# On Stronglyb−θ-Continuous Mappings in Fuzzifying Topology

Authors: Ting Yang; Ahmed Mostafa Khalil
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, we will define the new notions (e.g.,b−θ-neighborhood system of point, b−θ-closure (interior) of a set, and b−θ-closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b−θ-closed set is fuzzifying γ-closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b−θ-derived set and fuzzifying b−θ-boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b−θ-continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b−θ-continuous mapping, fuzzifying strongly b−θ-irresolute mapping, and fuzzifying weakly b−θ-irresolute mapping along with different conditions for their existence are obtained.


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## Body

## 1. Introduction and Preliminaries

In classical topology, the notions ofb-open set, b-closed set, and strongly θ−b-continuous mapping are presented in [1, 2]. After that, Hanafy [3] used the term γ-open sets instead of b-open sets and studied the notions of γ-open sets and γ-continuous mapping in fuzzy topology [4]. Benchalli and Karnel [5] presented a novel form of fuzzy subset named fuzzy b-open (closed) set, and some basic properties are proved and also their relations with different fuzzy sets in fuzzy topological spaces are investigated. In 2017, Dutta and Tripathy [6] introduced a new kind of open set named fuzzy b−θ open set (i.e., which is a generalization of b−θ open set). Ying [7] extended the basic notions in classical topology to fuzzifying topology based on fuzzy logic (i.e., as considered a novel approach of fuzzy topology, which depends on the various basic relations of topological spaces and the logical analysis of topological axioms). Many researchers are interested in fuzzifying topology (such as fuzzifying semiopen sets [8], fuzzifying preopen sets [9], fuzzifying α-open sets [10], fuzzifying β-open sets [11], and fuzzifying γ-open sets [12]). Therefore, in this article, we will extend the notions of b−θ-neighborhood system of a point, b−θ-closure (interior) of a set, b−θ-open (closed) set, b−θ-derived sets, and b−θ-boundary sets in fuzzifying topology. Also, we introduce the notion of fuzzifying strongly b−θ-continuous mapping, fuzzifying strongly b−θ-irresolute mapping, and fuzzifying weakly b−θ-irresolute mapping between two fuzzifying topological spaces.The rest of this article is arranged as follows. In this section, we briefly recall several notions: closed (open) set, closure (interior) of a set, neighborhood system of point,γ-closed (open) set, γ-closure (interior) of a set, γ-neighborhood system of point, continuous mapping, and γ-continuous mapping in fuzzifying topology which are used in the sequel. In Section 2, we define the notions of b−θ-neighborhood system of a point, b−θ-closure (interior) of a set, and b−θ-open (closed) set in fuzzifying topology. The interesting relation properties of the above notions are explained in detail. In Section 3, we present the notions b−θ-derived set and b−θ-boundary set in fuzzifying topology and introduce the characterizations of interesting properties between fuzzifying b−θ-derived set and fuzzifying b−θ-closure of a set. In Section 4, we define the fuzzifying strongly b−θ-continuous mapping, fuzzifying strongly b−θ-irresolute mapping, and fuzzifying weakly b−θ-irresolute mapping between two fuzzifying topological spaces and investigate some properties of them.Firstly, we give the notions of a fuzzy logical [7, 13] as follows:(1)
We give the symbolϑ which is the truth value of ϑ (i.e., the set of truth values means the unit interval 0,1).(2)
We can write⊨φ⇔φ=1 (i.e., φ is valid).(3)
ForΦ,Ψ∈0,1X (i.e., 0,1X mean the whole of fuzzy subsets of a set X) have the following for x∈X:(1)¬ϑ=1−ϑ,ϑ1∧ϑ2=minϑ1,ϑ2,ϑ1⟶ϑ2=min1,1−ϑ1+ϑ2,∀xϑx=infx∈Xϑx,x∈Φ=Φx,∃xϑx≔¬∀x¬ϑx,ϑ1∨ϑ2≔¬¬ϑ1∧¬ϑ2,ϑ1↔ϑ2≔ϑ1⟶ϑ2∧ϑ2⟶ϑ1,Φ⊑Ψ≔∀xx∈Φ⟶x∈Ψ=infx∈Xmin1,1−Φx+Ψx,Φ≡Ψ≔Φ⊑Ψ∧Ψ⊑Φ,ϑ1∨•ϑ2≔¬ϑ1⟶¬ϑ2=min1,ϑ1+ϑ2,ϑ1∧•ϑ2≔¬ϑ1⟶ϑ2=max0,ϑ1+ϑ2−1.Secondly, we present the basic notions related to fuzzifying topological space as follows.Definition 1.
(see [7]). τ (i.e., τ∈F^2X, 2X the set of all subsets of a set X) is called a fuzzifying topological space (for short, FTSX,τ) if we have the following three conditions:(i)
τX=τ∅=1(ii)
∀Φ,Ψ,τΦ⊓Ψ≥τΦ∧τΨ(iii)
∀Φλ:λ∈Λ,τ⊔λ∈ΛΦλ≥infλ∈ΛτΦλDefinition 2.
(see [7, 13]). The several notions of FTSX,τ are given as follows ∀Φ,Ψ∈2X:(i)
F¯ (i.e., F¯∈F^2X) is called the set of all fuzzifying closed sets if Φ∈F≔Φc∈τ, where Φc=X−Φ is the complement of Φ(ii)
Nx (i.e., Nx∈F^2X,x∈X) is called a fuzzifying neighborhood system of x if NxΦ=supx∈Ψ⊑ΦτΨ(iii)
C¯Φ is called a fuzzifying closure of Φ if C¯Φx=1−NxΦc(iv)
ℐ¯Φ is called a fuzzifying interior of Φ if ℐ¯Φx=NxΦNoiri and Sayed [12] presented and studied the following notions in FTSX,τ as indicated below.Definition 3.
(see [12]). (i)
τγ (i.e.τγ∈F^2X) is called the set of all fuzzifying γ-open sets if(2)Φ∈τγ≔∀xx∈Φ⟶x∈C¯ℐ¯Φ⊔ℐ¯C¯Φ,i.e.,(3)τγΦ=infx∈ΦmaxC¯ℐ¯Φx,ℐ¯C¯Φx.(ii)
F¯γ (i.e., F¯γ∈F^2X) is called the set of all fuzzifying γ-closed sets if Φ∈Fγ≔Φc∈τγ, where Φc is the complement of Φ.(iii)
Nxγ (i.e., Nxγ∈F^2X,x∈X) is called a fuzzifying γ-neighborhood system of x if NxγΦ=supx∈Ψ⊑ΦτγΨ.(iv)
C¯γΦ is called a fuzzifying γ-closure of Φ if C¯γΦx=1−NxγΦc.(v)
ℐ¯γΦ is called a fuzzifying γ-interior of Φ if ℐ¯γΦx=NxγΦ.Definition 4.
(see [14]). Co (i.e., Co∈F^YX) (a unary fuzzy predicate) is called fuzzifying continuous mappings between FTSX,τ and FTSY,φ if(4)Coψ≔∀OO∈φ⟶ψ−1O∈τ,i.e.,(5)Coψ=infO∈2Ymin1,1−φO+τψ−1O.Definition 5.
(see [12]). Coγ (i.e., Coγ∈F^YX) (a unary fuzzy predicate) is called fuzzifying γ-continuous mappings between FTSX,τ and FTSY,φ if(6)Coγψ≔∀OO∈φ⟶ψ−1O∈τ,i.e.,(7)Coγψ=infO∈2Ymin1,1−φO+τγψ−1O.


## 2. On Fuzzifyingb−θ-Neighborhood System

Definition 6.
Nxbθ (i.e., Nxbθ∈F^2X,x∈X) is called a fuzzifying b−θ-neighborhood system of x if(8)NxbθΦ=supΨ∈2Xmax0,NxγΨ+infy∉ΦNyγΨc−1.Theorem 1.
LetNbθ be a mapping from X to F^N2X s.t. x↦Nxbθ and F^N is a family of N-fuzzy (Normal fuzzy) subsets of 2X) having the following three properties ∀x∈X,∀Φ,Ψ∈2X:(i)
⊨Φ∈Nxbθ⟶x∈Φ(ii)
⊨Φ⊑Ψ⟶Φ∈Nxbθ⟶Ψ∈Nxbθ(iii)
⊨Φ∈Nxbθ∧Ψ∈Nxbθ↔Φ⊓Ψ∈NxbθProof.
(i)
Case 1: ifΦ∈Nxbθ=0, then Φ∈Nxbθ≤x∈Φ.Case 2: ifΦ∈Nxbθ>0, then supΨ∈2Xmax0,NxγΨ+C¯γΨ⊑Φ−1>0. There exists Ω∈2X such that NxγΩ+C¯γΩ⊑Φ−1>0. From Theorems 4.2 (1) and 5.2 (3) in [12], we obtainConsequently,C¯γΩ⊑Φ=infy∉Φ1−C¯γΩy>1−C¯γΩx, and so x∈Φ. If x∉Φ, then we have infy∉Φ1−C¯γΩy>1−C¯γΩx; it is contradiction. Thus, x∈Φ=1, and hence, Φ∈Nxbθ≤x∈Φ.(ii)
Case 1: ifΦ⊑Ψ=0, then NxbθΦ≤NxbθΨ.Case 2: ifΦ⊑Ψ=1 and Ω∈2X, thenand soNxγΩ+C¯γΩ⊑Ψ−1≥NxγΩ+C¯γΩ⊑Φ−1. Thus, supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Ψ−1≥supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ−1. Hence, NxbθΦ≤NxbθΨ.(iii)
Φ⊓Ψ∈Nxbθ=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ⊓Ψ−1. Since(9)C¯γΩ⊑Φ⊓Ψ=∀xx∈C¯γΩ⟶x∈Φ⊓Ψ=infx∈Xmin1,1−C¯γΩx+Φ⊓Ψx=infx∉Φ⊓Ψ1−C¯γΩx=infx∈Φc⊔Ψc1−C¯γΩx=infx∈Φc1−C¯γΩx∧infx∈Ψc1−C¯γΩx=C¯γΩ⊑Φ∧C¯γΩ⊑Ψ.
Thus,(10)NxbθΦ⊓Ψ=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ⊓Ψ−1=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ∧C¯γΩ⊑Ψ−1=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ−1∧NxγΩ+C¯γΩ⊑Ψ−1=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ−1∧max0,NxγΩ+C¯γΩ⊑Ψ−1=supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Φ−1∧supΩ∈2Xmax0,NxγΩ+C¯γΩ⊑Ψ−1=NxbθΦ∧NxbθΨ.Next, we will generalize the notion ofb−θ-closure [2] in FTSX,τ as follows.Definition 7.
C¯bθΦΦ∈2X is called a fuzzifying b−θ-closure of Φ if(11)x∈C¯bθΦ≔∀Ψ∈2XΨ∈Nxγ⟶¬Φ⊓C¯γΨ≡∅.Theorem 2.
The following relation is holding inFTSX,τ:(12)C¯bθΦx≔infΨ∈2Xmin1,1−NxγΨ+supy∈ΦC¯γΨy.Proof.
(13)∀ΨNxγΨ⟶¬Φ⊓C¯γΨ≡∅=infΨ∈2Xmin1,1−NxγΨ+¬Φ⊓C¯γΨ≡∅=infΨ∈2Xmin1,1−NxγΨ+1−Φ⊓C¯γΨ≡∅=infΨ∈2Xmin1,1−NxγΨ+1−Φ⊓C¯γΨ⊑∅∧∅⊑Φ⊓C¯γΨ=infΨ∈2Xmin1,1−NxγΨ+1−infy∈Φmin1,1−Φ⊓C¯γΨy+0=infΨ∈2Xmin1,1−NxγΨ+supy∈Φmax0,Φ⊓C¯γΨy=infΨ∈2Xmin1,1−NxγΨ+supy∈ΦminΦy,C¯γΨy=infΨ∈2Xmin1,1−NxγΨ+supy∈ΦC¯γΨy.Theorem 3.
The following relations are holding inFTSX,τ:(i)
C¯bθΦx=1−NxbθΦc(ii)
⊨C¯bθ∅≡∅(iii)
⊨Φ⊑C¯bθΦ(iv)
⊨x∈C¯bθΦ↔∀ΨΨ∈Nxbθ⟶Φ⊓Ψ≠∅(v)
⊨Φ⊑Ψ⟶C¯bθΦ⊑C¯bθΨ(vi)
⊨C¯γΦ⊑C¯bθΦ(vii)
⊨C¯bθΦ⊔Ψ≡C¯bθΦ⊔C¯bθΨProof.
(i)
(14)C¯bθΦx=infΨ∈2Xmin1,1−NxγΨ+supy∈ΦC¯γΨy=1−supΨ∈2Xmax0,NxγΨ−supy∈ΦC¯γΨy=1−supΨ∈2Xmax0,NxγΨ+1−supy∈ΦC¯γΨy−1=1−supΨ∈2Xmax0,NxγΨ+infy∈Φ1−C¯γΨy−1=1−∃ΨΨ∈Nxγ∧•C¯γΨ⊑Φc=1−NxbθΦc.(ii)
From (i) above and sinceNxbθ is normal, we have C¯bθ∅x=1−Nxbθ∅c=0.(iii)
Case 1: ifx∉Φ, then NxbθΦ=0.Case 2: ifx∈Φ, then C¯bθΦx=1−NxbθΦc=1−0=1. Thus, Φ⊑C¯bθΦ=1.(iv)
∀ΨΨ∈Nxbθ⟶Φ⊓Ψ≠∅=infΨ⊑Φc1−NxbθΨ=1−NxbθΦc=x∈C¯bθΦ.(v)
Case 1: ifΦ⊑Ψ=0, then C¯bθΦx≤C¯bθΨx.Case 2: ifΦ⊑Ψ=1, then C¯bθΦx=1−NxbθΦc≤1−NxbθΨc=C¯bθΨx.(vi)
From Theorems 5.2 (3) and 5.3 (2) in [12], we have C¯γΦx=infΨ∈2Xmin1,1−NxγΨ+supy∈AΨy≤infΨ∈2Xmin1,1−NxγΨ+supy∈ΦC¯γΨy=C¯bθΦx.(vii)
It is easy to getC¯bθΦ⊔C¯bθΨ⊑C¯bθΦ⊔Ψ. Conversely, for every x∈X,(15)C¯bθΦ⊔Ψx=infΩ∈2Xmin1,1−NxγΩ+supy∈Φ⊔ΨC¯γΩy=infΩ∈2Xmin1,1−NxγΩ+supy∈ΦC¯γΩy∨supy∈ΨC¯γΩy≤infΩ∈2Xmin1,1−NxγΩ+supy∈ΦC¯γΩy∨infΩ∈2Xmin1,1−NxγΩ+supy∈ΨC¯γΩy=C¯bθΦx∨C¯bθΨx.Lemma 1.
⊨Φ∈Nxbθ⟶Φ∈Nxγ.Proof.
It follows from Theorem3 (vi).Clearly,Nuγv=0≤Nubθv=1 if we consider X=u,v and τX=τ∅=τu=1,τu=0 in FTSX,τ.Definition 8.
(i)
F¯bθ (i.e., F¯bθ∈F^2X) is called the set of all fuzzifying b−θ-closed sets if(16)Φ∈F¯bθ≔Φ≡C¯bθΦ,i.e.,(17)F¯bθΦ=infx∈Φc1−C¯bθΦx.(ii)
τbθ (i.e., τbθ∈F^2X is called the set of all fuzzifying b−θ-open sets if Φ∈τbθ≔Φc∈F¯bθ, where Φc is the complement of Φ.By Definition8, we can conclude NxbθΦ=supx∈Ψ⊑ΦτbθΨ.Definition 9.
ℐ¯bθΦΦ∈2X is called a fuzzifying b−θ-interior of Φ if(18)x∈ℐ¯bθΦ≔Φ∈Nxbθ,i.e.,(19)ℐ¯bθΦx=NxbθΦ.Theorem 4.
The following relations are holding inFTSX,τ:(i)
⊨ℐ¯bθΦ≡C¯bθΦcc(ii)
⊨ℐ¯bθX≡X(iii)
⊨ℐ¯bθΦ⊑Φ(iv)
⊨Φ∈τbθ↔∀xx∈Φ⟶Φ∈Nxbθ(v)
⊨Φ∈τbθ∧Φ⊑Ψ⟶Φ⊑ℐ¯bθΨ(vi)
⊨Φ⊑Ψ⟶ℐ¯bθΦ⊑ℐ¯bθΨ(vii)
⊨ℐ¯bθΦ⊑ℐ¯γΦ(viii)
⊨ℐ¯bθΦ⊓Ψ≡ℐ¯bθΦ⊓ℐ¯bθΨProof.
(i)
By Theorem3 (i), we obtain C¯bθΦcx=1−NxbθΦ=1−ℐ¯bθΦx. Hence, ℐ¯bθΦ≡C¯bθΦcc=1.(ii)
By (i) above, we obtainℐ¯bθX=C¯bθXcc=C¯bθ∅c=∅c=X.(iii)
By Theorem3 (iii), we obtain ℐ¯bθΦ≡C¯bθΦcc⊑Φcc≡Φ.(iv)
(20)Φ∈τbθ=infx∈Φ1−C¯bθΦcx=infx∈Φ1−infΨ∈2Xmin1,1−NxγΨ+supy∈ΦcC¯γΨy=infx∈ΦsupΨ∈2Xmax0,NxγΨ−supy∈ΦcC¯γΨy=infx∈ΦsupΨ∈2Xmax0,NxγΨ+1−supy∈ΦcC¯γΨy−1=infx∈ΦsupΨ∈2Xmax0,NxγΨ+infy∈Φc1−C¯γΨy−1=infx∈ΦsupΨ∈2XNxγΦ∧•infy∈Φc1−C¯γΨy=∀xx∈Φ⟶∃ΨΨ∈Nxγ∧•C¯γΨ⊑Φ=∀xx∈Φ⟶Φ∈Nxbθ.(v)
Case 1: ifΦ⊑Ψ=0, then Φ⊑ℐ¯bθΨ≥Φ∈τbθ∧Φ⊑Ψ.Case 2: ifΦ⊑Ψ=1, then Φ⊑ℐ¯bθΨ=infx∈Φℐ¯bθΨx=infx∈ΦNxbθΨ≥infx∈ΦNxbθΦ=τbθΦ=Φ∈τbθ∧Φ⊑Ψ from Theorem 1 (ii) and by (v) above.(vi)
Similar to Theorem3 (vi).(vii)
By Lemma1, we obtain ℐ¯bθΦx=NxbθΦ≤NxγΦ=ℐ¯γΦx.(viii)
By Theorem3 (vii), we obtain(21)ℐ¯bθΦ⊓Ψx=1−C¯bθΦ⊓Ψcx=1−C¯bθΦc⊔Ψcx=1−C¯bθΦcx⊔C¯bθΨcx=1−C¯bθΦcx⊓1−C¯bθΨcx=ℐ¯bθΦx⊓ℐ¯bθΨx.Theorem 5.
The following two relations are holding inFTSX,τ:(i)
⊨Φ∈τbθ↔Φ≡ℐ¯bθΦ(ii)
⊨Φ∈F¯bθ↔Φ≡C¯bθΦProof.
(i)
(22)Φ∈τbθ=Φc∈F¯bθ=infx∈X∼Φc1−C¯bθΦcx=infx∈Φℐ¯bθΦx=Φ⊑ℐ¯bθΦ=Φ⊑ℐ¯bθΦ∧ℐ¯bθΦ⊑Φ=Φ≡ℐ¯bθΦ.(ii)
Similar to (i).Theorem 6.
τbθ (i.e., τbθ∈F^2X,X is a set) is a FTSX,τ if we have the following three conditions:(i)
τbθX=τbθ∅=1(ii)
∀Φ,Ψ,τbθΦ⊓Ψ≥τbθΦ∧τbθΨ(iii)
∀Φλ:λ∈Λ,τbθ⊔λ∈ΛΦλ≥infλ∈ΛτbθΦλProof.
(i)
Clear.(ii)
By Theorem4 (v) and Theorem 1 (iii), we obtain(23)τbθΦ∧τbθΨ=infx∈Φℐ¯bθΦx∧infx∈Bℐ¯bθΨx≤infx∈Φ⊓Ψℐ¯bθΦx∧ℐ¯bθΨx=infx∈Φ⊓Ψℐ¯bθΦ⊓Ψx=τbθΦ⊓Ψ.(iii)
By Theorem4 (v), we obtain(24)τbθ⊔λ∈ΛΦλ=infx∈⊔λ∈ΛΦλℐ¯bθ⊔λ∈ΛΦλx=infλ∈λinfx∈Φλℐ¯bθ⊔λ∈ΛΦλx≥infλ∈λinfx∈Φλℐ¯bθΦλx=infλ∈λτbθΦλ.
This completes the proof.Theorem 7.
The following two relations are holding inFTSX,τ:(i)
⊨Φ∈τbθ↔∀xx∈Φ⟶∃ΨΨ∈τbθ∧x∈Ψ⊑Φ(ii)
⊨Φ∈τbθ↔∀xx∈Φ⟶∃ΨΨ∈Nxbθ∧Ψ⊑ΦProof.
(i)
∀xx∈Φ⟶∃ΨΨ∈τbθ∧x∈Ψ⊑Φ=infx∈Φsupx∈Ψ⊑ΦτbθΨ. Firstly, we obtain infx∈Φsupx∈Ψ⊑ΦτbθΨ≥τbθΦ. Further, assume that Ψx=Ψ:x∈Ψ⊑Φ. Hence, f∈∏x∈AΨx, we obtain ⊔x∈Afx=Φ,τbθΦ=τbθ⊔x∈Φfx≥infx∈Φτbθfx and τbθΦ≥supf∈∏x∈ΦΨxinfx∈Φτbθfx=infx∈Φsupx∈Ψ⊑ΦτbθΨ.(ii)
By (i) we obtain(25)∀xx∈Φ⟶∃ΨΨ∈Nxbθ∧Ψ⊑Φ=infx∈ΦsupΨ⊑ΦNxbθΨ=infx∈ΦsupΨ⊑Φsupx∈Ω⊑ΨτbθΩ=infx∈Φsupx∈Ω⊑ΨτbθΩ=Φ∈τbθ.Theorem 8.
The following two relations are holding inFTSX,τ:(i)
⊨τbθ⊑τγ(ii)
⊨F¯bθ⊑F¯γProof.
(i)
By Corollary 4.1 in [12] and Theorem 3 (vi), we obtain Φ∈τbθ=infx∈Φ1−C¯bθΦcx≤infx∈Φ1−C¯γΦcx=infx∈Φ1−1+NxγΦ=infx∈ΦNxγΦ=Φ∈τγ.(ii)
Follows from (i) above.Next, we will generalize Theorem 3.8 (a) in [2] in FTSX,τ by the following theorem.Theorem 9.
The following relation is holding inFTSX,τ:(26)⊨Φ∈τγ⟶C¯bθΦ≡C¯γΦ.Proof.
Firstly, we prove that⊨Φ∈τγ⟶¬Φ⊓C¯γΨ≡∅⟶¬Φ⊓Ψ≡∅. By Corollary 4.1 and Theorem 5.3 (1) in [12], we obtain(27)τγΦ∧•¬Φ⊓C¯γΨ≡∅=max0,infx∈ΦNxγΦ+supy∈ΦC¯γΨy−1=max0,supy∈ΦC¯γΨy−supx∈Φ1−NxγΦ≤max0,supx∈ΦC¯γΨx−1+NxγΦ=max0,supx∈ΦΨ⊔DγΨx−1+NxγΦ=max0,supx∈ΦmaxΨx,DγΨx−1+NxγΦ.Case 1: ifx∈Ψ, then(28)τγΦ∧•¬Φ⊓C¯γΨ≡∅=max0,supx∈ΦΨx−1+NxγΦ≤supx∈ΦNxγΦx≤supx∈ΦΦx≤supx∈ΦΦx∧Ψx=¬Φ⊓Ψ≡∅.Case 2: ifx∉Ψ, then(29)τγΦ∧•¬Φ⊓C¯γΨ≡∅=max0,supx∈ΦDγΨx−1+NxγΦ=max0,supx∈ΦinfΩ∈2Xmin1,1−NxγΩ+supy∈ΩΩ−xx−1+NxγΦ≤max0,supx∈Φ1−NxγΩ+supy∈ΦΨ−xy−1+NxγΦ=supy∈ΦΨ−xy=supy∈ΦΨy=¬Φ⊓Ψ≡∅.
Consequently,¬Φ⊓C¯γΨ≡∅⟶¬Φ⊓Ψ≡∅≥τγΦ.
Thus,(30)C¯bθΦ≡C¯γΦ=C¯bθΦ⊑C¯γΦ∧C¯γΦ⊑C¯bθΦ=C¯bθΦ⊑C¯γΦ=∀ΨΨ∈Nxγ⟶¬Φ⊓C¯γΨ≡∅⟶∀ΩΩ∈Nxγ⟶¬Φ⊓Ω≡∅≥∀ΨΨ∈Nxγ⟶¬Φ⊓C¯γΨ≡∅⟶Ψ∈Nxγ⟶¬Φ⊓Ψ≡∅=∀Ψ¬Φ⊓C¯γΨ≡∅⟶¬Φ⊓Ψ≡∅≥infΨ∈2XτγΦ=τγΦ.
This completes the proof.Corollary 1.
The following relation is holding inFTSX,τ:(31)⊨Φ∈τ⟶C¯bθΦ≡C¯γΦ.Theorem 10.
The following relation is holding inFTSX,τ:(32)⊨Φ∈F¯bθ⟶C¯bθΦ≡C¯γΦ.Proof.
(33)C¯bθΦ≡C¯γΦ=min1,1−C¯bθΦx+C¯γΦx=infx∈Xmin1,1−C¯bθΦx+1−NxγΦc=infx∈Φcmin1,1−C¯bθΦx+1−NxγΦc≥infx∈Φcmin1,1−C¯bθΦx=infx∈Φc1−C¯bθΦx=Φ∈F¯bθ.


## 3. Fuzzifyingb−θ-Derived Sets and Fuzzifying b−θ-Boundary Sets

Definition 10.
DbθΦΦ∈2X is called a fuzzifying b−θ-derived set of Φ if(34)x∈DbθΦ≔∀ΨΨ∈Nxbθ⟶Ψ⊓Φ−x≠∅,i.e.,(35)DbθΦx=infΨ⊓Φ−x=∅1−NxbθΨ.Theorem 11.
The following relations are holding inFTSX,τ:(i)
⊨DbθΦx=1−NxbθΦc⊔x(ii)
⊨Dbθ∅≡∅(iii)
⊨Φ⊑Ψ⟶DbθΦ⊑DbθΨProof.
(i)
DbθΦx=infΨ⊓Φ∼x=∅1−NxbθΨ=1−supΨ⊓Φ−x=∅NxbθΨ=1−supΨ⊑Φc⊔xNxbθΨ=1−NxbθΦc⊔x.(ii)
By (i) above, we obtainDbθ∅x=1−Nxbθ∅c⊔x=1−NxbθX=1−1=0 (i.e., Nxbθ is normal).(iii)
Case 1:Φ⊑Ψ=0, then DbθΦ≤DbθΨ.Case 2: ifΦ⊑Ψ=1 and by Theorem 1 (ii), we obtain DbθΦx=1−NxbθΦc⊔x≤1−NxbθΨc⊔x=DbθΨx.Theorem 12.
The following relation is holding inFTSX,τ:(36)⊨Φ∈F¯bθ↔DbθΦ⊑Φ.Proof.
By Theorem7, we obtain(37)DbθΦ⊑Φ=infx∈Xmin1,1−DbθΦx+x∈Φ=infx∈Φc1−DbθΦx=infx∈Φc1−1−NxbθΦc⊔x=infx∈ΦcNxbθΦc⊔x=infx∈ΦcNxbθΦc=infx∈Φcsupx∈Ψ⊑ΦcτbθΨ=τbθΦc=Φ∈F¯bθ.Theorem 13.
The following two relations are holding inFTSX,τ:(i)
⊨DbθΦ⊔DbθΨ≡DbθΦ⊔Ψ(ii)
⊨x∈ℐ¯bθΦ↔x∈Φ∧x∈DbθΦccProof.
(i)
By Theorem11 (i) and Theorem 1 (iii), we obtain(38)DbθΦ⊔Ψx=1−NxbθΦ⊔Ψc⊔x=1−NxbθΦc⊓Ψc⊔x=1−NxbθΦc⊔x⊓Ψc⊔x=1−NxbθΦc⊔x∧NxbθΨc⊔x=1−NxbθΦc⊔x∨1−NxbθΨc⊔x=DbθΦx∨DbθΨx.(ii)
Case 1: ifx∉Φ, then by Theorem 1 (i), NxbθΦ=0. Hence, x∈ℐ¯bθΦ=0=x∈Φ∧x∈DbθΦcc.Case 2: ifx∈Φ, then x∈DbθΦcc=1−DbθΦcx=1−1−NxbθΦ⊔x=NxbθΦ=x∈ℐ¯bθΦ.Theorem 14.
The following two relations are holding inFTSX,τ:(i)
⊨C¯bθΦ≡Φ⊔DbθΦ(ii)
⊨Ψ≡⋅C¯bθΦ⟶Ψ∈F¯bθProof.
(i)
Case 1: ifx∈Φ, Φ⊔DbθΦx=C¯bθΦx.Case 2: ifx∉Φ, then(ii)
IfΦ⊑Ψ=0, then Ψ≡⋅C¯bθΦ=0. Now, we suppose that Φ⊑Ψ=1 and have Ψ⊑C¯bθΦ=1−supx∈Ψ−ΦNxbθΦc, C¯bθΦ⊑Ψ=infx∈ΨcNxbθΦc. So Ψ≡.C¯bθΦ=max0,infx∈ΨcNxbθΦc−supx∈Ψ−ΦNxbθΦc. If Ψ≡⋅C¯bθΦ>t, then infx∈ΨcNxbθΦc>t+supx∈Ψ−ΦNxbθΦc. For any x∈Ψc, we get supx∈Ω⊑ΦcτbθΩ>t+supx∈Ψ−ΦNxbθΦc, i.e., there exists Ωx such that x∈Ωx⊑Φc and τbθΩx>t+supx∈Ψ−ΦNxbθΦc. Next, we will show that Ωx⊑Ψc. If not, then there is x′∈Ψ−Φ with x′∈Ωx. Thus, we get supx∈Ψ−ΦNxbθΦc≥Nx′bθΦc≥τbθΩx≥t+supx∈Ψ−ΦNxbθΦc; it is contradiction. Therefore, F¯bθΨ=τbθΨc=infx∈ΨcNxbθΨc≥infx∈ΨcτbθΩx>t+supx∈Ψ−ΦNxbθΦc>t. Since t is arbitrary, it holds that Ψ≡⋅C¯bθΦ≤Ψ∈F¯bθ.Definition 11.
ℬbθΦΦ∈2X is called a fuzzifying b−θ-boundary set of Φ if(39)x∈ℬbθΦ≔x∈C¯bθΦ∧x∈C¯bθΦc,i.e.,(40)ℬbθΦx=minC¯bθΦx,C¯bθΦcx.Theorem 15.
The following relation is holding inFTSX,τ:(41)⊨x∈ℬbθΦ↔∀ΨΨ∈Nxbθ⟶Ψ⊓Φ≠∅∧Ψ⊓Φc≠∅.Proof.
Similar to Lemma 2.1 in [13].Theorem 16.
The following two relations are holding inFTSX,τ:(i)
⊨C¯bθΦ≡Φ⊔ℬbθΦ and so ⊨Φ∈F¯bθ↔ℬbθΦ⊑Φ(ii)
⊨ℐ¯bθΦ≡Φ⊓ℬbθcΦ and so ⊨Φ∈τbθ↔ℬbθΦ⊓Φ≡∅Proof.
(i)
Case 1: ifx∈Φ, then C¯bθΦx=Φ⊔ℬbθΦx=1 (by Theorem 3 (iii)).Case 2: ifx∉Φ, then Φ⊔ℬbθΦx=ℬbθΦx=minC¯bθΦx,C¯bθΦcx=C¯bθΦx. Therefore, by Theorem 12 and Theorem 14 (i), we get(ii)
By Theorem4 (i) and (i) above, we get ℐ¯bθΦ=Φc⊔ℬbθΦcc=Φ⊓ℬbθcΦ. Also, from Theorem 5 (i), we obtain ⊨Φ∈τbθ↔ℐ¯bθΦ≡Φ↔Φ⊓ℬbθcΦ≡Φ↔Φ⊑ℬbθΦc↔ℬbθΦ⊓Φ≡∅.


## 4. Fuzzifying Stronglyb−θ-Continuous Mapping, Fuzzifying Strongly b−θ-Irresolute Mapping, and Fuzzifying Weakly b−θ-Irresolute Mapping

In the following section, we define a fuzzifying stronglyb−θ-continuous mapping, fuzzifying strongly b−θ-irresolute mapping, and fuzzifying weakly b−θ-irresolute mapping between FTSX,τ and FTSY,φ.Definition 12.
SCobθ (i.e., SCobθ∈F^YX (a unary fuzzy predicate) is called fuzzifying strongly b−θ-continuous mappings between FTSX,τ and FTSY,φ if(42)SCobθψ≔∀OO∈φ⟶ψ−1O∈τbθ,i.e.,(43)SCobθψ=infO∈2Ymin1,1−φO+τbθψ−1O.Definition 13.
Assuming thatFTSX,τ and FTSY,φ and ∀ψ∈YX, we put(i)
ζ1ψ≔∀ΨΨ∈F¯Y⟶ψ−1Ψ∈F¯bθX, where F¯Y is a family of fuzzifying closed subset of Y and also F¯bθX is a family of fuzzifying b−θ-closed subset of X(ii)
ζ2ψ≔∀x∀OO∈Nψx⟶ψ−1O∈Nxbθ, where Nψx is a fuzzifying neighborhood system of ψx of Y and Nxbθ is a fuzzifying b−θ-neighborhood system of x of X(iii)
ζ3ψ≔∀x∀OO∈Nψx⟶∃PψP⊑O∧P∈Nxbθ(iv)
ζ4ψ≔∀ΦψC¯bθXΦ⊑C¯YψΦ(v)
ζ5ψ≔∀ΨC¯bθXψ−1Ψ⊑ψ−1C¯YΨ(vi)
ζ6ψ≔∀Φψ−1ℐ¯Φ⊑ℐ¯bθψ−1ΦTheorem 17.
⊨ψ∈SCobθ↔ψ∈ζk,k=i,…,6.Proof.
(i)
We show that⊨ψ∈SCobθ↔ψ∈ζ6.(44)ψ∈ζ1=infΨ∈2Ymin1,1−F¯YΨ+F¯bθXψ−1Ψ=infΨ∈2Ymin1,1−φY−Ψ+F¯bθX−ψ−1Ψ=infΨ∈2Ymin1,1−φY−Ψ+τbθψ−1Y−Ψ=infO∈2Ymin1,1−φO+τbθψ−1O=SCobθψ.(ii)
We prove that⊨ψ∈SCobθ↔ψ∈ζ2. Firstly, we show that ψ∈ζ2≥ψ∈SCobθ. If NψxO≤Nxbθψ−1O, then min1,1−NψxO+Nxbθψ−1O=1≥ψ∈SCobθ. Assume that NψxO>Nxbθψ−1O. Thus, if ψx∈Φ⊑O, then x∈ψ−1Φ⊑ψ−1O. Hence,(45)NψxO−Nxbθψ−1O=supψx∈Φ⊑OφΦ−supx∈Ψ⊑ψ−1OτbθΦ≤supψx∈Φ⊑OφΦ−supψx∈Φ⊑Oτbθψ−1Φ≤supψx∈Φ⊑OφΦ−τbθψ−1Φ.Consequently,1−NψxO+Nxbθψ−1O≥supψx∈Φ⊑O1−φΦ+τbθψ−1Φ. Thus,(46)min1,1−NψxO+Nxbθψ−1O≥infψx∈Φ⊑O1−φΦ+τbθψ−1Φ≥infP∈2Ymin1−φP+τbθψ−1P=ψ∈SCobθ.Therefore,infx∈XinfO∈2Ymin1−φO+τbθψ−1O≥ψ∈SCobθ.Secondly, we show thatψ∈SCobθ≥ψ∈α2. From Corollary 4.1 in [12] and Theorem 4 (iv), we have(47)ψ∈SCobθ=infO∈2Ymin1,1−φO+τbθψ−1O≥infO∈2Ymin1,1−infψx∈ONψxO+infx∈ψ−1ONxbθψ−1O≥infO∈2Ymin1,1−infx∈ψ−1ONψxO+infx∈ψ−1ONxbθψ−1O≥infx∈XinfO∈2Ymin1−NψxO+Nxbθψ−1O=ψ∈ζ2.(iii)
We show that⊨ψ∈ζ2↔ψ∈ζ3. By Theorem 1 (ii), we obtain(48)supP∈2X,ψP⊑ONxbθP=supP∈2X,P⊑ψ−1ONxbθP=Nxbθψ−1O.Thus,(49)ψ∈α3=infx∈XinfO∈2Ymin1−NψxO+supP∈2X,ψP⊑ONxbθP=infx∈XinfO∈2Ymin1−NψxO+Nxbθψ−1O=ψ∈ζ2.(iv)
We show that⊨ψ∈ζ4↔ψ∈ζ5. Firstly, we prove that ψ∈ζ4≤ψ∈ζ5. For every Ψ∈2Y, we have that ψ−1ψC¯bθXψ−1Ψ⊒C¯bθXψ−1Ψ=1,C¯Yψψ−1Ψ⊑C¯YΨ=1 and ψ−1C¯Yψψ−1Ψ⊑ψ−1C¯YΨ=1. By Lemma 1.2 (2) in [14], we obtain(50)C¯bθXψ−1Ψ⊑ψ−1C¯YΨ≥ψ−1ψC¯bθXψ−1Ψ⊑ψ−1C¯YΨ≥ψ−1ψC¯bθXψ−1Ψ⊑ψ−1C¯Yψψ−1Ψ≥ψC¯bθXψ−1Ψ⊑C¯Yψψ−1Ψ.Thus,(51)ψ∈ζ5=infΨ∈2YC¯bθXψ−1Ψ⊑ψ−1C¯YΨ≥infΨ∈2YψC¯bθXψ−1Ψ⊑C¯Yψψ−1Ψ≥infΦ∈2YψC¯bθXΦ⊑C¯YψΦ=ψ∈ζ4.Secondly,∀Φ∈2X, there exists Ψ∈2Y s.t. ψΦ=Ψ and ψ−1Ψ⊒Φ. By Lemma 1.2 (1) in [14], we obtain(52)ψ∈α4=infΦ∈2YψC¯bθXΦ⊑C¯YψΦ≥infΦ∈2YψC¯bθXΦ⊑ψψ−1C¯YψΦ≥infΦ∈2YC¯bθXΦ⊑ψ−1C¯YψΦ≥infΨ∈2Y,Ψ=ψΦC¯bθXψ−1Ψ⊑ψ−1C¯YΨ≥infΨ∈2YC¯bθXψ−1Ψ⊑ψ−1C¯YΨ=ψ∈ζ5.(v)
We show that⊨ψ∈ζ2↔ψ∈ζ5. Thus,(53)ψ∈ζ5=infΨ∈2YC¯bθXψ−1Ψ⊑ψ−1C¯YΨ=infΨ∈2Yinfx∈Xmin1,1−1−NxbθX−ψ−1Ψ+1−NψxY−Ψ=infΨ∈2Yinfx∈Xmin1,1−NψxY−Ψ+Nxbθψ−1Y−Ψ=infO∈2Yinfx∈Xmin1,1−NψxO+Nxbθψ−1O=ψ∈ζ2.(vi)
We prove that⊨ψ∈ζ6↔ψ∈ζ2. Thus,(54)ψ∈ζ6=infΦ⊑Yinfx∈Xmin1,1−ℐ¯Φψx+ℐ¯bθψ−1Φx=infΦ⊑Yinfx∈Xmin1,1−NψxΦ+Nxbθψ−1Φ=ψ∈ζ2.This completes the proof.Remark 1.
The following relation⊨ψ∈SCbθ⟶ψ∈Coγ is holding in crisp setting, while in FTSX,τ does not hold by the next example.Example 1.
If we considerX=u,v,w and τX=τ∅=τu=τu,w=1,τv=τu,v=0, and τw=τv,w=1/6 in FTSX,τ and defined the identity mapping between FTSX,τ and FTSX,φ by(55)φΦ=1,Φ∈X,∅,u,v,0,otherwise.
Thus,SCobθψ=1>5/6=Coγψ.Definition 14.
SIrbθ (i.e., SIrbθ∈F^YX (a unary fuzzy predicate) is called fuzzifying strongly b−θ-irresolute mapping between FTSX,τ and FTSY,φ if(56)SIrbθψ≔∀OO∈φγ⟶ψ−1O∈τbθ,i.e.,(57)SIrbθψ=infO∈2Ymin1,1−φγO+τbθψ−1O.Definition 15.
Assuming thatFTSX,τ and FTSY,φ and ∀ψ∈YX, we put(i)
ϵ1ψ≔∀ΨΨ∈F¯γY⟶ψ−1Ψ∈F¯bθX, where F¯γY is a family of fuzzifying γ-closed subset of Y and F¯bθX is a family of fuzzifying b−θ-closed subset of X(ii)
ϵ2ψ≔∀x∀OO∈Nψxγ⟶ψ−1O∈Nxbθ, where Nψxγ is a fuzzifying γ-neighborhood system of ψx of Y and Nxbθ is a fuzzifying b−θ-neighborhood system of x of X(iii)
ϵ3ψ≔∀x∀OO∈Nψxγ⟶∃PψP⊑O∧P∈Nxbθ(iv)
ϵ4ψ≔∀ΦψC¯bθXΦ⊑C¯γYψΦ(v)
ϵ5ψ≔∀ΨC¯bθXψ−1Ψ⊑ψ−1C¯γYΨ(vi)
ϵ6ψ≔∀Φψ−1ℐ¯γΦ⊑ℐ¯bθψ−1ΦTheorem 18.
⊨ψ∈SIrbθ↔ψ∈ϵk,k=i,…,6.Proof.
Similar to Theorem17.Definition 16.
WIrbθ (i.e., WIrbθ∈F^YX) (a unary fuzzy predicate) is called fuzzifying weakly e−θ-irresolute mapping between FTSX,τ and FTSY,φ if(58)WIrbθψ≔∀OO∈φbθ⟶ψ−1O∈τγ,i.e.,(59)WIreθψ=infO∈2Ymin1,1−φbθO+τγψ−1O.Definition 17.
Assuming thatFTSX,τ and FTSY,φ and ∀ψ∈YX, we put(i)
η1ψ≔∀ΨΨ∈F¯bθY⟶ψ−1Ψ∈F¯γX, where F¯bθY is a family of fuzzifying b−θ-closed subset of Y and F¯γX is a family of fuzzifying b−θ-closed subset of X(ii)
η2ψ≔∀x∀OO∈Nψxbθ⟶ψ−1O∈Nxγ, where Nψxγ is a fuzzifying b−θ-neighborhood system of ψx of Y and Nxγ is a fuzzifying γ-neighborhood system of x of X(iii)
η3ψ≔∀x∀OO∈Nψxbθ⟶∃PψP⊑O∧P∈Nxγ(iv)
η4ψ≔∀ΦψC¯γXΦ⊑C¯bθYψΦ(v)
η5ψ≔∀ΨC¯γXψ−1Ψ⊑ψ−1C¯bθYΨ(vi)
η6ψ≔∀Φψ−1ℐ¯bθΦ⊑ℐ¯γψ−1Φ.Theorem 19.
⊨ψ∈WIrbθ↔ψ∈ηk,k=i,…,6.Proof.
Similar to Theorem17.Next, we will generalize Theorem 5.8 (a) in [2] in FTSX,τ by the following theorem.Theorem 20.
Assume thatFTSX,τ, FTSY,φ, FTSZ,ϑ, ψ∈YX, and χ∈ZY. The following two relations are holding:(i)
⊨SCobθψ⟶Coχ⟶SCobθχoψ(ii)
⊨Coχ⟶SCobθψ⟶SCobθχoψProof.
(i)
We will show thatSCobθψ≤Coχ⟶SCobθχoψ.Case 1: ifCoχ≤SCobθχoψ, the results holds.Case 2: ifCoχ≥SCobθχoψ, then(60)Coχ−SCobθχoψ=infP∈2Zmin1,1−ϑP+φχ−1P−infP∈2Zmin1,1−ϑP+τbθχoψ−1P≤supP∈2Zφχ−1P−τbθχoψ−1P=supP∈2Zφχ−1P−τbθψ−1χ−1P≤supO∈2YφO−τbθψ−1O.Therefore,Coχ⟶SCobθχoψ=min1,1−Coχ+SCobθχoψ≥infO∈2Ymin1,1−φO+τbθψ−1O=SCobθψ.(ii)(61)Coχ⟶SCobθψ⟶SCobθχoψ=Coχ⟶¬SCobθψ∧•¬SCobθχoψ=¬Coχ∧•¬¬SCobθψ∧•¬SCobθχoψ=¬Coχ∧•SCobθψ∧•¬SCobθχoψ=¬SCobθψ∧•Coχ∧•SCobθχoψ=¬SCobθψ∧•¬¬Coχ∧•¬SCobθχoψ=SCobθψ⟶¬Coχ∧•¬SCobθχoψ=SCobθψ⟶Coχ⟶SCobθχoψ.


## 5. Conclusions

The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in the semantic method of continuous-valued logic). It continues various investigations into fuzzy topology in a legitimate way and extends some fundamental results in general topology to fuzzifying topology. An important virtue of our approach (in which we follow Ying) is that we define topological notions as fuzzy predicates (by formulae of Łukasiewicz fuzzy logic) and prove the validity of fuzzy implications (or equivalences). Unlike the (more widespread) style of defining notions in fuzzy mathematics as crisp predicates of fuzzy sets, fuzzy predicates of fuzzy sets provide a more genuine fuzzification; furthermore, the theorems in the form of valid fuzzy implications are more general than the corresponding theorems on crisp predicates of fuzzy sets. The main contributions of the present paper are to define fuzzifyingb−θ-neighborhood system of a point, fuzzifying b−θ-closure of a set, fuzzifying b−θ-interior of a set, fuzzifying b−θ-open sets, fuzzifying b−θ-closed sets, fuzzifying b−θ-derived sets, and fuzzifying b−θ-boundary sets in the setting fuzzifying topological space. Also, we define the concepts of fuzzifying strongly b−θ-continuous mapping, fuzzifying strongly b−θ-irresolute mapping, and fuzzifying weakly b−θ-irresolute mapping of fuzzifying topological spaces and obtain some basic properties of such spaces. There are some problems for further study:(1)
One obvious problem is our results are derived in the Łukasiewicz continuous logic. It is possible to generalize them to a more general logic setting, like resituated lattice-valued logic considered in [13, 14].(2)
What is the justification for fuzzifying stronglyb−θ-continuous functions in the setting of 2,L topologies?(3)
Obviously, fuzzifying topological spaces in [15] form a fuzzy category. Perhaps, this will become a motivation for further study of the fuzzy category.(4)
What is the justification for fuzzifying stronglyb−θ-continuous functions in M,L-topologies, etc.?Furthermore, the future possible research of the authors will be to give several new results (e.g., fuzzifying semi-θ-open sets and fuzzifying α−θ-open sets), which are similar to the results of fuzzifying b−θ-open sets, fuzzifying semipre-θ-open sets [16] and fuzzifying pre-θ-open sets [17] based on fuzzifying topology.



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