃

−1(𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃(𝐴))). Therefore, 

ℎ𝑃(𝑝𝑓𝑀𝑐𝑙(𝐴)) ≥ 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃(𝐴)). 

    2.  By replacing 𝐴 with 𝐵 in (i), we obtain ℎ𝑃(𝑝𝑓𝑀𝑐𝑙(ℎ𝑃
−1(𝐵))) ≥ 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃(ℎ𝑃

−1(𝐵))) ≥

𝑝𝑓𝑖𝑛𝑡(𝐵). Hence, 𝑝𝑓𝑀𝑐𝑙(ℎ𝑃
−1(𝐵)) ≥ ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵)).              

Remark 3.2  Theorems 3.3 and 3.4 are true for 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠.  

4  Pythagorean fuzzy contra 𝑴 -irresolute maps in 𝒑𝒇𝒕𝒔 

 In this section, we introduce 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟) maps and study some of its 

characterizations. 

Definition 4.1  A map ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) is called a Pythagorean fuzzy contra 𝑀 (resp. 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮) -

irresolute (briefly, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟)) map if ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 

(resp. 𝑝𝑓𝒮𝑐𝑠, 𝑝𝑓𝛿𝑐𝑠, 𝑝𝑓𝛿𝒮𝑐𝑠, 𝑝𝑓𝛿𝒫𝑐𝑠, 𝑝𝑓𝜃𝑐𝑠, 𝑝𝑓𝑒𝑐𝑠 and 𝑝𝑓𝜃𝒮𝑐𝑠) in (𝑋1, 𝛤𝑃) for every 𝑝𝑓𝑀𝑜𝑠 

(resp. 𝑝𝑓𝒮𝑜𝑠, 𝑝𝑓𝛿𝑜𝑠, 𝑝𝑓𝛿𝒮𝑜𝑠, 𝑝𝑓𝛿𝒫𝑜𝑠, 𝑝𝑓𝜃𝑜𝑠, 𝑝𝑓𝑒𝑜𝑠 and 𝑝𝑓𝜃𝒮𝑜𝑠) 𝐵 of (𝑋2, 𝛹𝑃).  

Theorem 4.1  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟), then ℎ𝑃 is 

a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 ) map. But not conversely.  

  Proof. (i) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝑜𝑠 is a 𝑝𝑓𝒮𝑜𝑠, 𝐵 

is a 𝑝𝑓𝒮𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝒮𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠 map. 

(ii) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝑜𝑠 is a 𝑝𝑓𝑀𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑀𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map. 

(iii) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝛿𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝛿𝑜𝑠 is a 𝑝𝑓𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

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(iv) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝛿𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝛿𝑜𝑠 is a 𝑝𝑓𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝛿𝒮𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠 map. 

(v) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝛿𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝛿𝑜𝑠 is a 𝑝𝑓𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝛿𝒫𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 map. 

(vi) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝜃𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝜃𝑜𝑠 is a 𝑝𝑓𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝜃𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠 map. 

(vii) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝑜𝑠 is a 𝑝𝑓𝑒𝑜𝑠, 𝐵 is a 

𝑝𝑓𝑒𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑒𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 map. 

(viii) Let ℎ𝑃 be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟 map. Let 𝐵 be any 𝑝𝑓𝜃𝑜𝑠 in 𝑋2. Since every 𝑝𝑓𝜃𝑜𝑠 is a 𝑝𝑓𝑜𝑠, 𝐵 is 

a 𝑝𝑓𝑜𝑠 in 𝑋2. By hypothesis ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝜃𝒮𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠 map.            

Example 4.1   Let 𝑋 = {𝑥1, 𝑥2} = 𝑌 = {𝑦1, 𝑦2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3  & 𝐴4 in 𝑋 and 𝐵1 &𝐵2 in 𝑌 are 

defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐵1 = {< 𝑦1, 0.10,0.90 >, < 𝑦2, 0.30,0.70 >} 

 𝐵2 = {< 𝑦1, 0.20,0.40 >, < 𝑦2, 0.40,0.40 >}. 

 Here, 𝜏1 = {0𝑃, 1𝑃, 𝐴1, 𝐴2, 𝐴3, 𝐴4}, 𝜏2 = {0𝑃, 1𝑃, 𝐵1}, 𝜏3 = {0𝑃, 1𝑃, 𝐴2
𝑐}, 𝜏4 = {0𝑃, 1𝑃, 𝐴4

𝑐} and 𝜏5 =
{0𝑃, 1𝑃, 𝐴2

𝑐 , 𝐴3
𝑐}.  

(i) Let 𝑓1: (𝑋, 𝜏1) → (𝑌, 𝜏2) be an identity mapping. Then, 𝑓1 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 but not 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟, because the set 𝐵2 is a 𝑝𝑓𝑀𝑜𝑠 in 𝑌 but 𝑓−1(𝐵2) = 𝐵2 is not 𝑝𝑓𝑀𝑐𝑠 in 𝑋. (ii) Let 

𝑓2: (𝑋, 𝜏1) → (𝑌, 𝜏3) be an identity mapping. Then, 𝑓2 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝒮𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐼𝑟𝑟, because 

the set 𝐴2
𝑐 is a 𝑝𝑓𝒮𝑜𝑠 in 𝑌 but 𝑓2

−1(𝐴2
𝑐) = 𝐴2

𝑐 is not 𝑝𝑓𝒮𝑐𝑠 in 𝑋. (iii) Let 𝑓3: (𝑋, 𝜏1) → (𝑌, 𝜏4) be an 

identity mapping. Then, 𝑓3 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠) but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟 (resp. 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟), because the set 𝐴4
𝑐 is a 𝑝𝑓𝛿𝑜𝑠 (resp. 𝑝𝑓𝛿𝒮𝑜𝑠) in 𝑌 but 𝑓3

−1(𝐴4
𝑐) = 𝐴4

𝑐 is not 𝑝𝑓𝛿𝑐𝑠 

(resp. 𝑝𝑓𝛿𝒮𝑐𝑠) in 𝑋. (iv) Let 𝑓4: (𝑋, 𝜏1) → (𝑌, 𝜏5) be an identity mapping. Then, 𝑓4 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠 

but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, because the set 𝐴1 is a 𝑝𝑓𝛿𝒫𝑜𝑠 in 𝑌 but 𝑓4
−1(𝐴1) = 𝐴1 is not 𝑝𝑓𝛿𝒫𝑐𝑠 in 𝑋.  

Example 4.2   Let 𝑋 = {𝑥1, 𝑥2} = 𝑌 = {𝑦1, 𝑦2} and 𝑝𝑓𝑠’s 𝐴1, 𝐴2, 𝐴3, 𝐴4  & 𝐴5 in 𝑋 are defined as, 

 𝐴1 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.40,0.60 >} 

 𝐴2 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.70 >} 

 𝐴3 = {< 𝑥1, 0.90,0.10 >, < 𝑥2, 0.70,0.30 >} 

 𝐴4 = {< 𝑥1, 0.20,0.80 >, < 𝑥2, 0.30,0.70 >} 

 𝐴5 = {< 𝑥1, 0.10,0.90 >, < 𝑥2, 0.30,0.60 >}. 

 Here, 𝜏1 = {0𝑃, 1𝑃, 𝐴2, 𝐴4}, 𝜏2 = {0𝑃, 1𝑃, 𝐴1
𝑐, 𝐴2

𝑐, 𝐴3
𝑐 , 𝐴4

𝑐}. Let ℎ𝑃: (𝑋, 𝜏1) → (𝑌, 𝜏2) be an identity 

mapping. Then, ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 but not 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟, because the set 𝐴5
𝑐 is a 𝑝𝑓𝑒𝑜𝑠 in 𝑌 but 

ℎ𝑃
−1(𝐴5

𝑐) = 𝐴5
𝑐 is not 𝑝𝑓𝑒𝑐𝑠 in 𝑋.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

337 
https://internationalpubls.com 

 Theorem 4.2  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟), then ℎ𝑃 is a 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map if 𝑋1 is a 𝑝𝑓𝑀𝑈1/2 (resp. 𝑝𝑓𝛿𝑈1/2, 𝑝𝑓𝛿𝒮𝑈1/2, 𝑝𝑓𝛿𝒫𝑈1/2, 𝑝𝑓𝜃𝑈1/2, 𝑝𝑓𝑒𝑈1/2 and 

𝑝𝑓𝜃𝒮𝑈1/2 )-space.  

  Proof. Let 𝐵 be a 𝑝𝑓𝑜𝑠 in 𝑋2. Then, 𝐵 is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋2. Therefore ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1, by 

hypothesis. Since 𝑋1 is a 𝑝𝑓𝑀𝑈1/2 -space, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑐𝑠 in 𝑋1. Hence, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝐶𝑡𝑠 map. 

The proof of other cases is similar.              

Theorem 4.3  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) and 𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 (resp. 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟) maps, then 𝑔𝑃 ∘ ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋3, 𝛷𝑃) is a 𝑝𝑓𝑀𝐼𝑟𝑟 (resp. 𝑝𝑓𝛿𝐼𝑟𝑟, 𝑝𝑓𝛿𝒮𝐼𝑟𝑟, 

𝑝𝑓𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝜃𝐼𝑟𝑟, 𝑝𝑓𝑒𝐼𝑟𝑟 and 𝑝𝑓𝜃𝒮𝐼𝑟𝑟) map.  

  Proof. Let 𝐾 be a 𝑝𝑓𝑀𝑜𝑠 in 𝑋3. Then, 𝑔𝑃
−1(𝐾) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋2. Since ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 map, 

ℎ𝑃
−1(𝑔𝑃

−1(𝐾)) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1. Hence 𝑔𝑃 ∘ 𝑓𝑃 is a 𝑝𝑓𝑀𝐼𝑟𝑟 map. The proof of other cases is similar.              

Theorem 4.4  Let ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟) map and 

𝑔𝑃: (𝑋2, 𝛹𝑃) → (𝑋3, 𝛷𝑃) be a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 (resp. 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐶𝑡𝑠, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐶𝑡𝑠, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐶𝑡𝑠 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐶𝑡𝑠) map, then 𝑔𝑃 ∘
ℎ𝑃: (𝑋, 𝜏1) → (𝑍, 𝜏3) is a 𝑝𝑓𝑀𝐶𝑡𝑠 (resp. 𝑝𝑓𝛿𝐶𝑡𝑠, 𝑝𝑓𝛿𝒮𝐶𝑡𝑠, 𝑝𝑓𝛿𝒫𝐶𝑡𝑠, 𝑝𝑓𝜃𝐶𝑡𝑠, 𝑝𝑓𝑒𝐶𝑡𝑠 and 

𝑝𝑓𝜃𝒮𝐶𝑡𝑠) map.  

  Proof. Let 𝐾 be a 𝑝𝑓𝑜𝑠 in 𝑋3. Then, 𝑔𝑃
−1(𝐾) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋2. Since, ℎ𝑃 is a 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟, ℎ𝑃
−1(𝑔𝑃

−1(𝐾)) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1. Hence, 𝑔𝑃 ∘ ℎ𝑃 is a 𝑝𝑓𝑀𝐶𝑡𝑠 map. The proof of other 

cases is similar.              

Theorem 4.5   Let a map ℎ𝑃: (𝑋1, 𝛤𝑃) → (𝑋2, 𝛹𝑃). Then the following conditions are equivalent if 𝑋1 

and 𝑋2 are 𝑝𝑓𝑀𝑈1/2 -spaces.   

    1.  ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 map. 

    2.  ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1, for each 𝑝𝑓𝑀𝑐𝑠 𝐵 in 𝑋2. 

    3.  𝑝𝑓𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵)), for each 𝑝𝑓𝑠 𝐵 of 𝑋2.  

   Proof. (i) → (ii): Let 𝐵 be any 𝑝𝑓𝑀𝑐𝑠 in 𝑋2. Then, 𝐵𝑐 is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋2. Since ℎ𝑃 is 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟, 

ℎ𝑃
−1(𝐵𝑐) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1. But ℎ𝑃

−1(𝐵𝑐) = (ℎ𝑃
−1(𝐵))𝑐. Therefore, ℎ𝑃

−1(𝐵) is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋1. 

(ii) → (iii) : Let 𝐵 be any 𝑝𝑓𝑠 in 𝑋2 and 𝑝𝑓𝑖𝑛𝑡(𝐵) ≤ 𝐵. Then, ℎ𝑃
−1(𝑝𝑓𝑐𝑙(𝐵))ℎ𝑃

−1(𝐵) ≤ ℎ𝑃
−1(𝐵). Since 

𝑝𝑓𝑖𝑛𝑡(𝐵) is a 𝑝𝑓𝑜𝑠 in 𝑋2, 𝑝𝑓𝑖𝑛𝑡(𝐵) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋2. Therefore, (𝑝𝑓𝑖𝑛𝑡(𝐵))𝑐 is a 𝑝𝑓𝑀𝑐𝑠 in 𝑋2. By 

hypothesis, ℎ𝑃
−1((𝑝𝑓𝑖𝑛𝑡(𝐵))𝑐) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1. Since, ℎ𝑃

−1((𝑝𝑓𝑖𝑛𝑡(𝐵))𝑐) =

(ℎ𝑃
−1(𝑝𝑓𝑖𝑛𝑡(𝐵)))𝑐, ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵)) is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1. Since, 𝑋1 is a 𝑝𝑓𝑀𝑈1/2 -space, ℎ𝑃
−1(𝑝𝑓𝑖𝑛𝑡(𝐵)) 

is a 𝑝𝑓𝑜𝑠 in 𝑋1. Hence, 𝑝𝑓𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ 𝑝𝑓𝑐𝑙(ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵))) = ℎ𝑃
−1(𝑝𝑓𝑖𝑛𝑡(𝐵)). That is, 

𝑝𝑓𝑐𝑙(ℎ𝑃
−1(𝐵)) ⊇ ℎ𝑃

−1(𝑝𝑓𝑖𝑛𝑡(𝐵)). 

(iii) → (i) : Let 𝐵 be any 𝑝𝑓𝑀𝑐𝑠 in 𝑋2. Since 𝑋2 is a 𝑝𝑓𝑀𝑈1/2-space, 𝐵 is a 𝑝𝑓𝑐𝑠 in 𝑋2 and 𝑝𝑓𝑐𝑙(𝐵) =

𝐵. Hence, ℎ𝑃
−1(𝐵) = ℎ𝑃

−1(𝑝𝑓𝑀𝑐𝑙(𝐵)) ⊇ 𝑝𝑓𝑀𝑖𝑛𝑡(ℎ𝑃
−1(𝐵)). But clearly, ℎ𝑃

−1(𝐵) ⊇ 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐵)). 

Therefore, 𝑝𝑓𝑖𝑛𝑡(ℎ𝑃
−1(𝐵)) = ℎ𝑃

−1(𝐵). This implies, ℎ𝑃
−1(𝐵) is a 𝑝𝑓𝑜𝑠 and hence, it is a 𝑝𝑓𝑀𝑜𝑠 in 𝑋1. 

Thus, ℎ𝑃 is a 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐼𝑟𝑟 map.              



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 6s (2025) 

 

338 
https://internationalpubls.com 

Remark 4.1  Theorem 4.5 is true for 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒮𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝛿𝒫𝐼𝑟𝑟, 

𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝐼𝑟𝑟, 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑒𝐼𝑟𝑟 and 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝜃𝒮𝐼𝑟𝑟.  

5  Conclusion 

 In this paper, using 𝑝𝑓𝑀𝑜𝑠 we have defined 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀𝐶𝑡𝑠 map and analyzed its properties. After 

that we have defined 𝑝𝑓𝑐𝑜𝑛𝑡𝑟𝑎𝑀-irresolute maps. In future, these concepts can be extended to some 

mathematical applications. 

References 

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