Throughout this paper, we follow the notions and terminologies as used in [1] and [2], and for simplicity, STS stands for soft topological space. For the purpose of dealing with uncertain objects, Molodtsov [3] introduced soft sets in 1999. Let Y be a universal set and B be a set of parameters. A soft set over Y relative to B is a function H : B → ℘(Y). The family of all soft sets over Y relative to B is denoted by SS (Y, B). Let H ∈ SS (Y, B), then H is a countable soft set if H (b) is a countable set for all b ∈ B. The family of all members of SS (Y, B) that are countable is denoted by CSS (Y, B). Furthermore, the null soft set and the absolute soft set are denoted by 0_B and 1_B, respectively. The authors in [4] define the notion of STSs as follows: The triplet (Y, τ, B), where τ ⊆ SS (Y, B) is a STS if 0_B and 1_B ∈ τ, τ is closed under finite soft intersection, and τ is closed under arbitrary soft union. If (Y, τ, B) is a STS, then the members of τ are called soft open sets and their soft complements are called soft closed sets. Topologists recently have applied various topological concepts to soft topological spaces ([1,2,5–25]).

Chen [26] introduced the concept of soft semi-open sets, a weaker form of soft open sets in STSs. This paper was followed by many more papers about soft semi-open sets and their modifications. In this paper, we introduce soft ω_s-open sets as a class of soft sets that lies strictly between soft open sets and soft semi-open sets. We present the natural properties of soft ω_s-open sets. Furthermore, using ω_s-open sets, we define and investigate soft ω_s-closure and ω_s-interior as new soft topological operators. We also compare our proposed concepts and their corresponding previous analogous concepts in Theorems 2.2, 2.17, 3.3, and 3.19, with the comparisons supported by examples. We present several sufficient conditions for the equivalence between the proposed concepts and their corresponding previous analogous concepts. We next discuss relationships between relative topological spaces and generated soft topological spaces. Finally, we raise an open question related to soft ω_s-open.

In [27,28], the authors showed that soft sets are a class of special information systems. This insight constitutes our motivation to study the structures of soft sets for information systems. Therefore, this paper does not only form the theoretical basis for further applications of soft topology, such as soft ω_s-mappings and soft ω_s-compactness, but also comments on the future development of information systems.

The following definitions and results will be used in the sequel:

Definition 1.1

Let (X, ℑ) be a topological space and let D ⊆ X. The D is called

(a) [29] semi-open if there is V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑ(V). SO(X, ℑ) will denote the family of all semi-open sets in (X, ℑ).

(b) [30] ω_s-open if there is V ∈ ℑ such that V ⊆ D ⊆ Cl_ℑω (V). ω_s(X, ℑ) will denote the family of all ω_s-open sets in (X, ℑ).

Definition 1.2 [26]

Let (X, τ, A) be a STS and let K ∈ SS(X, A). Then

(a)K is called a soft semi-open set in (X, τ, A) if there exists F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τ (F). SO(X, τ, A) will denote the family of all soft semi-open sets in (X, τ, A).

(b)K is called a soft semi-closed set in (X, τ, A) if 1_A – K ∈ SO(X, τ, A).

Definition 1.3 [26]

Let (X, τ, A) be a STS and H ∈ SS(X, A).

(a) The soft semi-closure of H in (X, τ, A) is denoted by S-Cl_τ (H) and defined by S-Cl_τ (H) = ∩̃ {M : M is soft semi-closed in (X, τ, A) and H ⊆̃ M}.

(b) The soft semi-interior of H in (X, τ, A) is denoted by S-int_τ (H) and defined by S-int_τ (H) = ∪̃ {K : K is soft semi-open in (X, τ, A) and K ⊆̃ H}.

As defined in [2], a STS (X, τ, A) is called soft anti-locally countable if for every F ∈ τ – {0_A}, F ∉ CSS(X, A).

Proposition 1.4 [2]

Let (X, τ, A) be soft anti-locally countable. Then for all G ∈ τ_ω, Cl_τ (G) = Cl_τω (G). As defined in [2], a STS (X, τ, A) is called soft locally countable if for a_x ∈ SP(X, A) there exists G ∈ CSS(X, A) ∩ τ such that a_x ∊̃ G.

Proposition 1.5 [2]

A STS (X, τ, A) is soft locally countable if and only if (X, τ_ω, A) is a discrete STS.

Proposition 1.6 [2]

For any STS (X, τ, A) we have τ_ω = (τ_ω)_ω.

Proposition 1.7 [26]

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then

(a) S-Cl_τ (H) is the smallest soft semi-closed in (X, τ, A) containing H.

(b) H is soft semi-closed in (X, τ, A) if and only if H = S-Cl_τ (H).

Definition 2.1

Let (X, τ, A) be a STS and let K ∈ SS(X, A). Then K is called a soft ω_s-open set in (X, τ, A) if there exists F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). ω_s(X, τ, A) will denote the family of all soft ω_s-open sets in (X, τ, A).

Theorem 2.2

Let (X, τ, A) be a STS. Then τ ⊆ ω_s(X, τ, A) ⊆ SO(X, τ, A).

Proof

Let G ∈ τ, choose F = G. Then F ∈ τ and F ⊆̃ G ⊆̃ Cl_τω (F) which implies that G ∈ ω_s(X, τ, A). This shows that τ ⊆ ω_s(X, τ, A). Let K ∈ ω_s(X, τ, A), then by definition we find F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F) ⊆̃ Cl_τ (F) which implies that K ∈ SO(X, τ, A). It follows that ω_s(X, τ, A) ⊆ SO(X, τ, A).

The following three examples show that none of the two inclusions in Theorem 2.2 is equality in general:

Example 2.3

Let X = ℝ and A = [0, 1]. Let F, G ∈ SS(X, A) such that for every a ∈ A, F(a) = ℕ and G(a) = ℚ^c. Let τ = {0_A, 1_A, F, G, F ∪̃ G}. One can easily check that Cl_τω (F) = F, Cl_τ (F) = 1_A – G and Cl_τω (G) = 1_A – F. Hence 1_A – G ∈ SO(X, τ, A) – ω_s(X, τ, A) and 1_A – F ∈ ω_s(X, τ, A) – τ.

Example 2.4

Let X = ℝ and A = ℕ. Let τ = {F ∈ SS(X, A) : F(a) belongs to the usual topology on ℝ}.

Let G ∈ SS(X, A) such that for every a ∈ A, G(a) = [0, 1). Then G ∈ ω_s(X, τ, A) – τ.

Example 2.5

Let X = ℚ and A = ℝ. Let F, G ∈ SS(X, A) such that for every a ∈ A, F(a) = {1} and G(a) = {1, 2}. Let τ = {0_A, 1_A, F}. Then G ∈ SO(X, τ, A) – ω_s(X, τ, A).

Theorem 2.6

If (X, τ, A) is a soft anti-locally countable STS, then ω_s(X, τ, A) = SO(X, τ, A).

Proof

By Theorem 2.2, we only need to see that SO(X, τ, A) ⊆ ω_s(X, τ, A). Let K ∈ SO(X, τ, A), then there is F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τ (F). By Proposition 1.4, Cl_τ (F) = Cl_τω (F). Therefore, K ∈ ω_s(X, τ, A). This shows that SO(X, τ, A) ⊆ ω_s(X, τ, A).

Theorem 2.7

For any STS (X, τ, A), SO(X, τ_ω, A) = ω_s(X, τ_ω, A).

Proof

Let (X, τ, A) be STS. By Theorem 2.2, we only need to show that SO(X, τ_ω, A) ⊆̃ ω_s(X, τ_ω, A). Let K ∈ SO(X, τ_ω, A), then there is F ∈ τ_ω such that F ⊆̃ K ⊆̃ Cl_(τω)ω (F). As by Proposition 1.6, (τ_ω)_ω = τ_ω, then Cl_(τω)ω (F) = Cl_τω (F). Hence, K ∈ ω_s(X, τ_ω, A). This ends the proof that SO(X, τ_ω, A) ⊆ ω_s(X, τ_ω, A).

Theorem 2.8

If (X, τ, A) is a soft locally countable STS, then τ = ω_s(X, τ, A).

Proof

By Theorem 2.2, we only need to see that ω_s(X, τ, A) ⊆ τ. Let K ∈ ω_s(X, τ, A). Then there is F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). By Proposition 1.5, Cl_τω (F) = F which implies that K = F. Thus, K ∈ τ. This shows that τ = ω_s(X, τ, A).

Soft ω-open sets and soft ω_s-open sets are independent of each other:

Example 2.9

Let X = ℝ and A = ℕ. Let F, G, K ∈ SS(X, A) defined by F(a) = {a} ∪ (2, 3), G(a) = [0, ∞) and K(a) = (2, 3) for all a ∈ A. Let τ = {0_A, 1_A, F}. As (X, τ, A) is soft anti-locally countable, then by Proposition 1.5, Cl_τω (F) = Cl_τ (F) = 1_A. So G ∈ ω_s(X, τ, A) – τ_ω and K ∈ τ_ω – ω_s(X, τ, A).

Theorem 2.10

Let (X, τ, A) be a STS and let K ∈ SS(X, A). Then K is soft ω_s-open in (X, τ, A) if and only if K ⊆̃ Cl_τω (int_τ (K)).

Proof

Necessity

Suppose that K is soft ω_s-open set in (X, τ, A). Then we find F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). As F ⊆̃ K, then F = int_τ (F) ⊆̃ int_τ (K), which implies that Cl_τω(F) ⊆̃ Cl_τω (int_τ (K)). It follows that K ⊆̃ Cl_τω (int_τ (K)).

Sufficiency

Suppose that K ⊆̃ Cl_τω (int_τ (K)). Put F = int_τ (K). Then F ∈ τ with F ⊆̃ K ⊆̃ Cl_τω (int_τ (K)). Hence, K is soft ω_s-open set in (X, τ, A).

Theorem 2.11

Let (X, τ, A) be a STS. If {K_α : α ∈ Δ} ⊆ ω_s(X, τ, A), then ∪ α ∈ Δ ˜ K α ∈ ω s ( X , τ , A ).

Proof

For every α ∈ Δ, we find F_α ∈ τ such that F_α ⊆̃ K_α ⊆̃ Cl_τω (F_α). Then ∪ α ∈ Δ ˜ F α ∈ τ and ∪ α ∈ Δ ˜ F α ⊆ ˜ ∪ α ∈ Δ ˜ K α ⊆ ˜ ∪ α ∈ Δ ˜ C l τ ω ( K α ) ⊆ ˜ C l τ ω ( ∪ α ∈ Δ ˜ K α ). Therefore, ∪ α ∈ Δ ˜ K α ∈ ω s ( X , τ , A ).

Soft intersection of two soft ω_s-open sets is not a soft ω_s open in general:

Example 2.12

Let X = ℝ and A = ℕ. Let F, G ∈ SS(X, A) defined by F(a) = (0, ∞) and G(a) = (–∞, 0) for all a ∈ A. Let τ = {0_A, 1_A, F, G, F ∪̃ G}. As (X, τ, A) is soft anti-locally countable, then by Proposition 1.4, Cl_τω (F) = Cl_τ (F) = 1_A – G and Cl_τω (G) = Cl_τ (G) = 1_A – F. Then 1_A – F, 1_A – G ∈ ω_s(X, τ, A) but ((1_A – F) ∩̃ (1_A – G))(a) = {0} for all a ∈ A and hence (1_A – F) ∩̃ (1_A – G) ∉ ω_s(X, τ, A).

Theorem 2.13

Let (X, τ, A) be a STS. If G ∈ τ and K ∈ ω_s(X, τ, A), then G ∩̃ K ∈ ω_s(X, τ, A).

Proof

Let G ∈ τ and K ∈ ω_s(X, τ, A). As K ∈ ω_s(X, τ, A), then there is F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). Then G ∩̃ F ∈ τ and G ∩̃ F ⊆̃ G ∩̃ K ⊆̃ G ∩̃ Cl_τω (F) ⊆̃ Cl_τω (G ∩̃ F). This ends the proof that G ∩̃ K ∈ ω_s(X, τ, A).

Theorem 2.14

Let (X, τ, A) be a STS. If K ∈ ω_s(X, τ, A) and K ⊆̃ G ⊆̃ Cl_τω (K), then G ∈ ω_s(X, τ, A).

Proof

Suppose that K ∈ ω_s(X, τ, A) and K ⊆̃ G ⊆̃ Cl_τω (K). As K ∈ ω_s(X, τ, A), then there is F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). As K ⊆̃ Cl_τω (F), then Cl_τω (K) ⊆̃ Cl_τω (F). As G ⊆̃ Cl_τω (K), then G ⊆̃ Cl_τω (F). Thus, we have F ∈ τ and F ⊆̃ K ⊆̃ G ⊆̃ Cl_τω (K) ⊆̃ Cl_τω (F).

Therefore, G ∈ ω_s(X, τ, A).

Theorem 2.15

For any STS (X, τ, A),

Proof

As int_τ (K) ∈ τ for every K ∈ ω_s(X, τ, A), then {int_τ (K) : K ∈ ω_s(X, τ, A)} ⊆ τ. Conversely, let F ∈ τ, then F = int_τ (F). On the other hand, by Theorem 2.2, F ∈ ω_s(X, τ, A). It follows that τ ⊆ {int_τ (K) : K ∈ ω_s(X, τ, A)}.

Definition 2.16

Let (X, τ, A) be a STS and let M ∈ SS(X, A). Then M is called soft ω_s-closed set in (X, τ, A) if 1_A – M is soft ω_s-open set in (X, τ, A).

Theorem 2.17

Let (X, τ, A) be a STS. Then

(a) Every soft closed set in (X, τ, A) is soft ω_s-closed set in (X, τ, A).

(b) Every soft ω_s-closed set in (X, τ, A) is soft semi-closed set in (X, τ, A).

(c) 0_A and 1_A are soft ω_s-closed sets in (X, τ, A).

Proof

(a) Let M be a soft closed set in (X, τ, A), then 1_A – M ∈ τ and by Theorem 2.2 1_A – M ∈ ω_s(X, τ, A). Thus, M is a soft ω_s-closed set in (X, τ, A).

(b) Let M be a soft ω_s-closed set in (X, τ, A), then 1_A – M ∈ ω_s-(X, τ, A) and by Theorem 2.2 1_A – M ∈ SO(X, τ, A). Thus, M is soft semi-closed set in (X, τ, A).

(c) Follows by (a).

One can use Example 2.3 to show that none of the implications in Theorem 2.15 (a), (b) is reversible in general.

Theorem 2.18

Let (X, τ, A) be a STS. If M_α is soft ω_s-closed in (X, τ, A) for all α ∈ Δ, then ∩ α ∈ Δ ˜ M α is soft ω_s-closed in (X, τ, A).

Proof

Suppose that M_α is soft ω_s-closed in (X, τ, A) for all α ∈ Δ. Then for every α ∈ Δ, 1_A – M_α ∈ ω_s(X, τ, A). Thus, by Theorem 2.9, we have 1 A - ( ∩ α ∈ Δ ˜ M α ) = ∪ α ∈ Δ ˜ ( 1 A - M α ) ∈ ω s ( X , τ , A ). It follows that ∩ α ∈ Δ ˜ M α is soft ω_s-closed in (X, τ, A).

Soft ω_s-closedness is not closed under finite soft union:

Example 2.19

Consider (X, τ, A), F, and G as in Example 2.12. As 1_A – F, 1_A – G ∈ ω_s(X, τ, A), then F and G are soft ω_s-closed in (X, τ, A). As 1_A – (F ∪̃ G) = (1_A – F) ∩̃ (1_A – G) ∉ ω_s(X, τ, A), then F ∪̃ G is not soft ω_s-closed in (X, τ, A).

Theorem 2.20

Let (X, τ, A) be a STS. If M is soft closed in (X, τ, A) and N is soft ω_s-closed in (X, τ, A), then M ∪̃ N is soft ω_s-closed in (X, τ, A).

Proof

Suppose that M is soft closed in (X, τ, A) and N is soft ω_s-closed in (X, τ, A). Then 1_A – M ∈ τ and 1_A – N ∈ ω_s(X, τ, A). So by Theorem 2.13, 1_A – (M ∪̃ N) = (1_A – M) ∩̃ (1_A – N) ∈ ω_s(X, τ, A). It follows that M ∪̃ N is soft ω_s-closed in (X, τ, A).

Theorem 2.21

Let (X, τ, A) be a STS and let M ∈ SS(X, A). Then M is soft ω_s-closed in (X, τ, A) if and only if

Proof

Necessity

Assume that M is soft ω_s-closed in (X, τ, A). Then 1_A – M is soft ω_s-open in (X, τ, A). Theorem Theorem 2.10 implies that 1_A – M ⊆̃ Cl_τω (int_τ (1_A – M)), and so

Sufficiency

Suppose that int_τω (Cl_τ (M)) ⊆̃ M. Then

Therefore, by Theorem 2.10, 1_A – M is soft ω_s-open in (X, τ, A) and so M is soft ω_s-closed in (X, τ, A).

Theorem 2.22

If f_pu : (X, τ, A) → (Y, σ, B) is a soft open mapping such that f_pu : (X, τ_ω, A) → (Y, σ_ω, B) is soft continuous, then for every K ∈ ω_s(X, τ, A), we have f_pu(K) ∈ ω_s(Y, σ, B).

Proof

Let K ∈ ω_s(X, τ, A). Then there exists F ∈ τ such that F ⊆̃ K ⊆̃ Cl_τω (F). Thus we have f_pu (F) ⊆̃ f_pu(K) ⊆̃ f_pu (Cl_τω (F)). As f_pu : (X, τ, A) → (Y, σ, B) is soft open mapping, then f_pu (F) ∈ σ. As f_pu : (X, τ_ω, A) → (Y, σ_ω, B) is soft continuous, then f_pu (Cl_τω (F)) ⊆̃ Cl_τω (f_pu (F)). Therefore, f_pu(K) ∈ ω_s(Y, σ, B).

In Theorem 2.22, we cannot drop the condition ‘soft open mapping’:

Example 2.23

Let X = ℝ, A = ℕ, F, G ∈ SS(X, A) with F(a) = ℝ and G(a) = {0} for all a ∈ A, τ = SS(X, A), and σ = {0_B, 1_B, F}. Define p : X → X and u : A → A by p (x) = 0 for all x ∈ X and u(a) = a for all a ∈ A. Then clearly that f_pu : (X, τ_ω, A) → (X, τ_ω, A) is soft continuous, but G ∈ ω_s (X, τ_ω, A) while f_pu (G) = G ∉ ω_s (X, τ_ω, A).

Definition 3.1

Let (X, τ, A) be a STS and H ∈ SS(X, A). The soft ω_s-closure of H in (X, τ, A) is denoted by ω_s-Cl_τ (H) and defined by ω_s-Cl_τ (H) = ∩̃ {M : M is soft ω_s-closed in (X, τ, A) and H ⊆̃ }.

Theorem 3.2

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then

(a) ω_s-Cl_τ (H) is the smallest soft ω_s-closed in (X, τ, A) containing H.

(b) H is soft ω_s-closed in (X, τ, A) if and only if H = ω_s-Cl_τ (H).

Proof

(a) Follows from Definition 3.1 and Theorem 2.18.

(b) Follows immediately by (a).

Theorem 3.3

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then H ⊆̃ S-Cl_τ (H) ⊆̃ ω_s-Cl_τ (H) ⊆̃ Cl_τ (H).

Proof

Follows from the definitions and parts (a) and (b) of Theorem 2.17.

In Theorem 3.3 S-Cl_τ (H) ≠ ω_s-Cl_τ (H) and ω_s-Cl_τ (H) ≠ Cl_τ (H), in general:

Example 3.4

Consider Example 2.3. We proved that 1_A – G ∈ SO(X, τ, A) – ω_s(X, τ, A) and 1_A – F ∈ ω_s(X, τ, A) – τ. So, we have F as an ω_s-closed set in (X, τ, A) that is not closed set in (X, τ, A), and G as a semi-closed set in (X, τ, A) that is not ω_s-closed set in (X, τ, A). Thus, by Proposition 3.9 (2) of [26] and Theorem 3.2 (b), S-Cl_τ (G) = G ≠ ω_s-Cl_τ (H) and ω_s-Cl_τ (F) = F ≠ Cl_τ (F).

Theorem 3.5

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then

(a) If H is soft closed in (X, τ, A), then H = S-Cl_τ (H) = ω_s-Cl_τ (H) = Cl_τ (H).

(b) If H is soft ω_s-closed in (X, τ, A), then H = S-Cl_τ (H) = ω_s-Cl_τ (H).

Proof

(a) Suppose that soft H is closed in (X, τ, A). Then H = Cl_τ (H) and by Theorem 3.3 we have H = S-Cl_τ (H) = ω_s-Cl_τ (H) = Cl_τ (H).

(b) Suppose that H is soft ω_s-closed in (X, τ, A). Then by Theorem 3.2 (b) we have H = ω_s-Cl_τ (H), and by Theorem 3.3 we conclude that H = S-Cl_τ (H) = ω_s-Cl_τ (H).

Theorem 3.6

If (X, τ, A) is a soft anti-locally countable STS then for every H ∈ SS(X, A), S-Cl_τ (H) = ω_s-Cl_τ (H).

Proof

As (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 we have ω_s(X, τ, A) = SO(X, τ, A) and hence S-Cl_τ (H) = ω_s-Cl_τ (H).

Theorem 3.7

If (X, τ, A) is a soft locally countable STS then for every H ∈ SS(X, A), ω_s-Cl_τ (H) = Cl_τ (H).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 2.8 we have τ = ω_s(X, τ, A) and hence ω_s-Cl_τ (H) = Cl_τ (H).

Theorem 3.8

If (X, τ, A) is a STS and H ∈ SS(X, A), then S-Cl_τω (H) = ω_s-Cl_τω (H).

Proof

By Theorem 2.7 we have SO(X, τ_ω, A) = ω_s(X, τ_ω, A) and hence S-Cl_τω (H) = ω_s-Cl_τω (H).

Theorem 3.9

Let (X, τ, A) be a STS and H, K ∈ SS(X, A). Then

(a) If H ⊆̃ K, then ω_s-Cl_τ (H) ⊆̃ ω_s-Cl_τ (K).

(b) ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K) ⊆̃ ω_s-Cl_τ (H ∪̃ K).

(c) ω_s-Cl_τ (H ∩̃ K) ⊆̃ ω_s-Cl_τ (H) ∩̃ ω_s-Cl_τ (K).

Proof

(a) Suppose that H ⊆̃ K. As K ⊆̃ ω_s-Cl_τ (K), then H ⊆̃ ω_s-Cl_τ (K). By Theorem 3.2 (a), ω_s-Cl_τ (K) is soft ω_sclosed in (X, τ, A). Again by Theorem 3.2 (a), ω_s-Cl_τ (H) ⊆̃ ω_s-Cl_τ (K).

(b) As H ⊆̃ H ∪̃ K and K ⊆̃ H ∪̃ K, then by (a), ω_s-Cl_τ (H) ⊆̃ ω_s-Cl_τ (H ∪̃ K) and ω_s-Cl_τ (K) ⊆̃ ω_s-Cl_τ (H ∪̃ K). Hence ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K) ⊆̃ ω_s-Cl_τ (H ∪̃ K).

(c) As H ∩̃ K ⊆̃ H and H ∩̃ K ⊆̃ K, then by (a), ω_s-Cl_τ (H ∩̃ K) ⊆̃ ω_s-Cl_τ (H) and ω_s-Cl_τ (H ∩̃ K) ⊆̃ ω_s-Cl_τ (K). Hence ω_s-Cl_τ (H ∩̃ K) ⊆̃ ω_s-Cl_τ (H) ∩̃ ω_s-Cl_τ (K).

The soft inclusion in Theorem 3.9 (b) cannot be replaced by soft equality in general:

Example 3.10

Consider (X, τ, A) as in Example 2.19. We proved that F and G are soft ω_s-closed in (X, τ, A) while F ∪̃ G is not soft ω_s-closed in (X, τ, A). Therefore, by Theorem 3.2 (b), ω_s-Cl_τ (F) ∪̃ ω_s-Cl_τ (G) = F ∪̃ G while ω_s-Cl_τ (F ∪̃ G) ≠ F ∪̃ G.

Theorem 3.11

If (X, τ, A) is a soft locally countable STS, then for any H, K ∈ SS(X, A) we have ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K) = ω_s-Cl_τ (H ∪̃ K).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 3.7,

Theorem 3.12

Let (X, τ, A) be a STS and let H, K ∈ SS(X, A). If H is soft closed in (X, τ, A), then ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K) = ω_s-Cl_τ (H ∪̃ K).

Proof

By Theorem 3.9 (b) we only need to show that ω_s-Cl_τ (H ∪̃ K) ⊆̃ ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K). As H is soft closed in (X, τ, A), then by Theorem 3.5 (a), H = ω_s-Cl_τ (H). So by Theorem 2.20, ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K) is ω_s-closed in (X, τ, A). As H ∪̃ K ⊆̃ ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K), then by Theorem 3.2 (a) we have ω_s-Cl_τ (H ∪̃ K) ⊆̃ ω_s-Cl_τ (H) ∪̃ ω_s-Cl_τ (K).

The soft inclusion in Theorem 3.9 (c) cannot be replaced by soft equality in general:

Example 3.13

Let X = ℤ × ℤ, A = ℝ and τ = {F ∈ SS(X, A) : F(a) ∈ τ_cof for every a ∈ A}. Let H, K ∈ SS(X, A) defined by H(a) = ℤ × {0} and K(a) = {0} × ℤ for all a ∈ A. Then (H ∩̃ K) (a) = {(0, 0)} for all a ∈ A. Note that M ∈ SS(X, A) is soft closed in (X, τ, A) if and only if for all a ∈ A, either M(a) is finite or M (a) = ℤ × ℤ. Then Cl_τ (H) = Cl_τ (K) = 1_A and Cl_τ (H ∩̃ K) = H ∩̃ K. As (X, τ, A) is soft locally countable, then by Theorem 3.7, ω_s-Cl_τ (H) = Cl_τ (H) = 1_A, ω_s-Cl_τ (K) = Cl_τ (K) = 1_A, and ω_s-Cl_τ (H ∩̃ K) = H ∩̃ K. Thus, ω_s-Cl_τ (H) ∩̃ ω_s-Cl_τ (K) = 1_A while ω_s-Cl_τ (H ∩̃ K) ω_s-Cl_τ (H ∩̃ K) ≠ 1_A.

Theorem 3.14

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then S-Cl_τ (Cl_τ (H)) = Cl_τ (S-Cl_τ (H)) = ω_s-Cl_τ (Cl_τ (H)) = Cl_τ (ω_s-Cl_τ (H)) = Cl_τ (H).

Proof

As Cl_τ (H) is soft closed in (X, τ, A), then, by Theorem 3.5 (a) we have Cl_τ (H) = S-Cl_τ (Cl_τ (H)) = ω_s-Cl_τ (Cl_τ (H)).

By Theorem 3.3, H ⊆̃ S-Cl_τ (H) ⊆̃ ω_s-Cl_τ (H) ⊆̃ Cl_τ (H). So, Cl_τ (H) ⊆̃ Cl_τ (S-Cl_τ (H)) ⊆̃ Cl_τ (ω_s-Cl_τ (H)) ⊆̃ Cl_τ (Cl_τ (H)) = Cl_τ (H). Hence, we have Cl_τ (H) = Cl_τ (S-Cl_τ (H)) = Cl_τ (ω_s-Cl_τ (H)).

Theorem 3.15

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then ω_s-Cl_τ (S-Cl_τ (H)) = S-Cl_τ (ω_s-Cl_τ (H)) = ω_s-Cl_τ (H).

Proof

As ω_s-Cl_τ (H) is soft ω_s-closed in (X, τ, A), then by Theorem 3.5 (b) we have ω_s-Cl_τ (H) = S-Cl_τ (ω_s-Cl_τ (H)).

By Theorem 3.3, H ⊆̃ S-Cl_τ (H) ⊆̃ ω_s-Cl_τ (H). So by Theorem 3.9 (a), ω_s-Cl_τ (H) ⊆̃ ω_s-Cl_τ (S-Cl_τ (H)) ⊆̃ ω_s-Cl_τ (ω_s-Cl_τ (H)) = ω_s-Cl_τ (H). Therefore,

Theorem 3.16

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then a_x ∊̃ ω_s-Cl_τ (H) if and only if for all K ∈ ω_s(X, τ, A) with a_x ∊̃ K we have K ∩̃ H ≠ 0_A.

Proof

Necessity

Suppose that a_x ∊̃ ω_s-Cl_τ (H) and suppose to the contrary that there is K ∈ ω_s(X, τ, A) such that a_x ∊̃ K and K ∩̃ H = 0_A. So we have H ⊆̃ 1_A – K with 1_A – K is soft ω_s-closed set in (X, τ, A), and hence ω_s-Cl_τ (H) ⊆̃ 1_A – K. As a_x ∊̃ ω_s-Cl_τ (H), then a_x ∊̃ 1_A – K, a contradiction.

Sufficiency

By contradiction. Then we have a_x ∊̃ (1_A – ω_s-Cl_τ (H)) ∈ ω_s(X, τ, A). Then by assumption (1_A – ω_s-Cl_τ (H)) ∩̃ H = 0_A, a contradiction.

Definition 3.17

Let (X, τ, A) be a STS and H ∈ SS(X, A). The soft ω_s-interior of H in (X, τ, A) is denoted by ω_s-int_τ (H) and defined by ω_s-int_τ (H) = ∪̃ {K : K is soft ω_s-open in (X, τ, A) and K ⊆̃ H}.

Theorem 3.18

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then

(a) ω_s-int_τ (H) is the largest soft ω_s-open in (X, τ, A) contained in H.

(b) H is soft ω_s-open in (X, τ, A) if and only if H = ω_s int_τ (H).

Proof

(a) Follows from Definition 3.17 and Theorem 2.11.

(b) Follows immediately by (a).

Theorem 3.19

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then int_τ (H) ⊆̃ ω_s-int_τ (H) ⊆̃ S-int_τ (H) ⊆̃ H.

Proof

Follows from the definitions and Theorem 2.2.

In Theorem 3.19 int_τ (H) ≠ ω_s-int_τ (H) and ω_s-int_τ (H) ≠ S-int_τ (H), in general:

Example 3.20

Consider Example 2.3. We proved that 1_A – G ∈ SO(X, τ, A) – ω_s(X, τ, A) and 1_A – F ∈ ω_s(X, τ, A) – τ. Thus, by Proposition 3.9 (2) and Theorem 3.2 (b), S-int_τ (1_A – G) = 1_A – G ≠ ω_s-int_τ (1_A – G) and ω_s-int_τ (1_A – F) = 1_A – F ≠ int_τ (1_A – F).

Theorem 3.21

Let (X, τ, A) be a STS and H ∈ SS(X, A). Then

(a) If H is soft open in (X, τ, A), then H = S-int_τ (H) = ω_s-int_τ (H) = int_τ (H).

(b) If H is soft ω_s-open in (X, τ, A), then H = S-int_τ (H) = ω_s-int_τ (H).

Proof

(a) Suppose that H is soft open in (X, τ, A). Then H = int_τ (H) and by Theorem 3.19 we have H = S-int_τ (H) = ω_s-int_τ (H) = int_τ (H).

(b) Suppose that H is soft ω_s-open in (X, τ, A). Then by Theorem 3.2 (b), H = ω_s-int_τ (H). Thus, by Theorem 3.3 we ω_s-int_τ (H) ⊆̃ S-int_τ (H) ⊆̃ ω_s-int_τ (H). Therefore, H = S-int_τ (H) = ω_s-int_τ (H).

Theorem 3.22

If (X, τ, A) is a soft anti-locally countable STS then for every H ∈ SS(X, A), S-int_τ (H) = ω_s int_τ (H).

Proof

As (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 we have ω_s(X, τ, A) = SO(X, τ, A) and hence S-int_τ (H) = ω_s-int_τ (H).

Theorem 3.23

If (X, τ, A) is a soft locally countable STS then for every H ∈ SS(X, A), ω_s-int_τ (H) = int_τ (H).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 2.8 we have τ = ω_s(X, τ, A) and hence ω_s-int_τ (H) = int_τ (H).

Theorem 3.24

If (X, τ, A) is a STS and H ∈ SS(X, A), then S-int_τω (H) = ω_s-int_τω (H).

Proof

By Theorem 2.7 we have SO(X, τ_ω, A) = ω_s(X, τ_ω, A) and hence S-int_τω (H) = ω_s-int_τω (H).

Theorem 3.25

Let (X, τ, A) be a STS and H, K ∈ SS(X, A). Then

(a) If H ⊆̃ K, then ω_s-int_τ (H) ⊆̃ ω_s-int_τ (K).

(b) ω_s-int_τ (H) ∪̃ ω_s-int_τ (K) ⊆̃ ω_s-int_τ (H ∪̃ K).

(c) ω_s-int_τ (H ∩̃ K) ⊆̃ ω_s-int_τ (H) ∩̃ ω_s-int_τ (K).

Proof

(a) Suppose that H ⊆̃ K. Then we have ω_s-int_τ (H) ⊆̃ H ⊆̃ K. By Theorem 3.18 (a), ω_s-int_τ (H) is soft ω_s-open in (X, τ, A). Again by Theorem 3.18 (a), ω_s-int_τ (H) ⊆̃ ω_s-int_τ (K).

(b) As H ⊆̃ H ∪̃ K and K ⊆̃ H ∪̃ K, then by (a), ω_s-int_τ (H) ⊆̃ ω_s-int_τ (H ∪̃ K) and ω_s-int_τ (K) ⊆̃ ω_s-int_τ (H ∪̃ K). Hence ω_s-int_τ (H) ∪̃ ω_s-int_τ (K) ⊆̃ ω_s-int_τ (H ∪̃ K).

(c) As H ∩̃ K ⊆̃ H and H ∩̃ K ⊆̃ K, then by (a), ω_s-int_τ (H ∩̃ K) ⊆̃ ω_s-int_τ (H) and ω_s-int_τ (H ∩̃ K) ⊆̃ ω_s-int_τ (K). Hence ω_s-int_τ (H ∩̃ K) ⊆̃ ω_s-int_τ (H) ∩̃ ω_s-int_τ (K).

The soft inclusion in Theorem 3.26 (c) cannot be replaced by soft equality in general:

Example 3.27

Consider (X, τ, A) as in Example 2.12. We proved that 1_A – F and 1_A – G are soft ω_s-open in (X, τ, A) while (1_A – F) ∩̃ (1_A – G) is not soft ω_s-open in (X, τ, A). Therefore, by Theorem 3.18 (b),

while

Theorem 3.28

If (X, τ, A) is a soft locally countable STS, then for any H, K ∈ SS(X, A) we have ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) = ω_s-int_τ (H ∩̃ K).

Proof

As (X, τ, A) is soft locally countable, then by Theorem 3.23,

Theorem 3.29

Let (X, τ, A) be a STS and let H, K ∈ SS(X, A). If H is soft open in (X, τ, A), then ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) = ω_s-int_τ (H ∩̃ K).

Proof

By Theorem 3.25 (c) we only need to show that ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) ⊆̃ ω_s-int_τ (H ∩̃ K). As H is soft open in (X, τ, A), then by Theorem 3.18 (b), H = ω_s-int_τ (H). So by Theorem 2.13, ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) is soft ω_s-open in (X, τ, A). As ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) ⊆̃ H ∩̃ K, then by Theorem 3.18 (a) we have ω_s-int_τ (H) ∩̃ ω_s-int_τ (K) ⊆̃ ω_s-int_τ (H ∩̃ K).

The soft inclusion in Theorem 3.25 (b) cannot be replaced by soft equality in general:

Example 3.30

Consider (X, τ, A) in Example 3.13 and let B = {–n : n ∈ ℕ}. Let H, K ∈ SS(X, A) defined by H(a) = ℤ × ℕ and K(a) = ({0} ∪ B) × ℤ for all a ∈ A. As (X, τ, A) is soft locally countable, then by Theorem 3.23, ω_s-int_τ (H) = int_τ (H) = 0_A, ω_s-int_τ (K) = int_τ (K) = 0_A, and ω_s-Cl_τ (H ∪̃ K) = int_τ (H ∪̃ K) = int_τ (1_A) = 1_A. Thus, ω_s-int_τ (H) ∪̃ ω_s-int_τ (K) = 0_A while ω_s-Cl_τ (H ∪̃ K) ≠ 0_A.

Theorem 3.31

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then int_τ (H) = ω_s-int_τ (int_τ (H)) = S-int_τ (int_τ (H)) = int_τ (ω_s-int_τ (H)) = int_τ (S-int_τ (H)).

Proof

As int_τ (H) is soft open in (X, τ, A), then by Theorem 3.21 (a) we have int_τ (H) = S-int_τ (int_τ (H)) = ω_s-int_τ (int_τ (H)).

By Theorem 3.19 we have int_τ (H) ⊆̃ ω_s-int_τ (H) ⊆̃ S-int_τ (H) ⊆̃ H. So, int_τ (H) = int_τ (int_τ (H)) ⊆̃ int_τ (ω_s-int_τ (H)) ⊆̃ int_τ (S-int_τ (H)) ⊆̃ int_τ (H).

Hence,

Theorem 3.32

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then ω_s-int_τ (S-int_τ (H)) = S-int_τ (ω_s-int_τ (H)) = ω_s-int_τ (H).

Proof

As ω_s-int_τ (H) is soft ω_s-open in (X, τ, A), then by Theorem 3.21 (b) we have ω_s-int_τ (H) = S-int_τ (ω_s-int_τ (H))

By Theorem 3.19 we have

So by Theorem 3.25 (a) we have ω_s-int_τ (H) = ω_s-int_τ (ω_s-int_τ (H)) ⊆̃ ω_s-int_τ (S-int_τ (H)) ⊆̃ ω_s-int_τ (H). Therefore, ω_s-int_τ (H) = ω_s-int_τ (S-int_τ (H)).

Theorem 3.33

Let (X, τ, A) be a STS and let H ∈ SS(X, A). Then

(a) ω_s-int_τ (1_A – H) ∩̃ ω_s-Cl_τ (H) = 0_A.

(b) 1_A = ω_s-int_τ (1_A – H) ∪̃ ω_s-Cl_τ (H).

(c) 1_A – ω_s-Cl_τ (H) = ω_s-int_τ (1_A – H) and 1_A – ω_s-int_τ (1_A – H) = ω_s-Cl_τ (H).

Proof

(a) Suppose to the contrary that there exists a_x ∊̃ (ω_s-int_τ (1_A – H) ∩̃ ω_s-Cl_τ (H)). Then we have a_x ∊̃ ω_s-int_τ (1_A – H) ∈ ω_s(X, τ, A) and a_x ∊̃ ω_s-Cl_τ (H). Thus by Theorem 3.16 ω_s-int_τ (1_A – H) ∩̃ H ≠ 0_A. But ω_s-int_τ (1_A – H) ∩̃ H ⊆̃ (1_A – H) ∩̃ H = 0_A, a contradiction.

(b) It is sufficient to show that

Let a_x ∊̃ (1_A – ω_s-Cl_τ (H)), then by Theorem 3.16 there is K ∈ ω_s(X, τ, A) such that a_x ∊̃ K and K ∩̃ H = 0_A. So, we have a_x ∊̃ K ⊆̃ 1_A – H and hence a_x ∈ ω_s-int_τ (1_A – H).

(c) Follows from (a) and (b).

At this stage, we believe that the following four questions are natural:

Question 4.1

Let (X, τ, A) and let K ∈ ω_s (X, τ, A). Is it true that K(a) ∈ ω_s (X, τ_a) for all a ∈ A?

Question 4.2

Let (X, τ, A) and let K ∈ SS(X, A) such that K(a) ∈ ω_s (X, τ_a) for all a ∈ A. Is it true that K ∈ ω_s (X, τ, A)?

Question 4.3

Let (X, τ, A) and let K ∈ SO (X, τ, A). Is it true that K(a) ∈ SO (X, τ_a) for all a ∈ A?

Question 4.4

Let (X, τ, A) and let K ∈ SS(X, A) such that K(a) ∈ SO (X, τ_a) for all a ∈ A. Is it true that K ∈ SO (X, τ, A)?

We will leave Question 4.3 as an open question. However, in the following example we will give negative answers for Questions 4.1, 4.2, and 4.4:

Example 4.5

Let X = ℝ and A = {a, b}. Let F, G, H, K ∈ SS(X, A) defined by

Let τ = {0_A, 1_A, F, G, F ∪̃ G}. As clearly that (X, τ, A) is soft anti-locally countable, then by Theorem 2.6 ω_s(X, τ, A) = SO(X, τ, A). It is not difficult to check that H ∈ SO(X, τ, A) = ω_s (X, τ, A) while H (b) ∉ ω_s (X, τ_b), which gives a negative answer for Questions 4.1. Also it is not difficult to check that K(a) ∈ ω_s (X, τ_a) ⊆ SO(X, τ_a) and K (b) ∈ ω_s(X, τ_b) ⊆ SO(X, τ_b) while K ∉ ω_s(X, τ, A) = SO(X, τ, A), which gives negative answers for Questions 4.2 and 4.4.

If we add the condition ‘(X, τ, A) is soft locally countable’, then Question 4.1 will have a positive answer:

Theorem 4.6

Let (X, τ, A) be a soft locally countable and let K ∈ ω_s (X, τ, A). Then K(a) ∈ ω_s (X, τ_a) for every a ∈ A.

Proof

Let a ∈ A. As (X, τ, A) is soft locally countable and K ∈ ω_s (X, τ, A), then by Theorem 2.8, K ∈ τ and so K(a) ∈ τ_a ⊆ ω_s (X, τ_a).

Lemma 4.7

Let {(X, ℑ_a) : a ∈ A} be an indexed family of topological spaces and let τ = ⊕ a ∈ A ℑ a. Let H ∈ SS(X, A), then for every a ∈ A, Cl_(τa)ω (H (a)) = (Cl_τω (H)) (a).

Proof

Let a ∈ A. By Proposition 7 of [4], Cl_(τa)ω (H(a)) ⊆ (Cl_τω (H)) (a). To see that (Cl_τω (H)) (a) ⊆ Cl_(τa)ω (H (a)), let x ∈ (Cl_τω (H)) (a) and let U ∈ (τ_a)_ω such that x ∈ U. By Theorem 7 of [2], (τ_a)_ω = (τ_ω)_a. Then a_U ∈ τ_ω. As a_x ∈ a_U ∈ τ_ω and a_x ∈ Cl_τω (H), then by Lemma 5 of [2], H ∩̃ a_U ≠ 0_A. Thus, we must have (H ∩̃ a_U) (a) = H (a) ∩ U ≠ ∅︀. It follows that x ∈ Cl_(τω)a (H(a)).

If we add the condition ‘τ is a generated soft topology’, then Questions 4.1 and 4.2 will have positive answers:

Theorem 4.8

Let {(X, ℑ_a) : a ∈ A} be an indexed family of topological spaces and let τ = ⊕ a ∈ A ℑ a. Let H ∈ SS(X, A). Then H ∈ ω_s(X, τ, A) if and only if H(a) ∈ ω_s(X, τ_a) for every a ∈ A.

Proof

Necessity

Suppose that H ∈ ω_s(X, τ, A) and let a ∈ A. Choose F ∈ τ such that F ⊆̃ H ⊆̃ Cl_τω (F). So, F(a) ⊆ H(a) ⊆ (Cl_τω (F)) (a). As F(a) ∈ τ_a and by Lemma 4.7 we have (Cl_τω (F)) (a) = Cl_(τa)ω (F(a)), then H(a) ∈ ω_s(X, τ_a).

Sufficiency

Suppose that H(a) ∈ ω_s(X, τ_a) for every a ∈ A. Then for every a ∈ A, there exists U_a ∈ τ_a = ℑ_a such that U_a ⊆ H(a) ⊆ Cl_(τa)ω (U_a). Let K ∈ SS(X, A) with K(a) = U_a ∈ ℑ_a for every a ∈ A. Then K ∈ ⊕ a ∈ A ℑ a = τ

and by Lemma 4.7, (Cl_τω (K)) (a) = Cl_(τa)ω (K(a)) = Cl_(τa)ω (U_a) for all a ∈ A. As for every a ∈ A, K(a) = U_a ⊆ H(a) ⊆ Cl_(τa)ω (U_a) = (Cl_τω (K)) (a), then K ⊆̃ H ⊆̃ Cl_τω (K). It follows that H ∈ ω_s(X, τ, A).

Lemma 4.9

Let {(X, ℑ_a) : a ∈ A} be an indexed family of topological spaces and let τ = ⊕ a ∈ A ℑ a. Let H ∈ SS(X, A), then for every a ∈ A, Cl_τa (H(a)) = (Cl_τ (H)) (a).

Proof

Let a ∈ A. By Proposition 7 of [4], Cl_τa (H(a)) ⊆ (Cl_τ (H)) (a). To see that (Cl_τ (H)) (a) ⊆ Cl_τa (H(a)), let x ∈ (Cl_τ (H)) (a) and let U ∈ τ_a = ℑ_a such that x ∈ U. Then we have a_x ∊̃ a_U ∈ τ and a_x ∊̃ Cl_τ (H), and by Lemma 5 of [2], H ∩̃ a_U ≠ 0_A. Thus, we must have (H ∩̃ a_U) (a) = H(a) ∩ U ≠ ∅︀. It follows that x ∈ Cl_τa (H (a)).

If we add the condition ‘τ is a generated soft topology’, then Questions 4.3 and 4.4 will have positive answers:

Theorem 4.10

Let {(X, ℑ_a) : a ∈ A} be an indexed family of topological spaces and let τ = ⊕ a ∈ A ℑ a. Let H ∈ SS(X, A). Then H ∈ SO(X, τ, A) if and only if H(a) ∈ SO(X, τ_a) for every a ∈ A.

Proof

Necessity

Suppose that H ∈ SO(X, τ, A). Then there exists F ∈ τ such that F ⊆̃ H ⊆̃ Cl_τ (F). So, F(a) ⊆ H(a) ⊆ (Cl_τ (F)) (a). As F(a) ∈ τ_a and by Lemma 4.9 we have (Cl_τ (F)) (a) = Cl_τa (F (a)), then H(a) ∈ SO(X, τ_a).

Sufficiency

Suppose that H(a) ∈ SO(X, τ_a) for every a ∈ A. Then for every a ∈ A, there exists U_a ∈ τ_a = ℑ_a such that U_a ⊆ H(a) ⊆ Cl_τa (U_a). Let K ∈ SS(X, A) with K(a) = U_a ∈ ℑ_a for every a ∈ A. Then K ∈ ⊕ a ∈ A ℑ a = τ and by Lemma 4.9, (Cl_τ (K)) (a) = Cl_τa (K (a)) = Cl_τa (U_a) for all a ∈ A. As for every a ∈ A, K(a) = U_a ⊆ H(a) ⊆ Cl_τa (U_a) = (Cl_τ (K)) (a), then K ⊆̃ H ⊆̃ Cl_τ (K). It follows that H ∈ SO(X, τ, A).

We introduced soft ω_s-open sets as a new class of soft sets. We then proved that the class of soft ω_s-open sets lies strictly between the classes of soft open sets and soft semi-open sets. Furthermore, we studied two new soft operators, soft ω_s-closure and soft ω_s-interior, as well as relationships regarding relative topological spaces and generated soft topological. Finally, we end by raising an open question related to soft ω_s-open. We believe the proposed concept can open up doors to research areas such as soft ω_s-mappings and soft ω_s-compactness.

No potential conflict of interest relevant to this article was reported.