Abstract
In this work, we introduce new types of soft separation axioms calledpt-softαregular andpt-softαTi-spacesi=0,1,2,3,4using partial belong and total nonbelong relations between ordinary points and softα-open sets. These soft separation axioms enable us to initiate new families of soft spaces and then obtain new interesting properties. We provide several examples to elucidate the relationships between them as well as their relationships withe-softTi, softαTi, andtt-softαTi-spaces. Also, we determine the conditions under which they are equivalent and link them with their counterparts on topological spaces. Furthermore, we prove thatpt-softαTi-spacesi=0,1,2,3,4are additive and topological properties and demonstrate thatpt-softαTi-spacesi=0,1,2are preserved under finite product of soft spaces. Finally, we discuss an application of optimal choices using the idea ofpt-softTi-spacesi=0,1,2on the content of soft weak structure. We provide an algorithm of this application with an example showing how this algorithm is carried out. In fact, this study represents the first investigation of real applications of soft separation axioms.
Highlights
Afterwards, Maji et al [2] started studying the operations between soft sets such as soft union and soft intersections
We continue studying soft topology using the definition given by Shabir and Naz. ey formulated the Mathematical Problems in Engineering notions of soft interior and soft closure operators and soft subspaces and shed light on soft separation axioms
Following Shabir and Naz’s work, many researchers explored the topological concepts on the domain of soft topology and examined the similarity and divergence between classical topology and soft topology
Summary
Definition 1 (see [1]). For a nonempty set X and a set of parameters E, a pair (G, E) is said to be a soft set over X provided that G is a map of E into the power set P(X). Each G(e) is called a component of GE (or e-approximate), and a family of all soft sets defined over X with E is denoted by S(XE). We say that a soft set is countable If we make all approximations of a soft set equal to a fixed subset S of the universal set X, we call it a stable soft set and denote it by S. (i) eir intersection, denoted by GE∩ HE, is a soft set UE, where a mapping U: E ⟶ 2X is given by U(e) G(e) ∩ H(e). (ii) eir union, denoted by GE∪ HE, is a soft set UE, where a mapping U: E ⟶ 2X is given by U(e) G(e) ∪ H(e). Definition 6 (see [4]). e Cartesian product of two soft sets GE and HF over X and Y, respectively, is a soft set G × HE×F over X × Y defined by (G × H)(e, f) G(e) × H(f) for each (e, f) ∈ E × F
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