# Boolean Algebra of SoftQ-Sets in Soft Topological Spaces

Authors: Samer Al Ghour
Journal: Applied Computational Intelligence and Soft Computing
 (2022)

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## Abstract

We define softQ-sets as soft sets whose soft closure and soft interior are commutative. We show that the soft complement, soft closure, and soft interior of a soft Q-set are all soft Q-sets. We show that a soft subset K of a given soft topological space is a soft Q-set if and only if K is a soft symmetric difference between a soft clopen set and a soft nowhere dense set. And as a corollary, the class of soft Q-sets contains simultaneously the classes of soft clopen sets and soft nowhere dense sets. Also, we prove that the class of soft Q-sets is closed under finite soft intersections and finite soft unions, and as a main result, we prove that the class of soft Q-sets forms a Boolean algebra. Furthermore, via soft Q-sets, we characterize soft sets whose soft boundaries and soft interiors are commutative. In addition, we investigate the correspondence between Q-sets in topological spaces and soft Q-sets in soft topological spaces.


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## Body

## 1. Introduction and Preliminaries

Some problems in medicine, engineering, the environment, economics, sociology, and other fields have their own doubts. Therefore, we are unable to deal with these problems by conventional methods. For more than thirty years, fuzzy set theory [1], rough set theory [2], and vague set theory [3] have played an essential role in dealing with these problems. Molodtsov [4] argues that each of these theories has its own set of problems. These difficulties mainly come from the inadequacy of the parameterization tool for the theories. Research through soft set theory has included almost all branches of science. Soft set theory has been applied to solve problems using Riemann integral, Beron’s integral, game theory, function smoothness, operations research, measure theory, probability, and decision-making problems [4–6].General topology, as one of the main branches of mathematics, is the branch of topology that deals with the basic definitions of set theory and structures used in topology. It is the foundation of most other branches of topology, including algebraic topology, geometric topology, and differential topology. Shabir and Naz [7] initiated soft topology, which is a new branch of topology that combines soft set theory and topology. Since then, numerous studies have appeared in soft topology [8–24] and others, and substantial contributions can still be made. A subset S of a given topological space is said to be a Q-set if the interior and closure operators of this subset are commute. Levine [25] discovered that a subset of a topological space is a Q-set if and only if it is the symmetric difference of a set that is clopen and a set that is nowhere dense. As an important application of Levine’s characterization of Q-sets, the authors in [26] have proved that a topological space is compact, Hausdorff, and metastonean if and only if its classes of Borel sets and Q-sets are equal. The aim of this paper is to extend the concept of Q-sets and their related properties and results to include soft topological spaces. In this paper, We define soft Q-sets as soft sets whose soft closure and soft interior are commutative. We show that the soft complement, soft closure, and soft interior of a soft Q-set are all soft Q-sets. We show that a soft subset K of a given soft topological space is a soft Q-set if and only if K is a soft symmetric difference between a soft clopen set and a soft nowhere dense set. In addition, as a corollary, the class of soft Q-sets contains simultaneously the classes of soft clopen sets and soft nowhere dense sets. Also, we prove that the class of soft Q-sets is closed under finite soft intersections and finite soft unions, and as a main result, we prove that the class of soft Q-sets forms a Boolean algebra. Furthermore, via soft Q-sets, we characterize soft sets whose soft boundaries and soft interiors are commutative. In addition, we investigate the correspondence between Q-sets in topological spaces and soft Q-sets in soft topological spaces.Boolean algebra constitutes the basis for the design of circuits used in electronic digital computers. In addition, it is of significance to the theory of probability, the geometry of sets, and information theory. So, this paper not only forms the theoretical basis for further applications of soft topology, but it also leads to the development of the theory of probability, the geometry of sets, and information theory.In this paper, we follow the notions and terminologies that appeared in [27, 28]. Throughout this paper, ST and STS will denote topological space and soft topological space, respectively. Let Y,σ,B be a STS, Y,μ be a TS, H∈SSY,B, and U⊆Y. ClσH, IntσH, BdσH, ClμU, and IntμU will be used throughout this paper to denote the soft closure of H in Y,σ,B, the soft interior of H in Y,σ,B, the soft boundary of H in Y,σ,B, the closure of U in Y,ℵ, and the interior of U in Y,ℵ, respectively.The following definitions will be used in the sequel:Definition 1 (see [25]).
LetY,μbe a TS and letU⊆Y. ThenUis called aQ-set inY,μifIntμClμU=ClμIntμU.Definition 2 (see [29]).
A TSY,μis called extremally disconnected ifClμU∈μfor eachU∈μ.Definition 3 (see [27]).
LetYbe a universal set andBbe a set of parameters. ThenG∈SSY,Bdefined by(a)
Gb=Zifb=e∅ifb≠e will be denoted by eZ.(b)
Gb=Z for all b∈B will be denoted by CZ.Theorem 1 (see [7]).
LetY,σ,Bbe a STS. Then the collectionGb:G∈σdefines a topology onYfor everyb∈B. This topology will be denoted byσb.Theorem 2 (see [27]).
LetYbe an initial universe and letBbe a set of parameters. Letμb:b∈Bbe an indexed family of topologies onYand let(1)σ=G∈SSY,B:Gb∈μb,for allb∈B.
Thenσ defines a soft topology on Y relative to B. This soft topology will be denoted by ⊕b∈Bμb.Definition 4 (see [30]).
LetY,σ,Bbe a STS and letM∈SSY,B. ThenMis called soft nowhere dense inY,σ,BifIntσClσM=0B.Definition 5 (see [31]).
A TSY,σ,Bis called soft extremally disconnected ifClμM∈σfor eachM∈σ.Theorem 3 (see [32]).
LetY,σ,Bbe a STS and letM∈SSY,B. Then(a)
Intσ1B−M=1B−ClσM.(b)
Clσ1B−M=1B−IntσM.Theorem 4.
LetY,σ,Bbe a STS. IfFandGare soft nowhere dense sets inY,σ,B, thenF∪˜Gis soft nowhere dense inY,σ,B.Proof.
Similar to that used in Theorem 3.8 of [33].


## 2. SoftQ-Sets

Definition 6.
LetY,σ,Bbe a STS and letM∈SSY,B. ThenMis called a softQ-set inY,σ,BifIntσClσM=ClσIntσM.Theorem 5.
LetY,σ,Bbe a STS and letM∈SSY,B. ThenMis a softQ-set inY,σ,Bif and only if1B−Mis a softQ-set inY,σ,B.Proof.
Necessity. Suppose that M is a soft Q-set in Y,σ,B. Then IntσClσM=ClσIntσM, and so
1B−IntσClσM=1B−ClσIntσM. Now, by Theorem 3, we have,(2)1B−IntσClσM=Clσ1B−ClσM=ClσIntσ1B−M,and1B−ClσIntσM=Intσ1B−IntσM=IntσClσ1B−M.
Therefore,IntσClσ1B−M=ClσIntσ1B−M. Hence, 1B−M is a soft Q-set in Y,σ,B.
Sufficiency. Suppose that 1B−M is a soft Q-set in Y,σ,B. Then, by the above, 1B−1B−M=M is a soft Q-set in Y,σ,B.Theorem 6.
LetY,σ,Bbe a STS and letM∈SSY,B. IfMis a softQ-set inY,σ,B, thenClσMis a softQ-set inY,σ,B.Proof.
Suppose thatM is a soft Q-set in Y,σ,B. Then IntσClσM=ClσIntσM, and so(3)ClσIntσClσM=ClσClσIntσM=ClσIntσM.
On the other hand,IntσClσClσM=IntσClσM. Thus,
IntσClσClσM=ClσIntσClσM, and hence ClσM is a soft Q-set in Y,σ,B.Theorem 7.
LetY,σ,Bbe a STS and letM∈SSY,B. IfMis a softQ-set inY,σ,B, thenIntσMis a softQ-set inY,σ,B.Proof.
Suppose thatM is a soft Q-set in Y,σ,B. Then IntσClσM=ClσIntσM, and so(4)IntσClσIntσM=IntσIntσClσM=IntσClσM.
On the other hand,ClσIntσIntσM=ClσIntσM. Thus,
IntσClσIntσM=ClσIntσIntσM, and hence IntσM is a soft Q-set in Y,σ,B.
The converse of Theorem7 is false:Example 1.
LetY=ℝ, B=ℕ, and M,G∈SSY,B defined by(5)Mb=ℚ,ifbis even,ℚc,ifbis odd,andGb=ℚc,ifbis even,ℚ,ifbis odd.
Letσ=0B,1B,G. Then IntσM=0B, and so IntσM is a soft Q-set in Y,σ,B. On the other hand, since IntσClσM=Intσ1B=1B while ClσIntσM=Clσ0B=0B, then M is not a soft Q-set in Y,σ,B.Theorem 8.
LetY,σ,Bbe a STS and letM∈SSY,B. IfMor1B−Mis soft nowhere dense inY,σ,B, thenMis a softQ-set inY,σ,B.Proof.
Suppose thatM is soft nowhere dense. Then IntσClσM=0B. On the other hand, since IntσM⊆˜IntσClσM=0B, then IntσM=0B and so ClσIntσM=0B. Therefore, IntσClσM=ClσIntσM, and hence M is a soft Q-set in Y,σ,B.
Suppose that1B−M is soft nowhere dense. Then by the above, 1B−M is a soft Q-set in Y,σ,B. So, by Theorem 5, 1B−1B−M=M is a soft Q-set in Y,σ,B.
The converse of Theorem8 is false:Example 2.
LetY=1,2, B=a,b , and M∈SSY,B defined by Ma=1, Mb=2. Let σ=0B,1B,M,1B−M. Then IntσM=ClσM=M and Intσ1B−M=Clσ1B−M=1B−M. Thus, IntσClσM=ClσIntσM=M≠0B and IntσClσ1B−M=1B−M≠0B. Hence, M is a soft Q-set in Y,σ,B but neither M nor 1B−M is soft nowhere dense in Y,σ,B.Lemma 1.
IfY,σ,Bis a STS andM,N,K∈SSY,Bsuch thatMis soft clopen inY,σ,B,N⊆˜MandK⊆˜1B−M, then(6)IntσN∪˜K=IntσN∪˜IntσK.Proof.
IntσN∪˜IntσK⊆˜IntσN∪˜K is always true. To see that IntσN∪˜K⊆˜IntσN∪˜IntσK, let by∈˜IntσN∪˜K.Case 1.by∈˜M. Then by∈˜M∩˜IntσN∪˜K∈σ and(7)M∩˜IntσN∪˜K⊆˜M∩˜N∪˜K=M∩˜N∪˜M∩˜K=N∪˜0B=N.This implies thatby∈˜IntσN⊆˜IntσN∪˜IntσK.Case 2.by∈˜1B−M. Then by∈˜1B−M∩˜IntσN∪˜K∈σ and(8)1B−M∩˜IntσN∪˜K⊆˜1B−M∩˜N∪˜K=1B−M∩˜N∪˜1B−M∩˜K=0B∪˜K=K.This implies thatby∈˜IntσK⊆˜IntσN∪˜IntσK.Lemma 2.
LetY,σ,Bbe a STS andK,S∈SSY,B. IfKis soft clopen inY,σ,B, then(9)ClσK∩˜S=K∩˜ClσS.Proof.
SinceK is soft clopen in Y,σ,B, then ClσK∩˜S⊆˜ClσK∩˜ClσS=K∩˜ClσS. To see that K∩˜ClσS⊆˜ClσK∩˜S, suppose to the contrary that there exists by∈˜K∩˜ClσS−ClσK∩˜S. Since by∈˜1B−ClσK∩˜S, then there exists F∈σ such that by∈˜F and K∩˜S∩˜F=0B. On the other hand, since by∈˜K∩˜F∈σ and by∈˜ClσS, then K∩˜F∩˜S=K∩˜S∩˜F≠0B. This is a contradiction.
Now we state the main result of this section.Theorem 9.
LetY,σ,Bbe a STS andM∈SSY,B. ThenMis a softQ-set inY,σ,Bif and only ifM=K−N∪˜N−KwhereKis soft clopen inY,σ,BandNis soft nowhere dense inY,σ,B.Proof.
Necessity. Suppose that M is a soft Q-set in Y,σ,B. Let K=IntσClσM=ClσIntσM. Then K is soft clopen in Y,σ,B. Let N=M−K∪˜K−M.Claim 1.
(a)
M−K is soft nowhere dense in Y,σ,B.(b)
K−M is soft nowhere dense in Y,σ,B.(c)
N is soft nowhere dense in Y,σ,B.(d)
M=K−N∪˜N−K.Proof.
of Claim
(a). SinceIntσClσM−K⊆˜IntσClσM=K and(10)IntσClσM−K=IntσClσM∩˜1B−K⊆˜IntσClσ1B−K=Intσ1B−K=1B−K,
thenIntσClσM−K⊆˜K∩˜1B−K=0B. Hence, M−K is soft nowhere dense in Y,σ,B.
(b) Note thatIntσClσK−M⊆˜IntσClσK=K. Also, we have,(11)IntσClσK−M=IntσClσK∩˜1B−M⊆˜IntσClσ1B−M.
But by Theorem3,(12)IntσClσ1B−M=1B−Clσ1B−Clσ1B−M=1B−ClσIntσM=1B−K.
Therefore,IntσClσK−M⊆˜1B−K. Hence, IntσClσK−M⊆˜K∩˜1B−K=0B. It follows that K−M is soft nowhere dense in Y,σ,B.
(c) Follows from (a) and (b) and the fact that the soft union of two soft nowhere dense sets is soft nowhere dense.
(d)K−N∪˜N−K = K∩˜1B−N∪˜N∩˜1B−K = K∩˜1B−M∩˜1B−K∪˜K∩˜1B−M∪˜M∩˜1B−K∪˜K∩˜1B−M∩˜1B−K = K∩˜1B−M∩˜1B−K∩˜1B−K∩˜1B−M∪˜M∩˜1B−K = K∩˜1B−M∪˜K∩˜1B−K∪˜M∪˜M∩˜1B−K = K∩˜M∩˜1B−M∪˜K∪˜M∩˜1B−K = K∩˜M∪˜M∩˜1B−K = M∩˜K∪˜1B−K = M∩˜1B = M.
The above claim ends the proof.
Sufficiency. Suppose that M=K−N∪˜N−K where K is soft clopen in Y,σ,B and N is soft nowhere dense in Y,σ,B. Since N is soft nowhere dense, then IntσN−K⊆˜IntσN⊆˜IntσClσN=0B, and so IntσN−K=0B.
SinceK is soft clopen in Y,σ,B, K−N⊆˜K and N−K⊆˜1B−K, then by Lemma 1,(13)IntσK−N∪˜N−K=IntσK−N∪˜IntσN−K=IntσK−N∪˜0B=IntσK−N=IntσK∩˜1B−N=IntσK∩˜Intσ1B−N=K∩˜Intσ1B−N.
Also, by Lemma2, ClσK∩˜Intσ1B−N=K∩˜ClσIntσ1B−N. Moreover, by Theorem 8, ClσIntσ1B−N=IntσClσ1B−N. Therefore,(14)ClσIntσM=ClσIntσK−N∪˜N−K=ClσK∩˜Intσ1B−N=K∩˜ClσIntσ1B−N=K∩˜IntσClσ1B−N.
Now,(15)IntσClσM=IntσClσK−N∪˜N−K=IntσClσK−N∪˜ClσN−K.
SinceK is soft clopen in Y,σ,B, ClσK−N⊆˜ClσK=K, and ClσN−K⊆˜Clσ1B−K=1B−K, then by Lemma 1,(16)IntσClσK−N∪˜ClσN−K=IntσClσK−N∪˜IntσClσN−K.
Also, sinceN is soft nowhere dense in Y,σ,B, then IntσClσN−K⊆˜IntσClσN=0B and hence, IntσClσN−K=0B . Moreover, by Lemma 2,(17)ClσK−N=ClσK∩˜1B−K=K∩˜Clσ1B−K.
Therefore,(18)IntσClσM=IntσClσK−N∪˜ClσN−K=IntσClσK−N∪˜IntσClσN−K=IntσClσK−N∪˜0B=IntσClσK−N=IntσClσK∩˜1B−N=IntσK∩˜Clσ1B−N=IntσK∩˜IntσClσ1B−N=K∩˜IntσClσ1B−N=ClσIntσM.
It follows thatM is a soft Q-set in Y,σ,B.Corollary 1.
LetY,σ,Bbe a STS. IfKis a soft clopen set inY,σ,B, thenKis a softQ-set inY,σ,B.Proof.
LetN=0B. Then N is soft soft nowhere dense in Y,σ,B. Since K=K−N∪˜N−K, then by Theorem 9, K is a soft Q-set in Y,σ,B.Corollary 2.
LetY,σ,Bbe a soft connected STS andM∈SSY,B. ThenMis a softQ-set inY,σ,Bif and only ifMis soft nowhere dense inY,σ,Bor1B−Mis soft nowhere dense inY,σ,B.Proof.
By Theorem9, M can be written as K−N∪˜N−K where K is soft clopen in Y,σ,B and N is soft nowhere dense in Y,σ,B. Since Y,σ,B is soft connected, then the only soft clopen sets in Y,σ,B are 0B and 1B . Thus, M is a soft Q-set in Y,σ,B if and only if M=0B−N∪˜N−0B=N or M=1B−N∪˜N−1B=1B−N. In the case M=N, M is soft nowhere dense in Y,σ,B, and in the case M=1B−N, 1B−M=N is soft nowhere dense in Y,σ,B.


## 3. Boolean Algebra of SoftQ-Sets

Definition 7.
LetYbe a universal set andBbe a set of parameters. LetS,T∈SSY,B. The soft setS−T∪˜T−Sis called the soft symmetric difference ofSandTand will be denotedSΔT.Lemma 3.
LetYbe a universal set andBbe a set of parameters. LetF,G,H∈SSY,B. Then(1)
FΔG=GΔF.(2)
FΔGΔH=FΔGΔH.(3)
F∩˜GΔH=F∩˜GΔF∩˜H.Proof.
Straightforward.Lemma 4.
LetY,σ,Bbe a STS. IfFandGare soft nowhere dense sets inY,σ,B, thenFΔGis soft nowhere dense inY,σ,B.Proof.
LetF and G be two soft nowhere dense sets in Y,σ,B. Since F−G⊆˜F and G−F⊆˜G, then F−G and G−F are soft nowhere dense sets in Y,σ,B. Thus, by Theorem 4, FΔG=F−G∪˜G−F is soft nowhere dense sets in Y,σ,B.Theorem 10.
LetY,σ,B be a STS. If M1 and M2 are soft Q-sets in Y,σ,B, then M1∩˜M2 is a soft Q-set in Y,σ,B.Proof.
By Theorem9, M1=K1ΔN1 and M2=K2ΔN2 where K1 and K2 are both soft clopen in Y,σ,B, and N1 and N2 are both soft nowhere dense in Y,σ,B. Then by Lemma 3,(19)M1∩˜M2=M1∩˜K2△N2=M1∩˜K2ΔM1∩˜N2=K1△N1∩˜K2ΔK1△N1∩˜N2=K1∩˜K2ΔN1∩˜K2ΔK1∩˜N2ΔN1∩˜N2=K△N,Where K=K1∩˜K2 and N=N1∩˜K2ΔK1∩˜N2ΔN1∩˜N2. By Corollary 1, K is a soft Q-set in Y,σ,B. Since N1 and N2 are soft nowhere dense sets in Y,σ,B, then N1∩˜K2, K1∩˜N2, and N1∩˜N2 are soft nowhere dense sets in Y,σ,B. Thus, by Lemma 4, N is soft nowhere dense in Y,σ,B. Therefore, by Theorem 9, M1∩˜M2 is a soft Q-set in Y,σ,B.Theorem 11.
LetY,σ,Bbe a STS. IfMandGare softQ-sets inY,σ,B, then(a)
M∪˜G is a soft Q-set in Y,σ,B.(b)
M−G is a soft Q-set in Y,σ,B.(c)
BdσM is a soft Q-set in Y,σ,B.Proof.
(a) SinceM and G are soft Q-sets in Y,σ,B, then by Theorem 5, 1B−M and 1B−G are soft Q-sets in Y,σ,B. Since 1B−M∪˜G=1B−M∩˜1B−G, then by Theorem 10, 1B−M∪˜G is a soft Q-set in Y,σ,B. Thus, again by Theorem 5, M∪˜G is a soft Q-set in Y,σ,B.
(b) SinceG is a soft Q-set in Y,σ,B, then by Theorem 5, 1B−G is a soft Q-set in Y,σ,B. Thus, by Theorem 10, M−G=M∩˜1B−G is a soft Q-set in Y,σ,B.
(c) SinceM is a soft Q-set in Y,σ,B, then by Theorem 5, 1B−M is a soft Q-set in Y,σ,B. Thus, by Theorem 6, ClσM and Clσ1B−M are soft Q-sets in Y,σ,B. Hence, by Theorem 10, ClσM∩˜Clσ1B−M=BdσM is a soft Q-set in Y,σ,B.Theorem 12.
LetY,σ,Bbe a STS and letQbe the class of all softQ-sets ofY,σ,B. ThenQis a Boolean algebra with respect to the distinguished elements and Boolean operations defined by(1)
0=0B(2)
1=1B(3)
M∧N=M∩˜N(4)
M∨N=M∪˜N(5)
M′=1B−M.Proof.
We need to show that the(a)
right sides of (1)–(5) are softQ-sets; and(b)
Boolean axioms are satisfied by definition.
(a) SinceClσIntσ0B=IntσClσ0B=0B and ClσIntσ1B=IntσClσ1B=1B, then the right sides of (1) and (2) are soft Q-sets. Also, by Theorems 10, 11 (a), and 5, the right sides of (3)–(5) are soft Q-sets.
(b) Straightforward.
The following example shows that the Boolean algebra of all softQ-sets of a STS need not be complete, in general:Example 3.
LetY=ℤ,B=a, and(20)σ=F∈SSY,B:Fa=∅orY−Fais finite.
It is not difficult to check thatY,σ,B is soft connected and a soft set M is soft nowhere dense in Y,σ,B if and only if Ma is finite. Thus, by Corollary 2, M is a soft Q-set in Y,σ,B if and only if Ma is finite or Y−Ma is finite. For each n∈ℕ, let Mn∈SSY,B such that Mna=n. Then Mn is a soft Q-set in Y,σ,B for each n∈ℕ.Claim 2.
The setMn:n∈ℕhas no supremum.Proof of Claim. Suppose to the contrary that Mn:n∈ℕ has a supremum S. Then for each n∈ℕ, Mn⊆˜S and so, ∪n∈ℕ˜Mn⊆˜S. Thus, ℕ=∪n∈ℕ˜Mna⊆Sa. Hence, ℤ−Sa is finite. Since ℤ−Sa⊆ℤ−ℕ and ℤ−Sa is finite while ℤ−ℕ is infinite, then there exists x∈ℤ−ℕ∩Sa. Let T∈SSY,B with Ta=Sa−x. Then for each n∈ℕ, Mn⊆˜T. On the other hand, T⊆˜S and T≠S, a contradiction.


## 4. Correspondence

We start this section with the following natural question:Question 1.
LetY,σ,Bbe a STS andMbe a softQ-set inY,σ,B. Is it true thatMbis aQ-set inY,σbfor allb∈B?
The following example gives a negative answer to Question1.Example 4.
LetY=ℝ,B=a,b, andF,G,H∈SSY,Bdefined byFa=−∞,0,Fb=∅,Ga=Gb=0,∞,Ha=∅, andHb=−∞,0. Letσ=0B,1B,F,G,F∪˜G. SinceH=1B−F∪˜G, thenIntσClσH=IntσH=0BandClσIntσH=Clσ0B=0B. Therefore,His a softQ-set inY,σ,B. Note thatσb=∅,Y,0,∞. SinceIntσbClσbHb=IntσbClσb−∞,0=Intσbℝ=ℝ, while,ClσbIntσbHb=ClσbIntσb−∞,0=Clσb∅=∅, thenHbis not aQ-set inY,σb.Theorem 13.
LetY,μb:b∈B be a family of STSs and let M∈SSY,B. Then M is a soft Q-set in Y,⊕b∈Bμb,B if and only if Ma is a Q -set in Y,μa for all a∈B.Proof.
Necessity. Suppose thatM is a soft Q-set in Y,⊕b∈Bμb,B and let a∈B. Let σ=⊕b∈Bμb. Since M is a soft Q-set in Y,⊕b∈Bμb,B , then IntσClσM=ClσIntσM, and so IntσClσMa=ClσIntσMa. But by Lemma 4.9 of [34],(21)IntσClσMa=IntμaClσMa=IntμaClμaMa,andClσIntσMa=ClμaIntσMa=ClμaIntμaMa.
Therefore,IntμaClμaMa=ClμaIntμaMa. Hence, Ma is a Q-set in Y,μa.
Sufficiency. Suppose that Ma is a Q-set in Y,μa for all a∈B. Then for each a∈B, IntμaClμaMa=ClμaIntμaMa. Let σ=⊕b∈Bμb. Then by Lemma 4.9 of [34], for every a∈B,(22)IntμaClμaMa=IntμaClσMa=IntσClσMa,andClμaIntμaMa=ClμaIntσMa=ClσIntσMa.
Thus,IntσClσM=ClσIntσM. Hence, M is a soft Q-set in Y,⊕b∈Bμb,B.Corollary 3.
LetY,μbe a TS andBbe a set of parameters. ThenMis a softQ-set inY,τμ,Bif and only ifMbis aQ-set inY,μfor allb∈B.Proof.
For eachb∈B, let μb=μ. Then τμ=⊕b∈Bμb. So, by Theorem 13, we get the result.Proposition 1.
LetY,μb:b∈B be a family of STSs and let M∈SSY,B. Then Y,⊕b∈Bμb,B is soft extremally disconnected if and only if Y,μb is extremally disconnected for all b∈B.Proof. Necessity. Suppose that Y,⊕b∈Bμb,B is soft extremally disconnected and let b∈B. Let U∈μb. Then bU∈⊕b∈Bμb and so Cl⊕b∈BμbbU=bClμbU∈⊕b∈Bμb. Hence, bClμbUb=ClμbU∈μb. Therefore, Y,μb is extremally disconnected.Sufficiency. Suppose that Y,μb is extremally disconnected for all b∈B. Let F∈⊕b∈Bμb. Then for all b∈B , Fb∈μb and so ClμbFb∈μb. Thus, by Lemma 4.9 of [34], Cl⊕b∈BμbFb=ClμbFb∈μb for all b∈B. Therefore, Cl⊕b∈BμbF∈⊕b∈Bμb. Hence, Y,⊕b∈Bμb,B is soft extremally disconnected.Corollary 4.
LetY,μ be a TS and B be a set of parameters. Then Y,τμ,B is soft extremally disconnected if and only if Y,μ is extremally disconnected.Proof.
For eachb∈B, let μb=μ. Then τμ=⊕b∈Bμb. So, by Proposition 1, we get the result.Theorem 14.
LetY,σ,B be a soft extremally disconnected STS. Then every soft open set in Y,σ,B is a soft Q-set in Y,σ,B.Proof.
LetM∈σ. Since Y,σ,B is soft extremally disconnected, then ClσM∈σ and so IntσClσM=ClσM. On the other hand, since M∈σ, then IntσM=M and so ClσIntσM=ClσM. Therefore, IntσClσM=ClσIntσM. Thus, M is a soft Q-set in Y,σ,B.
The following example shows that softQ-sets in a soft extremally disconnected STS need not be soft open sets in general:Example 5.
LetYbe any infinite set andBbe any set of parameters. Letμbe the cofinite topology onY. Consider the STSY,τμ,B. SinceY,μis extremally disconnected, then by Corollary4,Y,τμ,Bis soft extremally disconnected. ChooseU⊆Ysuch thatUis a non-empty finite set. ThenIntμClμU=ClμIntμU=∅, and henceUis aQ-set inY,μ. Since for eachb∈B,CUb=U, then by Corollary3,CUis a softQ-set inY,τμ,B. On the other hand, it is clear thatCU∉τμ.
The following example shows in Theorem14 that the condition “soft extremally disconnected” cannot be dropped:Example 6.
LetY=ℝandBbe any set of parameters. Letμbe the usual topology onY. Consider the STSY,τμ,B. ThenC0,1∈τμ. On the other hand, sinceIntτμClτμC0,1=C0,1butClτμIntτμC0,1=C0,1, thenC0,1is not a softQ-set inY,τμ,B.


## 5. The Commutativity of the Soft Boundary and Soft Interior of a STS

Lemma 5.
LetY,σ,Bbe a STS. IfM∈σandNis soft dense inY,σ,B, thenClσM∩˜N=ClσM∩˜ClσN=ClσM.Proof.
We only need to show thatClσM∩˜ClσN⊆˜ClσM∩˜N. Let by∈˜ClσM∩˜ClσN and let G∈σ such that by∈˜G. Since by∈˜ClσM, then G∩˜M≠0B. Since G∩˜M∈σ−0B and N is soft dense in Y,σ,B, then G∩˜M∩˜N=G∩˜M∩˜N≠0B. It follows that by∈˜ClσM∩˜N.Lemma 6.
LetY,σ,Bbe a STS and letM∈SSY,B. ThenIntσBdσM∩˜BdσIntσM=0B.Proof.
By Theorem3, we have
IntσClσ1B−M=Intσ1B−IntσM=1B−ClσIntσM. Thus,(23)IntσBdσM∩˜BdσIntσM=IntσClσM∩˜Clσ1B−M∩˜ClσIntσM∩˜Clσ1B−IntσM=IntσClσM∩˜IntσClσ1B−M∩˜ClσIntσM∩˜Clσ1B−IntσM=IntσClσM∩˜1B−ClσIntσM∩˜ClσIntσM∩˜Clσ1B−IntσM=0B.Theorem 15.
LetY,σ,Bbe a STS and letM∈SSY,B. ThenIntσBdσM=BdσIntσMif and only ifIntσClσM=ClσIntσM=IntσM.Proof.
Necessity. Suppose that IntσBdσM=BdσIntσM. Then by Lemma 6, IntσBdσM=BdσIntσM=0B. Since(24)IntσBdσM=IntσClσM∩˜1B−ClσIntσM
thenIntσClσM⊆˜ClσIntσM.
Also, applying Theorem3, we have(25)BdσIntσM=ClσIntσM∩˜Clσ1B−IntσM=ClσIntσM∩˜ClσClσ1B−M=ClσIntσM∩˜ClσClσ1B−M=ClσIntσM∩˜Clσ1B−M=0B,
and hence,ClσIntσM⊆˜1B−Clσ1B−M=IntσM. Therefore, IntσClσM⊆˜ClσIntσM⊆˜IntσM. But IntσM⊆˜IntσClσM is always true. Therefore, IntσM=IntσClσM=ClσIntσM.
Sufficiency. Suppose that IntσClσM=ClσIntσM=IntσM. Then(26)IntσBdσM=IntσClσM∩˜Clσ1B−M=IntσClσM∩˜IntσClσ1B−M=IntσM∩˜Intσ1B−IntσM⊆˜IntσM∩˜1B−IntσM=0B,andBdσIntσM=ClσIntσM∩˜Clσ1B−IntσM=ClσIntσM∩˜ClσClσ1B−M=IntσM∩˜Clσ1B−M=IntσM∩˜1B−IntσM=0B.
Thus,IntσBdσM=BdσIntσM.Theorem 16.
LetY,σ,Bbe a STS and letM∈SSY,B. ThenIntσBdσM=BdσIntσMif and only ifM=K∪˜N, whereKis soft clopen inY,σ,B,Nis soft nowhere dense inY,σ,B, andK∩˜N=0B.Proof.
By Theorem15, it is sufficient to show that IntσClσM=ClσIntσM=IntσM if and only if M=K∪˜N, where K is soft clopen in Y,σ,B, N is soft nowhere dense in Y,σ,B, and K∩˜N=0B.
Necessity. Suppose that IntσClσM=ClσIntσM=IntσM. Let K=IntσM and N=M−K. Then K is soft clopen in Y,σ,B, M=K∪˜N, and K∩˜N=0B. The proof that N is soft nowhere dense in Y,σ,B is similar to that used in the proof of Claim (a) in Theorem 9.
Sufficiency. Suppose that M=K∪˜N, where K is soft clopen in Y,σ,B, N is soft nowhere dense in Y,σ,B, and K∩˜N=0B. By Theorem 8 and Corollary 1, we have N and K are soft Q-sets in Y,σ,B. So, by Theorem 11 (a), M=K∪˜N is a soft Q-set in Y,σ,B. Hence, IntσClσM=ClσIntσM. Since N is soft nowhere dense in Y,σ,B, then 1B−ClσN is soft dense in Y,σ,B, and so 1B−N is soft dense. Thus, by Lemma 5, Clσ1B−K∩˜1B−N=Clσ1B−K=1B−K. Therefore, by Theorem 3, we have(27)IntσM=IntσK∪˜N=1B−Clσ1B−K∪˜N=1B−Clσ1B−K∩˜1B−N=1B−1B−K=K.
Hence,IntσM is soft closed in Y,σ,B. Thus,(28)IntσM⊆˜IntσClσM=ClσIntσM=IntσM.
This shows thatIntσClσM=ClσIntσM=IntσM.


## 6. Conclusion

In this paper, softQ-sets as a new class of sets which contains both soft clopen and soft nowhere dense sets are introduced. Soft Q-sets have been characterized in terms of soft clopen sets and soft nowhere dense sets. It is proved that soft Q-sets form a Boolean algebra that is not complete, in general. Furthermore, soft sets whose soft boundaries and soft interiors are commutative are characterized. In addition, the correspondence between Q-sets in topological spaces and soft Q-sets in soft topological spaces is investigated.Boolean algebra constitutes the basis for the design of circuits used in electronic digital computers. In addition, it is of significance to the theory of probability, the geometry of sets, and information theory. So, this paper not only forms the theoretical basis for further applications of soft topology, but it also leads to the development of the theory of probability, the geometry of sets, and information theory.In the upcoming work, we plan to: (1) find sufficient condition for the Boolean algebra of softQ-sets to be complete; and (2) investigate the behavior of soft Q-open sets under product soft topological spaces.



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*Source: 5200590-2022-08-28.xml*