EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3370-3385 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Generalization for Somewhere Dense Sets with Some Applications Amani Rawshdeh1,∗, Heyam H. Al-Jarrah2, Khalid Y. Al-Zoubi2 1 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Alsalt, Jordan 2 Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan Abstract. In this paper, we present a new generalization for somewhere dense of a topological space (Z, τ), namely ωSWD-open subsets. We introduce the concept of this family and discuss some of their properties with the help of illustrative example. Moreover, we will show if the space (Z, τ) is anti-locally countable and τ is finer than the cocountable topology then the class of ωSWD-open and somewhere dense subsets of (Z, τ) will be equivalent. Moreover, we present more properties for the class of somewhere dense subsets of (Z, τ), the most important of which is a generalization for a theorem in [1]. Furthermore, we finish this work by shedding light on one type of covering properties where we study the notion of almost ωSWD-compact spaces with some of their properties. 2020 Mathematics Subject Classifications: 54A05, 54A10, 54C10, 54D20 Key Words and Phrases: SWD-open subsets, ω−open subsets, ωSWD-open subsets, ωSWD- compact spaces 1. Introduction In recent decades, a major area of study for general topology researchers has been the study of various kinds of generalized open sets. Mathematicians examine various topological notions, such as continuity, compactness, etc. In 1937, stone [18] introduced the concept of regular open sets. In 1963, Levine [13] presented the notion of semi-open sets. In 1965, Njasted [16] introduced α-open sets. In 1982, Mashhour et al [15] introduced the concepts of pre-open and studied their topological properties. In 1983, Abd El-Monsef et al [9] studied the notion of β-open sets. In 1996, Andrijevic [6] defined and explored the idea of b-open sets. A subset H of a space (Z, τ) is called a regular open (semi-open, α-open, pre-open, β-open , b-open) sets if H = Int(Cl(H))(resp., H ⊆ Cl(Int(H)), H ⊆ Int(Cl(Int(H))), H ⊆ Int(Cl(H)), H ⊆ Cl(Int(Cl(H))), H ⊆ Int(Cl(H))∪Cl(Int(H))). ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5486 Email addresses: amanirawshdeh@bau.edu.jo (A. Rawshdeh), heyam@yu.edu.jo (H. H. Al-Jarrah), khalidz@yu.edu.jo (K. Y. Al-Zoubi) https://www.ejpam.com 3370 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3371 Another form of generalization of open sets that we need in this work is ω−open sets. A subset H of a space (Z, τ) is called an ω−closed [11] if it contains all its condensation points, where a point x ∈ Z is called a condensation point of H [10] if for each G ∈ τ with x ∈ G, the set G ∩H is uncountable. The complement of an ω−closed is called ω−open. Moreover, in [5] the authors introduced an equivalent definition of ω−open subsets, where H ⊆ Z is an ω−open subsets of (Z, τ) if for each x ∈ H there is G ∈ τ with x ∈ G such that G−H is countable. The study and exploitation of these generalizations have become very widespread and many works have been presented based on these sets, for example in [12] the authors presented some applications of pre-open sets, where they introduced and studied topological properties of pre-limit points, pre-interior and pre-closure and other topological notions [ see [8],[14]]. In 2017, Al-Shami [1] examined and studied some main properties of somewhere dense sets on topological spaces where a subset H ⊆ Z is a somewhere dense set of (Z, τ) if there is a non-empty open set G with G ⊆ Cl(H) which is equivalent to say Int(Cl(H)) is a non-empty set. Moreover, he showed that, with the expectation of the empty set, all semi-open, α-open sets, pre-open, β-open, and b-open sets are contained in the class of somewhere dense sets. Then, Al-Shami and Noiri [3] used the class of somewhere dense sets to define the concept of SWD-continuous and SWD-homeomorphism functions. Then in [4], they introduced and investigated the notions of almost SWD-compact, almost SWD- lindelöf spaces, nearly SWD-compact, nearly SWD-lindelöf, mildly SWD-compact and mildly SWD-lindelöf spaces and they studied the relationships between them. Moreover, in [2] the author contributed to this area and used the notion of somewhere dense sets to improve the approximations and accuracy measure in rough set theory. In this work, we study and present more properties of somewhere dense of (Z, τ). One of the most important of these properties is a generalization for a theorem that was introduced in [1]. Then, based on the class of all somewhere dense and ω−open subsets of (Z, τ), we introduce and study the class of ωSWD-open subsets, which is a new generalization for somewhere dense of a topological space (Z, τ) and hence it is a new kind of generalized open sets. We organize this work as follows: In Sec. 2, we give more properties of somewhere dense sets of (Z, τ). In Sec. 3, we introduce the notion of ωSWD-open subsets and we verify some of basic properties of this class with the help of illustrative examples. Also, we investigate what are the conditions to become the class of ωSWD-open and somewhere dense subsets of (Z, τ) are equivalent. Then we show that this family is not a topology through an example that shows this family is not closed under finite intersection. Moreover, we use ωSWD-open subsets to generalize the notions of interior and closure and define ωSWD-continuous and ωSWD-irresolute. In Sec.4, we use ωSWD-open subsets and the closure operator which are discussed in Sec.3 to study one type of covering properties, namely almost ωSWD-compact spaces and study some of its properties. Throughout this work the family of all somewhere dense sets of (Z, τ)(resp., the family of all closed somewhere dense which is equivalent to the family of the complement of all somewhere dense of (Z, τ)) is denoted by SWD(Z, τ) (resp.,SWDC(Z, τ)). Moreover, SWD interior of H (is denoted by, IntSWD(H)) is given by IntSWD(H) = ∪{G : G ⊆ H A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3372 and G ∈ SWD(Z, τ) } and the SWD closure of H (is denoted by, ClSWD(H)) is given by ClSWD(H) = ∩{F : H ⊆ F and F ∈ SWDC(Z, τ) }. Also, the family of all ω−open subsets of a space (Z, τ) forms a topology on Z finer than τ and denoted by τω[5]. The ω−interior( resp., ω−closure) of a subset H of a space (Z, τ) is the interior (resp., closure) of H in the space (Z, τω) and it is denoted by Intω(H)(resp., Clω(H)). In this paper, we will write T S instead of topological space. The sets R and Q, respectively the set of real numbers and rational numbers. The cofinite topology, the cocountable topology, the indiscrete topology, and the usual topology are denoted by τcof , τcoc, τind and τu respectively. Also, if H is a subset of a space (Z, τ), then the relative topology on H in (Z, τ) will be denoted by τH . Definition 1. [10] A filter on Z is a family F ⊆ P(Z) which satisfies the following: (i) ϕ /∈ F . (ii) If H,G ∈ F , then H ∩G ∈ F . (iii) If H ∈ F and H ⊆ G ⊆ Z, then G ∈ F . Moreover, A filter F on Z is said to be a maximal filter on Z if each filter H on Z that contains F we have F = H. Also, a family F ⊆ P(Z) is said to be a filter base on Z if it is a non-empty such that ϕ /∈ F and if H,G ∈ F then there is V ∈ F with V ⊆ H ∩G. Definition 2. Let (Z, τ) be a T S. Then (Z, τ) is said to be: (i) Hyperconnected [17] if no mutually disjoint non-empty open sets. (ii) Strongly hyperconnected [1] if a subset of Z is dense iff it is non-empty and open. Theorem 1. [1] Let (Z, τ) be a T S and H,G ⊆ Z. If H ∈ SWD(Z, τ) and (Z, τ) is: (i) hyperconnected, then H ∩G ∈ SWD(Z, τ) whenever G ∈ τ . (ii) strongly hyperconnected, then H ∩G ∈ SWD(Z, τ) whenever G ∈ SWD(Z, τ). Definition 3. [10] Let {(Zα, τα) : α ∈ ∆} be a family of topological spaces with Zα∩ Zβ = ϕ for each α ̸= β. Let Z = ∪ α∈∆ Zα with the topology τs = {G ⊆ Z : G ∩ Zα ∈ τα for each α ∈ ∆}. Then (Z, τs) is called the sum of the spaces {(Zα, τα) : α ∈ ∆}and denoted by Z = ⊕ α∈∆ Zα. Theorem 2. [1] Let ( n Π α=1 Zα, τ) be a finite product T S. Then Hα ∈ SWD(Zα, τα) for each α = 1, 2, ..., n, iff n Π α=1 Hα ∈ SWD( n Π α=1 Zα, τ). Theorem 3. Let (Z, τ) be a T S and H,G ⊆ Z. Then: (i) If H ⊆ G and H ∈ SWD(Z, τ), then G ∈ SWD(Z, τ) [1]. (ii) If E ∈ τ and H ⊆ E, then H ∈ SWD(Z, τ) whenever H ∈ SWD(E, τE) [7]. Definition 4. [5] Let (Z, τ) be a T S. Then (Z, τ) is said to be anti-locally countable if each non-empty open subset of (Z, τ) is uncountable. Note that, if (Z, τ) is an anti locally countable space, then (Z, τω) is also anti-locally countable. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3373 Definition 5. [4] Let (Z, τ) be a T S. Then: (i) A family H = {Hα : α ∈ ∆} is said to be SWD(Z, τ)-cover of Z if Z = ∪ α∈∆ Hα with Hα ∈ SWD(Z, τ). (ii) (Z, τ) is said to be almost SWD-compact if for each SWD(Z, τ)-cover H = {Hα : α ∈ ∆} of Z there is a finite subset ∆◦ ⊆ ∆ with Z = ∪ α∈∆◦ ClSWD(Hα). 2. More Properties of Somewhere Dense sets In this section, we examine further properties of somewhere dense of a topological space (Z, τ). Proposition 1. Let (Z, τ) and (K, σ) be two T Ss and Γ : (Z, τ)→ (K, σ) be a continuous, open and surjective function. If H ⊆ Z and H ∈ SWD(Z, τ), then Γ(H) ∈ SWD(K, σ). Proof. Since H ∈ SWD(Z, τ), then there is G ∈ τ with ϕ ̸= G ⊆ Cl(H). Therefore, ϕ ̸= Γ(Int(Cl(G)) ⊆ Int(Γ(Cl(H)) ⊆ Int(Cl((Γ(H)) and hence Γ(H) ∈ SWD(K, σ). Theorem 4. Let {(Zα, τα) : α ∈ ∆} be a family of topological spaces with Zα ∩ Zβ = ϕ for each α ̸= β. For each α ∈ ∆, let ϕ ̸= Hα ⊆ Zα and put H = ∪ α∈∆ Hα. Then: (i) H ∈ SWD(Z, τs) iff there is α◦ ∈ ∆ with Hα◦ ∈ SWD(Zα◦ , τα◦). (ii) If Hα ∈ SWD(Zα, τα) for each α ∈ ∆, then H ∈ SWD(Z, τs). Proof. (i) First, note that for all α ∈ ∆, Clα(Hα) = Cl(Hα) where Clα(Hα) is the closure of Hα in Zα while Cl(Hα) is the closure of Hα in Z. Now, choose α◦ ∈ ∆ with Hα◦ ∈ SWD(Zα◦ , τα◦). Then there is Gα◦ ∈ τα◦ with ϕ ̸= Gα◦ ⊆ Clα◦(Hα◦) = Cl(Hα◦) ⊆ Cl(H) and hence H ∈ SWD(Z, τs). Conversely, since H ∈ SWD(Z, τs) and the family {Hα : α ∈ ∆} is locally finite in (Z, τs), then there is G ∈ τs with ϕ ̸= G ⊆ Cl(H) = Cl( ∪ α∈∆ Hα) = ∪ α∈∆ Cl(Hα) = ∪ α∈∆ Clα(Hα). Since ϕ ̸= G, choose xα◦ ∈ G for some α◦ ∈ ∆. Then G ∩ Zα◦ is a non-empty set in (Zα◦ , τα◦) such that G ∩ Zα◦ ⊆ ∪ α∈∆ Clα(Hα)∩ Zα◦ = Clα◦(Hα◦). Therefore, Hα◦ ∈ SWD(Zα◦ , τα◦). (ii) Follows from part (i). The following theorem is one of the most important results that we present in this section. Since Theorem 5 ( Part ii) is a generalization of Theorem 2. Theorem 5. Let Z = Π α∈∆ Zα be the product space of the spaces (Zα, τα), α ∈ ∆ with the Tychonoff topology τp. Let Hα ⊆ Zα for each α ∈ ∆. Then the following are equivalent: (i) Hα ∈ SWD(Zα, τα) for each α ∈ ∆. (ii) For each finite subset ∆∗ ⊆ ∆, the set H = Π α∈∆∗ Hα × Π β∈∆−∆∗ Zβ ∈ SWD(Z, τp). (iii) For each α ∈ ∆, the set H = Hα × Π β∈∆ β ̸=α Zβ ∈ SWD(Z, τp). Proof. The implication (ii −→ iii) is obvious. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3374 (i −→ ii) For each α ∈ ∆∗, there is Gα ∈ τα with ϕ ̸= Gα ⊆ Zα and Gα ⊆ Cl(Hα). Then G = Π α∈∆∗ Gα × Π β∈∆−∆∗ Zβ is a non-empty open set in (Z, τp) such that G ⊆ Π α∈∆∗ Cl(Hα) × Π β∈∆−∆∗ Zβ = Cl(Π( α∈∆∗ Hα) × Π β∈∆−∆∗ Zβ) = Cl(H). Therefore, H ∈ SWD(Z, τp). (iii → i) Since for each α ∈ ∆, the projection function πα : (Z, τp) → (Zα, τα) is continuous, open and surjective such that πα(Hα × Π β∈∆ β ̸=α Zβ) = Hα, then by Proposition 1, Hα ∈ SWD(Zα, τα) for each α ∈ ∆. Theorem 6. Let Z = Π α∈∆ Zα be the product space of the spaces (Zα, τα), α ∈ ∆ with the Tychonoff topology τp. Let Hα ⊆ Zα for each α ∈ ∆. Then : (i) If Hα ∈ SWD(Zα, τα) for each α ∈ ∆, then ∪ α∈∆ ( Hα ×Π β∈∆ β ̸=α Zβ) ∈ SWD(Z, τp). (ii) If Π α∈∆ Hα ∈ SWD(Z, τp), then Hα ∈ SWD(Zα, τα) for each α ∈ ∆. Proof. (i) Follows from Theorem 5 (Part iii) and Theorem 3 (Part i). (ii) Follows from Proposition 1 ( see the proof of the implication (iii→ i) in Theorem 5). The following example shows that the converses of Theorem 6 need not be true in general: Example 1. (i) Let Z = R and consider the spaces (Z1, τ1) = (R, τu) and (Z2, τ2) = (R, τind). Then the set ({2} × Z2) ∪ (Z1 × {1}) is somewhere dense of (R, τu) × (R, τind) while {2} /∈ SWD((R, τu). (ii) For each α ∈ ∆ with ∆ is an infinite set, consider (Zα, τα) = (Kα, τind) where Kα any set with |Kα| > 1. For each α ∈ ∆, choose xα ∈ Kα, then for each α ∈ ∆, Hα = {xα} ∈ SWD(Kα, τind) while Π α∈∆ Hα /∈ SWD(Z, τp). Theorem 7. Let Z = Π α∈∆ Zα be the Cartesian product of the spaces (Zα, τα) with the topology τb which is generated by the base { Π α∈∆ Vα : Vα ∈ τα for each α ∈ ∆}(τb is called the box topology). Then Hα ∈ SWD(Zα, τα) for each α ∈ ∆ iff Π α∈∆ Hα ∈ SWD(Z, τb). Proof. For each α ∈ ∆, there is Gα ∈ τα with ϕ ̸= Gα ⊆ Zα and Gα ⊆ Cl(Hα). Then G = Π α∈∆ Gα is a non-empty open set of Z such that G ⊆ Π α∈∆ Cl(Hα) = Cl(Π( α∈∆ Hα)). Conversely, let α◦ ∈ ∆. Then Hα◦ ×Π α∈∆ α ̸=α◦ Hα ∈ SWD(Z, τb) and hence there is G ∈ τb such that ϕ ̸= G ⊆ Cl(Hα◦ ×Π α∈∆ α ̸=α◦ Hα) and so there is a basic open set V = Π α∈∆ Vα with Vα◦ × Π α∈∆ α ̸=α◦ Vα ⊆ Cl(Hα◦ ×Π α∈∆ α ̸=α◦ Hα). Therefore, Vα◦ ⊆ Cl(Hα◦) and thus Hα◦ ∈ SWD(Zα◦ , τα◦). A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3375 Corollary 1. (i) Let Z = Π α∈∆ Zα be the product space of the spaces (Zα, τα), α ∈ ∆ with the topology τp and Fα ⊆ Zα for each α ∈ ∆. If ∪ α∈∆ ( Fα ×Π β∈∆ β ̸=α Zβ) ∈ SWDC(Z, τp), then Fα ∈ SWDC(Zα, τα). (ii) Let Z = Π α∈∆ Zα be the product space of the spaces (Zα, τα), α ∈ ∆ with the topology τb and Fα ⊆ Zα. Then Fα ∈ SWDC(Zα, τα) iff ∪ α∈∆ ( Fα ×Π β∈∆ β ̸=α Zβ) ∈ SWDC(Z, τb). (iii) Let Z = n Π α=1 Zα be the finite product space of the spaces (Zα, τα) and Fα ⊆ Zα for each α ∈ {1, 2.., n}. Then for each α ∈ {1, 2.., n}, Fα ∈ SWDC(Zα, τα) iff n ∪ α=1 ( Fα ×Π β∈{1,2...,n} β ̸=α Zβ) is closed somewhere dense of Z. Note that, in Example 1 (Part (ii)), Fα = {xα} ∈ SWDC(Zα, τα) while ∪ α∈∆ ( Fα ×Π β∈∆ β ̸=α Zβ) /∈ SWDC(Z, τp), since Z − ∪ α∈∆ ( Fα ×Π β∈∆ β ̸=α Zβ) = Π α∈∆ (Zα − Fα) /∈ SWD(Z, τ) and so the converse of Corollary 1 (Part i) is not true in general. 3. ω−Somewhere Dense Open Sets With Some Applications In this section we introduce the notion of ωSWD-open subsets, denoted by ωSWD(Z, τ), as a new generalization for somewhere dense of a topological space (Z, τ). Then we discuss the sufficient conditions for the equivalence between the classes SWD(Z, τ), ωSWD(Z, τ) and ωSWD(Z, τω). Also, we study some applications by using ωSWD(Z, τ). Definition 6. Let (Z, τ) be a T S with H ⊆ Z. A point x ∈ Z is an ωSWD−condensation point of H if for each S ∈ SWD(Z, τ) with x ∈ S, the set S ∩ H is uncountable. If H contains all its ωSWD−condensation points, then H is said to be ωSWD-closed subset of (Z, τ) and its complement is an ωSWD-open subset of (Z, τ). The collection of all ωSWD-closed (resp., ωSWD-open) subsets of (Z, τ) will be denoted by ωSWDC(Z, τ) (resp., ωSWD(Z, τ)). The proofs of the following results are straightforward and thus are omitted. Proposition 2. Let (Z, τ) be a T S with H ⊆ Z. Then H ∈ ωSWD(Z, τ) iff for each x ∈ H there is S ∈ SWD(Z, τ) with x ∈ S and S −H is countable. Corollary 2. Let (Z, τ) be a T S with H ⊆ Z. Then H ∈ ωSWD(Z, τ) iff for each x ∈ H there is S ∈ SWD(Z, τ) and a countable set G with x ∈ S −G ⊆ H. Theorem 8. Let (Z, τ) be a T S.Then ωSWD(Z, τω) ⊆ ωSWD(Z, τ). Proof. Let H ∈ ωSWD(Z, τω) and x ∈ H. Then there is S ∈ SWD(Z, τω) with x ∈ S and C = S − H is countable. Then there is G ∈ τω with ϕ ̸= G ⊆ Clω(S). A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3376 Choose t ∈ G. Then, there is V ∈ τ with t ∈ V and C1 = V − G is countable. Thus, V ⊆ G ∪ C1 ⊆ Clω(S) ∪ Clω(C1) = Clω(S ∪ C1) ⊆ Cl(S ∪ C1). Therefore, S ∪ C1 ∈ SWD(Z, τ) with x ∈ S ∪ C1 and (S ∪ C1) − H = (S − H) ∪ (C1 − H) is countable and hence H ∈ ωSWD(Z, τ). By using Definition 6 and Theorem 8 we generate the following diagram where none of these implications being reversible. τ → τω ↙ ↘ SWD(Z, τ) SWD(Z, τω) ↓ ↓ ωSWD(Z, τ) ←− ωSWD(Z, τω) Example 2. (i) Consider Z = {1, 2, 3} with τ = {ϕ,Z, {1, 2}}. Then {3} ∈ τω − {ϕ} ⊆ SWD(Z, τω) while {3} /∈ SWD(Z, τ). (ii) Consider (R, τu) with H = Q. Then H ∈ SWD(R, τu) while H /∈ SWD(R, (τu)ω) since IntωClω(H) = Intω(H) = Int(H) = ϕ. (iii) Consider (R, τcof ) with H = {1}. Then H ∈ ωSWD(R, τcof ) while H /∈ SWD(R, τcof ). (iv) Consider Z = R with τ = {R} ∪ {G ⊆ R : G ⊆ R − Q} and H = Q. Then H ∈ ωSWD(R, τω) while H /∈ SWD(R, τω).To show that, H ∈ ωSWD(R, τω), let x ∈ H. Choose r ∈ R − Q. Then G = {r} ∈ τ such that ϕ ̸= G ⊆ H ∪{r} ⊆ Clω(H ∪{r}). So H ∪{r} ∈ SWD(R, τω) with x ∈ H ∪{r} and (H ∪{r}) − H is countable. Therefore, H ∈ ωSWD(R, τω). (v) Consider (R, τind) with H = {1}. Since ωSWD(R, τind) = P(R), then {1} ∈ ωSWD(R, τind) while {1} /∈ ωSWD(R, (τind)ω). Since if there is S ∈ SWD(R, (τind)ω) with 1 ∈ S and S−{1} is countable, then S is countable and hence IntωClω(S) = Intω(S) = ϕ. Therefore, {1} /∈ ωSWD(R, (τind)ω). Theorem 9. Let (Z, τ) be a T S. If (Z, τ) is an anti-locally countable and τ is finer than the cocountable topology, then the following families are equivalent: (i) SWD(Z, τ). (ii) ωSWD(Z, τ). (iii) ωSWD(Z, τω). Proof. (i→ ii) Trivial. (ii → iii) Let H ∈ ωSWD(Z, τ) and x ∈ H. Then there is S ∈ SWD(Z, τ) with x ∈ S and C = S − H is countable. Hence S ⊆ H ∪ C. Now, choose G ∈ τ with ϕ ̸= G ⊆ Cl(S) ⊆ Cl(H ∪C) ⊆ Cl(H)∪Cl(C) = Cl(H)∪C (since τcoc ⊆ τ). Since (Z, τ) is an anti-locally countable and ϕ ̸= G ∈ τ , then G−C ̸= ϕ. Now, we claim G−C ⊆ Clω(S). Suppose not, then there is t ∈ G − C and t /∈ Clω(S) and so there is V ∈ τω with t ∈ V and V ∩ S = ϕ. Now, choose, O ∈ τ with t ∈ O and C1 = O − V is countable. Then we have t ∈ O−C1 ∈ τ with O−C1 ⊆ V . Finally, put W = (G−C)∩ (O−C1). Then W ∈ τ A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3377 with t ∈W and hence W ∩S ̸= ϕ. On the other hand, W ∩S ⊆ (O−C1)∩S ⊆ V ∩S = ϕ, which is a contradiction. Therefore, S ∈ SWD(Z, τω) and hence H ∈ ωSWD(Z, τω). (iii) → (i) Let H ∈ ωSWD(Z, τω) and x ∈ H. Then there is S ∈ SWD(Z, τω) with x ∈ S and C = S −H is countable. Choose G ∈ τω with t ∈ G ⊆ Clω(S) ⊆ Clω(H ∪ C) ⊆ Clω(H) ∪ Clω(C) = Clω(H) ∪ C. Now, choose O ∈ τ with t ∈ O and O − G = C1 is countable. Since, O − C1 ⊆ G ⊆ Clω(H) ∪ C, then ϕ ̸= (O − C1)− C ⊆ Clω(H) ⊆ Cl(H). Therefore, H ∈ SWD(Z, τ). Corollary 3. Let (Z, τ) be a T S. If (Z, τ) is an anti-locally countable, then: (i) SWD(Z, τω) = ωSWD(Z, τω). (ii) SWD(Z, τ) = SWD(Z, τω) whenever τcoc ⊆ τ . Proof. (i) Since (Z, τω) is anti locally countable and τcoc ⊆ τω, then by Theorem 9, SWD(Z, τω) = ωSWD(Z, τω). (ii) From part(i) and Theorem 9. Note that from Example 2 (Part iii) imposing the condition of anti locally countable on (Z, τ) alone in Theorem 9 is not enough and hence we looked for another condition on (Z, τ). Theorem 10. Let (Z, τ) be a T S. Then ∪ α∈∆ Hα ∈ ωSWD(Z, τ) whenever Hα ⊆ Z and Hα ∈ ωSWD(Z, τ) for each α ∈ ∆. Proof. If ∪ α∈∆ Hα = ϕ, then ∪ α∈∆ Hα ∈ ωSWD(Z, τ). Now, let x ∈ ∪ α∈∆ Hα. Then there is α(x) ∈ ∆ such that x ∈ Hα(x) and hence there is S ∈ SWD(Z, τ) with x ∈ S and S − Hα(x) is countable. Since S − ∪ α∈∆ Hα ⊆ S − Hα(x), then S − ∪ α∈∆ Hα is countable. Therefore, ∪ α∈∆ Hα ∈ ωSWD(Z, τ). Corollary 4. Let (Z, τ) be a T S.Then ∩ α∈∆ Hα ∈ ωSWDC(Z, τ) whenever Hα ⊆ Z and Hα ∈ ωSWDC(Z, τ) for each α ∈ ∆. The next example shows that the intersection of two ωSWD-open subsets of (Z, τ) is not an ωSWD-open in general and hence we can conclude for any topology τ on Z, ωSWD(Z, τ) may not be a topology on Z. Example 3. Consider the space (R, τcoc). Take H = [0, 1] and G = [1, 2]. Then H,G ∈ SWD(R, τcoc) ⊆ωSWD(R, τcoc), while H ∩ G = {1} /∈ ωSWD(R, τcoc), since there is no S ∈ SWD(R, τcoc) with 1 ∈ S and S − {1} is countable. Proposition 3. Let (Z, τ) be a T S. Then the family ωSWD(Z, τ) is a topology on Z if one of the following hold: (i) (Z, τ) is strongly hyperconnected. (ii) Z is countable or τ is the indiscrete topology. Proof. Straightforward. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3378 Theorem 11. Let (Z, τ) be a hyperconnected T S and H,G ⊆ Z. If H ∈ τω and G ∈ ωSWD(Z, τ), then H ∩G ∈ ωSWD(Z, τ). Proof. Let x ∈ H ∩ G. Then there are V ∈ τ with x ∈ V and S ∈ SWD(Z, τ) with x ∈ S such that V −H and S−G are countable sets. By Theorem1, V ∩S ∈ SWD(Z, τ) with x ∈ V ∩S and since (V ∩S)− (H ∩G) ⊆ (V −H)∪ (S−G), then (V ∩S)− (H ∩G) is countable. Therefore, H ∩G ∈ ωSWD(Z, τ). Corollary 5. Let (Z, τ) be hyperconnected T S and H,G ⊆ Z. If Z − H ∈ τω and G ∈ ωSWDC(Z, τ), then H ∪G ∈ ωSWDC(Z, τ). From Example 3 we note that in Theorem 11 is not enough to be (Z, τ) is hyper- connected and hence we looked for another condition on the sets. Also, note that this example (Example 3), Q ∩ (R−Q) ∈ ωSWD(R, τcoc) and (R−Q) ∈ ωSWD(R, τcoc) but Q /∈ τω and hence the converse of Theorem 11 is not true in general. Theorem 12. Let (Z, τ) be a T S and H,E ⊆ Z. If E ∈ τ and H ⊆ E, then: (i) If H ∈ ωSWD(E, τE), then H ∈ ωSWD(Z, τ). (ii) If H ∈ ωSWD(Z, τ), then H ∈ ωSWD(E, τE) provided that E is dense. Proof. (i) Obvious by using Theorem 3 (Part (ii)). (ii) Let x ∈ H. Then there is S ∈ SWD(Z, τ) with x ∈ S and S − H is countable. Since E is an open dense subset, then for some G ∈ τ we can have ϕ ̸= G∩E ⊆ Cl(S)∩E ⊆ Cl(S∩E) and hence G∩E ⊆ Cl(S∩E)∩E = ClE(S∩E)(The closure of S∩E in (E, τE)). Therefore, S ∩ E ∈ SWD(E, τE) and hence H ∈ ωSWD(E, τE). Remark 1. Let (Z, τ) and (K, σ) be two T Ss. Then: (i) If ωSWD(Z, τ) ⊆ωSWD(K, σ), then it is not true in general τ ⊆ σ. (ii) If τ ⊆ σ, then it is not true in general ωSWD(Z, τ) ⊆ ωSWD(K, σ). The following example illustrates Remark 1. Example 4. (i) Note that, ωSWD(R, τcoc) ⊆ωSWD(R, τind) while τcoc ⊈ τind. (ii) Note that, τind ⊆ τcoc while ωSWD(R, τind) ⊈ ωSWD(R, τcoc). Theorem 13. Let (Z, τ) be a T S. Then: (i) τ = Int(ωSWD(Z, τ)) = {Int(H) : H ∈ ωSWD(Z, τ)}. (ii) τω = Intω(ωSWD(Z, τ)) = {Intω(H) : H ∈ ωSWD(Z, τ)}. Proof. LetG ∈ τ . ThenG ∈ ωSWD(Z, τ) and hence Int(G) = G ∈ Int(ωSWD(Z, τ)). Conversely, is obvious since {Int(H) : H ∈ ωSWD(Z, τ)} ⊆ τ . (ii)The proof is similar technique in part (i). Definition 7. Let (Z, τ) be a T S and H ⊆ Z. Then: (i) IntωSWD(H) = ∪{G : G ⊆ H and G ∈ ωSWD(Z, τ) }. (ii) ClωSWD(H) = ∩{F : H ⊆ F and F ∈ ωSWDC(Z, τ) }. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3379 Theorem 14. Let (Z, τ) be a T S and H,G ⊆ Z. Then: (i) IntSWD(H) ⊆ IntωSWD(H) and ClωSWD(H) ⊆ ClSWD(H). (ii) H ∈ ωSWD(Z, τ) iff H = IntωSWD(H). (iii) IntωSWD(IntωSWD(H)) = IntωSWD(H). (iv) H ∈ ωSWDC(Z, τ) iff H = ClωSWD(H). (v) x ∈ ClωSWD(H) iff for each G ∈ ωSWD(Z, τ) with x ∈ G we have G ∩H ̸= ϕ. (vi) IntωSWD(Z −H) = Z − ClωSWD(H). (vii) ClωSWD(Z −H) = Z−IntωSWD(H). Proof. Straightforward. In general, in Theorem 14 the reverse inclusion of (Part i) does not hold, since in Example 2 (Part iii), IntSWD({1}) = ϕ while IntωSWD({1}) ={1}. Also, if Z = {1, 2, 3} with the topology σ = {ϕ,Z, {1}}. Then ClSWD({1}) = Z while ClωSWD({1}) = {1}. Definition 8. Let (Z, τ) and (K, σ) be two T Ss. Then a function Γ : (Z, τ)→ (K, σ) is said to be: (i) An ωSWD-continuous iff Γ−1(G) ∈ ωSWD(Z, τ) for each G ∈ σ. (ii) An ωSWD-irresolute iff Γ−1(G) ∈ ωSWD(Z, τ) for each G ∈ ωSWD(K, σ). Proposition 4. Each ωSWD-irresolute function is ωSWD-continuous. The following example will show the converse of Proposition 4 is not true in general. Example 5. Consider the identity function Γ : (R, τcoc)→ (R, τcof ). Then Γis an ωSWD- continuous while it is not ωSWD-irresolute since Γ−1({1}) /∈ ωSWD(Z, τcoc). In the following theorem, we use the family ωSWD(Z, τ) to present a theorem similar to Theorem 4. Theorem 15. Let {(Zα, τα) : α ∈ ∆} be a family of topological spaces with Zα∩ Zβ = ϕ for each α ̸= β. For each α ∈ ∆, let ϕ ̸= Hα ⊆ Zα and put H = ∪ Hα α∈∆ . Then: (i) H ∈ ωSWD(Z, τs) iff there is α◦ ∈ ∆ with Hα◦ ∈ ωSWD(Zα◦ , τα◦). (ii) If Hα ∈ ωSWD(Zα, τα), then H ∈ ωSWD(Z, τs). Proof. (i) Let x ∈ H. Then there is S ∈ SWD(Z, τs) with x ∈ S and S − H is countable. Write S = ∪ α∈∆ Sα. Then by Theorem 4 (Part (i)) there is α◦ ∈ ∆ with Sα◦ ∈ SWD(Zα◦ , τα◦) such that Sα◦−Hα◦ ⊆ Sα◦− ∪ α∈∆ Hα ⊆ S−H. Therefore, Sα◦−Hα◦ is countable. Now, let x◦ ∈ Hα◦ . Then S∗ α◦ = Sα◦ ∪{x◦} ∈ SWD(Zα◦ , τα◦) with x◦ ∈ S∗ α◦ and S∗ α◦ − Hα◦ is countable. Therefore, Hα◦ ∈ ωSWD(Zα◦ , τα◦). Conversely, choose x◦ ∈ Hα◦ . Then there is Sα◦ ∈ SWD(Zα◦ , τα◦) with x◦ ∈ Sα◦ and Sα◦−Hα◦ is countable. Now, let x ∈ H and put S = Sα◦ ∪ {x}. Then by Theorem 4 (Part (i)), S ∈ SWD(Z, τs) with x ∈ S and S − H ⊆ S − Hα◦ ⊆ (Sα◦ − Hα◦) ∪ {x}. Since (Sα◦ − Hα◦) ∪ {x} is countable, then H ∈ ωSWD(Z, τs). (ii) Follows from part (i). In the next example we will show that the converse of part (ii) of Theorem 4 and Theorem 15, is not true in general. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3380 Example 6. Let (Z1, τ1) = ((0, 1), τind), (Z2, τ2) = ((2, 3), τcoc) and consider Z = Z1 ⊕Z2. Let H = {x1} ∪ E, where x1 ∈ Z1 and E is countable subset of Z2. Note that, Cl(H) = Z1 ∪ E and Z1 ⊆ Int(Cl(H)). Therefore, H ∈ SWD(Z, τs) and hence H ∈ ωSWD(Z, τs). On the other hand E /∈ ωSWD(Z2, τ2). 4. Almost ωSWD-compact spaces In this section, we will introduce and study the notion of almost ωSWD-compact spaces with some of their properties. Definition 9. Let (Z, τ) be a T S and H ⊆ Z. A point x ∈ Z is said to be ωSWDθ- accumulation point of H if ClωSWD(G) ∩H ̸= ϕ for each G ∈ ωSWD(Z, τ) with x ∈ G. Furthermore, an ωSWDθ- closure of H is the set off all ωSWDθ- accumulation points of H and it is denoted by ClωSWDθ(H). If ClωSWDθ(H) = H, then H is said to be ωSWDθ- closed subset of (Z, τ) and its complement is said to be ωSWDθ- open subset of (Z, τ). The following Proposition can be easily constructed. Proposition 5. Let (Z, τ) be a T S and H ⊆ Z. Then : (i) H is ωSWDθ- open set iff for each x ∈ H there is G ∈ ωSWD(Z, τ) with x ∈ G ⊆ ClωSWD(G) ⊆ H. (ii) If H ∈ ωSWD(Z, τ)∩ωSWDC(Z, τ), then H is ωSWDθ- closed subset of (Z, τ). (iii) ClωSWD(H) ⊆ ClωSWDθ(H) and if H ∈ ωSWD(Z, τ), then ClωSWD(H) = ClωSWDθ(H). Definition 10. Let (Z, τ) be a T S and H ⊆ Z. Then: (i) A family H = {Hα : α ∈ ∆} is said to be ωSWD(Z, τ)-cover(resp., ωSWD(Z, τ)- θ-cover, τ -cover) of H if H ⊆ ∪ α∈∆ Hα and Hα is an ωSWD-open (resp., ωSWDθ- open, open) subset of (Z, τ) for each α ∈ ∆. (ii) (Z, τ) is said to be almost ωSWD-compact if for each ωSWD(Z, τ)-cover H = {Hα : α ∈ ∆} of Z there is a finite subset ∆◦ ⊆ ∆ with Z = ∪ α∈∆◦ ClωSWD(Hα). The following results follow immediately from Definitions 5 and 10 and the fact that τ ⊆ SWD(Z, τ) ⊆ ωSWD(Z, τ) for any space (Z, τ). Theorem 16. If a topological space (Z, τ) is almost ωSWD-compact, then the following hold: (i) (Z, τ) is almost SWD-compact. (ii) If H = {Hα : α ∈ ∆} is τ -cover of Z, then there is a finite subset ∆◦ ⊆ ∆ with Z = ∪ α∈∆◦ ClωSWD(Hα). (iii) If H = {Hα : α ∈ ∆} is an ωSWD(Z, τ)-cover of Z, then there is a finite subset ∆◦ ⊆ ∆ with Z = ∪ α∈∆◦ Cl(Hα). The converses of Theorem 16 is not true in general as will see in the following examples. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3381 Example 7. (i) Consider (R, τind). Then (R, τind) satisfied (Parts ii and iii) of The- orem 16, but (R, τind) is not almost ωSWD-compact since H = {{x} : x ∈ R} is an ωSWD(Z, τ)-cover of R which has no finite subset ∆◦ = {x1, x2, ......xn} ⊆ R with R = ∪ xi∈∆◦ ClωSWD{xi} = ∪ xi∈∆◦ {xi}. (ii) Let Z = R with the topology τ = {ϕ,R, {1}}.Then (R, τ) is almost SWD-compact but it is not almost ωSWD-compact since H = {{1, x} : x ∈ R} is an ωSWD(Z, τ)-cover of R which has no finite subset ∆◦ = {x1, x2, ......xn} ⊆ R with R = ∪ x∈∆◦ ClωSWD({1, x}) = ∪ x∈∆◦ {1, x}. Theorem 17. Let (Z, τ) be a T S. If (Z, τ) is almost ωSWD-compact, then each ωSWD(Z, τ)- θ-cover of Z has a finite subcover. Proof. Let H ={Hα : α ∈ ∆} be ωSWD(Z, τ)-θ-cover of Z. For each x ∈ Z there is Hα(x) ∈ H with x ∈ Hα(x) for some α(x) ∈ ∆. By Proposition 5 there is Gα(x) ∈ ωSWD(Z, τ) with x ∈ Gα(x) ⊆ ClωSWD(Gα(x)) ⊆ Hα(x).Therefore, G = {Gα(x) : x ∈ Z} is an ωSWD(Z, τ)-cover of Z and hence there is a finite subset ∆◦ ⊆ ∆ with Z = ∪ x∈∆◦ ClωSWD(Gα(x)) ⊆ ∪ x∈∆◦ Hα(x). Definition 11. Let (Z, τ) be a T S and x ∈ Z. A filter base F on (Z, τ) is said to be: (i) ωSWDθ- converge to x if for each G ∈ ωSWD(Z, τ) with x ∈ G there is F ∈ F such that F ⊆ ClωSWD(G). (ii) ωSWDθ- accumulate at x if ClωSWD(G) ∩ F ̸= ϕ for each F ∈ F and for each G ∈ ωSWD(Z, τ) with x ∈ G. Note that, if a filter base F ωSWDθ- converges to a point x, then F ωSWDθ- accu- mulates at x. Also, it is obvious to show that a maximal filter base F ωSWDθ- converges to a point x iff F ωSWDθ- accumulates at x. Theorem 18. Let (Z, τ) be a T S . Then the following are equivalent: (i) (Z, τ) is almost ωSWD-compact. (ii) Each maximal filter base ωSWDθ- converges to some point of Z. (iii) Each filter base ωSWDθ- accumulates at some point of Z. (iv) For each family {Hα : α ∈ ∆} of ωSWD-closed subsets of (Z, τ) and ∩ α∈∆ Hα = ϕ, there is a finite subset ∆◦ ⊆ ∆ with ∩ α∈∆◦ IntωSWD(Hα) = ϕ. (v) For each family {Hα : α ∈ ∆} of ωSWD-closed subsets of (Z, τ) and ∩ α∈∆◦ IntωSWD(Hα) ̸= ϕ for each finite subset ∆◦ ⊆ ∆, then ∩ α∈∆ Hα ̸= ϕ. Proof. The implication (iv → v) is obvious. (i → ii) Let F be a maximal filter base on Z and suppose that it does not ωSWDθ- converge to any point of Z. Since F does not ωSWDθ- accumulate at any point x ∈ Z, there is Fx ∈ F and Gx ∈ ωSWD(Z, τ) with x ∈ Gx such that ClωSWD(Gx) ∩ Fx = ϕ. Therefore, {Gx : x ∈ Z} is an ωSWD(Z, τ)-cover of Z and so there is a finite subset A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3382 ∆◦ = {x1, x2, ......xn} ⊆ Z with Z = ∪ x∈∆◦ ClωSWD(Gx). But F is a filter base on Z and hence there is F◦ ∈ F with F◦ ⊆ ∩{Fxi : i = 1, 2, ...n}. Since Fxi∩ ClωSWD(Gxi) = ϕ, then F◦ = ϕ which is a contradiction. (ii → iii) Let F be a filter base on Z. Then there is F◦ a maximal filter base with F ⊆ F◦. Since F◦ ωSWDθ- converges to x for some x ∈ Z, then for each G ∈ ωSWD(Z, τ) with x ∈ G there is F◦ ∈ F◦ such that F◦ ⊆ ClωSWD(G). Therefore for each F ∈ F , ϕ ̸= F◦ ∩ F ⊆ ClωSWD(G) ∩ F and hence F ωSWDθ- accumulates at x. (iii → iv) Let {Hα : α ∈ ∆} be a family of ωSWDC(Z, τ) with ∩ α∈∆ Hα = ϕ. If ∩ α∈∆◦ IntωSWD(Hα) ̸= ϕ for each finite subset ∆◦ ⊆ ∆. Then F = { ∩ α∈∆◦ IntωSWD(Hα) : ∆◦ ⊆ ∆ and ∆◦ is finite} is a filter base on Z and hence F ωSWDθ- accumulates at x for some x ∈ Z. Since {Z −Hα : α ∈ ∆} is an ωSWD(Z, τ)-cover of Z, then x ∈ Z −Hα◦ for some α◦ ∈ ∆. Therefore, ClωSWD(Z−Hα◦)∩IntωSWD(Hα◦) = ϕ which is a contradiction. (v → i) Let H = {Hα : α ∈ ∆} be an ωSWD(Z, τ)-cover of Z. Then Z − Hα ∈ ωSWDC(Z, τ) for each α ∈ ∆ and ∩ α∈∆ (Z − Hα) = ϕ. Hence there is a finite subset ∆◦ ⊆ ∆ with ∩ α∈∆◦ IntωSWD(Z −Hα) = ϕ.Therefore Z = ∪ x∈∆◦ ClωSWD(Hα). Theorem 19. Let (Z, τ) be almost ωSWD-compact T S and E ⊆ Z. If E ∈ τ ∩ ωSWDC(Z, τ), then (E, τE) is almost ωSWD-compact. Proof. Let H = {Hα : α ∈ ∆} be a family with E = ∪ α∈∆ Hα and Hα ∈ ωSWD(E, τE) for each α ∈ ∆. Then by Theorem 12, Hα ∈ ωSWD(Z, τ) for each α ∈ ∆ and hence {Hα : α ∈ ∆}∪{Z−H} = Z. Since (Z, τ) is almost ωSWD-compact, then there is a finite subset ∆◦ ⊆ ∆ with Z = [∪ x∈∆◦ ClωSWD(Hα)]∪{Z−H}.Therefore, E = ∪ α∈∆◦ ClωSWD(Hα) ⊆ ∪ α∈∆◦ ClωSWDE (Hα). Definition 12. Let (Z, τ) be a T S and H ⊆ Z. A subset H is said to be almost ωSWD- compact relative to Z (in Z) if whenever H = {Hα : α ∈ ∆} is an ωSWD(Z, τ)-cover of H, then there is a finite subset ∆◦ of ∆ with H ⊆ ∪ x∈∆◦ ClωSWD(Hα). The following Theorem can be easily constructed. Theorem 20. Let (Z, τ) be a T S and H ⊆ Z. The following are equivalent: (i) H is almost ωSWD-compact relative to Z. (ii) If F is a maximal filter base on Z and meets H, then it ωSWDθ- converges to some point of H. (iii) If F is a filter base on Z and meets H, then it ωSWDθ- accumulates at some point of H. (iv) If {Hα : α ∈ ∆} is a family of ωSWD-closed subsets of (Z, τ) and [∩ α∈∆ Hα]∩H = ϕ, then there is a finite subset ∆◦ ⊆ ∆ with [∩ α∈∆◦ IntωSWD(Hα)] ∩H = ϕ. A. Rawshdeh, H. H. Al-Jarrah, K. Y. Al-Zoubi / Eur. J. Pure Appl. Math, 17 (4) (2024), 3370-3385 3383 Proposition 6. Let (Z, τ) be a T S and H,G ⊆ Z. If H is an ωSWDθ- closed subset of (Z, τ) and G is almost ωSWD-compact relative to Z, then H∩G is almost ωSWD-compact relative to Z. Proof. Let H ={Hα : α ∈ ∆} be ωSWD(Z, τ)-cover of H ∩ G. Then by Propo- sition 5 (Part i) for each x ∈ Z − H there is Wx ∈ ωSWD(Z, τ) with x ∈ Wx ⊆ ClωSWD(Wx) ⊆ Z −H. Therefore, H∪{Wx : x ∈ G −H} is an ωSWD(Z, τ)-cover of G and hence there are a finite subset ∆◦ ⊆ ∆ and a finite subset {x1, x2, ....xn} ⊆ G − H with G ⊆ [∪ x∈∆◦ ClωSWD(Hα)] ∪ [ n ∪ i=1 ClωSWD(Wxi)]. Since ClωSWD(Wx) ⊆ Z − H, then H ∩G ⊆ ∪ x∈∆◦ ClωSWD(Hα). Corollary 6. If (Z, τ) is an almost ωSWD-compact and H ⊆ Z is an ωSWDθ- closed subset of (Z, τ), then H is almost ωSWD-compact relative to Z. Theorem 21. Let (Z, τ) be a T S. If there is a non-empty proper subset E ∈ ωSWD(Z, τ)∩ ωSWDC(Z, τ) of Z, then (Z, τ) is almost ωSWD-compact iff each H ∈ ωSWD(Z, τ) ∩ ωSWDC(Z, τ) is an almost ωSWD-compact relative to Z. Proof. Let H ∈ ωSWD(Z, τ) ∩ ωSWDC(Z, τ). Then by Proposition 5 (Part ii) H is an ωSWDθ- closed subset of (Z, τ) and hence by Corollary 6, H is almost ωSWD-compact relative to Z. Conversely, Let H ={Hα : α ∈ ∆} be ωSWD(Z, τ)-cover of Z. Then E and (Z − E) are almost ωSWD-compact relative to Z and hence there are finite subset ∆◦ ⊆ ∆ with Z = ∪ x∈∆◦ ClωSWD(Hα). Proposition 7. A finite union of almost ωSWD-compact subsets relative to Z is almost ωSWD-compact relative to Z. Proof. Let n ∪ i=1 Gi be a finite union of almost ωSWD-compact subsets relative to Z and H ={Hα : α ∈ ∆} be ωSWD(Z, τ)-cover of n ∪ i=1 Gn. Then for each i ∈ {1, 2, 3...n} there is a finite subset ∆i ⊆ ∆ with Gi ⊆ ∪ α∈∆i ClωSWD(Hα). It is clear that n ∪ i=1 ∆i is finite set. Therefore, n ∪ i=1 Gi ⊆ ∪ α∈ n ∪ i=1 ∆i ClωSWD(Hα). The proofs of the following results are obvious and hence they are omitted. Theorem 22. Let Γ : (Z, τ) → (K, σ) be ωSWD-irresolute. If H is almost ωSWD- compact relative to Z, then Γ(H) is almost ωSWD-compact relative to K. Corollary 7. Let Γ : (Z, τ) → (K, σ) be surjective ωSWD-irresolute. If (Z, τ) is almost ωSWD-compact, then (K, σ) is almost ωSWD-compact. Corollary 8. If ΠZα∈∆ is almost ωSWD-compact, then Zα is almost ωSWD-compact for each α ∈ ∆. REFERENCES 3384 5. Conclusions The study of different types of generalized open sets has been one of the main areas of research in general topology during the last several decades. Mathematicians investi- gate the properties of various broad topological concepts using generalized open sets. To continue this line of research, this manuscript has been written. The main achievements of this work are: (i) We present a generalization for Theorem 2 which was introduced in [1] and provide additional features of SWD(Z, τ). (ii) We introduce the notion of ωSWD(Z, τ) which is a new generalization for some- where dense subsets of a topological space (Z, τ) and hence it is a new generalization for open subsets of a topological space (Z, τ). (iii) We verify some fundamental features of ωSWD(Z, τ) and study the requirements for the equivalence between the classes SWD(Z, τ), ωSWD(Z, τ) and ωSWD(Z, τω). (iv) We study the notions of the interior, closure, ωSWD-continuous and ωSWD- irresolute via ωSWD(Z, τ). (v) We study the notion of almost ωSWD-compact spaces with some of their proper- ties. This work can be considered as a starting point for many topics and studies in topology since ωSWD(Z, τ) forms a generalization of open sets. Therefore, in upcoming papers, we plan to study the notion of connected, separation axioms and other types of covering such as paracompact spaces via the class ωSWD(Z, τ). Acknowledgements The publication of this paper was supported by Yarmouk University Research Council. References [1] T.M. Al-Shami. Somewhere dense sets and st1-spaces. Punjab Univ. J. Math., 49(2):101–111, 2017. [2] T.M. Al-Shami. Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets. Soft Comput., 25:14449–14460, 2021. [3] T.M. Al-Shami and T. Noiri. More notions and mappings via somewhere dense sets. Afr. Mat., 30(7):1011–1024, 2019. [4] T.M. Al-Shami and T. Noiri. Compactness and lindelöfness using somewhere dense and cs-dense sets. Novi Sad J. 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