# (3, 2)-Fuzzy Sets and Their Applications to Topology and Optimal Choices

Authors: Hariwan Z. Ibrahim; Tareq M. Al-shami; O. G. Elbarbary
Journal: Computational Intelligence and Neuroscience
 (2021)

---

## Abstract

The purpose of this paper is to define the concept of (3, 2)-fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on (3, 2)-fuzzy sets. (3, 2)-Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of (3, 2)-fuzzy topological space and discuss the master properties of (3, 2)-fuzzy continuous maps. Then, we introduce the concept of (3, 2)-fuzzy points and study some types of separation axioms in (3, 2)-fuzzy topological space. Moreover, we establish the idea of relation in (3, 2)-fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed (3, 2)-fuzzy relation to ascertain the suitability of colleges to applicants.


---

## Body

## 1. Introduction

The concept of fuzzy sets was proposed by Zadeh [1]. The theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. After the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. The integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [2–4].The idea of intuitionistic fuzzy sets suggested by Atanassov [5] is one of the extensions of fuzzy sets with better applicability. Applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multicriteria decision making [6–8]. Yager [9] offered a new fuzzy set called a Pythagorean fuzzy set, which is the generalization of intuitionistic fuzzy sets. Fermatean fuzzy sets were introduced by Senapati and Yager [10], and they also defined basic operations over the Fermatean fuzzy sets.The concept of fuzzy topological spaces was introduced by Chang [11]. He studied the topological concepts like continuity and compactness via fuzzy topological spaces. Then, Lowen [12] presented a new type of fuzzy topological spaces. Çoker [13] subsequently initiated a study of intuitionistic fuzzy topological spaces. Recently, Olgun et al. [14] presented the concept of Pythagorean fuzzy topological spaces and Ibrahim [15] defined the concept of Fermatean fuzzy topological spaces.The main purpose of this paper is to introduce the concept of (3, 2)-fuzzy sets and compare them with the other types of fuzzy sets. We introduce the set of operations for the (3, 2)-fuzzy sets and explore their main features. Following the idea of Chang, we define a topological structure via (3, 2)-fuzzy sets as an extension of fuzzy topological space, intuitionistic fuzzy topological space, and Pythagorean fuzzy topological space. We discuss the main topological concepts in (3, 2)-fuzzy topological spaces such as continuity and compactness. In addition, the concept of relation to (3, 2)-fuzzy sets is investigated. Finally, an improved version of max-min-max composite relation for (3, 2)-fuzzy sets is proposed.


## 2. (3, 2)-Fuzzy Sets

In this section, we initiate the notion of (3, 2)-fuzzy sets and study their relationship with other kinds of fuzzy sets. Then, we furnish some operations to (3, 2)-fuzzy sets.Definition 1.
LetX be a universal set. Then, the (3, 2)-fuzzy set (briefly, (3, 2)-FS) D is defined by the following:(1)D=r,αDr,βDr:r∈X,where αDr:X⟶0,1 is the degree of membership and βDr:X⟶0,1 is the degree of non-membership of r∈X to D, with the condition(2)0≤αDr3+βDr2≤1.
The degree of indeterminacy ofr∈X to D is defined by(3)πDr=1−αDr3+βDr25.
It is clear thatαDr3+βDr2+πDr5=1, and πDr=0 whenever αDr3+βDr2=1. In the interest of simplicity, we shall mention the symbol D=αD,βD for the (3, 2)-FS D=r,αDr,βDr:r∈X.Definition 2.
LetX be a universal set. Then, the intuitionistic fuzzy set (IFS) [5] (resp. Pythagorean fuzzy set (PFS) [9] and Fermatean fuzzy set (FFS) [10]) is defined by the following:(4)K=r,αKr,βKr:r∈X,with the condition 0≤αKr+βKr≤1 (resp. 0≤αKr2+βKr2≤1, 0≤αKr3+βKr3≤1), where αKr:X⟶0,1 is the degree of membership and βKr:X⟶0,1 is the degree of non-membership of every r∈X to K.To illustrate the importance of (3, 2)-FS to extend the grades of membership and non-membership degrees, assume thatαDr=0.9 and βDr=0.5 for X=r. We obtain 0.9+0.5=1.40>1 and 0.92+0.52=1.06>1 which means that D=0.9,0.5 neither follows the condition of IFS nor follows the condition of PFS. On the other hand, 0.93+0.52=0.979<1 which means we can apply the (3, 2)-FS to control it. That is, D=0.9,0.5 is a (3, 2)-FS.Theorem 1.
The set of (3, 2)-fuzzy membership grades is larger than the set of intuitionistic membership grades and Pythagorean membership grades.Proof.
It is well known that for any two numbersr1,r2∈0,1, we have(5)r13≤r12≤r1,r22≤r2.
Then, we get(6)r1+r2≤1⇒r12+r22≤1⇒r13+r22≤1.
Hence, the space of (3, 2)-fuzzy membership grades is larger than the space of intuitionistic membership grades and Pythagorean membership grades. This development can be evidently recognized in Figure1.Figure 1
Comparison of grade space of IFSs, PFSs, and (3, 2)-FSs.Lemma 1.
LetX=rj:j=1,…,k be a universal set and D be (3, 2)-FS. If πDrj=0, then αDrj=βDrj−1βDrj+13.Proof.
Presume thatD is (3, 2)-FS and πDrj=0 for rj∈X; then,(7)αDrj3+βDrj2=1⇒−αDrj3=βDrj2−1⇒−αDrj3=βDrj−1βDrj+1⇒αDrj3=βDrj−1βDrj+1⇒αDrj3=βDrj−1βDrj+1⇒αDrj=βDrj−1βDrj+13.Example 1.
LetD be (3, 2)-FS and r∈X such that βDr=0.82 and πDr=0. Then, αDr=βDr−1βDr+13=−0.181.823=0.32763.Definition 3.
Letδ be a positive real number δ>0. If D1=αD1,βD1 and D2=αD2,βD2 are two (3, 2)-FSs, then their operations are defined as follows:(1)
D1∩D2=minαD1,αD2,maxβD1,βD2.(2)
D1∪D2=maxαD1,αD2,minβD1,βD2.(3)
D1c=βD1,αD1.(4)
δD1=1−1−αD13δ5,βD1δ.(5)
D1δ=αD1δ,1−1−βD12δ5.Remark 1.
We will use supremum “sup” instead of maximum “max” and infimum “inf” instead of minimum “min” if the union and the intersection are infinite.Example 2.
Assume thatD1=αD1=0.9,βD1=0.5 and D2=αD2=0.89,βD2=0.49 are both (3, 2)-FSs. Then,(1)
D1∩D2=minαD1,αD2,maxβD1,βD2=min0.9,0.89,max0.5,0.49=0.89,0.5.(2)
D1∪D2=maxαD1,αD2,minβD1,βD2=max0.9,0.89,min0.5,0.49=0.9,0.49.(3)
D1c=0.5,0.9.(4)
δD1=1−1−αD13δ5,βD1δ=1−1−0.9345,0.54≈0.99892,0.06250, for δ=4.(5)
D1δ=αD1δ,1−1−βD12δ5=0.94,1−1−0.5245≈0.65610,0.92674, for δ=4.Theorem 2.
LetL1=αL1,βL1 and L2=αL2,βL2 be two (3, 2)-FSs; then, the following properties hold:(1)
L1∩L2=L2∩L1.(2)
L1∪L2=L2∪L1.(3)
L1∩L2∪L2=L2.(4)
L1∪L2∩L2=L2.Proof.
From Definition3, we can obtain(1)
L1∩L2=minαL1,αL2,maxβL1,βL2=minαL2,αL1,maxβL2,βL1=L2∩L1.(2)
The proof is similar to (1).(3)
L1∩L2∪L2=minαL1,αL2,maxβL1,βL2∪αL2,βL2=maxminαL1,αL2,αL2,minmaxβL1,βL2,βL2=αL2,βL2=L2.(4)
The proof is similar to (3).Theorem 3.
LetL1=αL1,βL1, L2=αL2,βL2 and L3=αL3,βL3 be three (3, 2)-FSs and δ>0; then,(1)
L1∩L2∩L3=L1∩L2∩L3.(2)
L1∪L2∪L3=L1∪L2∪L3.(3)
δL1∪L2=δL1∪δL2.(4)
L1∪L2δ=L1δ∪L2δ.Proof.
For the three (3, 2)-FSsL1,L2, and L3 and δ>0, according to Definition 3, we can obtain(1)
(8)L1∩L2∩L3=αL1,βL1∩minαL2,αL3,maxβL2,βL3=minαL1,minαL2,αL3,maxβL1,maxβL2,βL3=minminαL1,αL2,αL3,maxmaxβL1,βL2,βL3=minαL1,αL2,maxβL1,βL2∩αL3,βL3=L1∩L2∩L3.(2)
The proof is similar to (1).(3)
(9)δL1∪L2=δmaxαL1,αL2,minβL1,βL2=1−1−maxαL13,αL23δ5,minβL1δ,βL2δ,δL1∪δL2=1−1−αL13δ5,βL1δ∪1−1−αL23δ5,βL2δ=max1−1−αL13δ5,1−1−αL23δ5,minβL1δ,βL2δ=1−1−maxαL13,αL23δ5,minβL1δ,βL2δ=δL1∪L2.(4)
The proof is similar to (3).In the following result, we claim thatLc is (3, 2)-FS for any (3, 2)-FS L.Theorem 4.
LetL1=αL1,βL1 and L2=αL2,βL2 be two (3, 2)-FSs such that L1c and L2c are (3, 2)-FSs. Then,(1)
L1∩L2c=L1c∪L2c.(2)
L1∪L2c=L1c∩L2c.Proof.
For the two (3, 2)-FSsL1 and L2, according to Definition 3, we can obtain(1)
(10)L1∩L2c=minαL1,αL2,maxβL1,βL2c=maxβL1,βL2,minαL1,αL2=βL1,αL1∪βL2,αL2=L1c∪L2c.(2)
The proof is similar to (1).Definition 4.
LetD1=αD1,βD1 and D2=αD2,βD2 be two (3, 2)-FSs; then,(1)
D1=D2 if and only if αD1=αD2 and βD1=βD2.(2)
D1≥D2 if and only if αD1≥αD2 and βD1≤βD2.(3)
D2⊂D1 or D1⊃D2 if D1≥D2.Example 3.
(1)
IfD1=0.9,0.5 and D2=0.9,0.5 for X=x, then D1=D2.(2)
IfD1=0.9,0.5 and D2=0.81,0.61 for X=x, then D2≤D1 and D2⊂D1.


## 3. Topology with respect to (3, 2)-Fuzzy Sets

In this section, we formulate the concept of (3, 2)-fuzzy topology on the family of (3, 2)-fuzzy sets whose complements are (3, 2)-fuzzy sets and scrutinize main properties. Then, we define (3, 2)-fuzzy continuous maps and give some characterizations. Finally, we establish two types of (3, 2)-fuzzy separation axioms and reveal the relationships between them.


### 3.1. (3, 2)-Fuzzy Topology

Definition 5.
Letτ be a family of (3, 2)-fuzzy subsets of a non-empty set X. If(1)
1X,0X∈τ where 1X=1,0 and 0X=0,1,(2)
D1∩D2∈τ, for any D1,D2∈τ,(3)
∪i∈IDi∈τ, for any Dii∈I⊂τ,then τ is called a (3, 2)-fuzzy topology on X and X,τ is a (3, 2)-fuzzy topological space. We call D an open (3, 2)-FS if it is a member of τ and call its complement a closed (3, 2)-FS.Remark 2.
We callτ=1X,0X the indiscreet (3, 2)-fuzzy topology on X. If τ contains all (3, 2)-fuzzy subsets, then we call τ the discrete (3, 2)-fuzzy topology on X.Example 4.
Letτ=1X,0X,D1,D2,D3,D4,D5 be the family of (3, 2)-fuzzy subsets of X=x1,x2, where(11)D1=x1,αD1x1=0.8,βD1x1=0.62,x2,αD1x2=0.81,βD1x2=0.61,D2=x1,αD2x1=0.83,βD2x1=0.53,x2,αD2x2=0.82,βD2x2=0.62,D3=x1,αD3x1=0.79,βD3x1=0.63,x2,αD3x2=8,βD3x2=0.63,D4=x1,αD4x1=0.83,βD4x1=0.53,x2,αD4x2=0.82,βD4x2=0.61,D5=x1,αD5x1=0.8,βD51=0.62,x2,αD5x2=0.81,βD5x2=0.62.
Hence,τ is (3, 2)-fuzzy topology on X.Remark 3.
We showed that every fuzzy setD on a set X is a (3, 2)-fuzzy set having the form D=r,αDr,1−αDr:r∈X. Then, every fuzzy topological space X,τ1 in the sense of Chang is obviously a (3, 2)-fuzzy topological space in the form τ=D:αD∈τ1 whenever we identify a fuzzy set in X whose membership function is αD with its counterpart D=r,αDr,1−αDr:r∈X. Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is (3, 2)-fuzzy topology. The following examples explain this note.Example 5.
Considerτ=1X,0X,D1,D2 as family of fuzzy subsets of X=x, where(12)1X=c,α1Xx=1,1−α1Xx=β1Xx=0,0X=c,α0Xx=0,1−α0Xx=β0Xx=1,D1=c,αD1x=0.7,1−αD1x=βD1x=0.3,D2=c,αD2x=0.2,1−αD2x=βD2x=0.8.
Then,τ is fuzzy topology on X, and hence it is (3, 2)-fuzzy topology.Example 6.
Letτ=1X,0X,D1,D2 be the family of (3, 2)-fuzzy subsets on X=x1,x2 where(13)D1=x1,αD1x1=0.76,βD1x1=0.74,x2,αD1x2=0.6,βD1x2=0.83,D2=x1,αD2x1=0.75,βD2x1=0.74,x2,αD2x2=0.59,βD2x2=0.83.
Hence,τ is (3, 2)-fuzzy topology. On the other hand, τ is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.Definition 6.
LetX,τ be a (3, 2)-fuzzy topological space and D=x,αDx,βDx:x∈X be a (3, 2)-FS in X. Then, the (3, 2)-fuzzy interior and (3, 2)-fuzzy closure of D are, respectively, defined by(1)
clD=∩H:H is a closed (3, 2)-FS in X and D⊂H.(2)
intD=∪G:G is an open (3, 2)-FS in X and G⊂D.Remark 4.
LetX,τ be a (3, 2)-fuzzy topological space and D be any (3, 2)-FS in X. Then,(1)
intD is an open (3, 2)-FS.(2)
clD is a closed (3, 2)-FS.(3)
int1X=cl1X=1X and int0X=cl0X=0X.Example 7.
Consider the (3, 2)-fuzzy topological spaceX,τ in Example 4. If D=c1,0.67,0.81,c2,0.75,0.74, then intD=0X and clD=1X.Theorem 5.
LetX,τ be a (3, 2)-fuzzy topological space and D1,D2 be (3, 2)-FSs in X. Then, the following properties hold:(1)
intD1⊂D1 and D1⊂clD1.(2)
IfD1⊂D2, then intD1⊂intD2 and clD1⊂clD2.(3)
D1 is an open (3, 2)-FS if and only if D1=intD1.(4)
D1 is a closed (3, 2)-FS if and only if D1=clD1.Proof.
(1) and (2) are obvious.
(3) and (4) follow from Definition6.Corollary 1.
LetX,τ be a (3, 2)-fuzzy topological space and D1,D2 be (3, 2)-FSs in X. Then, the following properties hold:(1)
intD1∪intD2⊂intD1∪D2.(2)
clD1∩D2⊂clD1∩clD2.(3)
intD1∩D2=intD1∩intD2.(4)
clD1∪clD2=clD1∪D2.Proof.
(1) and (2) follows from (1) of the above theorem.
(3): sinceintD1∩D2⊂intD1 and intD1∩D2⊂intD2, we obtain intD1∩D2⊂intD1∩intD2. On the other hand, from the facts intD1⊂D1 and intD2⊂D2, we have intD1∩intD2⊂D1∩D2 and intD1∩intD2∈τ; we see that intD1∩intD2⊂intD1∩D2, and hence intD1∩D2=intD1∩intD2.
(4) can be proved similar to (3).Theorem 6.
LetX,τ be a (3, 2)-fuzzy topological space and D be (3, 2)-FS in X. Then, the following properties hold:(1)
clDc=intDc.(2)
intDc=clDc.(3)
clDcc=intD.(4)
intDcc=clD.Proof.
We only prove (1); the other parts can be proved similarly.
LetD=x,αDx,βDx:x∈X and suppose that the family of open (3, 2)-fuzzy sets contained in D is indexed by the family x,αUix,βUix:i∈J. Then, intD=x,∨αUix,∧βUix. Therefore, intDc=x,∧βUix,∨αUix. Now, Dc=x,βDx,αDx such that αUi≤αD, βUi≥βD for each i∈J. This implies that x,βUix,αUix:i∈J is the family of all closed (3, 2)-fuzzy sets containing Dc. That is, clDc=x,∧βUix,∨αUix. Hence, clDc=intDc.


### 3.2. (3, 2)-Fuzzy Continuous Maps

Definition 7.
Letf:X⟶Y be a map and A and B be (3, 2)-fuzzy subsets of X and Y, respectively. The functions of membership and non-membership of the image of A, denoted by fA, are, respectively, calculated by(14)αfAy≔supz∈f−1yαAz,iff−1y≠ϕ,0,otherwise,βfAy≔infz∈f−1yβAz,iff−1y≠ϕ,1,otherwise.
The functions of membership and non-membership of preimage ofB, denoted by f−1B, are, respectively, calculated by(15)αf−1Bx≔αBfx,βf−1Bx≔βBfx.Remark 5.
To show thatfA and f−1B are (3, 2)-fuzzy subsets, consider γAz5=αAz3+βAz2. If f−1y is non-empty, then we obtain(16)αfAy3+βfAy2=supz∈f−1yαAz3+infz∈f−1yβAz2=supz∈f−1yαAz3+infz∈f−1yβAz2=supz∈f−1yγAz5−βAz2+infz∈f−1yβAz2≤supz∈f−1y1−βAz2+infz∈f−1yβAz2=1.
In contrast,f−1y=ϕ leads to the fact that αfAy3+βfAy2=1.
It is easy to prove the case off−1B.Theorem 7.
Letf:X⟶Y be a map s.t. A and B are (3, 2)-fuzzy subsets of X and Y, respectively. Then, we have(1)
f−1Bc=f−1Bc.(2)
fAc⊆fAc.(3)
IfB1⊆B2, then f−1B1⊆f−1B2 where B1 and B2 are (3, 2)-fuzzy subsets of Y.(4)
IfA1⊆A2, then fA1⊆fA2 where A1 and A2 are (3, 2)-fuzzy subsets of X.(5)
ff−1B⊆B.(6)
A⊆f−1fA.Proof.
(1)
Considerv∈X and let B be a (3, 2)-fuzzy subset of Y. Then,(17)αf−1Bcv=αBcfv=βBfv=βf−1Bv=αf−1Bcv.Similarly, one can haveβf−1Bcv=βf−1Bcv. Therefore, f−1Bc=f−1Bc, as required.(2)
For anyw∈Y such that f−1w≠ϕ and for any (3, 2)-fuzzy subset A of X, we can write(18)γfAw5=αfAw3+βfAw2=supz∈f−1wαAz3+infz∈f−1wβAz2=supz∈f−1wγAz5−βAz2+infz∈f−1wβAz2≤supz∈f−1wγAz5−infz∈f−1wβAz2+infz∈f−1wβAz2=supz∈f−1wγAz5.Now from (18), we have(19)αfAcw=supz∈f−1wαAcz=supz∈f−1wβAz=supz∈f−1wγAz5−αAz3≥supz∈f−1wγAz5−supz∈f−1wαAz3≥γfAw5−αfAw3=βfAw=αfAcw.The proof is easy whenf−1w=ϕ. Following a similar technique, we obtain βfAcw≤βfAcw, which means that fAc⊆fAc.(3)
Assume thatB1⊆B2. Then, for each v∈X, αf−1B1v=αB1fv≤αB2fv=αf−1B2v. Also, βf−1B1v≥βf−1B2v. Hence, we obtain the desired result.(4)
Assume thatA1⊆A2 and w∈Y. The proof is easy when fw=ϕ. So, presume that fw≠ϕ. Then,(20)αfA1w=supz∈f−1wαA1z≤supz∈f−1wαA2z=αfA2w.Thus,αfA1≤αfA2 follows. Similarly, we have βfA1≥βfA2.(5)
For anyw∈Y s.t. fw≠ϕ, we find that(21)αff−1Bw=supz∈f−1wαf−1Bz=supz∈f−1wαBfz≤αBw.On the other hand, we haveαff−1Bw=0≤αBw when fw=ϕ. Similarly, we have βff−1Bw=0≥βBw.(6)
For anyv∈X, we have(22)αf−1fAv=αfAfv=supz∈f−1fvαAz≥αAv.Similarly, we haveβf−1fA≤βA.The proof of the following result is easy, and hence it is omitted.Theorem 8.
LetX and Y be two non-empty sets and f:X⟶Y be a map. Then, the following statements are true:(1)
f∪i∈IAi=∪i∈IfAi for any (3, 2)-fuzzy subset Ai of X.(2)
f−1∪i∈IBi=∪i∈If−1Bi for any (3, 2)-fuzzy subset Bi of Y.(3)
fA1∩A2⊂fA1∩fA2 for any two (3, 2)-fuzzy subsets A1 and A2 of X.(4)
f−1∩i∈IBi=∩i∈If−1Bi for any (3, 2)-fuzzy subset Bi of Y.Definition 8.
In a (3, 2)-fuzzy topological space, consider thatA and U are two (3, 2)-fuzzy subsets. We call U a neighborhood of A, briefly nbd, if there exists an open (3, 2)-fuzzy subset E such that A⊆E⊆U.Theorem 9.
A (3, 2)-fuzzy subsetA is open iff it contains a nbd of its each subset.Proof.
The proof is easy.Definition 9.
A mapf:X,τ1⟶Y,τ2 is said to be (3, 2)-fuzzy continuous if for any (3, 2)-fuzzy subset A of X and for any nbd V of fA there is a nbd U of A s.t. fU⊆V.Theorem 10.
The following statements are equivalent for a mapf:X,τ1⟶Y,τ2:(1)
f is (3, 2)-fuzzy continuous.(2)
For each (3, 2)-FSA of X and each nbd V of fA, there is a nbd U of A s.t. for each B⊆U, we obtain fB⊆V.(3)
For each (3, 2)-FSA of X and for each nbd V of fA, there is a nbd U of A s.t. U⊆f−1V.(4)
For each (3, 2)-FSA of X and for each nbd V of fA, f−1V is a nbd of A.Proof.
1⇒2: let f be a (3, 2)-fuzzy continuous map. Consider A as a (3, 2)-FS of X and V as a nbd of fA. Then, there is a nbd U of A s.t. fU⊆V. If B⊆U, we obtain fB⊆fU⊆V.2⇒3: assume A as a (3, 2)-FS of X and V as a nbd of fA. According to (2), there is a nbd U of A s.t. for each B⊆U, we find fB⊆V. Therefore, B⊆f−1fB⊆f−1V. Since B is chosen arbitrarily, U⊆f−1V.3⇒4: presume A is a (3, 2)-FS of X and V is a nbd of fA. According to (3), there is a nbd U of A s.t. U⊆f−1V. Since U is a nbd of A, there is an open (3, 2)-FS K of X s.t. A⊆K⊆U. On the other hand, we obtain A⊆K⊆f−1V because U⊆f−1V. This means that f−1V is a nbd of A.4⇒1: suppose that A is a (3, 2)-FS of X and V is a nbd of fA. By hypothesis, f−1V is a nbd of A. So, there is an open (3, 2)-FS K of X s.t. A⊆K⊆f−1V which means fK⊆ff−1V⊆V. Moreover, K is an open (3, 2)-FS, so it is a nbd of A. Hence, we obtain the proof that f is (3, 2)-fuzzy continuous.Theorem 11.
A mapf:X,τ1⟶Y,τ2 is (3, 2)-fuzzy continuous iff f−1B is an open (3, 2)-FS of X for each open (3, 2)-FS B of Y.Proof.
Necessity: presumef as a (3, 2)-fuzzy continuous map. Consider an open (3, 2)-FS B of Y s.t. A⊆f−1B. This directly gives that fA⊆B. It follows from Theorem 9 that there is a nbd V of fA satisfying V⊆B. Now, f is (3, 2)-fuzzy continuous, so by (4) of Theorem 10, we obtain that f−1V is a nbd of A. Also, it follows from (3) of Theorem 7 that f−1V⊆f−1B. So, f−1B is a nbd of A. Since A is an arbitrary subset of f−1B, then by Theorem 9, the (3, 2)-FS f−1B is open.


#### 3.2.1. Sufficiency

PresumeA is a (3, 2)-FS of X and V is a nbd of fA. Then, τ2 contains a (3, 2)-FS L of s.t. fA⊆L⊆V. By hypothesis, f−1L is an open (3, 2)-FS. Also, we have A⊆f−1fA⊆f−1L⊆f−1V. Thus, f−1V is a nbd of A which demonstrates that f is (3, 2)-fuzzy continuous.We build the following two examples such that the first one provides a (3, 2)-fuzzy continuous map, whereas the second one presents a fuzzy map that is not (3, 2)-fuzzy continuous.Example 8.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X,A1 and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where(23)A1=a1,0.7,0.78,a2,0.9,0.5,B1=b1,0.9,0.5,b2,0.7,0.78.
Letf:X⟶Y be defined as follows:(24)fx=b2,ifx=a1,b1,ifx=a2.
Since1Y,0Y, and B1 are open (3, 2)-fuzzy subsets of Y, then(25)f−11Y=a1,1,0,a2,1,0,f−10Y=a1,0,1,a2,0,1,f−1B1=a1,0.7,0.78,a2,0.9,0.5are open (3, 2)-fuzzy subsets of X. Thus, f is (3, 2)-fuzzy continuous.Example 9.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where B1=b1,0.82,0.62,b2,0.52,0.90.
Letf:X⟶Y be defined as follows:(26)fx=b1,ifx=a1,b2,ifx=a2.
SinceB1 is an open (3, 2)-fuzzy subset of Y, but f−1B1=a1,0.82,0.62,a2,0.52,0.90 is not an open (3, 2)-fuzzy subset of X, f is not (3, 2)-fuzzy continuous.Theorem 12.
The following are equivalent to each other:(1)
f:X,τ1⟶Y,τ2 is (3, 2)-fuzzy continuous.(2)
For each closed (3, 2)-fuzzy subsetB of Y we have that f−1B is a closed (3, 2)-fuzzy subset of X.(3)
clf−1B⊆f−1clB for each (3, 2)-fuzzy set in Y.(4)
f−1intB⊆intf−1B for each (3, 2)-fuzzy set in Y.Proof.
They can be easily proved using Theorems6, 7, and 11.Theorem 13.
LetY,τ be a (3, 2)-fuzzy topological space and f:X⟶Y be a map. Then, there is a coarsest (3, 2)-fuzzy topology τ1 over X such that f is (3, 2)-fuzzy continuous.Proof.
Let us define a class of (3, 2)-fuzzy subsetsτ1 of X by(27)τ1≔f−1V:V∈τ.
We prove thatτ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous.(1)
We can write for anyx∈X that(28)αf−10Yx=α0Yfx=0=α0Xx.Similarly, we immediately haveβf−10Yx=β0Xx for any x∈X which implies f−10Y=0X. Now, as 0Y∈τ, we have 0X=f−10Y∈τ1. In a similar manner, it is easy to see that 1X=f−11Y∈τ1.(2)
Assume thatD1,D2∈τ1. Then, for i=1,2, there exists Bi∈τ such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Thus, we obtain for any x∈X that(29)αD1∩D2x=minαD1x,αD2x=minαf−1B1x,αf−1B2x=minαB1fx,αB2fx=αB1∩B2fx=αf−1B1∩B2x.Similarly, it is not difficult to see thatβD1∩D2=βf−1B1∩B2. Hence, we get D1∩D2∈τ1.(3)
Assume thatDii∈I is an arbitrary subfamily of τ1. Then, for any i∈I, there exists Bi∈τ1 such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Therefore, one can get for any x∈X that(30)α∪i∈IDix=supi∈IαDix=supi∈Iαf−1Bix=supi∈IαBifx=α∪i∈IBifx=αf−1∪i∈IBix.
On the other hand, it is easy to see thatβ∪i∈IDi=βf−1∪i∈IBi. Thus, we have ∪i∈IDi∈τ1.
From Theorem11, the (3, 2)-fuzzy continuity of f is trivial. Now, we prove that τ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. Let τ2⊆τ1 be a (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. If B∈τ1, then there is V∈τ such that f−1V=B. Since f is (3, 2)-fuzzy continuous with respect to τ2, we have B=f−1V∈τ2. Hence, τ2=τ1, as required.


### 3.3. (3, 2)-Fuzzy Separation Axioms

Separation axioms are one of the most important and popular notions in topological studies. They have been studied and applied to model some real-life issues in soft setting as explained in [16, 17].Definition 10.
LetX≠∅ and x∈X be a fixed element in X. Suppose that r1∈0,1 and r2∈0,1 are two fixed real numbers such that r13+r22≤1. Then, a (3, 2)-fuzzy point pr1,r2x=x,αpx,βpx is defined to be a (3, 2)-fuzzy set of X as follows.(31)pr1,r2xy≔r1,r2,ify=x,0,1,otherwise,for y∈X. In this case, x is called the support of pr1,r2x. A (3, 2)-fuzzy point pr1,r2x is said to belong to a (3, 2)-fuzzy set D=x,αDx,βDx of X denoted by pr1,r2x∈D if r1≤αDx and r2≥βDx. Two (3, 2)-fuzzy points are said to be distinct if their supports are distinct.Remark 6.
LetD1=x,αD1x,βD1x and D2=x,αD2x,βD2x be two (3, 2)-fuzzy sets of X. Then, D1⊆D2 if and only if pr1,r2x∈D1 implies pr1,r2x∈D2 for any (3, 2)-fuzzy point pr1,r2x in X.Definition 11.
Letr1,r3∈0,1, r2,r4∈0,1, and x,y∈X. A (3, 2)-fuzzy topological space X,τ is said to be(1)
T0 if for each pair of distinct (3, 2)-fuzzy points pr1,r2x,pr3,r4y in X, there exist two open (3, 2)-fuzzy sets L and K such that(32)L=x,1,0,y,0,1,orK=x,0,1,y,1,0.(2)
T1 if for each pair of distinct (3, 2)-fuzzy points pr1,r2x,pr3,r4y in X, there exist two open (3, 2)-fuzzy sets L and K such that(33)L=x,1,0,y,0,1,K=x,0,1,y,1,0.Proposition 1.
LetX,τ be a (3, 2)-fuzzy topological space. If X,τ is T1, then X,τ is T0.Proof.
The proof is straightforward from Definition11.Here is an example which shows that the converse of above proposition is not true in general.Example 10.
ConsiderX=c1,c2 with the (3, 2)-fuzzy topology τ=1X,0X,D, where D=c1,1,0,c2,0,1. Then, X,τ is T0 but not T1 because there does not exist an open (3, 2)-fuzzy set K such that K=x,0,1,y,1,0.Theorem 14.
LetX,τ be a (3, 2)-fuzzy topological space, r1,r3∈0,1, and r2,r4∈0,1. If X,τ is T0, then for each pair of distinct (3, 2)-fuzzy points pr1,r2x, pr3,r4y of X, clpr1,r2x≠clpr3,r4y.Proof.
LetX,τ be T0 and pr1,r2x, pr3,r4y be any two distinct (3, 2)-fuzzy points of X. Then, there exist two open (3, 2)-fuzzy sets L and K such that(34)L=x,1,0,y,0,1,orK=x,0,1,y,1,0.
LetL=x,1,0,y,0,1 exist. Then, Lc=x,0,1,y,1,0 is a closed (3, 2)-fuzzy set which does not contain pr1,r2x but contains pr3,r4y. Since clpr3,r4y is the smallest closed (3, 2)-fuzzy set containing pr3,r4y, then clpr3,r4y⊆Lc, and therefore pr1,r2x∉clpr3,r4y. Consequently, clpr1,r2x≠clpr3,r4y.Theorem 15.
LetX,τ be a (3, 2)-fuzzy topological space. If p1,0x is closed (3, 2)-fuzzy set for every x∈X, then, X,τ is T1.Proof.
Supposep1,0x is a closed (3, 2)-fuzzy set for every x∈X. Let pr1,r2x, pr3,r4y be any two distinct (3, 2)-fuzzy points of X; then, x≠y implies that p1,0xc and p1,0yc are two open (3, 2)-fuzzy sets such that(35)p1,0yc=x,1,0,y,0,1,p1,0xc=x,0,1,y,1,0.
Thus,X,τ is T1.


## 3.1. (3, 2)-Fuzzy Topology

Definition 5.
Letτ be a family of (3, 2)-fuzzy subsets of a non-empty set X. If(1)
1X,0X∈τ where 1X=1,0 and 0X=0,1,(2)
D1∩D2∈τ, for any D1,D2∈τ,(3)
∪i∈IDi∈τ, for any Dii∈I⊂τ,then τ is called a (3, 2)-fuzzy topology on X and X,τ is a (3, 2)-fuzzy topological space. We call D an open (3, 2)-FS if it is a member of τ and call its complement a closed (3, 2)-FS.Remark 2.
We callτ=1X,0X the indiscreet (3, 2)-fuzzy topology on X. If τ contains all (3, 2)-fuzzy subsets, then we call τ the discrete (3, 2)-fuzzy topology on X.Example 4.
Letτ=1X,0X,D1,D2,D3,D4,D5 be the family of (3, 2)-fuzzy subsets of X=x1,x2, where(11)D1=x1,αD1x1=0.8,βD1x1=0.62,x2,αD1x2=0.81,βD1x2=0.61,D2=x1,αD2x1=0.83,βD2x1=0.53,x2,αD2x2=0.82,βD2x2=0.62,D3=x1,αD3x1=0.79,βD3x1=0.63,x2,αD3x2=8,βD3x2=0.63,D4=x1,αD4x1=0.83,βD4x1=0.53,x2,αD4x2=0.82,βD4x2=0.61,D5=x1,αD5x1=0.8,βD51=0.62,x2,αD5x2=0.81,βD5x2=0.62.
Hence,τ is (3, 2)-fuzzy topology on X.Remark 3.
We showed that every fuzzy setD on a set X is a (3, 2)-fuzzy set having the form D=r,αDr,1−αDr:r∈X. Then, every fuzzy topological space X,τ1 in the sense of Chang is obviously a (3, 2)-fuzzy topological space in the form τ=D:αD∈τ1 whenever we identify a fuzzy set in X whose membership function is αD with its counterpart D=r,αDr,1−αDr:r∈X. Similarly, one can note that every intuitionistic fuzzy topology (Pythagorean fuzzy topology) is (3, 2)-fuzzy topology. The following examples explain this note.Example 5.
Considerτ=1X,0X,D1,D2 as family of fuzzy subsets of X=x, where(12)1X=c,α1Xx=1,1−α1Xx=β1Xx=0,0X=c,α0Xx=0,1−α0Xx=β0Xx=1,D1=c,αD1x=0.7,1−αD1x=βD1x=0.3,D2=c,αD2x=0.2,1−αD2x=βD2x=0.8.
Then,τ is fuzzy topology on X, and hence it is (3, 2)-fuzzy topology.Example 6.
Letτ=1X,0X,D1,D2 be the family of (3, 2)-fuzzy subsets on X=x1,x2 where(13)D1=x1,αD1x1=0.76,βD1x1=0.74,x2,αD1x2=0.6,βD1x2=0.83,D2=x1,αD2x1=0.75,βD2x1=0.74,x2,αD2x2=0.59,βD2x2=0.83.
Hence,τ is (3, 2)-fuzzy topology. On the other hand, τ is neither intuitionistic fuzzy topology nor Pythagorean fuzzy topology.Definition 6.
LetX,τ be a (3, 2)-fuzzy topological space and D=x,αDx,βDx:x∈X be a (3, 2)-FS in X. Then, the (3, 2)-fuzzy interior and (3, 2)-fuzzy closure of D are, respectively, defined by(1)
clD=∩H:H is a closed (3, 2)-FS in X and D⊂H.(2)
intD=∪G:G is an open (3, 2)-FS in X and G⊂D.Remark 4.
LetX,τ be a (3, 2)-fuzzy topological space and D be any (3, 2)-FS in X. Then,(1)
intD is an open (3, 2)-FS.(2)
clD is a closed (3, 2)-FS.(3)
int1X=cl1X=1X and int0X=cl0X=0X.Example 7.
Consider the (3, 2)-fuzzy topological spaceX,τ in Example 4. If D=c1,0.67,0.81,c2,0.75,0.74, then intD=0X and clD=1X.Theorem 5.
LetX,τ be a (3, 2)-fuzzy topological space and D1,D2 be (3, 2)-FSs in X. Then, the following properties hold:(1)
intD1⊂D1 and D1⊂clD1.(2)
IfD1⊂D2, then intD1⊂intD2 and clD1⊂clD2.(3)
D1 is an open (3, 2)-FS if and only if D1=intD1.(4)
D1 is a closed (3, 2)-FS if and only if D1=clD1.Proof.
(1) and (2) are obvious.
(3) and (4) follow from Definition6.Corollary 1.
LetX,τ be a (3, 2)-fuzzy topological space and D1,D2 be (3, 2)-FSs in X. Then, the following properties hold:(1)
intD1∪intD2⊂intD1∪D2.(2)
clD1∩D2⊂clD1∩clD2.(3)
intD1∩D2=intD1∩intD2.(4)
clD1∪clD2=clD1∪D2.Proof.
(1) and (2) follows from (1) of the above theorem.
(3): sinceintD1∩D2⊂intD1 and intD1∩D2⊂intD2, we obtain intD1∩D2⊂intD1∩intD2. On the other hand, from the facts intD1⊂D1 and intD2⊂D2, we have intD1∩intD2⊂D1∩D2 and intD1∩intD2∈τ; we see that intD1∩intD2⊂intD1∩D2, and hence intD1∩D2=intD1∩intD2.
(4) can be proved similar to (3).Theorem 6.
LetX,τ be a (3, 2)-fuzzy topological space and D be (3, 2)-FS in X. Then, the following properties hold:(1)
clDc=intDc.(2)
intDc=clDc.(3)
clDcc=intD.(4)
intDcc=clD.Proof.
We only prove (1); the other parts can be proved similarly.
LetD=x,αDx,βDx:x∈X and suppose that the family of open (3, 2)-fuzzy sets contained in D is indexed by the family x,αUix,βUix:i∈J. Then, intD=x,∨αUix,∧βUix. Therefore, intDc=x,∧βUix,∨αUix. Now, Dc=x,βDx,αDx such that αUi≤αD, βUi≥βD for each i∈J. This implies that x,βUix,αUix:i∈J is the family of all closed (3, 2)-fuzzy sets containing Dc. That is, clDc=x,∧βUix,∨αUix. Hence, clDc=intDc.


## 3.2. (3, 2)-Fuzzy Continuous Maps

Definition 7.
Letf:X⟶Y be a map and A and B be (3, 2)-fuzzy subsets of X and Y, respectively. The functions of membership and non-membership of the image of A, denoted by fA, are, respectively, calculated by(14)αfAy≔supz∈f−1yαAz,iff−1y≠ϕ,0,otherwise,βfAy≔infz∈f−1yβAz,iff−1y≠ϕ,1,otherwise.
The functions of membership and non-membership of preimage ofB, denoted by f−1B, are, respectively, calculated by(15)αf−1Bx≔αBfx,βf−1Bx≔βBfx.Remark 5.
To show thatfA and f−1B are (3, 2)-fuzzy subsets, consider γAz5=αAz3+βAz2. If f−1y is non-empty, then we obtain(16)αfAy3+βfAy2=supz∈f−1yαAz3+infz∈f−1yβAz2=supz∈f−1yαAz3+infz∈f−1yβAz2=supz∈f−1yγAz5−βAz2+infz∈f−1yβAz2≤supz∈f−1y1−βAz2+infz∈f−1yβAz2=1.
In contrast,f−1y=ϕ leads to the fact that αfAy3+βfAy2=1.
It is easy to prove the case off−1B.Theorem 7.
Letf:X⟶Y be a map s.t. A and B are (3, 2)-fuzzy subsets of X and Y, respectively. Then, we have(1)
f−1Bc=f−1Bc.(2)
fAc⊆fAc.(3)
IfB1⊆B2, then f−1B1⊆f−1B2 where B1 and B2 are (3, 2)-fuzzy subsets of Y.(4)
IfA1⊆A2, then fA1⊆fA2 where A1 and A2 are (3, 2)-fuzzy subsets of X.(5)
ff−1B⊆B.(6)
A⊆f−1fA.Proof.
(1)
Considerv∈X and let B be a (3, 2)-fuzzy subset of Y. Then,(17)αf−1Bcv=αBcfv=βBfv=βf−1Bv=αf−1Bcv.Similarly, one can haveβf−1Bcv=βf−1Bcv. Therefore, f−1Bc=f−1Bc, as required.(2)
For anyw∈Y such that f−1w≠ϕ and for any (3, 2)-fuzzy subset A of X, we can write(18)γfAw5=αfAw3+βfAw2=supz∈f−1wαAz3+infz∈f−1wβAz2=supz∈f−1wγAz5−βAz2+infz∈f−1wβAz2≤supz∈f−1wγAz5−infz∈f−1wβAz2+infz∈f−1wβAz2=supz∈f−1wγAz5.Now from (18), we have(19)αfAcw=supz∈f−1wαAcz=supz∈f−1wβAz=supz∈f−1wγAz5−αAz3≥supz∈f−1wγAz5−supz∈f−1wαAz3≥γfAw5−αfAw3=βfAw=αfAcw.The proof is easy whenf−1w=ϕ. Following a similar technique, we obtain βfAcw≤βfAcw, which means that fAc⊆fAc.(3)
Assume thatB1⊆B2. Then, for each v∈X, αf−1B1v=αB1fv≤αB2fv=αf−1B2v. Also, βf−1B1v≥βf−1B2v. Hence, we obtain the desired result.(4)
Assume thatA1⊆A2 and w∈Y. The proof is easy when fw=ϕ. So, presume that fw≠ϕ. Then,(20)αfA1w=supz∈f−1wαA1z≤supz∈f−1wαA2z=αfA2w.Thus,αfA1≤αfA2 follows. Similarly, we have βfA1≥βfA2.(5)
For anyw∈Y s.t. fw≠ϕ, we find that(21)αff−1Bw=supz∈f−1wαf−1Bz=supz∈f−1wαBfz≤αBw.On the other hand, we haveαff−1Bw=0≤αBw when fw=ϕ. Similarly, we have βff−1Bw=0≥βBw.(6)
For anyv∈X, we have(22)αf−1fAv=αfAfv=supz∈f−1fvαAz≥αAv.Similarly, we haveβf−1fA≤βA.The proof of the following result is easy, and hence it is omitted.Theorem 8.
LetX and Y be two non-empty sets and f:X⟶Y be a map. Then, the following statements are true:(1)
f∪i∈IAi=∪i∈IfAi for any (3, 2)-fuzzy subset Ai of X.(2)
f−1∪i∈IBi=∪i∈If−1Bi for any (3, 2)-fuzzy subset Bi of Y.(3)
fA1∩A2⊂fA1∩fA2 for any two (3, 2)-fuzzy subsets A1 and A2 of X.(4)
f−1∩i∈IBi=∩i∈If−1Bi for any (3, 2)-fuzzy subset Bi of Y.Definition 8.
In a (3, 2)-fuzzy topological space, consider thatA and U are two (3, 2)-fuzzy subsets. We call U a neighborhood of A, briefly nbd, if there exists an open (3, 2)-fuzzy subset E such that A⊆E⊆U.Theorem 9.
A (3, 2)-fuzzy subsetA is open iff it contains a nbd of its each subset.Proof.
The proof is easy.Definition 9.
A mapf:X,τ1⟶Y,τ2 is said to be (3, 2)-fuzzy continuous if for any (3, 2)-fuzzy subset A of X and for any nbd V of fA there is a nbd U of A s.t. fU⊆V.Theorem 10.
The following statements are equivalent for a mapf:X,τ1⟶Y,τ2:(1)
f is (3, 2)-fuzzy continuous.(2)
For each (3, 2)-FSA of X and each nbd V of fA, there is a nbd U of A s.t. for each B⊆U, we obtain fB⊆V.(3)
For each (3, 2)-FSA of X and for each nbd V of fA, there is a nbd U of A s.t. U⊆f−1V.(4)
For each (3, 2)-FSA of X and for each nbd V of fA, f−1V is a nbd of A.Proof.
1⇒2: let f be a (3, 2)-fuzzy continuous map. Consider A as a (3, 2)-FS of X and V as a nbd of fA. Then, there is a nbd U of A s.t. fU⊆V. If B⊆U, we obtain fB⊆fU⊆V.2⇒3: assume A as a (3, 2)-FS of X and V as a nbd of fA. According to (2), there is a nbd U of A s.t. for each B⊆U, we find fB⊆V. Therefore, B⊆f−1fB⊆f−1V. Since B is chosen arbitrarily, U⊆f−1V.3⇒4: presume A is a (3, 2)-FS of X and V is a nbd of fA. According to (3), there is a nbd U of A s.t. U⊆f−1V. Since U is a nbd of A, there is an open (3, 2)-FS K of X s.t. A⊆K⊆U. On the other hand, we obtain A⊆K⊆f−1V because U⊆f−1V. This means that f−1V is a nbd of A.4⇒1: suppose that A is a (3, 2)-FS of X and V is a nbd of fA. By hypothesis, f−1V is a nbd of A. So, there is an open (3, 2)-FS K of X s.t. A⊆K⊆f−1V which means fK⊆ff−1V⊆V. Moreover, K is an open (3, 2)-FS, so it is a nbd of A. Hence, we obtain the proof that f is (3, 2)-fuzzy continuous.Theorem 11.
A mapf:X,τ1⟶Y,τ2 is (3, 2)-fuzzy continuous iff f−1B is an open (3, 2)-FS of X for each open (3, 2)-FS B of Y.Proof.
Necessity: presumef as a (3, 2)-fuzzy continuous map. Consider an open (3, 2)-FS B of Y s.t. A⊆f−1B. This directly gives that fA⊆B. It follows from Theorem 9 that there is a nbd V of fA satisfying V⊆B. Now, f is (3, 2)-fuzzy continuous, so by (4) of Theorem 10, we obtain that f−1V is a nbd of A. Also, it follows from (3) of Theorem 7 that f−1V⊆f−1B. So, f−1B is a nbd of A. Since A is an arbitrary subset of f−1B, then by Theorem 9, the (3, 2)-FS f−1B is open.


### 3.2.1. Sufficiency

PresumeA is a (3, 2)-FS of X and V is a nbd of fA. Then, τ2 contains a (3, 2)-FS L of s.t. fA⊆L⊆V. By hypothesis, f−1L is an open (3, 2)-FS. Also, we have A⊆f−1fA⊆f−1L⊆f−1V. Thus, f−1V is a nbd of A which demonstrates that f is (3, 2)-fuzzy continuous.We build the following two examples such that the first one provides a (3, 2)-fuzzy continuous map, whereas the second one presents a fuzzy map that is not (3, 2)-fuzzy continuous.Example 8.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X,A1 and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where(23)A1=a1,0.7,0.78,a2,0.9,0.5,B1=b1,0.9,0.5,b2,0.7,0.78.
Letf:X⟶Y be defined as follows:(24)fx=b2,ifx=a1,b1,ifx=a2.
Since1Y,0Y, and B1 are open (3, 2)-fuzzy subsets of Y, then(25)f−11Y=a1,1,0,a2,1,0,f−10Y=a1,0,1,a2,0,1,f−1B1=a1,0.7,0.78,a2,0.9,0.5are open (3, 2)-fuzzy subsets of X. Thus, f is (3, 2)-fuzzy continuous.Example 9.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where B1=b1,0.82,0.62,b2,0.52,0.90.
Letf:X⟶Y be defined as follows:(26)fx=b1,ifx=a1,b2,ifx=a2.
SinceB1 is an open (3, 2)-fuzzy subset of Y, but f−1B1=a1,0.82,0.62,a2,0.52,0.90 is not an open (3, 2)-fuzzy subset of X, f is not (3, 2)-fuzzy continuous.Theorem 12.
The following are equivalent to each other:(1)
f:X,τ1⟶Y,τ2 is (3, 2)-fuzzy continuous.(2)
For each closed (3, 2)-fuzzy subsetB of Y we have that f−1B is a closed (3, 2)-fuzzy subset of X.(3)
clf−1B⊆f−1clB for each (3, 2)-fuzzy set in Y.(4)
f−1intB⊆intf−1B for each (3, 2)-fuzzy set in Y.Proof.
They can be easily proved using Theorems6, 7, and 11.Theorem 13.
LetY,τ be a (3, 2)-fuzzy topological space and f:X⟶Y be a map. Then, there is a coarsest (3, 2)-fuzzy topology τ1 over X such that f is (3, 2)-fuzzy continuous.Proof.
Let us define a class of (3, 2)-fuzzy subsetsτ1 of X by(27)τ1≔f−1V:V∈τ.
We prove thatτ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous.(1)
We can write for anyx∈X that(28)αf−10Yx=α0Yfx=0=α0Xx.Similarly, we immediately haveβf−10Yx=β0Xx for any x∈X which implies f−10Y=0X. Now, as 0Y∈τ, we have 0X=f−10Y∈τ1. In a similar manner, it is easy to see that 1X=f−11Y∈τ1.(2)
Assume thatD1,D2∈τ1. Then, for i=1,2, there exists Bi∈τ such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Thus, we obtain for any x∈X that(29)αD1∩D2x=minαD1x,αD2x=minαf−1B1x,αf−1B2x=minαB1fx,αB2fx=αB1∩B2fx=αf−1B1∩B2x.Similarly, it is not difficult to see thatβD1∩D2=βf−1B1∩B2. Hence, we get D1∩D2∈τ1.(3)
Assume thatDii∈I is an arbitrary subfamily of τ1. Then, for any i∈I, there exists Bi∈τ1 such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Therefore, one can get for any x∈X that(30)α∪i∈IDix=supi∈IαDix=supi∈Iαf−1Bix=supi∈IαBifx=α∪i∈IBifx=αf−1∪i∈IBix.
On the other hand, it is easy to see thatβ∪i∈IDi=βf−1∪i∈IBi. Thus, we have ∪i∈IDi∈τ1.
From Theorem11, the (3, 2)-fuzzy continuity of f is trivial. Now, we prove that τ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. Let τ2⊆τ1 be a (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. If B∈τ1, then there is V∈τ such that f−1V=B. Since f is (3, 2)-fuzzy continuous with respect to τ2, we have B=f−1V∈τ2. Hence, τ2=τ1, as required.


## 3.2.1. Sufficiency

PresumeA is a (3, 2)-FS of X and V is a nbd of fA. Then, τ2 contains a (3, 2)-FS L of s.t. fA⊆L⊆V. By hypothesis, f−1L is an open (3, 2)-FS. Also, we have A⊆f−1fA⊆f−1L⊆f−1V. Thus, f−1V is a nbd of A which demonstrates that f is (3, 2)-fuzzy continuous.We build the following two examples such that the first one provides a (3, 2)-fuzzy continuous map, whereas the second one presents a fuzzy map that is not (3, 2)-fuzzy continuous.Example 8.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X,A1 and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where(23)A1=a1,0.7,0.78,a2,0.9,0.5,B1=b1,0.9,0.5,b2,0.7,0.78.
Letf:X⟶Y be defined as follows:(24)fx=b2,ifx=a1,b1,ifx=a2.
Since1Y,0Y, and B1 are open (3, 2)-fuzzy subsets of Y, then(25)f−11Y=a1,1,0,a2,1,0,f−10Y=a1,0,1,a2,0,1,f−1B1=a1,0.7,0.78,a2,0.9,0.5are open (3, 2)-fuzzy subsets of X. Thus, f is (3, 2)-fuzzy continuous.Example 9.
ConsiderX=a1,a2 with the (3, 2)-fuzzy topology τ1=1X,0X and Y=b1,b2 with the (3, 2)-fuzzy topology τ2=1Y,0Y,B1, where B1=b1,0.82,0.62,b2,0.52,0.90.
Letf:X⟶Y be defined as follows:(26)fx=b1,ifx=a1,b2,ifx=a2.
SinceB1 is an open (3, 2)-fuzzy subset of Y, but f−1B1=a1,0.82,0.62,a2,0.52,0.90 is not an open (3, 2)-fuzzy subset of X, f is not (3, 2)-fuzzy continuous.Theorem 12.
The following are equivalent to each other:(1)
f:X,τ1⟶Y,τ2 is (3, 2)-fuzzy continuous.(2)
For each closed (3, 2)-fuzzy subsetB of Y we have that f−1B is a closed (3, 2)-fuzzy subset of X.(3)
clf−1B⊆f−1clB for each (3, 2)-fuzzy set in Y.(4)
f−1intB⊆intf−1B for each (3, 2)-fuzzy set in Y.Proof.
They can be easily proved using Theorems6, 7, and 11.Theorem 13.
LetY,τ be a (3, 2)-fuzzy topological space and f:X⟶Y be a map. Then, there is a coarsest (3, 2)-fuzzy topology τ1 over X such that f is (3, 2)-fuzzy continuous.Proof.
Let us define a class of (3, 2)-fuzzy subsetsτ1 of X by(27)τ1≔f−1V:V∈τ.
We prove thatτ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous.(1)
We can write for anyx∈X that(28)αf−10Yx=α0Yfx=0=α0Xx.Similarly, we immediately haveβf−10Yx=β0Xx for any x∈X which implies f−10Y=0X. Now, as 0Y∈τ, we have 0X=f−10Y∈τ1. In a similar manner, it is easy to see that 1X=f−11Y∈τ1.(2)
Assume thatD1,D2∈τ1. Then, for i=1,2, there exists Bi∈τ such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Thus, we obtain for any x∈X that(29)αD1∩D2x=minαD1x,αD2x=minαf−1B1x,αf−1B2x=minαB1fx,αB2fx=αB1∩B2fx=αf−1B1∩B2x.Similarly, it is not difficult to see thatβD1∩D2=βf−1B1∩B2. Hence, we get D1∩D2∈τ1.(3)
Assume thatDii∈I is an arbitrary subfamily of τ1. Then, for any i∈I, there exists Bi∈τ1 such that f−1Bi=Di which implies αf−1Bi=αDi and βf−1Bi=βDi. Therefore, one can get for any x∈X that(30)α∪i∈IDix=supi∈IαDix=supi∈Iαf−1Bix=supi∈IαBifx=α∪i∈IBifx=αf−1∪i∈IBix.
On the other hand, it is easy to see thatβ∪i∈IDi=βf−1∪i∈IBi. Thus, we have ∪i∈IDi∈τ1.
From Theorem11, the (3, 2)-fuzzy continuity of f is trivial. Now, we prove that τ1 is the coarsest (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. Let τ2⊆τ1 be a (3, 2)-fuzzy topology over X such that f is (3, 2)-fuzzy continuous. If B∈τ1, then there is V∈τ such that f−1V=B. Since f is (3, 2)-fuzzy continuous with respect to τ2, we have B=f−1V∈τ2. Hence, τ2=τ1, as required.


## 3.3. (3, 2)-Fuzzy Separation Axioms

Separation axioms are one of the most important and popular notions in topological studies. They have been studied and applied to model some real-life issues in soft setting as explained in [16, 17].Definition 10.
LetX≠∅ and x∈X be a fixed element in X. Suppose that r1∈0,1 and r2∈0,1 are two fixed real numbers such that r13+r22≤1. Then, a (3, 2)-fuzzy point pr1,r2x=x,αpx,βpx is defined to be a (3, 2)-fuzzy set of X as follows.(31)pr1,r2xy≔r1,r2,ify=x,0,1,otherwise,for y∈X. In this case, x is called the support of pr1,r2x. A (3, 2)-fuzzy point pr1,r2x is said to belong to a (3, 2)-fuzzy set D=x,αDx,βDx of X denoted by pr1,r2x∈D if r1≤αDx and r2≥βDx. Two (3, 2)-fuzzy points are said to be distinct if their supports are distinct.Remark 6.
LetD1=x,αD1x,βD1x and D2=x,αD2x,βD2x be two (3, 2)-fuzzy sets of X. Then, D1⊆D2 if and only if pr1,r2x∈D1 implies pr1,r2x∈D2 for any (3, 2)-fuzzy point pr1,r2x in X.Definition 11.
Letr1,r3∈0,1, r2,r4∈0,1, and x,y∈X. A (3, 2)-fuzzy topological space X,τ is said to be(1)
T0 if for each pair of distinct (3, 2)-fuzzy points pr1,r2x,pr3,r4y in X, there exist two open (3, 2)-fuzzy sets L and K such that(32)L=x,1,0,y,0,1,orK=x,0,1,y,1,0.(2)
T1 if for each pair of distinct (3, 2)-fuzzy points pr1,r2x,pr3,r4y in X, there exist two open (3, 2)-fuzzy sets L and K such that(33)L=x,1,0,y,0,1,K=x,0,1,y,1,0.Proposition 1.
LetX,τ be a (3, 2)-fuzzy topological space. If X,τ is T1, then X,τ is T0.Proof.
The proof is straightforward from Definition11.Here is an example which shows that the converse of above proposition is not true in general.Example 10.
ConsiderX=c1,c2 with the (3, 2)-fuzzy topology τ=1X,0X,D, where D=c1,1,0,c2,0,1. Then, X,τ is T0 but not T1 because there does not exist an open (3, 2)-fuzzy set K such that K=x,0,1,y,1,0.Theorem 14.
LetX,τ be a (3, 2)-fuzzy topological space, r1,r3∈0,1, and r2,r4∈0,1. If X,τ is T0, then for each pair of distinct (3, 2)-fuzzy points pr1,r2x, pr3,r4y of X, clpr1,r2x≠clpr3,r4y.Proof.
LetX,τ be T0 and pr1,r2x, pr3,r4y be any two distinct (3, 2)-fuzzy points of X. Then, there exist two open (3, 2)-fuzzy sets L and K such that(34)L=x,1,0,y,0,1,orK=x,0,1,y,1,0.
LetL=x,1,0,y,0,1 exist. Then, Lc=x,0,1,y,1,0 is a closed (3, 2)-fuzzy set which does not contain pr1,r2x but contains pr3,r4y. Since clpr3,r4y is the smallest closed (3, 2)-fuzzy set containing pr3,r4y, then clpr3,r4y⊆Lc, and therefore pr1,r2x∉clpr3,r4y. Consequently, clpr1,r2x≠clpr3,r4y.Theorem 15.
LetX,τ be a (3, 2)-fuzzy topological space. If p1,0x is closed (3, 2)-fuzzy set for every x∈X, then, X,τ is T1.Proof.
Supposep1,0x is a closed (3, 2)-fuzzy set for every x∈X. Let pr1,r2x, pr3,r4y be any two distinct (3, 2)-fuzzy points of X; then, x≠y implies that p1,0xc and p1,0yc are two open (3, 2)-fuzzy sets such that(35)p1,0yc=x,1,0,y,0,1,p1,0xc=x,0,1,y,1,0.
Thus,X,τ is T1.


## 4. (3, 2)-Fuzzy Relations

A relation is a mathematical description of a situation where certain elements of sets are related to one another in some way. The system of fuzzy relation equations was first studied by Sanchez [18–21], who used it in medical research. Biswas [22] defined the method of intuitionistic medical diagnosis which involves intuitionistic fuzzy relations. Kumar et al. [23] used the applications of intuitionistic fuzzy set theory in diagnosis of various types of diseases. The notion of max-min-max composite relation for Pythagorean fuzzy sets was studied by Ejegwa [24], and the approach was improved and applied to medical diagnosis.In this section, we introduce the notions of max-min-max composite relation and improved composite relation for (3, 2)-FSs. Moreover, we provide a numerical example to elaborate on how we can apply the composite relations to obtain the optimal choices.Definition 12.
LetX and Y be two (crisp) sets. The (3, 2)-fuzzy relation R (briefly, (3, 2)-FR) from X to Y is a (3, 2)-FS of X×Y characterized by the degree of membership function αR and degree of non-membership function βR. The (3, 2)-FR R from X to Y will be denoted by RX⟶Y. If D is a (3, 2)-FS of X, then(1)
The max-min-max composition of the (3, 2)-FRRX⟶Y with D is a (3, 2)-FS C of Y denoted by C = R o D and is defined by(36)αRoDn=∨mαDm∧αRm,n,βRoDn=∧mβDm∨βRm,n,for alln∈Y.(2)
The improved composite relation ofRX⟶Y with D is a (3, 2)-FS C of Y denoted by C = R o D, such that(37)αRoDn=∨mαDm+αRm,n2,βRoDn=∧mβDm+βRm,n2,for alln∈Y.Definition 13.
LetQX⟶Y and RY⟶Z be two (3, 2)-FRs. Then, for all m,r∈X×Z and n∈Y,(1)
The max-min-max compositionR o Q is the (3, 2)-fuzzy relation from X to Z defined by(38)αRoQm,r=∨nαQm,n∧αRn,r,βRoQm,r=∧nβQm,n∨βRn,r.(2)
The improved composite relationR o Q is the (3, 2)-fuzzy relation from X to Z such that(39)αRoQm,r=∨nαQm,n+αRn,r2,βRoQm,r=∧nβQm,n+βRn,r2.Remark 7.
The improved composite and max-min-max composite relations for (3, 2)-fuzzy sets are calculated by the following:(40)SR=αRoQ−βRoQ⋅πRoQ.Example 11.
LetD1 and D2 be two (3, 2)-fuzzy sets for X=x1,x2,x3,x4. Assume that(41)D1=x1,0.8,0.61,x2,0.5,0.87,x3,0.85,0.55,x4,0.8,0.69,D2=x1,0.7,0.79,x2,0.78,0.73,x3,0.6,0.85,x4,0.89,0.54.
By using Definitions12 (1) and 13 (1), respectively, we find the max-min-max composite relation with application to D1 and D2 as follows:(42)αCd1i,d2k=∨xj0.7,0.5,0.6,0.8=0.8,βCd1i,d2k=∧xj0.79,0.87,0.85,0.69=0.69.
It is obvious that the minimum value of the membership values of the elements (that is,x1,x2,x3,x4) in D1 and D2, respectively, is 0.7,0.5,0.6, and 0.8. Also, the maximum value of the non-membership values of the elements (that is, x1,x2,x3,x4) in D1 and D2, respectively, is 0.79,0.87,0.85, and 0.69. From Remark 7, we can get(43)SR=0.8−0.69⋅50.0119≈0.52.
Again, by using Definitions12 (2) and 13 (2), respectively, we find the improved composite relation with application to D1 and D2 as follows:(44)αCd1i,d2k=∧xj0.75,0.64,0.725,0.845=0.845,βCd1i,d2k=∧xj0.7,0.8,0.7,0.615=0.615.
From Remark7, we can get(45)SR=0.845−0.615⋅50.018423875≈0.57.
Hence, from (43) and (45), we obtain that the improved composite relation produces better relation with greater relational value when compared to max-min-max composite relation.


## 5. Application of (3, 2)-Fuzzy Sets

We localize the idea of (3, 2)-FR as follows.LetS=r1,…,rl be a finite set of subjects related to the colleges, C=b1,…,bm be a finite set of colleges, and A=t1,…,tn be a finite set of students. Suppose that we have two (3, 2)-FRs, UA⟶S and RS⟶C, such that(46)U=t,r,αUt,r,βUt,r|t,r∈A×S,R=r,b,αRr,b,βRr,b|r,b∈S×C.whereαUt,r denotes the degree to which the student (t) passes the related subject requirement (r).βUt,r denotes the degree to which the student (t) does not pass the related subject requirement (r).αRr,b denotes the degree to which the related subject requirement (r) determines the college (b).βRr,b denotes the degree to which the related subject requirement (r) does not determine the college (b).T=RoU is the composition of R and U. This describes the state in which the students, ti, with respect to the related subject requirement, rj, fit the colleges, bk. Thus,(47)αTti,bk=∨rj∈SαUti,rj∧αRrj,bk,βTti,bk=∧rj∈SβUti,rj∨βRrj,bk,∀ti∈A and bk∈C, where i, j, and k take values from 1,…,n.The values ofαRoUti,bk and βRoUti,bk of the composition T = R o U are as follows (Table 1).Table 1
The compositionR o U.
R oUEMAESpSt1(0.81, 0.60)(0.81, 0.60)(0.81, 0.61)(0.81, 0.60)(0.81, 0.60)t2(0.82, 0.59)(0.82, 0.60)(0.82, 0.61)(0.82, 0.59)(0.82, 0.60)t3(0.82, 0.60)(0.82, 0.60)(0.82, 0.61)(0.82, 0.61)(0.82, 0.61)t4(0.82, 0.60)(0.83, 0.60)(0.82, 0.61)(0.82, 0.61)(0.82, 0.60)t5(0.83, 0.59)(0.83, 0.59)(0.83, 0.60)(0.83, 0.59)(0.83, 0.60)Table 2
The (3, 2)-fuzzy relationUA⟶S.
UA⟶SMathematicsComputer Sci.English Lang.BiologyPhysicsChemistryt1(0.81, 0.61)(0.80, 0.62)(0.81, 0.61)(0.80, 0.61)(0.71, 0.71)(0.81, 0.60)t2(0.80, 0.61)(0.81, 0.61)(0.80, 0.61)(0.62, 0.80)(0.82, 0.60)(0.82, 0.59)t3(0.82, 0.61)(0.82, 0.60)(0.82, 0.61)(0.80, 0.62)(0.62, 0.80)(0.81, 0.61)t4(0.81, 0.62)(0.83, 0.60)(0.81, 0.61)(0.81, 0.61)(0.80, 0.61)(0.82, 0.60)t5(0.83, 0.59)(0.82, 0.60)(0.83, 0.60)(0.82, 0.59)(0.81, 0.59)(0.83, 0.59)Table 3
The (3, 2)-fuzzy relationRS⟶C.
RS⟶CEMAESpSMathematics(0.83, 0.59)(0.84, 0.59)(0.80, 0.62)(0.82, 0.61)(0.83, 0.60)Computer Sci.(0.82, 0.60)(0.83, 0.59)(0.80, 0.61)(0.80, 0.62)(0.80, 0.61)English Lang.(0.84, 0.59)(0.83, 0.60)(0.84, 0.59)(0.83, 0.60)(0.84, 0.59)Biology(0.81, 0.61)(0.80, 0.609)(0.80, 0.62)(0.81, 0.61)(0.81, 0.60)Physics(0.83, 0.60)(0.82, 0.60)(0.82, 0.61)(0.82, 0.60)(0.82, 0.60)Chemistry(0.83, 0.59)(0.83, 0.60)(0.82, 0.61)(0.84, 0.59)(0.83, 0.60)Table 4
Greatest value given byT=αTti,bk−βTti,bk⋅πTti,bk.
TEMAESpSt10.4250.4250.4340.4250.425t20.4470.4500.4550.4470.450t30.4500.4500.4550.4550.455t40.4500.4790.4550.4550.450t50.4740.4740.4790.4740.479If the value ofT is given by the following:(48)T=αTti,bk−βTti,bk⋅πTti,bk,then the student placement can be achieved.


### 5.1. Application Example

By using a hypothetical case with quasi-real data, we apply this method. LetA=t1,t2,t3,t4,t5 be the set of students for the colleges; S = {English Lang., Mathematics, Biology, Physics, Chemistry, Computer Sci.} be the set of related subject requirement to the set of colleges; and C = {College of Engineering (E), College of Medicine (M), College of Agricultural Engineering Sciences (AE), College of Sport Sciences (Sp), College of Science (S)} be the set of colleges the students are vying for (Algorithm 1).Algorithm 1: Determination of the optimal college for students.
Step 1. The (3, 2)-fuzzy relation UA⟶S and the (3, 2)-fuzzy relation RS⟶C are given as in Tables 2 and 3, respectively. These data in (3, 2)-F values are assumably obtained after students finished from preparatory school.Step 2. Compute the composition R o U as in Table 1.Step 3. Calculate T=αTti,bk−βTti,bk⋅πTti,bk as in Table 4.Step 4. We present the decision making from Table 4. The greatest value of relation between students and colleges is taken for decisions.From Table4 and based on suitability of the students to the list of colleges, this decision making is made:(1)
t1 and t2 are suitable to study at College of Agricultural Engineering Sciences.(2)
t3 is suitable to study at College of Agricultural Engineering Sciences, College of Sport Sciences, and College of Science.(3)
t4 is suitable to study at College of Medicine.(4)
t5 is suitable to study at College of Agricultural Engineering Sciences and College of Science.


## 5.1. Application Example

By using a hypothetical case with quasi-real data, we apply this method. LetA=t1,t2,t3,t4,t5 be the set of students for the colleges; S = {English Lang., Mathematics, Biology, Physics, Chemistry, Computer Sci.} be the set of related subject requirement to the set of colleges; and C = {College of Engineering (E), College of Medicine (M), College of Agricultural Engineering Sciences (AE), College of Sport Sciences (Sp), College of Science (S)} be the set of colleges the students are vying for (Algorithm 1).Algorithm 1: Determination of the optimal college for students.
Step 1. The (3, 2)-fuzzy relation UA⟶S and the (3, 2)-fuzzy relation RS⟶C are given as in Tables 2 and 3, respectively. These data in (3, 2)-F values are assumably obtained after students finished from preparatory school.Step 2. Compute the composition R o U as in Table 1.Step 3. Calculate T=αTti,bk−βTti,bk⋅πTti,bk as in Table 4.Step 4. We present the decision making from Table 4. The greatest value of relation between students and colleges is taken for decisions.From Table4 and based on suitability of the students to the list of colleges, this decision making is made:(1)
t1 and t2 are suitable to study at College of Agricultural Engineering Sciences.(2)
t3 is suitable to study at College of Agricultural Engineering Sciences, College of Sport Sciences, and College of Science.(3)
t4 is suitable to study at College of Medicine.(4)
t5 is suitable to study at College of Agricultural Engineering Sciences and College of Science.


## 6. Discussion

The main idea of this work is to introduce a new type of fuzzy set called (3, 2)-FS. We illustrated that this type produces membership grades larger than intuitionistic and Pythagorean fuzzy sets which are already defined in the literature. However, Fermatean fuzzy sets give a larger space of membership grades than (3, 2)-FS. Figure2 illustrates the relationships between these types of fuzzy sets.Figure 2
Comparison of grade space of IFSs, PFSs, FFSs, and (3, 2)-FSs.We summarize the relationships in terms of the space of membership and non-membership grades in the following figure.Regarding topological structure, we illustrated that every fuzzy topology in the sense of Chang (intuitionistic fuzzy topology and Pythagorean fuzzy topology) is a (3, 2)-fuzzy topology. In contrast, every (3, 2)-fuzzy topological space is a Fermatean fuzzy topological space because every (3, 2)-fuzzy subset of a set can be considered as a Fermatean fuzzy subset. The next example elaborates that Fermatean fuzzy topological space need not be a (3, 2)-fuzzy topological space.Example 12.
LetX=x1,x2. Consider the following family of Fermatean fuzzy subsets τ=1X,0X,D1,D2, where(49)D1=x1,αD1x1=0.75,βD1x1=0.81,x2,αD1x2=0.85,βD1x2=0.7,D2=x1,αD2x1=0.76,βD2x1=0.81,x2,αD2x2=0.86,βD2x2=0.7.
Observe thatX,τ is a Fermatean fuzzy topological space, but X,τ is not a (3, 2)-fuzzy topological space.


## 7. Conclusions

In this paper, we have introduced a new generalized intuitionistic fuzzy set called (3, 2)-fuzzy sets and studied their relationship with intuitionistic fuzzy, Pythagorean fuzzy, and Fermatean fuzzy sets. In addition, some operators on (3, 2)-fuzzy sets are defined and their relationships have been proved. The notions of (3, 2)-fuzzy topology, (3, 2)-fuzzy neighborhood, and (3, 2)-fuzzy continuous mapping were studied. Furthermore, we introduced the concept of (3, 2)-fuzzy points and studied separation axioms in (3, 2)-fuzzy topological space. We also introduced the concept of relation to (3, 2)-fuzzy sets, called (3, 2)-FR. Moreover, based on academic performance, the application of (3, 2)-FSs was explored on student placement using the proposed composition relation.In future work, more applications of (3, 2)-fuzzy sets may be studied; also, (3, 2)-fuzzy soft sets may be studied. In addition, we will try to introduce the compactness and connectedness in (3, 2)-fuzzy topological spaces. The motivation and objectives of this extended work are given step by step in this paper.



---

*Source: 1272266-2021-10-27.xml*