Strictly Wider Class of Soft Sets via Supra Soft δ -Closure Operator

. In this work, we use the supra soft δ -closure operator to present a new notion of generalized closed sets in supra soft topological spaces (or SSTSs), named supra soft δ -generalized closed sets. We show that, this notion is more general than many of previous notions, which presented before in famous papers. We illustrate many of its essential properties in detail. Speciﬁcally, we illustrate that the new collection neither forms soft topology nor supra soft topology. Moreover, we study the behavior of the soft image and soft pre-images of supra soft δ -generalized closed sets under new types of soft mappings, named supra soft irresolute and supra soft δ -irresolute closed. In addition, we deﬁne the concept of supra soft δ -generalized open sets, as a complement of supra soft δ -generalized closed sets. Finally, the relationships with other forms of generalized open sets in SSTSs are explored, supported by concrete examples and counterexamples. Therefore, I think the development of the notions presented in this paper are su ﬃ ciently general relevance to allow for future extensions.


Introduction
In 1983, Mashhour et al. [1] generalized notions the topological spaces by presenting the concept of supra topological spaces.A. Alpers [2], in 2002, used these new notions to present applications to digital topologies.Kozae et al. [3], in 2002, presented new applications for theses notions in digital plane.Recently, in 2023 [4] Al-shami and Alshammari applied the supra topological spaces to information systems.Many applications based on closure operators [5,6] and novel types of open sets, named C-open sets [7], F-open [8] have been introduced.
In 2011, Ahmad and Kharal [9] applied the continuity notions to soft set theory [10,11].Several classes of generalized soft open sets and soft functions defined in [12][13][14].Zorlutuna [15] investigated more interesting properties of soft continuity.Wardowski [16] presented the notions of soft functions based on its fixed points.Alqahtani and Ameen [17,18] introduced Baire category soft sets in soft nodec spaces.
The definitions of soft ideal presented by Kandil et al. [19], in 2014.Theses approves have been investigated by using the concepts of soft semi-open sets [20,21].In 2018, applications in medical [22] via certain soft ideal rough topological spaces have been presented by Abd El-latif.
Several weaker classes of soft open sets have been provided by using the soft ideal notions [23][24][25].
Also, new versions of separation axioms [26,27] in STS have been studied.
In 2014, El-Sheikh and Abd El-latif [28] presented the notions of SSTSs by ignoring the condition of finite soft intersection in the concept of soft topological space (or STS) [29].So, the new collection became wider than the old one.It leading to investigate and generalize many topological properties to such spaces like different types of separation axioms [30,31] and irresolute functions bases on supra soft b-open sets [32,33].
Later, several generalizations and topological properties were provided; To name a few: Supra soft regular generalized closed sets [34,35], soft connected spaces and soft paracompact spaces [36], supra soft separated sets and supra soft connectedness based on supra soft b-open sets [37,38], supra soft compactness [39,40].Several types of weaker forms of supra soft separation axioms [41,42] based on supra semi open soft sets [43], supra β-open soft sets [44] ave been introduced and studied.Recently, Abd El-latif introduced the approach of supra soft somewhere dense sets [45].
He and his co-author [46] presented new categories of supra soft continuous functions based on this new notion.
Recently, [47], Abd El-latif defined the concept of supra soft δ-open sets in SSTSs.He introduced new soft operators named, supra soft δ-closure (interior, boundary, cluster) operator.Moreover, he applied these operators to introduce new weaker classes of supra soft continuity.
The concepts of soft generalized closed (or soft g-closed) sets in STS have been defined, in 2012, by Kannan [48].He and his co-authors [49] investigated this notion and presented the notion of soft strongly g-closed sets.Kandil et al. [50] presented the concepts of supra soft g-closed sets (based on soft ideals) [51].More application on weaker forms of generalized open sets were recently introduced in [52,53], to improve the measurement accuracy for information systems.
The purpose of this manuscript, is to present the concepts of supra soft δ-generalized closed sets in SSTSs by using the notion of supra soft δ-closure operator, in Section 3.With supporting by examples, we prove that our new notions are more general than many previous notions.More interesting properties such soft union (respectively, soft intersection) of finite numbers of supra soft δ-generalized closed sets are discussed.Moreover, the soft pre-images and soft images of supra soft δ-generalized closed sets are explored and studied under the supra soft irresolute and supra soft δ-irresolute closed functions.In Section 4, the concepts of supra soft δ-generalized open sets by using the supra soft δ-interior operator in SSTSs are presented.Furthermore, many of its basic properties are presented.
The elements of Θ are called supra soft open sets, and their soft complements are called supra soft closed sets.Definition 2.6.[28] Let (Z, τ, ∆) be an STS and (Z, Θ, ∆) be an SSTS.We say that, Θ is an SSTS associated with τ if τ ⊂ Θ. Definition 2.7.[28] Let (Z, Θ, ∆) be an SSTS over Z and (T, ∆) ∈ S(Z) ∆ .Then, the supra soft interior of (V, ∆), denoted by int s (V, ∆) is the soft union of all supra soft open subsets of (V, ∆).Also, the supra soft closure of (T, ∆), denoted by cl s (T, ∆) is the soft intersection of all supra soft closed supersets of (T, ∆).
Definition 2.9.[47] Let (Z, Θ, ∆) be an SSTS and (R, The collection of all supra soft δ-open sets will denoted by SOS δ (Z) and the collection of all supra soft δ-closed sets will denoted by SCS δ (Z).

Weaker class of supra soft closed sets via supra soft δ-closure operator
This section aims to define the concepts of supra soft δ-generalized closed sets (briefly, g s δ -closed sets) in SSTSs by using the supra soft δ-closure operator.We show that, this notion is wider than the notion of supra soft generalized closed sets [50], supra soft strongly generalized closed sets [34] and supra soft regular generalized closed sets [35].Moreover, we fond out that this notion does not success to form an STS or SSTS, since the soft intersection (respectively, soft union) of any two g s δ -closed sets need not to be an g s δ -closed, in general.In addition, we discuss the soft pre-images and soft images of g s δ -closed sets under supra soft irresolute and supra soft δ-irresolute closed functions.Furthermore, many examples and counterexamples are provided.
The family of all g s δ -closed sets will denoted by G s δ C(Z).

Proof.
Assume that (A, ∆) be a supra soft g-closed subset of an SSTS (X, Θ, ∆), Remark 3.2.The next example shall show that, the converse of the above theorem is not satisfied in general.

Conclusion and upcoming work
Our purpose of this project, is to generalize many kinds of supra soft generalizes closed sets in SSTSs by using the of supra soft δ-closure operator.We showed that the new class contains strictly the class of supra soft generalized closed sets [50], supra soft strongly generalized closed sets [34] and supra soft regular generalized closed sets [35].Also, the notions of g s δ -open sets were defined and several of its interested properties were studied.The presented results have strong depth that can herald future applications.So, our future work, is to generalize these notions by using the soft ideal notions [19] an soft semi open sets [57,58] and introduce many types of supra soft separation axioms, connectedness and compactness via the above-mentioned notions.

:
The soft union of arbitrary numbers of soft sets in τ belongs to τ,(3): The soft intersection of finite numbers of soft sets in τ belongs to τ.The triplet (Z, τ, ∆) is called an STS over Z. Also, the elements of τ are called soft open sets, and their soft complements are called soft closed sets.

Remark 3 . 1 .
The soft intersection (respectively, soft union) of any two g s δ -closed sets need not to be a g s δ -closed in general as shall shown in the following examples.Examples 3.1.