Soft supra strongly generalized closed sets

Abstract

The notions of supra soft topological space were first introduced by El-Sheikh et al. [5]. In this paper, we introduce the concept of soft supra strongly generalized closed sets (soft supra strongly g-closed for short) in a supra soft topological space (X, μ, E) and study their properties in detail, which is a generalization to the notions of soft strongly g-closed sets [15] and supra generalized closed sets [11]. Here, we used the concept of supra soft closure, supra soft interior and supra soft open sets to define soft supra strongly generalized closed sets. The relationship between soft supra strongly generalized closed sets and other existing soft sets have been investigated. Furthermore, the union and intersection of two soft supra strongly generalized closed (resp. open) sets have been obtained. It has been pointed out in this paper that many of these parameters studied have, in fact, applications in real world situations and therefore I believe that this is an extra justification for the work conducted in this paper.

Keywords

Soft sets soft topological space soft supra strongly generalized closed sets soft supra strongly generalized open sets and soft functions

1 Introduction

In 1970, Levine [16] introduced the notion ofg-closed sets in topological spaces as a generalization of closed sets. Indeed ideals are very important tools in general topology. In 1983, Mashhour et al. [17] introduced the supra topological spaces, not only, as a generalization to the class of topological spaces, but also, these spaces were easier in the application as shown in [3]. In 2001, Popa et al. [19] generalized the supra topological spaces to the minimal spaces and generalized spaces as a new wider classes. In 2007, Arpad Szaz [4] succeed to introduce an application on the minimal spaces and generalized spaces. In 1987, Abd El-Monsef et al. [1] introduced the fuzzy supra topological spaces. In 2001, El-Sheikh success to use the fuzzy supra topology to study some topological properties to the fuzzy bitopological spaces.

The notions of supra soft topological space were first introduced by El-Sheikh et al. [5]. Recently K. Kannan [14] introduced the concept of g-closed soft sets in a soft topological spaces which is generalized in [13, 15]. Here, we used the concept of supra soft closure, supra soft interior and supra soft open sets to define soft supra strongly generalized closed sets. The relationship between soft supra strongly generalized closed sets and other existing soft sets have been investigated. Furthermore, the union and intersection of two soft supra strongly generalized closed (resp. open) sets have been obtained.

2 Preliminaries and basic definitions

In this section, we present the basic definitions and results of soft set theory.

Definition 2.1. [18] Let X be an initial universe and E be a set of parameters. Let P (X) denote the power set of X and A be a non-empty subset of E. A pair (F, A) denoted by F_A is called a soft set over X, where F is a mapping given by F : A → P (X). In other words, a soft set over X is a parametrized family of subsets of the universe X. For a particular e ∈ A, F (e) may be considered the set of e-approximate elements of the soft set (F, A) and if e ∉ A, then F (e) = φ i.e F_A = {F (e) : e ∈ A ⊆ E, F : A → P (X)}. The family of all these soft sets denoted by SS (X) _A.

Definition 2.2. [21] Let τ be a collection of soft sets over a universe X with a fixed set of parameters E, then τ ⊆ SS (X) _E is called a soft topology on X if

$\tilde{X}, \tilde{φ} \in τ$ , where $\tilde{φ} (e) = φ$ and $\tilde{X} (e) = X, \forall e \in E$ ,

the union of any number of soft sets in τ belongs to τ,

the intersection of any two soft sets in τ belongs to τ.

The triplet (X, τ, E) is called a soft topological space over X.

Definition 2.3. [8] Let (X, τ, E) be a soft topological space. A soft set (F, A) over X is said to be closed soft set in X, if its relative complement (F, A) ^c is open soft set.

Definition 2.4. [8] Let(X, τ, E) be a soft topological space. The members of τ are said to be open soft sets in X. We denote the set of all open soft sets over X by OS (X, τ, E), or when there can be no confusion by OS (X) and the set of all closed soft sets by CS (X, τ, E), or CS (X).

Definition 2.5. [21] Let (X, τ, E) be a soft topological space, (F, E) ∈ SS (X) _E and Y be a non-null subset of X. Then the sub soft set of (F, E) over Y denoted by (F_Y, E), is defined as follows: $F_{Y} (e) = Y \cap F (e) \forall e \in E .$

In other words $(F_{Y}, E) = \tilde{Y} \tilde{\cap} (F, E)$ .

Definition 2.6. [21] Let (X, τ, E) be a soft topological space and Y be a non-null subset of X. Then $τ_{Y} = {(F_{Y}, E) : (F, E) \in τ}$ is said to be the soft relative topology on Y and (Y, τ_Y, E) is called a soft subspace of (X, τ, E).

Definition 2.7. [10] Let (X, τ, E) be a soft topological space and (F, E) ∈ SS (X) _E. Then (F, E) is said to be semi open soft set if $(F, E) \tilde{\subseteq} cl (int (F, E))$ . The set of all semi open soft sets is denoted by SOS (X) and the set of all semi closed soft sets is denoted by SCS (X).

Definition 2.8. [23] Let (X, τ₁, A) and (Y, τ₂, B) be soft topological spaces and f_pu : SS (X) _A → SS (Y) _B be a function. Then, the function f_pu is called:

Continuous soft if $f_{pu}^{- 1} (G, B) \in τ_{1} \forall (G, B) \in τ_{2}$ .

Open soft if f_pu (G, A) ∈ τ₂ ∀ (G, A) ∈ τ₁.

Closed soft if $f_{pu} (G, A) \in τ_{2}^{c} \forall (G, A) \in τ_{1}^{c}$ .

Theorem 2.9. [2] Let SS (X) _A and SS (Y) _B be families of soft sets. For the soft function f_pu : SS (X) _A → SS (Y) _B, the following statements hold,

$f_{pu}^{- 1} ((G, B)^{c}) = (f_{pu}^{- 1} (G, B))^{c} \forall (G, B) \in SS (Y)_{B}$ .

$f_{pu} (f_{pu}^{- 1} ((G, B))) \tilde{\subseteq} (G, B) \forall (G, B) \in SS (Y)_{B}$ . If f_pu is surjective, then the equality holds.

$(F, A) \tilde{\subseteq} f_{pu}^{- 1} (f_{pu} ((F, A))) \forall (F, A) \in SS (X)_{A}$ . If f_pu is injective, then the equality holds.

$f_{pu} (\tilde{X}) \tilde{\subseteq} \tilde{Y}$ . If f_pu is surjective, then the equality holds.

$f_{pu}^{- 1} (\tilde{Y}) = \tilde{X}$ and $f_{pu} ({\tilde{φ}}_{A}) = {\tilde{φ}}_{B}$ .

If $(F, A) \tilde{\subseteq} (G, A)$ , then $f_{pu} (F, A) \tilde{\subseteq} f_{pu} (G, A)$ .

If $(F, B) \tilde{\subseteq} (G, B)$ , then $f_{pu}^{- 1} (F, B) \tilde{\subseteq} f_{pu}^{- 1} (G, B) \forall (F, B), (G, B) \in SS (Y)_{B}$ .

$f_{pu}^{- 1} [(F, B) \tilde{\cup} (G, B)] = f_{pu}^{- 1} (F, B) \tilde{\cup} f_{pu}^{- 1} (G, B)$ and $f_{pu}^{- 1} [(F, B) \tilde{\cap} (G, B)] = f_{pu}^{- 1} (F, B) \tilde{\cap} f_{pu}^{- 1} (G, B) \forall (F, B), (G, B) \in SS (Y)_{B}$ .

$f_{pu} [(F, A) \tilde{\cup} (G, A)] = f_{pu} (F, A) \tilde{\cup} f_{pu} (G, A)$ and $f_{pu} [(F, A) \tilde{\cap} (G, A)] \tilde{\subseteq} f_{pu} (F, A) \tilde{\cap} f_{pu} (G, A) \forall (F, A), (G, A) \in SS (X)_{A}$ . If f_pu is injective, then the equality holds.

Definition 2.10. [14] A soft set F_E ∈ SS (X, E) is called soft generalized closed in a soft topological space (X, τ, E) if $cl (F_{E}) \tilde{\subseteq} G_{E}$ whenever $F_{E} \tilde{\subseteq} G_{E}$ and G_E ∈ τ.

Definition 2.11. [15] A soft set F_E ∈ SS (X, E) is called soft strongly generalized closed in a soft topological space (X, τ, E) if $cl (int (F_{E})) \tilde{\subseteq} G_{E}$ whenever $F_{E} \tilde{\subseteq} G_{E}$ and G_E ∈ τ.

Definition 2.12. [12] A soft set (F, E) is called soft regular closed set in a soft topological space (X, τ, E) if cl (int (F, E)) = (F, E).

Definition 2.13. [20] A soft set (F, E) is called soft regular generalized closed (soft rg-closed) in a soft topological space (X, τ, E) if $cl (F, E) \tilde{\subseteq} (G, E)$ whenever $(F, E) \tilde{\subseteq} (G, E)$ and (G, E) is regular open soft in X.

Definition 2.14. [13] A soft set (F, E) is called soft supra generalized closed set (soft suprag-closed) in a supra soft topological space (X, μ, E) if ${cl}^{s} (F, E) \tilde{\subseteq} (G, E)$ whenever $(F, E) \tilde{\subseteq} (G, E)$ and (G, E) is supra open soft in X.

Definition 2.15. [22] A soft set (F, E) is called soft supra regular generalized closed (soft supra rg-closed) in a supra soft topological space (X, μ, E) if ${cl}^{s} (F, E) \tilde{\subseteq} (G, E)$ whenever $(F, E) \tilde{\subseteq} (G, E)$ and (G, E) is supra regular open soft in X.

3 Soft supra strongly generalized closed sets

El-Sheikh et al. [5] introduced the notions of supra soft topological space. Kannan et al. [15] introduced the notions of soft strongly generalized closed sets to soft topological spaces. In this section, we generalize these notions to supra soft topological spaces.

Definition 3.1. A soft set (F, E) is called soft supra strongly generalized closed set (soft supra stronglyg-closed) in a supra soft topological space (X, μ, E) if ${cl}^{s} ({int}^{s} (F, E)) \tilde{\subseteq} (G, E)$ whenever $(F, E) \tilde{\subseteq} (G, E)$ and (G, E) is supra open soft in X.

Example 3.2. Suppose that there are three cars in the universe X given by X = {a, b, c}. Let E = {e₁, e₂} be the set of decision parameters which are stands for “expensive” and “beautiful” respectively.

Let (G₁, E) , (G₂, E) be two soft sets over the common universe X, which describe the composition of the cars, where $\begin{matrix} G_{1} (e_{1}) & = & {b, c}, G_{1} (e_{2}) = {a, b}, \\ G_{2} (e_{1}) & = & {a, b}, G_{2} (e_{2}) = {a, c} . \end{matrix}$

Then, $μ = {\tilde{X}, \tilde{φ}, (G_{1}, E), (G_{2}, E)}$ is a supra soft topology over X. Hence, the soft sets (F₁, E) , (F₂, E), where $\begin{matrix} F_{1} (e_{1}) & = & {b, c}, F_{1} (e_{2}) = {a, c}, \\ F_{2} (e_{1}) & = & {a, b}, F_{2} (e_{2}) = {a, b} . \end{matrix}$ are supra g-closed soft sets in (X, μ, E), but the soft sets (G₁, E) , (G₂, E) are not soft supra stronglyg-closed in (X, μ, E).

Remark 3.3. The soft intersection (resp. soft union) of any two soft supra strongly g-closed sets is not soft supra strongly g-closed in general as shown in the following examples.

Examples 3.4.

In Example 3.2, (H₁, E) , (H₂, E) are soft supra strongly g-closed in (X, μ, E), where H₁ (e₁) = {b, c}, H₁ (e₂) = X, H₂ (e₁) = X, H₂ (e₂) = {a, b}. But, their soft intersection $(H_{1}, E) \tilde{\cap} (H_{2}, E) = (S, E)$ where S (e₁) = {b, c}, S (e₂) = {a, b} is not soft supra strongly g-closed.

Suppose that there are four alternatives in the universe of houses X = {a, b, c, d} and consider E = {e₁, e₂} be the set of decision parameters which are stands for “quality of houses” and “green surroundings” respectively. Let (F₁, E) , (F₂, E) , (F₃, E) , (F₄, E) , (F₅, E) , (F₆, E) , (F₇, E) , (F₈, E) , (F₉, E) , (F₁₀, E) , (F₁₁, E) , (F₁₂, E) be twelve soft sets over the common universe X which describe the goodness of the houses, where F₁ (e₁) = {a}, F₁ (e₂) = {d}, F₂ (e₁) = {a, d}, F₂ (e₂) = {a, d}, F₃ (e₁) = {d}, F₃ (e₂) = {a}, F₄ (e₁) = {a, b}, F₄ (e₂) = {b, d}, F₅ (e₁) = {b, d}, F₅ (e₂) = {a, b}, F₆ (e₁) = {a, b, c}, F₆ (e₂) = {a, b, c}, F₇ (e₁) = {b, c, d}, F₇ (e₂) = {b, c, d}, F₈ (e₁) = {a, b, d}, F₈ (e₂) = {a, b, d}, F₉ (e₁) = X, F₉ (e₂) = {a, b, c}, F₁₀ (e₁) = {b, c, d}, F₁₀ (e₂) = X, F₁₁ (e₁) = {a, b, c}, F₁₁ (e₂) = X, F₁₂ (e₁) = X, F₁₂ (e₂) = {b, c, d}. Hence, $μ = {\tilde{X}, \tilde{φ}, (F_{1}, E), (F_{2}, E), (F_{3}, E), (F_{4}, E), (F_{5}, E), (F_{6}, E), (F_{7}, E), (F_{8}, E), (F_{9}, E), (F_{10}, E), (F_{11}, E), (F_{12}, E)}$ is a supra soft topology over X. Therefore, the soft sets (G, E) , (H, E), where G (e₁) = {d}, G (e₂) = {d}, H (e₁) = {a}, H (e₂) = {a}, are soft supra strongly g-closed sets in (X, μ, E), but their soft union $(G, E) \tilde{\cup} (H, E) = (A, E)$ where A (e₁) = {a, d}, A (e₂) = {a, d} is not soft supra strongly g-closed.

Theorem 3.5. Every supra closed soft set is a soft supra strongly g-closed in a supra soft topological space.

Proof. Let $F_{E} \tilde{\subseteq} G_{E}$ and G_E ∈ μ. Since F_E is closed soft, then ${cl}^{s} ({int}^{s} (F_{E})) \tilde{\subseteq} {cl}^{s} (F_{E}) = F_{E} \tilde{\subseteq} G_{E}$ . Therefore, F_E is soft supra strongly g-closed. □

Remark 3.6. The converse of the above theorem is not true in general as shall shown in the following example.

Example 3.7. In Example 3.2, the soft sets (F₁, E) , (F₂, E) are soft supra strongly g-closed in (X, μ, E), but not supra closed soft.

Theorem 3.8. Every soft supra g-closed is a soft supra strongly g-closed in a supra soft topological space.

Proof. Let $F_{E} \tilde{\subseteq} G_{E}$ and G_E ∈ μ. Since F_E is soft supra g-closed, then ${cl}^{s} ({int}^{s} (F_{E})) \tilde{\subseteq} {cl}^{s} (F_{E}) \tilde{\subseteq} G_{E}$ . Therefore, F_E is soft supra strongly g-closed. □

Remark 3.9. The converse of the above theorem is not true in general as shall shown in the following example.

Example 3.10. Suppose that there are two jobs in the universe X given by X = {a, b}. Let E = {e₁, e₂} be the set of decision parameters which are stands for “salary” and “position” respectively.

Let (G₁, E) , (G₂, E) , (G₃, E) be three soft sets over the common universe X, which describe the details of the jobs, where $\begin{matrix} G_{1} (e_{1}) & = & {a}, G_{1} (e_{2}) = X, \\ G_{2} (e_{1}) & = & {a}, G_{2} (e_{2}) = {a}, \\ G_{3} (e_{1}) & = & {a}, G_{2} (e_{2}) = {b} . \end{matrix}$

Then, $μ = {\tilde{X}, \tilde{φ}, (G_{1}, E), (G_{2}, E), (G_{3}, E)}$ is a supra soft topology over X. Hence, the soft set (F, E) is soft supra strongly g-closed in (X, μ, E), but not soft supra g-closed, where: $F (e_{1}) = φ, F (e_{2}) = {b} .$

Theorem 3.11. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra open soft and soft supra strongly g-closed, then it is supra closed soft.

Proof. Since F_E is supra open soft and soft supra strongly g-closed, then ${cl}^{s} (F_{E}) = {cl}^{s} ({int}^{s} (F_{E})) \tilde{\subseteq} F_{E} \tilde{\subseteq} {cl}^{s} (F_{E})$ . Therefore, F_E is supra closed soft. □

Corollary 3.12. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra open soft and soft supra strongly g-closed, then it is both supra soft regular open and supra soft regular closed set.

Proof. Since F_E is open soft and soft strongly g-closed set. By theorem 3, F_E is supra closed soft set. Thus, int^s (cl^s (F_E)) = F_E = cl^s (int^s (F_E)). Therefore, F_E is both supra soft regular open and supra soft regular closed set. □

Corollary 3.13. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra open soft and supra soft strongly g-closed, then it is supra soft rg-closed set.

Proof. Obvious from Theorem 3.11 and Corollary 3.12. □

Theorem 3.14. Let (X, μ, E) be a supra soft topological space and F_E be a soft supra strongly g-closed in X. If $F_{E} \tilde{\subseteq} H_{E} \tilde{\subseteq} {cl}^{s} ({int}^{s} (F_{E}))$ , then H_E is soft supra strongly g-closed.

Proof. Let $H_{E} \tilde{\subseteq} G_{E}$ and G_E ∈ μ. Since $F_{E} \tilde{\subseteq} H_{E} \tilde{\subseteq} G_{E}$ and F_E is soft supra strongly g-closed in X, then ${cl}^{s} ({int}^{s} (F_{E})) \tilde{\subseteq} G_{E}$ implies ${cl}^{s} ({int}^{s} (H_{E})) \tilde{\subseteq} {cl}^{s} ({int}^{s} (F_{E})) \tilde{\subseteq} G_{E}$ . Therefore, H_E is soft supra strongly g-closed. □

Theorem 3.15. A soft subset H_E of a supra soft topological space (X, μ, E) is soft supra strongly g-closed if and only if cl^s (int^s (H_E)) \ H_E contains only null supra closed soft set.

Proof. Necessity: Let H_E be a soft supra strongly g-closed set, F_E be a non null supra closed soft and $F_{E} \tilde{\subseteq} {cl}^{s} ({int}^{s} (H_{E})) \ H_{E}$ . Then, $F_{E} \tilde{\subseteq} H_{E}^{c}$ , implies that $H_{E} \tilde{\subseteq} F_{E}^{c}$ . Since H_E is soft supra strongly g-closed and $F_{E}^{c}$ is supra open soft, so ${cl}^{s} ({int}^{s} (H_{E})) \tilde{\subseteq} F_{E}^{c}$ . Hence, $F_{E} \tilde{\subseteq} [{cl}^{s} ({int}^{s} (H_{E}))]^{c}$ . Therefore, $F_{E} \tilde{\subseteq} {cl}^{s} ({int}^{s} (H_{E})) \tilde{\cap} [{cl}^{s} ({int}^{s} (H_{E}))]^{c} = \tilde{φ}$ . Thus, $F_{E} = \tilde{φ}$ which is a contradiction. Thus, cl^s (int^s (H_E)) \ H_E contains only null supra closed soft set.

Sufficient: Assume that cl^s (int^s (H_E)) \ H_E contains only null supra closed soft set, $H_{E} \tilde{\subseteq} G_{E}$ , G_E is supra open soft and suppose that ${cl}^{s} ({int}^{s} (H_{E})) \tilde{⊈} G_{E}$ . Then, ${cl}^{s} ({int}^{s} (H_{E})) \tilde{\cap} G_{E}^{c}$ is a non null supra closed soft subset of cl^s (int^s (H_E)) \ H_E, which is a contradiction. Thus, H_E is a soft supra strongly g-closed. □

Corollary 3.16. A soft supra strongly g-closed H_E is supra soft regular closed if and only if cl^s (int^s (H_E)) \ H_E is supra closed soft and $H_{E} \tilde{\subseteq} {cl}^{s} ({int}^{s} (H_{E}))$ .

Proof. Necessity: Sine H_E is supra soft regular closed, then H_E = cl^s (int^s (H_E)). Hence, ${cl}^{s} ({int}^{s} (H_{E})) \ H_{E} = \tilde{φ}$ is supra closed soft.

Sufficient: Assume that cl^s (int^s (H_E)) \ H_E is supra closed soft. Since H_E is soft supra strongly g-closed, so cl^s (int^s (H_E)) \ H_E contains only null supra closed soft from Theorem 3.15. Since cl^s (int^s (H_E)) \ H_E is supra closed soft, then it is soft supra strongly g-closed from Theorem 3.8. It follows that, ${cl}^{s} ({int}^{s} (H_{E})) \ H_{E} = \tilde{φ}$ , implies that ${cl}^{s} ({int}^{s} (H_{E})) \tilde{\subseteq} H_{E}$ . But, by hypothesis, $H_{E} \tilde{\subseteq} {cl}^{s} ({int}^{s} (H_{E}))$ . Therefore, H_E = cl^s (int^s (H_E)) and H_E is supra soft regular closed. □

4 Soft supra strongly generalized open sets

Definition 4.1.F_E is called soft supra strongly g-open set (soft supra strongly g-open) in a supra soft topological space (X, μ, E) if its relative complement $F_{E}^{c}$ is soft supra strongly g-closed.

Example 4.2. In Example 3.2, the soft sets (F₁, E) ^c, (F₂, E) ^c are soft supra strongly g-open, where (F₁, E) ^c, (F₂, E) ^c are defined by: $\begin{matrix} F_{1}^{c} (e_{1}) & = & {a} F_{1}^{c} (e_{2}) = {b}, \\ F_{2}^{c} (e_{1}) & = & {c} F_{2}^{c} (e_{2}) = {c} . \end{matrix}$

Theorem 4.3. A soft set G_E is soft supra strongly g-open set if and only if $F_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (G_{E}))$ whenever $F_{E} \tilde{\subseteq} G_{E}$ and F_E is supra closed soft.

Proof. Necessity: Let $F_{E} \tilde{\subseteq} G_{E}$ and F_E is supra closed soft in X, then $G_{E}^{c} \tilde{\subseteq} F_{E}^{c}$ and $F_{E}^{c}$ is supra open soft. Since $G_{E}^{c}$ is soft supra strongly g-closed, so ${cl}^{s} ({int}^{s} (G_{E}^{c})) \tilde{\subseteq} F_{E}^{c}$ . Consequently, $F_{E} \tilde{\subseteq} [{cl}^{s} ({int}^{s} (G_{E}^{c}))]^{c} = {int}^{s} ({cl}^{s} (G_{E}))$ .

Sufficient: Let $G_{E}^{c} \tilde{\subseteq} H_{E}$ and H_E is supra open soft in X. Then, $H_{E}^{c} \tilde{\subseteq} G_{E}$ and $H_{E}^{c}$ is supra closed soft in X. Hence, $H_{E}^{c} \tilde{\subseteq} {int}^{s} ({cl}^{s} (G_{E}))$ from the necessary condition. Thus, $[{int}^{s} ({cl}^{s} (G_{E}))]^{c} = {cl}^{s} ({int}^{s} (G_{E}^{c})) \tilde{\subseteq} H_{E}$ and H_E is supra open soft in X. This shows that, $G_{E}^{c}$ is soft supra strongly g-closed in X. Therefore, G_E is soft supra strongly g-open set. □

Remark 4.4. The soft intersection (resp. soft union) of any two soft supra strongly g-open sets is not soft supra strongly g-open in general as shown in the following examples.

Examples 4.5.

In Examples 3.4 (2), the soft sets (H₁, E) , (H₂, E) are soft supra strongly g-open in (X, μ, E), where $\begin{matrix} H_{1} (e_{1}) & = & {b, c}, H_{1} (e_{2}) = φ, \\ H_{2} (e_{1}) & = & φ, H_{2} (e_{2}) = {b, c} . \end{matrix}$ But, their soft union $(H_{1}, E) \tilde{\cup} (H_{2}, E) = (V, E)$ where V (e₁) = {b, c}, V (e₂) = b, c is not soft supra strongly g-open.

Suppose that there are two phones in the universe X given by X = {a, b}. Let E = {e₁, e₂} be the set of decision parameters which are stands for “expensive” and “system” respectively.

Let (G₁, E) , (G₂, E) , (G₃, E) , (G₄, E) be four soft sets over the common universe X, which describe the goodness of the phones, where $\begin{matrix} G_{1} (e_{1}) & = & X, G_{1} (e_{2}) = {b}, \\ G_{2} (e_{1}) & = & {a}, G_{2} (e_{2}) = X, \\ G_{3} (e_{1}) & = & {a}, G_{3} (e_{2}) = {b}, \\ G_{4} (e_{1}) & = & {b}, G_{4} (e_{2}) = {b}, \end{matrix}$ Then, $μ = {\tilde{X}, \tilde{φ}, (G_{1}, E), (G_{2}, E), (G_{3}, E), (G_{4}, E)}$ is a supra soft topology over X. Hence, the soft sets (H₁, E) , (H₂, E) are soft supra strongly g-open in (X, μ, E), where $\begin{matrix} H_{1} (e_{1}) & = & X, H_{1} (e_{2}) = {a}, \\ H_{2} (e_{1}) & = & {b}, H_{2} (e_{2}) = X . \end{matrix}$ But, their soft intersection $(H_{1}, E) \tilde{\cap} (H_{2}, E) = (A, E)$ where A (e₁) = {b}, A (e₂) = {a} is not soft supra strongly g-open.

Theorem 4.6. Every supra open soft is a soft supra strongly g-open.

Proof. Let A_E be a supra open soft set such that $F_{E} \tilde{\subseteq} A_{E}$ and F_E ∈ μ^c. Then, $F_{E} \tilde{\subseteq} A_{E} = {int}^{s} (A_{E}) \tilde{\subseteq} {int}^{s} ({cl}^{s} (A_{E}))$ . Therefore, A_E is soft supra strongly g-open. □

Remark 4.7. The converse of the above theorem is not true in general as shown in the following example.

Example 4.8. In Example 3.2, the soft sets (F₁, E) ^c, (F₂, E) ^c are soft supra strongly g-open in (X, μ, E), but not supra open soft over X, where: $\begin{matrix} F_{1}^{c} (e_{1}) & = & {a} F_{1}^{c} (e_{2}) = {b}, \\ F_{2}^{c} (e_{1}) & = & {c} F_{2}^{c} (e_{2}) = {c} . \end{matrix}$

Theorem 4.9. Every soft supra g-open is a soft supra strongly g-open.

Proof. Let A_E be soft supra g-open such that $F_{E} \tilde{\subseteq} A_{E}$ and A_E ∈ μ^c. Since A_E is soft supra g-open. Then, $F_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (A_{E})) = {int}^{s} (A_{E})$ . Therefore, A_E is soft supra strongly g-open. □

Remark 4.10. The converse of the above theorem is not true in general as shown in the following example.

Example 4.11. In Example 3.10, the soft set (F, E) ^c is soft supra strongly g-open, but not soft supra g-open, where: $F^{c} (e_{1}) = X, F^{c} (e_{2}) = {a} .$

Theorem 4.12. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra closed soft and soft supra strongly g-open, then it is supra open soft.

Proof. Since F_E is supra closed soft and soft supra strongly g-open, then $F_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (F_{E})) = {int}^{s} (F_{E}) \tilde{\subseteq} F_{E}$ . Therefore, F_E is supra open soft. □

Corollary 4.13. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra closed soft and soft supra strongly g-open, then it is both supra soft regular open and supra soft regular closedset.

Proof. Since F_E is closed soft and soft stronglyg-open. By Theorem 4.12, F_E is supra open soft. Thus, int^s (cl^s (F_E)) = F_E = cl^s (int^s (F_E)). Therefore, F_E is both supra soft regular open and supra soft regular closed set. □

Corollary 4.14. If a soft subset F_E of a supra soft topological space (X, μ, E) is both supra closed soft and supra soft strongly g-open, then it is supra soft rg-open set.

Proof. Obvious from Theorem 4.12 and Corollary 4.13. □

Theorem 4.15. A soft subset H_E of a supra soft topological space (X, μ, E) is soft supra strongly g-open if and only if H_E \ int^s (cl^s (H_E)) contains only null supra closed soft set.

Proof. It is similar to the proof of Theorem 3.15. □

Corollary 4.16. A soft supra strongly g-closed H_E is supra soft regular closed if and only if H_E \ ^ints (cl^s (H_E)) is supra closed soft and $H_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (H_{E}))$ .

Proof. Follows from Theorem 4.15. □

Theorem 4.17. Let (X, μ, E) be a supra soft topological space and F_E be a soft supra strongly g-open in X. If ${int}^{s} ({cl}^{s} (F_{E})) \tilde{\subseteq} H_{E} \tilde{\subseteq} F_{E}$ , then H_E is soft supra strongly g-open.

Proof. Let $G_{E} \tilde{\subseteq} H_{E}$ and G_E ∈ μ^c. Since $G_{E} \tilde{\subseteq} H_{E} \tilde{\subseteq} F_{E}$ and F_E is soft supra strongly g-open in X, then $G_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (F_{E}))$ . Hence, $G_{E} \tilde{\subseteq} {int}^{s} ({cl}^{s} (F_{E})) \tilde{\subseteq} {int}^{s} ({cl}^{s} (H_{E}))$ . Therefore, H_E is soft supra strongly g-open. □

5 Conclusion

In this work, the notions of supra strongly generalized closed sets and supra strongly generalized open sets have been introduced and investigated. In future, the generalization of these concepts to supra soft topological spaces [5] and soft ideals [11] will be introduced and the future research will be undertaken in this direction.

Footnotes

Acknowledgments

The author express his sincere thanks to the reviewers for their valuable suggestions. The author is also thankful to the editors-in-chief and managing editors for their important comments which helped to improve the presentation of the paper.

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