# The Role of Softθ-Topological Operators in Characterizing Various Soft Separation Axioms

Authors: Zanyar A. Ameen; Tareq M. Al-shami; Abdelwaheb Mhemdi; Mohammed E. El-Shafei
Journal: Journal of Mathematics
 (2022)

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## Abstract

This manuscript begins with an introduction to a softθ-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 θ-kernel operator is weaker than soft kernel operator but stronger than soft θ-closure. Both soft θ-closure and soft θ-kernel operators are equivalent on soft compact sets. Furthermore, the stated operators are utilized to obtain several new characterizations of soft Ri-topologies and soft Tj-topologies, for i=0,1 and j=0,1,2.


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## Body

## 1. Introduction

The majority of real-world problems in engineering, medicine, economics, the environment, and other professions are fraught with uncertainty. Molodtsov [1] 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 [2], rough set theory [3], 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 [4], and Çağman et al. [5], 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 [6] and soft compactness [7]. In [8], 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 Tj-separation axioms and [11] for soft Ri-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 Tj and soft Ti-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 Section2. The essential points of a soft θ-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 Ri and soft Tj topologies for i=0,1 and j=0,1,2, respectively. A brief summary and conclusions conclude Section 6 of our paper.


## 2. Preliminaries

LetX be a domain set, 2X be the set of all subsets of X, and E be a set of parameters. A pair F,E=e,Fe:e∈E is said to be a soft set over X, where F:E⟶2X is a set-valued mapping. The set of all soft sets over X parameterized by E is identified by SEX. We call a soft set F,E over X a soft point [14, 15], denoted by xe, if Fe=x and Fe′=∅ for every e′∈E with e′≠e, where e∈E and x∈X. An argument xe∈F,E means that x∈Fe. The set of all soft points over X is denoted by PEX. A soft set X,E−F,E (or simply F,Ec) is the complement of F,E, where Fc:E⟶2X is given by Fce=X−Fe for every e∈E. If F,E∈SEX, it is denoted by Φ if Fe=∅ for every e∈E and is denoted by X˜ if Fe=X for every e∈E. Evidently, X˜c=Φ and Φc=X˜. A soft set F,E is called degenerate if F,E=xe or F,E=Φ. It is said that A,E1 is a soft subset of B,E2 (written by A,E1⊆˜B,E2, [16]) if E1⊆E2 and Ae⊆Be for every e∈E1, and A,E1=B,E2 if A,E1⊆˜B,E2 and B,E2⊆˜A,E1. The union of soft sets A,E,B,E is represented by F,E=A,E∪˜B,E, where Fe=Ae∪Be for every e∈E, and intersection of soft sets A,E,B,E is given by F,E=A,E∩˜B,E, where Fe=Ae∩Be for every e∈E (see, [17]).Definition 1 (see [4, 5]).
A collectionT of SEX is said to be a soft topology on X if it satisfies the following axioms:(T.1)Φ,X˜∈T.(T.2) IfF1,E,F2,E∈T, then F1,E∩˜F2,E∈T.(T.3) IfFi,E:i∈I⊆˜T, then ∪˜i∈IFi,E∈T.
Terminologically, we callX,T,E a soft topological space on X. The elements of T are called soft open sets. The complements of every soft open or elements of Tc are called soft closed sets. The lattice of all soft topologies on X is referred to TEX (see, [18]).Definition 2 (see [4, 19]).
LetB,E∈SEX and T∈TEX.(1)
The soft closure ofB,E is clB,E:=∩˜F,E:B,E⊆˜F,E,F,E∈Tc.(2)
The soft interior ofB,E is intB,E:=∪˜F,E:F,E⊆˜B,E,F,E∈T.(3)
The soft kernel ofB,E is kerB,E:=∩˜G,E:B,E⊆˜G,E,G,E∈T.Definition 3.
[5] Let B,E∈SEX and T∈TEX. A point xe∈PEX is called a soft limit point of B,E if G,E∩˜B,E−xe≠Φ for all G,E∈T with xe∈G,E. The set of all soft limit points is symbolized by derB,E. Then, clF,E=F,E∪˜derF,E (see, Theorem 5 in [5])Definition 4 (see [20]).
LetT∈TEX. A set A,E∈SEX is called soft locally closed if there exist G,E∈T and F,E∈Tc such that A,E=G,E∩˜F,E. The family of all soft locally closed sets over X is referred to LCX.Definition 5.
[21] Let T∈TEX. A set A,E∈SEX is called soft θ-open if for every xe∈A,E, there exists G,E∈T such that xe∈G,E⊆˜clG,E⊆˜A,E. The set of all soft θ-open sets forms a soft topology on X and denoted by Tθ. The complement of soft θ-open sets are soft θ-closed and their family is denoted by Tθc.Remark 1.
One can easily check thatTθ⊆˜T.Definition 6 (see [21]).
LetB,E∈SEX and T∈TEX.(1)
The softθ-interior of B,E is intθB,E:=∪˜F,E:F,E⊆˜B,E,F,E∈Tθ.(2)
The softθ-closure of B,E is clθB,E:=∩˜F,E:B,E⊆˜F,E,F,E∈Tθc.Lemma 1 (see [19, 21]).
LetB,E∈SEX and T∈TEX. Then,(1)
B,E∈Tθc whenever clθB,E=B,E.(2)
clB,E⊆˜clθB,E.(3)
clθB,E∈Tc.(4)
clB,E=clθB,E whenever B,E∈T.(5)
kerB,E=xe∈PEX:clxe∩˜B,E≠ΦDefinition 7 (see [19]).
Forxe∈PEX and T∈TEX, we define(1)
the soft derived set ofxe as derxe:=clxe−xe.(2)
the soft shell ofxe as shelxe:=kerxe−xe.(3)
the soft setxe:=clxe∩˜kerxe.Lemma 2 (see [19]).
The following properties are valid for everyxe,ye′∈PEX and T∈TEX:(1)
ye′∈kerxe⇔xe∈clye′.(2)
ye′∈shelxe⇔xe∈derye′.(3)
ye′∈clxe⇒clye′⊆˜clxe.(4)
ye′∈kerxe⇒kerye′⊆˜kerxe.(5)
shelxe is degenerate iff for every ye′∈PEX with ye′≠xe, derxe∩˜derye′=Φ.(6)
derxe is degenerate iff for every ye′∈PEX with ye′≠xe, shelxe∩˜shelye′=Φ.(7)
Ifye′∈xe, then ye′=xe.(8)
Eitherye′=xe or ye′∩˜xe=Φ.Definition 8 (see [11, 22]).
A soft spaceX,E,T (or simply soft topology T∈TEX) is called(1)
softT0 if for every xe,ye′∈PEX with xe≠ye′, there exist U,E,V,E∈T such that xe∈U,E, ye′∉U,E or ye′∈V,E, xe∉V,E.(2)
softT1 if for every xe,ye′∈PEX with xe≠ye′, there exist U,E,V,E∈T such that xe∈U,E, ye′∉U,E and ye′∈V,E, xe∉V,E.(3)
softT2 if for every xe,ye′∈PEX with xe≠ye′, there exist U,E,V,E∈T such that xe∈U,E,ye′∈V,E, and U,E∩˜V,E=Φ.(4)
softR0 if for every xe∈PEX and every U,E∈T with xe∈U,E, we have clxe⊆˜U,E.(5)
softR1 if for every xe,ye′∈PEX with clxe≠clye′, there exist U,E,V,E∈T such that clxe∈U,E, clye′∈V,E and U,E∩˜V,E=Φ.Lemma 3 (see [22], Theorem 4.1).
LetT∈TEX. Then, T is soft T1 iff clxe=xe for every xe∈PEX.Theorem 1 (see [11], Theorem 3.5).
LetT∈TEX. The following properties are equivalent:(1)
T is soft R0.(2)
Eitherclxe=clye′ or clxe∩˜clye′=Φ for every xe,ye′∈PEX with xe≠ye′.(3)
For everyxe∈PEX and every F,E∈Tc with xe∉F,E, clxe∩˜F,E=Φ.(4)
For everyxe∈PEX and every F,E∈Tc with xe∉F,E, there is G,E∈T such that F,E⊆˜G,E and xe∉G,E.Theorem 2 (see [11], Theorem 3.13).
LetT∈TEX. The following properties are equivalent:(1)
T is soft R0.(2)
IfF,E∈Tc, then kerF,E=F,E.(3)
Ifxe∈F,E∈Tc, then kerxe⊆˜F,E.(4)
kerxe⊆˜clye′ for every xe∈PEX.Lemma 4 (see [11], Proposition 3.18).
LetT∈TEX. Then, T is soft T1 iff it soft T0 and soft R0.


## 3. Some Soft Topological Operators

In this section, we define “softθ-kernel” and “soft θ-derived set” as soft topological operators. Then, the connections between soft θ-kernel, soft kernel, soft closure, soft θ-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.
LetF,E∈SEX and let T∈TEX. The soft θ-kernel of F,E is defined by(1)kerθF,E:=∩˜G,E:G,E∈Tθ,F,E⊆˜G,E.Definition 10.
Forxe∈PEX and T∈TEX, we define the soft θ-derived set of xe as derθxe:=clθxe−xe.Lemma 5.
LetF,E,G,E∈SEX and T∈TEX. The following properties are valid:(1)
F,E⊆˜kerθF,E.(2)
kerθF,E⊆˜kerθkerθF,E.(3)
F,E⊆˜G,E⇒kerθF,E⊆˜kerθG,E.(4)
kerθF,E∩˜G,E⊆˜kerθF,E∩˜kerθG,E.(5)
kerθF,E∪˜G,E=kerθF,E∪˜kerθG,E.Proof.
Standard.
Recall that a soft spaceX,T,E is called soft compact [23] if every soft open cover of X˜ possesses a finite subcover.Lemma 6.
The following properties are valid for everyF,E∈PEX and T∈TEX:(1)
kerθF,E=xe∈PEX:clθxe∩˜F,E≠Φ.(2)
kerF,E⊆˜kerθF,E⊆˜clθF,E.(3)
IfF,E is soft compact, then clθF,E=kerθF,E.Proof.
(1)
Letxe∈kerθF,E. If clθxe∩˜F,E=Φ, then one can find G,E∈Tθ such that it contains F,E but not xe, a contradiction.Conversely, ifxe∉kerθF,E but clθxe∩˜F,E≠Φ, then there is G,E∈Tθ such that F,E⊆˜G,E but xe∉G,E and ye′∈clθxe∩˜F,E. Therefore, X˜−G,E∈Tθc including xe but not ye′. However, this contradicts to ye′∈clθxe∩˜F,E. Thus, xe∈kerθF,E.(2)
It follows from the fact thatTθ⊆˜T and clθF,E=xe∈PEX:clG,E∩˜F,E≠Φ,G,E∈T,xe∈G,E. That is, clθF,E can be seen as the intersection of the soft closure of every soft open set G,E that includes F,E. Equivalently, it is a soft closed set including kerF,E.(3)
From (2), it suffices to prove thatclθF,E⊆˜kerθF,E. Suppose F,E is a soft compact set. If xe∉kerθF,E, then clθxe∩˜F,E=Φ. Therefore, there exist Gye′,E,Hye′,E∈T such that xe∈Gye′,E, ye′∈Hye′,E, and Gye′,E∩˜Hye′,E=Φ for every ye′∈F,E. Thus, ℋ=Hye′,E:ye′∈F,E forms a soft open cover of F,E. Then, there is a finite subclass ye′1,ye′2,…,ye′n of ℋ such that F,E⊆˜∪˜i=1nHye′i,E. Set A,E=∩˜i=1nGye′i,E and B,E=∪˜i=1nHye′i,E. Therefore, A,E,B,E∈T such that xe∈A,E, F,E⊆˜B,E, and A,E∩˜B,E=Φ. This means that xe∉clθF,E. We are done.Lemma 7.
The following properties are valid for everyxe,ye′∈PEX and T∈TEX:(1)
xe=ye′⇔kerxe=kerye′⇔clxe=clye′.(2)
xe∈clθye′⇔ye′∈clθxe.(3)
kerθxe=clθxe.(4)
clxe=clxe.(5)
clθxe=clθxe.(6)
kerxe=kerxe.(7)
IfF,E∈T∪˜Tc and xe∈F,E, then xe⊆˜F,E.Proof.
(1)
It is enough to show thatkerxe=kerye′⇔clxe=clye′. If kerxe≠kerye′, then one can find ze∗∈kerxe but ze∗∉kerye′. From ze∗∈kerxe, we get xe∈clze∗ and then clxe⊆˜clze∗. Since ze∗∉kerye′, by Lemma 2. (1), clze∗∩˜ye′=Φ. Therefore, clze∗∩˜ye′=Φ implies ye′∉clxe. Hence, clye′≠clxe.(2)
Ifxe∉clθye′, then there are G,E,H,E∈T, respectively, containing xe, ye′ such that G,E∩˜H,E=Φ. This implies that ye′∉clθxe.(3)
It follows from Lemma6 (3) as every xe is soft compact.(4)
Sincexe∈xe, so clxe⊆˜clxe. On the other hand, clxe=clclxe∩˜clkerxe⊆˜clxe. Hence, clxe=clxe.
Other parts are similar or simple.


## 4. Characterizations of SoftRi-Spaces, i∈0,1

In this part, we obtain some characterizations of softR0 and soft R1-spaces via certain soft topological operators.Theorem 3.
LetT∈TEX. Then, T is soft R0 iff clθxe−clxe is a union of soft closed sets for every xe∈PEX.Proof.
Givenxe∈PEX. W.l.o.g, we let clθxe−clxe≠Φ, otherwise, the conclusion is trivially true. Suppose ye′∈clθxe−clxe. Then, clye′⊆˜clθxe. Since T is soft R0, by Theorem 1, clye′∩˜clxe=Φ. Thus, clye′⊆˜clθxe−clxe and so clθxe−clxe is a union of soft closed sets.
Conversely, letxe,ye′∈PEX. Suppose clθxe−clxe=∪˜F,E:F,E∈Tc. In order to prove that T is soft R0, we study the following cases:(i)
Assumeye′∈clθxe−clxe. Then, there is F,E∈Tc such that F,E⊆˜clθxe−clxe and ye′∈F,E. Therefore, clye′⊆˜F,E. This implies that clye′∩˜clxe=Φ.(ii)
Assumeye′∈clxe. Clearly, ye′∈clθxe and xe∈clθye′. If xe∉clye′, then xe∈clθye′−clye′. By (i), clye′∩˜clxe=Φ, which is a contradiction. Therefore, we must have xe∈clye′ and so clxe=clye′.(iii)
Assumeye′∉clθxe. Suppose if possible clxe∩˜clye′≠Φ, then there is ze∗∈clxe and ze∗∈clye′. By (ii), clxe=clye′=clze∗. Therefore, ye′∈clxe, a contradiction. Hence, clxe∩˜clye′=Φ.
In conclusion, we have shown that for everyxe,ye′∈PEX, either clxe=clye′ or clxe∩˜clye′=Φ. Thus, T is soft R0.Proposition 1.
LetT∈TEX. Then, T is soft R1 iff xe=clθxe for every xe∈PEX.Proof.
SupposeT is soft R1. The first direction is simple. That is xe=clxe∩˜kerxe⊆˜clxe⊆˜clθxe, see Lemma 2 (2). For the reverse, let ye′∉xe. By Lemma 2 (7), xe≠ye′. By Lemma 7 (1), clxe≠clye′. Since T is soft R1, there are G,E,H,E∈T such that xe∈G,E, ye′∈H,E, and G,E∩˜H,E=Φ. This implies ye′∉clθxe, and hence, clθxe⊆˜xe. Thus, xe=clθxe.
The converse can be proved similarly.Proposition 2.
LetT∈TEX. Then, T is soft R1 iff xe∈Tθc for every xe∈PEX.Proof.
SupposeT is soft R1. It suffices to show that clθxe⊆˜xe. Let ye′∈clθxe. By Lemma 7 (5), clθxe=clθxe, and so ye′∈clθxe. This means that for all G,E,H,E∈T containing xe,ye′, respectively, G,E∩˜H,E=Φ. We must have clxe=clye′, otherwise, we get a contradiction as clxe and clye′ can be separated by two disjoint soft open. Hence, ye′∈xe.
Assumexe∈Tθc for every xe∈PEX. That is, clθxe=xe. Since xe⊆˜kerxe and clxe⊆˜clθxe=clθxe, then we obtain that xe=clxe=clθxe=kerxe. For xe,ye′∈PEX, if clxe≠clye′, then ye′∉xe. Therefore, there are G,E,H,E∈T containing xe,ye′, respectively, G,E∩˜H,E=Φ. Obviously, clxe⊆˜G,E and clye′⊆˜H,E as clxe=kerxe, clye′=kerye′. Hence, T is soft R1.Proposition 3.
LetT∈TEX. The following properties are equivalent:(1)
T is soft R1.(2)
clθxe=clxe for every xe∈PEX.(3)
clθxe=kerxe for every xe∈PEX.(4)
clxe∈Tθc for every xe∈PEX.(5)
kerxe∈Tθc for every xe∈PEX.Proof.
IfT is soft R1, by Proposition 2, clθxe=xe. Since xe⊆˜kerxe and clxe⊆˜clθxe=clθxe, then we obtain that xe=clxe=kerxe=clθxe. The equivalence of these statements can be easily concluded.Proposition 4.
LetT∈TEX. The following properties are equivalent:(1)
T is soft R1.(2)
IfF,E∈Tc and xe∈F,E, then clθxe⊆˜F,E.(3)
IfG,E∈T and xe∈G,E, then clθxe⊆˜G,E.Proof.
It follows from Lemma7 (7) and Proposition 1.Proposition 5.
LetT∈TEX. Then, T is soft R1 iff either xe=ye′ or clθxe∩˜clθye′=Φ, for every xe,ye′∈PEX.Proof.
AssumeT is soft R1. Given xe,ye′∈PEX, then either clxe=clye′ or clxe≠clye′. If clxe=clye′, then by Lemma 7 (1), xe=ye′. If clxe≠clye′, since T is soft R1, then there exist disjoint U,E,V,E∈T such that clxe⊆˜U,E and clye′⊆˜V,E. By Proposition 3 (2), clθxe∩˜clθye′=Φ.
Conversely, letxe,ye′∈PEX such that clxe≠clye′. Since clxe⊆˜clθxe, by assumption, clxe∩˜clye′=Φ. Set U,E=X˜−clye′ and V,E=X˜−clxe. Therefore, U,E,V,E∈T such that clxe⊆˜U,E and clye′⊆˜V,E. Hence, T is soft R1.Proposition 6.
LetT∈TEX. The following properties are equivalent:(1)
T is soft R1.(2)
For everyxe,ye′∈PEX, either there exists G,E∈T such that xe∈G,E iff ye′∈G,E or there exist disjoint sets U,E,V,E∈T containing them.(3)
For everyxe,ye′∈PEX with clxe≠clye′, there exist F,E,D,E∈Tc such that xe∈F,E, ye′∈D,E, and X˜=F,E∪˜D,E.Proof.
It follows from the definition of a softR1-space and Proposition 5.
When all of the preceding propositions are added together, the following result arises:Theorem 4.
LetT∈TEX. The following properties are equivalent:(1)
T is soft R1.(2)
For everyxe∈PEX, xe=clθxe.(3)
For everyxe∈PEX, xe=clθxe.(4)
For everyxe∈PEX, clθxe=clxe.(5)
For everyxe∈PEX, clθxe=kerxe.(6)
For everyxe∈PEX, clxe∈Tθc.(7)
For everyxe∈PEX, kerxe∈Tθc.(8)
IfF,E∈Tc and xe∈F,E, then clθxe⊆˜F,E.(9)
IfG,E∈T and xe∈G,E, then clθxe⊆˜G,E.(10)
For everyxe,ye′∈PEX, either xe=ye′ or clθxe∩˜clθye′=Φ.(11)
For everyxe,ye′∈PEX with clxe≠clye′, there exist F,E,D,E∈Tc such that xe∈F,E, ye′∈D,E, and X˜=F,E∪˜D,E.Theorem 5.
ForT∈TEX, the following properties are equivalent:(1)
T is soft R1.(2)
kerθF,E=kerF,E for every F,E∈SEX.(3)
kerθF,E=clF,E for every soft compact F,E∈SEX.(4)
clθF,E=clF,E for every soft compact F,E∈SEX.Proof.
(1)⇒ (2) Suppose T is soft R1. By Theorem 4 (4), we have kerθF,E=xe∈PEX:clθxe∩˜F,E≠Φ=xe∈PEX:clxe∩˜F,E≠Φ=kerF,E.(2)⇒ (3) Given a soft compact set F,E, by (2) and Lemma 6 (3), clθF,E=kerθF,E=kerF,E. Since clF,E⊆˜clθF,E, so clF,E⊆˜kerθF,E. By Theorem 2, kerA⊆˜clF,E. Therefore, kerθA⊆˜clF,E. Hence, (3).(3)⇒ (4) It derives from Lemma 6 (3).(4)⇒ (1) It concludes from Theorem 4 (4).


## 5. Characterizations of SoftTi-Spaces, i∈0,1,2

In this section, we give characterizations of softT0, soft T1, and soft T2-spaces via the soft topological operators mentioned in Section 3.Theorem 6 (see [19]).
LetT∈TEX. Then, T is soft T0 iff derxe is a union of soft closed sets for every xe∈PEX.
Using the softθ-derived set operator, a conclusion similar to the above can be established for soft T1 topologies.Theorem 7.
LetT∈TEX. Then, T is soft T1 iff derθxe is a union of soft closed sets for every xe∈PEX.Proof.
SupposeT is soft T1. By Lemma 4, T is soft T0 and soft R0. By Theorems 3 and 6, we can easily conclude that derθxe is a union of soft closed sets for every xe∈PEX.
Conversely, givenxe∈PEX. If ye′∈derxe, then ye′∈derθxe and xe∈derθye′. Therefore, there exists F,E∈Tc with F,E⊆˜derθye′ such that xe∈F,E. Thus, clxe⊆˜derθye′. This implies that ye′∉clxe and hence derxe=clxe−xe=Φ. This proves that T is soft T1.Theorem 8.
LetT∈TEX. Then, T is soft T2 iff clθxe=xe for every xe∈PEX.Proof.
SupposeT is soft T2. Given xe∈PEX. Then, for every ye′∈PEX with ye′≠xe, there are G,E,H,E∈T such that xe∈G,E, ye′∈H,E, and G,E∩˜H,E=Φ. Therefore, G,E∩˜clH,E=Φ. This means that ye′∉clθxe. Hence, clθxe=xe.
Conversely, supposeclθxe=xe for every xe∈PEX. For any ye′∈PEX with ye′≠xe, ye′∉clθxe. This implies that there are disjoint G,E,H,E∈T such that xe∈G,E, ye′∈H,E. Thus, T is soft T2.
The next result is an immediate consequence of the above theorem:Corollary 1.
LetT∈TEX. Then, T is soft T2 iff Tθ is soft T1 iff Tθ is soft T0.Theorem 9.
ForT∈TEX, the following properties are equivalent:(1)
T is soft T2.(2)
clθxe∩˜clθye′=Φ for every xe,ye′∈PEX with xe≠ye′.(3)
kerθF,E=F,E for every F,E∈SEX.(4)
kerθF,E=F,E for every soft compact F,E∈SEX.(5)
clθF,E=F,E for every soft compact F,E∈SEX.Proof.
(1)⇒ (2) Let xe,ye′∈PEX with xe≠ye′. By (1) and Theorem 8, there exist disjoint G,E,H,E∈T such that clθxe⊆˜G,E and clθye′⊆˜H,E. Hence, clθxe∩˜clθye′=Φ.(2)⇒ (3) By Lemma 6 (1), kerθF,E=xe:clθxe∩˜F,E≠Φ. Since for every xe∈F,E and for every ye′≠xe, clθxe∩˜clθye′=Φ and xe∈clθxe. We must have that kerθF,E=F,E.(3)⇒ (4) Evident.(4)⇒ (1) Since each xe is soft compact, by Theorem 8, T is soft T2.


## 6. Conclusion

Soft separation axioms are collections of conditions for classifying a system of soft topological spaces according to particular soft topological properties. These axioms are usually described in terms of soft open or soft closed sets in a topological space. In this paper, we propose softθ-kernel and soft θ-derived set operators and make find their relationships with other soft topological operators. Then, the mentioned operators are used to characterize various soft separation axioms. We see that soft θ-kernel, soft θ-closure, and soft θ-derived set operators behave better than their corresponding soft operators for characterizing soft Ti and soft Rj-spaces, where i=0,1,2 and j=0,1 (c.f. [19]).



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*Source: 9073944-2022-07-07.xml*