A new approach for soft topology and soft function via soft element

Abstract

In this article, we give some new properties of elementary operations on soft sets and then we introduce a new soft topology by using elementary operations over a universal set with a set of parameters called elementary soft topology. Also, we define a topology, members of which are collections of the soft elements and give the relation between this topology and elementary soft topology. We show that this new soft topology is different from those previously defined soft topologies. We prove some of the properties of the topological concepts we investigate in this topology. Finally, we describe soft function and soft continuity and give an application of the soft function as soft set approach to the rotation in double-struckE3.

Compactness of soft cone metric space and fixed point theorems related to diametrically contractive mapping

“…The elementary union (F, P ) = (G, P ) ⋓ (H, P ) and elementary intersection (F, P ) = (G, P ) ⋒ (H, P ) of (G, P ), (H, P ) ∈ S(Ũ ) are defined by (F, P ) = SS(SE(G, P ) ∪ SE(H, P )) and (F, P ) = SS(SE(G, P ) ∩ SE(H, P )), respectively. (For details, see [16,20,28,42]).…”

“…Das and Samanta [20] introduced the soft metric concept via the notation of soft element and proved Banach fixed point theorem. Taşköprü and Altıntaş [42] established a new topological structure on soft sets using elementary operations. Nowadays, many researches make interesting studies by applying the soft set theory to almost every area.…”

“…12[42] Assume thatX is an absolute soft set. We call a soft sets class τ ⊂ S(X) an elementary soft topology onX and call the triple X , τ, P an elementary soft topological space if the following axioms are satisfied.…”

Countable and separable elementary soft topological space

“…The elementary union and elementary intersection of are defined by and , respectively. (For details, see 26 ).…”

“…concepts via soft elements 20‐25 . Taşköprü and Altıntaş 26 established a new topological structure on the soft sets using elementary operations ( ε ‐operations), which called elementary soft topology ( ε ‐soft topology). In recent years, many mathematical structures have been established on the soft sets and their applications have been given to different field of science 5,16,27‐43 …”