# Theta Omega Topological Operators and Some Product Theorems

Authors: Samer Al Ghour; Salma El-Issa
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We introduce and investigate the concepts ofθω-limit points and θω-interior points, and we use them to introduce two new topological operators. For a subset B of a topological space Y,σ, denote the set of all limit points of B (resp. θ-limit points of B, θω-limit points of B, interior points of B, θ-interior points of B, and θω-interior points of B) by DB (resp. DθB, DθωB, IntB, IntθB, and IntθωB). Several results regarding the two new topological operators are given. In particular, we show that DθωB lies strictly between DB and DθB and IntθωB lies strictly between IntθB and IntB. We show that DB=DθωB (resp. ClθB=ClθωB and DB=DθωB=DθB) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.


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## Body

## 1. Introduction

In 1943, Fomin [1] introduced the notion of θ-continuity. For the purpose of studying the important class of H-closed spaces in terms of arbitrary filterbases, the notions of θ-open subsets, θ-closed subsets, and θ-closure were introduced by Velicko [2] in 1966, in which he showed that the family of θ-open sets in a topological space Y,σ forms a topology on Y denoted by σθ (see also [3]). The work of Velicko is continued by [3–26] and others. Hdeib [27] introduced the class of ω-closed sets by which he introduced and investigated the notion of ω-continuity. The family of all ω-open sets in Y,σ is denoted by σω. It is known that σω is a topology on Y which is finer than σ. Research related to ω-open sets is still a hot area of research [28–36]. In 2017, Al Ghour and Irshidat [37] introduced θω-open subsets, θω-closed subsets, and θω-closure utilizing the topological spaces Y,σθ and Y,σω. It is proved in [37] that σθω forms a topology on Y which lies between σθ and σ, and that σθω=σ if and only if Y,σ is ω-regular. Also, in [37], ω−T2 topological spaces were characterized via θω-open sets. Authors in [35] introduced θω-connectedness and some new separation axioms. Also, research in [37] was continued by various researchers in [28–31]. The notion of interior operators is important in the axiomatization of modal logics. Judging from the importance of limit points in mathematical analysis, introducing a new limit point notion in any topological structure is still a hot area of research. The first goal of this paper is to introduce and investigate the concepts of θω-limit points and θω-interior points.In general topology, several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelöfness, and metacompactness. The area of research regarding the problem “What conditions onY,σ and Z,δ to insure that their product has property P”is still hot[38–45]. The second goal of this paper is to introduce several product theorems concerning metacompactness.


## 2. Preliminaries

From now on TS will denote topological space for simplicity. LetY,σ and Z,δ be TSs and let B⊆C⊆Y with C as nonempty. Then, B is called ω-open set in Y,σ [27] if for each y∈B, there is M∈σ and a countable set F⊆Y such that y∈M−F⊆B. The relative topology on C is denoted by σC, and the product topology on Y×Z is denoted by σ×δ. The closure of B in Y,σ (resp. C,σC, Y,σω) is denoted by B¯ (resp. B¯C, B¯ω). A point y∈Y is in θ-closure of B [2] (y∈ClθB) if for every G∈σ with y∈G, G¯∩B≠∅. B is called θ-closed [2] if ClθB=B. The complement of a θ-closed set is called a θ-open set. It is known that σθ=σ if and only if Y,σ is regular. A TS Y,σ is called ω-regular [37] if for each closed set C in Y,σ and y∈Y−C, there exist G∈σ and H∈σω such that y∈G, C⊆H, and G∩H=∅. In [37], the author defined θω-closure operator as follows: a point y∈Y is in θω-closure of By∈ClθωB if for any G∈σ with y∈G we have G¯ω∩B≠∅. G is called θω-closed if ClθωG=G. The complement of a θω-closed set is called a θω-open set. A TS Y,σ is called metacompact [46] if every open cover of Y,σ has a point-finite open refinement.The following sequence of definitions and theorems will be used in the sequel.Definition 1 (see [47]).
A TSY,σ is called locally countable if for each y∈Y, there is G∈σ such that G is countable and y∈G.Definition 2.
(see [48]). A TS Y,σ is called antilocally countable if each G∈σ−∅ is uncountable.Definition 3.
(see [9]). Let Y,σ be a TS B⊆Y. A point y∈Y is called θ-limit point of B if for each G∈σθ with y∈G, G∩B−y≠∅. The set of all θ-limit points of B is called the θ-derived set of B and is denoted by DθB.Definition 4.
(see [9]). Let Y,σ be a TS and B⊆Y. A point y∈Y is called a θ-interior point of B if there exists G∈σ such that y∈G⊆G¯⊆B. The set of all θ-interior points of B is called the θ-interior of B and is denoted by IntθB.Theorem 1 (see [37]).
IfY,σ is locally countable and B⊆Y, then B¯=ClθωB.Theorem 2 (see [37]).
IfY,σ is antilocally countable and B⊆Y, then ClθB=ClθωB.Theorem 3 (see [37]).
For any TSY,σ, σθ⊆σθω⊆σ.Theorem 4 (see [2]).
A TSY,σ is regular if and only if σ=σθ.Theorem 5 (see [37]).
LetY,σ be a TS and B⊆Y. Then, B is θω-open set if and only if for each y∈B, there exists G∈σ such that.Definition 5.
LetY,σ and Z,δ be TSs and let B⊆Y. Then,(a)
Y,σ is called C-scattered if every B∈σc−∅, there is b∈B and a compact set K such that b∈IntK⊆K⊆B [49](b)
B is called strongly placed in Y×Z if for every z∈Z and H∈σ×δ with B×z⊆H, there are V∈σ and W∈δ such that B×z⊆V×W⊆H [50](c)
Y is called scattered relative to Y×Z if for each B∈σc, there exists b∈B and V∈σA such that V¯ is Lindelöf and strongly placed in Y×Z [50]
It is well known that ifY,σ and Z,δ are TSs and Y is C-scattered, then Y is scattered relative to Y×Z but not conversely.Definition 6.
(see [51]). A Hausdorff TS Y,σ is called ultraparacompact if every open cover of Y has a locally finite clopen refinement.
Ellis [51] showed that a Hausdorff space Y,σ is ultraparacompact if every open cover has a pairwise disjoint open refinement.Theorem 6 (see [52]).
Letf:Y,σ⟶Z,δ be closed and continuous with Y,σ regular. If Z,δ is metacompact and f−1z is Lindelöf for each z∈Z, then Y,σ is metacompact.Theorem 7 (see [50]).
For any two TSsY,σ and Z,δ, Y is strongly placed in Y×Z if and only if the projection π:Y×Z,σ×δ⟶Z,δ is closed.Theorem 8 (see [35]).
LetY,σ and Z,δ be TSs and let B⊆Y. If B is strongly placed in Y×Z and C∈σ∩σc, then B∩C is strongly placed in Y×Z.


## 3. Theta Omega Limit Points

In this section, we explore the concept ofθω-limit points of a set and study its fundamental properties.Definition 7.
LetY,σ be a TS and B⊆Y. A point y∈Y is called θω-limit point of B if for each G∈σθω with y∈G, G∩B−y≠∅.
The set of allθω-limit points of B is called the θω-derived set of B and is denoted by DθωB.
The following result shows thatθω-derived set of a set B contains the derived set of B and contained in the θω-derived set of B.Theorem 9.
LetY,σ be a TS B⊆Y. The derived set of B is denoted by DB. Then, DB⊆DθωB⊆DθB.Proof.
To see thatDB⊆DθωB, let y∉DθωB, then there exists G∈σθω such that y∈G and G∩B−y=∅. By Theorem 3, G∈σ and so y∉DB. Therefore, we have DB⊆DθωB. To see that DθωB⊆DθB, let y∉DθB, then there exists G∈σθ such that y∈G and G∩B−y=∅. By Theorem 3, G∈σθω and so y∉DθωB. Therefore, we have DθωB⊆DθB.
The following example shows that the equality of each of the inclusions in Theorem9 does not hold in general.Example 1.
(Example 2.26 of [37]). Let X=ℝ and let σ=∅,ℝ,ℕ,ℚc,ℕ∪ℚc. It is proved in [37] that σθω=∅,ℝ,ℕ and σθ=∅,ℝ. Let B=−n:n∈ℕ∪0,1. Then, DθB=ℝ, DθωB=ℝ−1, and DB=ℚ−1.
Under the condition “regularity,” theθω-derived set, the derived set, and the θ-derived set are all equal.Theorem 10.
LetY,σ be a regular TS and B⊆Y. Then, DB=DθωB=DθB.Proof.
It follows from Theorems3, 4, and 9.
“Local countability” is a sufficient condition for theθω-derived set and the derived set to be equal to each other:Theorem 11.
LetY,σ be a locally countable TS and B⊆Y. Then, DB=DθωB.Proof.
By Theorem9, we have DB=DθωB. To see that DB=DθωB, suppose to the contrary that there is y∈DθωB−DB. Since x∉DB, there is G∈σ such that G∩B−y=∅. By Theorem 1, ClθωY−G=Y−G¯=Y−G and so G∈σθω. We conclude that y∉DθωB, a contradiction.
“Antilocal countability” is a sufficient condition for theθω-derived set and the θ-derived set to be equal to each other.Theorem 12.
LetY,σ be an antilocally countable TS and B⊆Y. Then, DθB=DθωB.Proof.
By Theorem9, we have DθωB⊆DθB. To see that DθB⊆DθωB, suppose to the contrary that there is y∈DθB−DθωB. Since x∉DθωB, there is G∈σθω such that G∩B−y=∅. By Theorem 2, ClθY−G=ClθωY−G=Y−G and so G∈σθ. We conclude that y∉DθB, a contradiction.
In Theorems13–16, we give some natural properties for θω-derived set.Theorem 13.
LetY,σ be a TS. If B⊆C⊆Y, then DθωB⊆DθωC.Proof.
Lety∉DθωC, there exists G∈σθω such that y∈G and G∩C−y=∅. Since B⊆C, then G∩B−x=∅ and hence y∉DθωB. It follows that DθωB⊆DθωC.Theorem 14.
LetY,σ be a TS, and let AandB be subsets of Y. Then, DθωA∪DθωB=DθωA∪B.Proof.
By Theorem13, DθωB⊆DθωA∪B and DθωB⊆DθωA∪B. Therefore, DθωA∪DθωB⊆DθωA∪B. Now, let y∉DθωA∪DθωB, then there exist θω-open sets G,H∈σθω such that y∈G∩H, G−y∩A=∅, and H−y∩B=∅. Let W=G∩H. Then, W∈σθω and(1)W−y∩A∪B=W−y∩A∪W−y∩B=∅∪∅=∅.
Thus,y∉DθωA∪B.Theorem 15.
LetY,σ be a TS, and let AandB be subsets of Y. Then, DθωA∩B⊆DθωA∩DθωB.Proof.
By Theorem13, DθωA∩B⊆DθωB and DθωA∩B⊆DθωB. Then, DθωA∩B⊆DθωA∩DθωB.
The following example shows that the inclusion in Theorem15 can not be replaced by equality in general.Example 2.
(Example 2.26 of [37]). Let Y=ℝ and σ=∅,ℝ,ℕ,ℚc,ℕ∪ℚc. It is proved in [37] that σθω=ℝ,∅,ℕ. Let A=3/4,3/2 and B=5/4,9/4. Then, DθωA=ℝ−1 and DθωB=ℝ−2, so DθωA∩DθωB=ℝ−1,2. On the other hand, DθωA∩B=Dθω5/4,3/2=ℝ−ℕ.Theorem 16.
LetY,σ be a TS and B⊆Y. Then, DθωDθωB−B⊆DθωB.Proof.
Lety∈DθωDθωB−B. Let G∈σθω with y∈G. Since y∈DθωDθωB, G∩DθωB−y≠∅. Choose z∈G∩DθωB−y. Since z∈DθωB and z∈G∈σθω, then G∩B−z≠∅. Choose w∈G∩B−z. Since w∈B and y∉B, then w≠y. Thus, G∩B−y≠∅ and hence y∈DθωB.
The following example shows that the inclusion in Theorem16 cannot be replaced by equality in general.Example 3.
LetY=ℝ and σ=ℝ,∅,ℕ,ℚc,ℕ∪ℚc. Let B=3/4,3/2. It is proved in [37] that σθω=ℝ,∅,ℕ. By Example 2, DθωB=ℝ−1. On the other hand,(2)DθωDθωB−B=Dθωℝ−1−34,32=ℝ−34,32.


## 4. Theta Omega Interior Points

In this section, we explore the concept ofθω-interior points of a set and study its fundamental properties.Definition 8.
LetY,σ be a TS and B⊆Y. A point y∈Y is called a θω-interior point of B if there exists G∈σ such that y∈G⊆G¯ω⊆B. The set of all θω-interior points of B is called the θω-interior of B and is denoted by IntθωB.
The following result shows that theθω-interior of a set B contains the θ-interior B and contained in the θω-interior of B.Theorem 17.
LetY,σ be a TS and B⊆Y. Then, IntθB⊆IntθωB⊆IntB.
The following example shows that each of the two inclusions in Theorem17 cannot be replaced by equality in general.Example 4.
(Example 2.26 of [37]). Let Y=ℝ and let σ=∅,ℝ,ℕ,ℚc,ℕ∪ℚc. Let A=ℕ and B=ℚc. It is proved in [37] that σθω=∅,ℝ,ℕ and σθ=∅,ℝ. We have IntθωA=ℕ but IntθA=∅. Also, we have IntθωB=∅ but IntB=ℚc.Theorem 18.
LetY,σ be a TS and B⊆Y. If G∈σ such that G⊆G¯ω⊆B, then G⊆IntθωB.Proof.
It follows directly from the definition ofIntθωB.
θω-interior is always open.Theorem 19.
LetY,σ be a TS and B⊆Y. Then, IntθωB is θω-open.Proof.
By the definition ofIntθωB and Theorem 18, for every y∈IntθωB, there exists Gy∈σ such that y∈Gy⊆Gy¯ω⊆IntθωB. By Theorem 5, it follows that IntθωB is θω-open.
The following is a characterization ofθω-open via θω-interior.Theorem 20.
LetY,σ be a TS and B⊆Y. Then, B is θω-open if and only if B=IntθωB.Proof.
Necessity: suppose thatB is a θω-open set. By the definition, we have IntθωB⊆B. To see that B⊆IntθωB, let y∈B. By Theorem 5, there exists G∈σ such that y∈G⊆G¯ω⊆B. Then, y∈IntθωB.
Sufficiency: suppose thatB=IntθωB. Then, by Theorem 19, B is θω-open.
The results in the rest of this section are some natural properties ofθω-interior.Theorem 21.
LetY,σ be a TS and B⊆Y. Then, IntθωIntθωB=IntθωB.Proof.
Follows from Theorem20.Theorem 22.
LetY,σ be a TS and B⊆Y. Then, Y−IntθωB=ClθωY−B.Proof.
To see thatY−IntθωB⊆ClθωY−B, let y∉ClθωY−B. Then, there is G∈σ such that y∈G and G¯ω∩Y−B=∅, so we have y∈G⊆G¯ω⊆B. This shows that y∉Y−IntθωB. To see that ClθωY−B⊆Y−IntθωB, let y∉Y−IntθωB. Then, y∈IntθωB, and so there is G∈σ such that y∈G⊆G¯ω⊆B. Therefore, we have G¯ω∩Y−B=∅, and so y∉ClθωY−B.Theorem 23.
LetY,σ be a TS and B⊆Y. Then, Y−ClθωB=IntθωY−B.Proof.
By Theorem22,(3)Y−ClθωB=Y−Y−IntθωY−B=IntθωY−B.Theorem 24.
LetY,σ be a TS and let A⊆B⊆Y. Then, IntθωA⊆IntθωB.Proof.
Lety∈IntθωA. Then, there exists G∈σ such that y∈G⊆G¯ω⊆A. Since A⊆B, then G⊆U¯ω⊆B. Thus, y∈IntθωB.Theorem 25.
LetY,σ be a TS and let A and B be subsets of Y. Then, IntθωA∪IntθωB⊆IntθωA∪B.Proof.
By Theorem24, we have IntθωA⊆IntθωA∪B and IntθωB⊆IntθωA∪B. Thus, IntθωA∪IntθωB⊆IntθωA∪B.Theorem 26.
LetY,σ be a TS, and let AandB be subsets of Y. Then, IntθωA∩B=IntθωA∩IntθωB.Proof.
By Theorem24, we have IntθωA∩B⊆IntθωA and IntθωA∩B⊆IntθωB. Thus, IntθωA∩B⊆IntθωA∩IntθωB. To see that IntθωA∩IntθωB⊆IntθωA∩B, let y∈IntθωA∩IntθωB. Then, there exist G,H∈σ such that y∈G⊆G¯ω⊆A and y∈G⊆H¯ω⊆B. Let W=G∩H. Then, W∈σ and y∈W⊆W¯ω=G∩H¯ω⊆G¯ω∩H¯ω⊆A∩B. It follows that y∈IntθωA∩B.


## 5. Metacompactness Product Theorems

In this section, we introduce several product theorems concerning metacompactness.The following result will be used in the proof of Theorems28 and 29.Theorem 27.
LetY,σ and Z,δ be metacompact TSs. If for every y∈Y there exists W∈σ such that y∈W and W¯×Z is metacompact, then Y×Z,σ×δ is metacompact.Proof.
LetA be an open cover of Y×Z,σ×δ. For every y∈Y, choose Wy∈σ such that y∈Wy and Wy¯×Z,σ×δWy¯×Z is metacompact. Since Wy:y∈Y is an open cover of the metacompact TS Y,σ, then it has a point-finite open refinement Vβ:β∈Γ. For each β∈Γ, Vβ¯×Z,σ×δVβ¯×Z is metacompact and has ℳβ=A∩Vβ¯×Z:A∈A as an open cover, and hence ℳβ has a point-finite open refinement ℋβ. It is not difficult to see that H∩Vβ×Z:H∈ℋβ,β∈Γ is a point-finite open refinement of A. It follows that Y×Z,σ×δ is metacompact.
The following two product theorems concerning metacompactness will be used in the proof of Theorem31 which is the main result of this section:Theorem 28.
LetY,σ and Z,δ be regular metacompact TSs. If for every y∈Y there exists W∈σ such that y∈W, W¯ is strongly placed in Y×Z, and W¯,σW¯ is Lindelöf, then Y×Z,σ×δ is metacompact.Proof.
For eachy∈Y, choose Wy∈σ such that y∈W, W¯ is strongly placed in Y×Z, and Wy¯,σWy¯ is Lindelöf. For every y∈Y, Wy¯ is strongly placed in Y×Z and so by Theorem 6, the projection function πy:Wy¯×Z,σ×δWy¯×Z⟶Z,δ is a closed function. For every y∈Y, Wy¯,σWy¯ is Lindelöf and since πy−1z=Wy¯×z, then Wy¯×z,σ×δWy¯×z is Lindelöf. For every y∈Y, Wy¯,σWy¯ is metacompact and so by Theorem 6, Wy¯×Z,σ×δWy¯×Z is metacompact. Thus, by Theorem 27, we have Y×Z,σ×δ is metacompact.Theorem 29.
LetY,σ and Z,δ be metacompact TSs and let B⊆Y such that B is closed in Y,σ, B,σB is Lindelöf, and B is strongly placed in Y×Z. If for all y∈Y−B there is M∈σY−B such that M¯Y−B×Z,σ×δM¯Y−B×Z is metacompact, then Y×Z,σ×δ is metacompact.Proof.
LetA be an open cover of Y×Z,σ×δ. For each z∈Z, M×z,σ×δM×z is Lindelöf with M×z⊂∪A, and so there exists Az⊆A such that A is countable and M×z⊆∪Az. Since M is strongly places in Y×Z, then for every z∈Z, there exist Uz∈σ and Vz∈δ such that M×z⊆Uz×Vz⊆∪Az. Since Vz:z∈Z is an open cover of the metacompact TS Z,δ, then it has a point-finite open refinement Gβ:β∈Γ. For each β∈Γ, choose zβ such that Gβ⊆Vzβ. Then, by Theorem 27 and the assumption, it is not difficult to see that Y−Uzβ×Z,σ×δY−Uzβ×Z is metacompact. Since A∩Y−Uzβ×Z:A∈A is an open cover of Y−Uzβ×Z, then it has a point-finite open refinement Aβ. It is not difficult to check that(4)A∩Uzβ×Gβ:A∈Azβ,β∈Γ∪G∩Y×Gβ:G∈Aβ,β∈Γ,is a point-finite open refinement of A. Therefore, Y×Z,σ×δ is metacompact.Theorem 30.
LetY,σ be ultraparacompact and Z,δ be metacompact. Suppose there exists D⊆Y such that D is closed in Y,σ and for every y∈D there exists W∈σD such that y∈W, W¯ is strongly placed in Y×Z, and W¯,σW¯ is Lindelöf, and for every y∈Y−D, there is K∈σY−D such that y∈K and K¯Y−D×Y is metacompact. Then, Y×Z,σ×δ is metacompact.Proof.
By assumption there existsA⊆σD such that for all A∈A, A¯ is strongly placed in Y×Z and A¯,σA¯ is Lindelöf, and ∪A=D. Since D,σD is ultraparacompact, then A has a pairwise disjoint open refinement Cβ:β∈Γ⊆σD. For every β∈Γ, choose Uβ∈σ such that Cβ=Uβ∩D. Put ℋ=Uβ:β∈Γ∪Y−D. Since Y,σ is ultraparacompact and ℋ is an open cover of Y,σ, then ℋ has a pairwise disjoint open refinement Mγ:γ∈Δ. For every γ∈Δ, Mγ meets at most one member of Cβ:β∈Γ. For every γ∈Δ, let Mγ∗=Mγ∩∪Cβ:β∈ΓandCβ∩Mγ≠∅, then Mγ∗=∅ or Mγ∗=Mγ∩A¯; for someA∈A, it follows thatMγ∗is closed inMγ,σMγandMγ∗,σMγ∗is Lindelöf and by Theorem 8; it is strongly placed inMγ×Y. By the assumption on Y−D and Theorem 29, we conclude that Mγ×Z,σ×δMγ×Y is metacompact. Since Y×Z=∪γ∈ΔMγ×Z is metacompact, then Y×Z,σ×δ is metacompact.
Now, we are ready to state the main result of this section.Theorem 31.
LetY,σ be ultraparacompact and Z,δ be regular and metacompact such that Y is scattered relative to the product Y×Z, then Y×Z,σ×δ is metacompact.Proof.
Denote byY0=Y and Y1 = {y∈Y: there is no U∈σ such that y∈U and U¯ is strongly placed in Y×Z and closure U¯,σU¯ is Lindelöf }. If there is an ordinal α>1 such that Yα has been defined and β=α+1, then Yβ=Yα1. If α is a limit ordinal, then Yα=∩β<αYβ. Since Y is scattered relative to Y×Z, then there exists an ordinal α such that Yα=∅.
The proof proceeds by transfinite induction onα. If Y1=∅, then for every y∈Y there exists Uy∈σ such that y∈U and U¯ is strongly placed in Y×Z and closure U¯,σU¯. And by Theorem 28, Y×Z,σ×δ is metacompact. If Yα+1=∅, then for every point y∈Yα there exists Uy∈σYα such that y∈Uy, Uy¯ is strongly placed in Y×Z, and Uy¯,σUy is Lindelöf and if y∈Y−Yα, choose a clopen set Cy such that y∈Cy⊆Y−Yα. Since Cyα⊆Y−Yαα=∅, then Cy is scattered relative to Cy×Z and Cy,σCy is ultraparacompact, and by the inductive assumption, it follows that Cy×Z,σ×δCy×Z is metacompact.
IfYα=∅ for the limit ordinal α, then the open cover Y−Yβ:β<α has a pairwise disjoint open refinement Oγ:γ∈Γ. For each γ∈Γ, choose β<α such that Oγ⊆Y−Yβ. Therefore, Oγβ=∅, and hence Oγ×Z,σ×δOγ×Z is metacompact. Since Y×Z=∪γ∈ΓOγ×Y, it follows that Y×Z,σ×δ is metacompact.Corollary 1.
The product of an ultraparacompactC-scattered TS with a metacompact regular TS is again metacompact.
By the end of this paper, the authors found it is suitable to raise the following open question.Question 1.
LetY,σ and Z,δ be regular and metacompact TSs such Y is scattered relative to the product Y×Z. Is Y×Z,σ×δ metacompact?


## 6. Conclusion

In this work, the research viaθω-open sets is continued by introducing the notions of θω-limit points and θω-interior points. Several relationships regarding these two notions are introduced. Moreover, several product theorems concerning metacompactness are given. In future studies, the following topics could be considered: (1) define θω-border, θω-frontier, and θω-exterior of a set using θω-open sets and (2) try to solve Question 1.



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