Two types of separation axioms on supra soft topological spaces

In 2011, Shabir and Naz [1] employed the notion of soft sets to introduce the concept of soft topologies; and in 2014, El-Sheikh and Abd El-Latif [2] relaxed the conditions of soft topologies to construct a wider and more general class, namely supra soft topologies. In this disquisition, we continue studying supra soft topologies by presenting two kinds of supra soft separation axioms, namely supra soft Ti-spaces and supra p-soft Ti-spaces for i = 0, 1, 2, 3, 4. These two types are formulated with respect to the ordinary points; and the difference between them is attributed to the applied non belong relations in their definitions. We investigate the relationships between them and their parametric supra topologies; and we provide many examples to separately elucidate the relationships among spaces of each type. Then we elaborate that supra p-soft Ti-spaces are finer than supra soft Ti-spaces in the case of i = 0, 1, 4; and we demonstrate that supra soft T3-spaces are finer than supra p-soft T3-spaces. We point out that supra p-soft Ti-axioms imply supra p-soft Ti−1, however, this characterization does not hold for supra soft Ti-axioms (see, Remark (3.30)). Also, we give a complete description of the concepts of supra p-soft Ti-spaces (i = 1, 2) and supra p-soft regular spaces. Moreover, we define the finite product of supra soft spaces andmanifest that the finite product of supra soft Ti (supra p-soft Ti) is supra soft Ti (supra p-soft Ti) for i = 0, 1, 2, 3. After investigating some properties of these axioms in relation with some of the supra soft topological notions such as supra soft subspaces and enriched supra soft topologies, we explore the images of these axioms under soft S*-continuousmappings. Ultimately, we provide an illustrative diagram to show the interrelations between the initiated supra soft spaces.


Introduction
The concept of soft sets [3] first appeared in 1999 as a virtual mathematical approach to overcome problems containing uncertainties or incomplete data. This approach is free from the difficulties that have troubled the existing mathematical approaches which utilized to solve problems associated with ambiguity such as fuzzy sets and rough sets. The theory of soft sets provides a general framework for modeling practical phenomena and brings about a rich potential for real life applications (see, for example, [4][5][6]).
Based on the soft set theory, the concept of soft topologies [1] was defined in 2011 as a new soft structure. Later on, many studies were carried out to study the properties of soft topologies and to compare them with those properties which exist via classical topologies (see, for example, [7][8][9][10]). In particular, the study of soft separation axioms was a great interest to researchers who studied this topic from many perspectives (see, for example, [11][12][13][14]). Recently, [15][16][17][18] corrected some errors related to soft separation axioms. The variety of the study of the soft separation axioms is attributed to twofold: One is the distinguished objects that we desire to separate: Are they soft points or ordinary points?, and the second one is the nature of the belong and non belong relations which we utilized in the definitions: Are they natural belong or natural non belong or partial belong or total non belong relations?
The concept of supra topologies [19] was introduced in 1983 as a wider class of general topologies. Since then several studies were done on the supra topologies and their applications (see, for example, [20][21][22][23][24][25]). In connection to generalizing the crisp mathematical structures to the soft mathematical structures, the concept of supra soft topologies [2] was established in 2014 as an extension of supra topologies. Some fundamental notions via supra soft topologies such as supra soft continuity [2], supra soft compactness [26], various types of generalized supra soft open sets and supra soft separation axioms [27][28][29][30][31] were introduced and investigated. In 2017, Hosny and Al-Kadi [32] generated a supra soft topology from a soft topology by utilizing a soft stack. It turns out that the study of the properties of supra soft topologies has not yet taken its due attention and many fundamental supra soft topological notions need to be formulated and discussed.
It should be noted that the existing supra soft separation axioms were studied using the notion of soft points, so the current work aims to introduce two types of supra soft separation axioms with respect to ordinary points, namely supra soft T i -spaces and supra p-soft T i -spaces for i = 0, 1, 2, 3, 4. One of the motivations to study supra soft T i -spaces is to establish a wider family which can be easily applied to classify the objects of study; and one of the motivations to study supra p-soft T i -spaces is to point out the significant role of a total non belong relation in obtaining results similar to the ones achieved via supra topologies, and relaxing the conditions on the concept of supra soft regular spaces.
The layout of this article is as follows: In Section 1, we recall some definitions and properties of soft sets and supra soft topologies. In Section 2, we introduce the notions of supra soft T i -spaces (i = 0, 1, 2, 3, 4); and study some properties. In Section 3, we introduce the notions of supra p-soft T i -spaces (i = 0, 1, 2, 3, 4); and investigate some properties. Also, we elucidate their relationships to supra soft separation axioms which are presented in Section 2. Finally, we give a conclusion and make a plan for future works.

Preliminaries
In this section, we recall some definitions and results as they are important for the material of our study. Definition 2.1. [3] A pair (G, E) is said to be a soft set over a non-empty set X provided that G is a mapping of a set of parameters E into 2 X . In this work, a soft set is denoted by G E instead of (G, E) and it is identified as G E = {(e, G(e)) : e ∈ E and G(e) ∈ 2 X }. And the set of all soft sets over X under a parameters set E, is denoted by S(X E ). Definition 2.2. [33] The relative complement of a soft set G E , denoted by G c E , is given by G c E = (G c ) E , where G c : E → 2 X is a mapping defined by G c (e) = X \ G(e) for each e ∈ E. Definition 2.3. [34] A soft set G E over X is said to be a null soft set, denoted bỹ︀ Φ, if G(e) = ∅ for each e ∈ E. And its relative complement is said to be an absolute soft set, denoted bỹ︀ X.  In the literature, there are many types of soft subsets, soft unions and soft intersections between two soft sets introduced and studied. For more details on this subject, we refer the reader to [36,37] and the references mentioned therein. Definition 2.15. [14] A soft set G E over X is said to be stable if there exists a subset S of X such that G(e) = S for each e ∈ E. Proposition 2.16. [14] Consider a soft mapping f ϕ : S(X A ) → S(Y B ) and let G A and H B be soft sets in S(X A ) and S(Y B ), respectively. Then the following statements hold. Proof. Straightforward.
The converse of the above proposition is not necessarily true as illustrated in the following example.  Proof. Straightforward.
Henceforth, we use the term a parametric supra topology for µe which given in the above proposition. Proof. Let x ≠ y ∈ X. Since (X, µ, E) is a supra soft T 2 -space, then there exist disjoint supra soft open sets F E and G E containing x and y, respectively. Therefore, F(e) and G(e) are disjoint supra soft open sets such that x ∈ F(e) and y ∈ G(e). This shows that (X, µe) is a supra T 2 -space.
To see that the converse of the above proposition is not necessarily true, we give the following example. Example 3.6. Assume that X and E are the same as in Example (3.3). Let the two soft sets F E , G E over X be given as follows: Then µ = {̃︀ Φ,̃︀ X, F E , G E } is a supra soft topology on X. Since there does not exist a proper supra soft open subset of̃︀ X containing x or y, then (X, µ, E) is not a supra soft T 0 -space. On the other hand, the two parametric supra topological spaces (X, µe 1 ) and (X, µe 2 ) are supra T 2 .
We draw the attention of the reader to the fact that there is no a relationship between supra soft T i -spaces and their parametric supra topological spaces in the case of i = 0, 1. This matter can be clarified in the following example.
Example 3.7. One of the implications is illustrated in the above example. In Example (3.3), it can be noted that (X, µ 1e 2 ) is not supra T 0 in spite of (X, µ 1 , E) being supra soft T 0 . And (X, µ 2e 1 ) and (X, µ 2e 2 ) are not supra T 1 in spite of (X, µ 2 , E) being supra soft T 1 .

Definition 3.8. A supra soft topological space (X, µ, E) is said to be stable if every supra soft open set is stable.
Proposition 3.9. If (X, µ, E) is stable, then it is supra soft T i if and only if a parametric supra topological space (X, µe) is supra T i for each i = 0, 1, 2.
Proof. We only prove the proposition in the case of i = 0. The other cases follow similar lines. Necessity: Suppose that x ≠ y in a supra soft T 0 -space (X, µ, E). Then there exists a supra soft open subset G E of (X, µ, E) such that is a supra open subset of (X, µe) and x ∈ G(e) for each e ∈ E. Since (X, µ, E) is stable, then y ∉ G(e) for each e ∈ E. Thus, (X, µe) is supra T 0 . Sufficiency: Let x ≠ y in a supra T 0 -space (X, µe). Then there exists a supra open subset G of (X, µe) such that x ∈ G, y ∉ G or y ∈ G, x ∉ G. Say, x ∈ G, y ∉ G. By the stability of (X, µ, E), there is a supra soft open subset M E of (X, µ, E) such that M(e) = G for each e ∈ E. Hence, (X, µ, E) is supra soft T 0 .
The following result can be proved easily. Example (3.6) clarifies that the converse of the above proposition is not necessarily true. Proposition 3.11. If x E is a supra soft closed subset of (X, µ, E) for each x ∈ X, then (X, µ, E) is a supra soft T 1 -space.
Proof. Let x ≠ y ∈ X and let the given condition be satisfied. Then x c E and y c E are supra soft open sets such that y ∈ x c E and x ∈ y c E . Clearly, x ∉ x c E and y ∉ y c E . Hence, (X, µ, E) is a supra soft T 1 -space. In the next example, we point out that the converse of the above proposition need not be true in general. Example 3.12. Assume that (X, µ 2 , E) is the same as in Example (3.3). It can be seen that x E and y E are not supra soft closed sets in spite of (X, µ, E) being a supra soft T 1 -space. Proposition 3.13. If (X, µ, E) is a supra soft T 2 -space, then x E is supra soft closed for each x ∈ X.
Proof. For each y i ∈ X∖{x}, it follows that there exist disjoint supra soft open sets G iE and F iE such that x ∈ G iE and y i ∈ F iE . Therefore, X∖{x} = ⋃︀ i∈I F i (e) for each e ∈ E. Thus,̃︀ The converse of the above proposition fails as demonstrated in the next example. Example 3.14. Assume that X = {x, y, z} and E = {e 1 , e 2 }; and let the three soft sets F E , G E , H E over X be given as follows: Hereafter, we define supra soft regular spaces and characterize them.  Proof. Necessity: In view of the Definition (2.13) and Definition (2.14), we find that x ∈ G E implies x G E for any soft set G E . Sufficiency: Consider G E is a supra soft open set and let x G E . Then we have two cases: The sufficient condition in the case of G E being a supra soft closed set can be proven in a similar way.
The converse of the above proposition need not be true in general as shown in the next example.
: Let x ≠ y ∈ X and (X, µ, E) be a supra soft T 0 -space. Then there exists a supra soft open set G E such that x ∈ G E and y ∉ G E , or y ∈ G E and x ∉ G E . Say, x ∈ G E and y ∉ G E . By Proposition (3.17), we find that y ̸ G E . Now, we have x ̸ G c E and y ∈ G c E . Since (X, µ, E) is supra soft regular, then there exist two disjoint supra soft open sets W 1 E and W 2 E such that x ∈ W 1 E and y ∈ G c Ẽ︀ ⊆W 2 E . This completes the proof.

Definition 3.24.
A supra soft topological space (X, µ, E) is said to be: (i) a supra soft T 3 if it is both supra soft regular and supra soft T 1 .
(ii) a supra soft T 4 if it is both supra soft normal and supra soft T 1 .
The converse of the above proposition need not true in general as shown in the next example.
Proof. Since (X, µ, E) is supra soft T 3 , then it is stable. So, from Proposition (3.9), we find that (X, µe) is supra T 1 . It remains to prove the supra regularity of (X, µe). Suppose that H is a supra closed subset of X such that x ∉ H. By Corollary (3.18), there exists a supra soft closed set F E such that F(e) = H for each e ∈ E.
Obviously, x ∉ F E . By hypothesis, there exist two disjoint supra soft open sets G E and W E such that x ∈ G E and F Ẽ︀ ⊆W E . This means that G(e) and W(e) are two disjoint supra open subsets of (X, µe) such that x ∈ G(e) and F(e) = H ⊆ W(e). Thus, (X, µe) is supra regular. Hence, (X, µe) is supra T 3 .
The converse of the above theorem is not necessarily true as illustrated in the following example.
Example 3.29. Assume that (X, µ, E) is the same as in Example (3.27). Then the two parametric supra topologies µe 1 (3.27). We clarify that (X, µ, E) is supra soft T 2 , but it is not supra soft T 3 . On the other hand, the collection of all supra soft closed subsets of (X, µ, E) is Example 3.32. Assume that X = {v, w, x, y, z} and E = {e 1 , e 2 }; and let the soft sets {F iE : i = 1, 2, ..., 23} over X be given as follows: Proof. Suppose that (X, µ, E) is a stable supra soft T 4 -space. Then it follows from Proposition (3.9), that (X, µe) is supra soft T 1 . Let F and H be two disjoint supra closed subsets of (X, µe). Since (X, µ, E) is stable, then there exist two disjoint supra soft closed subsets U E and V E of (X, µ, E) such that U(e) = F and V(e) = H for each e ∈ E. By hypothesis, there exist two disjoint supra soft open subsets G E and W E of (X, µ, E) such that U Ẽ︀ ⊆G E and V Ẽ︀ ⊆W E . This implies that G(e) and W(e) are disjoint supra open subsets of (X, µe) containing F and H, respectively. Thus, (X, µe) is supra normal. Hence, it is supra T 4 .
To prove the converse, let x ≠ y ∈ X. Since (X, µe) is supra T 4 , then there are two supra open sets G and W containing x and y, respectively, such that x ∉ W and y ∉ G; and since (X, µ, E) is stable, then there are two supra soft open sets M E and N E such that M(e) = G and N(e) = W for each e ∈ E. This means that (X, µ, E) is supra soft T 1 . Suppose that U E and V E are two non-null disjoint supra soft closed subsets of (X, µ, E). By the stability of (X, µ, E), it follows that U(e) and V(e) are two non-empty disjoint supra closed subsets of (X, µe) for each e ∈ E. Since (X, µe) is supra normal, then there are two disjoint supra open sets G and W containing U(e) and V(e), respectively. From Corollary (3.19), it follows that the soft sets M E and N E , which are defined as M(e) = G and N(e) = W for each e ∈ E, are disjoint supra soft open sets containing U E and V E , respectively. Thus, (X, µ, E) is supra soft normal. Hence, it is supra soft T 4 . Theorem 3.34. Every soft subspace (Y , µ Y , E) of a supra soft T i -space (X, µ, E) is a supra soft T i -space, for i = 0, 1, 2, 3.
Proof. We prove the theorem in the case of i = 3 and the other cases follow similar lines. To To prove the supra soft regularity of (Y , µ Y , E), let y ∈ Y and L E be a supra soft closed subset of (Y , µ Y , E) such that y ∉ L E . Then there exists a supra soft closed subset H E of (X, µ, E) such that L E =̃︀ Ỹ︀ ⋂︀ H E . Since y ∉ H E , then there exist disjoint supra soft open sets G E and F E such that H Ẽ︀ ⊆G E and y ∈ F E . Now, we find that Proof. By the above theorem, a supra soft closed subspace (Y , µ Y , E) of a supra soft T 4 -space (X, µ, E) is supra soft T 1 . To prove the supra soft normality of (Y , µ Y , E), let H 1 E and H 2 E be two disjoint supra soft closed subsets of (Y , µ Y , E). Since (Y , µ Y , E) is a supra soft closed subspace of (X, µ, E), then H 1 E and H 2 E are two disjoint supra soft closed subsets of (X, µ, E) as well. Since (X, µ, E) is supra soft normal, then there are two disjoint supra soft open subsets respectively. This completes the proof.
Proof. We prove the theorem for two supra soft topological spaces in the case of i = 0, 3. The other cases follow similar lines.
is supra soft regular. From (i) above and Theorem (3.23), the desired result is proved.

Remark 3.39. It is well known that a supra soft topological space (X, µ, E) is supra topological space if a set of parameters E is a singleton. We also know, from general topology, that a Sorgenfrey Line space is normal, however, the product of two Sorgenfrey Line spaces is not normal. This explains why supra soft T 4 -spaces are excluded from the above theorem.
For the sake of brevity, we omit the proofs of the following four results.

Supra p-soft T i -spaces (i = 0, 1, 2, 3, 4)
The main purpose of this section is to define and study the concepts of supra p-soft T i -spaces (i = 0, 1, 2, 3, 4) by using natural belong and total non belong relations. With the help of examples, the relationships between them, as well as between them and supra soft T i -spaces introduced in the previous section are illustrated.

Remark 4.2. For any two disjoint soft sets G E and F E such that x ∈ G E and y ∈ F E , we find that y ̸ G E and x ̸ F E if and only if y ∉ G E and x ∉ F E . So the definitions of supra p-soft T 2 -spaces and supra soft T 2 -spaces are equivalent. Hence, the properties of a supra soft T 2 -spaces are still valid for a supra p-soft T 2 -spaces.
Proposition 4.3. Every supra p-soft T i -space is a supra soft T i -space, for i = 0, 1.
Proof. It follows from the fact that a total non belong relation ̸ implies a non belong relation ∉ .
The next example points out that the converse of the above proposition is not true. Proof. In the case of i = 2, the proof follows from Proposition (3.5) and Remark (4.2).
In the case of i = 1, let x ≠ y ∈ X. Since (X, µ, E) is a supra soft T 1 -space, then there exist two supra soft open sets F E and G E containing x and y, respectively, such that y ̸ F E and x ̸ G E . Thus, F(e) and G(e) are non-empty proper supra soft open subsets of (X, µe) containing x and y, respectively, such that y ∉ F(e) and x ∉ G(e). This shows that (X, µe) is a supra T 2 -space.
One can similarly prove the proposition in the case of i = 0.
Example (3.6) obviously shows that the converse of the above proposition does not hold.
We discuss in the following results some properties related to a p-soft T 0 -space. Proof. Let x, y be two distinct points in a supra p-soft T 0 -space. Then there is a supra soft open set G E such that x ∈ G E and y ̸ G E or y ∈ G E and x ̸ G E . Say, x ∈ G E and y ̸ G E . Now, y Ẽ︀ ⋂︀ G E =̃︀ Φ. So, by the above lemma, x ̸ Clµ(y E ). But x ∈ Clµ(x E ). This shows that Clµ(x E ) ≠ Clµ(y E ).
Proof. From the proof of Theorem(4.7), we have x ̸ Clµ(y E ) or y ̸ Clµ(x E ) for each x ≠ y. Say, x ̸ Clµ(y E ).
As Clµ(P y β )̃︀ ⊆Clµ(y E ), then x ̸ Clµ(P y β ). Obviously, x Clµ(P x α ). Hence, Clµ(P x α ) ≠ Clµ(P y β ), as required. It can be seen from the next example that the converse of the above theorem fails. In the following results, we characterize a supra p-soft T 1 -space and investigate some of its features.

Theorem 4.10. (X, µ, E) is a supra p-soft T 1 -space if and only if x E is a supra soft closed set for all x ∈ X.
Proof. Necessity: For each y i ∈ X∖{x}, there is a supra soft open set G i E such that y i ∈ G i E and x ̸ G i E . Therefore, X∖{x} = ⋃︀ i∈I G i (e) and x ̸ ⋃︀ i∈I G i (e) for each e ∈ E. Thus,̃︀ ⋃︀ i∈I G i E = X∖{x} is a supra soft open set. Hence, x E is supra soft closed. Sufficiency: For each x ≠ y, we have x E and y E are supra soft closed sets. Now, ∈ (x E ) c and (y E ) c are two supra soft open sets containing y and x, respectively. Since x ̸ (X∖{x}) E and y ̸ (X∖{y}) E , then (X, µ, E) is a supra p-soft T 1 -space. Proof. Let X be the universe set with | X |≥ 2 and E be a set of parameters. Then the collection θ = {̃︀ Φ,̃︀ X, G iE such that G i (e) = X \ {x i }, for each e ∈ E : i = 1, 2, ..., | X |} forms a supra soft topology on X. The number of all non-null proper supra soft open subsets of (X, θ, E) is . Take arbitrary distinct points x i , x j ∈ X. The soft sets F E and H E defined as follows: x i ∈ X} are supra soft open sets containing x i and x j , respectively, such that x i ̸ H E and x j ̸ F E . This proves that (X, θ, E) is a supra soft T 1 -space. To prove that it is the smallest supra soft T 1 , remove any non-null proper supra soft open set. Say, G 1E . Then x 1 E is not a supra soft closed set. This contradicts that (X, θ, E) is supra soft T 1 . Hence, the desired result is proved.
Proof. For each y ∈ X such that x ≠ y, there exists a supra soft open set G E such that x ∈ G E and y ̸ G E . So x ∈̃︀ ⋂︀ G E and y ̸̃︀ ⋂︀ G E . Hence, the desired result is proved. Thus, ( X∖{x}̃︀ ⋃︀ H E ) c = P x e is a supra soft closed set.

Corollary 4.15. If (X, µ, E) is an enriched supra p-soft T 1 -space, then the intersection of all supra soft open sets containing U E is exactly U E for each U Ẽ︀ ⊆̃︀ X.
Proof. Let U E be a soft subset of̃︀ X and P x e ∈ U c E . Since P x e is supra soft closed, theñ︀ X \ P x e is a supra soft open set containing U E . By doing similarly for each P x e ∈ U c E , the corollary holds. The following two results demonstrate that the condition of supra soft regularity guarantees the equivalence between the introduced supra soft separation axioms for i = 0, 1, 2. Proof. The proof of the "if" part follows directly from Proposition(3.2) and Remark(4.2).
The proof of the "only if" part follows directly from Theorem(3.23) and Remark(4.2).

Corollary 4.17.
Let (X, µ, E) be a supra soft regular space. Then the following five statements are equivalent. Proof. The proof follows from the fact total non belong relation ̸ obviously implies natural non belong relation ∉ .
The next example elucidates that the converse of the above proposition fails.

Proposition 4.28. Every supra soft T 3 -space is a supra p-soft T 3 -space.
Proof. One can obtain the proof from Corollary (4.17) and Proposition (4.20).
It can be seen from Example (3.27) that the converse of the above proposition need not be true in general.

Proposition 4.29. Every supra p-soft T 4 -space is a supra soft T 4 -space.
Proof. Straightforward.
To show that the converse of the above proposition fails, we give the next example. Then (X, µ, E) is a supra soft T 4 -space. On the other hand, there does not exist a supra soft open set containing y such that x does not totally belong to it. So (X, µ, E) is not a supra p-soft T 1 -space, hence, it is not a supra p-soft T 4 -space. Now, we elucidate a relationship between supra p-soft T i -spaces and deduce some results which associate them with some soft topological notions such as supra soft subspaces and product supra soft spaces. Proof. We prove the proposition in the case of i = 3, 4. The other cases follow similar lines. For i = 3, let x, y be two distinct points in a supra p-soft T 3 -space (X, µ, E). Then (X, µ, E) is supra p-soft T 1 . So x E is supra soft closed. Since y ̸ x E and (X, µ, E) is supra p-soft regular, then there are disjoint supra soft open sets G E and F E such that x Ẽ︀ ⊆G E and y ∈ F E . Therefore, (X, µ, E) is a supra p-soft T 2 -space. For i = 4, let x ∈ X and H E be a supra soft closed set such that x ̸ H E . Since (X, µ, E) is supra p-soft T 1   Proof. We prove the theorem for two supra soft topological spaces in the case of i = 0, 3. The other cases follow similar lines.
(i) Consider two supra p-soft T 0 -spaces (X 1 , µ 1 , A 1 ) and (X 2 , µ 2 , A 2 ) and let (x 1 , y 1 ) ≠ (x 2 , y 2 ) in (X 1 × X 2 , µ 1 × µ 2 , A 1 × A 2 ). Without loss of generality, let x 1 ≠ x 2 . Then there exists a supra soft open subset G A1 of (X 1 , µ 1 , A 1 ) such that x 1 ∈ G A1 and x 2 ̸ G A1 or x 2 ∈ G A1 and x 1 ̸ G A1 . Say, x 1 ∈ G A1 and x 2 ̸ G A1 . Therefore, (x 1 , y 1 ) ∈ G A1 ×̃︁ X 2 and (x 2 , y 2 ) ̸ G A1 ×̃︁ X 2 . Thus, (X 1 × X 2 , µ 1 × µ 2 , A 1 × A 2 ) is a supra p-soft T 0 -space. (ii) Let H A1×A2 be a supra soft closed subset of a supra soft space ( Proof. We prove the theorem in the case of i = 3 and the other cases follow similar lines. To prove that (Y , µ Y , E) is supra p-soft T 1 , let x ≠ y ∈ Y. Since (X, µ, E) is a supra p-soft T 1 -space, then there exist supra soft open sets G E and F E such that x ∈ G E , y ̸ G E , y ∈ F E , and x ̸ F E . Therefore, To prove the supra p-soft regularity of (Y , µ Y , E), let y ∈ Y and L E be a supra soft closed subset of (Y , µ Y , E) such that y ̸ L E . Then there exists a supra soft closed subset H E of (X, µ, E) such that L E =̃︀ Ỹ︀ ⋂︀ H E and y ̸ H E . Therefore, there exist disjoint supra soft open sets G E and F E such that H Ẽ︀ ⊆G E and y ∈ F E . Now, we find that  Proof. We only prove the proposition in the case of i = 1. The other cases follow similar lines.
Let x ≠ y ∈ X. Since f is injective, then f (x) ≠ f (y) ∈ Y. By hypothesis, there are two supra soft open subsets G B and W B of (Y , θ, B) such that f (x) ∈ G B , f (y) ̸ G B and f (y) ∈ W B , f (x) ̸ W B . It follows from Proposition (2.16), that ). Since f ϕ is soft S * -continuous, then f −1 ϕ (G B ) and f −1 ϕ (W B ) are two supra soft open subsets of (X, µ, A). Hence, (X, µ, A) is supra p-soft T 1 For the sake of brevity, we omit the proofs of the following four results.    In the end of this work, we summarize the relationships between the two introduced supra soft separation axioms in the following diagram.

Conclusion and future work
Soft separation axioms are among the most widespread, important and interesting concepts in soft topology. They can be utilized to construct more restricted classes of soft topological spaces. This, as well as the reasons mentioned in the penultimate paragraph of the introduction section, prompt us to study separation axioms of supra soft topologies in this paper. So we define the concepts of supra soft T i -spaces (i = 0, 1, 2, 3, 4). The fact that we cannot adapt all claims concerning supra topologies to supra soft topologies by using supra soft T i -spaces (see, Example (3.12), Example (3.22), Remark (3.30)) leads us to define supra p-soft T i -spaces (i = 0, 1, 2, 3, 4) which preserve more properties of supra topologies on supra soft topologies (see, Theorem (4.10), Theorem (4.22), Proposition (4.31)). With the help of examples, we explore the relationships between supra   soft T i and between supra p-soft T i ; and we study the relationships between supra soft T i and supra p-soft T i . Also, we explain the relationships between them and their parametric supra topologies. We further discuss some essential properties of the initiated supra soft axioms. In particular, we characterize the concepts of supra p-soft T i -spaces (i = 1, 2) and supra p-soft regular spaces. We define new notions such as enriched supra soft topologies, supra soft basis and soft S * -continuous mappings; and conclude some results which relate them to some supra soft separation axioms initiated herein. Eventually, we plan to do the following in the upcoming papers: (i) Define a concept of supra soft topological ordered spaces in a similar way to the concept of soft topological ordered spaces [38]. (ii) Study a concept of soft lattices [39] with respect to the partial order relations given in [38]. (iii) Investigate the possibility for applications of these two types of supra soft separation axioms in digital and approximation spaces and decision making problems. (iv) Define a notion soft somewhere dense sets [40] via supra soft topological spaces and generalize supra soft separation axioms initiated herein with respect to this.