A novel approach to study soft preopen sets inspired by classical topologies

Abstract

Our target in the present work, is presenting the idea of weakly soft preopen (ws-preopen) subsets and studying some of its characterizations. With the assistance of some elucidative examples, the interrelationships between ws-preopen sets and some extensions of soft open sets are studied. Under some conditions such as extended and hyperconnected soft topologies, several motivating results and relationships are acquired. The interior and closure operators that built through ws-preopen and ws-preclosed subsets are introduced. Their main features that construe the relations among them are established. Soft continuity with respect to theses classes of soft subsets are studied and their substantial characteristics are investigated. Generally, the systematic relations and outcomes that are lost through the scope of this study are discussed. The proposed line in the current study will present new ways to discover novel concepts in the field of soft topology.

Keywords

extended soft topology

1 Introduction

The goal of defining a soft structure over a set is to form an appointed discretization of these major mathematical notions with an efficiently continuous nature, thus supplying new methods for the usage of the technique of mathematical analysis in true applications that included incomplete data. This is done through a specific parameterization of an assumed set. As always, this novel perspective on conceptions is attracting the attention of researchers and mathematicians in several related fields as well. Undoubtedly, the notion of a soft set is well consistent with some other novel mathematical notions like fuzzy sets, rough sets and etc. Additionally, this resulted in a series of articles, where soft versions of mathematical notions were achieved. The soft set theory (SST) was initiated by Molodtsov [44] for dealing with ambiguous objects. He offered the main findings of this theory and successfully used in distinct fields. Soft system provides general frame with the engagement of parameters. Maji and colleagues [41, 42] issued extensive theoretical articles on soft sets, where they utilized rough set model to solve a decision-making problem and then applied SST to it. Then, the researchers of [4] displayed the confusion of these ideas and updated to be proportionate with their analogues in the crisp set theory. Further, characteristic of the parameterization classes of soft sets have been taken to institute many sorts of operators and operations between soft sets as explained in [16, 46]. To increase the performance of SST to cope with complicated problems, it has been hybridized with other ambiguous instruments like fuzzy and rough sets as discussed in [15, 48].

At the same time in 2011, the idea of the birth of the soft topological spaces (soft_TS) emerged independently by Shabir and Naz [49] and Çagman et al. [26]. Their techniques of defining soft_TSs are different in the way of selecting the parameters set as constant or variable. This work follows the approach of constant parameters set as stipulated by Shabir and Naz. It was introduced and studied various notions in soft_TSs such soft closure, soft interior, soft continuity, soft separability, etc. by many scholars and researchers like Hussain and Ahmad [37]. Thereafter, Min [43] characterized the form of soft open and closed sets of soft regular spaces and showed the relation between soft T₂ and soft T₃-spaces. In [1, 29], the authors studied more characteristics of the soft β-separation axioms and soft b-separation axioms in soft_TSs. Zorlutuna [52] proved that a fuzzy topological space is a special case of soft_TSs, and ordinary topological space can be considered a soft_TS. El-Shafei et al. [31] established a powerful family of soft T_i-spaces that preserves more characteristics of classical separation axioms. Then, it was established new families of soft separation axioms by [17, 30]. These axioms were introduced and explored in the environments of infra-soft topologies [9], supra-soft topologies [18] and soft ordered structures [19, 20]. Remarkable modifications for the former studies of soft separation axioms have been proceeded by some researchers [7 , 50]. In 2021, Al-shami [12] examined how soft separation axioms are applied to choose the optimal alternatives of tourism programs. The notions of compact and Lindelöf spaces were presented by Aygünoglu and Aygün [25] and another kinds of compactness were introduced by Hida [33]. These concepts of covering properties have been generalized with respect to soft regular closed [5] and soft somewhere dense sets [6]. A motivating application of soft compactness to the information systems was prepared by Al-shami [10]. Also, the agreement between enriched and extended soft topologies by Al-shami and Kočinac [22], where they indicated that many topological properties navigate between this kind of soft topologies and their parametric topologies, was proved. Chen [27] put forward the concept of soft semi-open, Kandil et al. [38] contributed by soft preopen notion, Hosny [35] investigated a lot of properties of soft b-open sets, Al-shami [6] and Al-Ghour [3] offered soft somewhere dense sets and soft Q-sets, respectively. Al-shami [13] introduced four classes of separation axioms utilizing the concept of soft somewhat open sets and then displayed an interesting application to the nutrition systems. Authors in [34, 36] discussed some of near soft open sets with soft grills and soft ideals. In [39], the concept of functions between soft_TSs set up by Kharal and Ahmad, which it enhanced utilizing the crisp functions and soft points by Al-shami [11]. Certain kinds of soft functions such as soft continuous, open, and closed functions were investigated in [14, 53]. Moreover, the notions of Vietoris topology [28], Menger spaces [40], almost Menger [23], nearly Menger [24] and expandable spaces [47] via soft settings were integrated.

The organization of this study is given as following. The primary definitions and properties that are necessary to understand the obtained findings are gathered in Section 2. In the main part, Section 3, we provide a novel extension of soft open sets, namely ws-preopen sets. The major characteristics of this class are exhibited with needful counterexamples. In Section 4, we benefit from ws-preopen sets to present the soft version of interior, closure, boundary and limit soft points. Then, we allocate Section 5 to discuss the notion of w-soft pre-continuity and show the divergences between w-soft pre-continuity and soft continuity. In Section 6, with some consequences and upcoming contributions, the current paper is terminated.

2 Preliminaries

In the present section, the key concepts required for researchers to be aware with the manuscript’s context are summarized.

Suppose that $U$ is a nonempty universe set, $2^{U}$ represents the power set of $U$ , and Ω is the family of all possible parameters under consideration with respect to a universe. Usually, parameters represent properties, or attributes of objects in $U$ . In [44] Molodtsov introduced the concept of a soft set in the next manner:

Definition 2.1. [44] (S, Ω) is said to be a soft set over $U$ in which S is a mapping defined as

$S : Ω ⟶ 2^{U}$ , which means that, a soft set (S, Ω) may described by (S, Ω) = {(ω, S (ω)) : ω ∈ Ω and $S (ω) \in 2^{U}}$ ;

where S (ω) represents an ω-component of (S, Ω). The family of all soft sets specified on the universal set $U$ with a class of parameters Ω is designated by $2^{U_{Ω}}$ .

Henceforward, the notations (S, Ω) , (G, Ω) refer to soft sets with respect to $U$ , except as otherwise provided.

If a soft set (G, Ω) defined by $G (ω) = U - S (ω) \forall ω \in Ω$ , then (G, Ω) is said to be the complement of (S, Ω). That is, (S, Ω) ^c = (G, Ω).

To denote the complement we use (S, Ω) ^c or (S^c, Ω).

Definition 2.2. [42, 45] (S, Ω) is said to be:

(i) absolute soft set on condition that $S (ω) = U$ ∀ω ∈ Ω; that symbolized by $\tilde{U}$ .

(ii) null soft set, if it is complement of an absolute soft set, which symbolized by φ.

(iii) a soft point on condition that there exist ω ∈ Ω and $u \in U$ with S (ω) = {u} and S (a) =∅ ∀a ∈ Ω - {ω}. To refer a soft point u_ω is used. Terminologically, we write u_ω ∈ (S, Ω) providing u ∈ S (ω).

(iii) pseudo constant on condition that for every ω ∈ Ω we have $S (ω) = U$ or ∅.

Definition 2.3. [32] (S, Ω) is a soft subset of (G, Ω) (or (G, Ω) is a soft superset of (S, Ω)), indicated by $(S, Ω) \tilde{\subseteq} (G, Ω)$ , if S (ω) ⊆ G (ω) for all ω ∈ Ω.

Definition 2.4. [4] If $G (ω) = U - S (ω)$ for all ω ∈ Ω, then (G, Ω) is called a complement of (S, Ω). The complement of (S, Ω) is indicated by (S, Ω) ^c = (S^c, Ω).

Definition 2.5. [16] If (S, Ω) and (G, Ω) are soft sets, then:

(i) $(S, Ω) ⋃^{˜} (G, Ω) = (H, Ω),$ in which H (ω) = S (ω) ⋃ G (ω) for every ω ∈ Ω.

(ii) $(S, Ω) ⋂^{˜} (G, Ω) = (H, Ω),$ in which H (ω) = S (ω) ⋂ G (ω) for all ω ∈ Ω.

(iii) (S, Ω) \ (G, Ω) = (H, Ω) , in which H (ω) = S (ω) \ G (ω) for every ω ∈ Ω.

(iv) (S, Ω) × (G, Ω) = (H, Ω) , in which H (ω₁, ω₂) = S (ω₁) × G (ω₂) for every (ω₁, ω₂) ∈ Ω × Ω.

In the next, the modified issuance of soft functions definition is offered.

Definition 2.6. [11] Suppose that $M : U \to V$ and O : Ω → Σ are crisp functions. A soft function M_O of $2^{U_{Ω}}$ into $2^{V_{Σ}}$ is a relation satisfies that each $u_{ω} \in 2^{U_{Ω}}$ is related to one and only one $v_{σ} \in 2^{V_{Σ}}$ such that M_O (u_ω) = M (u) _O(ω) for all $u_{ω} \in 2^{U_{Ω}}$ .

In addition, $M_{O}^{- 1} (v_{σ}) = \underset{ω \in O^{- 1} (σ)}{u \in M^{- 1} (v) ⋃^{˜}} u_{ω}$ for each $v_{σ} \in 2^{V_{Σ}}$ .

That is, if $(S, Ω) \in 2^{U_{Ω}}$ and $(G, Ω) \in 2^{V_{Ω}}$ , M_O (S, Ω) and $M_{O}^{- 1} (G, Ω)$ , in which $M_{O} : 2^{U_{Ω}} \to 2^{V_{Ω}}$ a soft function, are given by $M_{O} (S, Ω) = {⋃^{˜}}_{u_{ω} \in (S, Ω)} M_{O} (u_{ω})$ , and

$M_{O}^{- 1} (G, Ω) = {⋃^{˜}}_{v_{σ} \in (G, Ω)} M_{O}^{- 1} (v_{σ})$ .

A soft function M_O is said to be surjective (resp., injective, bijective) if M, O are surjective (resp., injective, bijective).

Proposition 2.7. [39] Suppose that $M_{O} : 2^{U_{Ω}} \to 2^{V_{Σ}}$ is a soft function. If (S, Ω) and (G, Ω) are soft subsets of $\tilde{U}$ and $\tilde{V}$ , respectively, then:

(i) $(S, Ω) \tilde{\subseteq} M_{O}^{- 1} (M_{O} (S, Ω))$ .

(ii) If M_O is injective, then $(S, Ω) = M_{O}^{- 1} (M_{O} (S, Ω))$ .

(iii) $M_{O} (M_{O}^{- 1} (G, Σ)) \tilde{\subseteq} (G, Σ)$ .

(iv) If M_O is surjective, then $M_{O} (M_{O}^{- 1} (G, Σ)) = (G, Σ)$

Definition 2.8. [44] A subfamily $T$ of $2^{U_{Ω}}$ is said to be a soft topology of $U$ , if the next axioms hold:

(i) $\tilde{U}$ and φ are elements of $T$ .

(ii) $T$ is closed under the arbitrary unions.

(iii) $T$ is closed under the finite intersections.

The triplet $(U, T, Ω)$ is said to be a soft_TS. Each member in $T$ is said to be soft open (shortly, s-open) and its complement is said to be soft closed (shortly, s-closed).

Definition 2.9. [21] A soft_TS $(U, T, Ω)$ is said to be full provided that every non-null s-open set has no empty component.

Proposition 2.10. [44] Suppose that $(U, T, Ω)$ is a soft_TS. Then the collection $T_{ω} = {S (ω) : (S, Ω) \in T}$ for each ω ∈ Ω, specifies a topology on $U$ . This topology is said to be a parametric topology.

Definition 2.11. [44] Suppose that (S, Ω) is a soft subset of a soft_TS $(U, T, Ω)$ . Then (int (S) , Ω) and (cl (S) , Ω) are respectively specified by int (S) (ω) = int (S (ω)) and cl (S) (ω) = cl (S (ω)), where int (S (ω)), cl (S (ω)) are respectively the interior, closure of S (ω) in $(U, T_{ω})$ .

Definition 2.12. [25, 45] Suppose that $(U, T, Ω)$ is a soft_TS. Then,

(i) $T$ is said to be an enriched soft topology, if all pseudo constant soft sets are elements of $T$ .

(ii) $T$ with the property “ $(S, Ω) \in T$ iff $S (ω) \in T_{ω}$ for each ω ∈ Ω" is said to be an extended soft topology.

A deep examination on the enriched and extended soft topologies was behaved on [22]. The corresponding between these sorts of soft topologies was one of the significant and interesting results achieved in [22]. Henceforward, this kind of soft topology will be named an extended soft topology. Under this soft topology, it was demonstrated several consequences that connected soft topology with its parametric topologies. As a matter of fact, the next result represents a key point in the proof of many outcomes.

Theorem 2.13. [22] A soft_TS $(U, T, Ω)$ is extended iff (int (S) , Ω) = int (S, Ω) , (cl (S) , Ω) = cl (S, Ω) for any soft subset (S, Ω).

Theorem 2.14. [38] Suppose that (S, Ω) is a soft subset of a soft_TS $(U, T, Ω)$ . If (G, Ω) is s-open, then $(G, Ω) ⋂^{˜} cl (S, Ω) \tilde{\subseteq} cl ((G, Ω) ⋂^{˜} (S, Ω))$ .

Definition 2.15. A soft subset (S, Ω) of $(U, T, Ω)$ is called:

(i) soft α-open [2], if $(S, Ω) \tilde{\subseteq} int (cl (int (S, Ω)))$ .

(ii) soft preopen [38], if $(S, Ω) \tilde{\subseteq} int (cl (S, Ω))$ (or (S, Ω) is soft preopen, if there exists a s-open set (U, Ω) s.t. $(S, Ω) \tilde{\subseteq} (U, Ω) \tilde{\subseteq} cl (S, Ω)$ ).

(iii) soft somewhere dense [6], if (S, Ω) = φ or int (cl (S, Ω)) ≠ φ.

Definition 2.16. [53] A soft function $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is called soft continuous if $M_{O}^{- 1} (S, Ω)$ is a s-open set where (S, Ω) is s-open.

Theorem 2.17. [22] If $M_{O} : (U, T, Ω) \to (V, S, Ω)$ is a soft continuous, then $h : (U, T_{ω}) \to (V, S_{O (ω)})$ is continuous for all ω ∈ Ω.

3 Weakly soft preopen sets and their basic properties

As a novel extension of s-open subsets, the notion of ws-preopen sets will be introduced, where it exists between soft preopen and soft somewhere dense subsets of extended soft topology. Further, several counterexamples to indicate some divergences between this class of ws-preopen sets and other extensions will be constructed.

Definition 3.1. A soft subset (S, Ω) of $(U, T, Ω)$ is said to be ws-preopen, if it is a null soft set or there exists a component of it which is a nonempty preopen set. That is, S (ω) =∅ for every ω ∈ Ω or ∅ ≠ S (ω) ⊆ int (cl (S (ω))) for some ω ∈ Ω.

(S, Ω) is said to be a ws-preclosed set, if its complement is ws-preopen.

Proposition 3.2. (S, Ω) is ws-preclosed iff $(S, Ω) = \tilde{U}$ or $cl (int (S (ω))) \subseteq S (ω) \neq U$ for some ω ∈ Ω.

Proof. (⇒): Let (S, Ω) be a ws-preclosed set. Hence, (S^c, Ω) = φ or ∅ ≠ F^c (ω) ⊆ int (cl (S^c (ω))) for some ω ∈ Ω. This means that $(S, Ω) = \tilde{U}$ or $cl (int (S (ω))) \subseteq S (ω) \neq U$ for some ω ∈ Ω.

(⇐): Suppose that (S, Ω) is a soft set s.t. $(S, Ω) = \tilde{U}$ or $cl (int (S (ω))) \subseteq S (ω) \neq U$ for some ω ∈ Ω. Then, (S^c, Ω) = φ or ∅ ≠ F^c (ω) ⊆ int (cl (S^c (ω))) for some ω ∈ Ω. Which implies that (S^c, Ω) is ws-preopen. So, (S, Ω) is ws-preclosed. □ The next example explains that the class of ws-preopen (ws-preclosed) subsets is not closed under soft union or soft intersection.

Example 3.3. Let $ℝ$ be the set of real numbers and Ω = {ω₁, ω₂} be a set of parameters. Suppose that $T$ is the soft topology on $ℝ$ generated by ${(ω_{i}, S (ω_{i})) : S (ω_{i}) = (a_{i}, b_{i}); a_{i}, b_{i} \in ℝ; a_{i} \leq b_{i}$ and i = 1, 2}. Set (S, Ω) = {(ω₁, (0, 1)) , (ω₂, [0, 1])} and (G, Ω) = {(ω₁, [0, 1]) , (ω₂, (0, 1))} over $ℝ$ . It is clear that (S, Ω) , (G, Ω) are both ws-preopen and ws-preclosed. On the other side, their soft union is not ws-preopen and their soft intersection is not ws-preclosed. Also, (H, Ω) = {(ω₁, (1, 6)) , (ω₂, [2, 3])} and (K, Ω) = {(ω₁, [2, 3]) , (ω₂, (1, 6))} are both ws-preopen and ws-preclosed sets over $ℝ$ . But their soft intersection is not ws-preopen and their soft union is not ws-preclosed.

Proposition 3.4. Suppose that $(U, T, Ω)$ is a full soft_TS with the property of soft hyperconnected. Then the soft intersection of soft α-open and ws-preopen subsets is ws-preopen.

Proof. Assume that (S, Ω) , (G, Ω) are respectively soft α-open and ws-preopen sets. Then there exists a s-open subset (U, Ω) of (S, Ω) and ω ∈ Ω s.t. G (ω) is a nonempty preopen subset of $(U, T_{ω})$ . Since $T$ is full, so U (ω) is nonempty for each ω ∈ Ω. Since $T_{ω}$ is hyperconnected, then G (ω) is dense. Consequentially, U (ω) ∩ G (ω) is a nonempty preopen subset of $(U, T_{ω})$ . Which implies that U (ω) ∩ G (ω) is a dense subset. So that, S (ω) ∩ G (ω), which contains U (ω) ∩ G (ω), is a nonempty preopen subset of $(U, T_{ω})$ . Hence, $(S, Ω) \tilde{\cap} (G, Ω)$ is a ws-preopen set. □

Corollary 3.5. Suppose that $(U, T, Ω)$ is a full soft_TS with the property of soft hyperconnected. Then the soft intersection of s-open subsets and ws-preopen is ws-preopen.

Remark 3.6. Every pseudo constant soft subset (S, Ω) is a ws-preopen subset because S (ω) =∅ for all ω ∈ Ω or $int (cl (S (ω))) = U$ for some ω ∈ Ω.

The next propositions are understandable.

Proposition 3.7. Each s-open set is ws-preopen.

Proposition 3.8. Any soft subset (S, Ω) of $(U, T, Ω)$ with $S (ω) = U$ (resp. S (ω) =∅) is ws-preopen (resp. ws-preclosed).

In the following propositions, a condition that guarantees the relation between ws-preopen sets and soft preopen (or soft α-open) sets will be provided.

Proposition 3.9. If $(U, T, Ω)$ is extended, then every soft preopen (or soft α-open) set is ws-preopen.

Proof. Assume that (S, Ω) is a non-null soft preopen set. Then $(S, Ω) \tilde{\subseteq} int (cl (S, Ω))$ . Since $T$ is an extended soft topology, so S (ω) ⊆ int (cl (S (ω))) for all ω ∈ Ω. Which implies that there exists a component of (S, Ω) that is a nonempty preopen subset. Hence, (S, Ω) is ws-preopen.

Following similar argument one can prove the other case. □

Proposition 3.10. If $(U, T, Ω)$ is extended, then each ws-preopen set is soft somewhere dense.

Proof. Let (S, Ω) be a non-null ws-preopen set. Then there exists a component of (S, Ω) which is a nonempty preopen set. Since $T$ is extended, so int (cl (S, Ω)) = (int (cl (S)) , Ω)≠ ∅. Hence, (S, Ω) is soft somewhere dense. □ The following example exposes that a condition of “extended soft topology" equipped in Propositions 3.9, 3.10 is irreplaceable.

Example 3.11. Suppose that $U = {u_{1}, u_{2}, u_{3}}$ is a universe and Ω = {ω₁, ω₂} is a class of parameters. Choose the class $T$ consisting of φ, $\tilde{U}$ and the following soft subsets over $U$ with Ω

(S₁, Ω) = {(ω₁, {u₁}) , (ω₂, ∅)};

(S₂, Ω) = {(ω₁, ∅) , (ω₂, {u₂})};

(S₃, Ω) = {(ω₁, {u₁}) , (ω₂, {u₂, u₃})};

(S₄, Ω) = {(ω₁, {u₁}) , (ω₂, {u₃})};

(S₅, Ω) = {(ω₁, {u₃}) , (ω₂, {u₂})};

(S₆, Ω) = {(ω₁, {u₁, u₃}) , (ω₂, {u₂})};

(S₇, Ω) = {(ω₁, {u₁, u₃}) , (ω₂, {u₂, u₃})} and

(S₈, Ω) = {(ω₁, {u₁}) , (ω₂, {u₂})}.

Then, $(U, T, Ω)$ is a soft_TS. Note that a soft set (H, Ω) = {(ω₁, {u₁, u₂}) , (ω₂, {u₁, u₂})} is soft preopen because $int (cl (H, Ω)) = \tilde{U}$ . But it is not a ws-preopen set because int (cl (H (ω₁))) = {u₁} ⊉H (ω₁) and int (cl (H (ω₂))) = {u₂} ⊉H (ω₂). Also, a soft set (G, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, {u₃})} is a ws-preopen set because int (cl (G (ω₂))) = G (ω₂). But it is not a soft somewhere dense set because int (cl (G, Ω)) = φ.

To elucidate that the converse of Proposition 3.9 and Proposition 3.10 fail, the following examples are seen.

Example 3.12. Suppose that $U = {u_{1}, u_{2}, u_{3}}$ is a universe and Ω = {ω₁, ω₂} is a class of parameters. Choose the class $T$ consisting of φ, $\tilde{U}$ and the following soft subsets over $U$ with Ω

$(S_{1}, Ω) = {(ω_{1}, U), (ω_{2}, \emptyset)}$ ;

$(S_{2}, Ω) = {(ω_{1}, \emptyset), (ω_{2}, U)}$ ;

(S₃, Ω) = {(ω₁, {u₁}) , (ω₂, ∅)};

(S₄, Ω) = {(ω₁, {u₁}) , (ω₂, {u₃})};

(S₅, Ω) = {(ω₁, {u₁}) , (ω₂, {u₁, u₃})};

(S₆, Ω) = {(ω₁, {u₁}) , (ω₂, {u₂, u₃})};

$(S_{7}, Ω) = {(ω_{1}, {u_{1}}), (ω_{2}, U)}$ ;

(S₈, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, ∅)};

(S₉, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, {u₃})};

(S₁₀, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, {u₁, u₃})};

(S₁₁, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, {u₂, u₃})};

$(S_{12}, Ω) = {(ω_{1}, {u_{2}, u_{3}}), (ω_{2}, U)}$ ;

(S₁₃, Ω) = {(ω₁, ∅) , (ω₂, {u₃})};

$(S_{14}, Ω) = {(ω_{1}, U), (ω_{2}, {u_{3}})}$ ;

(S₁₅, Ω) = {(ω₁, ∅) , (ω₂, {u₁, u₃})};

$(S_{16}, Ω) = {(ω_{1}, U), (ω_{2}, {u_{1}, u_{3}})}$ ;

(S₁₇, Ω) = {(ω₁, ∅) , (ω₂, {u₂, u₃})} and

$(S_{18}, Ω) = {(ω_{1}, U), (ω_{2}, {u_{2}, u_{3}})}$ .

Then, $(U, T, Ω)$ is an extended soft_TS. Observe that a soft set (H, Ω) = {(ω₁, {u₃}) , (ω₂, {u₂})} is ws-preopen because H (ω₁) ⊆ int (cl (H (ω₁))) = {u₂, u₃}. But it is not a soft preopen set because int (cl (H, Ω))) = {(ω₁, {u₂, u₃}) , (ω₂, ∅)} ⊉ (H, Ω).

Example 3.13. It can be checked that a soft topology given in Example 3.3 is extended. Take a soft set (G, Ω) = {(ω₁, [1, 3]) , (ω₂, ∅)}. It is obvious that (G, Ω) is a soft somewhere dense subset because int (cl (G, Ω)) = {(ω₁, (1, 3)) , (ω₂, ∅)} ≠ φ. On the other hand, it is not a ws-preopen set because int (cl (G (ω₁))) = (1, 3) ⊉G (ω₁) and G (ω₂) is empty.

Proposition 3.14. The image and pre-image of ws-preopen set under a soft bi-continuous function (soft open and continuous) is ws-preopen.

Proof. To show the case of image, assume that $M_{O} : (U, T, Ω) \to (V, S, Σ)$ is a soft bi-continuous function. Let (S, Ω) be a ws-preopen set, then there exists ω ∈ Ω s.t. S (ω) is a nonempty preopen subset. Let O (ω) = σ. Regarding to Theorem 2.17, it follows from the soft bicontinuity of M_O that $M : (U, T_{ω}) \to (V, S_{O (ω) = σ})$ is a bicontinuous function. It is clear that a continuity of M implies that M (cl (V)) ⊆ cl (M (V)), and an openness of M implies that M (int (V)) ⊆ int (M (V)) for each subset V of $U$ . Thus $M (S (ω)) \tilde{\subseteq} M (int (cl (S (ω))) \tilde{\subseteq} int (cl (M (S (ω))))$ . So that M (S (ω)) is a nonempty preopen component of M_O (S, Ω); hence, M_O (S, Ω) is a ws-preopen subset of $(U, T, Ω)$ . □

Corollary 3.15. The property of being a ws-preopen set is a topological property.

Proposition 3.16. The product of two ws-preopen sets is ws-preopen.

Proof. Suppose that (S, Ω) , (G, Ω) are ws-preopen subsets. Let (H, Ω × Ω) = (S, Ω) × (G, Ω), then there are ω₁, ω₂ ∈ Ω s.t. S (ω₁) , G (ω₂) are nonempty preopen sets. Now, (ω₁, ω₂) ∈ Ω × Ω s.t. H (ω₁, ω₂) = S (ω₁) × G (ω₂). According to the classical topology, the product of two nonempty preopen sets is still a nonempty preopen; therefore, H (ω₁, ω₂) is a nonempty preopen set. Consequently, (H, Ω × Ω) is a ws-preopen set. □

4 Weakly pre-interior and weakly pre-closure operators

The next section is devoted to establish the operators of interior, closure, boundary, and limit by using the class of ws-preopen and ws-preclosed sets and then discuss their prime properties and relationships among them. The weakly pre-interior (resp., weakly pre-closure) of soft set need not be weakly preopen (resp., weakly preclosed) sets as presented with some counterexamples.

Definition 4.1. The weakly pre-interior points of a set (S, Ω), denoted by int_wp (S, Ω), is defined as the union of all ws-preopen sets contained in (S, Ω).

By Example 3.3, the weakly pre-interior points of a set need not be a weakly preopen is showed. That is, int_wp (S, Ω) = (S, Ω) does not imply that (S, Ω) is a weakly preopen set.

One can easily prove the next propositions.

Proposition 4.2. Assume that (S, Ω) is a subset of $(U, T, Ω)$ and $u_{ω} \in \tilde{U}$ . Then u_ω ∈ int_wp (S, Ω) iff there exists a ws-preopen set (G, Ω) contains u_ω s.t. $(G, Ω) ⋂^{˜} (S, Ω)$ .

Proposition 4.3. Let (S, Ω), (G, Ω) be soft sets. Then,

(i) ${int}_{wp} (S, Ω) \tilde{\subseteq} (S, Ω)$ .

(ii) ${int}_{wp} (S, Ω) \tilde{\subseteq} {int}_{wp} (G, Ω)$ , if $(S, Ω) \tilde{\subseteq} (G, Ω)$ .

Corollary 4.4. For any sets (S, Ω), (G, Ω), the next results are held:

(i) ${int}_{wp} [(S, Ω) ⋂^{˜} (G, Ω)] \tilde{\subseteq} {int}_{wp} (S, Ω) ⋂^{˜} {int}_{wp} (G, Ω)$ .

(ii) ${int}_{wp} (S, Ω) ⋃^{˜} {int}_{wp} (G, Ω) \tilde{\subseteq} {int}_{wp} [(S, Ω) ⋃^{˜} (G, Ω)]$ .

Proof. It automatically comes from the following:

(i): $(S, Ω) ⋂^{˜} (G, Ω) \tilde{\subseteq} (S, Ω)$ and $(S, Ω) ⋂^{˜} (G, Ω) \tilde{\subseteq} (G, Ω)$ .

(ii): $(S, Ω) \tilde{\subseteq} [(S, Ω) ⋃^{˜} (G, Ω)]$ and $(G, Ω) \tilde{\subseteq} [(S, Ω) ⋃^{˜} (G, Ω)]$

□ Let (E, Ω) = {(ω₁, {5}) , (ω₂, {6, 7})}, (S, Ω) = {(ω₁, (1, 2)) , (ω₂, [1, 2])}, (G, Ω) = {(ω₁, (2, 3)) , (ω₂, [2, 3])}, (H, Ω) = {(ω₁, ∅) , (ω₂, (0, 1])}, and (J, Ω) = {(ω₁, (0, 1)) , (ω₂, ∅)} be soft subsets given in Example 3.3. The following properties are remarked:

1. $(E, Ω) \tilde{⊈} {int}_{wp} (E, Ω) = φ$ .

2. ${int}_{wp} (E, Ω) \tilde{\subseteq} {int}_{wp} (H, Ω)$ whereas $(E, Ω) \tilde{⊈} (H, Ω)$ .

3. ${int}_{wp} [(S, Ω) ⋂^{˜} (G, Ω)] = φ$ whereas ${int}_{wp} (S, Ω) ⋂^{˜} {int}_{wp} (G, Ω) = {(ω_{1}, \emptyset), (ω_{2}, {2})}$ .

4. ${int}_{wp} (H, Ω) ⋃^{˜} {int}_{wp} (J, Ω) = {(ω_{1}, (0, 1)), (ω_{2}, (0, 1))}$ whereas ${int}_{wp} [(H, Ω) ⋃^{˜} (J, Ω)] = {(ω_{1}, (0, 1)), (ω_{2}, (0, 1])}$ .

So, the inclusion relations of Proposition 4.3 and Corollary 4.4 are proper.

Definition 4.5. The weakly pre-closure points of a subset (S, Ω) of $(U, T, Ω)$ , denoted by cl_wp (S, Ω), is designated as the intersection of all ws-preclosed sets containing (S, Ω).

According to Example 3.3, the weakly pre-closure points of a subset need not be a weakly preclosed set are obtained. Which means that, cl_wp (S, Ω) = (S, Ω) does not imply that (S, Ω) is a weakly preclosed set.

Proposition 4.6. Suppose that (S, Ω) is a subset of $(U, T, Ω)$ and $u_{ω} \in \tilde{U}$ . Then u_ω ∈ cl_wp (S, Ω) iff $(G, Ω) ⋂^{˜} (S, Ω) \neq φ$ for every ws-preopen set (G, Ω) contains u_ω.

Proof. (⇒) Let u_ω ∈ cl_wp (S, Ω). Suppose that there is ws-preopen set (G, Ω) containing u_ω s.t. $(G, Ω) ⋂^{˜} (S, Ω) = φ$ . Hence, $(S, Ω) \tilde{\subseteq} (G^{c}, Ω)$ . Therefore, ${cl}_{wp} (S, Ω) \tilde{\subseteq} (G^{c}, Ω)$ . Thus u_ω ∉ cl_wp (S, Ω). This is a contradiction, that is $(G, Ω) ⋂^{˜} (S, Ω) \neq φ$ , as required.

(⇐) Let $(G, Ω) ⋂^{˜} (S, Ω) \neq φ$ for each ws-preopen set (G, Ω) contains u_ω. Suppose that u_ω ∉ cl_wp (S, Ω). Then there is a soft preclosed set (H, Ω) containing (S, Ω) with u_ω ∉ (H, Ω). So u_ω ∈ (H^c, Ω) and $(H^{c}, Ω) ⋂^{˜} (S, Ω) = φ$ . This is a contradiction. Hence, the desired result is obtained. □

Corollary 4.7. If $(S, Ω) ⋂^{˜} (G, Ω) = φ$ s.t. (S, Ω) is a ws-preopen set and (G, Ω) is a soft set in $(U, T, Ω)$ , then $(S, Ω) ⋂^{˜} {cl}_{wp} (G, Ω) = φ$ .

Proof. Direct to prove. □

Proposition 4.8. The next properties hold for a subset (S, Ω) of $(U, T, Ω)$ .

(i) [int_wp (S, Ω)] ^c = cl_wp (S^c, Ω).

(ii) [cl_wp (S, Ω)] ^c = int_wp (S^c, Ω).

Proof. (i): If u_ω ∉ [int_wp (S, Ω)] ^c, then there exists a ws-preopen set (G, Ω) s.t. $u_{ω} \in (G, Ω) \tilde{\subseteq} (S, Ω)$ . Therefore, $(S^{c}, Ω) ⋂^{˜} (G, Ω) = φ$ and so u_ω ∉ cl_wp (S^c, Ω). Conversely, if u_ω ∉ cl_wp (S^c, Ω), then by following the previous steps u_ω ∉ [int_wp (S, Ω)] ^c can be verified.

(ii): Following similar approach given in (i). □

The next proposition is evident and its proof is simple, so it was omitted.

Proposition 4.9. Assume that (S, Ω), (G, Ω) are soft subsets of $(U, T, Ω)$ . Then

(i) $(S, Ω) \tilde{\subseteq} {cl}_{wp} (S, Ω)$ .

(ii) if $(S, Ω) \tilde{\subseteq} (G, Ω)$ , then ${cl}_{wp} (S, Ω) \tilde{\subseteq} {cl}_{wp} (G, Ω)$ .

Corollary 4.10. For any subsets (S, Ω), (G, Ω) of $(U, T, Ω)$ , the next results are achieved.

(i) ${cl}_{wp} [(S, Ω) ⋂^{˜} (G, Ω)] \tilde{\subseteq} {cl}_{wp} (S, Ω) ⋂^{˜} {cl}_{wp} (G, Ω)$ .

(ii) ${cl}_{wp} (S, Ω) ⋃^{˜} {cl}_{wp} (G, Ω) \tilde{\subseteq} {cl}_{wp} [(S, Ω) ⋃^{˜} (G, Ω)]$ .

□

Proof. It automatically comes from the following:

(i): $(S, Ω) ⋂^{˜} (G, Ω) \tilde{\subseteq} (S, Ω)$ and $(S, Ω) ⋂^{˜} (G, Ω) \tilde{\subseteq} (G, Ω)$ .

(ii): $(S, Ω) \tilde{\subseteq} [(S, Ω) ⋃^{˜} (G, Ω)]$ and $(G, Ω) \tilde{\subseteq} [(S, Ω) ⋃^{˜} (G, Ω)]$ . □ Let $(E, Ω) = {(ω_{1}, ℝ), (ω_{2}, [0, 1))}$ , $(S, Ω) = {(ω_{1}, ℝ), (ω_{2}, [1, 2))}$ , $(G, Ω) = {(ω_{1}, ℝ), (ω_{2}, [2, 3])}$ , and $(H, Ω) = {(ω_{1}, ℝ), (ω_{2}, (0, 1])}$ be soft subsets of a soft_TS given in Example 3.3. The following properties are demonstrated:

1. ${cl}_{wp} (E, Ω) = {(ω_{1}, ℝ), (ω_{2}, [0, 1])} \tilde{⊄} (E, Ω)$ .

2. ${cl}_{wp} (E, Ω) \tilde{\subseteq} {cl}_{wp} (H, Ω)$ whereas $(E, Ω) \tilde{⊈} (H, Ω)$ .

3. ${cl}_{wp} [(S, Ω) ⋂^{˜} (G, Ω)] = {(ω_{1}, ℝ), (ω_{2}, \emptyset)}$ whereas ${cl}_{wp} (S, Ω) ⋂^{˜} {cl}_{wp} (G, Ω) = {(ω_{1}, ℝ), (ω_{2}, {2})}$ .

Hence, the inclusion relations of Proposition 4.3 and Corollary 4.4 are proper.

Definition 4.11. A soft point u_ω is named a weakly pre-boundary point of a subset (S, Ω) of $(U, T, Ω)$ if u_ω belongs to the complement of ${int}_{wp} (S, Ω) ⋃^{˜} {int}_{wp} (S^{c}, Ω)$ .

All pre-boundary points of (S, Ω), denoted by b_wp (S, Ω), is said to bea weakly pre-boundary set.

Proposition 4.12. $b_{wp} (S, Ω) = {cl}_{wp} (S, Ω) ⋂^{˜} {cl}_{wp} (S^{c}, Ω)$ for every subset (S, Ω) of $(U, T, Ω)$ .

Proof. $b_{wp} (S, Ω) = [{int}_{wp} (S, Ω) ⋃^{˜} {int}_{wp} (S^{c}, Ω)]^{c}$

$= [{int}_{wp} (S, Ω)]^{c} ⋂^{˜} [{int}_{wp} (S^{c}, Ω)]^{c}$ (De Morgan’s law)

$= {cl}_{wp} (S^{c}, Ω) ⋂^{˜} {cl}_{wp} (S, Ω)$ (Proposition 4.8(2)) □

Corollary 4.13. For every subset (S, Ω) of $(U, T, Ω)$ , the following properties hold.

(i) b_wp (S, Ω) = b_wp (S^c, Ω).

(ii) b_wp (S, Ω) = cl_wp (S, Ω) \ int_wp (S, Ω).

(iii) ${cl}_{wp} (S, Ω) = {int}_{wp} (S, Ω) ⋃^{˜} b_{wp} (S, Ω)$ .

(iv) int_wp (S, Ω) = (S, Ω) \ b_wp (S, Ω).

Proof. (i): Obvious.

(ii): $b_{wp} (S, Ω) = {cl}_{wp} (S, Ω) ⋂^{˜} {cl}_{wp} (S^{c}, Ω)$

= cl_wp (S, Ω) \ [cl_wp (S^c, Ω)] ^c. According to (ii) of Proposition 4.8 the required relation is obtained.

(iii): ${int}_{wp} (S, Ω) ⋃^{˜} b_{wp} (S, Ω)$

$= {int}_{wp} (S, Ω) ⋃^{˜} [{cl}_{wp} (S, Ω) \ {int}_{wp} (S, Ω)] = {cl}_{wp} (S, Ω)$ .

(iv): (S, Ω) \ b_wp (S, Ω)

= (S, Ω) \ [cl_wp (S, Ω) \ int_wp (S, Ω)]

$= (S, Ω) ⋂^{˜} [{cl}_{wp} (S, Ω) ⋂^{˜} ({int}_{wp} (S, Ω))^{c}]^{c}$

$= (S, Ω) ⋂^{˜} [({cl}_{wp} (S, Ω))^{c} ⋃^{˜} {int}_{wp} (S, Ω)]$

$= [(S, Ω) ⋂^{˜} ({cl}_{wp} (S, Ω))^{c}] ⋃^{˜} [(S, Ω) ⋂^{˜} {int}_{wp} (S, Ω)]$

= int_wp (S, Ω). □

Proposition 4.14. For every subsets (S, Ω) , (G, Ω) of $(U, T, Ω)$ , the following properties hold.

(i) $b_{wp} ({int}_{wp} (S, Ω)) \tilde{\subseteq} b_{wp} (S, Ω)$ .

(ii) $b_{wp} ({cl}_{wp} (S, Ω)) \tilde{\subseteq} b_{wp} (S, Ω)$ .

Proof. By substituting in the formula (iii) of Corollary 4.13, the proof follows. □

Proposition 4.15. Let (S, Ω) be a subset of $(U, T, Ω)$ . Then

(i) (S, Ω) = int_wp (S, Ω) iff $b_{wp} (S, Ω) ⋂^{˜} (S, Ω) = φ$ .

(ii) (S, Ω) = cl_wp (S, Ω) iff $b_{wp} (S, Ω) \tilde{\subseteq} (S, Ω)$ .

Proof. (i): Suppose that (S, Ω) = int_wp (S, Ω). Then by (iv) of Corollary 4.13, (S, Ω) = int_wp (S, Ω) = (S, Ω) \ b_wp (S, Ω) and hence $b_{wp} (S, Ω) ⋂^{˜} (S, Ω) = φ$ . Conversely, let u_ω ∈ (S, Ω). Since u_ω ∉ b_wp (S, Ω) and u_ω ∈ cl_wp (S, Ω), by (iii) of Corollary 4.13, u_ω ∈ int_wp (S, Ω). Therefore, int_wp (S, Ω) = (S, Ω), as required.

(ii): Assume that (S, Ω) = cl_wp (S, Ω). Then $b_{wp} (S, Ω) = {cl}_{wp} (S, Ω) ⋂^{˜} {cl}_{wp} (S^{c}, Ω)$ $\tilde{\subseteq}$ cl_wp (S, Ω) = (S, Ω), as required. Conversely, if $b_{wp} (S, Ω) \tilde{\subseteq} (S, Ω)$ , then by (iii) of Corollary 4.13, ${cl}_{wp} (S, Ω) \tilde{\subseteq} {int}_{wp} (S, Ω) ⋃^{˜} (S, Ω) = (S, Ω)$ and hence cl_wp (S, Ω) = (S, Ω), as required. □

Corollary 4.16. int_wp (S, Ω) = (S, Ω) = cl_wp (S, Ω) iff b_wp (S, Ω) = φ.

Definition 4.17. A soft point u_ω is named a weakly pre-limit point of a set (S, Ω), if [(G, Ω) ∖ u_ω] ⋂ (S, Ω)≠φ for every ws-preopen set (G, Ω) containing u_ω.

All weakly pre-limit points of (S, Ω) is said to be a weakly pre-derived set and symbolized by l_wp (S, Ω).

Proposition 4.18. Suppose that (S, Ω) , (G, Ω) are soft sets. If $(S, Ω) \tilde{\subseteq} (G, Ω)$ , then $l_{wp} (S, Ω) \tilde{\subseteq} l_{wp} (G, Ω)$ .

Proof. Obvious by Definition 4.17. □

Corollary 4.19. Consider (S, Ω) and (G, Ω) are soft sets. Then:

(i) $l_{wp} [(S, Ω) ⋂^{˜} (G, Ω)] \tilde{\subseteq} l_{wp} (S, Ω) ⋂^{˜} l_{wp} (G, Ω)$ .

(ii) $l_{wp} (S, Ω) ⋃^{˜} l_{wp} (G, Ω) \tilde{\subseteq} l_{wp} [(S, Ω) ⋃^{˜} (G, Ω)]$ .

Theorem 4.20. ${cl}_{wp} (S, Ω) = (S, Ω) ⋃^{˜} l_{wp} (S, Ω)$ , for any soft set (S, Ω).

Proof. The side $(S, Ω) ⋃^{˜} l_{wp} (S, Ω) \tilde{\subseteq} {cl}_{wp} (S, Ω)$ is obvious. To prove the other side let $u_{ω} \notin [(S, Ω) ⋃^{˜} l_{wp} (S, Ω)]$ . Then u_ω ∉ (S, Ω) and u_ω ∉ l_wp (S, Ω). Therefore, there is ws-preopen (G, Ω) containing u_ω with $(G, Ω) \tilde{\cap} (S, Ω) = φ$ . Thus, u_ω ∉ cl_wp (S, Ω). Hence, ${cl}_{wp} (S, Ω) = (S, Ω) ⋃^{˜} l_{wp} (S, Ω)$ . □

Corollary 4.21. If (S, Ω) be a ws-preclosed set, then $l_{wp} (S, Ω) \tilde{\subseteq} (S, Ω)$ .

5 Continuity via ws-preopen sets

This section is dedicated to tackling the idea of soft continuity in terms of ws-preopen sets. Their main properties will be established and an illustrative example will be provided. The loss of the property says that “weakly pre-interior of soft subset is weakly preopen" leads to disappearing several characterizations of this kind of soft continuity will be shown.

Definition 5.1. A soft function $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is named ws-precontinuous, if the inverse image of each s-open set is ws-preopen.

It is easy to prove the next result, so delete its proof.

Proposition 5.2. If $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is a ws-precontinuous function and $N_{I} : (V, T_{V}, Ω) \to (W, T_{W}, Ω)$ is a soft continuous function, then N_I ∘ M_O is ws-precontinuous.

Proposition 5.3. Every soft continuous function is ws-precontinuous.

Proof. According to Proposition 3.7, the proof is understandable. □

Proposition 5.4. Let $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ be a soft function s.t. $T_{U}$ is extended. Then

(i) M_O is ws-precontinuous, if M_O is soft pre-continuous (soft α-continuous).

(ii) M_O is soft SD-continuous, if M_O is ws-precontinuous.

Proof. It follows from Propositions 3.9, 3.10. □

Theorem 5.5. A soft function $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is ws-precontinuous iff the inverse image of every s-closed subset is ws-preclosed.

Proof. Necessity: Consider (S, Ω) is a s-closed subset of $(V, T_{V}, Ω)$ . Then (S^c, Ω) is s-open. Therefore, $M_{O}^{- 1} (S^{c}, Ω) = \tilde{U} - M_{O}^{- 1} (S, Ω)$ is ws-preopen. Thus, $M_{O}^{- 1} (S, Ω)$ is a ws-preclosed set.

One can prove the sufficient part, by following similar argument. □

Theorem 5.6. If $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is ws-precontinuous, then the next properties are equivalent.

(i) $M_{O}^{- 1} (S, Ω) = {int}_{wp} (M_{O}^{- 1} (S, Ω))$ , for each s-open subset (S, Ω) of $(V, T_{V}, Ω)$ .

(ii) $M_{O}^{- 1} (S, Ω) = {cl}_{wp} (M_{O}^{- 1} (S, Ω))$ , for each s-closed subset (S, Ω) of $(V, T_{V}, Ω)$ .

(iii) ${cl}_{wp} (M_{O}^{- 1} (S, Ω)) \tilde{\subseteq} M_{O}^{- 1} (cl (S, Ω))$ , for each $(S, Ω) \tilde{\subseteq} \tilde{V}$ .

(iv) $M_{O} ({cl}_{wp} (G, Ω)) \tilde{\subseteq} cl (M_{O} (G, Ω))$ , for each $(G, Ω) \tilde{\subseteq} \tilde{U}$ .

(v) $M_{O}^{- 1} (int (S, Ω)) \tilde{\subseteq} {int}_{wp} (M_{O}^{- 1} (S, Ω))$ , for each $(S, Ω) \tilde{\subseteq} \tilde{V}$ .

Proof. (i) → (ii): Suppose that (S, Ω) is a s-closed subset of $(V, T_{V}, Ω)$ . Then (S^c, Ω) is s-open. Therefore, $M_{O}^{- 1} (S^{c}, Ω) = {int}_{wp} (M_{O}^{- 1} (S^{c}, Ω))$ . According to Proposition 4.8, $M_{O}^{- 1} (S, Ω) = {cl}_{wp} (M_{O}^{- 1} (S, Ω))$ .

(ii) → (iii): For any soft set $(S, Ω) \tilde{\subseteq} \tilde{V}$ , $M_{O}^{- 1} (cl (S, Ω)) = {cl}_{wp} (M_{O}^{- 1} (cl (S, Ω)))$ . Then

${cl}_{wp} (M_{O}^{- 1} (S, Ω)) \tilde{\subseteq} {cl}_{wp} (M_{O}^{- 1} (cl (S, Ω))) = M_{O}^{- 1} (cl (S, Ω))$ .

(iii) → (iv): It is obvious that ${cl}_{wp} (G, Ω) \tilde{\subseteq} {cl}_{wp} (M_{O}^{- 1} (M_{O} (G, Ω)))$ for each $(G, Ω) \tilde{\subseteq} \tilde{U}$ . By (iii), we get ${cl}_{wp} (M_{O}^{- 1} (M_{O} (G, Ω))) \tilde{\subseteq}$ $M_{O}^{- 1} (cl (M_{O} (G, Ω)))$ . Therefore, $M_{O} ({cl}_{wp} (G, Ω) \tilde{\subseteq}$ $M_{O} (M_{O}^{- 1} (cl (M_{O} (G, Ω))))$ $\tilde{\subseteq}$ cl (M_O (G, Ω)).

(iv) → (v): Let (S, Ω) be an arbitrary soft set in $(V, T_{V}, Ω)$ . Then $M_{O} ({cl}_{wp} (M_{O}^{- 1} (S^{c}, Ω))$ $\tilde{\subseteq} cl (M_{O} (M_{O}^{- 1} (S^{c}, Ω)))$ $\tilde{\subseteq} cl (S^{c}, Ω)$ . So that, ${cl}_{wp} ((M_{O}^{- 1} (S, Ω))^{c}) \tilde{\subseteq} M_{O}^{- 1} ((int (S, Ω))^{c}$ . Hence, $M_{O}^{- 1} (int (S, Ω))$ $\tilde{\subseteq} {int}_{wp} (M_{O}^{- 1} (S, Ω))$ .

(v) → (i): Suppose that (S, Ω) is a s-open subset in $(V, T_{V}, Ω)$ . By (v), $M_{O}^{- 1} (S, Ω) = M_{O}^{- 1} (int (S, Ω)) \tilde{\subseteq}$ ${int}_{wp} (M_{O}^{- 1} (S, Ω))$ . But ${int}_{wp} (M_{O}^{- 1} (S, Ω)) \tilde{\subseteq} M_{O}^{- 1} (S, Ω)$ , so $M_{O}^{- 1} (S, Ω) = {int}_{wp} (M_{O}^{- 1} (S, Ω))$ , as required. □ The converse of the Theorem 5.6 fails. To demonstrate that the next example is furnished.

Example 5.7. Assume that $U = {u_{1}, u_{2}, u_{3}}$ , $V = {υ_{1}, υ_{2}, υ_{3}}$ with Ω = {ω₁, ω₂}. Let $T_{U} = {φ, \tilde{U}, (S_{i}, Ω) : i = 1, 2, 3, 4}$ and $T_{V} = {φ, \tilde{V}, (H, Ω)}$ be two soft topologies defined on $U$ and $V$ , respectively, with the same set of parameters Ω, where

(S₁, Ω) = {(ω₁, {u₁}) , (ω₂, {u₁}};

(S₂, Ω) = {(ω₁, {u₂}) , (ω₂, {u₂}};

(S₃, Ω) = {(ω₁, {u₁, u₂}) , (ω₂, {u₁, u₂}};

(S₄, Ω) = {(ω₁, {u₂, u₃}) , (ω₂, {u₂, u₃}} and

(H, Ω) = {(ω₁, {υ₁}) , (ω₂, {υ₁}}.

Consider $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is a soft function, where $M : U \to V$ is designated as M (u₁) = M (u₃) = υ₁ and M (u₂) = υ₃.

and O : Ω → Ω is the identity function.

Now, $M_{O}^{- 1} (H, Ω) = {(ω_{1}, {u_{1}, u_{3}}), (ω_{2}, {u_{1}, u_{3}}}$ which is not a ws-preopen subset because int (cl ({u₁, u₃}) = {u₁}. Then M_O is not ws-precontinuous. Moreover, $M_{O}^{- 1} (φ) = {int}_{wp} (M_{O}^{- 1} (φ))$ ; $M_{O}^{- 1} (\tilde{V}) = {int}_{wp} (M_{O}^{- 1} (\tilde{V}))$ and $M_{O}^{- 1} (H, Ω) = {int}_{wp} (M_{O}^{- 1} (H, Ω))$ , That is all properties given in Theorem 5.6 hold true.

Now, the concepts of ws-preopen, ws-preclosed and w-soft pre-homeomorphism functions will be introduced.

Definition 5.8. A soft function $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is said to be:

(i) ws-preopen provided that the image of each s-open set is a ws-preopen set.

(ii) ws-preclosed provided that the image of each s-closed set is a ws-preclosed set.

(iii) ws-prehomeomorphism provided that M_O is bijective, ws-preopen and ws-preclosed.

Theorem 5.9. Let $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ be a soft function and (S, Ω) be any soft subset of $\tilde{U}$ . Then

(i) $M_{O} (int (S, Ω)) \tilde{\subseteq} {int}_{wp} (M_{O} (S, Ω))$ , if M_O is ws-preopen.

(ii) cl_wp (M_O (S, Ω)) $\tilde{\subseteq} M_{O} (cl (S, Ω))$ , if M_O is ws-preclosed.

Proof. (i): Consider (S, Ω) is a soft subset of $\tilde{U}$ . Then M_O (int (S, Ω)) is a ws-preopen subset of $(V, T_{V}, Ω)$ and so $M_{O} (int (S, Ω)) = {int}_{wp} (M_{O} (int (S, Ω))) \tilde{\subseteq}$

int_wp (M_O (S, Ω)).

(ii): The proof is similar to that of (i). □

Proposition 5.10. A bijective soft function $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is ws-preopen iff it is ws-preclosed.

Proof. Necessity: Consider (S, Ω) is a ws-preclosed subset of $(U, T_{U}, Ω)$ . Since M_O is ws-preopen, M_O (S^c, Ω) is ws-preopen. By bijectiveness of M_O, then M_O (S^c, Ω) = (M_O (S, Ω)) ^c. So that, M_O (S, Ω) is a ws-preclosed set. Consequently, M_O is ws-preclosed. By following a similar approach, the sufficient part will be proven. □

Proposition 5.11. Consider $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ is a w-soft preclosed function and $\tilde{Γ}$ be a s-closed subset of $\tilde{U}$ . Then $M_{O} ∣_{Γ} : (Γ, T_{Γ}, Ω) \to (V, T_{V}, Ω)$ is ws-preclosed.

Proof. Assume that (S, Ω) is a s-closed subset of $(Γ, T_{Γ}, Ω)$ . Then there exists a s-closed subset (G, Ω) of $(U, T_{U}, Ω)$ s.t. $(S, Ω) = (G, Ω) ⋂^{˜} \tilde{Γ}$ . Since $\tilde{Γ}$ is a s-closed set, then (S, Ω) is also a s-closed subset of $(U, T_{U}, Ω)$ . Since M_O ∣ _Γ (S, Ω) = M_O (S, Ω), then M_O ∣ _Γ (S, Ω) is a w-soft preclosed set. Thus, M_O ∣ _Γ is a ws-preclosed. □

Proposition 5.12. The next four statements hold for soft functions $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ and $N_{I} : (V, T_{V}, Ω) \to (W, T_{W}, Ω)$ .

(i) If M_O is soft open and N_I is soft α-open (soft preopen) s.t. $T_{W}$ is extended, then N_I ∘ M_O is ws-preopen.

(ii) If N_I ∘ M_O is ws-preopen and M_O is surjective soft continuous, then N_I is ws-preopen.

(iii) If N_I ∘ M_O is s-open and N_I is injective ws-precontinuous, then M_O is ws-preopen.

Proof. (i): Let N_I be soft α-open (soft preopen) and (S, Ω) ≠ φ as a s-open subset of $\tilde{U}$ . Then M_O (S, Ω) ≠ φ is a soft α-open (soft preopen) subset of $\tilde{V}$ . Thus, N_I (M_O (S, Ω)) is a soft α-open (soft preopen) subset. Since $T_{W}$ is extended, it follows from Proposition 3.9 that N_I (M_O (S, Ω)) is a ws-preopen subset. Hence, N_I ∘ M_O is ws-preopen.

(ii): Assume that (S, Ω) ≠ φ is a s-open subset of $\tilde{V}$ . Then $M_{O}^{- 1} (S, Ω) \neq φ$ is a s-open subset of $\tilde{U}$ . Therefore, $(N_{I} \circ M_{O}) (M_{O}^{- 1} (S, Ω))$ is a ws-preopen subset of $\tilde{W}$ . Since M_O is surjective, then $(N_{I} \circ M_{O}) (M_{O}^{- 1} (S, Ω)) = N_{I} (M_{O} (M_{O}^{- 1} (S, Ω))) = N_{I} (S, Ω)$ . Thus N_I is ws-preopen.

(iii): Let (S, Ω) ≠ φ be a s-open subset of $\tilde{U}$ . Then (N_I ∘ M_O) (S, Ω) ≠ φ is a s-open subset of $\tilde{W}$ . Therefore, $N_{I}^{- 1} (N_{I} \circ M_{O} (S, Ω))$ is a ws-preopen subset of $\tilde{V}$ . Since N_I is injective, $N_{I}^{- 1} (N_{I} \circ M_{O} (S, Ω)) = (N_{I}^{- 1} N_{I}) (M_{O} (S, Ω)) = M_{O} (S, Ω)$ . Thus, M_O is ws-preopen. □ One can prove the next finding following similar argument given above.

Proposition 5.13. The next four statements hold for soft functions $M_{O} : (U, T_{U}, Ω) \to (V, T_{V}, Ω)$ and $N_{I} : (V, T_{V}, Ω) \to (W, T_{W}, Ω)$ .

(i) If M_O is soft closed and N_I is soft α-closed (soft preclosed) s.t. $T_{W}$ is extended, then N_I ∘ M_O is ws-preclosed.

(ii) If N_I ∘ M_O is ws-preclosed and M_O is surjective soft continuous, then N_I is ws-preclosed.

(iii) If N_I ∘ M_O is soft closed and N_I is injective ws-precontinuous, then M_O is ws-preclosed.

6 Conclusion

In this study, the definition of “ws-preopen sets" as a novel class of generalizations of s-open subsets has been given, and their basic properties have been verified. This type of soft set has been constructed using its corresponding concept via parametric topologies. At First, we have demonstrated that the property of closing under the arbitrary soft unions and soft intersections of the families of ws-preopen and ws-preclosed sets is lost, whereas it is fulfilled with the prior known generalizations. With respect to its relationship with the previous generalizations, it has been elucidated that it lies between soft preopen and soft somewhere dense subsets of an extended soft topology. Then, the concepts of interior, closure, boundary, and limit soft points via ws-preopen and ws-preclosed sets have been introduced. The basic characterizations and the inferred formulas that connected each other have been scrutinized. Finally, the notions of soft continuity, openness and closeness defined by ws-preopen and ws-preclosed sets have been discussed. Among the unique properties obtained in this study, is that most descriptions of soft continuity have been evaporated for this type of continuity, which is due to the loss of the properties report that “ a soft subset (S, Ω) is ws-preopen iff int_wp (S, Ω) = (S, Ω)" and “ a soft subset (S, Ω) is ws-preclosed iff cl_wp (S, Ω) = (S, Ω)". To illustrate these divergences between this class and other generalizations, some counterexamples have been provided. To avoid this irregular behavior, it is planned to produce another type of soft continuity inspired by ws-preopen sets. Additionally, it is intended to study other topological ideas that can be formulated using this class of soft sets, e.g., covering property and separation axioms.

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgments

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

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