ThetaOmega Topological Operators and Some Product Theorems

We introduce and investigate the concepts of θω-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 D(B) (resp. Dθ(B), Dθω(B), Int(B), Intθ(B), and Intθω(B)). Several results regarding the two new topological operators are given. In particular, we show that Dθω(B) lies strictly between D(B) and Dθ(B) and Intθω(B) lies strictly between Intθ(B) and Int(B). We show that D(B) � Dθω(B) (resp. Clθ(B) � Clθω(B) and D(B) � Dθω(B) � Dθ(B)) 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.


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 σ θ (see also [3]). e work of Velicko is continued by  and others. Hdeib [27] introduced the class of ω-closed sets by which he introduced and investigated the notion of ω-continuity. e family of all ω-open sets in (Y, σ) is denoted by σ ω . It is known that σ ω is a topology on Y which is finer than σ. Research related to ω-open sets is still a hot area of research [28][29][30][31][32][33][34][35][36]. In 2017, Al Ghour and Irshidat [37] introduced θ ω -open subsets, θ ω -closed subsets, and θ ω -closure utilizing the topological spaces (Y, σ θ ) and (Y, σ ω ). It is proved in [37] that σ θ ω forms a topology on Y which lies between σ θ and σ, and that σ θ ω � σ if and only if (Y, σ) is ω-regular. Also, in [37], ω − T 2 topological spaces were characterized via θ ω -open sets. Authors in [35] introduced θ ω -connectedness and some new separation axioms. Also, research in [37] was continued by various researchers in [28][29][30][31]. e 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. e first goal of this paper is to introduce and investigate the concepts of θ ω -limit points and θ ω -interior points.
In general topology, several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelöfness, and metacompactness. e area of research regarding the problem "What conditions on (Y, σ) and (Z, δ) to insure that their product has property P"is still hot [38][39][40][41][42][43][44][45]. e second goal of this paper is to introduce several product theorems concerning metacompactness.

Preliminaries
From now on TS will denote topological space for simplicity. Let (Y, σ) and (Z, δ) be TSs and let B⊆C⊆Y with C as nonempty. en, 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. e relative topology on C is denoted by σ C , and the product topology on Y × Z is denoted by called θ-closed [2] if Cl θ (B) � B.
e complement of a θ-closed set is called a θ-open set. It is known that σ θ � 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 ∈ σ ω such that y ∈ G, C⊆H, and G ∩ H � ∅. In [37], the author defined θ ω -closure operator as follows: [46] if every open cover of (Y, σ) has a point-finite open refinement.
e following sequence of definitions and theorems will be used in the sequel.
Definition 1 (see [47]). A TS (Y, 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] e set of all θ-limit points of B is called the θ-derived set of B and is denoted by D θ (B).
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. e set of all θ-interior points of B is called the θ-interior of B and is denoted by Int θ (B).
Theorem 4 (see [2] It is well known that if (Y, 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.

Theta Omega Limit Points
In this section, we explore the concept of θ ω -limit points of a set and study its fundamental properties.
e set of all θ ω -limit points of B is called the θ ω -derived set of B and is denoted by D θ ω (B).
e following result shows that θ ω -derived set of a set B contains the derived set of B and contained in the θ ω -derived set of B.
e following example shows that the equality of each of the inclusions in eorem 9 does not hold in general. □ Example 1 (Example 2.26 of [37]). Let X � R and let Under the condition "regularity," the θ ω -derived set, the derived set, and the θ-derived set are all equal.
"Local countability" is a sufficient condition for the θ ω -derived set and the derived set to be equal to each other: □ Theorem 11. Let (Y, σ) be a locally countable TS and B⊆Y.
Proof. By eorem 9, we have "Antilocal countability" is a sufficient condition for the θ ω -derived set and the θ-derived set to be equal to each other.

be a TS, and let A and B be subsets of
Proof.
en, W ∈ σ θ ω and (1) □ Theorem 15. Let (Y, σ) be a TS, and let A and B be subsets of Proof.
en, en, e following example shows that the inclusion in eorem 16 cannot be replaced by equality in general. □ On the other hand, (2)

Theta Omega Interior Points
In this section, we explore the concept of θ ω -interior points of a set and study its fundamental properties.  Proof. By the definition of Int θ ω (B) and eorem 18, for every y ∈ Int θ ω (B), there exists G y ∈ σ such that y ∈ G y ⊆G y ω ⊆Int θ ω (B). By eorem 5, it follows that e following is a characterization of θ ω -open via θ ω -interior.

Theorem 20. Let (Y, σ) be a TS and B⊆Y. en, B is θ ω -open if and only if B � Int θ ω (B).
Proof. Necessity: suppose that B is a θ ω -open set. By the definition, we have Int θ ω (B)⊆B. To see that B⊆Int θ ω (B), let y ∈ B. By eorem 5, there exists G ∈ σ such that y ∈ G⊆G ω ⊆B. en, y ∈ Int θ ω (B).
Sufficiency: suppose that B � Int θ ω (B). en, by eo- e results in the rest of this section are some natural properties of θ ω -interior.

Metacompactness Product Theorems
In this section, we introduce several product theorems concerning metacompactness.
e following result will be used in the proof of eorems 28 and 29.

Proof. Let
A be an open cover of (Y × Z, σ × δ). For every y ∈ Y, choose W y ∈ σ such that y ∈ W y and (W y × Z, (σ × δ) W y ×Z ) is metacompact. Since W y : y ∈ Y is an open cover of the metacompact TS (Y, σ), then it has a point-finite open refinement V β : β ∈ Γ . For each β ∈ Γ, e following two product theorems concerning metacompactness will be used in the proof of eorem 31 which is the main result of this section: □ Theorem 28. Let (Y, 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, Proof. For each y ∈ Y, choose W y ∈ σ such that y ∈ W, W is strongly placed in Y × Z, and (W y , σ W y ) is Lindelöf. For every y ∈ Y, W y is strongly placed in Y × Z and so by eorem 6, the projection function π y : (W y × Z, (σ ×δ) W y ×Z ) ⟶ (Z, δ) is a closed function. For every y ∈ Y, is Lindelöf. For every y ∈ Y, (W y , σ W y ) is metacompact and so by eorem 6, (W y × Z, (σ × δ) W y ×Z ) is metacompact. us, by eorem 27, we have (Y × Z, σ × δ) is metacompact. □ Theorem 29. Let (Y, σ) and (Z, δ) be metacompact TSs and let B⊆Y such that B is closed in (Y, σ), (B, σ B ) is Lindelöf, and is an open cover of the metacompact TS (Z, δ), then it has a point-finite open refinement G β : β ∈ Γ . For each β ∈ Γ, choose z(β) such that G β ⊆V z(β) . en, by eorem 27 and the assumption, it is not difficult to see that is a point-finite open refinement of A. erefore, (Y × Z, σ × δ) is metacompact. □ Theorem 30. Let (Y, 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 ∈ σ D such that y ∈ W, W is strongly placed in Y × Z, and (W, σ W ) is Lindelöf, and for every y Proof. By assumption there exists A⊆σ D such that for all Now, we are ready to state the main result of this section. □ Theorem 31. Let (Y, 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 by Y (0) � Y and Y (1) � {y ∈ Y: there is no U ∈ σ such that y ∈ U and U is strongly placed in Y × Z and closure (U, σ U ) is Lindelöf }. If there is an ordinal α > 1 such that Y (α) has been defined and β � α + 1, then Y (β) � (Y (α) ) (1) . If α is a limit ordinal, then Y (α) � ∩ β<α Y (β) . Since Y is scattered relative to Y × Z, then there exists an ordinal α such that Y (α) � ∅. e proof proceeds by transfinite induction on α. If Y (1) � ∅, then for every y ∈ Y there exists U y ∈ σ such that y ∈ U and U is strongly placed in Y × Z and closure (U, σ U ). And by eorem 28, (Y × Z, σ × δ) is metacompact. If Y (α+1) � ∅, then for every point y ∈ Y (α) there exists U y ∈ σ Y (α) such that y ∈ U y , U y is strongly placed in Y × Z, and (U y , σ U y ) is Lindelöf and if y ∈ Y − Y (α) , choose a clopen set C y such that y ∈ C y ⊆Y − Y (α) . Since C (α) y ⊆(Y − Y (α) ) (α) � ∅, then C y is scattered relative to C y × Z and (C y , σ C y ) is ultraparacompact, and by the inductive assumption, it follows that (C y × Z, (σ × δ) C y ×Z ) is metacompact.
If  e product of an ultraparacompact C-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. Let (Y, 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?

Conclusion
In this work, the research via θ ω -open sets is continued by introducing the notions of θ ω -limit points and θ ω -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 θ ω -border, θ ω -frontier, and θ ω -exterior of a set using θ ω -open sets and (2) try to solve Question 1.

Data Availability
No data were used to support this study.
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