1 Introduction and preliminaries

Molodtsov [1] proposed soft sets as a beneficial alternative to existing mathematical methodologies for dealing with uncertainty. In contrast to probability theory, fuzzy set theory, and rough set theory, soft sets do not rely on precise values such as membership grade and probability. This is because, in most real cases, the genuine possibilities and membership grades are not well-known enough to assign precise values. This property of soft sets allows them to be used in various circumstances. Since its debut, the concept of soft sets has received significant attention and has been successfully used in various applications (for example, see [2–6]).

Several researchers have extended soft set theory to other mathematical structures, such as soft ideal theory [7], soft group theory [8], soft ring theory [9], soft σ-algebras [10, 11], and others.

Soft topology, which was presented in [12] as a merger of classical topology and soft set theory is one such structure. Noteworthy contributions to the development of soft topology in which authors modified and applied numerous traditional topological ideas to the setting of soft sets, for instance, soft compact spaces [13–15], soft metric [16], soft connected [17], soft separation axioms [18–22], and soft extremally disconnected spaces [23].

Both weak and strong kinds of soft open sets are essential in the study of soft topology because they provide a more general and flexible framework for investigating soft topological spaces and their features. They also provide a versatile framework for capturing uncertainty and vagueness in soft topology, allowing for variable levels of accuracy and ambiguity in diverse applications. Furthermore, they enable greater freedom in designing new classes of continuous functions, allowing us to construct functions that are better suited to specific applications and challenges. Therefore, this research area was and still is attractive to researchers. For instance, “soft semi-open” [24], “soft pre-open” [25], “soft β-open” [25], “soft θ-open” [26], “soft α-open” [27], “soft regular open” [28], “soft somewhere dense” [29], “soft δ-open” [30], “soft ω-open” [31], “weakly soft -open” [32], “soft γ-open” [33], “cluster soft sets” [34], “weakly soft β-open” [35], “weakly soft pre-open” [36], “soft parametric somewhat-open” [37], “weakly soft α-open” [38], and so on.

Soft continuity was defined as a main concept in the context of soft topological spaces [39]. Afterward, many forms of soft continuity appeared in the literature. For instance, “soft semi-continuous” [40], “soft pre-continuous” [27], “soft β-continuous” [41], “soft θ-continuous” [26], “soft α-continuous” [27], “soft δ-continuous” [30], “soft ω-continuous” [42], “soft ω_s-continuous” [42], “weakly soft β-continuous” [35], “weakly soft pre-continuous” [36], “weakly soft α-continuous” [38], “soft complete continuous” [43], “soft strongly continuous” [43], “soft C-continuous” [44], “soft almost C-continuous” [44], “soft ω-θ-continuous” [45], “soft weakly θ_ω-continuous” [45], and so on. The authors in [46] proved that soft continuity is useful in developing computational topological applications and digital images.

In this paper, we define soft super-continuity as a new form of soft mapping. We present various characterizations of this soft concept. Also, we show that soft super-continuity lies strictly between soft continuity and soft complete continuity and that soft super-continuity is a strong form of soft δ-continuity. In addition, we give some sufficient conditions for the equivalence between soft super-continuity and other related concepts. Moreover, we characterize soft semi-regularity in terms of super-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and products by soft super-continuity. In addition, we study the relationship between each soft super-continuity soft δ-continuity and their analogous notions in general topology. Finally, we give several sufficient conditions on the soft mapping to have a soft δ-closed graph.

This article is organized as follows: In Section 2, we introduce the concept of “soft super-continuity,” a new type of soft mapping. We present several characterizations of them. Also, we study their relationships with some other soft continuity types. Moreover, we use them to characterize soft semi-regularity. Furthermore, we provide several results on soft composition, restrictions, preservation, and products related to soft super-continuity. In addition to these, we investigate the correspondence between soft super-continuity and soft δ-continuity and their analogous notions in general topology. In Section 3, we introduce several sufficient conditions for a soft mapping to have a soft δ-closed graph.

In the rest of this section, we introduce some basic definitions and terminology we will use in the sequel:

Let E be an initial universe and Z be a set of parameters. A soft set over Z relative to E is a function , where is the power set of Z. The collection of soft sets over E relative to Z is denoted by SS (Z, E). Let H ∈ SS (Z, E). If H (a) = ∅ for each a ∈ E, then H is called the null soft set over Z relative to E and denoted by 0_E. If H (a) = Z for all a ∈ E, then H is called the absolute soft set over Z relative to E and denoted by 1_E. If there exist b ∈ E and y ∈ Z such that H (b) = {y} and H (a) = ∅ for all a ∈ E − {b}, then H is called a soft point over E relative to Z and denoted by b_y. The collection of all soft points over E relative to Z is denoted by SP (Z, E). If for some b ∈ E and X ⊆ Z, H (b) = X and H (a) = ∅ for all a ∈ L − {b}, then K will be denoted by a_X. If for some X ⊆ Z, H (a) = X for all a ∈ E, then H will be denoted by C_X. If H ∈ SS (Z, E) and a_x ∈ SP (Z, E), then a_x is said to belong to H (notation: ) if x ∈ H (a). Soft topological spaces were defined in [12] as follows: A triplet , where , is called a soft topological space if 0_E, 1_A ∈ ℜ, and is closed under finite soft intersections and arbitrary soft unions.

Throughout this paper, we will use concepts and phrases as they appear in [31, 47].

Let be a soft topological space, (Z, ξ) be a topological space, M ∈ SS (Z, E), and A ⊆ Z. Then , , Int_ξ(A), and Cl_ξ(A), , and ξ^c will denote the soft interior of M in , the soft closure of M in , the interior of A in (Z, ξ), and the closure of A in (Z, ξ), the collection of all soft closed sets in , and the family of all closed sets in (Z, ξ).

Definition 1.1. Let (Z, ξ) be a topological space, and let A ⊆ Z. Then A is called a

1. (a) [48] “regular open set in (Z, ξ)” if A = Int_ξ(Cl_ξ(A)). RO (ξ) will denote the collection of all regular open sets in (Z, ξ).
2. (b) [49] “δ-open set in (Z, ξ)” if for each z ∈ A, we find U ∈ λ such that z ∈ U ⊆ Int_ξ(Cl_ξ(U)) ⊆ A. ξ_δ will denote the collection of all δ-open sets in (Z, ξ).

It is well known that ξ_δ is a topology having RO (ξ) as a base.

Definition 1.2. A mapping g : (Z, ξ) → (Z, ϕ) is called

1. (a) [50] “δ-continuous” if for every z ∈ Z and V ∈ RO (ϕ) such that g(z) ∈ V, we find W ∈ RO (ξ) such that z ∈ W and g(W) ⊆ V.
2. (b) [51] “super-continuous (Notation: SC)” if g⁻¹(V) ∈ ξ_δ for every V ∈ ϕ.

Definition 1.3. Let be a soft topological space, and let K ∈ SS (Z, E). Then K is called a

1. (a) [28] “soft regular open set in ” if . The soft complement of a soft δ-open set in is called a “soft δ-closed set in ”. (resp. ) will denote the collection of all soft regular open (resp. soft regular open) sets in .
2. (b) [30] “soft δ-open set in ” if for each , we find H ∈ ℜ such that . ℜ_δ will denote the collection of all δ-open sets in .

It is well known that ℜ_δ is a soft topology having as a soft base.

Definition 1.4. A soft mapping is called

1. (a) [26] “soft θ-continuous” if for each and each G ∈ ℘ such that , there exists K ∈ ℜ such that and .
2. (b) [30] “soft δ-continuous” if for each and each G ∈ RO (℘) such that , there exists such that and .
3. (c) [52] “soft almost continuous” if for each and each G ∈ ℘ such that , there exists K ∈ ℜ such that and .
4. (d) [53] “soft almost open” if f_qv(K) ∈ ℘ for every .
5. (e) [43] “soft complete continuous” if for every G ∈ ℘.

Definition 1.5. A soft topological space is said to be

1. (a) [53] “soft Hausdorff” if for every two soft points e_x, s_y ∈ SP (Z, E) with e_x ≠ s_y, there exist M, N ∈ ℜ such that , , and .
2. (b) [53] “soft regular” if for every e_z ∈ SP (Z, E) and every T ∈ ℜ such that , there exists K ∈ ℜ such that .
3. (c) [54] “soft semi-regular” if for each and each K ∈ ℜ, there exists such that .
4. (d) [55] “soft almost regular” if for each and each , there are M, N ∈ ℜ such that , , and .
5. (e) [56] “soft nearly compact” (resp. “soft nearly Lindelof”) if for each such that , there exists a finite (resp. countable) subcollection such .

Definition 1.6. [56] Let be a soft topological space, and let K ∈ SS (Z, E). Then K is called a “soft nearly compact set relative to ” (resp. “soft nearly Lindelof set relative to ”) if for each such that , there exists a finite (resp. countable) subcollection such .

Definition 1.7. [13] Let and (Y, ℘, F) be two soft topological spaces, and let . Then the soft topology over Z × Y relative to E × F having as a soft base is called the product soft topology and is denoted by .

2 Soft super-continuity

In this section, we introduce the concept of “soft super-continuity,” a new type of soft mapping. We present several characterizations of them. Also, we study their relationships with some other soft continuity types. Moreover, we use them to characterize soft semi-regularity. Furthermore, we provide several results on soft composition, restrictions, preservation, and products related to soft super-continuity. In addition to these, we investigate the correspondence between soft super-continuity and soft δ-continuity and their analogous notions in general topology.

Definition 2.1. A soft mapping is called soft super-continuous (notation: SC) if ℜ_δ for every K ∈ ℘.

The following result gives several characterizations of soft super-continuous mappings:

Theorem 2.2. For a soft mapping , the following are equivalent:

1. (1). is soft SC.
2. (2). for every T ∈ ℘^c.
3. (3). for each .
4. (4). for each .
5. (5). f_qv : (Z, ℜ_δ, E) → (Y, ℘, F) is soft continuous.
6. (6). For a soft base for (Y, ℘, F), for every .
7. (7). For a soft subbase for (Y, ℘, F), for every .
8. (8). For each e_z ∈ SP (Z, E) and each G ∈ ℘ such that , we find H ∈ ℜ_δ such that and .
9. (9). For each e_z ∈ SP (Z, E) and each G ∈ ℘ such that , we find K ∈ ℜ such that and .

Proof. (1) → (2): Let T ∈ ℘^c. Then 1_F − T ∈ ℘. So, by (1), . Hence, .

(2) → (3): Let . Then Cl_℘(A) ∈ ℘^c. So, by (2), . Since , then .

(3) → (4): Let . Then, by (3), and so .

(4) → (5): Let K ∈ ℘. Then Int_℘(K) = K, and by (4), . Thus, . Hence, . This shows that f_qv : (Z, ℜ_δ, E) → (Y, ℘, F) is soft continuous.

(5) → (6) and (6) → (7) are obvious.

(7) → (8): Let K ∈ ℘. To show that , let . Then we find such that and so . Let . Then , , and by (7), G ∈ ℜ_δ. This ends the proof.

(8) → (9): Let e_z ∈ SP (M, Z) and G ∈ ℘. Then, by (8), H ∈ ℜ_δ such that and . Choose such that . Therefore, we have and .

(9) → (1): Let G ∈ ℘ and let . Then f_qv (e_z) , and by (9), there exists K ∈ ℜ such that and . Then we have and . This shows that .

Theorem 2.3. If is soft SC, then is SC for every a ∈ E.

Proof. Suppose that is soft SC, and let a ∈ E. By Theorem 4.9(5), f_qv : (Z, ℜ_δ, E) → (Y, ℘, F) is soft continuous. So, by Proposition 3.8 of [58], q : (Z, (ℜ_δ)_a) → (Y, ℘_v(a)) is continuous. Since by Theorem 30 of [57], , then is continuous. Hence, is SC.

The following two results discuss the relationships between soft super-continuity and its analogous concept in general topology:

Theorem 2.4. Let {(Z, β_e) : e ∈ E} and {(Y, α_f) : f ∈ F} be two collections of TSs. Let q : Z → Y and v : E → F be mappings where v is bijective. Then f_qv : (Z, ⊕_e∈Eβ_e, E) → (Y, ⊕_f∈Fα_f, F) is soft SC if and only if q : (Z, β_e) → (Y, α_v(e)) is SC for all e ∈ E.

Proof. Necessity. Let f_qv : (Z, ⊕_{e ∈ E}β_e, E) → (Y, ⊕_{f ∈ F}α_f, F) be soft SC. Let e ∈ E. Then, by Theorem 2.3, q : (Z, (⊕_e∈Eβ_z)_e) → (Y, (⊕_f∈Fα_f)_v(e)) is SC. But by Theorem 3.11 of [47], (⊕_{e ∈ E}β_e)_e = β_e and (⊕_f∈Fα_f)_v(e) = α_v(e). Hence, q : (Z, β_e) → (Y, α_v(e)) is SC.

Sufficiency. Let q : (Z, β_e) → (Y, α_v(e)) be SC for all e ∈ E. Let K ∈ ⊕_{f ∈ F}α_f. By Theorem 31 of [57], it is sufficient to show that for all e ∈ E. Let e ∈ E. Since q : (Z, β_e) → (Y, α_v(e)) is SC and K (v(e)) ∈ α_v(e), then .

Corollary 2.5. Let q : (Z, ξ) → (Y, ϕ) and v : E → F be two mappings where v is a bijection. Then q : (Z, ξ) → (Y, ϕ) is SC if and only if f_qv : (Z, τ(ξ), E) → (Y, τ(ϕ), F) is soft SC.

Proof. For each e ∈ E and f ∈ F, put β_e = ξ and α_f = ϕ. Then τ(α) = ⊕_e∈Eβ_e and τ(ϕ) = ⊕_f∈Fα_f. By using Theorem 2.4, we get the result.

In Theorem 2.6 and Example 2.7, we discuss the relationships between the classes of soft SC mappings and soft continuous mappings:

Theorem 2.6. Every soft SC mapping is soft continuous.

Proof. Let be soft SC. Let K ∈ ℘. Then, by Theorem 2.2(5), . Hence, f_qv is soft continuous.

Theorem 2.6 is not reversible.

Example 2.7. Let , E = {a, b, d}, and . Suppose that . Then we find x ∈ (0, 1) such that . So, we find K ∈ ℜ such that . Thus, K = a_(0,1), and so . Hence, . Let q : Z → Z and v : E → E be the identity mappings. Since , then is soft continuous but not soft SC.

In Theorem 2.8 and Example 2.9, we discuss the relationships between the classes of soft complete continuous mappings and soft SC mappings:

Theorem 2.8. Every soft complete continuous mapping is soft SC.

Proof. Let be soft complete continuous. Let K ∈ ℘. Then . Hence, f_qv is soft SC.

Theorem 2.8 is not reversible.

Example 2.9. Let Z = {1, 2, 3}, ξ = {∅, Z, {1}, {3}, {1, 3}}, and E = {a, b}. Consider the identity mappings q : (Z, ξ) → (Z, ξ) and v : E → E. It is not difficult to see that RO (ξ) = {∅, Z, {1}, {3}}. Then q is SC. On the other hand, since {1, 3} ∈ ξ while q⁻¹({1, 3}) = {1, 3} ∉ RO (ξ), then q is not complete continuous. Therefore, by Corollary 2.5 and Corollary 1 of [43], f_qv : (Z, τ(ξ), E) → (Z, τ(ξ), E) is soft SC but not soft complete continuous.

Theorem 2.10. If is soft δ-continuous, then is δ-continuous for every a ∈ E.

Proof. Suppose that is soft δ-continuous. Then, by Theorem 6.2(6) of [30], f_qv : (Z, ℜ_δ, E) → (Y, ℘_δ, F) is soft continuous. So, by Proposition 3.8 of [58], f_qv : (Z, (ℜ_δ)_a) → (Y, (℘_δ)_v(a)) is soft continuous. Since by Theorem 30 of [57], and (℘_δ)_v(a) = (℘_v(a))_δ, then is continuous. Hence, by Theorem 2.2(7) of [50], is δ-continuous.

The following two results discuss the relationships between soft δ-continuity and its analogous concept in general topology:

Theorem 2.11. Let {(Z, β_e) : e ∈ E} and {(Y, α_f) : f ∈ F} be two collections of TSs. Let q : Z → Y and v : E → F be mappings where v is bijective. Then f_qv : (Z, ⊕_{e ∈ E}β_e, E) → (Y, ⊕_{f ∈ F}α_f, F) is soft δ-continuous if and only if q : (Z, β_e) → (Y, α_v(e)) is δ-continuous for all e ∈ E.

Proof. Necessity. Let f_qv : (Z, ⊕_{e ∈ E}β_e, E)→ (Y, ⊕_{f ∈ F}α_f, F) be soft δ-continuous. Let e ∈ E. Then by Theorem 2.10, q : (Z, (⊕_{e ∈ E}β_z)_e)→ (Y, (⊕_{f ∈ F}α_f)_v(e)) is δ-continuous. But by Theorem 3.11 of [47], (⊕_{e ∈ E}β_e)_e = β_e and (⊕_{f ∈ F}α_f)_v(e) = α_v(e). Hence, q : (Z, β_e) → (Y, α_v(e)) is δ-continuous.

Sufficiency. Let q : (Z, β_e) → (Y, α_v(e)) be δ-continuous for all e ∈ E. Let K ∈ (⊕_{f ∈ F}α_f)_δ. Then by Theorem 31 of [57], K (f) ∈ (α_f)_δ for all f ∈ F. By Theorem 31 of [57] and Theorem 6.2(7) of [30], it is sufficient to show that for all e ∈ E. Let e ∈ E. Since q : (Z, β_e) → (Y, α_v(e)) is δ-continuous and K (v(e)) ∈ (α_v(e))_δ, then .

Corollary 2.12. Let q : (Z, ξ) → (Y, ϕ) and v : E → F be two mappings where v is a bijection. Then q : (Z, ξ) → (Y, ϕ) is δ-continuous if and only if f_qv : (Z, τ(ξ), E) → (Y, τ(ϕ), F) is soft δ-continuous.

Proof. For each e ∈ E and f ∈ F, put β_e = ξ and α_f = ϕ. Then τ(α) = ⊕_{e ∈ E}β_e and τ(ϕ) = ⊕_f∈Fα_f. By using Theorem 2.11, we get the result.

In Theorem 2.13 and Example 2.14, we discuss the relationships between the classes of soft SC mappings and soft δ-continuous mappings:

Theorem 2.13. Every soft SC mapping is soft δ-continuous.

Proof. Let be soft SC. Let G ∈ RO (℘) ⊆ ℘. Then . Thus, by Theorem 6.2(7) of [30], f_qv soft δ-continuous.

Theorem 2.13 is not reversible.

Example 2.14. Let , ξ the usual topology on Z, ϕ the co-countable topology on Z, and E = {a, b}. Consider the identity mappings q : (Z, ξ) → (Z, ϕ) and v : E → E. Since RO (ϕ) = {∅, Z}, then q is δ-continuous. On the other hand, since while , then q is not SC. Therefore, by Corollaries 2.5 and 2.12, f_qv : (Z, τ(ξ), E) → (Z, τ(ϕ), F) is soft δ-continuous but not soft SC.

The following result provides sufficient conditions for the equivalence between soft SC mappings and soft δ-continuous mappings:

Theorem 2.15. If is a soft δ-continuous mapping such that (Y, ℘, F) is soft semi-regular, then f_qv is soft SC.

Proof. Let f_qv be soft δ-continuous such that (Y, ℘, F) is soft semi-regular. Let e_z ∈ SS (Z, E) and let G ∈ ℘ such that . Since (Y, ℘, F) is soft semi-regular, then there exists K ∈ ℘ such that . Since f_qv is a soft δ-continuous, then there exists H ∈ ℜ such that and . This shows that f_qv is soft SC.

Soft semi-regularity condition cannot be removed from Theorem 2.15, as it can be deduced from Example 2.14.

Corollary 2.16. If is a soft δ-continuous mapping such that (Y, ℘, F) is soft regular, then f_qv is soft SC.

The following result introduces a new characterization of soft semi-regularity:

Theorem 2.17. The following are equivalent for any STS .

1. (a) is soft semi-regular.
2. (b) Every soft continuous mapping is also soft SC.

Proof. (a) ⇒ (b): Let be soft continuous. By (a), . Thus, f_qv : (Z, ℜ_δ, E) → (Y, ℘, F) is soft continuous. Hence, by Theorem 2.2(5), is also soft SC.

(b) ⇒ (a): We will show that . Let G ∈ ℜ. Take and f_qv the identity soft mapping . Since is soft continuous, then by (b), is soft SC. Thus, by Theorem 2.2(5), . This shows that .

Corollary 2.18. If is a soft δ-continuous mapping such that and (Y, ℘, F) are both soft semi-regular, then the following are equivalent:

1. (a) f_qv is soft SC.
2. (b) f_qv is soft δ-continuous.
3. (c) f_qv is soft continuous.

Proof. Follows from Theorems 2.15 and 2.17.

Corollary 2.19. If is a soft continuous mapping such that is soft regular, then f_qv is soft SC.

Proof. The proof follows from Theorem 2.17.

In Theorems 2.20, 2.21, and 2.22, we discuss the behavior of soft SC mappings under soft composition:

Theorem 2.20. If is soft SC and is soft continuous, then is soft SC.

Proof. Let be soft SC and be soft continuous. Let K ∈ ℜ. Since is soft continuous, . Since is soft SC, then . This shows that is soft SC.

Theorem 2.21. If is a soft almost open, soft SC, and surjective, and if is a soft mapping, then is soft SC if and only if is soft continuous.

Proof. Necessity. Let be soft SC. Let K ∈ ℜ. Then . Since is soft almost open, . Since is surjective, . This shows that is soft continuous.

Sufficiency. Let be soft continuous. Let K ∈ ℜ. Then . Thus, by soft super-continuity of , . This shows that is soft SC.

Theorem 2.22. If is soft almost continuous and is soft SC, then is soft continuous.

Proof. Let be soft almost continuous and be soft SC. Let K ∈ ℜ. Since is soft SC, . Since is soft almost continuous, then by Theorem 3.8(b) of [52], . This shows that is soft continuous.

The following question is natural:

Problem 2.23. Let and be two soft mappings such that is soft almost continuous and is soft continuous. Is it true that is soft SC?

The following example gives a negative answer to Problem 2.23:

Example 2.24. Let , Y = {a, b}, Z = {1, 2}, ξ the co-countable topology on X, ϕ = {∅, Y, {a}}, γ = {∅, Z, {2}}, and . Define q₁ : X → Y, q₂ : Y → Z, and v₁, v₂ : E → E by q₂(a) = 2, q₂(b) = 1, and v₁, v₂ : E → E are the identity mappings. Then q₁ : (X, ξ) → (Y, ϕ) is almost continuous, and q₂ ∘ q₁ : (X, ξ) → (Z, γ) is continuous, but q₂ : (Y, ϕ) → (Z, γ) is not SC.

Therefore, by Corollary 3.4 of [44], Theorem 5.31 of [47], and Corollary 2.5, is soft almost continuous, and : (X, τ(ξ), E) → (Z, τ(γ), E) is soft continuous, but is not soft SC.

In Theorems 2.25, 2.27, 2.30, and Corollaries 2.26, and 2.28, we establish various preservation theorems using soft SC mappings.

Theorem 2.25. Let be soft SC. If K is a soft nearly compact set relative to , then f_qv(K) is a soft compact subset of (Y, ℘, F).

Proof. Let such that . Then . Since f_qv is soft SC, then . Since K is a soft nearly compact set relative to , then we find a finite subfamily such that . So, . This shows that f_qv(K) is a soft nearly compact set relative to (Y, ℘, F).

Corollary 2.26. Let be soft SC and surjective. If is soft nearly compact, then (Y, ℘, F) is soft compact.

Theorem 2.27. Let be soft SC. If K is a soft nearly Lindelof set relative to , then f_qv(K) is a soft Lindelof subset of (Y, ℘, F).

Proof. Let such that . Then . Since f_qv is soft SC, then . Since K is a soft nearly Lindelof set relative to , then we find a countable subfamily such that . So, . This shows that f_qv(K) is a soft nearly Lindelof set relative to (Y, ℘, F).

Corollary 2.28. Let be soft SC and surjective. If is soft nearly Lindelof, then (Y, ℘, F) is soft Lindelof.

Theorem 2.29. Let be soft SC and bijective such that is soft nearly compact and (Y, ℘, F) is soft Hausdorff. Then f_qv is soft almost open.

Proof. Let K ∈ ℜ_δ. Then 1_E − K ∈ (ℜ_δ)^c. Since is soft nearly compact, then 1_E − K is a soft nearly compact subset of . So, by Theorem 2.25, f_qv(1_E − K) is a soft compact subset of (Y, ℘, F). Since (Y, ℘, F) is soft Hausdorff, then f_qv(1_E − K) ∈ ℘^c. Since f_qv is bijective, f_qv(1_E − K) = 1_F− f_qv(K). Thus, f_qv(K) ∈ ℘. This shows that f_qv is soft almost open.

Theorem 2.30. Let be soft SC, soft almost open, and bijective. If is soft almost regular, then (Y, ℘, F) is soft regular.

Proof. Let G ∈ ℘^c and b_y ∈ 1_F − G. Since f_qv is soft SC, then by Theorem 2.2(2), . Since f_qv is bijective and b_y ∈ 1_F − G, then . Choose such that . Put S = 1_E−H. Then with . Since is soft almost regular, then there are M, N ∈ ℜ such that , , and . Now implies . Since f_qv is soft almost open, then , . Since f_qv is surjective, then and . Since f_qv is injective, then . This shows that (Y, ℘, F) is soft regular.

Theorem 2.31. If is a soft continuous mapping such that (Y, ℘, F) is soft regular, then f_qv is soft SC.

Proof. Let G ∈ ℘. Then, by soft continuity of f_qv, . To show that , let . Then , and by soft regularity of (Y, ℘, F), there exists K ∈ ℘ such that , and so, . Since f_qv is soft continuous, we have and . Thus, we have and

This shows that . Therefore, by Theorem 2.2(5), f_qv is soft SC.

For any mapping g : A → B, the mapping h : A → A × B defined by H(a) = (a, H (a)) will be denoted by g^#. The following result explains the relationships between a soft SC mapping and its soft graph:

Theorem 2.32. Let be a soft mapping. Then is soft SC if and only if f_qv is soft SC and is soft semi-regular.

Proof. Necessity. Suppose that is soft SC. Let p : Z × Y → Y, s : Z × Y → X, u : E × F → E, and t : E × F → F be the projection mappings. Then and are soft continuous. So, by Theorem 2.20, and the soft identity mapping are soft SC. To show that is soft semi-regular, we will show that . Let G ∈ ℜ. Since is soft SC, then .

Sufficiency. Suppose that f_qv is soft SC and is soft semi-regular. Since f_qv is soft SC, then f_qv is soft continuous. So, is soft continuous. Since is soft semi-regular, then by Theorem 2.17, is soft SC.

Lemma 2.33. Let and (Y, ℘, F) be two STSs, A ∈ SS (Z, E) − {0_E}, and B ∈ SS (Y, F) − {0_F}. Then

1. (a) if and only if and B ∈ RO (℘).
2. (b) if and only if A ∈ ℜ_δ and B ∈ ℘_δ.

Proof. (a) Necessity. Let . Then

Thus, and . Hence, and B ∈ RO (℘).

Sufficiency. Let and B ∈ RO (℘). Then and . So,

(b) Necessity. Let . Let and . Then . So, there exists such that . Choose G ∈ ℜ and K ∈ ℘ such that . Thus,

Therefore, and . This shows that A ∈ ℜ_δ and B ∈ ℘_δ.

Sufficiency. Let A ∈ ℜ_δ and B ∈ ℘_δ, and let . Then and . Choose and H ∈ RO (℘) such that and . Thus, we have , and by (a), . This shows that .

The following result shows that the soft product of two soft SC mappings is soft SC:

Theorem 2.34. Let and be two soft mappings. Let q* : X × Z → Y × W and v* : E × S → F × T be the mappings defined by q*(x, z) = (q₁(x), q₂(z)) and v*(e, s) = (v₁(e), v₂(s)). Then is soft SC if and only if and are soft SCs.

Proof. Necessity. Suppose that f_q*v* is soft SC. Let G ∈ ℘ and H ∈ ℜ. Then and so, . So, by Lemma 2.33(b), and . This shows that and are soft SCs.

Sufficiency. Let and be soft SC. Let G ∈ ℘ and H ∈ ℜ. Then and . Since , then by Lemma 2.33(b), . Therefore, by Theorem 2.2(6), f_q*v* is soft SC.

Theorem 2.35. If are soft SC mappings and (Y, ℘, F) is soft Hausdorff, then .

Proof. Let . Then f_qv(e_z) ≠ f_pu(e_z). Since (Y, ℘, F) is soft Hausdorff, then there exist G, H ∈ ℘ such that , , and . Since f_qv, f_pu are soft SC, then .

Claim. .

Proof of Claim. Suppose, to the contrary, that there exists b_t such that , , and f_qv(b_t) = f_pu(b_t). Then , which is a contradiction.

Therefore, we have and by the above claim, . This shows that 1_E− . Hence, .

In the following result, we discuss the behavior of soft SC mappings under soft subspaces:

Theorem 2.36. Let be a soft SC mapping. Then

1. (a) If U ⊆ Z such that C_U ∈ ℜ, then the soft restriction is soft SC.
2. (b) If {U_α : α ∈ Δ} is a cover of Z such that and is soft SC for all α ∈ Δ, then is SC.

Proof. (a) Let be soft SC, and let U ⊆ Z such that C_U ∈ ℜ. Let and let G ∈ ℘ such that . Since f_qv is soft SC, then there exists H ∈ ℜ such that . Then we have , and

This shows that is soft SC.

(b) Let {U_α : α ∈ Δ} be a cover of Z such that and is soft SC for all α ∈ Δ. Let G ∈ ℘. Then . For each α ∈ Δ, is soft SC, and so, . For each α ∈ Δ, , and so . It follows that .

3 Soft mappings with soft δ-closed graphs

In this section, we introduce several sufficient conditions for a soft mapping to have a soft δ-closed graph.

For a given soft mapping f_qv : (Z, E) → (Y, T), the soft set is called the soft graph of f_qv and is denoted by G(f_qv). Thus, if and only if f_qv(e_z) = t_y if and only if q(z) = y and v(e) = t.

We start this section with a result that implies that a soft θ-continuous mapping takes values in a soft Hausdorff space once its values on a soft dense set are known.

Theorem 3.1. If is soft θ-continuous and (Y, ℘, T) is soft Hausdorff, then .

Proof. Let be soft θ-continuous, and (Y, ℘, T) is soft Hausdorff. Let . Then f_qv(e_z) ≠ t_y. Since (Y, ℘, T) is soft Hausdorff, then there exist G, H ∈ ℘ such that , , and . Now, implies that . By soft θ-continuity of f_qv, there exists K ∈ ℜ such that and , and thus, . Since and Int_℘(Cl_℘(H)) ∈ RO (℘), then by Lemma 2.33(a), . Since and , then .

Claim. .

Proof of Claim. Suppose, to the contrary, that there exists . Then , , and f_qv (a_z) = b_w. Thus, , which is a contradiction.

This claim ends the proof.

Theorem 3.2. For a soft mapping , the following are equivalent:

1. (a) .
2. (b) For every , there exist and H ∈ RO (℘) such that , , and .

Proof. (a) → (b): . Since by (a), , there exist such that

Choose A ∈ ℜ and B ∈ ℘ such that . Thus,

Let and H = Int_℘(Cl_℘(B)). Then and . To see that , suppose, to the contrary, that there exists such that . So, we have and , which is a contradiction. Therefore, .

(b) → (a): Let . Then, by (b), there exists and H ∈ RO (℘) such that , , and . By Lemma 2.33(a), . To show that , suppose, to the contrary, that there exists . Then , , and b_w = f_qv (a_n). Thus, , which is a contradiction.

Theorem 3.3. If is soft δ-continuous and (Y, ℘, T) is soft Hausdorff, then .

Proof. Let be soft δ-continuous, and (Y, ℘, T) is soft Hausdorff. Let . Then f_qv(e_z) ≠ t_y. Since (Y, ℘, T) is soft Hausdorff, then there exist G, H ∈ ℘ such that , , and . Now, implies that . By soft δ-continuity of f_qv, there exists K ∈ ℜ such that and . Therefore, we have , , and .

Theorem 3.2(b) ends the proof.

Theorem 3.4. Let be a soft mapping such that .

1. (a) If L is soft nearly compact relative to (Y, ℘, T), then .
2. (b) If M is soft nearly compact relative to , then f_qv(M) ∈ (℘_δ)^c.
3. (c) If L is a soft compact subset of (Y, ℘, T), then .
4. (d) If M is a soft compact subset of , then f_qv(M) ∈ (℘_δ)^c.

Proof. (a) Let L be nearly compact relative to (Y, ℘, T). We will show that . Let . Then for each , and by Theorem 3.2(b), there exist and H (t_y) ∈ RO (℘) such that , , and . Since L is soft nearly compact relative to (Y, ℘, T) and , then there exists a finite soft subset such that . Put . Then and . This shows that . Hence, .

(b) Let M be soft nearly compact relative to . We will show that 1_T − f_qv(M) ∈ ℘_δ. Let . Then for each , and by Theorem 3.2(b), there exist and H (e_z) ∈ RO (℘) such that , , and . Since M is soft nearly compact relative to and , then there exists a finite soft subset such that . Put . Then and . This shows that 1_T − f_qv(M) ∈ ℘_δ. Hence, f_qv(M) ∈ (℘_δ)^c.

(c) If L is a soft compact subset of (Y, ℘, T), then L is soft nearly compact relative to (Y, ℘, T) and, by (a), .

(d) If M is a soft compact subset of , then M is soft nearly compact relative to and, by (b), f_qv(M) ∈ (℘_δ)^c.

Lemma 3.5. If is soft nearly compact and , then L is soft nearly compact relative to .

Proof. Let be soft nearly compact, and . Let such that . Let . Then and . Since is soft nearly compact, there exists a finite subcollection such that . Let . Then is a finite subcollection of such that . This shows that L is soft nearly compact relative to .

Theorem 3.6. If is a soft mapping such that and (Y, ℘, T) is soft nearly compact, then f_qv is soft δ-continuous.

Proof. Let L ∈ RC(℘). Since (Y, ℘, T) is soft nearly compact, by Lemma 3.5, L is soft nearly compact relative to (Y, ℘, T). So, by Theorem 3.4(a), . Therefore, by Theorem 6.2(8) of [30], f_qv is soft δ-continuous.

Theorem 3.7. If is a soft mapping such that and (Y, ℘, T) is soft compact, then f_qv is soft SC.

Proof. Let L ∈ ℘^c. Since (Y, ℘, T) is soft compact, then L is a soft compact subset of (Y, ℘, T). So, by Theorem 3.4(c), . Therefore, by Theorem 2.2(2), f_qv is soft SC.

Theorem 3.8. If is soft δ-continuous and M is soft nearly compact relative to , then f_qv (M) is soft nearly compact relative to (Y, ℘, T).

Proof. Let is soft δ-continuous and M is soft nearly compact relative to . Let such that . Then . Since f_qv is soft δ-continuous, then by Theorem 6.2(7) of [30], . Since M is soft nearly compact relative to , then there exists a finite subcollection such that . Thus, . This shows that f_qv (M) is soft nearly compact relative to (Y, ℘, T).

Corollary 3.9. Soft near compactness is preserved under soft δ-continuous surjections.

4 Conclusion

Numerous aspects of everyday life are uncertain. The soft set theory is one idea put out to deal with uncertainty. This work focuses on soft topology, a special mathematical framework that topologists have created by using soft sets.

The uncertain versions of topology, like soft topology, are vital tools to transact with many impediments that we face in different situations of our lives; this matter can be noted from the published manuscripts that exploited topological concepts such as compactness, separation axioms, and generalizations of open sets to address these situations as illustrated [21, 59, 60].

In this paper, soft super-continuity is defined as a new type of soft mapping that is a strong form of each of soft continuity and soft δ-continuity and a weaker form of soft complete continuity. Several characterizations (Theorems 2.2, 2.17), relationships (Theorems 2.3, 2.4, 2.6, 2.8, 2.13, 2.15, 3.6, 3.7, Examples 2.7, 2.9, 2.14, Corollary 2.16), compositions (Theorems 2.20, 2.21, 2.22, Example 2.24), restrictions (Theorem 2.36), preservations (Theorems 2.25, 2.27, 2.30, 3.4, 3.8, Corollaries 2.26, 2.28, 3.9), and products (Theorems 2.32, 2.34) of soft super-continuity are introduced. In addition, the relationships between soft super-continuity and soft δ-continuity and their analogous notions in general, topology are studied (Theorems 2.3, 2.4, 2.10, 2.11, Corollaries 2.5, 2.12). Finally, several sufficient conditions for the soft mapping to have a soft δ-closed graph are given (Theorems 3.1, 3.2, 3.3).

Soft super-continuous functions are anticipated to find use in several fields, including robotics, soft data analysis, soft image processing, and soft decision-making. They offer a framework for representing and evaluating ambiguous and imprecise data, resulting in algorithms and systems that are more adaptable and versatile.

In the future, we may look at the following topics: (1) studying further properties of soft super-continuous functions; (2) defining soft strongly super-continuous mappings; and (3) finding an application for our new results in the “decision-making problem,” “information systems,” or “expert systems”.

Acknowledgments

The authors wish to thank the referees for their useful comments and suggestions.