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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 350-365

Published online December 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.4.350

© The Korean Institute of Intelligent Systems

## n Power Root Fuzzy Sets and Its Topology

Tareq M. Al-shami1 , Hariwan Z. Ibrahim2, Abdelwaheb Mhemdi3, and Radwan Abu-Gdairi4

1Department of Mathematics, Sana’a University, Sana’a, Yemen Future University, Egypt
2Department of Mathematics, Faculty of Education, University of Zakho, Zakho, Kurdistan Region-Iraq
3Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia
4Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan

Correspondence to :
Abdelwaheb Mhemdi (mhemdiabd@gmail.com)

Received: October 15, 2022; Revised: November 23, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

One of the most useful expansions of fuzzy sets for coping with information uncertainties is the q-rung orthopair fuzzy sets. In such circumstances, in this article, we define a novel extension of fuzzy sets called nth power root fuzzy set (briefly, nPR-fuzzy set) and elucidate their relationship with intuitionistic fuzzy sets, SR-fuzzy sets, CR-fuzzy sets and q-rung orthopair fuzzy sets. Then, we provide the necessary set of operations for nPR-fuzzy sets as well as study their various features. Furthermore, we familiarize the concept of nPR-fuzzy topology and investigate the basic aspects of this topology. In addition, we define separated nPR-fuzzy sets and then present the concept of disconnected nPR-fuzzy sets. Moreover, we study and characterize nPR-fuzzy continuous maps in great depth. Finally, we establish 𝕿0 and 𝕿1 in nPR-fuzzy topologies and discover the links between them.

Keywords: nPR-fuzzy sets, Operations, nPR-fuzzy topology, Separated nPR-FSs, Connected nPR-FS, nPR-fuzzy continuous maps, 𝕿0, 𝕿1

Zadeh [39] established the concept of fuzzy sets to deal with imprecise data, and various studies on generalizations of the concept of fuzzy set were undertaken after that. From health sciences to computer science, from physical sciences to arts, and from engineering and humanities to life sciences, it is seen to have numerous applications connected to fuzzy set theory in both theoretical and practical investigations. As examples of theoretical studies, Kirisci [28] discussed triangular fuzzy numbers, and Zarasiz studied algebraic fuzzy structure [40] and fuzzy matrices [41]. Pawlak [32] was the first one who proposed the concept of rough sets, and Molodtsov [31] introduced the concept of soft sets as a generic mathematical tool for dealing with ambiguous objects. These two types of non-crisp sets contribute to coping with uncertain environments in different fields; see, for example [5, 6, 810, 1215]. The merging between fuzzy sets and some doubtfulness techniques such as rough sets and soft sets have been studied; in this regard, Ahmad and Kharal [1] combined fuzzy sets with soft sets, then Cağman et al. [19] showed how fuzzy soft sets applied to optimal choices. Alcantud [2] explored the relationships between the topological structures in fuzzy and soft settings. Al-shami et al. [11] presented generalized umbral for orthopair fuzzy soft sets called (a, b)-Fuzzy soft sets. Investigation of covering-based rough set from fuzzy view was the main goal of some manuscripts; see, for example, [18, 37, 38]. In fuzzy algebra, Gulzar with his coauthors debated some types of fuzzy subgroups [25] as well as displayed interesting applications of complex IFSs in group theory [26]. New hybridizations of fuzzy set theory with some types of soft sets were established in [3, 33].

Atanassov [17] defined intuitionistic fuzzy sets as one of the intriguing generalizations of fuzzy sets with excellent application. Applications of intuitionistic fuzzy sets can be found in a variety of domains, including optimization issues, decisionmaking, and medical diagnostics [2224]. However, in many circumstances, the decision maker may assign degrees of membership and non-membership to a given attribute such that their aggregate is larger than one. As a result, Yager [30] introduced the Pythagorean fuzzy set notion, which is a generalization of intuitionistic fuzzy sets and a more effective tool for solving uncertain issues. Pythagorean fuzzy sets are a type of fuzzy set that can be used to characterize uncertain data more effectively and precisely than intuitionistic fuzzy sets. Senapati et al. [35] explored Fermatean fuzzy sets and introduced fundamental procedures on them. Another form of generalized Pythagorean fuzzy set named (3,2)-Fuzzy sets and SR-Fuzzy sets was defined by [27] and [4], respectively. Yager [29] developed the concept of the q-rung orthopair fuzzy sets and because of their greater range of depicting membership grades, they are more likely to be used in uncertain situations than other types of fuzzy sets. Al-shami et al. [4] presented another kind of fuzzy set called SR-Fuzzy set and discussed their properties in details. Saliha et al. [34] proposed CR-fuzzy sets and found the fundamental set of operations for them. Recently, Al-shami [7] has defined a new class of IFSs called “(2,1)-Fuzzy sets” and investigated main properties. He also has generated new weighted aggregated operators using this class and applied to multi-criteria decision-making problems.

One of the interesting themes is studying topology with respect to fuzzy sets. This line of research began by Chang [20] in 1968. He explored the main properties of some topological concepts and ideas via fuzzy set environment. In 1997, Çoker [21] defined the concept of an intuitionistic fuzzy topological space. He defined the main topological concepts via this space such as continuity, compactness, connectedness, and separation axioms. Recently, Turkarslan et al. [36] have familiarized a q-rung orthopair fuzzy topological space and studied some its properties. Recently, Ameen et al. [16] have initiated the concept of infra-fuzzy topological structures and explored main features.

In this paper, we propose a new environment of fuzzy sets so-called “nth power root fuzzy set” and compare it with the other types of fuzzy sets. The fundamental benefit of nth power root fuzzy sets is that they can be used in a wide range of decision-making situations. Then, we propose the set of operations for the nth power root fuzzy set and explore their main features. Also, the concept of topology for nth power root fuzzy sets is investigated. Thereafter, we describe separated and disconnected nth power root fuzzy sets under this topology, and also we study nth power root fuzzy continuous maps in details. Ultimately, we study two types of separation axioms in nth power root fuzzy topology.

In this section, we study the notion of nth power root fuzzy sets, and establish main properties. For computations, we use only three decimal places in whole paper.

### Definition 2.1

Let ℕ be the set of natural numbers and A be a universal set. Then, the nth power root fuzzy set (briefly, nPR-FS) ξ, which is a set of ordered pairs over A, is defined as following:

ξ={w,ϖξ(w),ωξ(w):wA},

where ϖξ:A[0,1] is the degree of membership and ωξ:A[0,1] is the degree of non-membership functions such that

0(ϖξ(w))n+ωξ(w)n1,

for each wA and n ∈ ℕ \ {1}.

Then, there is a degree of indeterminacy of wA to ξ defined by

πξ(w)=1-[(ϖξ(w))n+ωξ(w)n].

It is clear that (ϖξ(w))n+ωξ(w)n+πξ(w)=1. Otherwise, πξ(w)=0 whenever (ϖξ(w))n+ωξ(w)n=1.

For the sake of simplicity, we shall mention the symbol ξ = (ω̄ξ, ωξ) for the nPR-FS ξ={w,ϖξ(w),ωξ(w):wA}. We noticed that 2PR-fuzzy is called SR-FS in [?], and 3PR-fuzzy is called CR-FS in [34]. If 0ϖξ(w)+ωξ(w)1 (resp. 0ϖξ(w)n+ωξ(w)n1), then ξ = (ω̄ξ, ωξ) is called intuitionistic fuzzy set (IFS) [17](resp. n-rung orthopair fuzzy set (n-ROFS) [29]).

The spaces of some kinds of nPR-fuzzy membership grades are displayed in Figure 1.

### Remark 2.2

From Figure 2, we get that

• the space of 4-rung orthopair fuzzy membership grades is larger than the space of 4PR-fuzzy membership grades.

• ξ = (ω̄ξ ≈ 0.724, ωξ ≈ 0.276) is a point of intersection between 4PR-fuzzy and intuitionistic fuzzy sets.

• for ω̄ξ ∈ (0, 0.724) and ωξ ∈ (0.276, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of intuitionistic membership grades.

• for ω̄ξ ∈ (0.724, 1) and ωξ ∈ (0, 0.276) the space of 4PR-fuzzy membership grades starts to be smaller than the space of intuitionistic membership grades.

• ξ=(ϖξ=5-12,ωξ=7-352) is a point of intersection between 4PR-fuzzy and SR-fuzzy.

• for ϖξ(0,5-12) and ωξ(7-352,1) the space of 4PR-fuzzy membership grades starts to be larger than the space of SR-fuzzy membership grades.

• for ϖξ(5-12,1) and ωξ(0,7-352) the space of 4PR-fuzzy membership grades starts to be smaller than the space of SR-fuzzy membership grades.

• ξ = (ω̄ξ ≈ 0.819, ωξ ≈ 0.091) is a point of intersection between 4PR-fuzzy and CR-fuzzy sets.

• for ω̄ξ ∈ (0, 0.819) and ωξ ∈ (0.091, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of CR-fuzzy membership grades.

• for ω̄ξ ∈ (0.819, 1) and ωξ ∈ (0, 0.091) the space of 4PR-fuzzy membership grades starts to be smaller than the space of CR-fuzzy membership grades.

### Remark 2.3

It is clear that for any nPR-FS ξ = (ω̄ξ, ωξ) we have 0ϖξn+ωξnϖξn+ωξn1, then ξ is an n-rung orthopair fuzzy set. Therefore, every nPR-fuzzy set is an n-rung orthopair fuzzy set.

### Definition 2.4

Let ξ = (ω̄ξ, ωξ), ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be three nPR-fuzzy sets (nPR-FSs), then

• ξ1ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2}).

• ξ1ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2}).

• ξc=(ωξn2,(ϖξ)n2).

### Example 2.5

Suppose that ξ1 = (0.6, 0.5) and ξ2 = (0.5, 0.6) are both 4PR-FSs for A={w}. Then,

• ξ1ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{0.6,0.5},max{0.5,0.6})=(0.5,0.6).

• ξ1ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})=(max{0.6,0.5},min{0.5,0.6})=(0.6,0.5).

• ξ1c(0.958,0.000).

### Theorem 2.6

If ξ = (ω̄ξ, ωξ) is a nPR-FS, then ξc is also a nPR-FS and (ξc)c=ξ.

Proof

Since 0ϖξn1,0ωξn1 and 0(ϖξ)n+ωξn1, then

0(ωξn2)n+(ϖξ)n2n=(ϖξ)n+ωξn1,

and hence

0(ωξn2)n+(ϖξ)n2n1.

Thus, ξc is a nPR-FS and it is obvious that (ξc)c=(ωξn2,(ϖξ)n2)c=(ϖξ,ωξ).

### Theorem 2.7

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• ξ1ξ2 = ξ2ξ1.

• ξ1ξ2 = ξ2ξ1.

Proof

From Definition 2.4, we can obtain:

• ξ1ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{ϖξ2,ϖξ1},max{ωξ2,ωξ1})=ξ2ξ1.

• The proof is similar to (1).

### Theorem 2.8

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• (ξ1ξ2) ∪ ξ2 = ξ2.

• (ξ1ξ2) ∩ ξ2 = ξ2.

Proof

From Definition 2.4, we can obtain:

• (ξ1ξ2)ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})(ϖξ2,ωξ2)=(max{min{ϖξ1,ϖξ2},ϖξ2},min{max{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.

• (ξ1ξ2)ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})(ϖξ2,ωξ2)=(min{max{ϖξ1,ϖξ2},ϖξ2},max{min{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.

### Theorem 2.9

Let ξ1 = (ω̄ξ1, ωξ1), ξ2 = (ω̄ξ2, ωξ2) and ξ3 = (ω̄ξ3, ωξ3) be three nPR-FSs, then

• ξ1 ∩ (ξ2ξ3) = (ξ1ξ2) ∩ ξ3.

• ξ1 ∪ (ξ2ξ3) = (ξ1ξ2) ∪ ξ3.

Proof

For the three nPR-FSs ξ1, ξ2 and ξ3, according to Definition 2.4, we can obtain

• 1. ξ1(ξ2ξ3)=(ϖξ1,ωξ1)(min{ϖξ2,ϖξ3},max{ωξ2,ωξ3})=(min{ϖξ1,min{ϖξ2,ϖξ3}},max{ωξ1,max{ωξ2,ωξ3}})=(min{min{ϖξ1,ϖξ2},ϖξ3},max{max{ωξ1,ωξ2},ωξ3})=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})(ϖξ3,ωξ3)=(ξ1ξ2)ξ3.

• 2. The proof is similar to (1).

### Theorem 2.10

Let ξ1 = (ω̄ξ1, ωξ1), ξ2 = (ω̄ξ2, ωξ2) and ξ3 = (ω̄ξ3, ωξ3) be three nPR-FSs, then

• (ξ1ξ2) ∩ ξ3 = (ξ1ξ3) ∪ (ξ2ξ3).

• (ξ1ξ2) ∪ ξ3 = (ξ1ξ3) ∩ (ξ2ξ3).

Proof

It can be proved following similar arguments given to the proof of its counterpart in [7].

### Theorem 2.11

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• (ξ1ξ2)c=ξ1cξ2c.

• (ξ1ξ2)c=ξ1cξ2c.

Proof

For the two nPR-FSs ξ1 and ξ2, according to Definition 2.4, we can obtain:

• 1. (ξ1ξ2)c=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})c=(max{ωξ1n2,ωξ2n2},min{(ϖξ1)n2,(ϖξ2)n2})=(ωξ1n2,(ϖξ1)n2)(ωξ2n2,(ϖξ2)n2)=ξ1cξ2c.

• 2. The proof is similar to (1).

### Definition 2.12

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• ξ1 = ξ2 if and only if ω̄ξ1 = ω̄ξ2 and ωξ1 = ωξ2.

• ξ1ξ2 if and only if ω̄ξ1ω̄ξ2 and ωξ1ωξ2.

• ξ2ξ1 or ξ1ξ2 if ξ1ξ2.

### Example 2.13

Let A={w1,w2}, then

• ξ1 = ξ2 for two 4PR-FSs ξ1={w1,0.62,0.51,w2,0.45,0.76} and ξ2={w1,0.62,0.51,w2,0.45,0.76}.

• ξ2ξ1 and ξ2ξ1 for two 4PR-FSs ξ1={w1,0.62,0.51,w2,0.45,0.76} and ξ2={w1,0.61,0.52,w2,0.44,0.77}.

In this section, we define a new class of fuzzy topology using nPR-fuzzy subsets. We initiate the main concepts via this class such as connectedness, continuity and separation axioms. We describe these concepts and illustrate the relations between them with aid of counterexamples.

### 3.1 nPR-Fuzzy Topology

Definition 3.1

Let be a family of nPR-fuzzy subsets of a non-empty set A. If

• 1A,0A where 1A=(1,0) and0A=(0,1),

• ξ1ξ2, for any ξ1, ξ2,

• iIξi, for any {ξi}iN,

then is called an nPR-fuzzy topology (briefly, nPR-FT) on A and (A,) is an nPR-fuzzy topological space (briefly, nPR-FTS). Each member of is called an open nPR-FS. We call ξ a closed nPR-FS if its complement is an open nPR-FS.

Now, we give an example of 7PR-FTS.

Example 3.2

Let A={w1,w2} and ξi be 7PR-fuzzy subsets of A such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2, 3, 4, 5. If

ϖξ1(w1)=0.52,ωξ1(w1)=0.72,ϖξ1(w2)=0.61,ωξ1(w2)=0.72,ϖξ2(w1)=0.55,ωξ2(w1)=0.68,ϖξ2(w2)=0.62,ωξ2(w2)=0.73,ϖξ3(w1)=0.51,ωξ3(w1)=0.73,ϖξ3(w2)=0.59,ωξ3(w2)=0.74,ϖξ4(w1)=0.55,ωξ4(w1)=0.68,ϖξ4(w2)=0.62,ωξ4(w2)=0.72,ϖξ5(w1)=0.52,ωξ5(w1)=0.72,ϖξ5(w2)=0.61,ωξ5(w2)=0.73,

then the family of 7PR-fuzzy sets ={1A,0A,ξ1,ξ2,ξ3,ξ4,ξ5} is 7PR-FT on A.

Remark 3.3

Every nPR-FT is an n-rung orthopair fuzzy topology because every nPR-fuzzy subset of a set can be considered as an n-rung orthopair fuzzy subset. The next example elaborates that n-rung orthopair fuzzy topological space need not be an nPR-FTS.

Example 3.4

Let A={w1,w2} and ξi be 2-rung orthopair fuzzy subsets of A such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2. If

ϖξ1(w1)=0.96,ωξ1(w1)=0.25,ϖξ1(w2)=0.92,ωξ1(w2)=0.35,ϖξ2(w1)=0.95,ωξ2(w1)=0.25,ϖξ2(w2)=0.91,ωξ2(w2)=0.35,

then ={1A,0A,ξ1,ξ2} is 2-rung orthopair fuzzy topology but is not 2PR-FT.

Remark 3.5
• We call ={1A,0A} the indiscrete nPR-FT on A.

• If contains all nPR-fuzzy subsets, then we call the discrete nPR-FT on A.

Definition 3.6

Let ξ1 and ξ2 be two nPR-fuzzy subsets in an nPR-FTS (A,). Then, ξ2 is called a neighbourhood of ξ1 if there exists an open nPR-fuzzy subset ξ3 such that ξ1ξ3ξ2.

Theorem 3.7

An nPR-fuzzy set ξ1 is open in an nPR-FTS (A,) if and only if it contains a neighbourhood of its each subset.

Proof

Let ξ1 be an open nPR-fuzzy set and ξ2 be an nPR-fuzzy set such that ξ2ξ1. Since ξ2ξ1ξ1 and ξ1 is an open nPR-fuzzy set, then ξ1 is a neighborhood of ξ2. On the other hand, assume that ξ1 is a neighborhood of its each subset. From Definition 3.6, for arbitrary ξ2ξ1 there exists an open nPR-fuzzy set ξξ2 such that ξ2ξξ2ξ1. Thus, we have ξ1 ⊂ ∪ξ2ξ1ξξ2 and since for all ξ2ξ1 and ξξ2ξ1, we get ∪ξ2ξ1ξξ2ξ1. Consequently, we obtain ξ1 = ∪ξ2ξ1ξξ2 which implies ξ1 is an open nPR-fuzzy set.

### Definition 3.8

Let (A,) be an nPR-FTS and ξ = (ω̄ξ, ωξ) be an nPR-FS in A. Then, the nPR-fuzzy interior and nPR-fuzzy closure of ξ are, respectively, defined by

• int(ξ)={ξ2:ξ2is open nPR-FS in Aand ξ2ξ}.

• cl(ξ)={ξ1:ξ1is closed nPR-FS in Aand ξξ1}.

### Remark 3.9

Let ( A,) be an nPR-FTS and ξ be any nPR-FS in A. Then,

• int(ξ) is an open nPR-FS.

• cl(ξ) is a closed nPR-FS.

• int(1A)=cl(1A)=1A and int(0A)=cl(0A)=0A.

### Example 3.10

Consider the 7PR-FTS (A,) in Example 3.2. If ξ={w1,0.53,0.73,w2,0.63,0.74}, then int(ξ)=ξ3 and cl(ξ)=1A.

### Theorem 3.11

Let (A,) be an nPR-FTS and ξ1, ξ2 be two nPR-FSs in A. Then,

• int(ξ1)ξ1cl(ξ1).

• int(ξ1)int(ξ2) and cl(ξ1)cl(ξ2) if ξ1ξ2.

• ξ1 is an open nPR-FS if and only if ξ1=int(ξ1).

• ξ1 is a closed nPR-FS if and only if ξ1=cl(ξ1).

Proof

(1) and (2) are obvious.

(3) and (4) follow from Definition 3.8.

### Theorem 3.12

Let (A,) be an nPR-FTS and ξ1, ξ2 be two nPR-FSs in A. Then,

• int(ξ1)int(ξ2)int(ξ1ξ2).

• cl(ξ1ξ2)cl(ξ1)cl(ξ2).

• int(ξ1ξ2)=int(ξ1)int(ξ2).

• cl(ξ1)cl(ξ2)=cl(ξ1ξ2).

Proof
• Since ξ1ξ1ξ2 and ξ2ξ1ξ2, then by Theorem 3.11 (2), we have int(ξ1)int(ξ1ξ2) and int(ξ2)int(ξ1ξ2). Therefore, int(ξ1)int(ξ2)int(ξ1ξ2)int(ξ1ξ2)=int(ξ1ξ2).

• Since ξ1ξ2ξ1 and ξ1ξ2ξ2, then by Theorem 3.11 (2), we have cl(ξ1ξ2)cl(ξ1) and cl(ξ1ξ2)cl(ξ2). Therefore, cl(ξ1ξ2)=cl(ξ1ξ2)cl(ξ1ξ2)cl(ξ1)cl(ξ2).

• Since int(ξ1ξ2)int(ξ1) and int(ξ1ξ2)int(ξ2), we obtain int(ξ1ξ2)int(ξ1)int(ξ2). On the other hand, from the facts int(ξ1)ξ1 and int(ξ2)ξ2 we have int(ξ1)int(ξ2)ξ1ξ2 and int(ξ1)int(ξ2); we see that int(ξ1)int(ξ2)int(ξ1ξ2), and hence int(ξ1ξ2)=int(ξ1)int(ξ2).

• Can be proved similar to (3).

### Theorem 3.13

Let (A,) be an nPR-FTS and ξ be nPR-FS in A. Then,

• cl(ξc)=int(ξ)c.

• cl(ξc)c=int(ξ).

• int(ξc)c=cl(ξ).

• int(ξc)=cl(ξ)c.

Proof

Let ξ={w,ϖξ(w),ωξ(w):wA} and suppose that the family of open nPR-fuzzy sets contained in ξ are indexed by the family {w,ϖξi(w),ωξi(w):iJ}. Then, int(ξ)={w,ϖξi(w),ωξi(w)} and hence

• int(ξ)c={w,ωξi(w)n2,(ϖξi(w))n2}. Since ξc={w,ωξ(w)n2,(ϖξ(w))n2} and ω̄ξiω̄ξ, ωξiωξ for each iJ, so we obtain that w,ωξi(w)n2,(ϖξi(w))n2 is the family of closed nPR-fuzzy sets containing ξc, that is,

cl(ξc)={w,ωξi(w)n2,(ϖξi(w))n2}.

Thus, cl(ξc)=int(ξ)c.

• {w,ωξi(w)n2,(ϖξi(w))n2} is the family of closed nPR-fuzzy sets containing ξc, so,

cl(ξc)={m,ωξi(m)n2,(ϖξi(w))n2}

and

cl(ξc)c={m,(ϖξi(m))n2n2,(ωξi(m)n2)n2}={m,ϖξi(m),ωξi(m)}=int(ξ).

• from (1), we can getcl(ξ)=int(ξc)c.

• from (2), we can get cl(ξ)c=int(ξc).

### Definition 3.14

Two nPR-FSs ξ1 and ξ2 of an nPR-FTS (A,) are called separated nPR-FSs if cl(ξ1)ξ2=0A and ξ1cl(ξ2)=0A.

If ξ1 and ξ2 are separated nPR-FSs, then ξ1ξ2=0A. The converse may not be true in general, as it is shown in the following example.

### Example 3.15

Consider the 7PR-FTS (A,) in Example 3.2. If ξ1={w1,0,1,w2,0.64,0.71} and ξ2={w1,0.52,0.87,w2,0,1}, then ξ1ξ2=0A but ξ1 and ξ2 are not separated nPR-FSs.

### Corollary 3.16

If ξ1 and ξ2 are two nPR-FSs in an nPR-FTS (A,) such that ξ1ξ2=0A, then ξ1ξ2c.

Proof

Fromξ1ξ2=0A, we get min{ϖξ1,ϖξ2}=0 and max{ωξ1,ωξ2}=1. If the inequality ϖξ1ωξ2n2 were not true, then there would exist woA such that ϖξ1(wo)>ωξ2(wo)n2, then, since ϖξ1(wo)>0 and hence, ωξ1(wo)<1, we obtain ωξ2(wo)=1 which is an obvious contradiction. Similarly, if the inequality ωξ1 ≥ (ω̄ξ2)n2 were not true, then there would exist wooA such that ωξ1(woo)<(ϖξ2(woo))n2, then, since ϖξ2(woo)>0 and hence, ωξ2(woo)<1, we obtain ωξ1(woo)=1which is an obvious contradiction. So, we get ωξ1 ≥ (ω̄ξ2 )n2, and thus ξ1ξ2c.

### Theorem 3.17

If ξ10A and ξ20A are any two nPR-FSs in an nPR-FTS (A,), then the following statements are true:

• If ξ1 and ξ2 are separated nPR-FSs, ξ11ξ1 and ξ21ξ2, then ξ11 and ξ21 are also separated nPR-FSs.

• If ξ1ξ2=0A such that ξ1 and ξ2 are both closed nPR-FSs, then ξ1 and ξ2 are separated nPR-FSs.

Proof
• Since ξ11ξ1, then cl(ξ11)cl(ξ1). Then, ξ2cl(ξ1)=0A implies ξ21cl(ξ1)=0A and so ξ21cl(ξ11)=0A. Similarly ξ11cl(ξ21)=0A. Hence, ξ11 and ξ21 are separated nPR-FSs.

• Since ξ1=cl(ξ1),ξ2=cl(ξ2) and ξ1ξ2=0A, then cl(ξ1)ξ2=0A and cl(ξ2)ξ1=0A. Hence, ξ1 and ξ2 are separated nPR-FSs.

### Remark 3.18

If ξ1ξ2=0A such that ξ1 and ξ2 are both open nPR-FSs in an nPR-FTS (A,), then ξ1 and ξ2 are not separated nPR-FSs in general.

### Example 3.19

Let A={w1,w2} and ξi be 4PR-fuzzy subsets of A such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2, 3. If

ϖξ1(w1)=0,ωξ1(w1)=1,ϖξ1(w2)=0.25,ωξ1(w2)=0.86,ϖξ2(w1)=0.26,ωξ2(w1)=0.85,ϖξ2(w2)=0,ωξ2(w2)=1,ϖξ3(w1)=0.26,ωξ3(w1)=0.85,ϖξ3(w2)=0.25,ωξ3(w2)=0.86,

then ={1A,0A,ξ1,ξ2,ξ3} is 4PR-FT and ξ1ξ2=0A, but ξ1 and ξ2 are not separated 4PR-FSs.

### Corollary 3.20

If ξ1 and ξ2 are separated nPR-FSs in an nPR-FTS (A,), then there exist two open nPR-FSs ξ3 and ξ4 in (A,) such that ξ1ξ3 and ξ2ξ4.

Proof

Let ξ1 and ξ2 be separated nPR-FSs. Set ξ4=cl(ξ1)c and ξ3=cl(ξ2)c, then ξ3 and ξ4 are open nPR-FSs such that ξ1ξ3 and ξ2ξ4.

### Theorem 3.21

Let the following statements in an nPR-FTS (A,),

• 0A and 1A are the only open nPR-FSs and closed nPR-FSs in (A,).

• there are no two open nPR-FSs ξ1 and ξ2 such that ξ10A,ξ20A,ξ1ξ2=0A and 1A=ξ1ξ2.

• there are no two closed nPR-FSs ξ1 and ξ2 such that ξ10A,ξ20A,ξ1ξ2=0A and 1A=ξ1ξ2.

• there are no two separated nPR-FSs ξ1 and ξ2 such that ξ10A,ξ20A and 1Aξ1ξ2.

Then, (1) ⇒ (2) ⇔ (3) ⇔ (4) hold.

Proof

(1) ⇒ (2): Suppose (2) is false and that 1A=ξ1ξ2, where ξ1 and ξ2 are two open nPR-FSs such that ξ10A,ξ20A and ξ1ξ2=0A. By Theorems 2.6, 2.11 (2) and Corollary 3.16, we have ξ1c=ξ2 and hence ξ2 is open nPR-FS and closed nPR-FS in (A,) such that 1Aξ20A, which shows that (1) is false.

(2) ⇔ (3): This is clear.

(3) ⇒ (4): If (4) is false, then 1A=ξ1ξ2, where ξ1 and ξ2 are two separated nPR-FSs such that ξ10A and ξ20A. By Theorems 2.11 (2), we have 0A=1Ac=(ξ1ξ2)c=ξ1cξ2c. By Corollary 3.16 and Theorem 2.6, we get ξ1c(ξ2c)c=ξ2. Since cl(ξ2)ξ1=0A, we conclude that cl(ξ2)ξ2 so ξ2 is closed nPR-FS. Similarly, ξ1 must be closed nPR-FS. Therefore, (3) is false.

(4) ⇒ (3): If (3) is false, then 1A=ξ1ξ2, where ξ1 and ξ2 are two closed nPR-FSs such that ξ10A,ξ20A and ξ1ξ2=0A. Then, ξ1 and ξ2 are separated nPR-FSs by Theorem 3.11 (4). Therefore, (4) is false.

### Definition 3.22

Let (A,) be an nPR-FTS and ξ be an nPR-FS in A, then ξ is called disconnected nPR-FS if there are two separated nPR-FSs ξ1 and ξ2 in A such that ξ10A,ξ20A and ξ = ξ1ξ2. If ξ is not disconnected nPR-FS, then ξ is called connected nPR-FS.

### Example 3.23

Let A={w1,w2} and ξi be nPR-fuzzy subsets of A such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2. If

ϖξ1(w1)=0,ωξ1(w1)=1,ϖξ1(w2)=1,ωξ1(w2)=0,ϖξ2(w1)=1,ωξ2(w1)=0,ϖξ2(w2)=0,ωξ2(w2)=1,

then ={1A,0A,ξ1,ξ2} is nPR-FT and hence 1A is disconnected nPR-FS because there are two nPR-FSs ξ1 and ξ2 such that ξ10A,ξ20A,cl(ξ1)ξ2=0A,ξ1cl(ξ2)=0A and 1A=ξ1ξ2.

### Remark 3.24

An nPR-FS 1A

is connected in every indiscrete nPR-FTS (A,).

### Remark 3.25

An nPR-FS 1A is connected if any (therefore all) of the conditions (2) – (4) in Theorem 3.21 hold.

### Remark 3.26

According to the Definition 3.22 and Remark 3.25, 1A is disconnected nPR-FS if we can write 1A=ξ1ξ2, where the following (equivalent) statements are true:

• ξ1 and ξ2 are open nPR-FSs such that ξ10A,ξ20A and ξ1ξ2=0A.

• ξ1 and ξ2 are closed nPR-FSs such that ξ10A,ξ20A and ξ1ξ2=0A.

• ξ1 and ξ2 are separated nPR-FSs such that ξ10A and ξ20A.

### Corollary 3.27

Let (A,) be an nPR-FTS. Then, 1A is connected nPR-FS if 0A and 1A are the only open nPR-FSs and closed nPR-FSs in (A,).

Proof

Follows from Theorem 3.21 and Remark 3.25.

### Lemma 3.28

Suppose ξ1 and ξ2 are separated nPR-FSs in an nPR-FTS (A,). If ξξ1ξ2 and ξ is connected nPR-FS, then ξξ1 or ξξ2.

Proof

Since ξξ1ξ1 and ξξ2ξ2, then ξξ1 and ξξ2 are separated nPR-FSs and by Theorem 2.10 (1) we have ξ = ξ ∩ (ξ1ξ2) = (ξξ1) ∪ (ξξ2). Since ξ is connected nPR-FS, hence either ξξ1=0A, so ξξ2 or ξξ2=0A, so ξξ1.

### Theorem 3.29

Suppose ξ and ξi (i ∈ ℕ) are connected nPR-FSs in an nPR-FTS (A,) and that for each i, ξi and ξ are not separated nPR-FSs. Then, ξξi is connected nPR-FS.

Proof

Suppose that ξξi = ξ11ξ12, where ξ11 and ξ12 are separated nPR-FSs such that ξ110A and ξ120A. By Lemma 3.28, either ξξ11 or ξξ12. Without loss of generality, assume ξξ11. By the same reasoning we conclude that for each i, either ξiξ11 or ξiξ12. If some ξiξ12, then ξ and ξi would be separated nPR-FSs. Thus every ξiξ11 and so ξξiξ11. Therefore, ξ12ξ11ξ12 = ξξiξ11 and hence ξ12ξ11cl(ξ11), so ξ12=0A which is contradiction, therefore ξξi is connected nPR-FS.

In this section we study the continuity of a function defined among two nPR-FTSs.

Let f:AB be a function. Let ξ1 and ξ2 be nPR-fuzzy subsets of A and B, respectively. Then, the membership and non-membership functions of the image of ξ1 denoted by f[ξ1] are, respectively, calculated by

ϖf[ξ1](b)={supaf-1(b)ϖξ1(a),if f-1(b)φ,0,otherwise,

and

ωf[ξ1](b)={infaf-1(b)ωξ1(a),if f-1(b)φ,1,otherwise.

The membership and non-membership functions of pre-image of ξ2 denoted by f-1[ξ2], are, respectively, calculated by ϖf-1[ξ2](w)=ϖξ2(f(w)), and ϖf-1[ξ2](w)=ϖξ2(f(w)).

### Remark 3.30

To show that f[ξ1] and f-1[ξ2] are nPR-fuzzy sets. Consider θξ1(a)=(ϖξ1(a))n+(ωξ1(a))1n and bB, then we obtain

(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=(supaf-1(b)ϖξ1(a))n+(infaf-1(b)ωξ1(a))1n=supaf-1(b)(ϖξ1(a))n+infaf-1(b)(ωξ1(a))1n=supaf-1(b)(θξ1(a)-(ωξ1(a))1n)+infaf-1(b)(ωξ1(a))1nsupaf-1(b)(1-(ωξ1(a))1n)+infaf-1(b)(ωξ1(a))1n=1,

whenever f-1(b) is non-empty. On the other hand if f-1(b)=φ, then we have (ϖf[ξ1](b))n+(ωf[ξ1](b))1n=1. Thus, f[ξ1] is an nPR-fuzzy set of B.

Now, let wA, then (ϖf-1[ξ2](w))n+(ωf-1[ξ2](w))1n=(ϖξ2(f(w)))n+(ωξ2(f(w)))1n1. Therefore, f-1[ξ2] is an nPR-fuzzy set of A.

Following theorems give some properties of image and pre-image.

### Theorem 3.31

Let f:AB be a function such that ξ1 and ξ2 are nPR-fuzzy subsets of A and B, respectively. Then,

• f-1[ξ2c]=f-1[ξ2]c.

• f[ξ1]cf[ξ1c].

• f[f-1[ξ2]]ξ2.

• ξ1f-1[f[ξ1]].

### Proof

• For any wA and for any nPR-fuzzy subset ξ2 of B we get from the definition of the complement that

ϖf-1[ξ2c](w)=ϖξ2c(f(w))=ωξ2(f(w))n2=ωf-1[ξ2](w)n2=ϖf-1[ξ2]c(w).

Similarly we have

ωf-1[ξ2c](w)=ωξ2c(f(w))=(ϖξ2(f(w)))n2=(ϖf-1[ξ2](w))n2=ωf-1[ξ2]c(w).

Thus, we have f-1[ξ2c]=f-1[ξ2]c.

• For any bB such that f(b)=φ and for any nPR-fuzzy subset ξ1 of A, we can write

θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supaf-1(b)(ϖξ1(a))n+infaf-1(b)(ωξ1(a))1n=supaf-1(b)(θξ1(a)-(ωξ1(a))1n)+infaf-1(b)(ωξ1(a))1nsupaf-1(b)θξ1(a)-infaf-1(b)(ωξ1(a))1n+infaf-1(b)(ωξ1(a))1n=supaf-1(b)θξ1(a).

Now from (*), we have

ϖf[ξ1c](b)=supaf-1(b)ϖξ1c(a)=supaf-1(b)ωξ1(a)n2=supaf-1(b)θξ1(a)-(ϖξ1(a))nnsupaf-1(b)θξ1(a)-supaf-1(b)(ϖξ1(a))nnθf[ξ1](b)-(ϖf[ξ1](b))nn=ωf[ξ1](b)n2=ϖf[ξ1]c(b).

Hence, we obtain ϖf[ξ1]c(b)ϖf[ξ1c](b).

Similarly we have

θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supaf-1(b)(ϖξ1(a))n+infaf-1(b)(ωξ1(a))1n=supaf-1(b)(ϖξ1(a))n+infaf-1(b)(θξ1(a)-(ϖξ1(a))n)supaf-1(b)(ϖξ1(a))n+infaf-1(b)θξ1(a)+infaf-1(b)(-(ϖξ1(a))n)=infaf-1(b)θξ1(a).

Now, (**) which implies that

ωf[ξ1c](b)=infaf-1(b)ωξ1c(a)=infaf-1(b)(ϖξ1(a))n2=infaf-1(b)(θξ1(a)-(ωξ1(a))1n)n(infaf-1(b)θξ1(a)-infaf-1(b)(ωξ1(a))1n)n(θf[ξ1](b)-(ωf[ξ1](b))1n)n=(ϖf[ξ1](b))n2=ωf[ξ1]c(b).

Hence, we obtain ωf[ξ1c](b)ωf[ξ1]c(b).

On the other hand, the proof is trivial for each bB such that f(b)=φ.

Therefore, we obtain f[ξ1]cf[ξ1c].

• For any bB such that f(b)=φ, we can write

ϖf[f-1[ξ2]](b)=supaf-1(b)ϖf-1[ξ2](a)=supaf-1(b)ϖξ2(f(a))ϖξ2(b).

Similarly, we have

ωf[f-1[ξ2]](b)=infaf-1(b)ωf-1[ξ2](a)=infaf-1(b)ωξ2(f(a))ωξ2(b).

On the other hand if f(b)=φ, then we have ϖf[f-1[ξ2]](b)=0ϖξ2(b), and ωf[f-1[ξ2]](b)=1ωξ2(b). Therefore, f[f-1[ξ2]]ξ2.

• For any wA, we have ϖf-1[f[ξ1]](w)=ϖf[ξ1](f(w))=supaf-1(f(w))ϖξ1(a)ϖξ1(w).

Similarly, we have ωf-1[f[ξ1]](w)ωξ1(w).

Therefore, ξ1f-1[f[ξ1]].

### Theorem 3.32

Let f:AB be a function such that ξ1i and ξ2i are nPR-fuzzy subsets of A and B, respectively, for i = 1, 2. If

• ξ21ξ22, then f-1[ξ21]f-1[ξ22].

• ξ11ξ12, then f[ξ11]f[ξ12].

Proof
• Assume that ξ21ξ22. Then, for any wA we have

ϖf-1[ξ21](w)=ϖξ21(f(w))ϖξ22(f(w))=ϖf-1[ξ22](w).

Thus, ϖf-1[ξ21](w)ϖf-1[ξ22](w). Similarly, we have

ωf-1[ξ22](w)=ωξ22(f(w))ωξ21(f(w))=ωf-1[B1ξ21](w).

Thus, ωf-1[ξ21](w)ωf-1[ξ22](w). Therefore, f-1[ξ21]f-1[ξ22].

• Assume that ξ11ξ12 and bB. If f(b)=φ, then the proof is trivial. Assume that f(b)φ. Then, we have

ϖf[ξ11](b)=supaf-1(b)ϖξ11(a)supaf-1(b)ϖξ12(a)=ϖf[ξ12](b),

and

ωf[ξ12](b)=infaf-1(b)ωξ12(a)infaf-1(b)ωξ11(a)=ωf[ξ11](b).

Therefore, f[ξ11]f[ξ12].

The proof of the following result is easy and hence it is omitted.

### Corollary 3.33

Let f:AB be a function. Then, the following statements are true:

• f[iIξ1i]=iIf[ξ1i] for any nPR-fuzzy subset ξ1i of A.

• f-1[iIξ2i]=iIf-1[ξ2i] for any nPR-fuzzy subset ξ2i of B.

• f[ξ11ξ12]f[ξ11]f[ξ12] for any two nPR-fuzzy subsets ξ11 and ξ12 of A.

• f-1[iIξ2i]=iIf-1[ξ2i] for any nPR-fuzzy subset ξ2i of B.

### Definition 3.34

Let (A,1) and (B,2) be two nPR-FTSs and f:AB be a function. Then, f is said to be nPR-fuzzy continuous if for any nPR-fuzzy subset ξ1 of A and for any neighbourhood ξ4 of f[ξ1] there exists a neighbourhood ξ3 of ξ1 such that f[ξ3]ξ4.

### Theorem 3.35

Let (A,1) and (B,2) be two nPR-FTSs and f:AB be a function. Then, the following statements are equivalent:

• f is nPR-fuzzy continuous.

• For any nPR-fuzzy subset ξ1 of A and for any neighbourhood ξ4 of f[ξ1], there exists a neighbourhood ξ3 of ξ1 such that for any ξ2ξ3 we have f[ξ2]ξ4.

• For any nPR-fuzzy subset ξ1 of A and for any neighbourhood ξ4 of f[ξ1], there exists a neighbourhood ξ3 of ξ1 such that ξ3f-1[ξ4].

• For any nPR-fuzzy subset ξ1 of A and for any neighbourhood ξ4 of f[ξ1],f-1[ξ4] is a neighbourhood of ξ1.

Proof

(1) ⇒ (2): Assume that f is nPR-fuzzy continuous. Let ξ1 be an nPR-fuzzy subset of A and ξ4 be a neighbourhood of f[ξ1]. Then by (1), there exists a neighbourhood ξ3 of ξ1 such that f[ξ3]ξ4. Now, if ξ2ξ3, then we get f[ξ2]f[ξ3]ξ4. (2) ⇒ (3): Let ξ1 be an nPR-fuzzy set of A and ξ4 be a neighbourhood of f[ξ1]. From (2), there exists a neighbourhood ξ3 of ξ1 such that for any ξ2ξ3 we have f[ξ2]ξ4. Then, we can write ξ2f-1[f[ξ2]]f-1[ξ4]. As ξ2 is an arbitrary subset of ξ3, we have ξ3f-1[ξ4].

(3) ⇒ (4): Let ξ1 be an nPR-fuzzy subset of A and ξ4 be a neighbourhood of f[ξ1]. Then from (3), there exists a neighbourhood ξ3 of ξ1 such that ξ3f-1[ξ4]. Since ξ3 is a neighbourhood of ξ1 there exists an open nPR-fuzzy subset ξ5 of A such that ξ1ξ5ξ3. On the other hand as ξ3f-1[ξ4], one can get ξ1ξ5f-1[ξ4] which implies f-1[ξ4] is a neighbourhood of ξ1.

(4) ⇒ (1): Let ξ1 be an nPR-fuzzy subset of A and ξ4 be a neighbourhood of f[ξ1]. From the hypothesis, we have f-1[ξ4] is a neighbourhood of ξ1. Therefore, there exists an open nPR-fuzzy subset ξ5 of A such that ξ1ξ5f-1[ξ4] which implies f[ξ5]f[f-1[ξ4]]ξ4. Moreover, as ξ5 is open it is a neighbourhood of ξ1. Hence, f is nPR-fuzzy continuous.

### Theorem 3.36

Let (A,I1) and (B,I2) be two nPR-FTSs. A function f:AB is nPR-fuzzy continuous if and only if for each open nPR-fuzzy subset ξ2 of B we have f-1[ξ2] is an open nPR-fuzzy subset of A.

Proof

Assume that f is nPR-fuzzy continuous. Let ξ2 be an open nPR-fuzzy subset of B and ξ1f-1[ξ2]. Then, we get f[ξ1]ξ2. Since ξ2 is open nPR-fuzzy, then by Theorem 3.7, there exists a neighbourhood ξ4 of f[ξ1] such that ξ4ξ2. Thus, nPR-fuzzy continuity of f and (4) of Theorem 3.35 imply that f-1[ξ4] is a neighbourhood of ξ1. On the other hand from (1) of Theorem 3.32 we have f-1[ξ4]f-1[ξ2]. Therefore, f-1[ξ2] is a neighbourhood of ξ1 as well. As ξ1 is an arbitrary subset of f-1[ξ2], then by Theorem 3.7, the nPR-fuzzy subset f-1[ξ2] is open.

Conversely, let ξ1 be an nPR-fuzzy subset of A and ξ4 be a neighbourhood of f[ξ1]. Then, there exists an open nPR-fuzzy subset ξ6 of B such that f[ξ1]ξ6ξ4. Now, from the hypothesis f-1[ξ6] is open. On the other hand, we can write ξ1f-1[f[ξ1]]f-1[ξ6]f-1[ξ4]. Hence, f-1[ξ4] is a neighbourhood of ξ1. Then by Theorem 3.35 (4), f is nPR-fuzzy continuous.

### Proposition 3.37

Let (B,I*) be an nPR-FTS and f:AB be a function such that I1 and I2 are two nPR-fuzzy topologies on A. If I2I1 and f:(A,I2)(B,I*) is nPR-fuzzy continuous, then f:(A,I1)(B,I*) is nPR-fuzzy continuous.

Proof

From Theorem 3.36 and the fact I2I1.

We build the following two examples such that the first one provides an 6PR-fuzzy continuous map, whereas the second one presents a fuzzy map that is not 6PR-fuzzy continuous.

### Example 3.38

Consider A={w1,w2} with the 6PR-FT I2={1A,0A,ξ1} and B={b1,b2} with the 6PR-FT I2={1B,0B,ξ2}, where ξ1={w1,0.61,0.71,w2,0.73,0.31} and ξ2 = {〈b1, 0.73, 0.31〉, 〈b2, 0.61, 0.71〉}.

Let f:AB defined as follows:

f(w)={b2,if w=w1,b1,if w=m2.

Since 1B,0B and ξ2 are open 6PR-fuzzy subsets of B, then f-1[1B]={w1,1,0,w2,1,0},f-1[0B]={w1,0,1,w2,0,1} and f-1[ξ2]={w1,0.61,0.71,w2,0.73,0.31} are open 6PR-fuzzy subsets of A. Thus, f is 6PR-fuzzy continuous.

### Example 3.39

Consider A={w1,w2} with the 6PR-FT I1={1A,0A} and B={b1,b2} with the 6PR-FT I2={1B,0B,ξ2}, where ξ2 = {〈b1, 0.67, 0.54〉, 〈b2, 0.49, 0.89〉}.

Let f:AB defined as follows:

f(w)={b1,if w=w1,b2,if w=m2.

Since ξ2 is open 6PR-fuzzy subset of B, but f-1[ξ2]={w1,0.67,0.54,w2,0.49,0.89} is not open 6PR-fuzzy subset of A. Thus, f is not 6PR-fuzzy continuous.

### Corollary 3.40

A function f:AB is nPR-fuzzy continuous if and only if for each closed nPR-fuzzy subset ξ2 of B we have f-1[ξ2] is a closed nPR-fuzzy subset of A.

Proof

Assume that ξ2 is a closed nPR-fuzzy subset of B, then (ξ2)c is an open nPR-fuzzy set in B. Since f is nPR-fuzzy continuous, then from Theorem 3.36, we get f-1[(ξ2)c] is an open nPR-fuzzy subset of A. On the other hand, from Theorem 3.31 (1) we know that f-1[(ξ2)c]=(f-1[ξ2]c). Then (f-1[ξ2])c is an open nPR-fuzzy set. Thus, f-1[ξ2] is a closed nPR-fuzzy subset of A.

Conversely, let ξ be an open nPR-fuzzy subset of B, then ξc is a closed nPR-fuzzy set in B. By hypothesis, f-1[ξc] is a closed nPR-fuzzy subset of A and hence f-1[ξc]=(f-1[ξ])c is closed nPR-fuzzy. Therefore, f-1[ξ] is an open nPR-fuzzy subset of A and by Theorem 3.36 we get that f is nPR-fuzzy continuous.

### Corollary 3.41

The following are equivalent to each other:

• f:(A,I1)(B,I2) is nPR-fuzzy continuous.

• cl(f-1[ξ2])f-1[cl(ξ2)] for each nPR-fuzzy set ξ2 in B.

• f-1[int(ξ2)]int(f-1[ξ2]) for each nPR-fuzzy set ξ2 in B.

Proof

They can be easily proved using Remark 3.9, Theorems 3.11, 3.13, 3.31, 3.32, 3.36 and Corollary 3.40.

Now, we prove that an nPR-FT can be established on A, whenever an nPR-FT on B and a function f:AB are given. This topology is the coarsest topology on A.

### Theorem 3.42

Let (B,I) be an nPR-FTS and f:AB be a function. Then, there exists a coarsest nPR-FT I1 over A such that f is nPR-fuzzy continuous.

Proof

Let us define a class of nPR-fuzzy subsets I1 of A by I1={f-1[ξ4]:ξ4I}.

We prove that I1 is the coarsest nPR-FT over A such that f is continuous.

• We can write for any wA that

ϖf-1[0B](w)=ϖ0B(f(w))=0=ϖ0A(w).

Similarly, we immediately have ωf-1[0B](w)=ω0A(w) for any wA which implies f-1[0B]=0A. Now, as 0BI we have 0A=f-1[0B]I1. In similar manner, it is easy to see that 1A=f-1[1B]I1.

• Assume that ξ1,ξ2I1. Then, for i = 1, 2 there exists ξ2iI such that f-1[ξ2i]=ξi which implies ϖf-1[ξ2i]=ϖξi and ωf-1[ξ2i]=ωξi. Thus, we obtain for any wA that

ϖξ1ξ2(w)=min{ϖξ1(w),ϖξ2(w)}=min{ϖf-1[ξ21](w),ϖf-1[ξ22](w)}=min{ϖξ21(f(w)),ϖξ22(f(w))}=ϖξ21ξ22(f(w))=ϖf-1[ξ21ξ22](w).

Similarly, it is not difficult to see that ωξ1ξ2=ωf-1[ξ21ξ22]. Hence, we get ξ1ξ2I1.

• Assume that {ξi}iI is an arbitrary sub-family of I1. Then for any iI, there exists ξ2iI1 such that f-1[ξ2i]=ξi which implies ϖf-1[ξ2i]=ϖξi and ϖf-1[ξ2i]=ωξi. Therefore, one can get for any wA that

ϖiIξi(w)=supiIϖξi(w)=supiIϖf-1[ξ2i](w)=supiIϖξ2i(f(w))=ϖiIξ2i(f(w))=ϖf-1[iIξ2i](w).

On the other hand, it is easy to see that ωiIξi=ωf-1[iIξ2i]. Thus, we have iIξiI1.

Hence, I1 is an nPR-FT on A, and from Theorem 3.36, f is nPR-fuzzy continuous. Now, we prove that I1 is the coarsest nPR-FT over A such that f is nPR-fuzzy continuous. Let I2I1 be an nPR-FT over A such that f is nPR-fuzzy continuous. If ξ2I1 then there exists ξ4I such that f-1[ξ4]=ξ2. Since f is nPR-fuzzy continuous with respect to I2 we have ξ2=f-1[ξ4]I2. Hence, we have I2=I1.

The following corollary says that we can construct an nPR-FT on a given set B whenever there exists an nPR-FTS A and a function f:AB. This topology is the finest topology that makes the function f is nPR-fuzzy continuous.

### Corollary 3.43

Let (A,I*) be an nPR-FTS and f:AB be a function. Then, there exists a finest nPR-FT I** over B such that f is nPR-fuzzy continuous.

Proof

Can be proved similar to Theorem 3.42.

### Definition 3.44

Let Aφ and wA be a fixed element. Suppose ε1 ∈ (0, 1] and ε2 ∈ [0, 1) are two fixed real numbers such that ɛ1n+ɛ21n1. Then, an nPR-fuzzy point χ(ɛ1,ɛ2)w={w,ϖχ(w),ωχ(w)} is defined to be an nPR-fuzzy set of A as follows:

χ(ɛ1,ɛ2)w(b)={(ɛ1,ɛ2),if b=w,(0,1),otherwise,

for bA. In this case, w is called the support of χ(ɛ1,ɛ2)w. An nPR-fuzzy point χ(ɛ1,ɛ2)w is said to belong to an nPR-fuzzy set ξ={w,ϖξ(w),ωξ(w)} of A denoted by χ(ɛ1,ɛ2)wξ if ɛ1ϖξ(w) and ɛ2ϖξ(w).

Two nPR-fuzzy points are said to be distinct if their supports are distinct.

### Remark 3.45

Let ξ1={w,ϖξ1(w),ωξ1(w)} and ɛ2={w,ϖξ2(w),ωξ2(w)} be two nPR-fuzzy sets of A. Then, ξ1ξ2 if and only if χ(ɛ1,ɛ2)wξ1 implies χ(ɛ1,ɛ2)wξ2 for any nPR-fuzzy point χ(ɛ1,ɛ2)w in A.

### Definition 3.46

Let ε1, ε3 ∈ (0, 1], ε2, ε4 ∈ [0, 1) and w,bA. An nPR-FTS (A,I) is said to be:

• T0 if for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b in A, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

ξ6={w,1,0,b,0,1},

or

ξ5={w,0,1,b,1,0}.

• T1 if for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b in A, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

ξ6={w,1,0,b,0,1},

and

ξ5={w,0,1,b,1,0}.

### Example 3.47

Consider A={w1,w2} with the nPR-FT I={1A,0A,ξ1,ξ2}, where ξ1={w1,1,0,w2,0,1} and ξ2={w1,0,1,w2,1,0}. Then, (A,I) is To and T1.

### Corollary 3.48

Let (A,I) be an nPR-FTS. If (A,I) is T1, then (A,I) is To.

Proof

The proof is straightforward from the Definition 3.46.

Here is an example which shows that the converse of above corollary is not true in general.

### Example 3.49

Consider A={w1,w2} with the nPR-FT I={1A,0A,ξ}, where ξ={w1,1,0,w2,0,1}. Then, (A,I) is T0 but not T1 because there exists no open nPR-fuzzy set ξ5 such that ξ5={w1,0,1,w2,1,0}.

### Theorem 3.50

Let (A,I) be an nPR-FTS, ε1, ε3 ∈ (0, 1] and ε2, ε4 ∈ [0, 1). If (A,I) is To, then for each pair of distinct nPR-fuzzy points χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b of A,cl(χ(ɛ1,ɛ2)w)cl(χ(ɛ3,ɛ4)b).

Proof

Let (A,I) be To and χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b be any two distinct nPR-fuzzy points of A. Then, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

ξ6={w,1,0,b,0,1},

or

ξ5={w,0,1,b,1,0}.

Let ξ6={w,1,0,b,0,1} be exists. Then, ξ6c={w,0,1,b,1,0} is a closed nPR-fuzzy set which does not contain χ(ɛ1,ɛ2)w but contains χ(ɛ3,ɛ4)b. Since cl(χ(ɛ3,ɛ4)b) is the smallest closed nPR-fuzzy set containing χ(ɛ3,ɛ4)b, then cl(χ(ɛ3,ɛ4)b)ξ6c and therefore χ(ɛ1,ɛ2)wcl(χ(ɛ3,ɛ4)b). Consequently cl(χ(ɛ1,ɛ2)x)cl(χ(ɛ3,ɛ4)b).

### Theorem 3.51

Let (A,I) be an nPR-FTS. Then, (A,I) is T1 if χ(1,0)w is closed nPR-fuzzy set for every wA.

Proof

Suppose χ(1,0)w is a closed nPR-fuzzy set for every wA. Let χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b be any two distinct nPR-fuzzy points of A, then wb implies χ(1,0)wc and χ(1,0)bc are two open nPR-fuzzy sets such that

χ(1,0)bc={w,1,0,b,0,1},

and

χ(1,0)wc={w,0,1,b,1,0}.

Thus, (A,I) is T1.

In this article, we have proposed a new generalized fuzzy set called “nPR-fuzzy sets” and compared their relationship with other kinds of generalizations of fuzzy sets. Moreover, some operators on nPR-fuzzy sets have been studied and their relationship have been reveled. The concepts of topology, neighborhood, connectedness, and continuous mapping via nPR-fuzzy sets have been investigated. Finally, we have presented the notion of nPR-fuzzy points and introduced separation axioms via nPR-fuzzy topological space.

In future works, more applications of nPR-fuzzy sets may be investigated, and also nPR-fuzzy soft sets may be introduced. In addition, we will try to present some types of weighted aggregated operators over nPR-FSs and study a MCDM methods depending on these operators.

This research has received no external funding.

The authors declare that they have no competing interests.

No data were used to support this study.

Fig. 1.

Grades spaces of some kinds of nPR-fuzzy sets.

Fig. 2.

Comparison of grades space of IFSs, SR-FSs, CR-FSs, 4-ROFSs and 4PR-FSs

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TareqM. Al-shami received the M.S. and Ph.D. degrees in pure mathematics from the Department of Mathematics, Faculty of Science, Mansoura University, Egypt. He is currently an Assistant Professor with the department of Mathematics, Sana’a University, Sana’a, Yemen. He has published more than 136 research articles in international peer-reviewed SCIE and ESCI journals such as Information Sciences, Knowledge-Based Systems, Applied and Computational Mathematics, Computational and Applied Mathematics, Artificial Intelligence Review, AIMS mathematics, Soft Computing journal among others. His research interests include pure mathematics, topology and its extensions, ordered topology, soft set theory, rough set theory and fuzzy set theory with applications in decision-making, medical diagnosis, information measures, and information aggregation. Dr. Tareq received Obada-Prize for postgraduate students in Feb. 2019. Also, he obtained the first rank at Sana’a University -Yemen according to Scopus data of the region of Yemen for the period from 2018 to 2121.

Hariwan Z. Ibrahim is currently working as an assistant professor in the Department of Mathematics at the University of Zakho, Kurdistan Region-Iraq. He received his Ph.D. degree from the same university. His area of interest includes Fuzzy Topology, Topological Algebra, Ditopology, Ideal Topology and Soft Topology. He has published more than 80 research papers in international journals.

Abdelwaheb Mhemdi is a doctor in pure Mathematics. He was awarded a PhD. from the Faculty of Sciences, Tunis Elmanar University. He was an assistant professor from 2011 to 2016 in Tunis Elmanar University and Gafsa University. Since 2016, he has been teaching at Prince Sattam bin Abdulaziz University in Saudi Arabia. Now, he is an associate professor, and he is the head of the department of Mathematics in the College of Sciences and Humanities in Al Aflaj. Mhemdi produced many original articles in general Topology, especially in Separation Axioms, Soft Topological spaces, and Category of Topological Spaces.

Radwan Abu-Gdairi is an assistant professor of Mathematics at Department of Mathematics, Faculty of Science, Zarqa University, Jordan. His research interests are in the areas of pure and applied mathematics including Topology, Fuzzy topology, Rough set theory and it’s applications. He has published research articles in reputed international journals of mathematical sciences. He is referee of some mathematical journals.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 350-365

Published online December 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.4.350

## n Power Root Fuzzy Sets and Its Topology

Tareq M. Al-shami1 , Hariwan Z. Ibrahim2, Abdelwaheb Mhemdi3, and Radwan Abu-Gdairi4

1Department of Mathematics, Sana’a University, Sana’a, Yemen Future University, Egypt
2Department of Mathematics, Faculty of Education, University of Zakho, Zakho, Kurdistan Region-Iraq
3Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia
4Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan

Correspondence to:Abdelwaheb Mhemdi (mhemdiabd@gmail.com)

Received: October 15, 2022; Revised: November 23, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

One of the most useful expansions of fuzzy sets for coping with information uncertainties is the q-rung orthopair fuzzy sets. In such circumstances, in this article, we define a novel extension of fuzzy sets called nth power root fuzzy set (briefly, nPR-fuzzy set) and elucidate their relationship with intuitionistic fuzzy sets, SR-fuzzy sets, CR-fuzzy sets and q-rung orthopair fuzzy sets. Then, we provide the necessary set of operations for nPR-fuzzy sets as well as study their various features. Furthermore, we familiarize the concept of nPR-fuzzy topology and investigate the basic aspects of this topology. In addition, we define separated nPR-fuzzy sets and then present the concept of disconnected nPR-fuzzy sets. Moreover, we study and characterize nPR-fuzzy continuous maps in great depth. Finally, we establish 𝕿0 and 𝕿1 in nPR-fuzzy topologies and discover the links between them.

Keywords: nPR-fuzzy sets, Operations, nPR-fuzzy topology, Separated nPR-FSs, Connected nPR-FS, nPR-fuzzy continuous maps, 𝕿,0, 𝕿,1

### 1. Introduction

Zadeh [39] established the concept of fuzzy sets to deal with imprecise data, and various studies on generalizations of the concept of fuzzy set were undertaken after that. From health sciences to computer science, from physical sciences to arts, and from engineering and humanities to life sciences, it is seen to have numerous applications connected to fuzzy set theory in both theoretical and practical investigations. As examples of theoretical studies, Kirisci [28] discussed triangular fuzzy numbers, and Zarasiz studied algebraic fuzzy structure [40] and fuzzy matrices [41]. Pawlak [32] was the first one who proposed the concept of rough sets, and Molodtsov [31] introduced the concept of soft sets as a generic mathematical tool for dealing with ambiguous objects. These two types of non-crisp sets contribute to coping with uncertain environments in different fields; see, for example [5, 6, 810, 1215]. The merging between fuzzy sets and some doubtfulness techniques such as rough sets and soft sets have been studied; in this regard, Ahmad and Kharal [1] combined fuzzy sets with soft sets, then Cağman et al. [19] showed how fuzzy soft sets applied to optimal choices. Alcantud [2] explored the relationships between the topological structures in fuzzy and soft settings. Al-shami et al. [11] presented generalized umbral for orthopair fuzzy soft sets called (a, b)-Fuzzy soft sets. Investigation of covering-based rough set from fuzzy view was the main goal of some manuscripts; see, for example, [18, 37, 38]. In fuzzy algebra, Gulzar with his coauthors debated some types of fuzzy subgroups [25] as well as displayed interesting applications of complex IFSs in group theory [26]. New hybridizations of fuzzy set theory with some types of soft sets were established in [3, 33].

Atanassov [17] defined intuitionistic fuzzy sets as one of the intriguing generalizations of fuzzy sets with excellent application. Applications of intuitionistic fuzzy sets can be found in a variety of domains, including optimization issues, decisionmaking, and medical diagnostics [2224]. However, in many circumstances, the decision maker may assign degrees of membership and non-membership to a given attribute such that their aggregate is larger than one. As a result, Yager [30] introduced the Pythagorean fuzzy set notion, which is a generalization of intuitionistic fuzzy sets and a more effective tool for solving uncertain issues. Pythagorean fuzzy sets are a type of fuzzy set that can be used to characterize uncertain data more effectively and precisely than intuitionistic fuzzy sets. Senapati et al. [35] explored Fermatean fuzzy sets and introduced fundamental procedures on them. Another form of generalized Pythagorean fuzzy set named (3,2)-Fuzzy sets and SR-Fuzzy sets was defined by [27] and [4], respectively. Yager [29] developed the concept of the q-rung orthopair fuzzy sets and because of their greater range of depicting membership grades, they are more likely to be used in uncertain situations than other types of fuzzy sets. Al-shami et al. [4] presented another kind of fuzzy set called SR-Fuzzy set and discussed their properties in details. Saliha et al. [34] proposed CR-fuzzy sets and found the fundamental set of operations for them. Recently, Al-shami [7] has defined a new class of IFSs called “(2,1)-Fuzzy sets” and investigated main properties. He also has generated new weighted aggregated operators using this class and applied to multi-criteria decision-making problems.

One of the interesting themes is studying topology with respect to fuzzy sets. This line of research began by Chang [20] in 1968. He explored the main properties of some topological concepts and ideas via fuzzy set environment. In 1997, Çoker [21] defined the concept of an intuitionistic fuzzy topological space. He defined the main topological concepts via this space such as continuity, compactness, connectedness, and separation axioms. Recently, Turkarslan et al. [36] have familiarized a q-rung orthopair fuzzy topological space and studied some its properties. Recently, Ameen et al. [16] have initiated the concept of infra-fuzzy topological structures and explored main features.

In this paper, we propose a new environment of fuzzy sets so-called “nth power root fuzzy set” and compare it with the other types of fuzzy sets. The fundamental benefit of nth power root fuzzy sets is that they can be used in a wide range of decision-making situations. Then, we propose the set of operations for the nth power root fuzzy set and explore their main features. Also, the concept of topology for nth power root fuzzy sets is investigated. Thereafter, we describe separated and disconnected nth power root fuzzy sets under this topology, and also we study nth power root fuzzy continuous maps in details. Ultimately, we study two types of separation axioms in nth power root fuzzy topology.

### 2. nPR-Fuzzy Sets

In this section, we study the notion of nth power root fuzzy sets, and establish main properties. For computations, we use only three decimal places in whole paper.

### Definition 2.1

Let ℕ be the set of natural numbers and $A$ be a universal set. Then, the nth power root fuzzy set (briefly, nPR-FS) ξ, which is a set of ordered pairs over $A$, is defined as following:

$ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A},$

where $ϖξ:A→[0,1]$ is the degree of membership and $ωξ:A→[0,1]$ is the degree of non-membership functions such that

$0≤(ϖξ(w))n+ωξ(w)n≤1,$

for each $w∈A$ and n ∈ ℕ \ {1}.

Then, there is a degree of indeterminacy of $w∈A$ to ξ defined by

$πξ(w)=1-[(ϖξ(w))n+ωξ(w)n].$

It is clear that $(ϖξ(w))n+ωξ(w)n+πξ(w)=1$. Otherwise, $πξ(w)=0$ whenever $(ϖξ(w))n+ωξ(w)n=1$.

For the sake of simplicity, we shall mention the symbol ξ = (ω̄ξ, ωξ) for the nPR-FS $ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A}$. We noticed that 2PR-fuzzy is called SR-FS in [?], and 3PR-fuzzy is called CR-FS in [34]. If $0≤ϖξ(w)+ωξ(w)≤1$ (resp. $0≤ϖξ(w)n+ωξ(w)n≤1$), then ξ = (ω̄ξ, ωξ) is called intuitionistic fuzzy set (IFS) [17](resp. n-rung orthopair fuzzy set (n-ROFS) [29]).

The spaces of some kinds of nPR-fuzzy membership grades are displayed in Figure 1.

### Remark 2.2

From Figure 2, we get that

• the space of 4-rung orthopair fuzzy membership grades is larger than the space of 4PR-fuzzy membership grades.

• ξ = (ω̄ξ ≈ 0.724, ωξ ≈ 0.276) is a point of intersection between 4PR-fuzzy and intuitionistic fuzzy sets.

• for ω̄ξ ∈ (0, 0.724) and ωξ ∈ (0.276, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of intuitionistic membership grades.

• for ω̄ξ ∈ (0.724, 1) and ωξ ∈ (0, 0.276) the space of 4PR-fuzzy membership grades starts to be smaller than the space of intuitionistic membership grades.

• $ξ=(ϖξ=5-12,ωξ=7-352)$ is a point of intersection between 4PR-fuzzy and SR-fuzzy.

• for $ϖξ∈(0,5-12)$ and $ωξ∈(7-352,1)$ the space of 4PR-fuzzy membership grades starts to be larger than the space of SR-fuzzy membership grades.

• for $ϖξ∈(5-12,1)$ and $ωξ∈(0,7-352)$ the space of 4PR-fuzzy membership grades starts to be smaller than the space of SR-fuzzy membership grades.

• ξ = (ω̄ξ ≈ 0.819, ωξ ≈ 0.091) is a point of intersection between 4PR-fuzzy and CR-fuzzy sets.

• for ω̄ξ ∈ (0, 0.819) and ωξ ∈ (0.091, 1) the space of 4PR-fuzzy membership grades starts to be larger than the space of CR-fuzzy membership grades.

• for ω̄ξ ∈ (0.819, 1) and ωξ ∈ (0, 0.091) the space of 4PR-fuzzy membership grades starts to be smaller than the space of CR-fuzzy membership grades.

### Remark 2.3

It is clear that for any nPR-FS ξ = (ω̄ξ, ωξ) we have $0≤ϖξn+ωξn≤ϖξn+ωξn≤1$, then ξ is an n-rung orthopair fuzzy set. Therefore, every nPR-fuzzy set is an n-rung orthopair fuzzy set.

### Definition 2.4

Let ξ = (ω̄ξ, ωξ), ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be three nPR-fuzzy sets (nPR-FSs), then

• $ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})$.

• $ξ1∪ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})$.

• $ξc=(ωξn2,(ϖξ)n2)$.

### Example 2.5

Suppose that ξ1 = (0.6, 0.5) and ξ2 = (0.5, 0.6) are both 4PR-FSs for $A={w}$. Then,

• $ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{0.6,0.5},max{0.5,0.6})=(0.5,0.6)$.

• $ξ1∪ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})=(max{0.6,0.5},min{0.5,0.6})=(0.6,0.5)$.

• $ξ1c≈(0.958,0.000)$.

### Theorem 2.6

If ξ = (ω̄ξ, ωξ) is a nPR-FS, then $ξc$ is also a nPR-FS and $(ξc)c=ξ$.

Proof

Since $0≤ϖξn≤1,0≤ωξn≤1$ and $0≤(ϖξ)n+ωξn≤1$, then

$0≤(ωξn2)n+(ϖξ)n2n=(ϖξ)n+ωξn≤1,$

and hence

$0≤(ωξn2)n+(ϖξ)n2n≤1.$

Thus, $ξc$ is a nPR-FS and it is obvious that $(ξc)c=(ωξn2,(ϖξ)n2)c=(ϖξ,ωξ)$.

### Theorem 2.7

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• ξ1ξ2 = ξ2ξ1.

• ξ1ξ2 = ξ2ξ1.

Proof

From Definition 2.4, we can obtain:

• $ξ1∩ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})=(min{ϖξ2,ϖξ1},max{ωξ2,ωξ1})=ξ2∩ξ1$.

• The proof is similar to (1).

### Theorem 2.8

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• (ξ1ξ2) ∪ ξ2 = ξ2.

• (ξ1ξ2) ∩ ξ2 = ξ2.

Proof

From Definition 2.4, we can obtain:

• $(ξ1∩ξ2)∪ξ2=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})∪(ϖξ2,ωξ2)=(max{min{ϖξ1,ϖξ2},ϖξ2},min{max{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.$

• $(ξ1∪ξ2)∩ξ2=(max{ϖξ1,ϖξ2},min{ωξ1,ωξ2})∩(ϖξ2,ωξ2)=(min{max{ϖξ1,ϖξ2},ϖξ2},max{min{ωξ1,ωξ2},ωξ2})=(ϖξ2,ωξ2)=ξ2.$

### Theorem 2.9

Let ξ1 = (ω̄ξ1, ωξ1), ξ2 = (ω̄ξ2, ωξ2) and ξ3 = (ω̄ξ3, ωξ3) be three nPR-FSs, then

• ξ1 ∩ (ξ2ξ3) = (ξ1ξ2) ∩ ξ3.

• ξ1 ∪ (ξ2ξ3) = (ξ1ξ2) ∪ ξ3.

Proof

For the three nPR-FSs ξ1, ξ2 and ξ3, according to Definition 2.4, we can obtain

• 1. $ξ1∩(ξ2∩ξ3)=(ϖξ1,ωξ1)∩(min{ϖξ2,ϖξ3},max{ωξ2,ωξ3})=(min{ϖξ1,min{ϖξ2,ϖξ3}},max{ωξ1,max{ωξ2,ωξ3}})=(min{min{ϖξ1,ϖξ2},ϖξ3},max{max{ωξ1,ωξ2},ωξ3})=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})∩(ϖξ3,ωξ3)=(ξ1∩ξ2)∩ξ3.$

• 2. The proof is similar to (1).

### Theorem 2.10

Let ξ1 = (ω̄ξ1, ωξ1), ξ2 = (ω̄ξ2, ωξ2) and ξ3 = (ω̄ξ3, ωξ3) be three nPR-FSs, then

• (ξ1ξ2) ∩ ξ3 = (ξ1ξ3) ∪ (ξ2ξ3).

• (ξ1ξ2) ∪ ξ3 = (ξ1ξ3) ∩ (ξ2ξ3).

Proof

It can be proved following similar arguments given to the proof of its counterpart in [7].

### Theorem 2.11

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• $(ξ1∩ξ2)c=ξ1c∪ξ2c$.

• $(ξ1∪ξ2)c=ξ1c∩ξ2c$.

Proof

For the two nPR-FSs ξ1 and ξ2, according to Definition 2.4, we can obtain:

• 1. $(ξ1∩ξ2)c=(min{ϖξ1,ϖξ2},max{ωξ1,ωξ2})c=(max{ωξ1n2,ωξ2n2},min{(ϖξ1)n2,(ϖξ2)n2})=(ωξ1n2,(ϖξ1)n2)∪(ωξ2n2,(ϖξ2)n2)=ξ1c∪ξ2c.$

• 2. The proof is similar to (1).

### Definition 2.12

Let ξ1 = (ω̄ξ1, ωξ1) and ξ2 = (ω̄ξ2, ωξ2) be two nPR-FSs, then

• ξ1 = ξ2 if and only if ω̄ξ1 = ω̄ξ2 and ωξ1 = ωξ2.

• ξ1ξ2 if and only if ω̄ξ1ω̄ξ2 and ωξ1ωξ2.

• ξ2ξ1 or ξ1ξ2 if ξ1ξ2.

### Example 2.13

Let $A={w1,w2}$, then

• ξ1 = ξ2 for two 4PR-FSs $ξ1={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉}$ and $ξ2={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉}$.

• ξ2ξ1 and ξ2ξ1 for two 4PR-FSs $ξ1={〈w1,0.62,0.51〉,〈w2,0.45,0.76〉}$ and $ξ2={〈w1,0.61,0.52〉,〈w2,0.44,0.77〉}$.

### 3. Some Topological Concepts via nPR-Fuzzy Topology

In this section, we define a new class of fuzzy topology using nPR-fuzzy subsets. We initiate the main concepts via this class such as connectedness, continuity and separation axioms. We describe these concepts and illustrate the relations between them with aid of counterexamples.

### 3.1 nPR-Fuzzy Topology

Definition 3.1

Let $ℑ$ be a family of nPR-fuzzy subsets of a non-empty set $A$. If

• $1A,0A∈ℑ$ where $1A=(1,0)$ and$0A=(0,1)$,

• $ξ1∩ξ2∈ℑ$, for any ξ1, $ξ2∈ℑ$,

• $∪i∈Iξi∈ℑ$, for any ${ξi}i∈N⊂ℑ$,

then $ℑ$ is called an nPR-fuzzy topology (briefly, nPR-FT) on $A$ and ($A,ℑ$) is an nPR-fuzzy topological space (briefly, nPR-FTS). Each member of $ℑ$ is called an open nPR-FS. We call ξ a closed nPR-FS if its complement is an open nPR-FS.

Now, we give an example of 7PR-FTS.

Example 3.2

Let $A={w1,w2}$ and ξi be 7PR-fuzzy subsets of $A$ such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2, 3, 4, 5. If

$ϖξ1(w1)=0.52, ωξ1(w1)=0.72,ϖξ1(w2)=0.61, ωξ1(w2)=0.72,ϖξ2(w1)=0.55, ωξ2(w1)=0.68,ϖξ2(w2)=0.62, ωξ2(w2)=0.73,ϖξ3(w1)=0.51, ωξ3(w1)=0.73,ϖξ3(w2)=0.59, ωξ3(w2)=0.74,ϖξ4(w1)=0.55, ωξ4(w1)=0.68,ϖξ4(w2)=0.62, ωξ4(w2)=0.72,ϖξ5(w1)=0.52, ωξ5(w1)=0.72,ϖξ5(w2)=0.61, ωξ5(w2)=0.73,$

then the family of 7PR-fuzzy sets $ℑ={1A,0A,ξ1,ξ2,ξ3,ξ4,ξ5}$ is 7PR-FT on $A$.

Remark 3.3

Every nPR-FT is an n-rung orthopair fuzzy topology because every nPR-fuzzy subset of a set can be considered as an n-rung orthopair fuzzy subset. The next example elaborates that n-rung orthopair fuzzy topological space need not be an nPR-FTS.

Example 3.4

Let $A={w1,w2}$ and ξi be 2-rung orthopair fuzzy subsets of $A$ such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2. If

$ϖξ1(w1)=0.96, ωξ1(w1)=0.25,ϖξ1(w2)=0.92, ωξ1(w2)=0.35,ϖξ2(w1)=0.95, ωξ2(w1)=0.25,ϖξ2(w2)=0.91, ωξ2(w2)=0.35,$

then $ℑ={1A,0A,ξ1,ξ2}$ is 2-rung orthopair fuzzy topology but $ℑ$ is not 2PR-FT.

Remark 3.5
• We call $ℑ={1A,0A}$ the indiscrete nPR-FT on $A$.

• If $ℑ$ contains all nPR-fuzzy subsets, then we call $ℑ$ the discrete nPR-FT on $A$.

Definition 3.6

Let ξ1 and ξ2 be two nPR-fuzzy subsets in an nPR-FTS ($A,ℑ$). Then, ξ2 is called a neighbourhood of ξ1 if there exists an open nPR-fuzzy subset ξ3 such that ξ1ξ3ξ2.

Theorem 3.7

An nPR-fuzzy set ξ1 is open in an nPR-FTS ($A,ℑ$) if and only if it contains a neighbourhood of its each subset.

Proof

Let ξ1 be an open nPR-fuzzy set and ξ2 be an nPR-fuzzy set such that ξ2ξ1. Since ξ2ξ1ξ1 and ξ1 is an open nPR-fuzzy set, then ξ1 is a neighborhood of ξ2. On the other hand, assume that ξ1 is a neighborhood of its each subset. From Definition 3.6, for arbitrary ξ2ξ1 there exists an open nPR-fuzzy set ξξ2 such that ξ2ξξ2ξ1. Thus, we have ξ1 ⊂ ∪ξ2ξ1ξξ2 and since for all ξ2ξ1 and ξξ2ξ1, we get ∪ξ2ξ1ξξ2ξ1. Consequently, we obtain ξ1 = ∪ξ2ξ1ξξ2 which implies ξ1 is an open nPR-fuzzy set.

### Definition 3.8

Let ($A,ℑ$) be an nPR-FTS and ξ = (ω̄ξ, ωξ) be an nPR-FS in $A$. Then, the nPR-fuzzy interior and nPR-fuzzy closure of ξ are, respectively, defined by

• $int(ξ)=∪{ξ2:ξ2 is open nPR-FS in A and ξ2⊂ξ}$.

• $cl(ξ)=∩{ξ1:ξ1 is closed nPR-FS in A and ξ⊂ξ1}$.

### Remark 3.9

Let ( $A,ℑ$) be an nPR-FTS and ξ be any nPR-FS in $A$. Then,

• $int(ξ)$ is an open nPR-FS.

• $cl(ξ)$ is a closed nPR-FS.

• $int(1A)=cl(1A)=1A$ and $int(0A)=cl(0A)=0A$.

### Example 3.10

Consider the 7PR-FTS ($A,ℑ$) in Example 3.2. If $ξ={〈w1,0.53,0.73〉,〈w2,0.63,0.74〉}$, then $int(ξ)=ξ3$ and $cl(ξ)=1A$.

### Theorem 3.11

Let ($A,ℑ$) be an nPR-FTS and ξ1, ξ2 be two nPR-FSs in $A$. Then,

• $int(ξ1)⊂ξ1⊂cl(ξ1)$.

• $int(ξ1)⊂int(ξ2)$ and $cl(ξ1)⊂cl(ξ2)$ if ξ1ξ2.

• ξ1 is an open nPR-FS if and only if $ξ1=int(ξ1)$.

• ξ1 is a closed nPR-FS if and only if $ξ1=cl(ξ1)$.

Proof

(1) and (2) are obvious.

(3) and (4) follow from Definition 3.8.

### Theorem 3.12

Let ($A,ℑ$) be an nPR-FTS and ξ1, ξ2 be two nPR-FSs in $A$. Then,

• $int(ξ1)∪int(ξ2)⊂int(ξ1∪ξ2)$.

• $cl(ξ1∩ξ2)⊂cl(ξ1)∩cl(ξ2)$.

• $int(ξ1∩ξ2)=int(ξ1)∩int(ξ2)$.

• $cl(ξ1)∪cl(ξ2)=cl(ξ1∪ξ2)$.

Proof
• Since ξ1ξ1ξ2 and ξ2ξ1ξ2, then by Theorem 3.11 (2), we have $int(ξ1)⊂int(ξ1∪ξ2)$ and $int(ξ2)⊂int(ξ1∪ξ2)$. Therefore, $int(ξ1)∪int(ξ2)⊂int(ξ1∪ξ2)∪int(ξ1∪ξ2)=int(ξ1∪ξ2)$.

• Since ξ1ξ2ξ1 and ξ1ξ2ξ2, then by Theorem 3.11 (2), we have $cl(ξ1∩ξ2)⊂cl(ξ1)$ and $cl(ξ1∩ξ2)⊂cl(ξ2)$. Therefore, $cl(ξ1∩ξ2)=cl(ξ1∩ξ2)∩cl(ξ1∩ξ2)⊂cl(ξ1)∩cl(ξ2)$.

• Since $int(ξ1∩ξ2)⊂int(ξ1)$ and $int(ξ1∩ξ2)⊂int(ξ2)$, we obtain $int(ξ1∩ξ2)⊂int(ξ1)∩int(ξ2)$. On the other hand, from the facts $int(ξ1)⊂ξ1$ and $int(ξ2)⊂ξ2$ we have $int(ξ1)∩int(ξ2)⊂ξ1∩ξ2$ and $int(ξ1)∩int(ξ2)∈ℑ$; we see that $int(ξ1)∩int(ξ2)⊂int(ξ1∩ξ2)$, and hence $int(ξ1∩ξ2)=int(ξ1)∩int(ξ2)$.

• Can be proved similar to (3).

### Theorem 3.13

Let ($A,ℑ$) be an nPR-FTS and ξ be nPR-FS in $A$. Then,

• $cl(ξc)=int(ξ)c$.

• $cl(ξc)c=int(ξ)$.

• $int(ξc)c=cl(ξ)$.

• $int(ξc)=cl(ξ)c$.

Proof

Let $ξ={〈w,ϖξ(w),ωξ(w)〉:w∈A}$ and suppose that the family of open nPR-fuzzy sets contained in ξ are indexed by the family {$〈w,ϖξi(w),ωξi(w)〉:i∈J$}. Then, $int(ξ)={〈w,⋁ϖξi(w),⋀ωξi(w)〉}$ and hence

• $int(ξ)c={〈w,⋀ωξi(w)n2,(⋁ϖξi(w))n2〉}$. Since $ξc={〈w,ωξ(w)n2,(ϖξ(w))n2〉}$ and ω̄ξiω̄ξ, ωξiωξ for each iJ, so we obtain that $〈w,ωξi(w)n2,(ϖξi(w))n2〉$ is the family of closed nPR-fuzzy sets containing $ξc$, that is,

$cl(ξc)={〈w,⋀ωξi(w)n2,(⋁ϖξi(w))n2〉}.$

Thus, $cl(ξc)=int(ξ)c$.

• ${〈w,ωξi(w)n2,(ϖξi(w))n2〉}$ is the family of closed nPR-fuzzy sets containing $ξc$, so,

$cl(ξc)={〈m,⋀ωξi(m)n2,(⋁ϖξi(w))n2〉}$

and

$cl(ξc)c={〈m, (⋁ϖξi(m))n2n2,(⋀ωξi(m)n2)n2〉}={〈m,⋁ϖξi(m),⋀ωξi(m)〉}=int(ξ).$

• from (1), we can get$cl(ξ)=int(ξc)c$.

• from (2), we can get $cl(ξ)c=int(ξc)$.

### Definition 3.14

Two nPR-FSs ξ1 and ξ2 of an nPR-FTS ($A,ℑ$) are called separated nPR-FSs if $cl(ξ1)⋀ξ2=0A$ and $ξ1⋁cl(ξ2)=0A$.

If ξ1 and ξ2 are separated nPR-FSs, then $ξ1⋀ξ2=0A$. The converse may not be true in general, as it is shown in the following example.

### Example 3.15

Consider the 7PR-FTS ($A,ℑ$) in Example 3.2. If $ξ1={〈w1,0,1〉,〈w2,0.64,0.71〉}$ and $ξ2={〈w1,0.52,0.87〉,〈w2,0,1〉}$, then $ξ1∩ξ2=0A$ but ξ1 and ξ2 are not separated nPR-FSs.

### Corollary 3.16

If ξ1 and ξ2 are two nPR-FSs in an nPR-FTS ($A,ℑ$) such that $ξ1∩ξ2=0A$, then $ξ1⊂ξ2c$.

Proof

From$ξ1∩ξ2=0A$, we get $min{ϖξ1,ϖξ2}=0$ and $max{ωξ1,ωξ2}=1$. If the inequality $ϖξ1≤ωξ2n2$ were not true, then there would exist $wo∈A$ such that $ϖξ1(wo)>ωξ2(wo)n2$, then, since $ϖξ1(wo)>0$ and hence, $ωξ1(wo)<1$, we obtain $ωξ2(wo)=1$ which is an obvious contradiction. Similarly, if the inequality ωξ1 ≥ (ω̄ξ2)n2 were not true, then there would exist $woo∈A$ such that $ωξ1(woo)<(ϖξ2(woo))n2$, then, since $ϖξ2(woo)>0$ and hence, $ωξ2(woo)<1$, we obtain $ωξ1(woo)=1$which is an obvious contradiction. So, we get ωξ1 ≥ (ω̄ξ2 )n2, and thus $ξ1⊂ξ2c$.

### Theorem 3.17

If $ξ1≠0A$ and $ξ2≠0A$ are any two nPR-FSs in an nPR-FTS ($A,ℑ$), then the following statements are true:

• If ξ1 and ξ2 are separated nPR-FSs, ξ11ξ1 and ξ21ξ2, then ξ11 and ξ21 are also separated nPR-FSs.

• If $ξ1∩ξ2=0A$ such that ξ1 and ξ2 are both closed nPR-FSs, then ξ1 and ξ2 are separated nPR-FSs.

Proof
• Since ξ11ξ1, then $cl(ξ11)⊂cl(ξ1)$. Then, $ξ2∩cl(ξ1)=0A$ implies $ξ21∩cl(ξ1)=0A$ and so $ξ21∩cl(ξ11)=0A$. Similarly $ξ11∩cl(ξ21)=0A$. Hence, ξ11 and ξ21 are separated nPR-FSs.

• Since $ξ1=cl(ξ1),ξ2=cl(ξ2)$ and $ξ1∩ξ2=0A$, then $cl(ξ1)∩ξ2=0A$ and $cl(ξ2)∩ξ1=0A$. Hence, ξ1 and ξ2 are separated nPR-FSs.

### Remark 3.18

If $ξ1∩ξ2=0A$ such that ξ1 and ξ2 are both open nPR-FSs in an nPR-FTS ($A,ℑ$), then ξ1 and ξ2 are not separated nPR-FSs in general.

### Example 3.19

Let $A={w1,w2}$ and ξi be 4PR-fuzzy subsets of $A$ such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2, 3. If

$ϖξ1(w1)=0, ωξ1(w1)=1,ϖξ1(w2)=0.25, ωξ1(w2)=0.86,ϖξ2(w1)=0.26, ωξ2(w1)=0.85,ϖξ2(w2)=0, ωξ2(w2)=1,ϖξ3(w1)=0.26, ωξ3(w1)=0.85,ϖξ3(w2)=0.25, ωξ3(w2)=0.86,$

then $ℑ={1A,0A,ξ1,ξ2,ξ3}$ is 4PR-FT and $ξ1∩ξ2=0A$, but ξ1 and ξ2 are not separated 4PR-FSs.

### Corollary 3.20

If ξ1 and ξ2 are separated nPR-FSs in an nPR-FTS ($A,ℑ$), then there exist two open nPR-FSs ξ3 and ξ4 in ($A,ℑ$) such that ξ1ξ3 and ξ2ξ4.

Proof

Let ξ1 and ξ2 be separated nPR-FSs. Set $ξ4=cl(ξ1)c$ and $ξ3=cl(ξ2)c$, then ξ3 and ξ4 are open nPR-FSs such that ξ1ξ3 and ξ2ξ4.

### Theorem 3.21

Let the following statements in an nPR-FTS ($A,ℑ$),

• $0A$ and $1A$ are the only open nPR-FSs and closed nPR-FSs in ($A,ℑ$).

• there are no two open nPR-FSs ξ1 and ξ2 such that $ξ1≠0A,ξ2≠0A,ξ1∩ξ2=0A$ and $1A=ξ1∪ξ2$.

• there are no two closed nPR-FSs ξ1 and ξ2 such that $ξ1≠0A,ξ2≠0A,ξ1∩ξ2=0A$ and $1A=ξ1∪ξ2$.

• there are no two separated nPR-FSs ξ1 and ξ2 such that $ξ1≠0A,ξ2≠0A$ and $1A≠ξ1∪ξ2$.

Then, (1) ⇒ (2) ⇔ (3) ⇔ (4) hold.

Proof

(1) ⇒ (2): Suppose (2) is false and that $1A=ξ1∪ξ2$, where ξ1 and ξ2 are two open nPR-FSs such that $ξ1≠0A,ξ2≠0A$ and $ξ1∩ξ2=0A$. By Theorems 2.6, 2.11 (2) and Corollary 3.16, we have $ξ1c=ξ2$ and hence ξ2 is open nPR-FS and closed nPR-FS in ($A,ℑ$) such that $1A≠ξ2≠0A$, which shows that (1) is false.

(2) ⇔ (3): This is clear.

(3) ⇒ (4): If (4) is false, then $1A=ξ1∪ξ2$, where ξ1 and ξ2 are two separated nPR-FSs such that $ξ1≠0A$ and $ξ2≠0A$. By Theorems 2.11 (2), we have $0A=1Ac=(ξ1∪ξ2)c=ξ1c∩ξ2c$. By Corollary 3.16 and Theorem 2.6, we get $ξ1c⊂(ξ2c)c=ξ2$. Since $cl(ξ2)∩ξ1=0A$, we conclude that $cl(ξ2)⊂ξ2$ so ξ2 is closed nPR-FS. Similarly, ξ1 must be closed nPR-FS. Therefore, (3) is false.

(4) ⇒ (3): If (3) is false, then $1A=ξ1∪ξ2$, where ξ1 and ξ2 are two closed nPR-FSs such that $ξ1≠0A,ξ2≠0A$ and $ξ1∩ξ2=0A$. Then, ξ1 and ξ2 are separated nPR-FSs by Theorem 3.11 (4). Therefore, (4) is false.

### Definition 3.22

Let ($A,ℑ$) be an nPR-FTS and ξ be an nPR-FS in $A$, then ξ is called disconnected nPR-FS if there are two separated nPR-FSs ξ1 and ξ2 in $A$ such that $ξ1≠0A,ξ2≠0A$ and ξ = ξ1ξ2. If ξ is not disconnected nPR-FS, then ξ is called connected nPR-FS.

### Example 3.23

Let $A={w1,w2}$ and ξi be nPR-fuzzy subsets of $A$ such that ω̄ξi and ωξi are corresponding membership and non-membership functions of ξi for each i = 1, 2. If

$ϖξ1(w1)=0, ωξ1(w1)=1, ϖξ1(w2)=1, ωξ1(w2)=0,ϖξ2(w1)=1, ωξ2(w1)=0, ϖξ2(w2)=0, ωξ2(w2)=1,$

then $ℑ={1A,0A,ξ1,ξ2}$ is nPR-FT and hence $1A$ is disconnected nPR-FS because there are two nPR-FSs ξ1 and ξ2 such that $ξ1≠0A,ξ2≠0A,cl(ξ1)∩ξ2=0A,ξ1∩cl(ξ2)=0A$ and $1A=ξ1∪ξ2$.

### Remark 3.24

An nPR-FS $1A$

is connected in every indiscrete nPR-FTS ($A,ℑ$).

### Remark 3.25

An nPR-FS $1A$ is connected if any (therefore all) of the conditions (2) – (4) in Theorem 3.21 hold.

### Remark 3.26

According to the Definition 3.22 and Remark 3.25, $1A$ is disconnected nPR-FS if we can write $1A=ξ1∪ξ2$, where the following (equivalent) statements are true:

• ξ1 and ξ2 are open nPR-FSs such that $ξ1≠0A,ξ2≠0A$ and $ξ1∩ξ2=0A$.

• ξ1 and ξ2 are closed nPR-FSs such that $ξ1≠0A,ξ2≠0A$ and $ξ1∩ξ2=0A$.

• ξ1 and ξ2 are separated nPR-FSs such that $ξ1≠0A$ and $ξ2≠0A$.

### Corollary 3.27

Let ($A,ℑ$) be an nPR-FTS. Then, $1A$ is connected nPR-FS if $0A$ and $1A$ are the only open nPR-FSs and closed nPR-FSs in ($A,ℑ$).

Proof

Follows from Theorem 3.21 and Remark 3.25.

### Lemma 3.28

Suppose ξ1 and ξ2 are separated nPR-FSs in an nPR-FTS ($A,ℑ$). If ξξ1ξ2 and ξ is connected nPR-FS, then ξξ1 or ξξ2.

Proof

Since ξξ1ξ1 and ξξ2ξ2, then ξξ1 and ξξ2 are separated nPR-FSs and by Theorem 2.10 (1) we have ξ = ξ ∩ (ξ1ξ2) = (ξξ1) ∪ (ξξ2). Since ξ is connected nPR-FS, hence either $ξ∩ξ1=0A$, so ξξ2 or $ξ∩ξ2=0A$, so ξξ1.

### Theorem 3.29

Suppose ξ and ξi (i ∈ ℕ) are connected nPR-FSs in an nPR-FTS ($A,ℑ$) and that for each i, ξi and ξ are not separated nPR-FSs. Then, ξξi is connected nPR-FS.

Proof

Suppose that ξξi = ξ11ξ12, where ξ11 and ξ12 are separated nPR-FSs such that $ξ11≠0A$ and $ξ12≠0A$. By Lemma 3.28, either ξξ11 or ξξ12. Without loss of generality, assume ξξ11. By the same reasoning we conclude that for each i, either ξiξ11 or ξiξ12. If some ξiξ12, then ξ and ξi would be separated nPR-FSs. Thus every ξiξ11 and so ξξiξ11. Therefore, ξ12ξ11ξ12 = ξξiξ11 and hence $ξ12⊂ξ11⊂cl(ξ11)$, so $ξ12=0A$ which is contradiction, therefore ξξi is connected nPR-FS.

### 3.4 nPR-Fuzzy Continuous Maps

In this section we study the continuity of a function defined among two nPR-FTSs.

Let $f:A→B$ be a function. Let ξ1 and ξ2 be nPR-fuzzy subsets of $A$ and $B$, respectively. Then, the membership and non-membership functions of the image of ξ1 denoted by $f[ξ1]$ are, respectively, calculated by

$ϖf[ξ1](b)={supa∈f-1(b)ϖξ1(a),if f-1(b)≠φ,0,otherwise,$

and

$ωf[ξ1](b)={infa∈f-1(b)ωξ1(a),if f-1(b)≠φ,1,otherwise.$

The membership and non-membership functions of pre-image of ξ2 denoted by $f-1[ξ2]$, are, respectively, calculated by $ϖf-1[ξ2](w)=ϖξ2(f(w))$, and $ϖf-1[ξ2](w)=ϖξ2(f(w))$.

### Remark 3.30

To show that $f[ξ1]$ and $f-1[ξ2]$ are nPR-fuzzy sets. Consider $θξ1(a)=(ϖξ1(a))n+(ωξ1(a))1n$ and $b∈B$, then we obtain

$(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=(supa∈f-1(b)ϖξ1(a))n+(infa∈f-1(b)ωξ1(a))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n≤supa∈f-1(b)(1-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n=1,$

whenever $f-1(b)$ is non-empty. On the other hand if $f-1(b)=φ$, then we have $(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=1$. Thus, $f[ξ1]$ is an nPR-fuzzy set of $B$.

Now, let $w∈A$, then $(ϖf-1[ξ2](w))n+(ωf-1[ξ2](w))1n=(ϖξ2(f(w)))n+(ωξ2(f(w)))1n≤1$. Therefore, $f-1[ξ2]$ is an nPR-fuzzy set of $A$.

Following theorems give some properties of image and pre-image.

### Theorem 3.31

Let $f:A→B$ be a function such that ξ1 and ξ2 are nPR-fuzzy subsets of $A$ and $B$, respectively. Then,

• $f-1[ξ2c]=f-1[ξ2]c$.

• $f[ξ1]c⊂f[ξ1c]$.

• $f[f-1[ξ2]]⊂ξ2$.

• $ξ1⊂f-1[f[ξ1]]$.

### Proof

• For any $w∈A$ and for any nPR-fuzzy subset ξ2 of $B$ we get from the definition of the complement that

$ϖf-1[ξ2c](w)=ϖξ2c(f(w))=ωξ2(f(w))n2=ωf-1[ξ2](w)n2=ϖf-1[ξ2]c(w).$

Similarly we have

$ωf-1[ξ2c](w)=ωξ2c(f(w))=(ϖξ2(f(w)))n2=(ϖf-1[ξ2](w))n2=ωf-1[ξ2]c(w).$

Thus, we have $f-1[ξ2c]=f-1[ξ2]c$.

• For any $b∈B$ such that $f(b)=φ$ and for any nPR-fuzzy subset ξ1 of $A$, we can write

$θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)+infa∈f-1(b)(ωξ1(a))1n≤supa∈f-1(b)θξ1(a)-infa∈f-1(b)(ωξ1(a))1n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)θξ1(a).$

Now from (*), we have

$ϖf[ξ1c](b)=supa∈f-1(b)ϖξ1c(a)=supa∈f-1(b)ωξ1(a)n2=supa∈f-1(b)θξ1(a)-(ϖξ1(a))nn≥supa∈f-1(b)θξ1(a)-supa∈f-1(b)(ϖξ1(a))nn≥θf[ξ1](b)-(ϖf[ξ1](b))nn=ωf[ξ1](b)n2=ϖf[ξ1]c(b).$

Hence, we obtain $ϖf[ξ1]c(b)≤ϖf[ξ1c](b)$.

Similarly we have

$θf[ξ1](b)=(ϖf[ξ1](b))n+(ωf[ξ1](b))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(ωξ1(a))1n=supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)(θξ1(a)-(ϖξ1(a))n)≥supa∈f-1(b)(ϖξ1(a))n+infa∈f-1(b)θξ1(a)+infa∈f-1(b)(-(ϖξ1(a))n)=infa∈f-1(b)θξ1(a).$

Now, (**) which implies that

$ωf[ξ1c](b)=infa∈f-1(b)ωξ1c(a)=infa∈f-1(b)(ϖξ1(a))n2=infa∈f-1(b)(θξ1(a)-(ωξ1(a))1n)n≤(infa∈f-1(b)θξ1(a)-infa∈f-1(b)(ωξ1(a))1n)n≤(θf[ξ1](b)-(ωf[ξ1](b))1n)n=(ϖf[ξ1](b))n2=ωf[ξ1]c(b).$

Hence, we obtain $ωf[ξ1c](b)≤ωf[ξ1]c(b)$.

On the other hand, the proof is trivial for each $b∈B$ such that $f(b)=φ$.

Therefore, we obtain $f[ξ1]c⊂f[ξ1c]$.

• For any $b∈B$ such that $f(b)=φ$, we can write

$ϖf[f-1[ξ2]](b)=supa∈f-1(b)ϖf-1[ξ2](a)=supa∈f-1(b)ϖξ2(f(a))≤ϖξ2(b).$

Similarly, we have

$ωf[f-1[ξ2]](b)=infa∈f-1(b)ωf-1[ξ2](a)=infa∈f-1(b)ωξ2(f(a))≥ωξ2(b).$

On the other hand if $f(b)=φ$, then we have $ϖf[f-1[ξ2]](b)=0≤ϖξ2(b)$, and $ωf[f-1[ξ2]](b)=1≥ωξ2(b)$. Therefore, $f[f-1[ξ2]]⊂ξ2$.

• For any $w∈A$, we have $ϖf-1[f[ξ1]](w)=ϖf[ξ1](f(w))=supa∈f-1(f(w))ϖξ1(a)≥ϖξ1(w)$.

Similarly, we have $ωf-1[f[ξ1]](w)≤ωξ1(w)$.

Therefore, $ξ1⊂f-1[f[ξ1]]$.

### Theorem 3.32

Let $f:A→B$ be a function such that ξ1i and ξ2i are nPR-fuzzy subsets of $A$ and $B$, respectively, for i = 1, 2. If

• ξ21ξ22, then $f-1[ξ21]⊂f-1[ξ22]$.

• ξ11ξ12, then $f[ξ11]⊂f[ξ12]$.

Proof
• Assume that ξ21ξ22. Then, for any $w∈A$ we have

$ϖf-1[ξ21](w)=ϖξ21(f(w))≤ϖξ22(f(w))=ϖf-1[ξ22](w).$

Thus, $ϖf-1[ξ21](w)≤ϖf-1[ξ22](w)$. Similarly, we have

$ωf-1[ξ22](w)=ωξ22(f(w))≤ωξ21(f(w))=ωf-1[B1ξ21](w).$

Thus, $ωf-1[ξ21](w)≥ωf-1[ξ22](w)$. Therefore, $f-1[ξ21]⊂f-1[ξ22]$.

• Assume that ξ11ξ12 and $b∈B$. If $f(b)=φ$, then the proof is trivial. Assume that $f(b)≠φ$. Then, we have

$ϖf[ξ11](b)=supa∈f-1(b)ϖξ11(a)≤supa∈f-1(b)ϖξ12(a)=ϖf[ξ12](b),$

and

$ωf[ξ12](b)=infa∈f-1(b)ωξ12(a)≤infa∈f-1(b)ωξ11(a)=ωf[ξ11](b).$

Therefore, $f[ξ11]⊂f[ξ12]$.

The proof of the following result is easy and hence it is omitted.

### Corollary 3.33

Let $f:A→B$ be a function. Then, the following statements are true:

• $f[∪i∈Iξ1i]=∪i∈If[ξ1i]$ for any nPR-fuzzy subset ξ1i of $A$.

• $f-1[∪i∈Iξ2i]=∪i∈If-1[ξ2i]$ for any nPR-fuzzy subset ξ2i of $B$.

• $f[ξ11∩ξ12]⊂f[ξ11]∩f[ξ12]$ for any two nPR-fuzzy subsets ξ11 and ξ12 of $A$.

• $f-1[∩i∈Iξ2i]=∩i∈If-1[ξ2i]$ for any nPR-fuzzy subset ξ2i of $B$.

### Definition 3.34

Let ($A,ℑ1$) and ($B,ℑ2$) be two nPR-FTSs and $f:A→B$ be a function. Then, $f$ is said to be nPR-fuzzy continuous if for any nPR-fuzzy subset ξ1 of $A$ and for any neighbourhood ξ4 of $f[ξ1]$ there exists a neighbourhood ξ3 of ξ1 such that $f[ξ3]⊂ξ4$.

### Theorem 3.35

Let ($A,ℑ1$) and ($B,ℑ2$) be two nPR-FTSs and $f:A→B$ be a function. Then, the following statements are equivalent:

• $f$ is nPR-fuzzy continuous.

• For any nPR-fuzzy subset ξ1 of $A$ and for any neighbourhood ξ4 of $f[ξ1]$, there exists a neighbourhood ξ3 of ξ1 such that for any ξ2ξ3 we have $f[ξ2]⊂ξ4$.

• For any nPR-fuzzy subset ξ1 of $A$ and for any neighbourhood ξ4 of $f[ξ1]$, there exists a neighbourhood ξ3 of ξ1 such that $ξ3⊂f-1[ξ4]$.

• For any nPR-fuzzy subset ξ1 of $A$ and for any neighbourhood ξ4 of $f[ξ1],f-1[ξ4]$ is a neighbourhood of ξ1.

Proof

(1) ⇒ (2): Assume that $f$ is nPR-fuzzy continuous. Let ξ1 be an nPR-fuzzy subset of $A$ and ξ4 be a neighbourhood of $f[ξ1]$. Then by (1), there exists a neighbourhood ξ3 of ξ1 such that $f[ξ3]⊂ξ4$. Now, if ξ2ξ3, then we get $f[ξ2]⊂f[ξ3]⊂ξ4$. (2) ⇒ (3): Let ξ1 be an nPR-fuzzy set of $A$ and ξ4 be a neighbourhood of $f[ξ1]$. From (2), there exists a neighbourhood ξ3 of ξ1 such that for any ξ2ξ3 we have $f[ξ2]⊂ξ4$. Then, we can write $ξ2⊂f-1[f[ξ2]]⊂f-1[ξ4]$. As ξ2 is an arbitrary subset of ξ3, we have $ξ3⊂f-1[ξ4]$.

(3) ⇒ (4): Let ξ1 be an nPR-fuzzy subset of $A$ and ξ4 be a neighbourhood of $f[ξ1]$. Then from (3), there exists a neighbourhood ξ3 of ξ1 such that $ξ3⊂f-1[ξ4]$. Since ξ3 is a neighbourhood of ξ1 there exists an open nPR-fuzzy subset ξ5 of $A$ such that ξ1ξ5ξ3. On the other hand as $ξ3⊂f-1[ξ4]$, one can get $ξ1⊂ξ5⊂f-1[ξ4]$ which implies $f-1[ξ4]$ is a neighbourhood of ξ1.

(4) ⇒ (1): Let ξ1 be an nPR-fuzzy subset of $A$ and ξ4 be a neighbourhood of $f[ξ1]$. From the hypothesis, we have $f-1[ξ4]$ is a neighbourhood of ξ1. Therefore, there exists an open nPR-fuzzy subset ξ5 of $A$ such that $ξ1⊂ξ5⊂f-1[ξ4]$ which implies $f[ξ5]⊂f[f-1[ξ4]]⊂ξ4$. Moreover, as ξ5 is open it is a neighbourhood of ξ1. Hence, $f$ is nPR-fuzzy continuous.

### Theorem 3.36

Let ($A,I1$) and ($B,I2$) be two nPR-FTSs. A function $f:A→B$ is nPR-fuzzy continuous if and only if for each open nPR-fuzzy subset ξ2 of $B$ we have $f-1[ξ2]$ is an open nPR-fuzzy subset of $A$.

Proof

Assume that $f$ is nPR-fuzzy continuous. Let ξ2 be an open nPR-fuzzy subset of $B$ and $ξ1⊂f-1[ξ2]$. Then, we get $f[ξ1]⊂ξ2$. Since ξ2 is open nPR-fuzzy, then by Theorem 3.7, there exists a neighbourhood ξ4 of $f[ξ1]$ such that ξ4ξ2. Thus, nPR-fuzzy continuity of $f$ and (4) of Theorem 3.35 imply that $f-1[ξ4]$ is a neighbourhood of ξ1. On the other hand from (1) of Theorem 3.32 we have $f-1[ξ4]⊂f-1[ξ2]$. Therefore, $f-1[ξ2]$ is a neighbourhood of ξ1 as well. As ξ1 is an arbitrary subset of $f-1[ξ2]$, then by Theorem 3.7, the nPR-fuzzy subset $f-1[ξ2]$ is open.

Conversely, let ξ1 be an nPR-fuzzy subset of $A$ and ξ4 be a neighbourhood of $f[ξ1]$. Then, there exists an open nPR-fuzzy subset ξ6 of $B$ such that $f[ξ1]⊂ξ6⊂ξ4$. Now, from the hypothesis $f-1[ξ6]$ is open. On the other hand, we can write $ξ1⊂f-1[f[ξ1]]⊂f-1[ξ6]⊂f-1[ξ4]$. Hence, $f-1[ξ4]$ is a neighbourhood of ξ1. Then by Theorem 3.35 (4), $f$ is nPR-fuzzy continuous.

### Proposition 3.37

Let ($B,I*$) be an nPR-FTS and $f:A→B$ be a function such that $I1$ and $I2$ are two nPR-fuzzy topologies on $A$. If $I2⊂I1$ and $f:(A,I2)→(B,I*)$ is nPR-fuzzy continuous, then $f:(A,I1)→(B,I*)$ is nPR-fuzzy continuous.

Proof

From Theorem 3.36 and the fact $I2⊂I1$.

We build the following two examples such that the first one provides an 6PR-fuzzy continuous map, whereas the second one presents a fuzzy map that is not 6PR-fuzzy continuous.

### Example 3.38

Consider $A={w1,w2}$ with the 6PR-FT $I2={1A,0A,ξ1}$ and $B={b1,b2}$ with the 6PR-FT $I2={1B,0B,ξ2}$, where $ξ1={〈w1,0.61,0.71〉,〈w2,0.73,0.31〉}$ and ξ2 = {〈b1, 0.73, 0.31〉, 〈b2, 0.61, 0.71〉}.

Let $f:A→B$ defined as follows:

$f(w)={b2,if w=w1,b1,if w=m2.$

Since $1B,0B$ and ξ2 are open 6PR-fuzzy subsets of $B$, then $f-1[1B]={〈w1,1,0〉,〈w2,1,0〉},f-1[0B]={〈w1,0,1〉,〈w2,0,1〉}$ and $f-1[ξ2]={〈w1,0.61,0.71〉,〈w2,0.73,0.31〉}$ are open 6PR-fuzzy subsets of $A$. Thus, $f$ is 6PR-fuzzy continuous.

### Example 3.39

Consider $A={w1,w2}$ with the 6PR-FT $I1={1A,0A}$ and $B={b1,b2}$ with the 6PR-FT $I2={1B,0B,ξ2}$, where ξ2 = {〈b1, 0.67, 0.54〉, 〈b2, 0.49, 0.89〉}.

Let $f:A→B$ defined as follows:

$f(w)={b1,if w=w1,b2,if w=m2.$

Since ξ2 is open 6PR-fuzzy subset of $B$, but $f-1[ξ2]={〈w1,0.67,0.54〉,〈w2,0.49,0.89〉}$ is not open 6PR-fuzzy subset of $A$. Thus, $f$ is not 6PR-fuzzy continuous.

### Corollary 3.40

A function $f:A→B$ is nPR-fuzzy continuous if and only if for each closed nPR-fuzzy subset ξ2 of $B$ we have $f-1[ξ2]$ is a closed nPR-fuzzy subset of $A$.

Proof

Assume that ξ2 is a closed nPR-fuzzy subset of $B$, then $(ξ2)c$ is an open nPR-fuzzy set in $B$. Since $f$ is nPR-fuzzy continuous, then from Theorem 3.36, we get $f-1[(ξ2)c]$ is an open nPR-fuzzy subset of $A$. On the other hand, from Theorem 3.31 (1) we know that $f-1[(ξ2)c]=(f-1[ξ2]c)$. Then $(f-1[ξ2])c$ is an open nPR-fuzzy set. Thus, $f-1[ξ2]$ is a closed nPR-fuzzy subset of $A$.

Conversely, let ξ be an open nPR-fuzzy subset of $B$, then ξc is a closed nPR-fuzzy set in $B$. By hypothesis, $f-1[ξc]$ is a closed nPR-fuzzy subset of $A$ and hence $f-1[ξc]=(f-1[ξ])c$ is closed nPR-fuzzy. Therefore, $f-1[ξ]$ is an open nPR-fuzzy subset of $A$ and by Theorem 3.36 we get that $f$ is nPR-fuzzy continuous.

### Corollary 3.41

The following are equivalent to each other:

• $f:(A,I1)→(B,I2)$ is nPR-fuzzy continuous.

• $cl(f-1[ξ2])⊂f-1[cl(ξ2)]$ for each nPR-fuzzy set ξ2 in $B$.

• $f-1[int(ξ2)]⊂int(f-1[ξ2])$ for each nPR-fuzzy set ξ2 in $B$.

Proof

They can be easily proved using Remark 3.9, Theorems 3.11, 3.13, 3.31, 3.32, 3.36 and Corollary 3.40.

Now, we prove that an nPR-FT can be established on $A$, whenever an nPR-FT on $B$ and a function $f:A→B$ are given. This topology is the coarsest topology on $A$.

### Theorem 3.42

Let ($B,I$) be an nPR-FTS and $f:A→B$ be a function. Then, there exists a coarsest nPR-FT $I1$ over $A$ such that $f$ is nPR-fuzzy continuous.

Proof

Let us define a class of nPR-fuzzy subsets $I1$ of $A$ by $I1={f-1[ξ4]:ξ4∈I}$.

We prove that $I1$ is the coarsest nPR-FT over $A$ such that $f$ is continuous.

• We can write for any $w∈A$ that

$ϖf-1[0B](w)=ϖ0B(f(w))=0=ϖ0A(w).$

Similarly, we immediately have $ωf-1[0B](w)=ω0A(w)$ for any $w∈A$ which implies $f-1[0B]=0A$. Now, as $0B∈I$ we have $0A=f-1[0B]∈I1$. In similar manner, it is easy to see that $1A=f-1[1B]∈I1$.

• Assume that $ξ1,ξ2∈I1$. Then, for i = 1, 2 there exists $ξ2i∈I$ such that $f-1[ξ2i]=ξi$ which implies $ϖf-1[ξ2i]=ϖξi$ and $ωf-1[ξ2i]=ωξi$. Thus, we obtain for any $w∈A$ that

$ϖξ1∩ξ2(w)=min{ϖξ1(w),ϖξ2(w)}=min{ϖf-1[ξ21](w),ϖf-1[ξ22](w)}=min{ϖξ21(f(w)),ϖξ22(f(w))}=ϖξ21∩ξ22(f(w))=ϖf-1[ξ21∩ξ22](w).$

Similarly, it is not difficult to see that $ωξ1∩ξ2=ωf-1[ξ21∩ξ22]$. Hence, we get $ξ1∩ξ2∈I1$.

• Assume that {ξi}iI is an arbitrary sub-family of $I1$. Then for any iI, there exists $ξ2i∈I1$ such that $f-1[ξ2i]=ξi$ which implies $ϖf-1[ξ2i]=ϖξi$ and $ϖf-1[ξ2i]=ωξi$. Therefore, one can get for any $w∈A$ that

$ϖ∪i∈Iξi(w)=supi∈Iϖξi(w)=supi∈Iϖf-1[ξ2i](w)=supi∈Iϖξ2i(f(w))=ϖ∪i∈Iξ2i(f(w))=ϖf-1[∪i∈Iξ2i](w).$

On the other hand, it is easy to see that $ω∪i∈Iξi=ωf-1[∪i∈Iξ2i]$. Thus, we have $∪i∈Iξi∈I1$.

Hence, $I1$ is an nPR-FT on $A$, and from Theorem 3.36, $f$ is nPR-fuzzy continuous. Now, we prove that $I1$ is the coarsest nPR-FT over $A$ such that $f$ is nPR-fuzzy continuous. Let $I2⊂I1$ be an nPR-FT over $A$ such that $f$ is nPR-fuzzy continuous. If $ξ2∈I1$ then there exists $ξ4∈I$ such that $f-1[ξ4]=ξ2$. Since $f$ is nPR-fuzzy continuous with respect to $I2$ we have $ξ2=f-1[ξ4]∈I2$. Hence, we have $I2=I1$.

The following corollary says that we can construct an nPR-FT on a given set $B$ whenever there exists an nPR-FTS $A$ and a function $f:A→B$. This topology is the finest topology that makes the function $f$ is nPR-fuzzy continuous.

### Corollary 3.43

Let ($A,I*$) be an nPR-FTS and $f:A→B$ be a function. Then, there exists a finest nPR-FT $I**$ over $B$ such that $f$ is nPR-fuzzy continuous.

Proof

Can be proved similar to Theorem 3.42.

### Definition 3.44

Let $A≠φ$ and $w∈A$ be a fixed element. Suppose ε1 ∈ (0, 1] and ε2 ∈ [0, 1) are two fixed real numbers such that $ɛ1n+ɛ21n≤1$. Then, an nPR-fuzzy point $χ(ɛ1,ɛ2)w={〈w,ϖχ(w),ωχ(w)〉}$ is defined to be an nPR-fuzzy set of $A$ as follows:

$χ(ɛ1,ɛ2)w(b)={(ɛ1,ɛ2),if b=w,(0,1),otherwise,$

for $b∈A$. In this case, $w$ is called the support of $χ(ɛ1,ɛ2)w$. An nPR-fuzzy point $χ(ɛ1,ɛ2)w$ is said to belong to an nPR-fuzzy set $ξ={〈w,ϖξ(w),ωξ(w)〉}$ of $A$ denoted by $χ(ɛ1,ɛ2)w∈ξ$ if $ɛ1≤ϖξ(w)$ and $ɛ2≤ϖξ(w)$.

Two nPR-fuzzy points are said to be distinct if their supports are distinct.

### Remark 3.45

Let $ξ1={〈w,ϖξ1(w),ωξ1(w)〉}$ and $ɛ2={〈w,ϖξ2(w),ωξ2(w)〉}$ be two nPR-fuzzy sets of $A$. Then, ξ1ξ2 if and only if $χ(ɛ1,ɛ2)w∈ξ1$ implies $χ(ɛ1,ɛ2)w∈ξ2$ for any nPR-fuzzy point $χ(ɛ1,ɛ2)w$ in $A$.

### Definition 3.46

Let ε1, ε3 ∈ (0, 1], ε2, ε4 ∈ [0, 1) and $w,b∈A$. An nPR-FTS ($A,I$) is said to be:

• $T0$ if for each pair of distinct nPR-fuzzy points $χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b$ in $A$, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

$ξ6={〈w,1,0〉,〈b,0,1〉},$

or

$ξ5={〈w,0,1〉,〈b,1,0〉}.$

• $T1$ if for each pair of distinct nPR-fuzzy points $χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b$ in $A$, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

$ξ6={〈w,1,0〉,〈b,0,1〉},$

and

$ξ5={〈w,0,1〉,〈b,1,0〉}.$

### Example 3.47

Consider $A={w1,w2}$ with the nPR-FT $I={1A,0A,ξ1,ξ2}$, where $ξ1={〈w1,1,0〉,〈w2,0,1〉}$ and $ξ2={〈w1,0,1〉,〈w2,1,0〉}$. Then, ($A,I$) is $To$ and $T1$.

### Corollary 3.48

Let ($A,I$) be an nPR-FTS. If ($A,I$) is $T1$, then ($A,I$) is $To$.

Proof

The proof is straightforward from the Definition 3.46.

Here is an example which shows that the converse of above corollary is not true in general.

### Example 3.49

Consider $A={w1,w2}$ with the nPR-FT $I={1A,0A,ξ}$, where $ξ={〈w1,1,0〉,〈w2,0,1〉}$. Then, ($A,I$) is $T0$ but not $T1$ because there exists no open nPR-fuzzy set ξ5 such that $ξ5={〈w1,0,1〉,〈w2,1,0〉}$.

### Theorem 3.50

Let ($A,I$) be an nPR-FTS, ε1, ε3 ∈ (0, 1] and ε2, ε4 ∈ [0, 1). If ($A,I$) is $To$, then for each pair of distinct nPR-fuzzy points $χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b$ of $A,cl(χ(ɛ1,ɛ2)w)≠cl(χ(ɛ3,ɛ4)b)$.

Proof

Let ($A,I$) be $To$ and $χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b$ be any two distinct nPR-fuzzy points of $A$. Then, there exist two open nPR-fuzzy sets ξ6 and ξ5 such that

$ξ6={〈w,1,0〉,〈b,0,1〉},$

or

$ξ5={〈w,0,1〉,〈b,1,0〉}.$

Let $ξ6={〈w,1,0〉,〈b,0,1〉}$ be exists. Then, $ξ6c={〈w,0,1〉,〈b,1,0〉}$ is a closed nPR-fuzzy set which does not contain $χ(ɛ1,ɛ2)w$ but contains $χ(ɛ3,ɛ4)b$. Since $cl(χ(ɛ3,ɛ4)b)$ is the smallest closed nPR-fuzzy set containing $χ(ɛ3,ɛ4)b$, then $cl(χ(ɛ3,ɛ4)b)⊂ξ6c$ and therefore $χ(ɛ1,ɛ2)w∉cl(χ(ɛ3,ɛ4)b)$. Consequently $cl(χ(ɛ1,ɛ2)x)≠cl(χ(ɛ3,ɛ4)b)$.

### Theorem 3.51

Let ($A,I$) be an nPR-FTS. Then, ($A,I$) is $T1$ if $χ(1,0)w$ is closed nPR-fuzzy set for every $w∈A$.

Proof

Suppose $χ(1,0)w$ is a closed nPR-fuzzy set for every $w∈A$. Let $χ(ɛ1,ɛ2)w,χ(ɛ3,ɛ4)b$ be any two distinct nPR-fuzzy points of $A$, then $w≠b$ implies $χ(1,0)wc$ and $χ(1,0)bc$ are two open nPR-fuzzy sets such that

$χ(1,0)bc={〈w,1,0〉,〈b,0,1〉},$

and

$χ(1,0)wc={〈w,0,1〉,〈b,1,0〉}.$

Thus, ($A,I$) is $T1$.

### 4. Conclusions

In this article, we have proposed a new generalized fuzzy set called “nPR-fuzzy sets” and compared their relationship with other kinds of generalizations of fuzzy sets. Moreover, some operators on nPR-fuzzy sets have been studied and their relationship have been reveled. The concepts of topology, neighborhood, connectedness, and continuous mapping via nPR-fuzzy sets have been investigated. Finally, we have presented the notion of nPR-fuzzy points and introduced separation axioms via nPR-fuzzy topological space.

In future works, more applications of nPR-fuzzy sets may be investigated, and also nPR-fuzzy soft sets may be introduced. In addition, we will try to present some types of weighted aggregated operators over nPR-FSs and study a MCDM methods depending on these operators.

### Fig 1.

Figure 1.

Grades spaces of some kinds of nPR-fuzzy sets.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 350-365https://doi.org/10.5391/IJFIS.2022.22.4.350

### Fig 2.

Figure 2.

Comparison of grades space of IFSs, SR-FSs, CR-FSs, 4-ROFSs and 4PR-FSs

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 350-365https://doi.org/10.5391/IJFIS.2022.22.4.350

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