Three new soft separation axioms in soft topological spaces

: Soft ω -almost-regularity, soft ω -semi-regularity, and soft ω - T 2 12 as three novel soft separation axioms are introduced. It is demonstrated that soft ω -almost-regularity is strictly between “soft regularity” and “soft almost-regularity”; soft ω - T 2 12 is strictly between “soft T 2 12 ” and “soft T 2 ”, and soft ω -semi-regularity is a weaker form of both “soft semi-regularity” and “soft ω -regularity”. Several su ﬃ cient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft ω -almost-regularity are obtained, and a decomposition theorem of soft regularity by means of “soft ω -semi-regularity” and “soft ω -almost- regularity” is obtained. Furthermore, it is shown that soft ω -almost-regularity is heritable for speciﬁc kinds of soft subspaces. It is also proved that the soft product of two soft ω -almost regular soft topological spaces is soft ω -almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed.


Introduction and preliminaries
The need for theories that cope with uncertainty emerges from daily experiences with complicated challenges requiring ambiguous facts.Molodstov's [1] soft set is a contemporary mathematical approach to coping with these difficulties.Soft collection logic is founded on the parameterization principle, which argues that complex things must be seen from several perspectives, with each aspect providing only a partial and approximate representation of the full item.Molodstov [1] was a pioneer in the application of soft sets in a variety of domains, emphasizing their advantages over probability theory and fuzzy set theory, which deal with ambiguity or uncertainty.
Following that, Maji et al. [2] began researching soft set operations such as soft unions and soft intersections.To overcome the shortcomings of these operations, Ali et al. [3] created and showed new operations such as limited union, intersection, and complement of a soft set.Babitha and Sunil [4] investigated numerous aspects of linkages and functions in a soft setting.Qin and Hong [5] developed novel kinds of soft equal relations and showed some algebraic properties of them.Their pioneering work paved the way for subsequent papers (for more detail, see [6,7] and the references listed therein).Soft set theory has lately been a popular method among academics for dealing with uncertainty in a wide range of fields, including information theory [8], computer sciences [9], engineering [10], and medical sciences [11].
Separation axioms provide a way to study certain properties of compact and Lindelof spaces, as well as a way to categorize spaces and mappings into distinct families.As a result, topological scholars who presented various kinds of soft separation axioms became interested in soft separation axioms.Generally speaking, they can be separated into two classes: Soft points and ordinary points, based on the subjects being studied.While the authors in [14][15][16]28] examined soft separation axioms using ordinary points, the authors in [13,[29][30][31][32][33] and others have applied the concept of soft points.In the present work, we introduce soft ω-almost-regularity, soft ω-semi-regularity, and soft ω-T 2 1  2 as three novel soft separation axioms.
This article is organized as follows: In Section 1, after the introduction, we provide a few definitions that are relevant to this paper.
In Section 2, we define soft ω-almost-regularity as a new soft separation axiom that lies between soft regularity and soft almost-regularity.We introduce many characterizations of this type of soft separation axiom.Also, we provide several sufficient conditions establishing the equivalence between this newly introduced axiom and its relevant counterparts.Moreover, we establish that soft ω-almostregularity is heritable for specific types of soft subspaces.Furthermore, we show that soft ω-almostregularity is a productive soft property.In addition, we investigated the links between this class of soft topological spaces and its analogs in general topology.
In Section 3, we define soft ω-semi-regularity and soft ω-T 2 1 2 as two new soft separation axioms.We show that soft ω-semi-regularity is a weaker form of both soft semi-regularity and soft ω-regularity, and soft ω-T 2 1  2 lies strictly between soft T 2 1 2 and soft T 2 .Also, we provide several sufficient conditions establishing the equivalence between these newly introduced axioms and their relevant counterparts.Moreover, a decomposition theorem for soft regularity through the interplay of soft ω-semi-regularity and soft ω-almost-regularity is obtained.In addition, we investigated the links between these classes of soft topological spaces and their analogs in general topology.
This paper follows the notions and terminologies as appear in [34][35][36].Topological spaces and soft topological spaces, respectively, shall be abbreviated as TS and STS.
The following definitions will be used in the remainder of the paper: Definition 1.1.A TS (H, β) is called (a) [37] almost-regular (A-R, for simplicity) if for every z ∈ H and every [38] semi-regular (S-R, for simplicity) if RO(H, β) forms a base for β; (c) [39] ω-almost-regular (ω-A-R, for simplicity) if for every z ∈ H and every N ∈ S ωC(H, β) such that z ∈ H − N, we find U, V ∈ β such that z ∈ U, N ⊆ V, and U ∩ V = ∅; (d) [39] ω-semi-regular (ω-S-R, for simplicity) if RωO(H, β) forms a base for β.Definition 1.2.A STS (H, ϕ, Σ) is called (a) [13] soft T 2 if for every two soft points a s , b t ∈ S P(H, Σ), we find K, W ∈ ϕ such that a s ∈K, b y ∈W, and K ∩W = 0 Σ ; (b) [13] soft regular if for every a z ∈ S P(H, Σ) and every K ∈ ϕ such that a z ∈K, we find G ∈ ϕ such that a z ∈G ⊆Cl ϕ (G) ⊆K; (c) [32] soft T 2 1 2 if for every two soft points a s , b t ∈ S P(H, Σ), we find K, W ∈ ϕ such that a s ∈K, b y ∈W, and Cl ϕ (K) ∩Cl ϕ (W) = 0 Σ ; (d) [31] soft almost-regular (soft A-R, for simplicity) if for every r z ∈ S P(H, Σ) and every G ∈ S C(H, ϕ, Σ) such that r z ∈1 Σ − G, we find S , T ∈ ϕ such that r z ∈S , G ⊆T , and S ∩T = 0 Σ .
(d) [33] soft ω-regular for every a z ∈ S P(H, Σ) and every K ∈ ϕ such that a z ∈K, we find G ∈ ϕ such that a z ∈G ⊆Cl ϕ ω (G) ⊆K.

Soft ω-almost-regular spaces
In this section, we define soft ω-almost-regularity as a new soft separation axiom that lies between soft regularity and soft almost-regularity.We introduce many characterizations of this type of soft separation axiom.Also, we provide several sufficient conditions establishing the equivalence between this newly introduced axiom and its relevant counterparts.Moreover, we establish that soft ω-almostregularity is heritable for specific types of soft subspaces.Furthermore, we show that soft ω-almostregularity is a productive soft property.In addition, we investigated the links between this class of soft topological spaces and its analogs in general topology.Definition 2.1.An STS (H, ϕ, Σ) is called soft ω-almost-regular (soft ω-A-R, for simplicity) if for every r z ∈ S P(H, Σ) and every G ∈ S ωC(H, ϕ, Σ) such that r z ∈1 Σ − G, we find S , T ∈ ϕ such that r z ∈S , G ⊆T , and S ∩T = 0 Σ .
(6) For every r z ∈ S P(H, Σ) and every K ∈ ϕ such that r z ∈K, there is L ∈ ϕ such that r z ∈L ⊆Cl ϕ (L) ⊆Int ϕ Cl ϕ ω (K) .
In Theorems 2.3, 2.4, 2.7, and Corollary 2.8, we discuss the connections between soft almost-regularity and its analog in traditional topological spaces.Also, in Theorems 2.5, 2.6, 2.9, and Corollary 2.10, we discuss the connections between soft ω-almost-regularity and its analog in traditional topological spaces.
Sufficiency.Let (H, L r ) be A-R for every r ∈ Σ.Let r z ∈ S P(H, Σ) and let K ∈ ⊕ r∈Σ L r such that r z ∈K.By Theorem 3.5 of [34], we find U ∈ L r such that r z ∈r U ⊆K.Then, we have z ∈ U ∈ L r .So, by Theorem 2.1 (d) of [39], we find V ∈ L r such that z ∈ V ⊆ Cl L r (V) ⊆ Int L r Cl L r (U) .Thus, we have r V ∈ ⊕ r∈Σ L r and Corollary 2.8.Let (D, L) be a TS.Then, for any set Σ, (D, and by Theorem 2.7 we get the result.Theorem 2.9.Let {(H, L r ) : r ∈ Σ} be a collection of TSs.Then, (H, In contrast, by Lemma 4.9 of [40] and Theorem 8 of [35], . Thus, by Theorem 2.1 (e) of [39] (H, L r ) is ω-A-R.
The previously mentioned theorems lead to the following implications, yet Examples 2.16 and 2.17 that follow demonstrate that the opposite of these implications is false.
The following lemma will be used in the next main result: Lemma 2.18.Let (H, ϕ, Σ) be an STS.If C Y is a soft dense subset of (H, ϕ ω , Σ), then for any soft subset Theorems 2.19 and 2.21 establish that soft ω-almost-regularity is heritable for specific types of soft subspaces.Theorem 2. 19.
The following lemma will be used in the next main result: Lemma 2.20.

Conclusions
Soft separation axioms are a collection of requirements for categorizing a system of STSs based on certain soft topological features.These axioms are often expressed in terms of classes of soft sets.
In the next work, we intend to: 1) Define and investigate soft ω-almost-normality; 2) investigate the behavior of these new soft separation ideas under various kinds of soft mappings; and 3) find an application for our new two conceptions in the "decision-making problem", "information systems", or "expert systems".
is full, then by Theorem 3.8 (D, (C (L)) r ) = (D, L) is S-R.Sufficiency.Let (D, L) be S-R.Let r z ∈ S P(D, Σ) and let C U ∈ C (L) such that r z ∈C U .Then, we have z ∈ U ∈ L. So, we find V ∈ S O (D, L) such that z ∈ Int L (Cl L (V)) ⊆ U. Thus, we have C V ∈ C (L) and r z ∈C Int L( Cl L (V)) = Int C(L) Cl C(L) (C V ) ⊆C U .This shows that (D, C (L) , Σ) is soft S-R.Theorem 3.10.Let (D, L) be a TS.Then, for any set Σ, (D, C