# A Novel Fuzzy Structure: Infra-Fuzzy Topological Spaces

Authors: Zanyar A. Ameen; Tareq M. Al-shami; A. A. Azzam; Abdelwaheb Mhemdi
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

Obtaining a weaker condition that preserves some inspired topological properties is always desirable. As a result, we introduce the concept of infra-fuzzy topology, which is a subset family that degrades the concept of fuzzy topology by omitting the condition of closedness under arbitrary unions. Fundamental properties of infra-fuzzy topological spaces are investigated, including infra-fuzzy open and infra-fuzzy closed sets, infra-fuzzy interior and infra-fuzzy closure operators, and the infra-fuzzy boundary of a fuzzy set. It is not possible to expect the latter concepts to have properties identical to those in ordinary fuzzy topological spaces. More precisely, the infra-fuzzy interior of a set need not be infra-fuzzy open, and the infra-fuzzy closure and boundary of a set may not be infra-fuzzy closed. Then, employing infra-fuzzy neighborhood systems, infra-fuzzy Q-neighborhood systems, the basis of infra-fuzzy topology, and infra-fuzzy relative topology, we propose several approaches for generating infra-fuzzy topologies. Finally, we define the notions of continuity, openness, closedness, and homeomorphism of mappings in the context of infra fuzziness and investigate some of their properties and characterizations. We show that the usual characterization of earlier notions in the infra-fuzzy structure is incorrect. We demonstrate that the family of all infra-fuzzy homeomorphisms on an infra-fuzzy topological space forms a group under mappings composition. We finish this work by proving that each infra-fuzzy homeomorphism between two infra-fuzzy topological spaces produces an isomorphism on groups of infra-fuzzy homeomorphisms of the corresponding spaces.


---

## Body

## 1. Introduction

General topology, it is also named point-set topology, is the branch of topology that deals with the fundamental set-theoretic notions and constructions. It is the basis for most other branches of topology, including differential, geometric, and algebraic topology. The growth of topology has been supported by the variety of classes of topological spaces, examples and their aspects and relationships. The conception of topological space was enlarged.In the 80s, Mashhour et al. [1] initiated a supra topological space that is not closed under finite intersections. This research has been supported by numerous authors from around the world (see [2–5]). In the following decade, Császár [6] started a comprehensive investigation of families that are closed solely by arbitrary unions under the name of a generalized topological space. This area of research is essential in computer science and its applications. In formal concept analysis and data clustering, Soldano [7] considered the generalized topological space and called it extensional abstraction. It was also used in Banach games and the entropy problem in [8, 9]. Additional topological space generalizations appeared after that. These structures include minimal structure [10, 11], weak structure [12], and generalized weak structure [13] and so on. In fact, the entire study field has developed during the last two decades.Fuzzy topology [14] and soft topology [15] have a role no less than the general topology and do not lack any of the abovementioned generalizations. Fuzzy topology was generalized to supra fuzzy topology in [16]. The relation between fuzzy soft and soft topological spaces has been recently investigated by [17].The concept of an infra topological space was explicitly established by Al-Odhari [18] in 2015, although it has attracted little attention from researchers. On the other hand, infra topological spaces are useful to examine, as Witczak demonstrated in [19]. Recently, Al-shami [20] has studied the main properties of infra soft topological spaces. Also, he with coauthors explored the main topological concepts in these structures such as continuity [21], compactness [22], connectedness [23], and separation axioms [24]. The words of Witczak and the work of the Al-shami motivate us to study the infra topological structure in fuzzy settings. We begin by discussing the main properties of infra-fuzzy topological spaces, proceeded by the definition of various operators using infra-fuzzy open and closed sets. Finally, we examine at infra-fuzzy continuous, infra-fuzzy open, and infra-fuzzy closed mappings and infra-fuzzy homeomorphisms.


## 2. Preliminaries

LetZ be a universe (domain), and let FZ be the class of all fuzzy sets in Z. The closed interval 0,1 is denoted by I. All undefined terminologies used in the manuscript can be found in [14, 25, 26].Definition 1 (see [25]).
A mappingμ from Z to the unit interval I is named a fuzzy set in Z. The value μz is called the degree of the membership of z in μ for each z∈Z. The support of μ is the set z∈Z:μz>0. The complement of μ is, symbolized by μc (or 1−μ if there is no confusion), given by μcz=1−μz for all z∈Z.Definition 2 (see [25]).
Letμj:j∈J⊆FZ, where J is any index set. Then,
(i)
∨μjz=supμjz:j∈J, for each z∈Z.(ii)
∧μjz=infμjz:j∈J, for each z∈Z.Definition 3 (see [26]).
Letμ be a fuzzy set in Z. Then, μ is said to be a fuzzy point, denoted by zp, with the support z∈Z and the value p∈0,1 if zp:Z⟶I is the function defined as follows: for each y∈Z,
(1)zpy=pify=z;0otherwise.
A fuzzy pointzp said to be in μ, denoted by zp∈μ, if p≤μz. We may write p∈μ if no confusion causes or by a letter p we mean a fuzzy point. We write FpZ for the set of all fuzzy points in Z.Definition 4 (see [26]).
LetZ≠∅, and let μ,ν∈FZ. Then, μ,ν intersect if there is z∈Z such that μz∧νz≠0.Definition 5 (see [26]).
LetZ≠∅, and let μ,ν∈FZ. Then,
(i)
zp∈FpZ is called quasi-coincident with μ if p+μz>1 and is denoted zp≺μ.(ii)
zp∈FpZ is called not quasi-coincident with μ if p+μz≤1 and is denoted zp⊀μ.(iii)
μ is called quasi-coincident with ν (at z) if there exists z∈Z such that μz+νz>1 and is denoted μ≺ν.Lemma 6 (see [26, Proposition 17]).
LetZ≠∅, μ,ν∈FZ, zp∈FpZ. Then,
(i)
ifμ≺ν at z, then μz∧νz≠0;(ii)
zp∈μ iff zp⊀1−μ; and(iii)
μ≤ν iff μ⊀1−ν.Definition 7 (see [14]).
LetY,Z be nonempty sets, let μ∈FZ and λ∈FY, and let π:Z⟶Y. The image of μ under π is defined by
(2)πμz=supz∈π−1yμz,ifπ−1y≠∅,0,ifπ−1y=∅,for each y∈Y, and the inverse image of λ under π is defined by π−1λz=λπz, for each z∈Z.


## 3. Infra-Fuzzy Topology

In this section, we familiarize the concept of infra-fuzzy topology and present some methods to produce this concept such as the basis of infra-fuzzy topology, infra-fuzzy neighborhood systems, and infra-fuzzy Q-neighborhood systems. To elucidate the obtained results and relationships, we display some examples.Definition 8.
A subcollectionT of FZ is said to be an infra-fuzzy topology on Z if
(i)
0,1∈T; and(ii)
μ1∧μ2∈T whenever μ1,μ2∈T.
The pairZ,T is called an infra-fuzzy topological space, and the set of all infra-fuzzy topologies on Z is denoted by IFTZ. The members of T are called infra-fuzzy open (or IF-open) subsets of Z, and their complements are called infra-fuzzy closed (or IF-closed) sets. The members of Tc are also called IF-closed sets.Remark 9.
Evidently, each fuzzy topology is an infra-fuzzy topology, but not conversely.Lemma 10.
LetTj:j∈J be a subclass of IFTZ, where J is any index set. Then, T=∧j∈JTj∈IFTZ.Proof.
Straightforward.Lemma 11.
LetC be a subclass of FZ. There exists a unique T∈IFTZ containing C, and if T∗∈IFTZ that includes C, then T≤T∗.Proof.
Note that such an infra-fuzzy topology always exists becauseFZ is the infra-fuzzy topology on Z which includes C. Consider T, the intersection of all those infra-fuzzy topologies on Z which include C. Then, it follows from Lemma 10 that T is the required infra-fuzzy topology.Definition 12.
LetC be a subclass of FZ. The unique T∈IFTZ obtained in the above lemma is called the infra-fuzzy topology on Z generated by the collection C and is denoted by TC, which is the smallest infra-fuzzy topology on Z containing C.IfT∗,T′∈IFTZ, then T∗∨T′∈IFTZ is false in general.Example 1.
Letμ1,μ2 and μ3 in FI be defined by
(3)μ1x=1,if0≤x≤1/2−4x+3,if1/2≤x≤3/40,if3/4≤x≤1,μ2x=0,if0≤x≤1/4−1+4x,if1/4≤x≤1/21,if1/2≤x≤1,μ3x=0,if0≤x≤1/4−1+4x,if1/4≤x≤1/2−4x+3,if1/2≤x≤3/40,if3/4≤x≤1.
Then,T∗=0,μ1,1, T′=0,μ2,1∈IFTI, but T∗∨T′=0,μ1,μ2,1∉IFTI, since μ1∧μ2=μ3∉T∗∨T′.Definition 13.
LetT∈IFTZ. A subcollection B of T is called a base for T, if for each μ∈T, there exists a finite B0≤B such that μ=∧B0.Remark 14.
Each subcollectionB of FZ containing 0,1 can be a base for some T∈IFTZ.Definition 15.
LetT∈IFTZ. Then, η∈FZ is called an infra-fuzzy neighborhood of a fuzzy point p∈FpZ if there is μ∈T such that p∈μ≤η. The set of all infra-fuzzy neighborhoods of p∈FpZ is called an infra-fuzzy neighborhood system of p and is denoted by IFNp.Lemma 16.
LetT∈IFTZ. If μ∈T, then for each p∈μ, μ∈IFNp.Proof.
Standard.The converse of the above result is generally false as shown below.Example 2 (see [27, Example 4]).
Considerμ1,μ2,μ3∈FI defined below:
(4)μ1x=0,if0≤x≤1/22x−1,if1/2≤x≤1,μ2x=1,if0≤x≤1/4−4x+2,if1/4≤x≤1/20,if1/2≤x≤1,μ3x=1,if0≤x≤1/4−4x+2,if1/4≤x≤1/22x−1,if1/2≤x≤1.
Then,T=0,μ1,μ2,1∈IFTI. Note that ∀p∈μ3, μ3∈IFNp, but μ3∉T.Proposition 17.
LetT∈IFTZ and let p∈FpZ. Then, IFNp satisfies the following properties:
(i)
IFNp≠∅;(ii)
p∈η for each η∈IFNp;(iii)
ifη∈IFNp and μ∈FZ with η≤μ, then μ∈IFNp;(iv)
ifμ,η∈IFNp, then μ∧η∈IFNp; and(v)
ifη∈IFNp, then there exists μ∈IFNp such that η∈IFNq for each q∈μ.Proof.
(i)
Since1∈T and 1 contains all p∈FpZ, then 1∈IFNp for each p.(ii)
It is clear from the definition.(iii)
Letη∈IFNp and μ∈FZ. There is ν∈T such that p∈ν≤η. Since η≤μ, so p∈ν≤μ. Hence μ∈IFNp.(iv)
Letμ,η∈IFNp. There exists ν,σ∈T such that p∈ν≤η and p∈σ≤μ. Therefore, p∈σ∧ν≤η∧μ. Since σ∧ν∈T, hence μ∧η∈IFNp.(v)
Letη∈IFNp. There is ν∈T such that p∈ν≤η. Since ν∈T, by Lemma 16, ν∈IFNq for each q∈ν. Therefore, for each q∈ν, there is σ∈T such that q∈σ≤ν≤η. Thus, η∈IFNq.Theorem 18.
LetZ≠∅ be a universe. If, for each p∈FpZ, Cp is a subclass of FZ that fulfills the properties given in Proposition 17, then there exists a unique T∗∈IFTZ such that Cp=IFNpT∗, where IFNpT∗ is the set of all infra-fuzzy neighborhoods of p in Z,T∗.Proof.
SupposeT∗=μ:μ∈Cp,∀p∈μ, where μ∈FZ. We first show that T∗∈IFTZ. Evidently, 0∈T∗. By (III), 1∈T∗. Let μ,ν∈T∗ and let p∈μ∧ν. Then, p∈μ, p∈ν, and so μ,ν∈Cp. By (IV), μ∧ν∈Cp. Therefore, μ∧ν∈T∗. Thus, T∗∈IFTZ.
We now prove thatCp=IFNpT∗. Let η∈IFNpT∗. Then, there exists μ∈T∗ such that p∈μ≤η. By definition of T∗, μ∈Cp for each p∈μ. By (III), η∈Cp. Thus, IFNpT∗≤Cp.
Supposeη∈Cp. Set μ≔∨q∈FpZ:η∈IFNqT∗. Obviously, μ∈FZ with p∈μ. By (II), μ≤η. It is enough to prove that μ∈T∗. That is μ∈Cq for each q∈μ. Assume q∈μ. By definition of μ, η∈IFNqT∗. Since IFNqT∗≤Cq, so η∈Cq. By (V), there exists σ∈Cq such that η∈Cr for each r∈σ. Again, by definition of μ, r∈μ, and so σ≤μ. By (III), μ∈Cq. Hence, Cp≤IFNpT∗.Definition 19.
LetT∈IFTZ. Then, η∈FZ is called an infra-fuzzy Q-neighborhood of a fuzzy point p∈FpZ if there is μ∈T such that p≺μ≤η. The set of all infra-fuzzy Q-neighborhoods of p∈FpZ is called an infra-fuzzy Q-neighborhood system of p and is denoted by IFQNp.Proposition 20.
LetT∈IFTZ and let p∈FpZ. Then, IFQNp satisfies the following properties:
(i)
IFQNp≠∅;(ii)
p≺η for each η∈IFQNp;(iii)
ifη∈IFQNp and μ∈FZ with η≤μ, then μ∈IFQNp;(iv)
ifμ,η∈IFQNp, then μ∧η∈IFQNp; and(v)
ifη∈IFQNp, then there exists μ∈IFQNp such that η∈IFQNq for each q≺μ.Proof.
Similar to Proposition17.Theorem 21.
LetZ≠∅ be a universe. If, for each p∈FpZ, Cp is a subclass of FZ that fulfills the properties given in Proposition 20, then there exists a unique T∗∈IFTZ such that Cp=IFQNpT∗, where IFQNpT∗ is the set of all infra-fuzzy Q-neighborhoods of p in Z,T∗.Proof.
Similar to Theorem18.Definition 22.
LetZ,T be an infra-fuzzy topological space and Y∈FZ. Then, TY=μ∧Y:μ∈T is said to be an infra-fuzzy topological space related to Y. The pair Y,TY is called an infra-fuzzy topological subspace of Z,T.The proofs of results given below are the same as in the case of ordinary fuzzy topology; thus, we leave them out.Proposition 23.
LetY,TY be an infra-fuzzy topological subspace of Z,T. Then, λ is an IF-closed subset of Y,TY iff there exists an IF-closed subset ν of Z,T such that λ=ν∧Y.Proposition 24.
LetY be an IF-open subset of Z,T. Then, μ is an IF-open set in Y,TY iff it is an IF-open in Z,T.Proposition 25.
LetY be an IF-closed subset of Z,T. Then, μ is an IF-closed set in Y,TY iff it is an IF-closed in Z,T.


## 4. Infra-Fuzzy Operators

In this part, we define infra-fuzzy open and closed operators and scrutinize their main characterizations. Also, we formulate the concept of infra-fuzzy boundary of a fuzzy set and deduce some formulations to calculate it. With the help of examples, we demonstrate that some properties of these concepts which exist in ordinary fuzzy topology evaporate in the frame of infra-fuzzy topology.Definition 26.
LetT∈IFTZ and let η∈FZ. The infra-fuzzy interior of η is defined as
(5)iη≔supμ:μ≤η,μ∈T.Lemma 27.
LetT∈IFTZ, η∈FZ, and p∈η. Then, p∈iη iff there is μ∈T such that p∈μ≤η.Proof.
Apply the above definition.Theorem 28.
LetT∈IFTZ, and let μ,ν∈FZ. Then, the following properties hold:
(i)
iμ≤μ;(ii)
ifμ≤ν, then iμ≤iν;(iii)
ifμ∈T, then iμ=μ;(iv)
iμ=iiν; and(v)
iμ∧ν=iμ∧iν.Proof.
We only prove (iv) and (v); the other parts can be followed from the definition:(i)
By (i), we have thatiiμ≤iμ. Let p∈iμ. There exists σ∈T such that p∈σ≤μ. By (ii) and (iii), p∈iσ=σ≤iμ. Therefore, p∈iiμ. Hence, iμ=iiν.(ii)
Sinceμ∧ν≤μ and μ∧ν≤ν, so by (ii), iμ∧ν≤iμ and iμ∧ν≤iν. Thus, iμ∧ν≤iμ∧iν. On the other hand, we let p∈iμ∧iν. Then, p∈iμ, p∈iν. Therefore, there exists σ,η∈T such that p∈σ≤μ and p∈η≤ν. Since σ∧η∈T and p∈σ∧η≤μ∧ν. Hence, p∈iμ∧ν and so iμ∧ν=iμ∧iν.It is worth remarking that the converse of (iii) in the above need not be true. On the other hand, it is true in ordinary fuzzy topological spaces.Example 3.
Considerμ3 and T given in Example 2. One can easily check that iμ3=μ3, but μ3∈T.The above example asserts that the infra-fuzzy interior of a fuzzy set need not be an infra-fuzzy open set. However, the following result still true.Theorem 29.
LetT∈IFTZ, p∈FpZ, and μ∈FZ. Then p∈iμ iff there exists η∈IFNp such that η≤μ.Proof.
Apply the definition ofiμ and Proposition 17.Definition 30.
LetT∈IFTZ and let η∈FZ. The infra-fuzzy closure of η is defined as cη≔infμ:η≤μ,1−μ∈T.Theorem 31.
LetT∈IFTZ, μ∈FZ, and a=zp∈FpZ. Then, a∈cμ iff for each η∈IFQNa, η≺μ.Proof.
Ifa∈cμ, then for all κ∈Tc with μ≤κ, a∈κ (i.e. p<κz). By taking the complement, if a∈cμ, then for all σ=κc∈T with σ≤1−μ, σz≤1−μz. Equivalently, for all σ=κc∈T with σz≤1−μz, σ≤1−μ. This implies that σ≺μ from Lemma 6. Hence, for each η∈IFQNp, we have η≺μ.By reversing the above steps, one can prove the converse.Theorem 32.
LetT∈IFTZ and let μ,ν∈FZ. Then the following properties hold:
(i)
μ≤cμ;(ii)
ifμ≤ν, then cμ≤cν;(iii)
ifμ∈Tc, then μ=cμ;(iv)
cμ=ccν; and(v)
cμ∨ν=cμ∨cν.Proof.
(i) and (ii) Follow from the definition.
(iii) Use Theorem31.
(iv) By (i),cμ≤ccμ. On the other hand, let p∈ccμ. By Theorem 31, for each η∈IFQNp, cη≺μ, but η≤cη. Therefore, η≺μ for each η∈IFQNp. Hence p∈cμ. This shows that cμ=ccν.
(v) The first direction is simple asμ≤μ∨ν and ν≤μ∨ν, then, by (ii), cμ≤cμ∨ν and cν≤cμ∨ν. Thus cμ∨cν≤cμ∨ν. For other direction, we have that μ≤cμ and ν≤cν and so μ∨ν≤cμ∨cν. By the definition of infra-fuzzy closure, we must have cμ∨ν≤cμ∨cν. Hence the result.Remark 33.
We shall state that the infra-fuzzy closure of a fuzzy set need not beIF-closed (c.f. Theorem 32 (iii)). The example showing this can be concluded from Example 3.Corollary 34.
LetT∈IFTZ and let μj:j∈J⊆FZ, where J is any index set. Then,
(i)
∨j∈Jcμj≤c∨j∈Jμj;(ii)
∨j∈Jiμj≤i∨j∈Jμj;(iii)
c∧j∈Jμj≤∧j∈Jcμj; and(iv)
i∧j∈Jμj≤∧j∈Jiμj;Proposition 35.
LetT∈IFTZ and let μ∈FZ. Then
(i)
iμ=1−c1−μ;(ii)
cμ=1−i1−μ;(iii)
1−cμ=i1−μ; and(iv)
c1−μ=1−iμ.Proof.
We know thatiμ=supϑ, where ϑ=ν:ν∈T,ν≤μ. Obviously, 1−ϑ=1−ν:ν∈ϑ=κ:κ∈Tc,1−μ≤κ, where κ=1−ν. Therefore, c1−μ=inf1−ϑ. Now, we have 1−c1−μ=1−inf1−ϑ=sup1−1−ϑ=supϑ=iμ.
The other parts can be deduced by a similar technique.Proposition 36.
LetY,TY be an infra-fuzzy subspace of Z,T and let q∈FpY. Then, η∈IFNqTY iff there exists λ∈IFNqT such that η=λ∧Y.Proof.
Apply Definition22.Theorem 37.
LetY,TY be an infra-fuzzy subspace of Z,T and let μ∈FY. Then
(i)
iμ=iYμ∧iY;(ii)
iμ≤iYμ; and(iii)
cYμ=cμ∧Y,where iYμ,cYμ are, respectively, the infra-fuzzy interior and the infra-fuzzy closure of μ in the subspace Y,TY.Proof.
(i)
Letp∈iμ. There exists σ∈T such that p∈σ≤μ≤Y. Therefore, p∈σ∧Y≤μ and so p∈iYμ and p∈iY. Thus, iμ≤iYμ∧iY. For another direction, we let p∈iYμ∧iY. There are α,β∈T such that p∈α∧Y≤μ and p∈β≤Y. But α∧β∈T with p∈α∧β≤μ. Therefore, p∈iμ, and hence, iYμ∧iY≤iμ. Hence, (i).(i)
From (i) and Proposition36.(ii)
By the definition,(6)cYμ=infκ:1−κ∈TY,μ≤κ=infλ∧Y:1−λ∈T,μ≤λ∧Yby Proposition23=infλ∧Y:1−λ∈T,μ≤λ=infλ:1−λ∈T,μ≤λ∧Y=cμ∧Y.Definition 38.
LetT∈IFTZ and let η∈FZ. The infra-fuzzy boundary of η is defined by bη=cη−iη.From the achieved results on infra-fuzzy interior and closure, one can prove the following result:Theorem 39.
LetT∈IFTZ and let μ∈FZ. Then,
(i)
bμ=cμ∧c1−μ;(ii)
bμ=iμ∨i1−μ;(iii)
bμ=b1−μ;(iv)
bμ∨μ≤cμ;(v)
bμ may not be in Tc;(vi)
ifμ∈T, then bμ∧μ=0;(vii)
if1−μ∈T, then bμ≤μ; and(viii)
ifμ,1−μ∈T, then bμ=0.


## 5. Infra-Fuzzy Continuous Mappings

We dedicate this section to introducing the notion of continuity between infra-fuzzy topological structures. We discuss some descriptions of continuity as well as establish some properties that hold true under continuity.Definition 40.
Letπ be a mapping from an infra-fuzzy topological space Y,R into another infra-fuzzy topological space Z,T. The mapping π is called infra-fuzzy continuous at r∈FpY if r∈π−1β∈T for each β∈T with πr∈β. It is said that π is infra-fuzzy continuous if it is infra-fuzzy continuous at each r∈FpY.Remark 41.
We shall note that the above definition is not equivalent to the usual definition of continuity at a single fuzzy point. Namely, a mappingπ is infra-fuzzy continuous at r∈FpY if for each β∈T with πr∈β, there exists α∈R such that r∈α and πα≤β. Example 4 justifies our claim.Theorem 42.
For a mappingπ:Y,R⟶Z,T, the following statements are equivalent:
(i)
π is infra-fuzzy continuous;(ii)
ifπ−1β∈R for each β∈T; and(iii)
ifπ−1η∈Rc for each η∈Tc.Proof. (i)⇔(ii).
From Definitions8–40.
(ii)⇒(iii) Let η∈Tc. Then 1−η∈T. By (ii), π−11−η∈R. By Theorem 4.1 in [14], 1−π−1η=π−11−η∈R. Hence, π−1η∈Rc.
(iii)⇒(ii) By reversing the above steps, the proof will be accomplished.Theorem 43.
Ifπ:Y,R⟶Z,T is an infra-fuzzy continuous mapping, then π−1iμ≤iπ−1μ for each μ∈FZ.Proof.
Letμ∈FZ and let p∈π−1iμ. Then, πp∈ππ−1iμ≤iμ. There exists σ∈T such that πp∈σ≤μ. Therefore, p∈π−1σ≤π−1μ. Since π is infra-fuzzy continuous, then π−1σ∈R, and so p∈iπ−1μ. Thus, π−1iμ≤iπ−1μ.The converse of the above theorem is generally false, as shown below:Example 4.
Letμ1,μ2,μ3 and μ4 in FI be defined by μ1x=x,μ2x=1−x,(7)μ3x=x,if0≤x≤1/21−x,if1/2≤x≤1,μ4x=1−x,if0≤x≤1/2x,if1/2≤x≤1,
Letπ:I,T⟶I,R be the identity mapping, where T=0,μ1,μ2,μ3,1 and R=0,μ1,μ2,μ3,μ4,1. By some computation, we can check that for each μ∈FZ, we have π−1iμ≤iπ−1μ. On the other hand, μ4∈R, but π−1μ4∉T.Theorem 44.
Ifπ:Y,R⟶Z,T is an infra-fuzzy continuous injection, then iπμ≤πiμ for each μ∈FY.Proof.
Letq∈FpZ. If q∈iπμ. Then there is σ∈T such that p∈σ≤πμ. Therefore, π−1q∈π−1σ≤π−1πμ. Since π is infra-fuzzy continuous injective, so π−1σ∈R, and hence π−1q∈iμ. Thus, q∈πiμ. This shows that iπμ≤πiμ.Lemma 45.
For a mappingπ:Y,R⟶Z,T, the following statements are equivalent:
(i)
π−1iμ≤iπ−1μ for each μ∈FZ;(ii)
cπ−1μ≤π−1cμ for each μ∈FZ; and(iii)
πcν≤cπν for each ν∈FY.Proof. (i)⇔(ii).
Letμ∈FZ. Then, 1−μ∈FZ. By (i), π−1i1−μ≤iπ−11−μ. Proposition 35 and Theorem 4.1 in [14] guarantee that π−11−cμ≤i1−π−1μ⇔1−π−1cμ≤1−cπ−1μ⇔cπ−1μ≤π−1cμ.
(ii)⇒(iii) Let ν∈FY. Then πν∈FZ, so by (ii), cπ−1πν≤π−1cπν. Therefore, cν≤π−1cπν. This implies that πcν≤ππ−1cπν≤cπν. Thus, we get (iii).
(iii)⇒(ii) This part can be completed using the same technique as the previous one.From Theorem43 and Lemma 45, we haveCorollary 46.
Ifπ:Y,R⟶Z,T is an infra-fuzzy continuous mapping, then cπ−1μ≤π−1cμ for each μ∈FZ.Corollary 47.
Ifπ:Y,R⟶Z,T is an infra-fuzzy continuous mapping, then πcμ≤cπμ for each μ∈FY.Theorem 48.
LetX,S,Y,R,Z,T be infra-fuzzy topological spaces. If A⊆Y and B⊆Z, then the following statements hold: [(i)]
(i)
the inclusion mappingj:A,RA⟶Y,R is infra-fuzzy continuous.(ii)
if a mappingπ:Y,R⟶Z,T is infra-fuzzy continuous, then πA:A,RA⟶Z,T is infra-fuzzy continuous.(iii)
letπ:Y,R⟶Z,T be infra-fuzzy continuous and let πY⊆B. If ϕ:Y,R⟶B,TB is a mapping given by ϕp=πp for all p∈FpY, then ϕ is infra-fuzzy continuous.(iv)
letπ:Y,R⟶B,TB be infra-fuzzy continuous. If ϕ:Y,R⟶Z,T is a mapping given by ϕp=πp for all p∈FpY, then ϕ is infra-fuzzy continuous.(v)
ifπ:Y,R⟶Z,T and ϕ:X,S⟶Y,R are infra-fuzzy continuous mappings, then π∘ϕ:X,S⟶Z,T is infra-fuzzy continuous.Proof.
(i)
For eachμ∈R, we have j−1μ=μ∧A∈RA by the definition of relative infra-fuzzy topology on A. Hence, j:A,RA⟶Y,R is infra-fuzzy continuous.(ii)
Assume thatμ∈T. Then πA−1μ=π−1μ∧A. By infra-fuzzy continuity of π, π−1μ∈R. Therefore, πA−1μ∈RA. Thus πA is infra-fuzzy continuous.(iii)
Letμ∈TB. There exists ν∈T such that μ=ν∧B. Since π:Y,R⟶Z,T is infra-fuzzy continuous, π−1ν∈R. Since πY⊆B, so we have that(8)ϕ−1μ=ϕ−1ν∧B=ϕ−1ν∧ϕ−1B=π−1ν∧Y=π−1ν.
Thus,π−1μ∈R and hence ϕ is infra-fuzzy continuous.
(iv)
Letμ∈T. Obviously, μ∧B∈TB. By infra-fuzzy continuity of π and by definition of ϕ, we have π−1μ∧B=π−1μ∧π−1B=π−1μ∧Y=π−1μ∈R, and π−1μ=ϕ−1μ. Thus ϕ is infra-fuzzy continuous.(v)
Letμ∈T. Then π∘ϕ−1μ=ϕ−1π−1μ. Since π is infra-fuzzy continuous, then π−1μ∈R. Therefore, by infra-fuzzy continuity of ϕ, we get ϕ−1π−1μ∈S. Thus π∘ϕ is infra-fuzzy continuous.Theorem 49.
Letπ:Y,R⟶Z,T be a mapping and let Y=∪i=1nFi, where each Fi∈Rc. Then π is infra-fuzzy continuous iff each πFi:Fi,RFi⟶Z,T is infra-fuzzy continuous.Proof.
The first direction is proved in Theorem48 (ii).
Conversely, letλ∈Tc. Since, for each i=1,2,⋯,n, πFi is an infra-fuzzy continuous mapping, πFi−1λ∈RFic and so πFi−1λ∈Rc, by Proposition 25, since each Fi∈Rc Therefore,
(9)π−1λ=πF1−1λ∪πF2−1λ∪⋯∪πFn−1λ,is IFIF-closed as it is a finite union of IF-closed sets. Hence, π is infra-fuzzy continuous.


## 6. Infra-Fuzzy Homeomorphisms

This section studies new types of mappings in the context of infra fuzziness called open, closed, and homomorphism mappings. Their main features and characterizations are explored. Also, it is proven that the family of all infra-fuzzy homeomorphisms on an infra-fuzzy topological space forms a group under mappings composition.Definition 50.
Letπ be a mapping from an infra-fuzzy topological space Y,R into another infra-fuzzy topological space Z,T. The mapping π is called
(i)
infra-fuzzy open ifπβ∈T for each β∈R;(ii)
infra-fuzzy closed ifπη∈Tc for each η∈Rc.Proposition 51.
LetY,R,Z,T be infra-fuzzy topological spaces and let A⊆Y. The following statements hold:
(i)
the inclusion mappingj:A,RA⟶Y,R is infra-fuzzy open resp.,closed iff A∈Rresp.,A∈Rc.(ii)
if a mappingπ:Y,R⟶Z,T is infra-fuzzy open resp.,closed and if A∈Rresp.,A∈Rc, then πA:A,RA⟶Z,T is infra-fuzzy open resp.,closed.Proof.
(i)
ifA∈R and μ∈RA, then by Proposition 24, jμ=μ∈T. Thus i is infra-fuzzy open. On the other hand, if j is infra-fuzzy open and since A∈RA, then jA=A∈T. Hence, the proof.(ii)
Assume thatν∈RA. By Proposition 24, ν∈R. Since π infra-fuzzy open, so πν∈T. But πAν=πν; thus, πA is infra-fuzzy open.
The proofs forπ is infra-fuzzy closed are analogous.Proposition 52.
Letπ:Y,R⟶Z,T be an infra-fuzzy open bijection. Then, π is infra-fuzzy open iff π is infra-fuzzy closed.Proof.
It can be followed from the fact thatπ1−μ=1−πμ for each μ∈FY whenever π is bijective.Theorem 53.
Ifπ:Y,R⟶Z,T is an infra-fuzzy open mapping, then πiμ≤iπμ for each μ∈FY.Proof.
Letq∈FpZ such that q∈πiμ. Then, there is some p∈FpY with p∈iμ such that πp=q. Therefore, there is σ∈R such that p∈σ≤μ. Evidently, q=πp∈πσ≤πμ. Since π is infra-fuzzy open, so πσ∈T and hence q∈iπμ. Thus, πiμ≤iπμ.Remark 54.
The converse of the preceding observation may not be true, and we can use the inverse of the mapping in Example4 as a counterexample.Lemma 55.
For a mappingπ:Y,R⟶Z,T, the following statements are equivalent:
(i)
πiμ≤iπμ for each μ∈FY; and(ii)
π−1cν≤cπ−1ν for each ν∈FZ.Proof.
By a similar technique used as in Lemma45.The following are the immediate consequences of Proposition17, Theorem 53, and Lemma 55.Corollary 56.
Ifπ:Y,R⟶Z,T is an infra-fuzzy open mapping, then π−1cν≤cπ−1ν for each ν∈FZ.Corollary 57.
Ifπ:Y,R⟶Z,T is an infra-fuzzy open mapping, then πη∈IFNπp if η∈IFNp for each p∈FpY.Theorem 58.
Ifπ:Y,R⟶Z,T is an infra-fuzzy closed mapping, then cπμ≤πcμ for each μ∈FY.Proof.
Letq∈FpZ. If q∉cπμ, then π−1q≰cμ. For each p∈π−1q, there exists η∈IFQNπp such that η⊀μ. From Lemma 6, we obtain that η+μ≤1 and so μ≤1−η. Since π is infra-fuzzy closed, then π1−η=1−πη∈Tc. Therefore, cπμ≤1−πη. But q=πp⊀1−πη, hence q∉cπμ. Thus cπμ≤πcμ.The opposite of the preceding theorem is false. The counterexample showing this can be concluded from Remark54 and Proposition 52.Corollary 59.
Ifπ:Y,R⟶Z,T is an infra-fuzzy closed mapping, then πη∈IFQNπp if η∈IFQNp for each p∈FpY.Proof.
It follows from Theorems58–31.Corollary 60.
Ifπ:Y,R⟶Z,T is an infra-fuzzy closed continuous mapping, then πcμ=cπμ for each μ∈FY.Proof.
It is a direct consequence of Theorem58 and Corollary 47.Corollary 61.
Ifπ:Y,R⟶Z,T is an infra-fuzzy open continuous injection, then πiμ=iπμ for each μ∈FY.Proof.
It follows from Theorems44–53.Theorem 62.
Letϕ:X,S⟶Y,R and π:Y,R⟶Z,T be mappings. The following statements are true: [(i)]
(i)
ifϕ and π are infra-fuzzy open, then π∘ϕ is infra-fuzzy open;(ii)
ifπ∘ϕ is infra-fuzzy open and ϕ is infra-fuzzy continuous surjective, then π is infra-fuzzy open; and(iii)
ifπ∘ϕ is infra-fuzzy open and π is infra-fuzzy continuous injective, then ϕ is infra-fuzzy open.Proof.
(i)
Letσ∈S. Since ϕ is infra-fuzzy open, then ϕσ∈R. By infra-fuzzy openness of π, πϕσ∈T. But π∘ϕ=πϕσ. Hence, π∘ϕ is infra-fuzzy open.(ii)
Letμ∈R. Since ϕ is infra-fuzzy continuous, then ϕ−1μ∈S. Therefore, π∘ϕϕ−1μ∈T. By surjectivity of ϕ, we have that π∘ϕϕ−1μ=πϕϕ−1μ=πμ and hence π is infra-fuzzy open.(iii)
Letν∈S. Since π∘ϕ is infra-fuzzy open, then π∘ϕν∈T. Since π is infra-fuzzy continuous, π−1π∘ϕν∈R. By injectivity of π, we get that π−1π∘ϕν=π−1πϕν=ϕν. This proves that ϕ is infra-fuzzy open.One can prove the next result by using the same arguments as in the above proof.Theorem 63.
Letϕ:X,S⟶Y,R and π:Y,R⟶Z,T be mappings. The following statements are true:
(i)
ifϕ and π are infra-fuzzy closed, then π∘ϕ is infra-fuzzy closed;(ii)
ifπ∘ϕ is infra-fuzzy closed and ϕ is infra-fuzzy continuous surjective, then π is infra-fuzzy closed; and(iii)
ifπ∘ϕ is infra-fuzzy closed and π is infra-fuzzy continuous injective, then ϕ is infra-fuzzy closed.Definition 64.
A mappingπ:Y,R⟶Z,T is called infra-fuzzy homeomorphism if it is an infra-fuzzy open and infra-fuzzy continuous bijection. We say that Y,R and Z,T are infra-fuzzy homeomorphic if there exists an infra-fuzzy homeomorphism between them and write Y,R≅Z,T.Theorem 65.
For an infra-fuzzy bijectionπ:Y,R⟶Z,T, the following properties are equivalent:
(i)
π is an infra-fuzzy homeomorphism;(ii)
π is infra-fuzzy continuous and π−1 is infra-fuzzy continuous;(iii)
π is infra-fuzzy continuous and infra-fuzzy open; and(iv)
π is infra-fuzzy continuous and infra-fuzzy closed.Proof.
Apply Definition64 and Proposition 52.Corollary 66.
Ifπ:Y,R⟶Z,T is an infra-fuzzy homeomorphism, then for each μ∈FY the following properties are true:
(i)
μ∈R iff πμ∈T;(ii)
1−μ∈R iff 1−πμ∈T; and(iii)
forp∈FpY, μ∈IFNp iff πμ∈IFNπp.Proof.
Apply Definition64 and Proposition 52.Theorem 67.
For an infra-fuzzy homeomorphismπ:Y,R⟶Z,T, the following properties hold:
(i)
πiμ=iπμ for each μ∈FY;(ii)
πcμ=cπμ for each μ∈FY; and(iii)
πbμ=bπμ for each μ∈FY.Proof.
Both proofs are the immediate consequences of Corollaries 60–61.Proposition 68.
Ifϕ:X,S⟶Y,R and π:Y,R⟶Z,T are infra-fuzzy homeomorphisms, then π∘ϕ is an infra-fuzzy homeomorphism.Proof.
It is a consequence of Theorems48–62.Remark 69.
One can use Definition64 and Proposition 68 to show that being a homeomorphism is an equivalence relation on the class of all infra-fuzzy topological spaces.Theorem 70.
The set of all infra-fuzzy homeomorphismsℍZ,T of an infra-fuzzy space Z,T forms a group under the composition of mappings.Proof.
Let∗:ℍZ,T×ℍZ,T⟶ℍZ,T be a binary operation defined by ϕ∗π=π∘ϕ for π,ϕ∈ℍZ,T. By Proposition 68, ℍZ,T is closed under the operation ∘. It is not difficult to show that ∘ is associative. The identity element exists in ℍZ,T which is the identity mapping i:Z,T⟶Z,T. For each π∈ℍZ,T, by Theorem 65, π−1∈ℍZ,T, and π∘π−1=π−1∘π=i. This shows that each element in ℍZ,T possesses an inverse, and hence ℍZ,T,∘ is a group.Theorem 71.
Ifh:Y,R⟶Z,T is an infra-fuzzy homeomorphism, then h generates an isomorphism between the groups ℍY,R and ℍZ,T.Proof.
Leth:Y,R⟶Z,T be an infra-fuzzy homeomorphism. Define a mapping ϕh:ℍY,R⟶ℍZ,T by ϕhπ=h∘π∘h−1 for π∈ℍY,R. Since h is an infra-fuzzy homeomorphism, then ϕh is bijective. We now show that ϕh is a homomorphism. Let κ,π∈ℍY,R. Then
(10)ϕhπ∘κ=h∘π∘κ∘h−1=h∘π∘h−1∘h∘κ∘h−1=ϕhπ∘ϕhκ.
Thus,ϕh is a homomorphism, and hence ϕh is an isomorphism generated by h.


## 7. Conclusion

The concept of infra-fuzzy topology was developed in this paper as a novel structure that is weaker than fuzzy topology. The main purpose of studying this notion is to preserve some fuzzy topological properties under requirements that are weaker than the conditions of topology. This part of the research has worked to improve knowledge in this field in the following ways: We have established the fundamental properties of infra-fuzzy topological spaces first. The majority of interior and closure properties are true in the setting of infra-fuzzy topological spaces, but others are not. The interior of each infra-fuzzy open set is the open set itself, but the interior of a fuzzy set does not have to be infra-fuzzy open. The infra-fuzzy closure would be in a similar condition. We have also mentioned that the boundary of a fuzzy set may not be infra-fuzzy closed. Second, we have suggested the infra-fuzzy bases, infra-fuzzy subspaces, infra-fuzzy neighborhood systems, and infra-fuzzy Q-neighborhood systems as ways to generate infra-fuzzy topologies. The infra-fuzzy topologies formed by the third and fourth notions are not the same. Finally, we have focused at the notion of mappings between infra-fuzzy topological spaces and their continuity, openness, closedness, and homomorphism. We have demonstrated that in the context of infra-fuzzy topologies, a full characterization of those notions as in fuzzy topologies is impossible. We have shown that the set of all homeomorphisms of an infra-fuzzy topological space is a group under the composition operation. Furthermore, we have proven that ifY,R and Z,T are infra-fuzzy homeomorphic by h, then it generates a mapping ϕh under which ℍY,R and ℍZ,T are isomorphic.In the upcoming work, we plan to investigate the concepts of separation axioms, compactness, and connectedness. Also, we study the concepts given herein with respect to some recent types of fuzzy sets such as (3, 2)-fuzzy sets [28] and SR-fuzzy sets [29]. Moreover, we will integrate rough set theory and infra-fuzzy topology to form a new frame aiming to handle uncertain situations.



---

*Source: 9778069-2022-04-07.xml*