Soft set theory and topology

Abstract

In this paper we study and discuss the soft set theory giving new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets. Furthermore, we investigate the theory of soft topological spaces and we present new definitions, characterizations, and properties concerning the soft closure, the soft interior, the soft boundary, the soft continuity, the soft open and closed maps, and the soft homeomorphism.

On Soft Topological Spaces

“…Proposition 2.18 (see [9]) Let (X, τ X , A) and (Y, τ Y , B) be two soft topological spaces and e : A → B. Then, the following statements are equivalent:…”

“…We say that the family τ defines a soft topology on X if the following axioms are true: (1) 0 0 0 A , 1 1 1 A ∈ τ. Definition 2.10 (see [9]) Let (X, τ, A) be a soft topological space, a ∈ A, and x ∈ X. We say that a soft set (G, A) ∈ τ is an a-soft open neighborhood of x in (X, τ, A) if x ∈ G(a).…”

“…Definition 2.17 (see [9]) Let (X, τ X , A) and (Y, τ Y , B) be two soft topological spaces, x ∈ X, and e : A → B. A map f : X → Y is called soft e-continuous at the point x if for every a ∈ A and every e(a)-soft open neighborhood…”

“…We say that a soft set (G, A) ∈ τ is an a-soft open neighborhood of x in (X, τ, A) if x ∈ G(a). Proposition 2.16 (see [9]) Let (X, τ, A) be a soft topological space. Then, a family B ⊆ τ is a base for (X, τ, A) if and only if for every a ∈ A, x ∈ X, and every…”

“…Furthermore, D. Pei and D. Miao [22] showed that soft sets are a class of special information systems. In [9] for the soft set theory: new definitions, examples, new classes of soft sets, and properties for mappings between different classes of soft sets are introduced and studied. Moreover, the theory of soft topological spaces is investigated.…”