# Some Topological Notations via Maki’s Λ-Sets

Authors: A. A. Azzam; A. A. Nasef
Journal: Complexity
 (2020)

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## Abstract

Our purpose is to present the notions of aβ-Λ-set and a β-V-sets in topological space. We discuss the basic properties of β-Λ-sets and β-V-sets. Also, the achievement of the topology defined by these families of sets is obtained. Finally, these results are applied to the case of X,τ which is the digital n-space Zn,Kn (cf. Section 4).


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## Body

## 1. Introduction and Preliminaries

In the first section, we survey some of the standard facts on the concept of near open sets in the topological spaces. A generalized open set which is called aβ-open set is introduced by Abd El-Monsef et al. [1]. Maki [2] developed the definition of Levin [3] and Dunham [4] to generalize closed sets, by introducing the concept of Λ-set in topological spaces as the set that coincides with their kernel. Also, he compares the topology generated by Λ-set and the given topology. The development continued at the hands of Calads and Dontchev [5]. They presented concepts of Λs-sets, Vs-sets, and g.Vs-sets. The new topology deduced by the concept of pre-Λ-sets, and pre-V-sets were introduced by Ganster et al. [6]. Also, Caldas et al. [7, 8] set up notation of Λp-set and Λp-continuous function. Λp-irresolute functions are defined as an identification of irresolute functions and Vp-closed functions by using Λp-sets and Vp-sets. Notions enable us to obtain conditions under which functions and inverse functions preserve Λp-sets and Vp-sets. Abd El-Monsef et al. [9] introduced the notion of b-Λ-sets and b-V-sets and obtained new topologies defined by these families of sets. Also, g⋅Λb-sets, g⋅Vb-sets, and some of its properties were introduced and studied. The digital line is a typical example of a T1/2 space proved by Khalimsky et al. [10]. The Λβ-sets and Vβ-sets concepts can be applied in a rough set theory of decreasing the boundary region of any subset of an approximation space. The notation of a Λβ, Vβ, g.Λβ-sets, and g.Vβ-sets shall be introduced. Properties of these sets and some related new separation axioms are investigated. Also, we introduce a new class of topological spaces called TVβ-spaces and as an application with the clarification that the image of TVβ-spaces under a homeomorphism is a TVβ-space and that a TVβ-space is equivalent to pre-R0 space.Via this paper,Y,σ(or purely Y) represent the topological spaces. For a subset B of Y, clB, intB, and Bc denote the closure of B, the interior of B, and the complement of B, respectively. Let us recall the following definitions, which are useful in the sequel.Definition 1.
A subsetB of the space Y, which carries topology σ is called(a)
Semiopen [11] if B⊆clintB(b)
Preopen [12] if B⊆intclB(c)
α-Open [13] if B⊆intclintB(d)
β-Open [1] (or semi-preopen [14]) if B⊆clintclB(e)
b-Open [15] if B⊆intclB∪clintB
The class of all semiopen sets (resp., preopen,α-open, β-open, and b-open) is denoted by SoY,σ (resp., PoY,σ,αoY,σ,βoY,σs, and BoY,σ).
The complement of these sets called semiclosed [16] (resp., preclosed [12], α-closed [13], β-closed [1], and b-closed [15]), and the classes of all these sets will be denoted by ScY,σ(resp., PcY,σ, αcY,σ,βcY,σ, and BcY,σ).Definition 2.
A subsetB of a topological space Y,σ is called(a)
Λ-set (resp., V-set) [2] if it is an intersection (resp., union) of open supersets of B(resp., closed sets contained in B)(b)
Λα-set (resp., Vα-set) [5] if it is an intersection (resp., union) of α-open supersets of B (resp.,α-closed sets contained in B)(c)
Λs-set (resp.Vs-set) [5] if it is an intersection (resp., union) of semiopen supersets of B(resp., semiclosed sets contained in B)(d)
Pre-Λ-set (resp., pre-V-set) [6] if it is an intersection (resp., union) of preopen supersets of B(resp., preclosed sets contained in B)(e)
b-Λ-set (resp.b-V-set) [9] if it is an intersection (resp., union) of b-open supersets of B(resp., b-closed sets contained in B)Remark 1.
For some kernels in an ordinary topological spaceY,σ, the following different notations may be used (e.g., [4, 5, 7, 9, 10, 12, 17–20]).
For example,σ−kerB=kerB=BΛ, where BΛ=∩J:J⊇B,J∈σ and B is called the kernel of B.
SoY,σ−kerB=SkerB=BΛs=ΛsB, where BΛs=∩J:J⊇B,J∈SoY,σ and BΛs is named the semikernel of B.
PoY,σ−kerB=PkerB=BΛp=ΛpB, where BΛp=∩J:J⊇B,J∈PoY,σ and BΛp is named the prekernel of B.
σα-kerA=αkerB=BΛα=ΛαB, where BΛα=∩J:J⊇B,J∈σα and BΛα is named the α-kernel of B.
SpoY,σ-kerB=SpkerB=BΛsp=ΛspB, where BΛsp=∩J:J⊇B,J∈SpoY,σ and BΛsp is named the semi-prekernel of B.
BoY,σ-kerB=BkerB=BΛB=ΛBB, where BΛB=∩J:J⊇B,J∈BoY,σ and BΛB is named the B-kernel of B.
RoY,σ-kerB=RkerB=BΛR=ΛRB, where BΛR=∩J:J⊇B,J∈RoY,σ and BΛR is named the R-kernel of B.Definition 3.
A subsetB of a topological space Y,σ is named(a)
GeneralizedΛ-set [2] if ΛB⊆F, whenever B⊆F, and F∈CY,σ(b)
Generalized semi-Λ-set [5] if ΛsB⊆F, whenever B⊆F, and F∈ScY,σ(c)
Generalized pre-Λ-set [6] if ΛpB⊆F, whenever B⊆F, and F∈pcY,σ(d)
Generalizedb-Λ-set [9] if ΛbB⊆F, whenever B⊆F, and F∈BcX,τDefinition 4.
A spaceY which carries topology σ is named a β-T1-space [21] if to each pair of distinct points y,z of Y,σ, there corresponds a β-open set B containing y but not z and a β-open set C containing z but not y. Clearly, a space Y,σ is β-T1 if and only if each singleton is β-closed.Definition 5.
A spaceY which carries topology σ is said to be(i)
β-Connected [22, 23] if Y cannot be expressed as the union of two nonempty disjoint β-open sets of Y(ii)
β-R0 [24] if for any β-open set U and any point y∈U,βCly⊆U


## 2.Λβ-Sets and Vβ-Sets

In the second part, we introduce the concept of aβ-Λ-set and a β-V-set in topological space which is denoted by Λβ,Vβ respectively, and we proceed with the study of its characterization.Definition 6.
LetB be a subset of a space Y which carries topology σ. We define the subsets ΛβB and VβB as follows:(a)
ΛβB=BΛβ=∩G:G⊇B,G∈βOY,σ(b)
VβB=BVβ=∪F:F⊆B,Fc∈βoY,σ
As an illustration, consider the following example.Example 1.
LetY=1,2,3,4 and σ=Y,ϕ,1,2,3,1,2,3; then, βOY,σ=Y,ϕ,1,2,3,1,2,1,3,1,4,2,3,2,4,3,4,1,2,3,1,2,4,2,3,4,1,3,4, and Λβ1=1,Λβ2,4=2,4,Λβ2=2,Λβ1,3,4=1,3,4, Vβ1=1,Vβ2,4=2,4,Vβ1,3=1,3, and Vβ1,2,3=1,2,3.
In the first result, we summarize the fundamental properties of the setβ-Λ-sets.Proposition 1.
LetB,C, and Cλ:λ∈Ω be subsets of a space Y which carries topology σ. Then, the following properties are valid:(a)
C⊆ΛβC(b)
IfB⊆C, then ΛβB⊆ΛβC(c)
ΛβΛβC=ΛβC(d)
IfB∈βOY,σ, then B=ΛβB (i.e., B is an Λβ-set)(e)
Λβ∪λ∈ΩCλ=∪λ∈ΩΛβCλ(f)
ΛβCc=VβCc(g)
Λβ∩λ∈ΩCλ⊆∩λ∈ΩΛβCλProof.
(a)
It is obvious.(b)
Suppose thaty∉ΛβC. Then, there exists a subset J∈βOY,σ such that J⊇C with y∉G since C⊇B; then y∉ΛβC, and thus, ΛβB⊆ΛβC.(c)
It follows from (a) and (b) thatΛβC⊆ΛβΛβC. If y∉ΛβC, then there exists J∈βoY,σ such that C⊇J, and y∉J. Hence, ΛβC⊆J, and so, we have y∉ΛβΛβC. Then, ΛβΛβC=ΛβC.(d)
By Definition6, and since B∈βOY,σ, we have ΛβB⊆B. By (a), we have that A=ΛβB.(e)
Suppose that there exists a pointy such that y∉Λβ∪λ∈ΩCλ. Then, there exists a β-open set J such that ∪λ∈ΩCλ⊆J and y∈J. Thus, for each λ∈Ω, we have y∉ΛβCλ. This implies that y∉∪λ∈ΩΛβCλ. Conversely, suppose that there exists a point y∈Y such that y∉∪λ∈ΩΛβCλ. Then, by Definition 6, there exists a subset Jλ∈βOY,σ (for all λ∈Ω such that y∉Jλ,Cλ⊆Jλ. Let J=∪λ∈ΩJλ. Then, we have that y∉∪λ∈ΩJλ,∪λ∈ΩCλ⊆J, and J∈βOY,σ. This implies that y∉Λβ∪λ∈ΩCλ. Then, the proof (e) is completed.(f)
VβCc=∩Fc:Cc⊆Fc,Fc∈βOY,σ=ΛβCc.(g)
Suppose that there exist a pointy∉∩λ∈ΩΛβCλ; then, there exists λ∈Ω such that y∉ΛβCλ; hence, there exist λ∈Ω and J∈βOY,σ such that J⊇Cλ and y∉J. Thus, y∉Λβ∪λ∈ΩCλ. By using Proposition 1(f), one can easily verify our result.Proposition 2.
For subsetsB,C, and Cλ:λ∈Ω of a topological space Y,σ, the following properties hold:(a)
VβC⊆C(b)
IfB⊆C, then VβB⊆VβC(c)
VβVβC=VβC(d)
IfC is β-closed in Y,σ, then C=VβC(e)
Vβ∩λ∈ΩCλ=∩λ∈ΩVβCλ(f)
Vβ∪λ∈ΩCλ⊇∪λ∈ΩCβCλProof.
(a), (b), and (c) are produced directly from Definition6 and Proposition 1. To prove (d), let C be β-closed in Y,σ; then, Cc∈βOY,σ. By (d) and (f) of Proposition 1, Cc=ΛβCc=VβCc. Hence, C=VβC.
(f) can be demonstrated by using (d) and Proposition1 (f). We have Vβ∪λ∈ΩCλ=Λβ∪λ∈ΩCλcc=Λβ∩λ∈ΩCλcc⊆∩λ∈ΩΛβCλcc=∩λ∈ΩVβCλcc=∪λ∈ΩVβCλ.Remark 2.
ΛβB1∩B2≠ΛβB1∩ΛβB2 is generally not true, as the following example shows.Example 2.
LetY=1,2,3 and σ=Y,ϕ,1. Let B1=2 and B2=3. Then, ΛβB1∩B2=ϕ, but ΛβB1∩ΛβB2=Y.Definition 7.
In a spaceY which carries topology σ, a subset C is called a Λβ-set (resp., Vβ-set) of Y,σ if C=ΛβC (resp., C=Vβ-set).Example 3.
LetY,σ be a topological space as in Example 1. Then, A=1 is Λβ-set and C=2,3 is Vβ-set.Remark 3.
By Proposition1 (d) and Proposition 2 (d), we have that(a)
IfC is a Λ-set or if C∈βOY,σ, then C is a Λβ-set(b)
IfC is a V-set or if C is β-closed set, then C is a Vβ-set
The converse of this remark is not true in general as shown in Example4.Example 4.
LetR,τ be a usual topology, and a singleton y is not β-open but Λβ-set.Proposition 3.
For a spaceY which carries topology σ, the following statements hold:(a)
ϕ and Y are Λβ-sets and Vβ-sets, where ϕ and Y are subsets of Y(b)
Union of any number ofΛβ-sets (resp., Vβ-sets) is Λβ-set (resp., Vβ-set)(c)
Intersection of any number ofΛβ-sets (resp., Vβ-sets) is Λβ-set (resp., Vβ-set)(d)
A subsetC is a Λβ-set if and only if Cc is a Vβ-setProof.
We will only take a case ofΛβ-sets.
(a) Obvious. To prove (b), letCλ:λ∈Ω be a family of Λβ-sets in Y,σ. If C=∪Cλ:λ∈Ω, then by Proposition 1C is Λβ-sets. To prove (c), let Cλ:λ∈Ω be a family of Λβ-set in Y,σ. Then, by Proposition 1 (g) and Definition 6, we have Λβ∪λ∈ΩBλ⊆∪λ∈ΩVβCλ=∩λ∈ΩCλ; hence, by Proposition 1 (a), ∩λ∈ΩCλ = ∩λ∈ΩCλ=Λβ∩λ∈ΩCλ.Remark 4.
LetσΛβ and σVβ be the family of all Λβ-sets and Vβ-sets from Y. Then, σΛβ and σVβ are a topology on Y containing all β-open and β-closed sets, respectively. Clearly, Y,σΛβ and Y,σVβ are Alexandroff Space [25], that is, arbitrary intersections of open sets are open.Proposition 4.
For a topological spaceY,σ, the following statements hold:(a)
σΛ⊆σΛβ(b)
σΛβ⊆σΛs,σΛβ⊆σΛP and σΛβ⊆σΛb(c)
σΛs=σβΛs and σΛs⊆σΛβΛsProof.
(a)
LetB be a subset of Y and B is Λ-set or B∈σΛ. Then, B=∩U:B⊆U,U∈σ. Since every open set is β-open, then B=∩U:B⊆U,Uisβ−open, so σΛ⊆σΛβ.(b)
SinceβOY,σ⊆SOY,σ [25], by similar way of (a), then σΛβ⊆σΛs. Also, since βOY,σ⊆POY,σ⊆BOY,σ, then σΛβ⊆σΛP⊆σΛb.(c)
SinceSOY,σ=SOY,σβ, then σΛs=σβΛs, and since σβ⊆σΛβ, so σΛs⊆σΛβΛs.Proposition 5.
IfY,ζ and Y,σ are two topological spaces such that βoY,ζ=βoY,σ, then σΛ⊆ζΛβ.Proof.
Since for a two spacesY,ζ and Y,σ, if βoY,ζ=βoY,σ, then σ⊆ζβ, and SoσΛ⊆ζΛβ.Remark 5.
We note that in the above proposition ifζΛβ≠σΛβ, it not necessarily leads to σΛ⊆ζΛβ, as the following example.Example 5.
LetY=1,2,3, ζ=Y,ϕ,1,1,2, and σ=Y,ϕ,1,1,3; then, ζΛβ≠σΛβ≠βoY and ζΛβ=Y,ϕ,1,1,2,1,3, but σΛ=Y,ϕ,1,1,3.Proposition 6.
IfB∈POY,σ(resp., B∈σΛP) and C∈βOY,σ (resp., C∈σΛβ, then B∩C∈σΛβB,σB).Proof.
IfB is a preopen set and C is a β-open set; then, B∩C∈βOB,σB. So B∩C∈σΛβB,σβ. If B∈σΛP, then B∈∩J:B⊆J,J∈POY,σ, and if C∈σβY,σ, then C=∩U:C⊆U,U∈βoY,σ. So B∩C=∩J∩U:B∩C⊆J∩U,J∈PoY,σ,U∈βoY,σ=∩S:S∈βOB,τ,B∩C⊂S. Then, B∩C∈σΛβB,σB.Proposition 7.
For a spaceY which carries topology σ, the following properties are equivalents:(a)
Y,σ is β-T1(b)
Every subsetB of Y is Λβ−set(c)
Every subsetB of Y is Vβ-setProof.
a⟹c: let B⊆Y. Since B=∪y:y∈B, B is a union of β-closed sets; hence, B is Vβ-setb⟹c: clearly given by Proposition 1c⟹a: Since by (c), we have that every singleton is the union of β-closed sets, that is, it is β-closed; then, Y,σ is a β-T1-space
Recall that a subsetB of a topological space Y,σ is said to be generalized closed (briefly g-closed) [3] if ClB⊂U whenever B⊂U and U∈σ.
A topological spaceY,σ is called T1/2 if every g-closed subset of Y is closed. Dunham and Maki [26, 27] pointed out that Y,σ is T1/2 if and only if for each y∈Y, the singleton y is open or closed.Theorem 1.
For a spaceY which carries topology σ, the following properties hold:(a)
Y,σΛβ and Y,σVβ are T1/2(b)
IfY,σ is β-T1, then both Y,σΛβ and Y,σVβ are discrete spacesProof.
(a)
Lety∈Y. Then, y is either preclosed or open and hence y is either β-closed or β-open. If y is β-open, y∈Λβ. If y is β-closed in Y,σ, then Y\y is β-open, and hence, Y\y∈Λβ. Therefore, y is closed in Y,σΛβ. Hence, Y,σΛβ and Y,σVβ are T1/2 spaces.(b)
This follows from Proposition7.
Relationship between some types ofΛ-sets is summarized in Figure 1.Figure 1
Relationship between some forms of nearΛ-sets.Proposition 8.
For a topological spaceZ,ζ, the following statements are equivalent:(a)
Z,ζ is a β-T1(b)
The singletonz is a Λβ-set, for each z∈Z(c)
The singletonz is a β-closed, for each z∈ZProof.
a⟹b: let z∈Z be an arbitrary point. For any point y in which y≠z, there exists J∈βoZ,ζ such that z∈J and y∉J. Therefore, we have y∉Λβz. This shows that Λβz⊆z. Since z⊆Λβz, we have Λβz=z.b⟹c: let z∈Z. For any y∈Z−z,y=Λβy, and hence, there exists Jy∈βOZ,ζ such that z∉Jy and y∈Jy. Therefore, we obtain y∈Jy⊂Z−z, and hence, Z−z=∪Jy:y∈Z−z which is β-open. Thus, z is β-closed.c⟹a: For any distinct points z and y of Z, the singletons z and y are β-closed which implies to Z,ζ which is β-T1.Proposition 9.
For topological spacesY,σ and Y,σΛβ, the following properties hold:(a)
A spaceY,σ is β-T1 if and only if Y,σΛβ is the discrete space(b)
The identity functionf:Y,σΛβ⟶Y,σ is strongly β-irresolute(c)
IfY,σΛβ is connected, then Y,σ is β-connectedProof.
(a)
Suppose thatY,σ is β-T1 and y be any point of Y. By Proposition 8, y is a Λβ-set and y∈σΛβ. For any subset B of Y, by Proposition 3, B∈σΛβ. This shows that Y,σΛβ is discrete. Conversely, for each y∈Y,y∈σΛβ, and hence, y is a Λβ-set. Proposition 8Y,σ is β-T1.(b)
LetJ be any β-open set of Y,σ. By Proposition 3, f−1J=J∈σΛβ, and hence, f is strongly β-irresolute.(c)
Suppose thatY,σ is not β-connected. There exist nonempty β-open sets J1,J2 of Y,σ such that J1∩J2=ϕ and J1∪J2=Y. Therefore, f−1J1 and f−1J2 are the disjoint nonempty open sets in Y,σΛβ and f−1J1∪f−1J2=Y. This shows that y,σΛβ is not connected.Theorem 2.
For a topological spaceY,σ, the following properties are equivalent:(a)
Y,σ is β-T1(b)
Y,σ is β-R0(c)
βcly=Λβy, for each y∈YProof.
(a)⟹ (b): for any β-open set G, and any point y∈J, by Proposition 8, βCly=y⊂J. Hence, Y,σ is β-R0.(b)⟹ (a): for each y∈Y, y is preopen or preclosed [21]. Let z∈Y such that y≠z. In case y is preopen, we have y∈βOY,σ and z∉y. In case y preclosed, since z∈Y-y∈βOY,σ, βClz⊂Y-y, and hence, y∈Y-βClz∈βOY,σ. Since z∉Y-βClz, then there exists G∈βoY,σ such that y∈G and z∉J; consequently, there exists H∈βOY,σ such that z∈H and y∉H. Hence,Y,σ is β-T1.(b)⟹ (c): let y∈Y and J∈βOY,σ containing y. Then, by (a), βcly⊂J and βcly⊂Λβy. Suppose that z∈Λβy. Then, z∈J for every β-open set J containing y, and hence, y∈βclz. Now, we show that y∈βclz implies z∈βcly. Assume that z∉βcly, then there exists H∈βOY,σ such that z∈H and y∉H. Since Y,σ is β-R0,βclz⊂H and y∉βclz. Therefore, we have Λβy⊂βcly. Consequently, we obtain βclx=Λβy.(c)⟹ (b): let J∈βOY,σ and y∈J. Then, we have Λβy⊂J; by (b), y∈βcly⊂Λβy⊂G, and hence, βcly⊂J. This shows that Y,σ is β-R0.


## 3. GeneralizedΛβ-Sets and Generalized Vβ-Sets

Section 3 is devoted to the study of generalizingΛβ-set (g⋅Λβ-sets) and Vβ-set (g⋅Λβ-sets) development of Maki’s work [2].Definition 8.
In a spaceY which carries topology σ, a subset B is called a generalized Λβ-set of Y,σg⋅Λβ if ΛβB⊆F whenever, B⊆F and F is β-closed. A subset B is g.Vβ-sets of Y,σ if Y-B is g.Λβ-set of Y,σ.
Now,EΛβ (resp., EVβ) will be denoted by the set of all g.Λβ-sets (resp., g.Vβ) in Y,σ.Proposition 10.
LetY,σ be a topological space and I be any index set. Then,(a)
EveryΛβ-set is a g⋅Λβ-set(b)
EveryVβ-set is a g⋅Vβ-set(c)
IfBi∈EΛβ for all i∈I, then ∪Bi∈EΛβ(d)
IfBi∈EVβ for all i∈I, then ∪Bi∈EVβProof.
(a) and (b) are proved by Definition6, Proposition 1, and Definition 8. (c) and (d) are proved by (e) of Proposition 1 and Definition 8.Remark 6.
The intersection of twog.Λβ-sets is generally not a g.Λβ-set. Also, the union of two g.Vβ-sets in generally not a g.Vβ-set as shown in the following example.Example 6.
LetY=1,2,3 with σ=Y,ϕ,1,2,1,2; then, σΛβ=Y,ϕ,1,2,3,1,2,1,3,2,3. If B=1,3 and C=2,3, then B and C are g.Λβ-sets but B∩C=3 is not g.Λβ-set. If B=1 and C=2, then B and C are g.Vβ-sets but B∪C=1,2 is not g.Vβ-set.Remark 7.
The converse of (a) and (b) of Proposition10 is not true in general as shown by the following example.Example 7.
LetY,σ be the space in Example 6; the subset B=1,3 is g.Λβ-set but it is not a Λβ-set. Also, the subset 1 is g.Vβ-set but is not Vβ-set.Proposition 11.
LetY,σ be a topological space. Then,(a)
Eachy∈Y is a β−open set or Y−y is a g.Λβ-set of X,τ(b)
Eachy∈Y is a β−open set or Y−y is a g.Vβ-set of Y,σProof.
(a)
Lety be not a β-open set; then, only β-closed set F containing Y−y is Y. Thus, Y−yΛβ⊆F=Y and Y−y is a g.Λβ-set of Y,σ.(b)
It follows from (a) and Definition8.
The known relationships between the types ofΛ-generalized closed sets are summarized in Figure 2.Figure 2
Relationship between some types ofΛ-generalized closed sets.Definition 9.
A functionξ:Y,σ⟶Z,η is said to be(a)
Stronglyβ-irresolute [19] if for each y∈Y and each β-open set J of Z containing ξy, there is an open set H containing y such that ξH⊆J, and equivalently, if the inverse image of each β-open set is open.(b)
β-Irresolute [10] if for each y∈Y, and each β-open set J of Z containing ξy, there is a β-open set H containing y such that ξH⊆J, and equivalently, if the inverse image of each β-open set is β-open.Theorem 3.
(a)
If a functionξ:Y,σ⟶Z,η is β-irresolute, then ξ:Y,σ∧β⟶Z,η∧β is continuous(b)
The identity functionidy:Y,σ⟶Z,η is strongly β-irresoluteProof.
(a)
LetJ be any Λβ-set of Z,η, i.e., J∈σΛβ. Then, J=ΛβJ=∩W:J⊆W and W is β-open in Z,η. Since ξ is β-irresolute, ξ−1W is β-open in Y,σ for each W. Hence, we have ξ−1Q⊇∩ξ−1W:ξ−1J⊆ξ−1W, and W is β-open in Z,η⊇∩H:ξ−1J⊆H and H is β-open in Y,σ=Λβξ−1J. On the other hand, by the definition, ξ−1J⊆Λβξ−1J. Hence, we obtain ξ−1J=Λβξ−1J. Therefore, ξ−1J∈τΛβ, and ξ:Y,σ∧β⟶Z,η∧β is continuous.(b)
LetJ be any β-open set of Y,σ. Since J is β-open, by Proposition 11, idy−1J=J∈σΛβ and hence, idx is strongly β-irresolute.


## 4. Application

The digital line or so-called Khalimsky lineZ,k is the set of all integers Z, equipped with the topology k generated by 2n−1,2n,2n+1n∈Z (cf.[10], [28, p. 175]), [29, p. 905, 908]. The digital line or an interval as its subspace is a topological model of a 1-dimensional computer screen and points are the pixels. In Z,k, each singleton 2n is closed and each singleton 2n+1 is open (=regular open), where n∈Z.The digital n-space (e.g., [19], Definition 4) is the topological product of n-copies of digital line Z,k and this space is denoted by Zn,kn. For n=2,Z2,k2 is called the digital plane. In Z2,k2, singletons whose coordinates are odd and are open; those with coordinates even are closed. There are points which are neither open nor closed.Theorem 4.
For the digitaln-space Zn,kn where n≥1, we have the following properties:(1)
For any pointx of Zn,kn, β-Λx=x(2)
For any subsetE of Zn,kn, β-ΛE=EProof.
(i) We first note that for the case wheren=1, Zn=Znkn∪Znfn (disjoint union) holds, where n=1, and for the case where n≥2, Zn=Znkn∪Znfn∪∪Znmixr1≤r≤n−1 (disjoint union) and Znmixr≠ϕ1≤r≤n−1 hold, where n≥2. Let x∈Zn. It is enough to consider the following three cases for the point x∈Zn.Case 1.
x∈Znkn: since x is open in Zn,kn, it is β-open. Then, it is obvious that β−Λx=x in Zn,kn. We note this result is true for the case where n≥1.Case 2.
x∈Znfn: we put x=2s1,2s2,…,2sn where si∈Z1≤i≤n. Note that, for the point x∈Znfn,Unx=∏i=1n2si−1,2si,2si+1 is the smallest open set containing x. Then, there exist 2nβ-open sets x∪px,u1≤u≤2n containing the point x∈Znfn such that px,u1≤u≤2n=Unxkn=2s1+i1,2s2+i2,…,2sn+inik∈+1,−11≤k≤n and the family of Unxkn=2n. Thus, we have β-Λx⊂∩x∪px,u1≤u≤2n; moreover, ∩x∪px,u1≤u≤2n=x, because ∩px,u1≤u≤2n=ϕ. We conclude that β-Λx=x holds for this case. We note the result above it is true for the case where n≥1.Case 3.
x∈Znmixr, where 1≤r≤n−1n≥2: for the point x, we set x=x1,x2,…,xn; then by definition, r=ixiis an even integer 1≤i≤n). We recall the following subsets Irx and Jn−rx as follows: Irx=kxkis even=e1,e2,…,ere1<e2<⋯<er; Jn−rx=jxjis odd=o1,o2,…,on−ro1<o2<⋯<on−r; and 1,2,…,n=Irx∪Jn−rx (disjoint union), Irx≠ϕ,Jn−rx≠ϕ.
For the pointx∈Znmixr,Uux=∏i=1nUxi is the smallest open set containing x, where Uxek=xek−1,xek+11≤k≤r and Uxoj=xoj1≤j≤n−r. Then, there exist 2rβ-open sets x∪px,u1≤u≤2r containing the point x∈Znmixr such that px,u1≤u≤2r=Unxkn=z1,z2,…,znzek∈xek+1,xek−11≤kr,zoj1≤j≤n−r and the family of Unxkn=2r. Thus, it is shown that β-Λx⊂∩x∪px,u1≤u≤2r=x∪∩px,u1≤u≤2r=x, because ∩px,u1≤u≤2r=ϕ. Then, we show that β-Λx=x holds for this case. Therefore, for all cases above, we have proved that β-Λx=x holds in Zn,Kn,n≥1.
(2) SinceE=∪xx∈E, by Proposition 1, it is shown that -β−ΛE=∪β−Λxx∈E=∪xx∈E=E.


## 5. Conclusion

Recently, the use of set and functions for topological spaces has significantly evolved in data analysis, information systems, digital topology, and others. By researching generalizations of closed sets,Λβ-sets, and Vβ-sets, some new separation axioms have been found, and they turn out to be useful in the study of digital topology. The notion of a kernel of a set has applications in computer science [10, 30]. This notion is used in most of this paper. Moreover, it also has applications in some crucial fields of science and technology.



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*Source: 4237462-2020-05-28.xml*