This study follows the concepts and terminology in [1, 2]. In this study, STS and TS denote the soft topological space and topological space, respectively. As a general mathematical tool for addressing uncertainty, Molodtsov [3] introduced the notion of soft sets in 1999. Let Y be a universal set and E be a set of parameters. A soft set over Y relative to E is a function G : E → ℘(Y ). The family of all soft sets over Y relative to E is denoted as SS(Y,E). In this study, the null soft set and the absolute soft sets are denoted by 0_E and 1_E. As a contemporary structure of mathematics, STSs were defined in [4] as follows: a STS is a triplet (Y, τ,E), where τ ⊆ SS (Y,E), τ contains 0_E and 1_E, τ is closed under finite soft intersection, and τ is closed under an arbitrary soft union. Let (Y, τ,E) be an STS and F ∈ SS(Y,E), then F is said to be a soft open set in (Y, τ,E) if F ∈ τ and F is said to be a soft closed set in (Y, τ,E) if 1_E – F is a soft open set in (Y, τ,E). The collection of all soft closed sets in (Y, τ,E) is denoted as τ^c. The concept of soft topology and its applications remain a popular research area [1, 2, 5–28].

As a weaker form of the semi-open and preopen sets, the notion of b-open sets were introduced in [29]. Over the years, several studies have been published related to b-open sets [30–33]. Noiri et al. [34] introduced ωb-open sets in TSs, which is a class of sets containing both ω-open sets and b-open sets. In this study, we obtained several results regarding the soft ωb-open sets. For example, we demonstrate that they form a soft supratopology that contains classes of soft ω-open sets and soft b-open sets. In addition, using soft ωb-open sets, we define and investigate soft ωb-closure and soft ωb-interior as two new operators. Furthermore, we introduce and investigate soft b-antilocal countability is a novel soft-topological property. In addition, we introduce quasi-soft b-openness and weakly quasi-soft b-openness are two new classes of soft functions. Finally, we investigate the relationships between the new concepts and their analogs in a general topology. In future work, we hope to find an application of our new soft topological notions in the decision-making problem, as previously studies [35–38].

The remainder of this paper is organized as follows.

In Section 2, we introduce several properties of soft ωb-open sets. For example, we demonstrate that a collection of soft ωb-open sets forms a soft supratopology containing classes of soft ω-open sets and soft b-open sets. We also define and investigate the soft ωb-closure and soft ωb-interiors as two new operators using soft ωb-open sets. Moreover, we define and investigate soft b-antilocal countability, quasi soft bopenness, and weakly quasi soft b-openness. In addition, we investigate the relationships between the new concepts and their analogs in a general topology.

In Section 3, we establish the characterization and preservation theorems for soft b-Lindelof STSs.

Let (Y, τ,E) be an STS, (Y, μ) be a TS, H ∈ SS(Y,E), and U ⊆ Y. Throughout this paper, Cl_τ (H), Int_τ (H), Cl_μ(U), and Int_μ(U) denote the soft closures of H in (Y, τ,E), the soft interior of H in (Y, τ,E), the closure of U in (Y, μ), and the interior of U in (Y, μ), respectively.

The following definitions and results were used:

Definition 1.1 ([29])

Let (Y, μ) be a TS and A ⊆ Y. Then A is called a

(a) b-open set in (Y, μ) if A ⊆ Int_μ (Cl_μ(A)) ∪̃ Cl_μ(Int_μ (A)).

(b) b-closed set in (Y, μ) if Y – A is a b-open in (Y, μ).

The families of all b-open sets (resp. b-closed sets) of TS (Y, μ) is denoted as BO(Y, μ) (resp. BC(Y, μ)).

Definition 1.2 ([34])

Let (Y, μ) be a TS and A ⊆ Y. Then A is called

(a) an ωb-open set in (Y, μ) if for any a ∈ A, there exists C ∈ BO(Y, μ) such that a ∈ C and C – A are countable sets.

(b) an ωb-closed set in (Y, μ) if Y – A is ωb-open in (Y, μ).

The families of all b-open sets (resp. b-closed sets) of TS (Y, μ) is denoted as ωBO(Y, μ) (resp. ωBC(Y, μ)).

Definition 1.3 ([39])

Let (Y, τ,E) be an STS and T ∈ SS(Y,E). Then T is called a

(a) soft b-open set in (Y, τ,E) if

(b) soft b-closed set in (Y, τ,E) if 1_E – T is a soft b-open set in (Y, τ,E).

The families of all soft b-open sets (resp. soft b-closed sets) of the STS (Y, τ,E) is denoted by BO(Y, τ,E) (resp. BC(Y, τ,E)).

Definition 1.4 ([39])

Let (Y, τ,E) be an STS and H ∈ SS(Y,E). Then

(a) The soft b-closure of H in (Y, τ,E) is denoted by b-Cl_τ (H) and defined by

(b) The soft ωb-interior of H in (Y, τ,E) is denoted by ωb-Int_τ (H) and defined by

Definition 1.5

The function g : (Y, μ) –→ (Z, λ) is said to be

(a) [34] quasi b-open if for every U ∈ BO(Y, μ), g (U) ∈ λ.

(b) [34] weakly quasi b-open if for every U ∈ BO(Y, μ), g (U) ∈ BO(Z, λ).

(c) [40] b-continuous provided that for every V ∈ λ, fpu-1(V)∈BO(Y,μ).

(d) [34] ωb-continuous provided that for every V ∈λ, fpu-1(V)∈ωBO(Y,μ).

Definition 1.6 ([41])

Let (Y, μ) be a TS and L ⊆ Y. Then

(a) L is said to be a b-Lindelof relative to (Y, μ) if every cover ℋ of Y withℋ ⊆ BO(Y, μ) has a countable subcover.

(b) (Y, μ) is said to be b-Lindelof if Y is b-Lindelof relative to (Y, μ).

Definition 1.7 ([39])

A soft function f_pu : (Y, τ,E) –→ (Z, δ,D) is said to be soft b-continuous provided that for every K ∈ δ, fpu-1(K)∈BO(Y,τ,E).

Here, we introduce several properties of soft ωb-open sets. For example, we demonstrate that a collection of soft ωb-open sets forms a soft supratopology containing classes of soft ω-open sets and soft b-open sets. We also define and investigate the soft ωb-closure and soft ωb-interior as two new operators using soft ωb-open sets. Moreover, we define and investigate soft b-antilocal countability, quasi soft b-openness, and weakly quasi soft b-openness. In addition, we investigate the relationships between the new concepts and their analogs in a general topology.

Definition 2.1

A soft set T of STS (Y, τ,E) is called a soft ωb-open set in (Y, τ,E) if for any e_y∊̃T, S ∈ BO(Y, τ,E) exists such that e_y∊̃S and S – T ∈ CSS(Y,E). The soft complement of a soft ωb-open set in (Y, τ,E) is called soft ωb-closed set.

The families of all soft ωb-open sets (resp. soft ωb-closed sets) of an STS (Y, τ,E) is denoted by ωBO(Y, τ,E) (resp. ωBC(Y, τ,E)).

Theorem 2.2

For any STS (Y, τ,E), τ_ω ⊆ ωBO(Y, τ,E).

Corollary 2.3

If (Y, τ,E) is a soft locally countable STS, then ωBO(Y, τ,E) = SS(Y,E).

Theorem 2.4

For any STS (Y, τ,E), {T – S : T ∈ BO(Y, τ,E) and S ∈ CSS(Y,E)} ⊆ ωBO(Y, τ,E).

Corollary 2.5

For any STS (Y, τ,E),BO(Y, τ,E) ⊆ ωBO(Y, τ,E).

The following example shows that inclusion in Corollary 2.5 cannot be replaced by equality. In general,

Example 2.6

Let Y = {1, 2, 3}, E = ℕ, and τ = {F ∈ SS(Y,E) : F(e) ∈ {∅︀, Y,{1}, {2}, {1, 2}} for all e ∈ E}. It is easy to verify that BO(Y, τ,E) = τ ∪ {F ∈ SS(Y,E) : F(e) ∈ {{1, 3}, {2, 3}}}.

Therefore, C_{3}/∈ BO(Y, τ,E). Conversely, from Corollary 2.3, C_{3} ∈ SS(Y,E) = ωBO(Y, τ,E).

The following example shows that inclusion in Theorem 2.2 cannot be replaced by equality. In general,

Example 2.7

Let Y = ℝ, B = ℤ, μ be the usual topology on Y, and τ = {F ∈ SS(Y,E) : F(b) ∈ μ for all e ∈ E}. Consider (Y, τ,E). Let T = C_ℚ. As Cl_τ (T) = C_Clμ(ℚ) = C_ℝ = 1_E, then T ⊆̃ Int_τ (Cl_τ (T)) ∪̃Cl_τ (Int_τ (T)). Thus, by Corollary 2.5, T ∈ ωBO(Y, τ,E). However, T /∈ τ_ω.

Theorem 2.8

Let (Y, τ,E) be an STS and T ∈ SS(Y,E). Then T ∈ ωBO(Y, τ,E) if and only if for every e_y∊̃T, there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and K ∈ CSS(Y,E) such that S – K ⊆̃ T.

Theorem 2.9

Let (Y, τ,E) be an STS and T ∈ SS(Y,E). If F ∈ ωBC(Y, τ,E), then there exist if for C ∈ BC(Y, τ,E) and K ∈ CSS(Y,E) such that F ⊆̃ C∪̃ K.

Proof

Suppose that F ∈ ωBC(Y, τ,E). Then 1_E – F ∈ ωBO(Y, τ,E). If F = 1_E, then we are done. If F ≠ 1_E, then there exists e_y∊̃1_E –F. Therefore, by Theorem 2.8, there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and K ∈ CSS(Y,E) such that S – K ⊆̃ 1_E – F and thus F⊆̃ 1_E – (S – K) = 1_E – (S∩̃ (1_E – K)) = (1_E – S)∪̃ K. Put C = 1_E – S. Then C ∈ BC(Y, τ,E) such that F ⊆̃C∪̃ K.

Theorem 2.10

Let (Y, τ,E) be an STS. If T ∈ ωBO(Y, τ,E) and H ∈ τ_ω, then T∩̃ H ∈ ωBO(Y, τ,E).

Proof

Let e_y∊̃T∩̃ H. As e_y∊̃T ∈ ωBO(Y, τ,E), then there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and S – T ∈ CSS(Y,E). As e_y∊̃H ∈ τ_ω, W ∈ τ then exists such that e_y ∊̃W and W – H ∈ CSS(Y,E). Therefore, we have e_y ∊̃S∩̃ W and S∩̃ W ∈ BO(Y, τ,E). As

then (S∩̃ W) – (T∩̃ H), ∈ CSS(Y,E). It follows that T∩̃ H ∈ ωBO(Y, τ,E).

Corollary 2.11

Let (Y, τ,E) be an STS. If T ∈ ωBO(Y, τ, E) and H ∈ τ, then T ∩̃H ∈ ωBO(Y, τ,E).

For a STS (Y, τ,E), the collection ωBO(Y, τ,E) is not closed under finite soft intersection. In general,

Example 2.12

Let Y = ℝ, E = {a, b, c}, μ be the usual topology on Y, and τ = {H ∈ SS(Y,E) : H(e) ∈ μ for every e ∈ E}.

Then C_ℚ, C_[0,1) ∈ ωBO(Y, τ,E). Conversely, if C_ℚ ∩̃ C_[0,1) = C_ℚ∩[0,1) ∈ωBO(Y, τ,E), then there existsH∈BO(Y, τ,E) such that a₀ ∊̃H and H–C_ℚ∩[0,1) ∈ CSS(Y,E). As C_ℚ∩[0,1) ∈ CSS(Y,E), then H ∈ CSS(Y,E), which is impossible. Therefore, C_ℚ ∩̃C_[0,1)/∈ ωBO(Y, τ,E).

Theorem 2.13

Let (Y, τ,E) be an STS and let {T_α : α ∈ Δ} ⊆ ωBO(Y, τ,E). Then ∪̃_α∈ΔT_α ∈ ωBO(Y, τ,E).

Proof

Let e_y∊̃∪̃ _α∈ΔT_α. Choose β ∈ Δ such that e_y∊̃T_β. S ∈ BO(Y, τ,E) then exists such that e_y∊̃S and S – T_β ∈ CSS(Y,E). Now,

As S – T_β ∈ CSS(Y,E), S – (∪̃ _α∈ΔT_α) ∈ CSS(Y,E). Hence, ∪̃ _α∈ΔT_α ∈ ωBO(Y, τ,E).

Theorem 2.14

Let (Y, τ,E) be an STS and let {C_α : α ∈ Δ} ⊆ ωBC(Y, τ,E). Then ∩̃ _α∈ΔC_α ∈ ωBC(Y, τ,E).

Proof

Let {C_α : α ∈ Δ} ⊆ ωBC(Y, τ,E). Then {1_E – C_α : α ∈ Δ} ⊆ ωBO(Y, τ,E). Thus, by Theorem 2.13, ∪̃_α∈Δ(1_E – C_α) = 1_E – ∩̃ _α∈ΔC_α ∈ ωBO(Y, τ,E). Hence, ∩̃_α∈ΔC_α ∈ ωBC(Y, τ,E).

Theorem 2.15

Let {(Y, μ_e) : e ∈ E} be an indexed family of TSs. Let T ∈ SS(Y,E). Then T ∈ BO(Y,⊕_e∈Eμ_e,E) if and only if T(e) ∈ BO(Y, μ_e) for all e ∈ E.

Proof

Necessity. Let T ∈ BO(Y,⊕_e∈Eμ_e,E). Let e ∈ E. Let τ = ⊕_e∈Eμ_e. As T ∈ BO(Y,⊕_e∈Eμ_e,E), then T ⊆̃ Int_τ (Cl_τ (T))∪̃ Cl_τ (Int_τ (T)). Therefore,

According to Lemma 4.9 of [19],

and

Therefore,

Hence, T(e) ∈ BO(Y, μ_e).

Corollary 2.16

Let (Y, μ) be a TS and E be a set of parameters. Let T ∈ SS(Y,E). Then T ∈ BO(Y, τ(μ),E) if T(e) ∈ BO(Y, μ) for all e ∈ E.

Proof

For each e ∈ E, put μ_e = μ. Subsequently, we have τ (μ) = ⊕_e∈Eμ_e, and by Theorem 2.15 we get the result.

Theorem 2.17

Let {(Y, μ_e) : e ∈ E} be an indexed family of TSs. Let T ∈ SS(Y,E). Then T ∈ ωBO(Y,⊕_e∈Eμ_e,E) if T(e) ∈ ωBO(Y, μ_e) for all e ∈ E.

Proof

Corollary 2.18

Let (Y, μ) be a TS and E be a set of parameters. Let T ∈ SS(Y,E). Then T ∈ ωBO(Y, τ(μ),E) if T(e) ∈ ωBO(Y, μ) for all e ∈ E.

Proof

For each e ∈ E, put μ_e = μ. Subsequently, we have τ (μ) = ⊕_e∈Eμ_e, and we obtain the result using Theorem 2.17.

Definition 2.19

Let (Y, τ,E) be an STS and let H ∈ SS(Y,E).

(a) The soft ωb-closure of H in (Y, τ,E) is denoted by ωb-Cl_τ (H) and defined by

(b) The soft ωb-interior of H in (Y, τ,E) is denoted by ωb-Int_τ (H) and defined by

Theorem 2.20

Let (Y, τ,E) be an STS and H ∈ SS(Y,E). Then

(a) H ∈ ωBC(Y, τ,E) if and only if H = ωb-Cl_τ (H).

(b) H ∈ ωBO(Y, τ,E) if and only if H = ωb-Int_τ (H).

Proof

Obvious.

Theorem 2.21

Let (Y, τ,E) be an STS and H ∈ SS(Y,E). Then

(a) ωb-Cl_τ (H) is the smallest soft ωb-closed set in (Y, τ,E), which contains H.

(b) ωb-Int_τ (H) is the largest soft ωb-open set in (Y, τ,E), which is contained in H.

(c) e_y∊̃ωb-Cl_τ (H) if for every F ∈ ωBO(Y, τ,E) with e_y∊̃F, we have F∩̃ H ≠ 0_E.

Proof

(a) and (b) are significant.

(c) Suppose that e_y∊̃ωb-Cl_τ (H) and suppose to the contrary that there exists F ∈ ωBO(Y, τ,E) such that e_y∊̃F and F∩̃ H = 0_E. Therefore, we have H⊆̃ 1_E –F ∈ ωBC(Y, τ,E) and hence ωb-Cl_τ (H)⊆̃ 1_E – F. As e_y∊̃ωb-Cl_τ (H), then e_y∊̃1_E – F. However, e_y∊̃F. Conversely, suppose that for every F ∈ ωBO(Y, τ,E) with e_y∊̃F, we have F∩̃ H ≠ 0_E and suppose to the contrary that e_y∊̃1_E – ωb-Cl_τ (H). As by (a) ωb-Cl_τ (H) ∈ ωBC(Y, τ,E), then 1_E – ωb-Cl_τ (H) ∈ ωBO(Y, τ,E). Therefore, by assumption, (1_E – ωb-Cl_τ (H)) ∩̃ H ≠ 0_E. However, (1_E – ωb-Cl_τ (H))∩̃ H⊆̃ (1_E – H)∩̃ H = 0_E.

Theorem 2.22

Let (Y, τ,E) be an STS and H ∈ SS(Y,E).

(a) ωb-Cl_τ (1_E – H) = 1_E – ωb-Int_τ (H).

(b) ωb-Int_τ (1_E – H) = 1_E – ωb-Cl_τ (H).

Proof

(a) As 1_E –H⊆̃ 1_E –ωb-Int_τ (H) ∈ ωBC(Y, τ,E), then by Theorem 2.21 (a), ωb-Cl_τ (1_E–H)⊆̃ 1_E–ωb-Int_τ (H). To show that 1_E – ωb-Int_τ (H)⊆̃ ωb-Cl_τ (1_E – H), suppose to the contrary that there exists e_y∊̃ (1_E – ωb-Int_τ (H)) – ωb-Cl_τ (1_E – H). As e_y∊̃1_E – ωb-Cl_τ (1_E – H), then by Theorem 2.21(c), there exists F ∈ ωBO(Y, τ,E) such that e_y∊̃F and F∩̃ (1_E – H) = 0_E. Thus, we have e_y∊̃F⊆̃ H with F ∈ ωBO(Y, τ,E), and hence e_y∊̃ωb-Int_τ (H). However, e_y∊̃ (1_E – ωb-Int_τ (H)).

(b) By (a), ωb-Int_τ (1_E–H) = 1_E–ωb-Cl_τ (1_E–(1_E – H)) = 1_E – ωb-Cl_τ (H).

Theorem 2.23

Let (Y, τ,E) be a soft, locally countable STS. Then for any H ∈ SS(Y,E), H = ωb-Cl_τ (H) = ωb-Int_τ (H).

Proof

Following Corollary 2.3.

Theorem 2.24

Let (Y, τ,E) be an STS and H ∈ SS(Y,E). Then

(a) ωb-Cl_τ (H)⊆̃ b-Cl_τ (H).

(b) b-Int_τ (H)⊆̃ ωb-Int_τ (H).

Proof

Following these definitions and Corollary 2.5.

The following examples show that each inclusion in Theorem 2.24 cannot be replaced by an equality.

Example 2.25

Let Y = {1, 2, 3}, E = {a}, and τ = {F ∈ SS(Y,E) : F(a) ∈ {∅︀, Y, {1}, {1, 2}}}. Let H = a_{2}. As Int_τ (Cl_τ (H)) = Cl_τ (Int_τ (H)) = 0_E, then H /∈ BO(Y, τ,E), and thus b-Int_τ (H) = 0_E. In addition, b-Cl_τ (1_E–H) = 1_E – b-Int_τ (H) = 1_E. Conversely, by Theorem 2.23, ωb-Int_τ (H) = H ≠ b-Int_τ (H) and ωb-Cl_τ (1_E – H) = 1_E – H ≠ b-Cl_τ (1_E – H).

Definition 2.26

STS (Y, τ,E) is called a soft b-antilocally countable if for every F ∈ BO(Y, τ,E)–{0_E},F /∈ CSS(Y, E).

Theorem 2.27

Every soft b-antilocal countable is soft antilocally countable.

Proof

Follows from the definitions and the fact that τ ⊆ BO(Y, τ,E).

The following example shows that the converse of Theorem 2.27 need not be true in general:

Example 2.28

Let (Y, τ,E) be as in Example 2.7. Let T = C_ℚ. It is proved in Example 2.7 that T ∈ BO(Y, τ,E). As T ∈ (BO(Y, τ,E) – {0_E}) ∩ CSS(Y,E), then (Y, τ,E) is not soft b-antilocally countable. Conversely, it is clear that (Y, τ,E) is soft antilocally countable.

Theorem 2.29

Let {(Y, μ_e) : e ∈ E} be an indexed family of TSs. Then (Y,⊕_e∈Eμ_e,E) is soft b-antilocally countable if and only if (Y, μ_e) is b-antilocally countable for all e ∈ E.

Proof

Corollary 2.30

Let (Y, μ) be a TS and E be a set of parameters. Then (Y, τ(μ),E) is soft b-antilocally countable if and only if (Y, μ) b-antilocally countable for all e ∈ E.

Proof

For each e ∈ E, put μ_e = μ. Subsequently, we have τ (μ) = ⊕_e∈Eμ_e, and we obtain the result using Theorem 2.29.

Theorem 2.31

Let (Y, τ,E) be a soft b-antilocally countable and let H ∈ τ. Then ωb-Cl_τ (H) = b-Cl_τ (H).

Proof

By Theorem 2.24 (a), ωb-Cl_τ (H)⊆̃ b-Cl_τ (H). To show that b-Cl_τ (H)⊆̃ ωb-Cl_τ (H), let e_y∊̃b-Cl_τ (H) and let F ∈ ωBO(Y, τ,E) with e_y∊̃F. By Theorem 2.8, there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and K ∈ CSS(Y,E) such that S – K ⊆̃ F. Thus, (S∩̃ H) – K ⊆̃ F∩̃ H; hence, as e_y∊̃b-Cl_τ (H) and e_y∊̃S ∈ BO(Y, τ,E), then S∩̃ H ≠ 0_E. Note that S∩̃ H ∈ BO(Y, τ,E). As (Y, τ,E) is soft b-antilocally countable, then S∩̃H /∈ CSS(Y,E). Because K ∈ CSS(Y,E) and S∩̃H /∈ CSS(Y,E), (S∩̃ H) –K /∈ CSS(Y,E). As (S∩̃ H) –K ⊆̃ F∩̃ H, then F∩̃H /∈ CSS(Y,E). Thus, F∩̃ H ≠ 0_E. Therefore, by Theorem 2.21(c), e_y∊̃ωb-Cl_τ (H).

Theorem 2.31

Let (Y, τ,E) be a soft b-antilocally countable and let H ∈ τ. Then ωb-Cl_τ (H) = b-Cl_τ (H).

Proof

By Theorem 2.24 (a), ωb-Cl_τ (H)⊆̃ b-Cl_τ (H). To show that b-Cl_τ (H)⊆̃ ωb-Cl_τ (H), let e_y∊̃b-Cl_τ (H) and let F ∈ ωBO(Y, τ,E) with e_y∊̃F. By Theorem 2.8, there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and K ∈ CSS(Y,E) such that S – K ⊆̃ F. Thus, (S∩̃ H) – K ⊆̃ F∩̃ H; hence, as e_y∊̃b-Cl_τ (H) and e_y∊̃S ∈ BO(Y, τ,E), then S∩̃ H ≠ 0_E. Note that S∩̃ H ∈ BO(Y, τ,E). As (Y, τ,E) is soft b-antilocally countable, then S∩̃H /∈ CSS(Y,E). Because K ∈ CSS(Y,E) and S∩̃H /∈ CSS(Y,E), (S∩̃ H) –K /∈ CSS(Y,E). As (S∩̃ H) –K ⊆̃ F∩̃ H, then F∩̃H /∈ CSS(Y,E). Thus, F∩̃ H ≠ 0_E. Therefore, by Theorem 2.21(c), e_y∊̃ωb-Cl_τ (H).

Corollary 2.32

Let (Y, τ,E) be a soft b-antilocally countable and let M ∈ τ^c. Then ωb-Int_τ (M) = b-Int_τ (M).

Proof

As M ∈ τ^c, 1_E – M ∈ τ. Therefore, by Theorem 2.31, ωb-Cl_τ (1_E –M) = b-Cl_τ (1_E –M) and hence 1_E –ωb-Cl_τ (1_E –M) = 1_E –b-Cl_τ (1_E –M). But 1_E –b-Cl_τ (1_E –M) = b-Int_τ (M). Additionally, by Theorem 2.22(b), 1_E–ωb-Cl_τ (1_E –M) = ωb-Int_τ (1_E – (1_E –M)) = ωb-Int_τ (M). It follows that ωb-Int_τ (M) = b-Int_τ (M).

Definition 2.33

A soft function f_pu : (Y, τ,E) –→ (Z, δ,D) is said to be

(a) quasi soft b-open if for every H ∈ BO(Y, τ,E), f_pu(H) ∈ δ.

(b) weakly quasi soft b-open if for every H ∈ BO(Y, τ,E), f_pu(H) ∈ BO(Z, δ,D).

Theorem 2.34

Every quasi soft b-open soft function is weakly quasi soft b-open.

Proof

Straightforward.

The following is an example of a quasi soft b-open soft function:

Example 2.35

Let (Y, τ,E) be as shown in Example 2.6. We define p : Y –→ Y and u : E –→ E as follows:

Subsequently, f_pu : (Y, τ,E) –→ (Y, τ,E) is quasi soft b-open.

The following example shows that the converse of Theorem 2.34 need not be true in general:

Example 2.36

Let (Y, τ,E) be as shown in Example 2.6. We define p : Y –→ Y and u : E –→ E as follows:

Let f_pu : (Y, τ,E) –→ (Y, τ,E). Then f_pu is weakly quasi soft b-open. Let H ∈ SS(Y,E), where H(e) ∈ {1, 3} for all e ∈ E. Then H ∈ BO(Y, τ,E); however, f_pu(H) = H /∈ δ. Therefore, f_pu is not quasi soft b-open.

Theorem 2.37

Let p : (Y, μ) –→ (Z, λ) be function between the two TSs and let u : E –→ D be a function between the two sets of parameters. Subsequently, f_pu : (Y, τ(μ),E)–→ (Z, τ(λ),D) is quasi soft b-open if and only if p : (Y, μ) –→ (Z, λ) is quasi b-open.

Proof

Example 2.38

Let p : ℝ –→ ℝ and u : ℤ –→ ℤ be the identities functions and let μ be the usual topology on ℝ. Then clearly that f_pu : (ℝ, τ(μ), ℤ) –→ (ℝ, τ(μ), ℤ) is soft open function. Let U = [0, 1] ∪ [(1, 2) ∩ ℚ]. Since U ∈ BO(ℝ, ℤ) while p(U) = U /∈ μ, then p : (ℝ, ℤ) –→ (ℝ, ℤ) is not quasi b-open. Hence, by Theorem 2.37, f_pu : (ℝ, τ(μ), ℤ) –→ (ℝ, τ(μ), ℤ) is not quasi soft b-open.

Theorem 2.39

Let p : (Y, μ) –→ (Z, λ) be a function between the two TSs and let u : E –→ D be a function between the two sets of parameters. Subsequently, f_pu : (Y, τ(μ),E) –→ (Z, τ(λ),D) is weakly quasi soft b-open if and only if p : (Y, μ) –→ (Z, λ) is weakly quasi b-open.

Proof

Theorem 2.40

Let f_pu : (Y, τ,E) –→ (Z, δ,D) be quasi soft b-open. Then for every H ∈ ωBO(Y, τ,E), f_pu(H) ∈ δ_ω.

Proof

Let H ∈ ωBO(Y, τ,E) and d_z∊̃f_pu(H). Choose e_y∊̃H such that f_pu(e_y) = d_z. Since H ∈ ωBO(Y, τ,E), then there exists S ∈ BO(Y, τ,E) such that e_y∊̃S and S – H ∈ CSS(Y,E). Since f_pu : (Y, τ,E) –→, (Z, δ,D) is quasi soft b-open, then f_pu(S) ∈ δ such that d_z = f_pu(e_y)∊̃f_pu(S) and f_pu(S) – f_pu(H)⊆̃ f_pu(S – H) ∈ CSS(Z,D). Thus, f_pu(H) ∈ δ_ω.

Here, we establish the characterization and preservation theorems for soft b-Lindelof STSs.

Definition 3.1

Let (Y, τ,E) be an STS, H ∈ SS(Y,E), and β ⊆ SS(Y,E). Then

(a) β is the soft cover of H if H⊆̃ ∪̃_F∈βF.

(b) β is the soft b-open cover of H in (Y, τ,E) if β is a soft cover of H and β ⊆ BO(Y, τ,E).

(c) H is the soft b-Lindelof relative to (Y, τ,E) if every soft b-open cover of H in (Y, τ,E) has a countable soft subcover.

(d) (Y, τ,E) is the soft b-Lindelof if 1_E is soft b-Lindelof relative to (Y, τ,E).

Theorem 3.2

If (Y, τ,E) is an STS such that H is a soft b-Lindelof relative to (Y, τ,E) for all H ∈ τ, then H is soft b-Lindelof relative to (Y, τ,E) for all H ∈ SS(Y,E).

Theorem 3.3

For any STS (Y, τ,E), the following are equivalent:

(a) (Y, τ,E) is a soft b-Lindelof.

(b) Every soft cover of 1_E of the members of ωBO(Y, τ,E) has a countable soft subcover.

Theorem 3.4

Let {(Y, μ_e) : e ∈ E} be an indexed family of TSs. If (Y,⊕_e∈Eμ_e,E) is soft b-Lindelof, then E is countable, and (Y, μ_e) is b-Lindelof for all e ∈ E.

Definition 3.5

A soft function f_pu : (Y, τ,E) –→ (Z, δ,D) is said to be soft ωb-continuous provided that for every K ∈ δ, fpu-1(K)∈ωBO(Y,τ,E).

Theorem 3.6

Let p : (Y, μ) –→ (Z, λ) be a function between the two TSs and let u : B –→ D be a function between the two sets of parameters. Then f_pu : (Y, τ(μ),B) –→ (Z, τ(λ),D) is soft b-continuous if p : (Y, μ) –→ (Z, λ) is b-continuous.