Boolean Algebra of Soft Q-Sets in Soft Topological Spaces

Abstract

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. 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.

Novel Approach to Study Soft α-Open Sets and Its Applications via Fuzzy Soft Topologies

“…The concept of soft β-open sets was defined and studied by the authors of [3,4]. Also, the concepts of soft semi-open, somewhere dense and Q-sets were studied by the authors of [5,6]. Al-shami et al [7] introduced the concept of weakly soft semi-open sets and obtained its main properties.…”

On Fuzzy Soft Beta-Continuity and Beta-Irresoluteness: Properties and Applications

“…The concept of soft β-open sets was studied by the authors of [13,14], and some properties of soft β-continuity are investigated. Also, the concepts of somewhere dense and Q-sets were defined and studied by the authors of [15,16]. Al-shami et al [17] initiated the concept of weakly soft semi-open sets and studied its main properties.…”

On soft parametric somewhat-open sets and applications via soft topologies

“…Al-shami [43] displayed a new class called somewhere dense sets and applied to investigate soft continuity and soft compactness. The notions of soft Q -sets [44] and minimal soft sets [45] were defined by Al-Ghour. Recently, Ameen et al [46] have provided several soft functions motivated by the family of soft somewhat-open sets, which were exploited to initiate some classes of separation axioms by Al-shami [2] .…”

“…Another method to produce classical topologies from a soft topology Θ is given in [13] as follows: . The links and navigation of the concepts and their properties from a soft topology to its classical topologies and vise versa have been revealed and discussed in some interesting studies [44] , [25] . Al-shami and Kočinac [50] demonstrated the interchangeable characteristic for the operators of interior and closure between specific kind of soft topology namely “extended soft topology” and their classical topologies.…”