Boolean Algebra of Soft Q-Sets in Soft Topological Spaces

We dene 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 softQ-set are all softQ-sets. We show that a soft subset K of a given soft topological space is a soft Qset if and only if K is a soft symmetric dierence 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 softQ-sets is closed under nite soft intersections and nite soft unions, and as a main result, we prove that the class of softQ-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.


Introduction and Preliminaries
Some problems in medicine, engineering, the environment, economics, sociology, and other elds have their own doubts. erefore, 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. ese di culties 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][5][6].
General topology, as one of the main branches of mathematics, is the branch of topology that deals with the basic de nitions of set theory and structures used in topology. It is the foundation of most other branches of topology, including algebraic topology, geometric topology, and di erential 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][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][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 di erence 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, Hausdor , and metastonean if and only if its classes of Borel sets and Q-sets are equal. e 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 de ne 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 di erence 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 nite soft intersections and nite 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]. roughout this paper, ST and STS will denote topological space and soft topological space, respectively. Let e following definitions will be used in the sequel: Definition 1 (see [25]). Let (Y, μ) be a TS and let U⊆Y . en Definition 2 (see [29] Definition 3 (see [27]). Let Y be a universal set and B be a set of parameters. en G ∈ SS(Y, B) defined by Theorem 1 (see [7]). Let (Y, σ, B) be a STS. en the collection G(b): G ∈ σ { } defines a topology on Y for every b ∈ B . is topology will be denoted by σ b .
Theorem 2 (see [27]). Let Y be an initial universe and let B be a set of parameters. Let μ b : b ∈ B be an indexed family of topologies on Y and let (1) Then σ defines a soft topology on Y relative to B. is soft topology will be denoted by ⊕ b∈B μ b .
Proof. Straightforward. □ Lemma 4. Let (Y, σ, B) be a STS. If F and G are soft nowhere dense sets in (Y, σ, B) , then FΔG is soft nowhere dense in (Y, σ, B).
(c) Since M is a soft Q-set in (Y, σ, B), then by eorem 5, 1 B − M is a soft Q-set in (Y, σ, B). us, by eorem 6, Cl σ (M) and Cl σ (1 B − M) are soft Q-sets in (Y, σ, B). Hence, by eorem 10, (Y, σ, B). □ Theorem 12. Let (Y, σ, B) be a STS and let Q be the class of all soft Q -sets of (Y, σ, B). en Q is a Boolean algebra with respect to the distinguished elements and Boolean operations defined by Proof. We need to show that the (a) right sides of (1)-(5) are soft Q-sets; and (b) Boolean axioms are satisfied by definition.

Correspondence
We start this section with the following natural question:

Conclusion
In this paper, soft Q-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 soft Q-sets to be complete; and (2) investigate the behavior of soft Q-open sets under product soft topological spaces.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.