On weakly soft β -open sets and weakly soft β -continuity

Abstract

This work introduces weakly soft β-open subsets, a new family of soft-open sets. By this family, we expand a soft topology to a soft structure which is neither supra-soft topology nor infra-soft topology. The connections between this class of soft sets and other celebrated classes via soft topology are examined with some elucidative examples. Also, it is established some relationships under conditions of extended and hyperconnected soft topologies. Furthermore, the interior and closure operators are structured along with weakly soft β-open and weakly soft β-closed sets. Finally, the class of weakly soft β-continuous functions is introduced and its main characterizations are studied. It is investigated the systematic relationships and findings that are lost for this kind of soft continuity as well as it is shown the conditions required to maintain some of these relationships such as full, extended and hyperconnected soft topologies.

Keywords

Extended soft topology weakly soft β-open set β-closure weakly soft β-interior and weakly soft β-continuous

1 Introduction

This paper studies a new family of generalizations of soft-open sets by introducing the concept of weakly soft β-open sets. Now, the development of soft set theory and soft topologies including generalizations of soft-open sets. What follows are instances that illustrate the current research’s motivations and contributions to the current manuscript:

1.1 A brief review in the soft set theory

The idea of soft sets (S-sets) was presented by Molodtsov [45] to coping with problems emerging from imperfect information. Molodtsov, in his pioneering work [45], explored and described some applications of S-sets in different disciplines. Then, it has been employed to address many real-life problems [32 , 43]. The first attempt to create the frame of S-set theory was carried on by Maji et al. [41]. They formulated some operations over S-sets (soft union and intersection, complement, null and absolute soft sets, etc.) and investigated their basic properties. Then, it was shown some shortcomings of these operations by [6] as well as it was embraced new sorts of these operations in order to be commensurate with their counterparts in the crisp set theory as displayed in [18]. To provide a complete environment to model complicated practical problems, it was integrated S-sets with other instruments of uncertainty such as fuzzy sets [42, 48] and rough sets [3, 43].

1.2 A brief review in soft topology

In 2011, it was used S-sets to construct the concept of soft topological space (STS) by Çağman et al. [26] and Shabir and Naz [49], who established the fundamentals of STSs like soft neighbourhoods, soft interior and closure operators, soft separation, etcetera. In [44], the author gave some descriptions of soft regular spaces and adjusted certain gaps of soft T_i-spaces (i = 2, 3). El-Shafei et al. [29] proposed new types of belong and non-belong relations that associate ordinary points with soft sets. They applied these relations to category STSs with respect to a strong family of soft T_i-spaces. Then, these relations were exploited to establish different types of soft spaces which were applied to address some practical issues [17, 28]. Result of inconsiderate the nature of topology in soft settings, some claimed results appeared they have been amended by [8–10 , 50]. Kharal and Ahmad [37] studied soft functions between STSs, and then Al-shami [12] adopted another approach to soft functions using soft points. Soft continuity, openness, and closedness were introduced in [51]. Aygünoglu and Aygün [25] familiarized the spaces of soft compact and Lindelöf. Hida [34] embraced another type of soft compactness. Recently, it has been studied the concepts of primal topology [16], Menger [38], almost Menger [39], nearly Menger [40] and expandable [47] spaces in soft settings.

Nowadays, it is well known the important of generalizations of soft-open sets to studying soft topological concepts in a wider class. In this line, Chen [27] offered the class of soft semi-open sets. Akdag [2] defined soft b-open sets and probed main features. The authors of [1 , 35] introduced the concept of soft β-open sets and examined soft β-separation axioms. Kandil et al. [36] initiated the notion of soft pre-open. Al-shami and Al-Ghour contributed by soft somewhere dense sets [24] and soft Q-sets [5]. Recently, Al-shami et al. [19, 20] have adopted a new technique inspired from parametric topologies to produce weaker forms of soft α-open and soft semi-open sets. The concepts of covering properties and soft mappings have been extended utilizing some generalizations of soft-open sets like soft regular closed [7] and soft somewhere dense sets [15 , 24].

Some applications of soft topological concepts to real-life problems have been presented. Al-shami [13] employed soft separation axioms in soft weak structures to opt for the most favorable tourism programs. He [11] also made use of soft compactness in ordered setting to expect the missing values of information systems. Moreover, he [14] debated nutrition followed by individuals using the idea of soft somewhat open sets. Topologists have also been drawn to the relationships between topological concepts and properties via classical and soft settings. This first contribution was done by Al-shami and Kočinac [23] who showed this navigation between two specific kinds of STSs namely enriched and extended STSs. Following that, the interchangeability of many topological concepts between classical and STSs was investigated, and the conditions under which they are valid were determined. In spite of many researchers and scholars who have discussed and expanded traditional topological concepts in STSs, paramount additions remain possible. Thus, among topological scholars, the study of STSs is a contemporary theme.

1.3 Motivations and layout of this work

The first reason for producing this paper is to propose a fresh method for generalizing soft topology that is motivated by its traditional topologies. Second, to provide an entirely novel structure for developing soft topological notions like soft operators and continuity. Of naturally, the researchers can use the suggested class of soft-open sets to investigate other concepts such as soft covering characteristics and separation axioms. Finally, different analogs for each traditional topological notion should be developed to emphasize the significance of the soft topological context. We organized the content of this work as following:

(i) Section 2 of this article assembles the key definitions and features needed to wrap up this subject.

(ii) In Section 3, we provide the concept of weakly soft β-open sets as a new generalization of soft-open sets and examine their main characterizations and relationships with the help of examples.

(iii) The concepts of a weakly soft β-limit soft point and the weakly soft β-interior, weakly soft β-closure, weakly soft β-boundary of a set are developed in Section 4 along with the accompanying formulas.

(iv) Weakly soft β-continuity is a concept that is introduced in Section 5 along with several similar definitions.

(v) Finally, conclusions and the potential future direction of this study, which is to apply the newly proposed class to study other topological concepts as well as investigate their features via the structures of supra-topologies and infra-topologies, will be outlined in Section 6.

2 Preliminaries

In the following section, we will review the main ideas of soft sets and soft topologies that the article’s content is based on it.

2.1 Soft sets

Let TDSFTDW≠ ∅ be a universe of objects and let A be the set of all potential parameters under study along with TDSFTDW. Ordinarily, parameters represent attributes, characteristics, or properties of objects in TDSFTDW.

Definition 2.1. [45] An ordered 2-tuple $(C, A)$ is said to be an S-set over TDSFTDW, whereas $C$ is a function given by $C : A ⟶ 2^{TDSFTD W}$ and 2^TDSFTDW is the power set of TDSFTDW. That is, an S-set $(C, A)$ over TDSFTDW is a representation given below, which offers a parameterized set of subsets of TDSFTDW. $(C, A) = {(a, C (a)) : a \in A$ and $C (a) \in 2^{TDSFTD W}}$ ;

where every $C (a)$ is termed an a-component of $(C, A)$ .

In the present work, we shall denote S-sets over TDSFTDW by $(C, A)$ , $(D, A)$ and so on, and denote the family of all S-sets over TDSFTDW along with A by 2^TDSFTDW_A.

Definition 2.2. [6] If $D (a) = TDSFTD W - C (a)$ for all a ∈ A, then we call $(D, A)$ a complement of $(C, A)$ . We use $(C, A)^{c} = (C^{c}, A)$ as a symbol for the complement of $(C, A)$ .

Clearly, $[(C, A)^{c}]^{c} = (C, A)$ .

Definition 2.3. [41, 46] An S-set $(C, A)$ is said to be:

(i) absolute, symbolised by $\tilde{TDSFTD W}$ , if $C (a) = TDSFTD W$ for all a ∈ A.

(ii) null, symbolised by φ, if its complement is an absolute soft set. That is, $C (a) = \emptyset$ for all a ∈ A.

(iii) a soft point, symbolised by TDSFTDw_a, if $C (b) = {\begin{matrix} TDSFTD w, & if b = a; \\ \emptyset, & otherwise . \end{matrix}$ We write $TDSFTD w_{a} \in (C, A)$ if $TDSFTD w \in C (a)$ .

(iv) pseudo constant if for all a ∈ A we have $C (a) = {\begin{matrix} TDSFTD W, or \\ \emptyset \end{matrix}$

Definition 2.4. [31] We say that $(C, A)$ is an S-subset of $(D, A)$ (or $(D, A)$ is an S-superset of $(C, A)$ ), symbolised by $(C, A) \tilde{\subseteq} (D, A)$ , if $C (a) \subseteq D (a)$ for each a ∈ A.

Definition 2.5. [18] Let $(C, A)$ and $(D, A)$ be soft sets. Then:

(i) $(C, A) ⋃^{˜} (D, A) = (F, A),$ where $F (a) = C (a) ⋃ D (a)$ for all a ∈ A.

(ii) $(C, A) ⋂^{˜} (D, A) = (F, A),$ where $F (a) = C (a) ⋂ D (a)$ for all a ∈ A.

(iii) $(C, A) \tilde{\} (D, A) = (F, A),$ where $F (a) = C (a) \ D (a)$ for all a ∈ A.

(iv) $(C, A) \tilde{\times} (D, A) = (F, A),$ where $F (a_{1}, a_{2}) = C (a_{1}) \times D (a_{2})$ for all (a₁, a₂) ∈ A × A.

Subsequently, the amended version of the definition of soft functions is exhibited.

Definition 2.6. [12] Given crisp functions λ : TDSFTDW → TDSFTDX and χ : A → E. A soft function λ_χ from 2^TDSFTDW_A into 2^TDSFTDX_E is defined to be a relation such that each TDSFTDw_a ∈ 2^TDSFTDW_A is linked to one and only one TDSFTDx_e ∈ 2^TDSFTDX_E such that λ_χ (TDSFTDw_a) = λ (TDSFTDw) _χ(a) for all TDSFTDw_a ∈ 2^TDSFTDW_A.

In addition, $λ_{χ}^{- 1} (TDSFTD x_{e}) = \underset{a \in χ^{- 1} (e)}{TDSFTD w \in λ^{- 1} (TDSFTD x) ⋃^{˜}} TDSFTD w_{a}$ for each TDSFTDx_e ∈ 2^TDSFTDX_E.

Under a soft function λ_χ : 2^TDSFTDW_A → 2^TDSFTDX_E, the image and pre-image of $(C, A)$ , $(D, E)$ is respectively calculated by the next formulas: $λ_{χ} (C, A) = {⋃^{˜}}_{TDSFTD w_{a} \in (C, A)} λ_{χ} (TDSFTD w_{a})$ , and

$λ_{χ}^{- 1} (D, E) = {⋃^{˜}}_{TDSFTD x_{e} \in (D, E)} λ_{χ}^{- 1} (TDSFTD x_{e})$ .

If crisp functions λ and χ are surjective, injective, or bijective, so is the soft function λ_χ.

Proposition 2.7. [37] Let λ_χ : 2^TDSFTDW_A → 2^TDSFTDX_E be a soft function. If $(C, A)$ and $(D, E)$ are respectively S-subsets of $\tilde{TDSFTD W}$ and $\tilde{TDSFTD X}$ , then

(i) $(C, A) \tilde{\subseteq} λ_{χ}^{- 1} (λ_{χ} (C, A))$ .

(ii) If λ_χ is injective, then $(C, A) = λ_{χ}^{- 1} (λ_{χ} (C, A))$ .

(iii) $λ_{χ} (λ_{χ}^{- 1} (D, E)) \tilde{\subseteq} (D, E)$ .

(iv) If λ_χ is surjective, then $λ_{χ} (λ_{χ}^{- 1} (D, E)) = (D, E)$

2.2 Soft topology

Definition 2.8. [45] A soft topology on TDSFTDW is a subcollection ϒ of 2^TDSFTDW_A that obeys the next axioms:

(i) ϒ is closed under finite intersections and arbitrary unions;

(ii) $\tilde{TDSFTD W}$ and φ are elements of ϒ.

The triplet (TDSFTDW, ϒ, A) is called a soft topological space (briefly, STS). The terminology of “soft-open set” is given for every member of ϒ and the term of “soft-closed set” is given for an S-set whose complement is a soft-open set.

Definition 2.9. [21] An STS (TDSFTDW, ϒ, A) is said to be full provided that each soft open set that is not null has no empty component.

Proposition 2.10. [45] Let (TDSFTDW, ϒ, A) be an STS. Then we obtain a classical topology (called a parametric topology) for each a ∈ A as follows

$ϒ_{a} = {C (a) : (C, A) \in ϒ}$

Definition 2.11. [45] Let $(C, A)$ be an S-subset of an STS (TDSFTDW, ϒ, A). Then

(i) $◊ (C, A)$ and $□ (C, A)$ are defined the soft interior and closure of $C (a)$ in (TDSFTDW, ϒ_a), respectively.

(ii) $(◊ (C), A)$ and $(□ (C), A)$ are respectively defined by $◊ (C) (a) = ◊ (C (a))$ and $□ (C) (a) = □ (C (a))$ , where $◊ (C (a))$ and $□ (C (a))$ are respectively the interior and closure of $C (a)$ in (TDSFTDW, ϒ_a).

Definition 2.12. [36] An STS (TDSFTDW, ϒ, A) is called soft hyperconnected if TDSFTDW and φ are the only soft-open sets that are soft-closed sets.

Definition 2.13. [25, 46] Let (TDSFTDW, ϒ, A) be an STS. Then

(i) ϒ is said to be an enriched soft topology on TDSFTDW whenever ϒ includes all pseudo constant S-sets.

(ii) ϒ is said to be an extended soft topology whenever it possesses the property that $(C, A) \in ϒ$ iff $C (a) \in ϒ_{a}$ for each a ∈ A.

In [23], it was conducted a comprehensive investigation on two types of soft topologies namely enriched and extended soft topologies. Henceforth, we name such a soft topology an extended soft topology. One of the valuable results demonstrated under this soft topology is the interchangeable property between crisp interior (resp., closure) and soft interior (resp., soft closure) defined over soft topologies and their parametric topologies. The following finding is a crucial step for proving various other results.

Theorem 2.14. [23] An STS (TDSFTDW, ϒ, A) is extended iff $(◊ (C), A) = ◊ (C, A)$ and $(□ (C), A) = □ (C, A)$ for any S-subset $(C, A)$ .

Theorem 2.15. [36] Let $(C, A)$ be an S-subset of an STS (TDSFTDW, ϒ, A). If $(D, A)$ is soft-open, then

$(D, A) ⋂^{˜} □ (C, A) \tilde{\subseteq} □ ((D, A) ⋂^{˜} (C, A))$ .

Definition 2.16. An S-subset $(C, A)$ of (TDSFTDW, ϒ, A) is said to be:

(i) soft pre-open [36] if $(C, A) \tilde{\subseteq} ◊ (□ (C, A))$

(ii) soft semi-open [27] if $(C, A) \tilde{\subseteq} □ (◊ (C, A))$

(iii) soft b-open [2] if $(C, A) \tilde{\subseteq} ◊ (□ (C, A)) ⋃^{˜} □ (◊ (C, A))$ .

(iv) soft β-open [4] if $(C, A) \tilde{\subseteq} □ (◊ (□ (C, A)))$ .

(v) soft somewhere dense [24] if $(C, A) = φ$ or $◊ (□ (C, A)) \neq φ$ .

Definition 2.17. [51] A soft function $λ_{χ} : (TDSFTD W, ϒ_{TDSFTD W}, A) \to (TDSFTD X, S, A)$ is termed soft-continuous whenever $λ_{χ}^{- 1} (C, A)$ is a soft-open set for each soft open $(C, A)$ over TDSFTDW.

Theorem 2.18. [23] If $λ_{χ} : (TDSFTD W, ϒ, A) \to (TDSFTD X, S, A)$ is soft-continuous, then $λ : (TDSFTD W, ϒ_{a}) \to (TDSFTD X, S_{χ (a)})$ is continuous for every a ∈ A.

3 Weakly soft β-open sets and their basic properties

In this segment, we will introduce the notion of weakly soft β-open sets as a new extension of soft-open subsets. We will show that it places between soft β-open and soft somewhere dense sets in an extended soft topological space. Also, we point out that the conditions of full and hyperconnected soft topology are indispensable to preserve some classical relationships. Illustrative examples are built to demonstrate the implementation of some findings and the invalidity of others.

Definition 3.1. An S-subset $(C, A)$ of (TDSFTDW, ϒ, A) is called weakly soft β-open if $C (a) = \emptyset$ for all a ∈ A or $\emptyset \neq C (a) \subseteq □ (◊ (□ (C (a))))$ for some a ∈ A. That is, $(C, A)$ is a null S-set or it contains a non-empty β-open component.

We name the complement of a weakly soft β-open set a weakly soft β-closed set.

Proposition 3.2. A subset $(C, A)$ of (TDSFTDW, ϒ, A) is weakly soft β-closed iff $(C, A) = \tilde{TDSFTD W}$ or $◊ (□ (◊ (C (a)))) \subseteq C (a) \neq TDSFTD W$ for some a ∈ A.

Proof. (⇒): Let $(C, A)$ be a weakly soft β-closed set. Then, $(C^{c}, A) = φ$ or $\emptyset \neq C^{c} (a) \subseteq □ (◊ (□ (C^{c} (a))))$ for some a ∈ A. This means that $(C, A) = \tilde{TDSFTD W}$ or $◊ (□ (◊ (C (a)))) \subseteq C (a) \neq TDSFTD W$ for some a ∈ A, as required.

(⇐): Let $(C, A)$ be an S-set such that $(C, A) = \tilde{TDSFTD W}$ or $◊ (□ (◊ (C (a)))) \subseteq C (a) \neq TDSFTD W$ for some a ∈ A. Then, $(C^{c}, A) = φ$ or $\emptyset \neq C^{c} (a) \subseteq □ (◊ (□ (C^{c} (a))))$ for some a ∈ A. This implies that $(C^{c}, A)$ is weakly soft β-open. Hence, $(C, A)$ is weakly soft β-closed, as required. □ The classes of weakly soft β-open and weakly soft β-closed subsets are not closed under soft union and soft intersection, as the below example illustrates.

Example 3.3. Let A = {a₁, a₂} be a set of parameters defined on the set of real numbers and $ℝ$ , and ϒ be the soft topology on $ℝ$ generated by ${(a_{i}, C (a_{i})) : C (a_{i}) = (x_{i}, y_{i}); x_{i}, y_{i} \in ℝ; x_{i} \leq y_{i}$ and i = 1, 2}. Set $(C, A) = {(a_{1}, (0, 1)), (a_{2}, [0, 1] ⋃ {5})}$ and $(D, A) = {(a_{1}, [0, 1] ⋃ {5}), (a_{2}, (0, 1))}$ over $ℝ$ . One can evidently check that $(C, A)$ and $(D, A)$ are weakly soft β-open sets. On the other hand, their soft union {(a₁, [0, 1] ⋃ {5}) , (a₂, [0, 1] ⋃ {5})} is not a weakly soft β-open set because $[0, 1] ⋃ {5} ⊈ □ (◊ (□ (C (a)))) = [0, 1]$ for each a ∈ A. Also, the S-sets $(F, A) = {(a_{1}, ℝ), (a_{2}, [0, 1] ⋃ {5})}$ and $(G, A) = {(a_{1}, [0, 1] ⋃ {5}), (a_{2}, ℝ)}$ over $ℝ$ are weakly soft β-open sets. But their soft intersection {(a₁, [0, 1] ⋃ {5}) , (a₂, [0, 1] ⋃ {5})} is not a weakly soft β-open set because $[0, 1] ⋃ {5} ⊈ □ (◊ (□ (C (a)))) = [0, 1]$ for each a ∈ A. By taking the complement of the S-sets $(C, A), (D, A), (F, A)$ , and $(G, A)$ , we have seen that the class of weakly soft β-closed sets need not be closed under soft intersections and unions.

Lemma 3.4. Let (TDSFTDW, τ) be a hyperconnected space. Then

(i) Every non-empty β-open subset of (TDSFTDW, τ) is dense.

(ii) Every superset of a β-open subset of (TDSFTDW, τ) is dense.

Proof. To show (i), let M be a non-empty β-open subset of (TDSFTDW, τ). Then, □ (◊ (□ (M))) ≠ ∅. Therefore, ◊ (□ (M)) is a non-empty open subset. It follows by the condition of hyperconnectedness that □ (◊ (□ (M))) = TDSFTDW. Since □ (◊ (□ (M))) ⊆ □ (M), we obtain the desired result.

The proof of (ii) follows from the fact that superset of a dense subset is also dense □

Proposition 3.5. Let $(C, A)$ and $(D, A)$ be respectively soft semi-open and weakly soft β-open subsets of a full and soft hyperconnected STS (TDSFTDW, ϒ, A), then $(C, A) ⋂^{˜} (D, A)$ is weakly soft β-open.

Proof. Assume that $(C, A)$ is a non-null soft semi-open set. Then there is a non-null soft-open subset $(F, A)$ of $(C, A)$ . Since ϒ is full, then $F (a)$ is a non-empty open set for every a ∈ A. Let $(D, A) \neq φ$ be a weakly soft β-open set, then there exists a ∈ A such that $D (a)$ is a non-empty β-open subset of (TDSFTDW, ϒ_a). So, by the condition of soft hyperconnected, we get $D (a)$ is dense. Consequentially, $F (a) ⋂ D (a) \neq \emptyset$ . It follows from the above lemma that $F (a) ⋂ D (a)$ is a non-empty β-open subset of (TDSFTDW, ϒ_a); therefore, its superset $C (a) ⋂ D (a)$ is dense as well. Hence, $(C, A) ⋂^{˜} (D, A)$ is a weakly soft β-open set. □

Corollary 3.6. Let $(C, A)$ be soft-open (soft α-open) and $(D, A)$ be weakly soft β-open subsets of a full and soft hyperconnected STS (TDSFTDW, ϒ, A), then $(C, A) ⋂^{˜} (D, A)$ is weakly soft β-open.

Remark 3.7.

(i)If $(C, A)$ is pseudo constant, then it is a weakly soft β-subset, because $C (a) = \emptyset$ for all a ∈ A or $□ (◊ (□ (C (a)))) = TDSFTD W$ for some a ∈ A.

(ii) Any S-subset $(C, A)$ of (TDSFTDW, T, A) with $C (a) = TDSFTD W$ (resp. $C (a) = \emptyset$ ) is weakly soft β-open (resp. weakly soft β-closed).

The subsequent proposition is intelligible.

Proposition 3.8. Any soft topology is a subclass of weakly soft β-open sets.

According to Example 3.3, $(C, A)$ is weakly soft β-open subset of $(R, ϒ, A)$ but not soft-open. Hence, the converse of Proposition 3.8 is false in general.

A criterion that ensures the relationship between weakly soft β-open sets and soft β-open (or soft α-open) is used in the following statements.

Proposition 3.9. If (TDSFTDW, ϒ, A) is extended, then every soft β-open set is a weakly soft β-open set.

Proof. Let $(C, A) \neq φ$ soft β-open set. Then $(C, A) \tilde{\subseteq}$ $□ (◊ (□ (C, A)))$ . Since ϒ is an extended soft topology, so $C (a) \subseteq □ (◊ (□ (C (a))))$ for all a ∈ A. This means that $(C, A)$ has a non-empty β-open component and hence, $(C, A)$ is weakly soft β-open. □

Corollary 3.10. If (TDSFTDW, ϒ, A) is extended, then the class of soft α-open (soft α-open, soft semi-open, soft pre-open, soft b-open) sets is a subset of the class of weakly soft β-open sets.

Proposition 3.11. If (TDSFTDW, ϒ, A) is extended, then the class of weakly soft β-open sets is a subset of the class of soft somewhere dense sets.

Proof. If $(C, A) \neq φ$ is a weakly soft β-open set, then there $(C, A)$ has a non-empty β-open set as a component. Since ϒ is extended, $◊ (□ (C, A)) \neq \emptyset$ or $□ (◊ (C, A)) \neq \emptyset$ for some a ∈ A. Now, if $◊ (□ (C, A)) \neq \emptyset$ , then $(C, A)$ is soft somewhere dense, and if $□ (◊ (C, A)) \neq \emptyset$ , then we get $◊ (□ (C, A)) \neq \emptyset$ , which means that $(C, A)$ is soft somewhere dense. □

The criterion of “extended soft topology” imposed in Propositions 3.9, 3.11 is irreplaceable, as the following example illustrates:

Example 3.12. Let the universal set be TDSFTDW = {TDSFTDw₁, TDSFTDw₂, TDSFTDw₃} and a set of parameters be the natural numbers set $ℕ$ . We define a soft topology over TDSFTDW with $ℕ$ as follows $ϒ = {φ, \tilde{TDSFTD W}, (F_{ρ}, ℕ) : F_{ρ} (n) = TDSFTD W$ for each $n \in ℕ \ A$ and $F_{ρ} (n) \subseteq {TDSFTD w_{1}}$ for each n ∈ A, where A is any finite subset of $ℕ}$ . This soft topology is not extended because an S-set ${(n, H (n)) : H (n) = {TDSFTD w_{1}}$ for each $n \in ℕ}$ does not belong to ϒ in spite of {TDSFTDw₁} belongs to every parametric topology ϒ_n. Now, notice that an S-set $(C, ℕ) = {(n, C (n)) : C (n) = {TDSFTD w_{2}, TDSFTD w_{3}}$ for each $n \in ℕ}$ is a soft β-open subset of $(TDSFTD W, ϒ, ℕ)$ because it is a soft dense set. In contrast, $□ (◊ (□ (C (n)))) = \emptyset$ for all $n \in ℕ$ , which means that $(C, ℕ)$ is not a weakly soft β-open set. Also, it can be remarked that an S-set $(G, ℕ) = {(n, C (n)) : C (1) = {TDSFTD w_{1}}$ and $C (n) = \emptyset$ for each $n \in ℕ \ {1}}$ is a weakly soft β-open subset of $(TDSFTD W, ϒ, ℕ)$ because $C (1)$ is a non-empty β-open subset of (TDSFTDW, ϒ₁). On the other hand, it is not a soft somewhere dense set as $◊ (□ (C, ℕ)) = φ$ .

To elucidate that the reverse of Propositions 3.9 & 3.11 fail, the following examples are proposed:

Example 3.13. Let TDSFTDW = {TDSFTDw₁, TDSFTDw₂, TDSFTDw₃} be a universe and A = {a₁, a₂} be a parameters set. Take the family ϒ consisting of φ, $\tilde{TDSFTD W}$ and the following S-subsets over TDSFTDW with A

$(C_{1}, A) = {(a_{1}, TDSFTD W), (a_{2}, \emptyset)}$ ;

$(C_{2}, A) = {(a_{1}, \emptyset), (a_{2}, TDSFTD W)}$ ;

$(C_{3}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, \emptyset)}$ ;

$(C_{4}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, {TDSFTD w_{3}})}$ ;

$(C_{5}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{3}})}$ ;

$(C_{6}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, {TDSFTD w_{2}, TDSFTD w_{3}})}$ ;

$(C_{7}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, TDSFTD W)}$ ;

$(C_{8}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, \emptyset)}$ ;

$(C_{9}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{3}})}$ ;

$(C_{10}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{3}})}$ ;

$(C_{11}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{2}, TDSFTD w_{3}})}$ ;

$(C_{12}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, TDSFTD W)}$ ;

$(C_{13}, A) = {(a_{1}, \emptyset), (a_{2}, {TDSFTD w_{3}})}$ ;

$(C_{14}, A) = {(a_{1}, TDSFTD W), (a_{2}, {TDSFTD w_{3}})}$ ;

$(C_{15}, A) = {(a_{1}, \emptyset), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{3}})}$ ;

$(C_{16}, A) = {(a_{1}, TDSFTD W), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{3}})}$ ;

$(C_{17}, A) = {(a_{1}, \emptyset), (a_{2}, {TDSFTD w_{2}, TDSFTD w_{3}})}$ and

$(C_{18}, A) = {(a_{1}, TDSFTD W), (a_{2}, {TDSFTD w_{2}, TDSFTD w_{3}})}$ .

Then, (TDSFTDW, ϒ, A) is an extended STS. Remark that an S-set $(F, A) = {(a_{1}, {TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{2}})}$ is weakly soft β-open because $F (a_{1}) \subseteq □ (◊ (□ (F (a_{1})))) = {TDSFTD w_{2}, TDSFTD w_{3}}$ . While $(F, A)$ is not a soft β-open set as $◊ (□ (◊ (F, A)))) = φ ⊈ (F, A)$ .

Example 3.14. It can be checked that a soft topology given in Example 3.3 is extended. If we take an S-set $(D, A) = {(a_{1}, [1, 3] ⋃ {4}), (a_{2}, \emptyset)}$ , then $(D, A)$ is soft somewhere dense because $◊ (□ (D, A)) = {(a_{1}, (1, 3)), (a_{2}, \emptyset)} \neq φ$ . On the other hand, it is not a weakly soft β-open set because $□ (◊ (□ (D (a_{1})))) = [1, 3] ⊈ D (a_{1})$ and $D (a_{2})$ is empty.

Proposition 3.15. If a soft function $λ_{χ} : (TDSFTD W, ϒ, A) \to (TDSFTD X, S, A)$ is both soft-open and soft continuous, then $λ_{χ} (C, A)$ and $λ_{χ}^{- 1} (C, A)$ are weakly soft β-open.

Proof. To prove the image case, we assume that a soft function $λ_{χ} : (TDSFTD W, ϒ, A) \to (TDSFTD X, S, A)$ soft-open and soft-continuous. Let $(C, A)$ be a weakly soft β-open subset of (TDSFTDW, ϒ, A). Suppose that $C (a)$ is non-empty β-open for some a ∈ A. Set χ (a) = a^*. By Theorem 2.18, $λ : (TDSFTD W, ϒ_{a}) \to (TDSFTD X, S_{χ (a) = a^{*}})$ is continuous. The same holds for open functions. We know that λ (□ (V)) ⊆ □ (λ (V)) by continuity of λ and λ (◊ (V)) ⊆ ◊ (λ (V)) by openness of λ, for each subset V of TDSFTDW. This means that $λ (C (a)) \tilde{\subseteq} λ (□ (◊ (□ (C (a))))$ $\tilde{\subseteq} □ (◊ (□ (λ (C (a)))))$ .

Therefore, $λ (C (a))$ is a non-empty β-open component of $λ_{χ} (C, A)$ ; hence, $λ_{χ} (C, A)$ is a weakly soft β-subset of (TDSFTDW, ϒ, A). □

Proposition 3.16. Let $(C, A)$ and $(D, A)$ be weakly soft β-open subsets. Then, $(C, A) \tilde{\times} (D, A)$ is weakly soft β-open.

Proof. Suppose that $(C, A)$ and $(D, A)$ are weakly soft β-open subsets and let $(F, A \times A) = (C, A) \times (D, A)$ . Then there are a₁, a₂ ∈ A such that $C (a_{1})$ and $D (a_{2})$ are non-empty β-open subsets. Now, (a₁, a₂) ∈ A × A such that $F (a_{1}, a_{2}) = C (a_{1}) \times D (a_{2})$ . It follows from the classical topology that $C (a_{1}) \times D (a_{2})$ is a non-empty β-open subset; so that $F (a_{1}, a_{2})$ is a non-empty β-open set. Thus, $(F, A \times A)$ is a weakly soft β-open subset. □

4 Weakly β-interior and Weakly β-closure operators

This part applies weakly soft β-open and weakly soft β-closed set to study the concepts of interior, limit points, closure, and boundary of an S-set in an STS. We will evince their fundamentals and look at the relationships among them. Also, we elucidate, with the help of some examples, that the weakly β-interior of an S-subset many not be weakly β-open. The same is true for weakly β-closure of S-sets and weakly β-closed set.

Definition 4.1. The weakly β-interior of an S-set $(C, A)$ in (TDSFTDW, ϒ, A) is defined to be the soft union of all weakly soft β-open sets that are included in $(C, A)$ and is symbolized by $◊_{wp} (C, A)$ .

According to Example 3.3, we have $◊_{w β} {(a_{1}, [0, 1] ⋃ {5}), (a_{2}, [0, 1] ⋃ {5})} =$ ${(a_{1}, [0, 1] ⋃ {5}), (a_{2}, [0, 1] ⋃ {5})}$ . It can be remarked that {(a₁, [0, 1] ⋃ {5}) , (a₂, [0, 1] ⋃ {5})} is not a weakly β-open set.

The following propositions have simple proofs, therefore we will skip them.

Proposition 4.2. Let $(C, A)$ be a subset of (TDSFTDW, ϒ, A) and $TDSFTD w_{a} \in \tilde{TDSFTD W}$ . Then $TDSFTD w_{a} \in ◊_{w β} (C, A)$ iff $TDSFTD w_{a} \in (D, A) ⋂^{˜} (C, A)$ for some weakly soft β-open set $(D, A)$ .

Proposition 4.3. Let $(C, A)$ , $(D, A)$ be S-subsets of (TDSFTDW, ϒ, A). Then

$◊_{w β} (C, A) \tilde{\subseteq} (C, A)$ .

if $(C, A) \tilde{\subseteq} (D, A)$ , then $◊_{w β} (C, A) \tilde{\subseteq} ◊_{w β} (D, A)$ .

Corollary 4.4. For any two subsets $(C, A)$ , $(D, A)$ of (TDSFTDW, ϒ, A), the next results hold true:

$◊_{w β} [(C, A) ⋂^{˜} (D, A)] \tilde{\subseteq} ◊_{w β} (C, A) ⋂^{˜} ◊_{w β} (D, A)$ .

$◊_{w β} (C, A) ⋃^{˜} ◊_{w β} (D, A) \tilde{\subseteq} ◊_{w β} [(C, A) ⋃^{˜} (D, A)]$ .

Proof. It naturally arises from the following:

1. $(C, A) ⋂^{˜} (D, A) \tilde{\subseteq} (C, A)$ and $(C, A) ⋂^{˜} (D, A) \tilde{\subseteq} (D, A)$ .

2. $(C, A) \tilde{\subseteq} [(C, A) ⋃^{˜} (D, A)]$ and $(D, A) \tilde{\subseteq} [(C, A) ⋃^{˜} (D, A)]$ □ Let $(C, A) = {(a_{1}, (1, 2)), (a_{2}, [1, 2])}$ , $(D, A) = {(a_{1}, (2, 3)), (a_{2}, [2, 3])}$ , $(F, A) = {(a_{1}, \emptyset), (a_{2}, ℚ)}$ and $(G, A) = {(a_{1}, {5}), (a_{2}, {6, 7})}$ be S-subsets of an STS given in Example 3.3. We observe the next properties:

1. $(G, A) \tilde{⊈} ◊_{w β} (G, A) = φ$ .

2. $◊_{w β} (G, A) \tilde{\subseteq} ◊_{w β} (F, A)$ whereas $(G, A) \tilde{⊈} (F, A)$ .

3. $◊_{w β} [(C, A) ⋂^{˜} (D, A)] = φ$ whereas $◊_{w β} (C, A) ⋂^{˜} ◊_{w β} (D, A) = {(a_{1}, \emptyset), (a_{2}, {2})}$ .

4. $◊_{w β} (F, A) ⋃^{˜} ◊_{w β} (G, A) = {(a_{1}, \emptyset), (a_{2}, ℚ)}$ whereas $◊_{w β} [(F, A) ⋃^{˜} (G, A)] = {(a_{1}, {5}), (a_{2}, ℚ)}$ .

This concludes that the equality of the statements in Proposition 4.3 and Corollary 4.4 may not be hold.

Definition 4.5. The weakly β-closure points of an S-set $(C, A)$ in (TDSFTDW, ϒ, A) is defined to be the intersection of all weakly soft β-closed sets containing $(C, A)$ and is symbolized by $□_{w β} (C, A)$ .

According to Example 3.3, it can be remarked that $□_{w β} {(a_{1}, ℝ \ {1}), (a_{2}, ℝ)} = {(a_{1}, ℝ \ {1}), (a_{2}, ℝ)}$ , whereas {(a₁, [0, 1] ⋃ {5}) , (a₂, [0, 1] ⋃ {5})} is not a weakly β-closed set.

Proposition 4.6. Let $(C, A)$ be an S-set in (TDSFTDW, ϒ, A) and $TDSFTD w_{a} \in \tilde{TDSFTD W}$ . Then $TDSFTD w_{a} \in □_{w β} (C, A)$ iff for each weakly soft β-open set $(D, A)$ contains TDSFTDw_a we have $(D, A) ⋂^{˜} (C, A) \neq φ$ .

Proof. [⇒] Let $TDSFTD w_{a} \in □_{w β} (C, A)$ . Suppose that there is weakly soft β-open set $(D, A)$ containing TDSFTDw_a with $(D, A) ⋂^{˜} (C, A) = φ$ . Then $(C, A) \tilde{\subseteq} (D^{c}, A)$ . Therefore, $□_{w β} (C, A) \tilde{\subseteq} (D^{c}, A)$ . Thus $TDSFTD w_{a} \notin □_{w β} (C, A)$ . This is a contradiction, which means that $(D, A) ⋂^{˜} (C, A) \neq φ$ , as required.

[⇐] Let $(D, A) ⋂^{˜} (C, A) \neq φ$ for each weakly soft β-open set $(D, A)$ contains TDSFTDw_a. Suppose that $TDSFTD w_{a} \notin □_{w β} (C, A)$ . Then there is a soft β-closed set $(F, A)$ containing $(C, A)$ with $TDSFTD w_{a} \notin (F, A)$ . So $TDSFTD w_{a} \in (F^{c}, A)$ and $(F^{c}, A) ⋂^{˜} (C, A) = φ$ . This is a contradiction. Hence, the desired result is obtained. □ Corollary 4.7. If $(C, A) ⋂^{˜} (D, A) = φ$ such that $(C, A)$ is a weakly soft β-open set and $(D, A)$ is an S-set in (TDSFTDW, ϒ, A), then $(C, A) ⋂^{˜} □_{w β} (D, A) = φ$ .

Proposition 4.8. The next statements hold true for an S-set $(C, A)$ in (TDSFTDW, ϒ, A).

$[◊_{w β} (C, A)]^{c} = □_{w β} (C^{c}, A)$ .

$[□_{w β} (C, A)]^{c} = ◊_{w β} (C^{c}, A)$ .

Proof. 1. If $TDSFTD w_{a} \notin [◊_{w β} (C, A)]^{c}$ , then there is a weakly soft β-open set $(D, A)$ with $TDSFTD w_{a} \in (D, A) \tilde{\subseteq} (C, A)$ . Therefore, $(C^{c}, A) ⋂^{˜} (D, A) = φ$ and hence $TDSFTD w_{a} \notin □_{w β} (C^{c}, A)$ . Conversely, if $TDSFTD w_{a} \notin □_{w β} (C^{c}, A)$ one follow the previous steps to show $TDSFTD w_{a} \notin [◊_{w β} (C, A)]^{c}$ .

2. Similar to 1. □ One can easily demonstrate the next result, so we will cancel its proof.

Proposition 4.9. Let $(C, A)$ , $(D, A)$ be S-subsets of (TDSFTDW, ϒ, A). Then

$(C, A) \tilde{\subseteq} □_{w β} (C, A)$ .

if $(C, A) \tilde{\subseteq} (D, A)$ , then $□_{w β} (C, A) \tilde{\subseteq} □_{w β} (D, A)$ .

$□_{w β} [(C, A) ⋂^{˜} (D, A)] \tilde{\subseteq} □_{w β} (C, A) ⋂^{˜} □_{w β} (D, A)$ .

$□_{w β} (C, A) ⋃^{˜} □_{w β} (D, A) \tilde{\subseteq} □_{w β} [(C, A) ⋃^{˜} (D, A)]$ .

To illustrate that the inclusion relations of Proposition 4.3 is proper, let $(C, A) = {(a_{1}, (1, 2)), (a_{2}, {1})}$ , $(D, A) = {(a_{1}, ℝ), (a_{2}, ℝ \ {1})}$ , and $(F, A) = {(a_{1}, ℝ), (a_{2}, (0, 1])}$ be S-subsets of an STS given in Example 3.3. We remark the following properties:

1. $□_{w β} (D, A) = \tilde{ℝ} \tilde{⊄} (D, A)$ .

2. $□_{w β} (F, A) \tilde{\subseteq} □_{w β} (D, A)$ whereas $(F, A) \tilde{⊈} (D, A)$ .

3. $□_{w β} [(C, A) ⋂^{˜} (D, A)] = {(a_{1}, (1, 2)), (a_{2}, \emptyset)}$

whereas $□_{w β} (C, A) ⋂^{˜} □_{w β} (D, A) = {(a_{1}, (1, 2)), (a_{2}, {1})}$ .

Definition 4.10. A soft point TDSFTDw_a is named a weakly β-boundary point of an S-set $(C, A)$ in (TDSFTDW, ϒ, A) if TDSFTDw_a belongs to the complement of $◊_{w β} (C, A) ⋃^{˜} ◊_{w β} (C^{c}, A)$ .

All β-boundary points of $(C, A)$ , denoted by $\partial_{w β} (C, A)$ , is called a weakly β-boundary set.

Proposition 4.11. $\partial_{w β} (C, A) = □_{w β} (C, A) ⋂^{˜} □_{w β} (C^{c}, A)$ for every subset $(C, A)$ of (TDSFTDW, ϒ, A).

Proof. $\partial_{w β} (C, A) = [◊_{w β} (C, A) ⋃^{˜} ◊_{w β} (C^{c}, A)]^{c}$

$= [◊_{w β} (C, A)]^{c} ⋂^{˜} [◊_{w β} (C^{c}, A)]^{c}$ (De Morgan’s law)

$= □_{w β} (C^{c}, A) ⋂^{˜} □_{w β} (C, A)$ (Proposition 4.8(2)) □

Corollary 4.12. For every subset $(C, A)$ of (TDSFTDW, ϒ, A), the following properties hold.

$\partial_{w β} (C, A) = \partial_{w β} (C^{c}, A)$ .

$\partial_{w β} (C, A) = □_{w β} (C, A) \ ◊_{w β} (C, A)$ .

$□_{w β} (C, A) = ◊_{w β} (C, A) ⋃^{˜} \partial_{w β} (C, A)$ .

$◊_{w β} (C, A) = (C, A) \ \partial_{w β} (C, A)$ .

Proof. 1. Obvious.

2. $\partial_{w β} (C, A) = □_{w β} (C, A) ⋂^{˜} □_{w β} (C^{c}, A)$

$= □_{w β} (C, A) \ [□_{w β} (C^{c}, A)]^{c}$ .

By 2 of Proposition 4.8 the required relation is obtained.

3. $◊_{w β} (C, A) ⋃^{˜} \partial_{w β} (C, A) = ◊_{w β} (C, A) ⋃^{˜} [□_{w β} (C, A) \ ◊_{w β} (C, A)] = □_{w β} (C, A)$ .

4. $(C, A) \ \partial_{w β} (C, A) = (C, A) \ [□_{w β} (C, A) \ ◊_{w β} (C, A)]$

$= (C, A) ⋂^{˜} [□_{w β} (C, A) ⋂^{˜} (◊_{w β} (C, A))^{c}]^{c}$

$= (C, A) ⋂^{˜} [(□_{w β} (C, A))^{c} ⋃^{˜} ◊_{w β} (C, A)]$

$= [(C, A) ⋂^{˜} (□_{w β} (C, A))^{c}] ⋃^{˜} [(C, A) ⋂^{˜} ◊_{w β} (C, A)]$

$= ◊_{w β} (C, A)$ . □

Proposition 4.13. For every subsets $(C, A), (D, A)$ of (TDSFTDW, ϒ, A), the following properties hold.

$\partial_{w β} (◊_{w β} (C, A)) \tilde{\subseteq} \partial_{w β} (C, A)$ .

$\partial_{w β} (□_{w β} (C, A)) \tilde{\subseteq} \partial_{w β} (C, A)$ .

Proof. By substituting in the formula No. 3 of Corollary 4.12, the proof follows. □

Proposition 4.14. Let $(C, A)$ be a subset of (TDSFTDW, ϒ, A). Then

$(C, A) = ◊_{w β} (C, A)$ iff $\partial_{w β} (C, A) ⋂^{˜} (C, A) = φ$ .

$(C, A) = □_{w β} (C, A)$ iff $\partial_{w β} (C, A) \tilde{\subseteq} (C, A)$ .

Proof. 1. Suppose that $(C, A) = ◊_{w β} (C, A)$ . Then by 4 of Corollary 4.12, $(C, A) = ◊_{w β} (C, A) = (C, A) \ \partial_{w β} (C, A)$ and hence $\partial_{w β} (C, A) ⋂^{˜} (C, A) = φ$ . Conversely, let $TDSFTD w_{a} \in (C, A)$ . Since $TDSFTD w_{a} \notin \partial_{w β} (C, A)$ and $TDSFTD w_{a} \in □_{w β} (C, A)$ , by 3 of Corollary 4.12, $TDSFTD w_{a} \in ◊_{w β} (C, A)$ . Therefore, $◊_{w β} (C, A) = (C, A)$ , as required.

2. Assume that $(C, A) = □_{w β} (C, A)$ . Then $\partial_{w β} (C, A) = □_{w β} (C, A) ⋂^{˜} □_{w β} (C^{c}, A)$ $\tilde{\subseteq}$ $□_{w β} (C, A) = (C, A)$ , as required. Conversely, if $\partial_{w β} (C, A) \tilde{\subseteq} (C, A)$ , then by 3 of Corollary 4.12, $□_{w β} (C, A) \tilde{\subseteq} ◊_{w β} (C, A) ⋃^{˜} (C, A) = (C, A)$ and hence $□_{w β} (C, A) = (C, A)$ , as required. □

Corollary 4.15. Let $(C, A)$ be a subset of (TDSFTDW, ϒ, A). Then $◊_{w β} (C, A) = (C, A) = □_{w β} (C, A)$ iff $\partial_{w β} (C, A) = φ$ .

Definition 4.16. ttyy1 A soft point TDSFTDw_a is said to be a weakly β-limit point of a subset $(C, A)$ of (TDSFTDW, ϒ, A) if $[(D, A) ∖ TDSFTD w_{a}] ⋂ (C, A) \neq φ$ for each weakly soft β-open set $(D, A)$ containing TDSFTDw_a.

All weakly β-limit points of $(C, A)$ is called a weakly β-derived set and denoted by $l_{w β} (C, A)$ .

Proposition 4.17. Let $(C, A)$ and $(D, A)$ be subsets of (TDSFTDW, ϒ, A). If $(C, A) \tilde{\subseteq} (D, A)$ , then $l_{w β} (C, A) \tilde{\subseteq} l_{w β} (D, A)$ .

Proof. Straightforward by Definition 4.16. □

Corollary 4.18. Consider $(C, A)$ and $(D, A)$ are subsets of (TDSFTDW, ϒ, A). Then:

$l_{w β} [(C, A) ⋂^{˜} (D, A)] \tilde{\subseteq} l_{w β} (C, A) ⋂^{˜} l_{w β} (D, A)$ .

$l_{w β} (C, A) ⋃^{˜} l_{w β} (D, A) \tilde{\subseteq} l_{w β} [(C, A) ⋃^{˜} (D, A)]$ .

Theorem 4.19. Let $(C, A)$ be a subset of (TDSFTDW, ϒ, A), then $□_{w β} (C, A) = (C, A) ⋃^{˜} l_{w β} (C, A)$ .

Proof. The side $(C, A) ⋃^{˜} l_{w β} (C, A) \tilde{\subseteq} □_{w β} (C, A)$ is obvious. To prove the other side let $TDSFTD w_{a} \notin [(C, A) ⋃^{˜} l_{w β} (C, A)]$ . Then $TDSFTD w_{a} \notin (C, A)$ and $TDSFTD w_{a} \notin l_{w β} (C, A)$ . Therefore, there is weakly soft β-open $(D, A)$ containing TDSFTDw_a with $(D, A) ⋂^{˜} (C, A) = φ$ . Thus, $TDSFTD w_{a} \notin □_{w β} (C, A)$ . Hence, we find that $□_{w β} (C, A) = (C, A) ⋃^{˜} l_{w β} (C, A)$ . □

Corollary 4.20. Let $(C, A)$ be a weakly soft β-closed subset of (TDSFTDW, ϒ, A), then $l_{w β} (C, A) \tilde{\subseteq} (C, A)$ .

5 Continuity via weakly soft β-open sets

This part is consecrated to treating with the notion of soft continuity via weakly soft β-open idea. Several characterizations of this type of soft-continuous functions will be studied. Also, we point out that some properties of this kind of soft continuity are disappeared.

Definition 5.1. A soft function λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is said to be weakly soft β-continuous if $λ_{χ}^{- 1} (C, A)$ is weakly soft β-open for each $(C, A) \in ϒ_{TDSFTD} X$ .

One can easily evidence the next result, so we will cancel its proof.

Proposition 5.2. If λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is a weakly soft β-continuous function and ξ_κ : (TDSFTDX, ϒ_TDSFTDX, A) → (TDSFTDY, ϒ_TDSFTDY, A) is a soft-continuous function, then ξ_κ ∘ λ_χ is weakly soft β-continuous.

Proposition 5.3. Let λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) be a soft function. Then

(i) If λ_χ is soft-continuous, then λ_χ is weakly soft β-continuous.

(ii) If λ_χ is soft β-continuous (soft α-continuous, soft semi-continuous, soft pre-continuous, soft b-continuous) such that ϒ_TDSFTDW is extended, then λ_χ is weakly soft β-continuous.

(iii) If λ_χ is weakly soft β-continuous such that ϒ_TDSFTDW is extended, then λ_χ is soft SD-continuous.

Proof. (i): It holds true by Proposition 3.8.

(ii): It holds true by Proposition 3.9 and Corollary 3.10.

(iii): It holds true by Proposition 3.11. □

Theorem 5.4. A soft function λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is weakly soft β-continuous iff $λ_{χ}^{- 1} (C, A)$ is weakly soft β-closed subset for each $(C, A)^{c} \in ϒ_{TDSFTD} X$ .

Proof. ⇒: Let $(C, A)$ be a soft-closed subset of (TDSFTDX, ϒ_TDSFTDX, A). Then $λ_{χ}^{- 1} (C^{c}, A) = \tilde{TDSFTD W} - λ_{χ}^{- 1} (C, A)$ is weakly soft β-open because $(C^{c}, A)$ is soft-open. Thus, $λ_{χ}^{- 1} (C, A)$ is a weakly soft β-closed set.

⇐: Following similar argument of the necessary part.

□

Theorem 5.5. If λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is weakly soft β-continuous, then the next properties are equivalent.

For each soft-open subset $(C, A)$ of (TDSFTDX, ϒ_TDSFTDX, A), $λ_{χ}^{- 1} (C, A) = ◊_{w β} (λ_{χ}^{- 1} (C, A))$ .

For each soft-closed subset $(C, A)$ of (TDSFTDX, ϒ_TDSFTDX, A), $λ_{χ}^{- 1} (C, A) = □_{w β} (λ_{χ}^{- 1} (C, A))$ .

$□_{w β} (λ_{χ}^{- 1} (C, A)) \tilde{\subseteq} λ_{χ}^{- 1} (□ (C, A))$ for each $(C, A) \tilde{\subseteq} \tilde{TDSFTD X}$ .

$λ_{χ} (□_{w β} (D, A)) \tilde{\subseteq} □ (λ_{χ} (D, A))$ for each $(D, A) \tilde{\subseteq} \tilde{TDSFTD W}$ .

$λ_{χ}^{- 1} (◊ (C, A)) \tilde{\subseteq} ◊_{w β} (λ_{χ}^{- 1} (C, A))$ for each $(C, A) \tilde{\subseteq} \tilde{TDSFTD X}$ .

Proof. (1 →2): Suppose that $(C, A)$ is a soft-closed subset of (TDSFTDX, ϒ_TDSFTDX, A). Then $(C^{c}, A)$ is soft-open. Therefore, $λ_{χ}^{- 1} (C^{c}, A) = ◊_{w β} (λ_{χ}^{- 1} (C^{c}, A))$ . According to Proposition 4.8, $λ_{χ}^{- 1} (C, A) = □_{w β} (λ_{χ}^{- 1} (C, A))$ .

(2 →3): For any S-set $(C, A) \tilde{\subseteq} \tilde{TDSFTD X}$ , $λ_{χ}^{- 1} (□ (C, A)) = □_{w β} (λ_{χ}^{- 1} (□ (C, A)))$ . Then

$□_{w β} (λ_{χ}^{- 1} (C, A)) \tilde{\subseteq} □_{w β} (λ_{χ}^{- 1} (□ (C, A))) = λ_{χ}^{- 1} (□ (C, A))$ .

(3 →4): It is obvious that $□_{w β} (D, A) \tilde{\subseteq} □_{w β} (λ_{χ}^{- 1} (λ_{χ} (D, A)))$ for each $(D, A) \tilde{\subseteq} \tilde{TDSFTD W}$ . By 3, we get $□_{w β} (λ_{χ}^{- 1} (λ_{χ} (D, A))) \tilde{\subseteq}$ $λ_{χ}^{- 1} (□ (λ_{χ} (D, A)))$ . Therefore, $λ_{χ} (□_{w β} (D, A) \tilde{\subseteq}$ $λ_{χ} (λ_{χ}^{- 1} (□ (λ_{χ} (D, A))))$ $\tilde{\subseteq}$ $□ (λ_{χ} (D, A))$ .

(4 →5): Let $(C, A)$ be an arbitrary S-set in (TDSFTDX, ϒ_TDSFTDX, A). Then $λ_{χ} (□_{w β} (λ_{χ}^{- 1} (C^{c}, A))$ $\tilde{\subseteq} □ (λ_{χ} (λ_{χ}^{- 1} (C^{c}, A)))$ $\tilde{\subseteq} □ (C^{c}, A)$ . So that, $□_{w β} ((λ_{χ}^{- 1} (C, A))^{c}) \tilde{\subseteq} λ_{χ}^{- 1} ((◊ (C, A))^{c}$ . Hence, $λ_{χ}^{- 1} (◊ (C, A))$ $\tilde{\subseteq} ◊_{w β} (λ_{χ}^{- 1} (C, A))$ .

(5 →1): Suppose that $(C, A)$ is a soft-open subset in (TDSFTDX, ϒ_TDSFTDX, A). By 5, $λ_{χ}^{- 1} (C, A) = λ_{χ}^{- 1} (◊ (C, A)) \tilde{\subseteq}$ $◊_{w β} (λ_{χ}^{- 1} (C, A))$ . But $◊_{w β} (λ_{χ}^{- 1} (C, A)) \tilde{\subseteq} λ_{χ}^{- 1} (C, A)$ , so $λ_{χ}^{- 1} (C, A) = ◊_{w β} (λ_{χ}^{- 1} (C, A))$ , as required. □ The converse of the above theorem fails. To demonstrate that the next example is furnished.

Example 5.6. Let TDSFTDW = {TDSFTDw₁, TDSFTDw₂, TDSFTDw₃} and TDSFTDX = {TDSFTDx₁, TDSFTDx₂, TDSFTDx₃} with A = {a₁, a₂}. Let $ϒ_{TDSFTD} W = {φ, \tilde{TDSFTD W}, (C_{i}, A) : i = 1, 2, 3, 4}$ and $ϒ_{TDSFTD} X = {φ, \tilde{TDSFTD X}, (F, A)}$ be two soft topologies defined on TDSFTDW and TDSFTDX, respectively, with the same set of parameters A, where

$(C_{1}, A) = {(a_{1}, {TDSFTD w_{1}}), (a_{2}, {TDSFTD w_{1}}}$ ;

$(C_{2}, A) = {(a_{1}, {TDSFTD w_{2}}), (a_{2}, {TDSFTD w_{2}}}$ ;

$(C_{3}, A) = {(a_{1}, {TDSFTD w_{1}, TDSFTD w_{2}}), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{2}}}$ ;

$(C_{4}, A) = {(a_{1}, {TDSFTD w_{2}, TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{2}, TDSFTD w_{3}}}$ and

$(F, A) = {(a_{1}, {TDSFTD x_{1}}), (a_{2}, {TDSFTD x_{1}}}$ .

Consider λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is a soft function, where λ : TDSFTDW → TDSFTDX is defined as follows λ (TDSFTDw₁) = λ (TDSFTDw₃) = TDSFTDx₁ and λ (TDSFTDw₂) = TDSFTDx₃.

and χ : A → A is the identity function.

Now, $λ_{χ}^{- 1} (F, A) = {(a_{1}, {TDSFTD w_{1}, TDSFTD w_{3}}), (a_{2}, {TDSFTD w_{1}, TDSFTD w_{3}}}$ which is not a weakly soft β-open subset because ◊ (□ ({TDSFTDw₁, TDSFTDw₃}) = {TDSFTDw₁}. Then λ_χ is not weakly soft β-continuous. On the other hand, $λ_{χ}^{- 1} (φ) = ◊_{w β} (λ_{χ}^{- 1} (φ))$ ; $λ_{χ}^{- 1} (\tilde{TDSFTD X}) = ◊_{w β} (λ_{χ}^{- 1} (\tilde{TDSFTD X}))$ and $λ_{χ}^{- 1} (F, A) = ◊_{w β} (λ_{χ}^{- 1} (F, A))$ , which implies that the characterizations displayed in Theorem 5.5 are satisfied.

In the remaining part of this segment, we define two soft functions namely weakly soft β-open and weakly soft β-closed functions.

Definition 5.7. If the image of each soft-open (resp., soft-closed) set under a soft function λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is weakly soft β-open (resp., weakly soft β-closed), then we call λ_χ a weakly soft β-open (resp., weakly soft β-closed) function.

Theorem 5.8. Let λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) be a soft function and $(C, A)$ be any S-subset of $\tilde{TDSFTD W}$ . Then

If λ_χ is weakly soft β-open, then $λ_{χ} (◊ (C, A)) \tilde{\subseteq}$

$◊_{w β} (λ_{χ} (C, A))$ .

If λ_χ is weakly soft β-closed, then $□_{w β} (λ_{χ} (C, A))$ $\tilde{\subseteq} λ_{χ} (□ (C, A))$ .

Proof. 1. Let $(C, A)$ be an S-subset of $\tilde{TDSFTD W}$ . Then $λ_{χ} (◊ (C, A))$ is a weakly soft β-subset of (TDSFTDX, ϒ_TDSFTDX, A) and so $λ_{χ} (◊ (C, A)) = ◊_{w β} (λ_{χ} (◊ (C, A))) \tilde{\subseteq} ◊_{w β} (λ_{χ} (C, A))$ .

2. Similar to 1. □

Proposition 5.9. The concepts of weakly soft β-open and weakly soft β-closed functions are identical if a soft function is bijective.

Proof. ⇒: Assume that λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) is a bijective soft function and let $(C, A)$ be a weakly soft β-closed subset of (TDSFTDW, ϒ_TDSFTDW, A). Since λ_χ is weakly soft β-open, $λ_{χ} (C^{c}, A)$ is weakly soft β-open. Then $λ_{χ} (C^{c}, A) = (λ_{χ} (C, A))^{c}$ . So that, $λ_{χ} (C, A)$ is a weakly soft β-closed set. Hence, λ_χ is weakly soft β-closed. The converse side is obtained by following a similar argument. □

Proposition 5.10. Let λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) be a weakly soft β-closed function and $\tilde{TDSFTD Y}$ be a soft-closed subset of $\tilde{TDSFTD W}$ . Then λ_χ ∣ _TDSFTDY : (TDSFTDY, ϒ_TDSFTDY, A) → (TDSFTDX, ϒ_TDSFTDX, A) is weakly soft β-closed.

Proof. By taking a soft-closed subset $(C, A)$ of (TDSFTDY, ϒ_TDSFTDY, A) we obtain a soft-closed subset $(D, A)$ of (TDSFTDW, ϒ_TDSFTDW, A) with $(C, A) = (D, A) ⋂^{˜} \tilde{TDSFTD Y}$ . Since $\tilde{TDSFTD Y}$ is a soft-closed subset of (TDSFTDW, ϒ_TDSFTDW, A), then $(C, A)$ is also a soft-closed subset of (TDSFTDW, ϒ_TDSFTDW, A). Since $λ_{χ} ∣_{TDSFTD} Y (C, A) = λ_{χ} (C, A)$ , then $λ_{χ} ∣_{TDSFTD} Y (C, A)$ is a weakly soft β-closed set. Thus, λ_χ ∣ _TDSFTDY is a weakly soft β-closed.

□

Proposition 5.11. The next three statements hold for soft functions λ_χ : (TDSFTDW, ϒ_TDSFTDW, A) → (TDSFTDX, ϒ_TDSFTDX, A) and ξ_κ : (TDSFTDX, ϒ_TDSFTDX, A) → (TDSFTDY, ϒ_TDSFTDY, A).

If λ_χ is soft-open (resp., soft-closed) and ξ_κ is soft j-open (resp., soft j-closed) such that ϒ_TDSFTDY is extended, then ξ_κ ∘ λ_χ is weakly soft β-open (resp., weakly soft β-closed), where j ∈ {α, semi, pre, b, β}.

If ξ_κ ∘ λ_χ is weakly soft β-open (resp., weakly soft β-closed) and λ_χ is soft-continuous surjective, then ξ_κ is weakly soft β-open (resp., weakly soft β-closed).

If ξ_κ ∘ λ_χ is soft-open (resp., soft-open) and ξ_κ is weakly soft β-continuous (resp., weakly soft β-closed) injective, then λ_χ is weakly soft β-open (resp., weakly soft β-closed).

Proof. We suffice by proving the cases outside the parentheses.

1. Without loss of generality, let j = α. Then consider $(C, A) \neq φ$ as a soft-open subset of $\tilde{TDSFTD W}$ . So $λ_{χ} (C, A) \neq φ$ is a soft-open subset of $\tilde{TDSFTD X}$ . Thus, $ξ_{κ} (λ_{χ} (C, A))$ is a soft α-open subset. According to Corollary 3.10, $ξ_{κ} (λ_{χ} (C, A))$ is a weakly soft β-open subset. Hence, ξ_κ ∘ λ_χ is weakly soft β-open.

2. Suppose that $(C, A) \neq φ$ is a soft-open subset of $\tilde{TDSFTD X}$ . Then $λ_{χ}^{- 1} (C, A) \neq φ$ is a soft-open subset of $\tilde{TDSFTD W}$ . Therefore, $(ξ_{κ} \circ λ_{χ}) (λ_{χ}^{- 1} (C, A))$ is a weakly soft β-open subset of $\tilde{TDSFTD X}$ . Since λ_χ is surjective, then $(ξ_{κ} \circ λ_{χ}) (λ_{χ}^{- 1} (C, A)) = ξ_{κ} (λ_{χ} (λ_{χ}^{- 1} (C, A))) = ξ_{κ} (C, A)$ . Thus ξ_κ is weakly soft β-open.

3. Let $(C, A) \neq φ$ be a soft-open subset of $\tilde{TDSFTD W}$ . Then $(ξ_{κ} \circ λ_{χ}) (C, A) \neq φ$ is a soft-open subset of $\tilde{TDSFTD X}$ . Therefore, $ξ_{κ}^{- 1} (ξ_{κ} \circ λ_{χ} (C, A))$ is a weakly soft β-open subset of $\tilde{TDSFTD X}$ . Since ξ_κ is injective, $ξ_{κ}^{- 1} (ξ_{κ} \circ λ_{χ} (C, A)) = (ξ_{κ}^{- 1} ξ_{κ}) (λ_{χ} (C, A)) = λ_{χ} (C, A)$ . Thus, λ_χ is weakly soft β-open. □

6 Conclusion

This article contributes to the area of soft topologies; especially, to the category of generalizations of soft-open sets. We have displayed the concept of “weakly soft β-open sets” as a novel family of extensions of soft-open subsets. This family is created by the classical topologies induced from the original soft topology. We have discussed the main properties of this family and elucidated its relationships with the celebrated existing generalizations. Some of these relationships are invalid in normal case, so we have provided the necessary conditions to keep them. In particular, we have demonstrated that this family lies between soft β-open and soft somewhere dense subsets provided that a given soft topology is extended. Then, we have studied the concepts of interior and closure operators, boundary, and limit soft points with respect to the families of weakly soft β-open and weakly soft β-closed sets. We have derived their main properties and the formulas that could be used to link them. Ultimately, we have studied the concepts of soft continuity, openness, and closeness inspired by weakly soft β-open and weakly soft β-closed sets.

We draw attention to that some characterizations of soft interior and closure operators and soft continuity have been lost for their analogs types introduced herein. For operators, the loss is due to the differences of components of two distinct soft sets that are in charge of possessing the properties of weakly soft β-open or weakly soft β-closed. Whereas, the loss that occurred for properties of the current soft continuity is due to the inability of determining whether the soft set is weakly soft β-open or not by its weakly soft β-interior soft points.

In upcoming work, we plan to to formulate another type of soft continuity induced from weakly soft β-open sets in a way that preserves the characterizations lost herein. We also want to look at how covering properties and separation axioms work in the context of weakly soft β-open sets. Moreover, the concepts proposed in this manuscript can be navigated to some frameworks such as supra-topology and infra-topology, fuzzy soft topology and multi soft topology.

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgments

This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project (PSAU-2022/01/19703)

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