1. 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 supra-topology. 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
-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
-spaces. Furthermore, the supra-topological frames show the easiness and diversity of building examples that satisfy supra
-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
-soft
-space implies that every soft point is supra-soft closed, as proven in [
33], whereas this characteristic does not hold for spaces of supra
-soft
and supra
-soft
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
-soft
-spaces
, and investigates their main properties. In
Section 4, we formulate the concepts of supra
-soft
-spaces
. Then, we disclose some relationships among them, as well as with respect to supra
-soft
-spaces
. To end this paper,
Section 5 contains some conclusions and hints at some directions for future research.
3. Supra -Soft -Spaces
In this section, we introduce the novel separation axioms that jointly form the family of supra -soft -spaces. We do this in our next definition:
Definition 22. The supra-soft is called:
- (i)
Supra -soft if there exists a member for every that satisfies , or .
- (ii)
Supra -soft if there exist two members for every that satisfy , and .
- (iii)
Supra -soft if there exist disjoint members for every that satisfy , and .
- (iv)
Supra -soft regular when for each supra-soft closed set such that , disjoint supra-soft open sets and exist with the properties and .
- (v)
Supra -soft (respectively, supra -soft ) if it is supra -soft regular (respectively, supra-soft normal) and supra -soft .
Firstly, in this section, we reveal the relationship between these separation axioms as well as their relationships with supra -soft and supra -soft -spaces.
Proposition 5. - (i)
Every supra -soft -space is supra -soft for .
- (ii)
Every supra -soft -space is supra -soft for .
Proof. We first prove (i). The proofs of the two cases and come directly from the definition above. To prove the case of , let . Since is supra -soft , then we have two members and of that satisfy , and , . Now, . Since is supra -soft regular, then we have disjoint supra-soft open sets and that satisfy , and . It is clear that . Since and are disjoint, then and ; therefore, is supra -soft .
Since implies , 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 1. Let denote a set of parameters. Consider a universe and the following soft sets over :
,
, and
.
Now, notice that the family defines a supra-soft topology on . Upon inspection, one can check that is supra -soft . However, disjoint supra-soft open sets do not exist. Therefore, is not supra -soft .
Example 2. For , let be a soft set on . Then, is a supra-soft topology on . It can be easily checked that is supra -soft ; however, it is not a supra -soft .
Example 3. Over the universal set with a parameters set , the next soft sets are defined.
and
.
The collection defines a supra-soft topology on . As a matter of fact, is both supra -soft and supra -soft . In contrast, there exist no supra open sets totally containing either a or q. Therefore, is not supra -soft ; also, it is not supra -soft .
The next two examples explain that supra -soft and supra -soft -spaces are independent of each other.
Example 4. Let the next soft sets be defined on the universal set and a set of parameters :
;
;
.
The collection defines a supra-soft topology on . Although is -soft , it is not supra -soft .
Example 5. A supra-soft topology can be identified with a supra topology when is a singleton, and then supra -soft -spaces coincide with supra -spaces. Therefore, Example 5.12 of [35] provides an example of a supra -soft -space that fails to be supra -soft . As a direct consequence of Propositions 3 and 5, one obtains the next proposition:
Proposition 6. Every supra -soft -space is a supra -soft -space for .
The two following examples guarantee that supra -soft and supra -soft -spaces are independent concepts.
Example 6. Reconsider studied in Example 3. Although is -soft , it is not supra -soft .
Example 7. Let the next soft sets be defined on the universal set and a set of parameters :
;
;
;
;
;
.
Then, defines a supra-soft topology on . It can be noted that is supra -soft . In contrast, is a supra-soft closed set such that . Because there do not exist two disjoint supra-soft open sets that satisfy the condition of a supra -soft regular space, we conclude that is not supra -soft .
The next conclusion can be easily proven. Its proof is omitted.
Proposition 7. When is a supra- soft such that , the following properties hold true:
- (i)
If is a supra -soft -space (supra -soft -space), then it includes one proper supra-soft open set at least.
- (ii)
If is a supra -soft -space (supra -soft -space), then it includes two proper supra-soft open sets at least.
Proposition 8. If is a supra-soft open set in such that and its complement are full soft sets, then is supra -soft .
Proof. Let . Since is a full soft set, then and ; and since is also full, then and . By hypothesis, is a supra-soft open set; hence, is a supra -soft -space. □
Corollary 1. If is a supra-soft open subset of such that is also a partition soft set, then is supra -soft .
Proposition 9. If and are full supra-soft clopen sets in , then is supra -soft .
Proof. Let . By hypothesis, since and are full soft sets, then and ; and and . By hypothesis, is a supra-soft clopen set; hence, is a supra -soft -space. □
Corollary 2. If is a supra-soft clopen subset of such that is partition soft set, then is supra -soft .
In what follows, we establish some results that associate supra -soft -spaces with supra -spaces in the parametric supra topological spaces.
Theorem 2. If is supra , then is a supra -soft -space for .
Proof. When , let . Since is supra . Then, there are two members in such that , and , . This necessitates the existence of two members in satisfy and . Note that , , and , . Hence, is supra -soft .
A similar technique is followed to prove . □
Remark 2. From Example 1, we see that is supra -soft , but , and , which represent parametric supra topological spaces, are not supra .
The spaces of supra -soft and supra (in the parametric supra topological spaces) are not navigated in the cases of . The following example explains this fact.
Example 8. Let there be the two next soft sets over with a parameter set as follows:
;
.
Then, is a supra-soft topology on . Now, is supra -soft and supra -soft . However, and are not supra .
Example 9. Assume that is the supra-soft of Example 7. It was demonstrated that is not supra -soft . Furthermore, it is not a supra -soft -space because disjoint supra-soft open sets containing the disjoint supra-soft closed sets and do not exist. On the other hand, and are supra .
Example 10. Consider the next four soft sets over with a parameter set as follows: Then, is a supra-soft topology on . It is clear that is not supra -soft ; however, and are supra .
Theorem 3. Let be extended. If all are supra , then is supra -soft for .
Proof. The theorem is demonstrated in Theorem 2 when . For the proof of the other cases, it suffices to prove the properties of supra -soft regular and supra-soft normal.
First, assume that is a supra-soft closed set and let . Then, for a parameter . Since is a supra closed set, and is supra regular, we find two disjoint members such that and . By the extension of , there exist two members and of defined as follows
and for all the others
and for all the others
This directly leads to that and being disjoint such that and , which ends the proof that is supra -soft regular.
Second, we show that is supra-soft normal. To do this, let be disjoint supra-soft closed sets. This leads to obtain and as disjoint supra closed sets for every . Since is supra normal, then there exist two disjoint supra open sets U and V such that and . Again, by the extension of , there exist two members of defined as follows
and for all the others
and for all the others
Now, and are disjoint supra-soft open sets such that and . Thus, is supra-soft normal. Hence, it is supra -soft . □
The converse of the above theorem does need to be not true as illustrated in Example 8.
Proposition 10. An extended supra-soft is supra -soft .
Proof. Let and let and be given by the following formulas
, when
for all and .
By the extension of , and are disjoint supra-soft open sets such that , and . Hence, we prove the desired result. □
Example 3 demonstrates that the converse of the above proposition fails.
Theorem 4. Let be stable. Then, is a supra -space if and only if is a supra -soft -space for all j.
Proof. By the stability of , we obtain U as a member of if and only if is a member of . Hence, the proof is complete. □
Corollary 3. Let be supra -soft regular. Then, is supra iff is supra -soft for all j.
Proposition 11. is a supra -soft regular space if and only if, when and is a supra-soft open set partially containing a, a supra-soft open set exists such that .
Proof. Necessity: Let and be a supra-soft open set that partially contains a. Then, is a supra-soft closed set and . By hypothesis, there are disjoint supra-soft open sets and such that and . Obviously, . Thus, .
Sufficiency: Let be a supra-soft closed set. Suppose that satisfies . Then, . By hypothesis, there exists a supra-soft open set such that . Obviously, . By the disjointness of and , the desired finding is obtained. □
Theorem 5. When a supra-soft is supra -soft regular, the following spaces are identical:
- (i)
is supra -soft .
- (ii)
is supra -soft .
- (iii)
is supra -soft .
Proof. By Proposition 5, the directions are proven.
To prove , let . Since is supra -soft , then we have a supra-soft open set such that and , or and . Say, and . It is clear that and . Since is supra -soft regular, we can assure the existence of two disjoint supra-soft open sets and that satisfy both and . Since and are disjoint, then and . This ends the proof that is supra -soft . □
Theorem 6. For each , supra-soft is a supra-soft hereditary property.
Proof. We prove the claim when . The other cases follow similar lines.
Let be a subspace of which is supra -soft . First, we show that is supra -soft . Let . Then, contains two members and such that and . According to the definition of a subspace, we have and . Furthermore, and . Thus, is supra -soft .
To prove the supra -soft regularity of , let and be a supra-soft closed subset of such that . Then, there exists a supra-soft closed subset of such that . Since , there exist disjoint members and of such that and . Now, we find that and and . Thus, is supra -soft regular. Hence, is supra -soft . □
Theorem 7. The property of being a supra-soft -space is a supra-soft closed hereditary property.
Theorem 8. The finite product of supra -soft -spaces is supra -soft for .
Proof. For . Without loss of generality, let and be supra -soft -spaces. Let us assume that in . Then, or . Say, . Then, there are two disjoint members of such that and ; and and . Now, and are supra-soft open sets such that and , and and . Since , then is supra -soft . □
Proposition 12. Let be a soft -continuous mapping such that f and ϕ are, respectively, injective and surjective. If is supra -soft , then is supra -soft for .
Proof. When . Let . Then, there are only two points with and because f is injective. Since is supra -soft , there are two disjoint members and of such that , and , . Since is surjective, Proposition 1 provides us with , and , . By the soft -continuity of , we obtained and as supra-soft open sets. Obviously, they are disjoint as well. Hence, is supra -soft . □
For all j, it can be proven that the next results follow a similar argument; hence, we omit their proofs.
Proposition 13. Let be a bijective soft -continuous. If is supra -soft , then is supra -soft .
Proposition 14. Let be a bijective soft -open. If is supra -soft , then is supra -soft .
Proposition 15. The property of being a supra -soft -space is preserved under an -homeomorphism map.
4. Supra -Soft -Spaces
This section introduces the novel separation axioms that jointly form the family of supra -soft -spaces. They are presented in Definition 23 below:
Definition 23. A supra-soft is said to be:
- (i)
Supra -soft if there exists a member for every satisfies , or .
- (ii)
Supra -soft if there exist two members for every that satisfy , and .
- (iii)
Supra -soft if there exist disjoint members for every that satisfy , and .
- (iv)
Supra -soft regular if for every supra-soft closed set such that , there exist disjoint members such that and .
- (v)
Supra -soft (resp. supra -soft ) if it is supra -soft regular (resp. supra-soft normal) and supra -soft .
Proposition 16. The next properties are satisfied:
- (i)
Supra -soft -spaces are supra -soft for .
- (ii)
Supra -soft -spaces are supra -soft for .
- (iii)
Supra -soft -spaces are supra -soft for all j.
Proof. (i) is a direct consequence of the above definition.
Since implies , 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 displayed in Example 2 is a supra -soft -space; however, it is not supra -soft .
Example 12. Let the next soft sets be defined over and , where and
;
;
;
;
and
;
Now, the above six soft sets plus the absolute and null soft sets form a supra-soft topology δ on . Note that is a supra -soft . In contrast, there are no disjoint members of δ satisfying a condition of supra -soft for , which means that is not supra -soft .
Example 13. Example 8 shows that is both supra -soft and supra -soft . On the other hand, there is no member in δ (except for the null soft set) that does not totally include k or q, which implies that is not supra -soft .
Example 14. Let be the same as in Example 4. It can be shown that is supra -soft for . However, there is no member in δ (except for the absolute soft set) that totally contains k or q; therefore, is not supra -soft .
In the next example, we clarify that the systematic relation implies that does not hold true for their counterparts: supra -soft and supra -soft .
Example 15. Define the soft sets over and , where and ,
;
;
;
;
Now, the above four soft sets plus the absolute and null soft sets form a supra-soft topology δ on . It can be noted that is supra -soft . In contrast, there are no disjoint members of δ satisfying a condition of supra -soft for any two distinct points.
The behaviors and features of supra -soft -spaces under different circumstances are investigated in the following.
Proposition 17. is a supra -soft -space if is a supra-soft closed set for all .
Proof. Let . By hypothesis, and are supra-soft closed sets. Then, and . Since and , then is -soft . □
The converse of this property fails to hold true, as shown in Example 12.
Theorem 9. If has a basis of supra-soft clopen sets, then is supra -soft regular.
Proof. Consider that is a supra-soft closed set and let . Then, which is a supra-soft open set. By hypothesis, the basis contains a supra-soft clopen set such that . Now, . Obviously, and are disjoint supra-soft open sets; hence, is supra -soft regular. □
Theorem 10. Let be a -soft regular space. Then, every supra -soft -space is supra -soft for .
Proof. When . Let . Then, contains a supra-soft open set such that and , or and . Say, and . Since , it follows by supra -soft regularity that contains two disjoint supra-soft open sets and such that and . Thus, and which completes the proof.
In a similar way, is proven. □
The spaces of supra -soft regular (supra -soft regular, supra-soft normal) need not be -soft , -soft and -soft for all j. The example below points out this fact.
Example 16. Consider to be defined by Example 3. One can observe that is supra -soft regular and supra-soft normal. Note that δ does not include a supra-soft open set (excepting the absolute soft set) totally including k or q. Furthermore, δ does not include a supra-soft open set (except for the null soft set) does not totally including k or q. This means that is not -soft , or -soft , or -soft for all j.
Theorem 11. Suppose that is extended. When or , is supra -soft iff it is supra -soft .
Proof. We proceed with the argument for , the case is analogous.
Necessity: Let . There must be a member of such that and , or and . Say, and . If for all , then the proof finishes. Otherwise, we do not lose generality if we consider that there is such that and for each . Since is extended, there must be a supra-soft open set that satisfies and for each . Note that and ; hence, is supra -soft .
Sufficiency: Let . There must exist a member of such that and , or and . Say, and . If for all , then the proof finishes. Otherwise, consider, without loss of generality, that there exists such that and for each . Since is extended, there must be a supra-soft open set that satisfies and for each . Note that and ; hence, is -soft . □
Corollary 4. Let be extended. Then, is supra -soft -space iff it is supra -soft .
Theorem 12. Suppose that a supra-soft is stable. The following statements hold true:
- (i)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (ii)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (iii)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (iv)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (v)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
Proof. By the stability of , we obtain iff , and iff . Hence, we obtain the required equivalences. □
Corollary 5. Let a supra-soft be a supra -soft regular. The following statements hold true:
- (i)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (ii)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (iii)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (iv)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
- (v)
is supra -soft ⇔ is supra -soft ⇔ is supra -soft ⇔ is supra -soft .
Now, we explain how supra -soft -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 is supra -soft for ; nevertheless, its parametric supra topological spaces fail to be supra . In addition, defined by Example 2 is supra -soft ; however, its parametric supra topological space is not supra .
In contrast, defined by Example 3 is not a supra -soft -space; however, its parametric supra topological spaces are supra .
Theorem 13. Let be an extended supra-soft. If there exists such that is supra , then is supra -soft for each .
Proof. We prove the claim when . The other cases follow similar lines.
Let and assume that is supra . There must exist two disjoint members of containing k and q, respectively. Since is extended, then there exist two members of such that , and for all . This implies that , , and , ; hence, is supra -soft . □
Theorem 14. Let be an extended supra-soft. If all is supra , then is supra -soft for all j.
Proof. The cases follow from the above theorem.
In the cases of , we are finished if we demonstrate the properties of supra -soft regular and supra-soft normal.
We first prove that is supra -soft regular. Suppose that is a supra-soft closed set and that . There must exist such that . Since is supra regular, there exist disjoint members of such that , . Since is extended, then there are members of given by
and for all the others
and for all the others
Now, and are disjoint as well as , . Thus, is supra -soft regular, and consequently, it is supra -soft .
With respect to the space of supra-soft normal, it is proven in Theorem 3; hence, is supra -soft . □
Remark 4. Example 3 confirms that the restriction to extended supra-soft in Theorems 13 and 14 is not redundant.
One can prove the following results in a similar way to those in the previous section:
Theorem 15. Suppose that is stable. When , is supra iff is supra -soft .
Corollary 6. Let be supra -soft regular. When , is supra iff is supra -soft .
Theorem 16. If is a soft subspace of a -soft -space , then is supra -soft for .
Theorem 17. If is a supra-soft closed subspace of a supra -soft -space , then is a supra -soft -space.
Theorem 18. When , the finite product of supra -soft -spaces is supra -soft .
Proposition 18. Let be a soft -continuous mapping such that f is injective and ϕ is surjective. If is supra -soft , then is supra -soft for .
Proposition 19. Let be a bijective soft -continuous map. If is supra -soft , then is supra -soft for all j.
Proposition 20. Let be a bijective soft -open map. If is supra -soft , then is supra -soft for all j.
Proposition 21. The property of being a supra -soft -space for all j is preserved under -homeomorphisms.