Two New Families of Supra-Soft Topological Spaces Deﬁned by Separation Axioms

: This paper contributes to the ﬁeld of supra-soft topology. We introduce and investigate supra pp -soft T j and supra pt -soft T j -spaces ( j = 0,1,2,3,4 ) . These are deﬁned in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the ﬁrst type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra tp -soft T j and supra tt -soft T j -spaces ( j = 0,1,2,3,4 ) . This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for i = 0,1, possessing the T i property by a parametric supra-topological space implies possessing the pp -soft T i property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of pp -soft T i -spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, ﬁnite product spaces, and soft S (cid:63) -continuous mappings preserve these axioms.


Introduction
This article is in continuation of earlier contributions in a fruitful area of research merging topology and soft set theory. The hybridization of the particular extension called supratopology with soft sets has produced an interesting structure called supra-soft topology, which is the exact focus for our research. Firstly, the state of the art in this field will be briefly summarized.
In order to characterize situations containing uncertainties, the structure called 'soft set' [1] was launched in 1999. Soon afterwards, some operations on soft sets (such as their union and intersection) and operators (like complements of soft sets) were formulated in [2]. This paper defined the null and absolute soft sets, which act as a soft counterpart of the universal and empty crisp sets, respectively. As to their applicability, Maji et al. [3], later improved by [4], succeeded in using soft sets for decision making. Despite the shortcomings of some results and concepts in these seminal works, they form the foundational basis of soft set theory, together with a remarkable recent addition: their semantical interpretation [5]. Actually, the literature has produced many types of operations between soft sets. More opportunities to make full use of soft set theory have been provided in both theoretical [6][7][8] and applied studies (in computer science [9], medical science [10,11], computational biology [12], etc.).
In addition, powerful mathematical structures have been exported to the soft set field [13], thus emphasizing its importance for the analysis of abstract theories. Particularly, topological notions have been hybridized with soft set theory since 2011. This year, Shabir and Naz [14] proposed the concept of soft topology. Zorlutuna et al. [15] set forth the idea of soft point. This is a good tool for the investigation of properties of soft interior points and soft neighborhood systems, and in fact, it was independently reformulated by both [16] and [17]. While [16] used the new phrasing of soft point to investigate soft metric spaces, Ref. [17] employed it to study soft neighborhood systems and uncover some relations of the soft limit points of a soft set. The comparative performance of soft topologies and standard topologies has been the subject of studies such as [18,19]. Recently, abstract soft topological concepts such as compactness, separation axioms, and generalized open sets were employed to address practical problems in the areas of economic [20] and nutrition systems [21].
In 1983, Mashhour et al. [22] proposed the concept of supratopological spaces. In their model, supratopologies are collections of subsets generalizing the axiomatics for a topological space by dispensing with the postulate that the collection is closed under finite intersections. Then, El-Sheikh and Abd El-Latif [23] exported that model to soft set theory in 2014, and they conceived of the concept of supra-soft topology. Quite naturally, supra-soft topological spaces contain soft topological spaces. Several researchers studied essential notions related to the new structure, inclusive of supra-soft continuity [23], new generalizations of supra-soft open sets, and various classes of supra-soft separation axioms [24,25]. By utilizing soft stacks, Ref. [26] conceived a supra-soft topology from a soft topology. However, we note that the research of the fundamental concepts and notions of supra-soft topological spaces has not yet received the required consideration. As a result, a number of interesting notions in this structure still need to be formulated and properly discussed. We contribute to this area with an original inspection of the class of 'separation axioms' that are meaningful for the investigation of supra-soft topology.
Several arguments justify the study of topological concepts within the frame of supratopology. First, this setting suffices to preserve some topological characteristics and properties under conditions that do not require a topology; for example, the supra-interior and supra-closure points of a set are, respectively, still supra-open and supra-closed sets; a supra-closed subset of a supra-compact space is supra-compact, etc. Second, studying topological ideas via supra-topology produces a richer variety of concepts, especially over a finite set; for instance, the only T 1 -topology defined on a finite set is the discrete topology (which is a trivial case, and hence meaningless in application areas), whereas there are several sorts of supra-topologies that produce T 1 -spaces. Furthermore, the supra-topological frames show the easiness and diversity of building examples that satisfy supra T i -spaces compared with their counterparts on classical topology-especially those related to strong types of separation axioms. Third, supra-topology provides an appropriate environment to describe many real-life problems, which can be noted via rough set approximation operators produced by topological approaches [27]. To illustrate this point, notice that most generalizations of open sets form a supra topology but not a topology (for they are only closed under arbitrary union), which affects the performance and properties of lower and upper approximation operators in rough set theory. It is worth noting that the abstract and theoretical extension of topology called "infra-topology" was also applied in the analysis of information systems by the theory of rough sets [27].
Soft separation axioms stand out among the most relevant characteristics of soft topology. This is also the case in point-set topology. With the help of these properties, one can establish the structured categories of well-behaved soft topological or just topological spaces. A large variety of soft separation axioms have been proposed (and some facts have been corrected [28]), which can be ascribed to the factors of two kinds. Firstly, the objects that we want to separate: one can opt for either ordinary points or soft points. Secondly, the belongingness and non-belongingness relations used in the definitions: here, one can opt for either partial or total versions. In this regard, we emphasize the fact that El-shafei et al. [28] introduced new types of relations between ordinary points and soft sets, namely partial belong and total non-belong relations; they add to the relations given in the foundational [14]. Further information on this issue can be found in [14,[28][29][30][31][32]. Likewise, in this paper, we investigate separation axioms in the context of supra-soft topology beyond the pioneering analysis in Al-shami and El-Shafei [33]. The two types of supra-soft separation axioms defined herein represent new classes of supra-soft topological spaces wider than those given in [33]. The relationships among these classes of soft axioms are elucidated (cf., Propositions 5, 6 and 16 plus some interesting counterexamples). We note that using a partial belong relation in the definitions of new two types of supra-soft separation axioms makes initiating examples that more easily prove certain relations among topological concepts. One of the divergences between them is the sufficient conditions that guarantee the existence of some soft spaces; for example, supra tt-soft T 1 -space implies that every soft point is supra-soft closed, as proven in [33], whereas this characteristic does not hold for spaces of supra pp-soft T 1 and supra pt-soft T 1 introduced herein (cf., Proposition 17 and Example 12).
This article is organized as follows. We mention some definitions and properties that should help the reader understand this research in Section 2. Then, Section 3 defines the concepts of supra pp-soft T j -spaces (j = 0, 1, 2, 3, 4), and investigates their main properties. In Section 4, we formulate the concepts of supra pt-soft T j -spaces (j = 0, . . . , 4). Then, we disclose some relationships among them, as well as with respect to supra pt-soft T j -spaces (j = 0, . . . , 4). To end this paper, Section 5 contains some conclusions and hints at some directions for future research.

Preliminary Concepts
This section recalls some notions that we need in this paper. The notation 2 O captures the set of all subsets of O.  In addition to the null/absolute soft sets and to soft points, other basic types of soft sets are given by the next two related definitions: Notice that, by definition, each k M is a stable soft set, its components being equal to the singleton {k}.

Elements of Soft-Set Theory
for each m ∈ M.
Definition 12 ([17]). We say that the soft mapping f φ : S(O M ) → S(Y N ) is injective (respectively, surjective or bijective) when both mappings f and φ are injective (respectively, surjective or bijective).
Then, we recall some belongingness relations between points in the reference set O and soft sets over O: ([14,28]). Fix k ∈ O and a soft set G M over O. Then, we represent: (i) k ∈ G M (it reads as k totally belongs to G M ) if k ∈ G(m) for all m ∈ M. (ii) k ∈ G M (it reads as k does not partially belong to G M ) when there is m ∈ M with k ∈ G(m). (iii) k G M (it reads as k partially belongs to G M ) when there is m ∈ M with k ∈ G(m). (iv) k G M (it reads as k, does not totally belong to G M ) when k ∈ G(m) for all m ∈ M.
The behavior of these relations under some types of soft mappings is described in the next proposition.

Elements of Supra-Soft Topology
Now, we proceed to give some introduction to the main notions in supra-soft topological spaces. We begin with the fundamental concept in this field that owes to the idea of a supratopological space: When exported to a soft topological context, we obtain: 23,24]). Let us fix a set of parameters M. A collection δ of soft sets over O under M is a supra-soft topology on O when O is a member of δ and the collection is closed under arbitrary soft unions. We say in this case that the triple (O, δ, M) is a supra-soft topological space (briefly, supra-soft TS ). We call the members of δ supra-soft open sets; and their complements are called supra-soft closed sets.
The concepts above are related by the following property: Relative topologies are defined in the following terms: A suitably defined idea of 'homeomorphism' allows one to conceive of properties that are supra-soft topological: A bijective soft mapping is called a soft S -homeomorphism provided it is soft S -continuous and soft S -open. A property that is preserved by a soft S -homeomorphisms is called a supra-soft topological property [33].
Finally, in this section, we recall some separation axioms pertaining to this context: is supra-soft normal provided that any two disjoint supra-soft closed sets can be separated by disjoint supra-soft open sets. (v) Supra tp-soft T 3 (respectively, a supra tp-soft T 4 ) when it is both supra tp-soft regular (respectively, supra-soft normal) and supra tp-soft T 1 .
Whilst the concepts of supra tp-soft T j -spaces and supra tt-soft T j -spaces were studied in [33] under the name of "supra-soft T i -spaces" and "supra p-soft T i -spaces", respectively, we believe that the adapted terminology that we employ here, namely supra tt-soft regular and supra tt-soft T j -spaces, is better, because the previous two types of soft separation axioms and the two types introduced in this study are distinguished in terms of the types of belongingness and non-belongingness relations between ordinary points and supra-soft open and supra-soft closed sets. Furthermore, supra tt-soft regular and supra tt-soft T jspaces were defined in [33] as an alternative to the corresponding tt-version in the above definition. They are, respectively, stated by a routine replacement of ∈ in Definition 20 by the relation .

Remark 1.
The spaces of supra tp-soft T 3 and supra tp-soft T 4 are independent of each other [33].
An equivalence between the concepts of extended (cf., Definition 21 below) and enriched soft topology was proven in [18], which also made use of this technique to establish many results that associated a soft topology with its parametric topologies. We can now define an extended supra-soft topology in a similar way: In this section, we introduce the novel separation axioms that jointly form the family of supra pp-soft T j -spaces. We do this in our next definition: Firstly, in this section, we reveal the relationship between these separation axioms as well as their relationships with supra tt-soft T j and supra tp-soft T j -spaces.
Proof. We first prove (i). The proofs of the two cases j = 1 and j = 2 come directly from the definition above. To prove the case of j = 3, let Since k ∈ G M implies k G M , we obtain the proof of (ii).
The three following examples point out that the converses of the statements in Proposition 5 fail to hold true.

Example 5.
A supra-soft topology can be identified with a supra topology when M is a singleton, and then supra pp-soft T i -spaces coincide with supra T i -spaces. Therefore, Example 5.12 of [35] provides an example of a supra pp-soft T 3 -space that fails to be supra pp-soft T 4 .
As a direct consequence of Propositions 3 and 5, one obtains the next proposition: Proposition 6. Every supra tt-soft T j -space is a supra pp-soft T j−1 -space for j = 0, 1, 2, 4.
The two following examples guarantee that supra tt-soft T 3 and supra pp-soft T 3spaces are independent concepts. The next conclusion can be easily proven. Its proof is omitted.     Proof. We prove the claim when j = 3. The other cases follow similar lines. Let (Y, δ Y , M) be a subspace of (O, δ, M) which is supra pp-soft T 1 . First, we show that (Y, δ Y , M) is supra pp-soft T 1 . Let k = q ∈ Y. Then, δ contains two members G M and F M such that k G M , q ∈ G M and q F M , k ∈ F M . According to the definition of a subspace, To prove the supra pp-soft regularity of (Y, δ Y , M), let k ∈ Y and L M be a supra-soft closed subset of (Y, δ Y , M) such that k ∈ L M . Then, there exists a supra-soft closed subset Theorem 7. The property of being a supra-soft T 4 -space is a supra-soft closed hereditary property.

Proof. Obvious.
Theorem 8. The finite product of supra pp-soft T j -spaces is supra pp-soft T j for j = 0, 1, 2.
Proof. For j = 2. Without loss of generality, let (O, δ, M) and (Y, υ, N ) be supra pp-soft T 2 -spaces. Let us assume that (k 1 , q 1 ) = (k 2 , q 2 ) in O × Y. Then, k 1 = k 2 or q 1 = q 2 . Say, k 1 = k 2 . Then, there are two disjoint members U M , V N of δ such that k 1 U M and k 2 ∈ U M ; and k 2 V N and Proposition 12. Let f φ : (O, δ, M) → (Y, υ, N ) be a soft S -continuous mapping such that f and φ are, respectively, injective and surjective. If (Y, υ, N ) is supra pp-soft T j , then (O, δ, M) is supra pp-soft T j for j = 2, 1, 0.
Proof. When j = 2. Let k = q ∈ O. Then, there are only two points x = y ∈ Y with f (k) = x and f (q) = y because f is injective. Since (Y, υ, N ) is supra pp-soft T 2 , there are two disjoint members G N and F N of δ such that x G N , y ∈ G N and y F N , x ∈ F N . Since φ is surjective, Proposition 1 provides us with k For all j, it can be proven that the next results follow a similar argument; hence, we omit their proofs.
Proposition 15. The property of being a supra pp-soft T j -space is preserved under an S -homeomorphism map. (j = 0, 1, 2, 3, 4) This section introduces the novel separation axioms that jointly form the family of supra pt-soft T j -spaces. They are presented in Definition 23 below: (v) Supra pt-soft T 3 (resp. supra pt-soft T 4 ) if it is supra pt-soft regular (resp. supra-soft normal) and supra pt-soft T 1 .

Supra pt-Soft T j -Spaces
Proposition 16. The next properties are satisfied: (i) Supra pt-soft T j -spaces are supra pt-soft T j−1 for j = 1, 2.
(iii) Supra tt-soft T j -spaces are supra pt-soft T j for all j.
Proof. (i) is a direct consequence of the above definition.
Since k G M implies k ∈ G M , we obtain the proofs of (ii) and (iii).
We provide the following examples to illustrate that the converse of the above proposition is not always true. Example 11. It can be easily checked that (O, δ, M) displayed in Example 2 is a supra pt-soft T 0 -space; however, it is not supra pt-soft T 1 . In the next example, we clarify that the systematic relation T 3 implies that T 2 does not hold true for their counterparts: supra pt-soft T 2 and supra pt-soft T 3 .   The spaces of supra pp-soft regular (supra pt-soft regular, supra-soft normal) need not be pt-soft T j , tp-soft T j and tt-soft T j for all j. The example below points out this fact. Proof. We proceed with the argument for j = 0, the case j = 1 is analogous. Necessity: Let k = q. There must be a member G M of δ such that k ∈ G M and q ∈ G M , or q ∈ G M and k ∈ G M . Say, k ∈ G M and q ∈ G M . If q ∈ G(m) for all m ∈ M, then the proof finishes. Otherwise, we do not lose generality if we consider that there is m ∈ M such that q ∈ G(m) and q ∈ G(m ) for each m = m. Since Proof. By the stability of G M , we obtain k ∈ G M iff k G M , and k ∈ G M iff k G M . Hence, we obtain the required equivalences.

Corollary 5.
Let a supra-soft TS (O, δ, M) be a supra tp-soft regular. The following statements hold true: Now, we explain how supra pt-soft T j -spaces behave in their parametric spaces and vice versa. In fact, the next remark shows that there is no navigation for these axioms between soft and classical frames.

Remark 3.
In Example 4, we showed that (O, δ, M) is supra pt-soft T j for j = 2, 3, 4; nevertheless, its parametric supra topological spaces fail to be supra T 1 . In addition, (O, δ, M) defined by Example 2 is supra pt-soft T 0 ; however, its parametric supra topological space (O, δ m 2 ) is not supra T 0 .
In contrast, (O, δ, M) defined by Example 3 is not a supra pt-soft T 0 -space; however, its parametric supra topological spaces are supra T 4 . Proof. The cases j = 0, 1, 2 follow from the above theorem. In the cases of j = 3, 4, we are finished if we demonstrate the properties of supra pt-soft regular and supra-soft normal. We first prove that (O, δ, M) is supra pt-soft regular. Suppose that H M is a supra-soft closed set and that k ∈ H M . There must exist m ∈ M such that k H(m).  Proposition 21. The property of being a supra pt-soft T j -space for all j is preserved under Shomeomorphisms.

Conclusions and Future Work
Supra-soft topology has become one of the most remarkable developments in soft topology. It was born from the principle that weaker conditions sometimes suffice to ensure some valid properties. Furthermore, it can be connected to intuitionistic fuzzy supratopologies [37], because each soft fuzzy set, that is extension of the soft set is representable by an intuitionistic fuzzy set [38]. Since we may need to relax certain soft topological conditions to model some known phenomena, we expanded this area of expertise with the concepts of supra pp-soft T j and supra pt-soft T j -spaces (j = 0, 1, 2, 3, 4). They afford new classifications of supra-soft TSs . They have been formulated with respect to the distinct ordinary points and they are different in terms of the relations that associate points with soft sets. The first class uses partial belong and partial non-belong relations, whereas the second utilizes partial belong and total non-belong relations. With the help of examples, we have shown that these two types of spaces defined by supra-soft separation axioms are wider than those given in [33]. We have also explained the interrelations between them and their parametric supra topological spaces. Moreover, we have investigated whether subspaces, finite product spaces, and soft S -continuous mappings preserve these axioms.
In future works, we plan to compare among the different types of supra-soft separation axioms that have been introduced in the literature. We will also try to benefit from the classification obtained by the current axioms in decision-making problems, following techniques that may be similar to those given in [20]. Moreover, we shall investigate the concepts given herein using some generalizations of supra-soft open sets such as supra- To conclude, we provide Table 1 to show whether some topological properties are valid or invalid via supra-soft and infra-soft topologies. Moreover, we demonstrate in Table 2 how the classical systematic relationships among T i -spaces (i = 0, 1, 2, 3, 4) behave with respect to supra pp-soft T i , pt-soft T i , supra tp-soft T i and supra tt-soft T i .  Finite space is compact Need not be true Need not be True Need not be true Table 2. Description of some properties via classical supra topology and supra-soft topology.

Classical Properties under Comparison Supra pp-Soft T i Supra pt-Soft T i Supra tp-Soft T i Supra tt-Soft T i
Supra T i -space is supra T i−1 -space (i = 1, 2) True True True True Supra T 3 -space is supra T 2 -space True Need not be true True True Supra T 4 -space is supra T 3 -space Need not be true Need not be true Need not be true True