In 1999, Molodtsov [1] initiated the notion of soft set theory as a new mathematical tool which is free from the complex problems. Later on Maji et al. [2] proposed several operations on soft sets and some basic properties and then Pei and Miao [3] investigated the relationships between soft sets and information systems.

In 2011, Shabir and Naz [4] introduced the notion of soft topological spaces and the author [5] corrected some their results. Zorlutuna et al. [6] continued to study the properties of soft topological spaces by defining the concepts of interior and soft neighborhoods in soft topological spaces. In 2011, Cagman et al. [7] defined soft topological spaces by modifying the soft set. Also, Roy and Samanta [8] strengthen the definition of the soft topological spaces presented in [7].

In 2017, with the aim of generalizing the notion of soft topology, Zakari et al. [9] introduced a soft weak structure. Recently, Al-Saadi and Min [10] investigated the notion of soft generalized closed sets in a soft weak structure.

Meanwhile, Min and Kim [11] introduced a new notion called weak structures as the following: Let X be a non-empty set and P(X) be its power set. A structure ω on X is called a weak structure on X if and only if (i) φ ∈ ω: (ii) For U, V ∈ ω, U ∩ V ∈ ω.

In this work, by applying the notion of weak structure in [11], we want to introduce the new notion of soft w-structure which is a generalized soft topological structure defined by Shabir and Naz [10] and a stronger structure than soft weak structure defined by Zakari et al. [9]. And we investigate some basic properties of this new class by using the notion of soft set. Moreover, we study the notions of soft w−T₀ (soft w−T₁, soft w−T₂) and some properties of such notions. From now on, let X be a non-empty common universe, E a set of parameters, and P(X) denote the power set of X.

Definition 1.1 ( [1])

For A ⊆ E, a pair (F, A) is called a soft set over X, where F is a mapping given by F : A → P(X). For e ∈ A, F(e) may be considered as the set of e-approximate elements of the soft set (F, A).

Definition 1.2 ( [2])

A soft set (F, A) over X is said to be:

• A null soft set denoted by ̃ if F(e) = ∅︀ for all e ∈ A.

• An absolute soft set denoted by X̃ if F(e) = X for all e ∈ A.

Definition 1.3 ( [2])

For any two soft sets (F, A) and (G, B) defined over a common universe X, we have:

• (F, A) ⊂̃ (G, B) iff A ⊆ B and F(e) ⊆ G(e) for all e ∈ A.

• (F, A)=(G, B) iff (F, A) ⊂̃ (G, B) and (G, B) ⊂̃ (F, A).

• (F, A)∪̃(G, B)=(H, C) where C = A ∪ B and

  H(e)={F(e),if e∈A-B,G(e),if e∈B-A,F(e)∪G(e),if e∈A∩B,

  for all e ∈ C.

• (F, A)∩̃(G, B) = (K, D) where D = A ∩ B and K(e) = F(e) ∩ G(e) for all e ∈ C.

• x ∈ (F, A) where x ∈ X iff x ∈ F(e) for all e ∈ A and x ∉ (F, A) whenever x ∉ F(e) for some e ∈ A.

Definition 1.4 ([12])

For a soft set (F, A) over X, the relative complement of (F, A) (denoted by (F, A)′) is defined by (F, A)′=(F′, A), where F′ : A → P(X) is given by F′(e) = X − F(e) for all e ∈ A.

Definition 1.5 ( [4])

Let τ be the collection of soft sets over X. Then τ is called a soft topology on X if τ satisfies the following axioms:

• ̃, X̃ belong to τ.

• The union of any number of soft sets in τ belong to τ.

• The intersection of any two soft sets in τ belong to τ.

The triple (X, τ, E) is called a soft topological space over X. The member of τ are said to be soft open in X. A soft set (F, E) over X is said to be soft closed in X if its relative complement (F, E)′ belong to τ.

Definition 2.1

Let sw be the collection of soft sets over X. Then sw is called a soft w-structure on X if sw satisfies the following axioms:

• ̃, X̃ belong to sw.

• The intersection of any two soft sets in sw belongs to sw.

The triple (X, sw, E) is called a soft w-space over X. The member of sw is said to be soft w-open in X. A soft set (F, E) over X is said to be soft w-closed in X if its relative complement (F, E)′ belongs to sw.

Remark 2.2

Let sw be a soft w-structure over X. The soft w-structure sw is a kind of generalized soft topology and a stronger structure than a soft weak structure defined by Zakari et al. [9] as the following: Let X be a non-empty set and E a set of parameters. A collection ω of soft sets defined over X with respect to E is called a soft weak structure [9] iff ̃ ∈ ω.

Example 2.3

Let X = {h₁, h₂, h₃, h₄}, E = {e₁, e₂} and sw = {̃, X̃, (F₁, E), (F₂, E), (F₃, E)}, where

Then sw is a soft w-structure over X with respect to E but not a soft topology.

Definition 2.4

Let sw be a soft w-structure over X with respect to E. For a soft set (F, E) over X, the soft w-closure of (F, E) (simply, c_sw(F, E)) and the soft w-interior of (F, E) (simply, c_sw(F, E)) are defined as the following:

• i_sw(F, E)= ∪̃{(G, E) : (G, E) ⊂̃ (F, E), (G, E) ∈ sw}.

• c_sw(F, E)= ∩̃{(H, E) : (F, E) ⊂̃ (H, E), (H, E)′ ∈ sw}.

Theorem 2.5

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If there exists a soft w-open set (G, E) such that x ∈ (G, E) ⊂̃ (F, E), then x ∈ i_sw(F, E)

Example 2.6

As in Example 2.3, consider the soft w-structure sw over X with respect to E and a soft set (F₄, E) as follows:

Then (F₄, E) = i_sw(F₄, E). For h₃ ∈ i_sw(F₄, E), there is no a soft w-open set containing h₃ in sw. So the converse of Theorem 2.5 is not always true.

Theorem 2.7

Let sw be a soft w-structure over X with respect to the parameters set E and (F, E) a soft set. If x ∈ c_sw(F, E), then (G, E)∩̃(F, E) ≠ ̃ for all (G, E) ∈ sw such that x ∈ (G, E).

Example 2.8

Let X = {h₁, h₂, h₃}, E = {e₁, e₂} and sw = {̃, X̃, (F₁, E), (F₂, E), (F₃, E)} where

Then sw is a soft w-structure over X with respect to E. Consider a soft set (F₄, E) defined as:

Since (F₄, E) is soft w-closed, (F₄, E) = c_sw(F₄, E). For h₁ ∈ X, (F₂, E) is the only soft w-open set and (F₄, E)∩̃(F₂, E) ≠ ̃, however, h₁ ∉̃ c_sw(F₄, E). So the converse of Theorem 2.7 is not always true.

Theorem 2.9

Let sw be a soft w-structure defined over X with respect to the parameters set E and (F, E) be a soft set.

• If (F, E) is a soft w-open set, then (F, E) = i_sw(F, E).

• If (F, E) is a soft w-closed set, then (F, E) = c_sw(F, E).

Example 2.10

Let X = {h₁, h₂, h₃}, E = {e₁, e₂} and sw = {̃, X̃, (F₁, E), (F₂, E), (F₃, E), (F₄, E), (F₅, E)}, where

Then sw is a soft w-structure over X with respect to E. For a soft set (F₅, E), c_sw(F₅, E) = (F₅, E) but (F₅, E) is not soft w-closed. And, for a soft set (F₆, E), i_sw(F₆, E) = (F₆, E) but (F₆, E) is not soft w-open.

Theorem 2.11

Let sw be a soft w-structure over X with respect to E. Let (F, E) and (G, E) be two soft sets over X. Then:

• i_sw(F, E) ⊂̃ (F, E).

• If (F, E) ⊂̃ (G, E), then i_sw(F, E) ⊂̃ i_sw(G, E).

• i_sw((F, E)∩̃(G, E)) = i_sw(F, E)∩̃i_sw(G, E).

• i_sw(i_sw(F, E)) = i_sw(F, E).

Theorem 2.12

Let sw be a soft w-structure defined over X with respect to E. If (F, E) and (G, E) are two soft sets over X, then:

• (F, E) ⊂̃ c_sw(F, E).

• If (F, E) ⊂̃ (G, E), then c_sw(F, E) ⊂̃ c_sw(G, E).

• c_sw(F, E)∪̃c_sw(G, E) = c_sw((F, E)∪̃(G, E)).

• c_sw(c_sw(F, E)) = c_sw(G, E).

• i_sw(F, E)′=(c_sw(F, E))′ and c_sw(F, E)′=(i_sw(F, E))′.

Definition 2.13

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called:

• w − T₀ if for each x, y ∈ X such that x ≠ y, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∉ (F, E) or x ∉ (F, E) and y ∈ (F, E).

• w − T₁ if for each x, y ∈ X such that x ≠ y, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E) and y ∉ (F, E) and x ∉ (G, E) and y ∈ (G, E).

• w − T₂ if for each x, y ∈ X such that x ≠ y, there exist soft w-open sets (F, E) and (G, E) such that x ∈ (F, E), y ∈ (G, E) and (F, E)∩̃(G, E) = ̃.

We have the following diagram:

Example 2.14

Let X = {h₁, h₂, h₃}, E = {e₁, e₂} and sw = {̃, X̃, (F₁, E), (F₂, E), (F₃, E), (F₄, E), (F₅, E), (F₆, E)}, where

Then sw is a soft w-structure over X with respect to E. It is obviously a soft w − T₁ space. For h₁, h₂ ∈ X, (F₂, E) and (F₃, E) are unique soft w-open sets of h₁, h₂, respectively. But (F₂, E)∩̃(F₃, E) ≠ ̃. So (X, sw, E) is not soft w − T₂.

Example 2.15

Let X = {h₁, h₂, h₃}, E = {e₁, e₂} and sw = {̃, X̃, (F₁, E), (F₂, E), (F₃, E), (F₄, E)}, where

Then sw is a soft w-structure over X with respect to E. It is obviously a soft w − T₀ space but it is not soft w − T₁.

Let sw be a soft w-structure over X with respect to E. A soft w-space (X, sw, E) is called relative soft w − T₀ if for each x, y ∈ X such that x ≠ y, there exists a soft w-open set (F, E) such that x ∈ (F, E) and y ∈ (F, E)′ or x ∈ (F, E)′ and y ∈ (F, E).

Theorem 2.16

Let sw be a soft w-structure on X. If X is a relative soft w−T₀ space, then for each x, y ∈ X such that x ≠ y, we have c_sw(x, E) ≠ c_sw(y, E).

Theorem 2.17

Let sw be a soft w-structure on X. If y ∈ c_sw(x, E), then for each soft w-open set (G, E) containing y, there exists a parameter e ∈ E such that x ∈ G(e).

Theorem 2.18

Let sw be a soft weak structure on X. A soft w-space (X, sw, E) is soft w − T₁ if (x, E) is soft w-closed set for all x ∈ X.

The author introduced the notion of soft w-structure and investigated some basic properties of this new structure. In the next research, the author will introduce the associated soft w-structures induced by soft topologies and study the relationship between soft w-structures and associated soft w-structure induced by soft topologies.