# New Types ofμ-Proximity Spaces and Their Applications

Authors: Rodyna A. Hosny; Tareq M. Al-shami; Abdelwaheb Mhemdi
Journal: Journal of Mathematics
 (2022)

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## Abstract

Near set theory supplies a major basis for the perception, differentiation, and classification of elements in classes that depend on their closeness, either spatially or descriptively. This study aims to introduce a lot of concepts; one of them isμ-clusters as the useful notion in the study of μ-proximity (or μ-nearness) spaces which recognize some of its features. Also, other types of μ-proximity, termed Rμ-proximity and Oμ-proximity, on X are defined. In a μ-proximity space X,δμ, for any subset K of X, one can find out nonempty collections δμK=G⊆X∣Kδ¯μG, which are hereditary classes on X. Currently, descriptive near sets were presented as a tool of solving classification and pattern recognition problems emerging from disjoint sets; hence, a new approach to basic μ-proximity structures, which depend on the realization of the structures in the theory of hereditary classes, is introduced. Also, regarding to specific options of hereditary class operators, various kinds of μ-proximities can be distinguished.


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## Body

## 1. Introduction

A proximity (or nearness) space is a sort of structured set, that consists of a nonempty setX and a binary relation between the subsets of X. In constructive mathematics, any one of these relations may be possessed as major, and the others defined utilizing it; thence, we can differentiate, constructively, among a set-set nearness space “δ,” a set-set apartness space “δ¯”(negation of δ or nonnear), and a set-set neighborhood space “≪.”Initiatively, descriptive near sets have established to be valuable in an assortment of applications as topology [1, 2], solving a lot of problems that rely on human perception [3, 4] that arises in fields of image analysis [5], image processing [6], face recognition [7], rough set [8, 9], environmental space [10], and information systems [11–13], as well as science problems [14]. Also, Peters and Wasilewski [15] put in an approach to the foundations of information science which are formulated in the context of near sets.The concept of proximity spaces was introduced by Naimpally and Warrack in [16]. A spatial nearness relation [1] δ is defined by δ=A,B∈PX×PX∣clA∩clB≠∅. It has ever after proven to be a valuable model in the rating of topological spaces. By introducing f-proximities that depends on certain functions f, Thron [17] created a framework for kinds of proximities. Generalized proximity structures have been widely investigated by several articles including [18, 19]. In [20, 21], several generalized proximities have been established utilizing Ef-proximity and ideals. In [22], Kandil et al. introduced an approach to proximity structures that depend on the recognition of many of the entities important in the theory of ideals. Also, they proposed the concept of g-proximities and showed that, for different choice of “g,” one can obtain many of the known types of generalized proximities. Kandil et al. [23] presented approach of proximity and generalized proximity based on the soft sets. Also, they generalized the notions of compact, proximity relation and proximal neighborhood in the multiset context [24].Many researchers have worked with weaker axioms than those of the fundamental concept ofEfremovic proximity space [25] enabling them to introduce an arbitrary topology on the underlying set with nice properties, and the theory possesses deep results, rich machinery, and tools. In 2019, Mukherjee et al. [26] constructed a generalized proximity structure, named μ-proximity on set X, which induces a generalized topology(GT) on X. Also, Yildirim [27] constructed a generalized μ–proximity structure by using hereditary class on a set. Császár studies attracted many researchers concentration, inducing their considerable studies which involve an extension of generalized topologies utilizing some specific sort of classes of sets called hereditary classes [28].In this study, the concept ofμ-clusters in the study of μ-proximity (or μ-nearness) spaces is presented and some of its features are investigated. As a generalization of [22], the theory of basic μ-proximities in terms of hereditary classes is developed. A new approach to basic μ-proximity structures, which depend on the realization of the structures in the theory of hereditary classes, is introduced. Also, regarding to specific options of hereditary class operators, various kinds of μ-proximities are distinguished.


## 2. Preliminaries

To outline this paper as self-sufficient as possible, we recall the next definitions and results which are due to different references.Definition 1 (see [28]).
A nonempty familyH of subsets of X is called hereditary class if it is closed under subsets. The set of all hereditary classes on X is denoted by H.Definition 2 (see [26]).
A binary relationδμ on the power set PX of a set X is called a μ-proximity (μ-nearness) on X and X,δμ is a μ-proximity (μ-nearness) space if, for all G,K⊆X, δμ satisfies the following axioms:(1)
GδμK⇒KδμG(2)
GδμK, G⊆U, and K⊆V⇒UδμV(3)
xδμx, ∀x∈X(4)
Gδ¯μK⇒∃E⊆X s.t. Gδ¯μE, X\Eδ¯μK
A relationδμ on X is called a basic μ-proximity if it satisfies only conditions (1), (2), and (3). We denote by ℘X the family of all basic μ-proximities on X. Henceforth, we write xδμG for xδμG.
Several properties of the relationδμ on X have been mentioned with details in [26].Remark 1 (see [26]).
A generalized topologyμ is compatible with the μ-proximity relation of sets δμ, denoted μ−∼δμ, τδμ=μ−.Definition 3 (see [16]).
Ifδμ1 and δμ2 are two μ-proximities on a set X, then δμ2 is called finer than δμ1 (in symbols δμ1≺δμ2) if Gδμ2K implies Gδμ1K.Definition 4 (see [26]).
A subsetG of a μ-proximity space X,δμ is a δμ nbhood. of a set K if Kδ¯μX\G. The set of all δμ nbhood. of K with respect to δμ is denoted by Nδμ,K or simply, NμK, i.e., NμK=G∣Kδ¯μX\G.Lemma 1 (see [26]).
For all subsetsG,K of a basic μ-proximity space X,δμ, then the following statements hold:(1)
IfG⊆K, then NμK⊆NμG(2)
K∈NμGiffX\G∈NμX\K(3)
Nμ∅=PXProposition 1 (see [26]).
LetX,δμ be a μ-proximity space and τδμ=μ−. Then, the μ-closure cμG of a set G in X,μ is given by cμG=x:xδμG.
In the following proposition, one can deduce some useful properties ofδμ.Proposition 2.
LetX be a nonempty set. For each x∈X and G,K⊆X,(1)
xδμG, xδμK⇒xδμG∪K(2)
x∈G⇒xδμG(3)
xδμG, yδμK∀y∈G⇒xδμK(4)
If there is a pointp∈Xs.t. Gδμp and pδμK, then GδμK


## 3. Onμ-Clusters

Let us consider the concept ofμ-cluster from μ-proximity spaces and explore some of its properties.Definition 5.
LetX,δμ be a μ-proximity space and G,K⊆X. A μ-cluster is a nonempty collection σμ of subsets of X s.t.(1)
IfG∈σμ, K∈σμ, then GδμK(2)
IfGδμK, for every K∈σμ, then G∈σμ
The family of allμ-clusters of X is denoted by ℵμX.Theorem 1.
Ifσ1μ,σ2μ are μ-clusters in a μ-proximity space X,δμ and if σ2μ⊆σ1μ, then σ1μ=σ2μ.Proof.
LetG∉σ2μ. Then, there exists B∈σ2μ s.t. Bδ¯μG. Since σ2μ⊆σ1μ, then there exists B∈σ1μ s.t. Bδ¯μG. By (1) of Definition 5, G∉σ1μ. Hence, σ1μ⊆σ2μ, so σ1μ=σ2μ.Remark 2.
IfℵμX is a family of finite nested μ-clusters of X, i.e.,σnμ⊆⋯⊆σ2μ⊆σ1μ, then ℵμX=σ1μ.
In view of Definition2 and Proposition 2, the following examples are given.Lemma 2.
LetX,δμ be a μ-proximity space. A collection σμ in X is a μ-cluster iff all sets G∈σμ are near point x. In other words, the collection σμx=G⊆X∣xδμG is a μ-cluster.σμx is called a principal μ-cluster or point μ-cluster.Remark 3.
G∈σμx iff x∈cμG.Example 1.
Ifδμ is defined on X, the family of all natural numbers, as GδμK iff both G and K are nonempty sets. Then, the collection σμ=G⊆X∣G≠∅ is a μ-cluster. Also, σμ=σμx, for every x∈X.Theorem 2.
Ifσμ is a μ-cluster in a μ-proximity space X,δμ, then(1)
σμx≠∅(2)
G∈σμ and G⊆K⇒K∈σμ(3)
G∈σμ iff cμG∈σμ(4)
If there is a pointx∈Xs.t. x∈σμ, then σμ=σμx(5)
A∈σμ iff for all G,K⊆X, G∈σμ and X\A∪K=X⇒GδμKProof.
(1)
Obvious.(2)
LetG∈σμ and G⊆K. Suppose K∉σμ; then, there exists B∈σμ s.t. Bδ¯μK. Since G,B∈σμ, then from (2) of Definition 2 and (1) of Definition 5, BδμG and so BδμK. This is a contradiction. Hence, K∈σμ.(3)
IfG∈σμ, then, from (1), cμG∈σμ. In the other side, suppose G∉σμ. Then, Bδ¯μG, for some B∈σμ. In view of Lemma 2.8 of [26], cμBδ¯μcμG. So, Bδ¯μcμG. Then, from (1) of Definition 5, cμG∉σμ. Hence, the proof has been completed.(4)
LetG∈σμ. Since x∈σμ, then, by (1) of Definition 5, xδμG. Hence, G∈σμx, i.e., σμ⊆σμx. On the contrary, let G∈σμx; then, xδμG. Suppose G∉σμ; then, Bδ¯μG for some B∈σμ. Since x,B∈σμ, then xδμB. According to (4) of Proposition 2, BδμG. This is a contradiction. Hence, G∈σμ. Then, σμx⊆σμ, so σμ=σμx.(5)
⇒ Let A∈σμ and G∈σμ; then, GδμA. Suppose X\A∪K=X; then, every element of A belongs to K. Hence, from (2), K∈σμ, so GδμK. ⇐ Assume that, for all G,K⊆X, G∈σμ and X\A∪K=X imply GδμK. Choose K=A; then, GδμA, for all G∈σμ. So, A∈σμ.
Now, we define a functionα from PX into a family ℵμX of all μ-clusters of X by(1)αG=σμ∈ℵμX∣G∈σμ.Theorem 3.
(1)
α∅=∅ and αX=ℵμX(2)
G⊆K⇒αG⊆αK(3)
αG∪αK⊆αG∪KandαG∩K⊆αG∩αK(4)
αG∩αK≠∅iffGδμKProof.
(1)
In view of (2) of Theorem 2, α∅=σμ∈ℵμX∣∅∈σμ=∅ and αX=σμ∈ℵμX∣X∈σμ=ℵμX.(2)
Letσμ∈αG; then, G∈σμ. Since G⊆K, hence, in view of (2) of Theorem 2, K∈σμ, so σμ∈αK, i.e., αG⊆αK.(3)
It is obvious from (2).(4)
⇒ Suppose αG∩αK≠∅; then, there exists σμ∈αG∩αK. Hence, σμ∈αG and σμ∈αK which imply that G and K∈σμ. From Definition 5, GδμK.⇐ Assume that GδμK and αG∩αK=∅; then, σμ∉αG for every σμ∈αK. Hence, GδμK, for every K∈σμ which imply that G∈σμ. It is a contradiction.
Next, we shall introduce an appropriate proximityΦ on ℵμX. Λ,Γ⊆ℵμXΛΦΓ⇔Λ⊆αG and Γ⊆αK imply GδμK, for all G,K⊆X.Theorem 4.
The structureℵμX,Φ is a μ-proximity space.Proof.
(1)
ΛΦΓ⇒ΓΦΛ.(2)
SupposeΩΦ¯Υ, Λ⊆Ω, and Γ⊆Υ; then, Ω⊆αG, Υ⊆αK, and Gδ¯μK, for some G,K⊆X. From hypothesis Λ⊆Ω and Γ⊆Υ, hence, Λ⊆αG, Γ⊆αK, and Gδ¯μK, for some G,K⊆X. It follows that ΛΦ¯Γ. Consequently, ΛΦΓ, Λ⊆Ω, and Γ⊆Υ imply ΩΦΥ(3)
Letσ be a μ-cluster of X. If σ⊆αG and σ⊆αK, then from (4) of Theorem 3, GδμK, for all G,K⊆X. Hence, σΦσ.(4)
SupposeΛΦ¯Γ; then, Λ⊆αG, Γ⊆αK, and Gδ¯μK, for some G,K⊆X.Corollary 1.
Λ∩Γ≠∅⇒ΛΦΓ.Proof.
LetΛ∩Γ≠∅; then, there exists σ s.t. σ∈Λ and σ∈Γ. Suppose ΛΦ¯Γ; then, Λ⊆αG, Γ⊆αK, and Gδ¯μK, for some G,K⊆X. Hence, σ⊆αG and σ⊆αK. According to (4) of Theorem 3, GδμK. It is a contradiction. So, ΛΦΓ.


## 4. Onμ-Proximity with Hereditary Class

In accordance with principalμ-clusters notion, we will turn to the concept of hereditary classes.Definition 6.
LetX,δμ be a μ-proximity space and x∈X; then, G⊆X:G∉σμx is a hereditary class on X.Remark 4.
LetX,δμbe aμ-proximity space,G⊆Xandx∈X. Then,(1)
δμx=G⊆X:xδ¯μG(2)
G∈δμxiffx∈δμG(3)
cμG=x∈X∣x∉δμG
In the next section, we introduce the notionδμK for any subset K of X as a generalization of δμx for any x∈X.Definition 7.
LetX,δμ be a μ-proximity space and k⊆X; then, we define(2)δμK=G⊆X∣Kδ¯μG.
Next example shows thatδμK is not an ideal on X, for any set K⊆X.Example 2.
LetX=a,b,c and let δμ be a μ-proximity on X defined as GδμK⇔G=K.
IfG1=a, G2=b, and K=a,b, then G1,G2∈δμK but G1∪G2∉δμK.Example 3.
Evidently,δμ∅=PX and δμX=∅.
Next, we reformulate Definition2 in terms of δμ. as follows:Definition 8.
A binary relationδμ on PX is called a μ-proximity on X if, for all G,K⊆X, δμ satisfies the following axioms:(1)
G∈δμK⇒K∈δμG(2)
U∈δμV, G⊆U, and K⊆V⇒G∈δμK(3)
x∉δμx, ∀x∈X(4)
G∈δμK⇒ there exists E⊆X s.t. G∈δμX\E and E∈δμK
A relationδμ is called a basic μ-proximity if it satisfies only conditions (1), (2), and (3). We write x∈δμK for x∈δμK and δμx for δμx.
It is clear that (2) in the axiomatical definition ofμ-proximity relation δμ can be equivalently replaced by 2⋆δμG∪K⊆δμG∩δμK, for every G,K⊆X.
In the following, we will display considerable of the properties ofδμ..
From Definition7, the next lemmas follow directly.Lemma 3.
LetX,δμbe aμ-proximity space. Then,(1)
G1∪G2∈δμK⇒G1∈δμKandG2∈δμK⇒G1∩G2∈δμK(2)
Gr∈δμK, r∈Λ⇒∩r∈ΛGr∈δμK(3)
G∈δμK⇒x∈δμK∀x∈G(4)
X∉δμxandx∉δμXLemma 4.
For all subsetsG,K of a μ-proximity space X,δμ, the following statements hold:(1)
G⊆K⇒δμK⊆δμG(2)
δμG∩δμK⊆δμG∩K(3)
A∈δμG, B∈δμK⇒A∩B∈δμG∩K
Regarding to hereditary classes onX, one can introduce μ-proximity relations on X as we show in the following examples.Example 4.
LetX be a set with any hereditary class H and x∉H. For any subsets G and K of X, we define(3)G∈δμK⇔G∩K∈H.
Then, the relationδμ is a μ-proximity on X.Example 5.
LetX be a set with any hereditary class H and x∉H. For any subsets G and K of X, define G∈δμK⇔G or K∈H.
Then, the relationδμ is a μ-proximity on X.Theorem 5.
LetX,δμ be a μ-proximity space and G,K⊆X. If μ-closures and μ-interiors are taken with respect to μ−=τδμ, then the following properties are true:(1)
G∈δμK implies G∩K=∅(2)
G∈δμK iff X\G∈NμK(3)
Kisδμ closed WRT μ−=τδμ if x∈δμK, ∀x∉K(4)
K∈τδμ WRT μ−=τδμ if x∈δμX\K, ∀x∈K(5)
G∈δμK implies cμG∈δμK and G∈δμcμK(6)
G∈δμK iff cμG∈δμcμK(7)
x∈δμK iff cμx∈δμcμKProof.
Direct to prove.Lemma 5.
Letδμ1,δμ2 be two μ-proximities on a set X. Then, δμ1≺δμ2 iff δμ1G⊆δμ2G, ∀G⊆X.Proof.
Accessible consequence of Definition3.Theorem 6.
Letμ1 and μ2 be two μ-completely regular generalized topologies on X and δμ1 and δμ2 be the μ-proximities on X defined as G∈δμjK⇔G and K are functionally distinguishable WRT μj, respectively, j=1,2.
Then,μ1⊆μ2 implies δμ1≺δμ2Proof.
IfG∈δμ1K, then there exists a μ1continuous function f:X,μ1⟶0,1, where 0,1 is endowed with the subspace generalized topology induced by κ on R (where κ is the generalized topology on the set R of reals generated by the base β=−∞,p:p∈R∪p,∞:p∈R) s.t. fG=0 and fK=1. Since μ1⊆μ2, then f is a μ2 continuous function from X,μ2 to 0,1 s.t. fG=0 and fK=1. So, G∈δμ2K. According to Lemma 5, δμ1≺δμ2.Theorem 7.
Letδμ1,δμ2 be two μ-proximities on a set X. Then, the following statements are equivalent:(1)
δμ1x=δμ2x, ∀x∈X(2)
cδμ1G=cδμ2G, ∀G⊆X(3)
Nδμ1x=Nδμ2x, ∀x∈XProof.
Easy to prove.Definition 9.
Aμ-proximity space X,δμ is T0 iff for any two distinct points x,y of X, xδ¯μy.
Utilizing hereditary classes, another equivalent definition ofT0-space is obtained.Theorem 8.
Aμ-proximity space X,δμ is T0 iff for any two distinct points x,y of X, δμx≠δμy.Proof.
Letx,y be any two distinct points in a T0-space X,δμ; then, xδ¯μy. In view of Definition 5, y∉σμx or x∉σμy. Suppose y∉σμx, which gives y∈δμx, but from (3) of Definition 2y∉δμy. Consequently, δμx≠δμy. Conversely, let x,y∈X and x≠y with xδμy. Suppose that δμx≠δμy; then, there is a subset G of X s.t. G∈δμx and G∉δμy. Then, G∉σμx and G∈σμy. Hence, xδ¯μG but yδμG. Since xδμy and yδμG, then in view of (3) of Proposition 2, xδμG. It is a contradiction. Thus, X,δμ is T0-space.Lemma 6.
LetX,μ be a μ-normal GTS. Then, cμG∩cμK=∅⇔cμG, cμK are functionally distinguishable.Proof.
SupposecμG∩cμK=∅; then, by Urysohn’s lemma, cμG and cμK are functionally distinguishable. In the other side, suppose cμG∩cμK≠∅. Then, there exists p∈X s.t. p∈cμG∩cμK. Since there is no function f s.t. fp has distinct values at p; hence, cμG and cμK are not functionally distinguishable.Theorem 9.
LetX,μ be a μ-normal GTS. For any subsets G and KofX, the relation δμ on X given by G∈δμK⇔cμG∩cμK=∅ is a compatible μ-proximity on X.Proof.
According to Lemma6, cμG∩cμK=∅ iff cμG and cμK are functionally distinguishable. From the features of a μ-continuous function, G and K are functionally distinguishable iff cμG and cμK are functionally distinguishable. So, G∈δμK iff cμG∩cμK=∅ iff G and K are functionally distinguishable. By Urysohn’s lemma, every μ-normal GTS is μ-completely regular; then, from Theorem 2.11 of [26], the relation δμ is a compatible μ-proximity on X.Theorem 10.
IfX,μ is a μ-completely regular, GTS has a compatib1e μ-proximity δμ defined by(4)G∈δμK⇔cμG∩cμK=∅
Then,X,μ is μ-normal GTS.Proof.
SupposecμG,cμK are disjoint μ-closed sets; then, G∈δμK. Hence, there exists E⊆X s.t. G∈δμX\E and E∈δμK. From Corollary 2.7 of [26] and Definition 7, G⊆X\cμX\E=iμE and K⊆iμXE. Since iμE and iμX\E are disjoint μ-open sets, so X,μ is μ-normal.


## 5. On Basicμ-Proximity with Hereditary Class

Definition 10.
A relationδμ is called Oμ-proximity on X if it is a basic μ-proximity on X, and it satisfies the following condition:(5)GδμAandaδμK∀a∈A⇒GδμK.Example 6.
In one of the schools, suppose thatG,A,K be parents’ council, set of students, and set of teachers, respectively. Evidently, δμ satisfies Oμ-proximity axioms on X, see Figures 1 and 2.Figure 1
GδμA and aδμK∀a∈A⇒GδμK.Figure 2
xδ¯μK⇒xδ¯μE and X∖Eδ¯μK, for some E⊆X.Definition 11.
A relationδμ is called Rμ-proximity on X if it is a basic μ-proximity on X, and it satisfies the following condition:(6)xδ¯μK⇒thereexistsE⊆Xs.t.xδ¯μEandXEδ¯μK.Theorem 11.
Letδμbe a basicμ-proximity onX. Then, cμK=∩A∣X\A∈δμK.Proof.
Suppose thatL=∩A∣X\A∈δμK. We shall prove L=cμK. Let x∉cμK; then, x∈δμK. Hence, x∉L, so L⊆cμK. On the contrary, let x∉L; then, there is a subset A of X s.t. x∉A and X\A∈δμK. According to (1) of Lemma 3, x∈δμK. Thus, x∉cμK, so cμK⊆L. Hence, L=cμKTheorem 12.
Letδμ∈℘X. Then, the following are equivalent:(1)
δμ is a μ-proximity on X(2)
IfG∈δμK, then NμG∩δμK≠∅(3)
IfG∈NμK, then there exists E∈NμKs.t. G∈NμEProof.
1⇒2 Let G∈δμK. In view of (1) and (4) of Definition 7, there exists E⊆X s.t. X\E∈δμG and E∈δμK. Hence, by (2) of Theorem 5, E∈NμG∩δμK. Consequently, NμG∩δμK≠∅.2⇒3 Let G∈NμK; then, X\G∈δμK. Hence, NμX\G∩δμK≠∅ which implies that there exists a set A s.t. A∈NμX\G and A∈δμK. In view of (2) of Lemma 1 and (2) of Theorem 5, G∈NμX\A and X\A∈NμK. Put X/A; then, (3) holds.3⇒1 Let G∈δμK; then, X\G∈NμK. According to (3), then there exists E∈NμK s.t. X\G∈NμE. Therefore, G∈δμE and X\E∈δμK, so δμ is a μ-proximity on X.Theorem 13.
Letδμ∈℘X. Then, the following are equivalent:(1)
δμ is a Rμ-proximity on X(2)
Ifx∈δμK, then Nμx∩δμK≠∅(3)
IfK∈Nμx, then there exists E∈Nμx s.t. K∈NμEProof.
1⇒2 Let x∈δμK. From Definition 11, there exists E⊆X s.t. x∈δμX\E and E∈δμK. So, X\E∈δμx and E∈δμK. Hence, E∈Nμx∩δμK. Consequently, Nμx∩δμK≠∅.2⇒3 Let K∈Nμx; then, X\K∈δμx. Hence, NμX\K∩δμx≠∅ which implies that there exists a set A of X s.t. A∈NμX\K and A∈δμx. In view of (2) of Lemma 1 and (2) of Theorem 5, K∈NμX\A and X\A∈Nμx. Put X/A; then, (3) holds.3⇒1 Let x∈δμK; then, X\K∈Nμx. According to (3), then there exists E∈Nμx s.t. X\K∈NμE. Therefore, K∈δμE and X\E∈δμx, so δμ is a Rμ-proximity on X.Definition 12.
LetH be a hereditary class on a basic μ-proximity space X,δμ. A mapping λ:℘X×H⟶H is called a hereditary class operator on X if it identifies to each pair δμ,H, a hereditary class λδμ,H on X, satisfying the following conditions: λδμ,H1⊆λδμ,H2 whenever H1⊆H2, for every H1, H2∈H.Definition 13.
Letλ be a hereditary class operator on X. Then, a basic μ-proximity δμ on X is called a λ-μ-proximity if, for every G⊆X, δμG⊆λδμ,δμG. The family of all λ-μ-proximities is denoted by μPλ.
In the next definition, several kinds of hereditary class operators are listed.Definition 14.
For a setX, for all δμ∈℘X and H∈H, we define(1)
eδμ,H=H(2)
λ0δμ,H=G⊆X∣NμG∩H≠∅(3)
λ1δμ,H=G⊆X∣cδμG∈H(4)
λ2δμ,H=G⊆X∣x∈δμG∪H,∀x∈X(5)
λ3δμ,H=G⊆X∣Nμa∩H≠∅,∀a∈G
When there is no ambiguity, we will writeλi for λiδμ,H, where i=0,1,2,3.Theorem 14.
For allδμ∈℘XandH∈Hand forλ∈e,λ0,λ1,λ2,λ3, we have thatλis a hereditary class operator onX.Proof.
(1)
It is understandable;eδμ,H=H is a hereditary class operator on X.(2)
Suppose thatδμ∈℘X and H∈H. Let G∈λ0=λ0δμ,H; then, NμG∩H≠∅. If A⊆G, then according to Lemma 1, NμA∩H≠∅ and so A∈λ0. Hence, λ0 is a hereditary class. Now, let H1⊆H2 and G∈λ0δμ,H1; then, NμG∩H1≠∅. Therefore, NμG∩H2≠∅ and so G∈λ0δμ,H2. Consequently, λ0 is a hereditary class operator on X.(3)
By usingδμclosure operator properties, λ1 is a hereditary class operator on X.(4)
In view of Lemma4 (1), λ2 is a hereditary class operator on X.(5)
By similar manner,λ3 is a hereditary class operator on X.Theorem 15.
Letλ be a hereditary class operator. If I=λ∣λδμ,∩r∈ΛHr=∩r∈Λλδμ,Hr,δμ∈℘X,Hr∈H,Hr∈H,r∈Λ, thene,λ1,λ2∈I.Proof.
Straightforward.Corollary 2.
Letλ be a hereditary class operator. If I˜=λ∣λδμ,H1∩H2=λδμ,H1∩λδμ,H2,δμ∈℘X,H1,H2∈H, thene,λ1,λ2∈I˜Proof.
The proof is obvious by usingI⊆I˜.Remark 5.
The following example illustrates thatλ0,λ3∉I˜, in general, if H1,H2 are hereditary classes.Example 7.
In Example2, suppose H1=∅,a and H2=∅,b. Then, λ0δμ,H1∩H2≠λ0δμ,H1∩λ0δμ,H2 and λ3δμ,H1∩H2≠λ3δμ,H1∩λ3δμ,H2. Hence, λ0,λ3∉I˜Theorem 16.
Letλ be a hereditary class operator. If T=λ∣λδμ1,H=λδμ2Hwithcδμ1.=cδμ2.,δμ1,δμ2∈℘X,H∈H, thene,λ1,λ2,λ3∈T.Proof.
We shall prove only forλ2 and the rest of the proof is similar. Let G∈λ2δμ1,H; then, x∈δμ1G∪H, ∀x∈X. Hence, G∈δμ1x. Since cδμ1.=cδμ2., then, by using Theorem 7, δμ1x=δμ2x. Consequently, G∈δμ2x and so G∈λ2δμ2,H, i.e., λ2δμ1,H⊆λ2δμ2,H. By the same manner, we can prove λ2δμ2,H⊆λ2δμ1,H. It follows that λ2∈T.
From Definition3, one can deduce the following results.Lemma 7.
Letδμ1,δμ2 be two μ-proximities on a set X and G⊆X. Ifδμ1≺δμ2, then(1)
cδμ2G⊆cδμ1G(2)
Nμ1G⊆Nμ2GTheorem 17.
Letλ be a hereditary class operator. If U=λ∣λδμ1,H⊆λδμ2Hwheneverδμ1≺δμ2,H∈H, then e,λ0,λ1,λ2,λ3∈U.Proof.
LetG∈λ0δμ1,H; then, Nμ1G∩H≠∅. Since δμ1≺δμ2, hence by Lemma 7 (2), Nμ1G⊆Nμ2G. Consequently, Nμ2G∩H≠∅. So, G∈λ0δμ2,H, i.e., λ2δμ1,H⊆λ2δμ2,H. It follows that λ2∈U. The rest of the proof is similar.Theorem 18.
X,δμis aλ0-μ-proximity space iffNμG∩δμK≠∅, for everyG∈δμK.Proof.
⇒ Let G∈δμK. Since δμ is a λ0-μ-proximity on X and δμK is a hereditary class on X, then G∈λ0δμ,δμK. Hence, by Theorem 12, NμG∩δμK≠∅. ⇐ Let G∈δμK; then, NμG∩δμK≠∅, which implies that G∈λ0δμ,δμK. Consequently, δμK⊆λ0δμ,δμK, so δμ is λ0-μ-proximity.
In view of Theorems12 and 18, the next corollary is verified.Corollary 3.
δμis aμ-proximity onX iff it isλ0-μ-proximity.Theorem 19.
Letδμ∈℘XandH∈H; then,λ0δμ,∪K∈HδμX\K.Proof.
Suppose thatG∈λ0δμ,H. Then, NμG∩H≠∅. So, there exists K∈H s.t. K∈NμG, i.e., X\K∈δμG, which leads to K∈H s.t. G∈δμX\K. Consequently, G∈∪K∈HδμX\K, so λ0δμ,H⊆∪K∈HδμX\K. By the same manner, ∪K∈HδμX\K⊆λ0δμ,H is obtained.Theorem 20.
Letδμ∈℘X. Ifδμ∈μPλ1, then cδμ is a μ-closure operator.Proof.
Certainly, fromcδμ properties, cδμ operator is monotone and extensive. So, we shall prove cδμ is an idempotent operator. Obviously, cδμG⊆cδμcδμG. Let G⊆X and x∈cδμcδμG; then, by Remark 4 (3), cδμG∉δμx. Since δμx is a hereditary class on X, so from Definition 14, G∉λ1δμ,δμx. Since δμ∈μPλ1, i.e., δμ on X is an λ1-μ-proximity or δμx⊆λ1δμ,δμx; then, G∉δμx, so x∈cδμG. Consequently, cδμcδμG⊆cδμG. So, cδμcδμG=cδμG. Hence, cδμ is a μ-closure operator.Theorem 21.
Letδμ∈℘X. Then, δμ is a λ1-μ-proximity iff cδμG∈δμK, for every G∈δμK.Proof.
. LetG∈δμK. Since δμ is a λ1 proximity. Then, G∈λ1δμ,δμK and so cδμG∈δμK. Conversely, let K⊆X and G∈δμK; then, cδμG∈δμK. Therefore, G∈λ1δμ,δμH, so δμK⊆λ1δμ,δμK. Hence, δμ is a λ1-μ-proximity.Theorem 22.
Letδμ∈℘X. Then, δμ is an Oμ-proximity on X iff it is λ1-μ-proximity.Proof.
⇒ Let G⊆X and K∉λ1δμ,δμG; then, cδμK∉δμG. In view of Definition 8, G∉δμcδμK. From the definition of cδμK, then x∉δμK, ∀x∈cδμK. So, G∉δμcδμK, x∉δμK, and ∀x∈cδμK. Since δμ is an Oμ-proximity relation on X, then G∉δμK. So, δμ is λ1-μ-proximity.
⇐ Let G∉δμA and a∉δμK∀a∈A. Obviously, A⊆cδμK, so G∉δμcδμK. Hence, cδμK∉δμG. Since δμ is a λ1-μ-proximity relation on X, then K∉δμG, i.e., G∉δμK, which induces to δμ is an Oμ-proximity on X.Theorem 23.
Letδμ∈℘XandH∈H. Ifλ∈e,λ0,λ1,λ3, thenλδμ,H⊆H.Proof.
We prove only forλ0The rest of the proof follow directly from definitions of e and λ1.
LetG∈λ0δμ,H; then, NμG∩H≠∅. Hence, there exists E∈H s.t. E∈NμG. Since E∈NμG, then G⊆E∈H, so G∈H, i.e., λ0δμ,H⊆H.Theorem 24.
Letδμ∈℘Xandx∈X. Then, δμ∈μPλ2iffG∈δμK⇒G∈δμxorK∈δμx.Proof.
⇒ Let G∈δμK. Since δμ∈μPλ2, then G∈λ2δμ,δμK, so x∈δμG∪δμK, ∀x∈X. Thus, G∈δμx or K∈δμx. ⇐ Let K∈δμG and x∈X; then, K∈δμx or G∈δμx. Hence, xδμG∪δμK, ∀x∈X; it follows that K∈h2δμ,δμG, ∀G⊆X. Hence, δμG⊆λ2δμ,δμG. Consequently, δμ∈μPλ2.Theorem 25.
X,δμis aλ3-μ-proximity space iffNμx∩δμK≠∅, for everyx∈δμK.Proof.
⇒ Let x∈δμK. Since δμ is a λ3-μ-proximity on X and δμK is a hereditary class on X, then x∈λ3δμ,δμK. Hence, by Theorem 13, Nμx∩δμK≠∅. ⇐ Let x∈δμK, then Nμx∩δμK≠∅, which implies that x∈λ0δμ,δμK. Consequently, δμK⊆λ3δμ,δμK, so δμ is λ3-μ-proximity.Corollary 4.
δμis aRμ-proximity onX iff it isλ3-μ-proximity.Theorem 26.
For allδμ∈℘Xand for allH∈H, we have(1)
μPλ0⊆μPλ1∩μPλ3(2)
μPλ1⊆μPλ2Proof.
(1)
Letδμ∈μPλ0, i.e., δμ is a λ0-μ-proximity on X. Suppose that B∈δμG; then, NμB∩δμG≠∅. Hence, there exists F⊆X s.t. F∈NμB and F∈δμG and so X\F∈δμB. According to Theorem 11, cδμB⊆F. Since F∈δμG, then cδμB∈δμG. Hence, B∈λ1δμ,δμG. Consequently, δμ∈μPλ1 and so μPλ0⊆μPλ1. Also, let δμ∈μPλ0. Suppose G∈δμK; then, NμG∩δμK≠∅, so Nμa∩δμK≠∅, for every a∈G. Hence, G∈λ3δμ,δμH. Hence, δμK⊆λ3δμ,δμH. Consequently, δμ∈μPλ3. So, μPλ0⊆μPλ3.(2)
Letδμ∈μPλ1 and let G∈δμK. Then, by Theorem 21, cδμG∈δμK. We claim that G∈λ2δμ,δμH. Suppose G∉h2δμ,δμH; then, there exists x∈X s.t. x∉δμG and x∉δμK; then, x∈cδμG, x∉δμK. However, x⊆cδμG and δμK∈H, so cδμG∉δμK, a contradiction. Hence, G∈h2δμ,δμH. It follows that δμK⊆h2δμ,δμH. Consequently, δμ∈μPh2 and so μPh1⊆μPh2.Theorem 27.
Letλbe a hereditary class operator. IfE=λ∣λδμ,H⊆λδμ,λδμ,H,δμ∈μPλ,H∈H, thene,λ0,λ1,λ2∈E.Proof.
LetG∈λ0δμ,H; then, NμG∩H≠∅ which implies that there exists F∈H s.t. F∈NμG. Since δμ∈μPλ0, then, by Theorem 18, there exists B∈NμG s.t. F∈NμB and so NμB∩H≠∅. So, B∈λ0δμ,H. However, B∈NμG; thus, NμG∩λ0δμ,H≠∅. Hence, G∈λ0δμ,λ0δμ,H. Consequently, λ0δμ,H⊆λ0δμ,λ0δμ,H. It follows that λ0∈E.
Next, letδμ∈μPλ1 and let G∈λ1δμ,H. Then, cδμG∈H and so cδμG=cδμcδμG∈H (by Theorem 20). Hence, cδμG∈λ1δμ,H. Hence, G∈λ1δμ,λ1δμ,H. Consequently, λ1δμ,H⊆λ1δμ,λ1δμ,H. It follows that λ1∈E.
Now, letG∉λ2δμ,λ2δμ,H. Since δμ∈μPλ2, hence, there exists x∈X s.t. x∉δμG∪λ2δμ,H. This implies that x∉δμG and x∉λ2δμ,H. It follows that there exists y∈X s.t. y∉δμx∪H. Then, y∉δμx and y∉H. Since x∉δμG, i.e., G∉δμx, then in view of Theorem 24, y∉δμG∪H. Hence, G∉λ2δμ,H. Hence, λ2δμ,H⊆λ2δμ,λ2δμ,H. Consequently, λ2∈E.
Finally, letG∈λ3δμ,H. Since δμ∈μPλ3, then Nμa∩H≠∅, for every a∈G, which implies that there exists F∈H s.t. F∈Nμa, for every a∈G. Therefore, in view of Theorem 13 and Theorem 25, there exists B∈Nμa s.t. F∈NμB, for every a∈G. Hence, NμB∩H≠∅, so Nμx∩H≠∅, for every x∈B. It follows thatB∈λ3δμ.SinceB∈Nμa,soNμa∩λ3δμ≠∅.Consequently,G∈λ3δμ,λ3δμ,H.Hence,λ3δμ,H⊆λ3δμ,λ3δμ,H and so λ3∈E.


## 6. Application

Near sets in mathematics are either spatially close or descriptively close. The classical idea of the nearness of sets is spatial, where sets are near, as long as the sets possess joint elements. Descriptively close sets consist of organs that have matching descriptions, i.e., the setX with descriptively close sets δ include some of sets that consist of elements, in which every element of them have position and measurable attributes as colour or frequency of apparition.In the next section, we will display an application about spatially close usingδμ. idea.Remark 6.
Obviously, a pointμ-cluster σμx is spatial nearness collection for any point x.Example 8.
In Example 2 of [29], suppose that X is the set of points in the picture (see Figure 3). Let G,K⊆X be the set of points in the knights’ horse and set of points in the suspended knight, respectively. G∈δμK, since there is no common element between cμG and cμK. So, the subsets G,K are spatially nonnear sets.Figure 3
A bit more, punch, 1845 [29].


## 7. Conclusion

In this work, we have introduced the concept ofμ-clusters to study μ-proximity (or μ-nearness) spaces and investigated main properties. Also, we have defined other types of μ-proximity called Rμ-proximity and Oμ-proximity on X. Furthermore, we have presented descriptive near sets as a tool of solving classification and pattern recognition problems emerging from disjoint sets; hence, a new approach to basic μ-proximity structures, which depend on the realization of the structures in the theory of hereditary classes, has been introduced.Finally, we hope this article helps to enrich the near set theory and opens up a door for researchers to conduct further studies in this interesting theory.



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