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The Role of Soft -Topological Operators in Characterizing Various Soft Separation Axioms
This manuscript begins with an introduction to a soft -kernel operator. Then, the main properties and connections of this soft topological operator with other known soft topological operators are examined. We show that soft -kernel operator is weaker than soft kernel operator but stronger than soft -closure. Both soft -closure and soft -kernel operators are equivalent on soft compact sets. Furthermore, the stated operators are utilized to obtain several new characterizations of soft -topologies and soft -topologies, for and .
The majority of real-world problems in engineering, medicine, economics, the environment, and other professions are fraught with uncertainty. Molodtsov  presented the soft set theory in 1999 as a mathematical model for reducing uncertainty. This is freed of the drawbacks of prior theories such as fuzzy set theory , rough set theory , and so on. The nature of parameter sets, particularly those connected to soft sets, provides a consistent foundation for modeling uncertain data. This leads to the rapid growth of soft set theory in a short amount of time, as well as a wide range of real-world applications of soft sets.
Influenced by the standard postulates of ordinary topological space, Shabir and Naz , and Çağman et al. , separately, established another branch of topology known as “soft topology,” which is a mixture of soft set theory and topology. It focuses on the development of the system of all soft sets. The study in [4, 5], in particular, was essential in building the subject of soft topology. Following these works, researchers have been discussed the topological concepts via soft topological spaces such as soft bases  and soft compactness . In , the authors applied some soft operators to generate soft topologies.
The separation axioms are simply axioms in the sense that these criteria could be added as additional hypotheses to the definition of topological space to create a more restricted description of what a topological space is. These axioms have a great role in developing (classical) topology. Correspondingly, soft separation axioms are a significant aspect in the late development of soft topology; see for example [9, 10] for soft -separation axioms and  for soft -separation axioms. Despite the fact that intensive studies have been conducted on these axioms, however, significant contributions can indeed be made. Hence, we characterize both soft and soft -separation axioms in terms of the discussed soft topological operators. It should be noted that some amendments for a number of properties of separation axioms in soft settings were given in [12, 13].
The following is how the paper’s body is organized: we provide an overview of the literature on soft set theory and soft topology in Section 2. The essential points of a soft -kernel operator and its link to the associated soft topological operators are discussed in Section 3. Sections 4 and 5 use soft operators provided in Section 3 to characterize soft and soft topologies for and , respectively. A brief summary and conclusions conclude Section 6 of our paper.
Let be a domain set, be the set of all subsets of , and be a set of parameters. A pair is said to be a soft set over , where is a set-valued mapping. The set of all soft sets over parameterized by is identified by . We call a soft set over a soft point [14, 15], denoted by , if and for every with , where and . An argument means that . The set of all soft points over is denoted by . A soft set (or simply ) is the complement of , where is given by for every . If , it is denoted by if for every and is denoted by if for every . Evidently, and . A soft set is called degenerate if or . It is said that is a soft subset of (written by , ) if and for every , and if and . The union of soft sets is represented by , where for every , and intersection of soft sets is given by , where for every (see, ).
Definition 1 (see [4, 5]). A collection of is said to be a soft topology on if it satisfies the following axioms: (T.1) . (T.2) If , then . (T.3) If , then .Terminologically, we call a soft topological space on . The elements of are called soft open sets. The complements of every soft open or elements of are called soft closed sets. The lattice of all soft topologies on is referred to (see, ).
Definition 4 (see ). Let . A set is called soft locally closed if there exist and such that . The family of all soft locally closed sets over is referred to .
Definition 5.  Let . A set is called soft -open if for every , there exists such that . The set of all soft -open sets forms a soft topology on and denoted by . The complement of soft -open sets are soft -closed and their family is denoted by .
Remark 1. One can easily check that .
Definition 6 (see ). Let and .(1)The soft -interior of is .(2)The soft -closure of is .
Definition 7 (see ). For and , we define(1)the soft derived set of as .(2)the soft shell of as .(3)the soft set .
Lemma 2 (see ). The following properties are valid for every and :(1).(2).(3).(4).(5) is degenerate iff for every with , .(6) is degenerate iff for every with , .(7)If , then .(8)Either or .
Definition 8 (see [11, 22]). A soft space (or simply soft topology ) is called(1)soft if for every with , there exist such that , or , .(2)soft if for every with , there exist such that , and , .(3)soft if for every with , there exist such that , and .(4)soft if for every and every with , we have .(5)soft if for every with , there exist such that , and .
Lemma 3 (see , Theorem 4.1). Let . Then, is soft iff for every .
Theorem 1 (see , Theorem 3.5). Let . The following properties are equivalent:(1) is soft .(2)Either or for every with .(3)For every and every with , .(4)For every and every with , there is such that and .
Theorem 2 (see , Theorem 3.13). Let . The following properties are equivalent:(1) is soft .(2)If , then .(3)If , then .(4) for every .
Lemma 4 (see , Proposition 3.18). Let . Then, is soft iff it soft and soft .
3. Some Soft Topological Operators
In this section, we define “soft -kernel” and “soft -derived set” as soft topological operators. Then, the connections between soft -kernel, soft kernel, soft closure, soft -derived set, and soft derived set operators are obtained. The results of the present part will be used to characterize several soft separation axioms.
Definition 9. Let and let . The soft -kernel of is defined by
Definition 10. For and , we define the soft -derived set of as .
Lemma 5. Let and . The following properties are valid:(1).(2).(3).(4).(5).
Recall that a soft space is called soft compact  if every soft open cover of possesses a finite subcover.
Lemma 6. The following properties are valid for every and :(1).(2).(3)If is soft compact, then .
Proof. (1)Let . If , then one can find such that it contains but not , a contradiction. Conversely, if but , then there is such that but and . Therefore, including but not . However, this contradicts to . Thus, .(2)It follows from the fact that and . That is, can be seen as the intersection of the soft closure of every soft open set that includes . Equivalently, it is a soft closed set including .(3)From (2), it suffices to prove that . Suppose is a soft compact set. If , then . Therefore, there exist such that , , and for every . Thus, forms a soft open cover of . Then, there is a finite subclass of such that . Set and . Therefore, such that , , and . This means that . We are done.
Lemma 7. The following properties are valid for every and :(1).(2).(3).(4).(5).(6).(7)If and , then .
Proof. (1)It is enough to show that . If , then one can find but . From , we get and then . Since , by Lemma 2. (1), . Therefore, implies . Hence, .(2)If , then there are , respectively, containing , such that . This implies that .(3)It follows from Lemma 6 (3) as every is soft compact.(4)Since , so . On the other hand, . Hence, .Other parts are similar or simple.
4. Characterizations of Soft -Spaces,
In this part, we obtain some characterizations of soft and soft -spaces via certain soft topological operators.
Theorem 3. Let . Then, is soft iff is a union of soft closed sets for every .
Proof. Given . W.l.o.g, we let , otherwise, the conclusion is trivially true. Suppose . Then, . Since is soft , by Theorem 1, . Thus, and so is a union of soft closed sets.
Conversely, let . Suppose . In order to prove that is soft , we study the following cases:(i)Assume . Then, there is such that and . Therefore, . This implies that .(ii)Assume . Clearly, and . If , then . By (i), , which is a contradiction. Therefore, we must have and so .(iii)Assume . Suppose if possible , then there is and . By (ii), . Therefore, , a contradiction. Hence, .In conclusion, we have shown that for every , either or . Thus, is soft .
Proposition 1. Let . Then, is soft iff for every .
Proof. Suppose is soft . The first direction is simple. That is , see Lemma 2 (2). For the reverse, let . By Lemma 2 (7), . By Lemma 7 (1), . Since is soft , there are such that , , and . This implies , and hence, . Thus, .
The converse can be proved similarly.
Proposition 2. Let . Then, is soft iff for every .
Proof. Suppose is soft . It suffices to show that . Let . By Lemma 7 (5), , and so . This means that for all containing , respectively, . We must have , otherwise, we get a contradiction as and can be separated by two disjoint soft open. Hence, .
Assume for every . That is, . Since and , then we obtain that . For , if , then . Therefore, there are containing , respectively, . Obviously, and as , . Hence, is soft .
Proposition 3. Let . The following properties are equivalent:(1) is soft .(2) for every .(3) for every .(4) for every .(5) for every .
Proof. If is soft , by Proposition 2, . Since and , then we obtain that . The equivalence of these statements can be easily concluded.
Proposition 4. Let . The following properties are equivalent:(1) is soft .(2)If and , then .(3)If and , then .
Proposition 5. Let . Then, is soft iff either or , for every .
Proof. Assume is soft . Given , then either or . If , then by Lemma 7 (1), . If , since is soft , then there exist disjoint such that and . By Proposition 3 (2), .
Conversely, let such that . Since , by assumption, . Set and . Therefore, such that and . Hence, is soft .
Proposition 6. Let . The following properties are equivalent:(1) is soft .(2)For every , either there exists such that iff or there exist disjoint sets containing them.(3)For every with , there exist such that , , and .
Proof. It follows from the definition of a soft -space and Proposition 5.
When all of the preceding propositions are added together, the following result arises:
Theorem 4. Let . The following properties are equivalent:(1) is soft .(2)For every , .(3)For every , .(4)For every , .(5)For every , .(6)For every , .(7)For every , .(8)If and , then .(9)If and , then .(10)For every , either or .(11)For every with , there exist such that , , and .
Theorem 5. For , the following properties are equivalent:(1) is soft .(2) for every .(3) for every soft compact .(4) for every soft compact .
Proof. (1) (2) Suppose is soft . By Theorem 4 (4), we have . (2) (3) Given a soft compact set , by (2) and Lemma 6 (3), . Since , so . By Theorem 2, . Therefore, . Hence, (3). (3) (4) It derives from Lemma 6 (3). (4) (1) It concludes from Theorem 4 (4).
5. Characterizations of Soft -Spaces,
In this section, we give characterizations of soft , soft , and soft -spaces via the soft topological operators mentioned in Section 3.
Theorem 6 (see ). Let . Then, is soft iff is a union of soft closed sets for every .
Using the soft -derived set operator, a conclusion similar to the above can be established for soft topologies.
Theorem 7. Let . Then, is soft iff is a union of soft closed sets for every .