Compactness and connectedness via the class of soft somewhat open sets

1.   Introduction

In our daily life, we face many problems involving uncertainty that we cannot cope with using traditional mathematical tools. To deal with these types of problems, it was proposed some scenarios such as fuzzy sets and soft sets. Molodtsov ^[35], in 1999, provided the concept of soft sets as a new mathematical approach to handle imprecise data and vague situations. Molodtsov elaborated how can be applied soft sets in several disciplines and gave their advantages compared to fuzzy sets. Then, this approach has attracted the attention of a huge number of researchers and scholars who are interested in uncertainty in both theoretical and applied issues. Maji et al. ^[34], in 2002, addressed decision-making problems by making use of soft sets, and then they ^[35] provided the initial conception for a set of operations between soft sets in 2003. Some of these operations suffered deficiencies which motivated some authors to adjust their definitions as well as adopted new types of them for different purposes as illustrated in the published literature ^[6,39].

In 2011, it was introduced two techniques to define a topology over a family of soft sets by Shabir and Naz ^[43] and Çaǧman et al. ^[27]. The difference between their definition was in terms of the form of a set of parameters; Shabir and Naz familiarized a soft topology over a constant set of parameters whereas Çaǧman et al. formulated a soft topology over different subsets of parameters. According to the definition of Shabir and Naz, it can be characterized soft topological concepts in a similar way to their counterparts in classical topology, which promotes and enhances researchers to follow this line.

After the advent of soft topology, many contributions to discuss topological concepts and notions via soft settings have been conducted. The basic topological ideas were put forward by Shabir and Naz ^[43]. Min ^[36] shed light on soft separation axioms and proved that every soft $T_3$ space is soft $T_2$. El-Shafei et al. ^[28] defined new types of belonging and non-belonging relations and applied to present novel kinds of soft $T_i$-spaces. As illustrated in recent studies ^[15,21,44] soft separation axioms are still a hot topic. Compactness is one of the celebrated idea in soft topologies; it was introduced by Aygünoǧlu and Aygün ^[25]. Hida ^[30] benefited from the variety of belonging relations in the soft set theory to initiate new kinds of soft compactness. Al-shami et al.^[14] discussed new sorts of compactness namely, almost soft compact and approximately soft compact spaces. The classical behaviour of compact subset of Hausdorff spaces is invalid via soft topology as demonstrated in ^[28]. Al-jarrah et al. ^[7] and Al-shami et al. ^[16] studied soft compact and Lindelöf spaces via soft regular closed and soft somewhere dense sets. Selection principles and covering properties have been recently explored via soft topologies in ^[18,19,33]. The concept of soft connected spaces was probed in ^[42,46]. Asaad ^[23] scrutinized the main features of soft extremally disconnected spaces. Recently, Al-Ghour and Ameen ^[5] have discussed the idea of maximal soft connected topologies.

Kharal and Ahmad ^[32] formulated the concept of soft mappings by using two crisp mappings, one between the universal set and the other between the sets of parameters. To reduce calculation burden and its difficulty that arises from the definition of soft mappings, Al-shami ^[9] updated this definition by using the soft version of ordinary point. In ^[47,48], the authors studied continuity between soft topological spaces and gave its characterizations. Alcantud ^[1] examined the properties of soft countability axioms. The concept of soft expandable spaces was introduced in ^[40]. Generating soft topologies by operators was the main goal of some articles ^[24,26]. The relationships between soft topological concepts and their classical counterparts were revealed by Al-shami and Kočinac ^[17]. They also researched sufficient conditions that guarantee keep properties between soft topologies and classical topologies. Alcantud ^[2] also discussed the relationships between soft topology and fuzzy topology. Topological-hybrid structures induced from fuzzy (intuitionistic) soft sets were discussed in ^[31,45].

Generalization of soft open sets is an interesting topic via soft topologies; they offered new frameworks to study the topological concepts such as separation axioms, compactness, etc. In ^[20], it was investigated some soft functions via soft $\gamma$-open sets. Al-Ghour ^[3] defined the concept of soft $Q$-sets which contains both soft nowhere dense and soft clopen sets by commutative between soft interior and closure operators. He also applied it to boolean algebra which is of importance to the information theory and theory of probability. He ^[4] also introduced the concepts of soft $\omega$-regular open sets and investigated their main features. In this regard, Ameen et al. ^[22] defined some soft functions via the class of soft somewhat open sets. Then, Al-shami ^[13] applied this class to introduce four types of soft separation axioms with interesting properties. He also analyzed the nutrition systems of individuals and examined their suitability depending on the class of soft somewhat open sets defined on minimal soft structures.

This manuscript is a continuation of the previous works that are based on soft somewhat open sets. After this introduction, it is organized this article as follows. In Section 2, we review the main definitions and results that make this work self-contained. In Section 3, we define the concepts of soft $sw$-compact and soft $sw$-Lindelöf spaces and then display three types of their generalizations. With the aid of interesting example, we elucidate the relationships between them and probed their essential features. We dedicate Section 4 to presenting the concept of soft $sw$-connected spaces and elaborating its main characterizations. We show the equivalence between this concept and soft hyperconnected spaces and discuss the conditions under which this concept is preserved via soft topology and their parametric topologies. Finally, we offer some conclusions and open a door for some future works in Section 5.

2.   Preliminaries

To make this manuscript self-contained, we will mention the concepts and findings introduced in published studies that we need to comprehend the results obtained herein.

2.1.   Soft set theory

Definition 2.1. ^[37] Let $\mathbf{A}$ be any nonempty set of parameters and $2^E$ be a power set of the universal set $E$. If $H:\mathbf{A} \rightarrow \ 2^E$ is a mapping, then the pair $(H, \mathbf{A})$ is called a soft set over $E$ and it can be written as follows: $(H, \mathbf{A}) = \{(a, H(a)):a\in \mathbf{A}$ and $H(a)\in 2^E\}$. Each $H(a)$ is called a component of $(H, \mathbf{A})$.

Through this manuscript, $(H, \mathbf{A})$ will denote a soft set over $E$.

Definition 2.2. ^[37,38] A soft set $(H, \mathbf{A})$ is said to be:

(i) absolute, denoted by $\widetilde{E}$, if $H(a) = E$ for each $a\in\mathbf{A}$;

(ii) null, denoted by $\widetilde{\phi}$, if $H(a) = \emptyset$ for each $a\in\mathbf{A}$;

(iii) a soft point if there are $a\in \mathbf{A}$ and $e\in E$ with $H(a) = \{e\}$ and $H(b) = \emptyset$ for each $b\in\mathbf{A}- \{a\}$. A soft point is briefly denoted by $P^e_a$;

(iv) pseudo constant if $H(a) = E$ or $\emptyset$ for each $a\in \mathbf{A}$;

(v) finite (resp., countable) if all components are finite (resp., countable); otherwise it is called infinite (resp., uncountable).

Definition 2.3. ^[29] We call $(H, \mathbf{A})$ a soft subset of $(F, \mathbf{A})$ (or $(F, \mathbf{A})$ a soft superset of $(F, \mathbf{A})$), denoted by $(H, \mathbf{A})\widetilde{\subseteq}(F, \mathbf{A})$ if $H(a)\subseteq F(a)$ for each $a\in \mathbf{A}$.

Definition 2.4. ^[6] If $F(a) = E - H(a)$ for each $a\in \mathbf{A}$, the we call $(F, \mathbf{A})$ a complement of $(H, \mathbf{A})$. The complement of $(H, \mathbf{A})$ is symbolized by $(H, \mathbf{A})^c = (H^c, \mathbf{A})$.

Definition 2.5. Let $(H, \mathbf{A})$ and $(K, \mathbf{A})$ be soft sets. Then:

(i) $(H, \mathbf{A})\widetilde{\bigcup}(K, \mathbf{A}) = (G, \mathbf{A}),$ where $G(a) = H(a)\bigcup K(a)$ for each $a\in \mathbf{A}$ ^[6].

(ii) $(H, \mathbf{A})\widetilde{\bigcap}(K, \mathbf{A}) = (G, \mathbf{A}),$ where $G(a) = H(a)\bigcap K(a)$, for each $a\in \mathbf{A}$ ^[35].

Definition 2.6. ^[28] Let $(H, \mathbf{A})$ be a soft set and $e\in E$. Then:

(i) $e\in (H, \mathbf{A})$ if $e\in H(a)$ for each $a\in \mathbf{A}$.

(ii) $e\Subset (H, \mathbf{A})$ if $e\in H(a)$ for some $a\in \mathbf{A}$.

Definition 2.7. Let $\Omega:E\rightarrow X$ and $\pi:\mathbf{A}\rightarrow \mathbf{B}$ be two crisp mappings. A soft mapping $\Omega_{\pi}$ from the domain (family of all soft points over $E$ with $\mathbf{A}$) to the codomain (family of all soft points over $X$ with $\mathbf{B}$) is a relation associated each soft point in the domain with one and only one soft point in codomain such that

In addition, $\Omega^{-1}_{\pi}(P^x_b) = \underset{e\in \Omega^{-1}(x)} {\underset{a\in \pi^{-1}(b)}{\sqcup}} P^e_a$ for each $P^x_b\in P(X_\mathbf{B})$.

2.2.   Soft topology

Definition 2.8. ^[43] A family $\mathcal{T}$ of soft sets defined over a universal set $E$ with a parameters set $\mathbf{A}$ which contains absolute soft set $\widetilde{E}$ and null soft set $\widetilde{\phi}$ is said to be a soft topology on $E$ provided that it is closed under arbitrary soft union and finite soft intersection.

We call the triplet $(E, \mathcal{T}, \mathbf{A})$ a soft topological space (briefly, soft$_{TS}$). A member of $\mathcal{T}$ is called soft open and its complement is called soft closed.

Proposition 2.9. ^[43] Let $(E, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$. Then $\mathcal{T}_{a} = \{H(a): (H, \mathbf{A})\in \mathcal{T}\}$ defines a classical topology on $E$ for each $a\in \mathbf{A}$. This topology is called a parametric topology.

Definition 2.10. ^[18] Let $(H, \mathbf{A})$ be a soft subset of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then $(cl(H), \mathbf{A})$ is defined by $cl(H)(a) = cl(H(a))$, where $cl(H(a))$ is the closure of $H(a)$ in $(E, \mathcal{T}_{a})$.

Definition 2.11. ^[25,38] Let $(E, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$. Then $\mathcal{T}$ is said to be:

(i) an enriched soft topology provided that $\mathcal{T}$ contains all pseudo constant soft sets;

(ii) an extended soft topology provided that $(H, \mathbf{A})\in \mathcal{T}$ iff $H(a)\in\mathcal{T}_{a}$ for each $a\in \mathbf{A}$.

It was proved in ^[18] the identity between enriched soft topology and extended soft topology. To unite terminology, we call this type of soft topology an extended soft topology, and call $(E, \mathcal{T}, \mathbf{A})$ an extended soft$_{TS}$. The next theorem is a key point to discovering the behaviours of soft topological concepts and keeping them via classical and soft topologies.

Theorem 2.12. ^[18] A soft subset of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is extended iff $(int(H), \mathbf{A}) = int(H, \mathbf{A})$ and $(cl(H), \mathbf{A}) = cl(H, \mathbf{A})$ for any subset $(H, \mathbf{A})$.

Definition 2.13. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be:

(i) soft compact (resp., soft Lindelöf) ^[25] if every soft open cover of $\widetilde{E}$ has a finite (resp., countable) subcover. If we replace "soft open cover" by "soft clopen cover", then we obtain the concepts of mildly soft compact and mildly soft Lindelöf;

(ii) almost soft compact (resp., almost soft Lindelöf) ^[14] if every soft open cover has a finite (resp., countable) subfamily such that the closure of whose members cover $\widetilde{E}$;

(iii) weakly soft compact (resp., weakly soft Lindelöf) ^[14] if every soft open cover has a finite (resp., countable) subfamily such that its closure covers $\widetilde{E}$.

Definition 2.14. ^[14] A family $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is said to have the finite (resp., countable) intersection property if $\widetilde{\bigcap}_{\rho\in\delta} (H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for any finite (resp., countable) subset $\delta$ of $I$;

Definition 2.15. ^[42] A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ which only contains the absolute and null soft sets as soft open and soft closed subsets is called soft connected.

Definition 2.16. ^[22] A soft subset $(H, \mathbf{A})$ of $(E, \mathcal{T}, \mathbf{A})$ is said to be soft somewhat open (briefly, soft $sw$-open) if $(H, \mathbf{A}) = \phi$ or $int(H, \mathbf{A})\neq \phi$. The complement of a soft $sw$-open set is said to be soft somewhat closed (briefly, soft $sw$-closed).

Definition 2.17. ^[22] Let $(H, \mathbf{A})$ be a soft subset of soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then

(i) The union of all soft somewhat open sets contained in $(H, \mathbf{A})$ is called the soft $sw$-interior and it is denoted by $swint(H, \mathbf{A})$.

(ii) The intersection of all soft $sw$-closed sets containing $(H, \mathbf{A})$ is called the soft $sw$-closure of $(H, \mathbf{A})$ and it is denoted by $swcl(H, \mathbf{A})$.

Definition 2.18. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be $tt$-soft $swT_2$ if for every $e\neq x \in E$, there are disjoint soft $sw$-open sets $(H, \mathbf{A})$ and $(F, \mathbf{A})$ such that $e\in (H, \mathbf{A})$ and $x\in(F, \mathbf{A})$.

Theorem 2.19. ^[13] A subset $(H, \mathbf{A})$ of an extended soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is soft $sw$-open (resp., soft $sw$-closed) iff there is an $sw$-open (resp., $sw$-closed) subset $G$ of $(E, \mathcal{T}_a)$ with $H(a) = G$.

The mentioned-above result describes a unique character of this generalization of soft open sets via parametric topologies. For the other celebrated generalizations of soft open sets, the sufficient part of the above theorem must hold true for all attributes of $\mathbf{A}$. The next example clarifies this fact.

Example 2.20. Let $\mathcal{T} = \{\phi, \widetilde{E}, (G_\rho, \mathbf{A}):\rho = 1, 2, ..., 7\}$ be a soft topology over the universal set $E = \{x, y\}$ with a set of parameters $\mathbf{A} = \{a, b\}$, where

$(G_1, \mathbf{A}) = \{(a, E), (b, \emptyset)\}$;

$(G_2, \mathbf{A}) = \{(a, \emptyset), (b, E)\}$;

$(G_3, \mathbf{A}) = \{(a, \{x\}), (b, \emptyset)\}$;

$(G_4, \mathbf{A}) = \{(a, \emptyset), (b, \{x\})\}$;

$(G_5, \mathbf{A}) = \{(a, \{x\}), (b, \{x\})\}$;

$(G_6, \mathbf{A}) = \{(a, \{x\}), (b, E)\}$, and

$(G_7, \mathbf{A}) = \{(a, E), (b, \{x\})\}$.

It is clear that $(E, \mathcal{T}, \mathbf{A})$ is an extended soft$_{TS}$. For a soft set $(H, \mathbf{A}) = \{(a, \{x\}), (b, \{y\})\}$, we find that $cl(int(H, \mathbf{A})) = int(cl(H, \mathbf{A})) = \{(a, E), (b, \emptyset)\}$ which means that $(H, \mathbf{A})$ is neither soft semi-open nor soft pre-open. However, $\{x\}$ is both semi-open and pre-open subset of $(E, \mathcal{T}_a)$.

Definition 2.21. ^[22] A soft mapping $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (X, \mathcal{T}_{X}, \mathbf{A})$ is said to be soft $sw$-continuous (resp., soft $sw$-irresolute) if $\Omega_\pi^{-1}(H, \mathbf{A})$ is a soft $sw$-open set where $(H, \mathbf{A})$ is a soft open (resp., soft $sw$-open) set.

Theorem 2.22. ^[22] Let $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (Z, \mathcal{T}_{Z}, \mathbf{A})$ be a soft mapping. Then the next properties are identical.

(i) $\Omega_\pi$ is soft $sw$-continuous.

(ii) $\Omega_\pi^{-1}(F, \mathbf{A})$ is soft $sw$-closed for every soft closed subset $(F, \mathbf{A})$ of $(Z, \mathcal{T}_{Z}, \mathbf{A})$.

(iii) $\Omega_\pi(swcl(H, \mathbf{A}))\widetilde{\subseteq} cl(\Omega_\pi(H, \mathbf{A}))$ for each $(H, \mathbf{A})\widetilde{\subseteq} \widetilde{E}$.

3.   Novel types of compactness and Lindelöfness via soft somewhat open sets

In this section, we apply the family of $sw$-open sets to display four novel kinds of soft compact and Lindelöf spaces namely, soft $sw$-compact, soft $sw$-Lindelöf, almost soft $sw$-compact, almost soft $sw$-Lindelöf, weakly soft $sw$-compact, weakly soft $sw$-Lindelöf, mildly soft $sw$-compact and mildly soft $sw$-Lindelöf spaces. We explore the main properties of these spaces and elucidate the relationships between them with the assistance of counterexamples. Also, we supply a complete description for each one of these spaces and investigate them under specific kinds of soft mappings. Finally, we scrutinize how these types of compactness and Lindelöfness navigate between soft topology and their parametric topologies and then discuss the role of full and extended soft topologies in keeping this property.

3.1.   Soft $sw$-compact and soft $sw$-Lindelöf spaces

Definition 3.1. A family of soft $sw$-open subsets of $(E, \mathcal{T}, \mathbf{A})$ is called a soft somewhat open cover (briefly, $sw$-open cover) of $\widetilde{E}$ if $\widetilde{E}$ is a soft subset of this family.

Definition 3.2. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be soft $sw$-compact (resp., soft $sw$-Lindelöf) if for every soft $sw$-open cover $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ of $(E, \mathcal{T}, \mathbf{A})$, there is a finite (resp., countable) subset $\delta$ of $I$ with $\widetilde{E} = \widetilde{\bigcup}_{\rho\in \delta}(H_\rho, \mathbf{A})$.

In the next examples, we present two soft topological spaces, one of them is soft $sw$-compact and the other is not soft $sw$-Lindelöf.

Example 3.3. Let $\mathcal{T} = \{\widetilde{\phi}, (E, \mathbf{A})\widetilde{\subseteq} \widetilde{\mathbb{R}}: (E, \mathbf{A})$ is finite$\}$ be a soft topology on the set of real numbers $\mathbb{R}$, where $\mathbf{A}$ is a finite set of parameters. It is clear that every soft subset of $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is soft $sw$-open iff it is soft open. Then, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact.

Example 3.4. Let $\mathbf{A} = \{a, b\}$ be a set of parameters and $(H, \mathbf{A}) = \{(a, \{1\}), (b, \mathbb{R}\setminus\{1\})\}$ be a soft set over the set of real numbers $\mathbb{R}$. Then $\mathcal{T} = \{\widetilde{\phi}, \widetilde{\mathbb{R}}, (H, \mathbf{A})\}$ is a soft topology on $\mathbb{R}$. Now, $\{\{(a, \{1\}), (b, \mathbb{R})\}, \{(a, \{1, r\}), (b, \mathbb{R}\setminus\{1\})\}:r\in \mathbb{R}\}$ is a soft $sw$-open cover of $\mathbb{R}$. It is clear that this cover has not a countable subcover; hence, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is not a soft $sw$-Lindelöf space.

Proposition 3.5. (i) Every soft $sw$-compact space is soft $sw$-Lindelöf.

(ii) Every soft $sw$-compact (resp., soft $sw$-Lindelöf) space is soft compact (resp., soft Lindelöf).

(iii) The family of soft $sw$-compact (resp., soft $sw$-Lindelöf) subsets is closed under finite (resp., countable) union.

Proof. Straightforward.

By replacing a word "finte" in Example 3.3 by "countable", it follows that the converse of (i) in Proposition 3.5 fails. Also, Example 3.4 elucidates that the converse of (ii) in Proposition 3.5 is not true in general.

Proposition 3.6. Let $(E, \mathcal{T}_1, \mathbf{A})$ and $(E, \mathcal{T}_2, \mathbf{A})$ be soft$_{TSs}$ such that $\mathcal{T}_1\subseteq \mathcal{T}_2$. If $(E, \mathcal{T}_2, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf), then $(E, \mathcal{T}_1, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Proof. Consider $\mathcal{C}$ is a soft $sw$-open cover of $(E, \mathcal{T}_1, \mathbf{A})$. It is clear that every soft $sw$-open subset of $\mathcal{T}_1$ is a soft $sw$-open subset of $\mathcal{T}_2$, so $\mathcal{C}$ is also a soft $sw$-open cover of $(E, \mathcal{T}_2, \mathbf{A})$. By soft $sw$-compactness (resp., soft $sw$-Lindelöfness) of $(E, \mathcal{T}_2, \mathbf{A})$ we obtain a finite (resp., countable) subcover for $\widetilde{E}$, as required.

The indiscrete soft$_{TS}$ defined over the set of real numbers $\mathbb{R}$ with a set of parameters $\mathbf{A} = \{a, b\}$ is soft $sw$-compact. Obviously, this indiscrete soft topology is a subfamily of soft topology given in Example 3.4, which is not a soft $sw$-Lindelöf. Hence, the converse of Proposition 3.6 need not be true.

Proposition 3.7. If a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf), then every soft $sw$-closed subset of $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Proof. Assume that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(F, \mathbf{A})$ which is a soft $sw$-closed subset of $(E, \mathcal{T}, \mathbf{A})$. Now, $\{(H_\rho, \mathbf{A}):\rho\in I\} \widetilde{\bigcup}(H^c, \mathbf{A})$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$ which is a soft $sw$-compact space. Therefore, $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n} (H_\rho, \mathbf{A}) \; \widetilde{\bigcup}(H^c, \mathbf{A})$. Thus, $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho = 1}^{n} (H_\rho, \mathbf{A})$, which ends the proof that $(F, \mathbf{A})$ is soft $sw$-compact. Similarly, it can be proved the case between parentheses.

Corollary 3.8. The soft intersection of soft $sw$-compact (resp., soft $sw$-Lindelöf) and soft $sw$-closed sets is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Theorem 3.9. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf) if and only if $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for every family $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ of soft $sw$-closed sets has a finite (resp., countable) intersection property.

Proof. Necessity: Let $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ be a family of soft $sw$-closed subsets of a soft $sw$-compact space $(E, \mathcal{T}, \mathbf{A})$. Suppose that $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A}) = \widetilde{\phi}$. Then, $\widetilde{E} = \widetilde{\bigcup}_{\rho\in I}(H_\rho^c, \mathbf{A})$. By assumption, $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n} (H_\rho^c, \mathbf{A})$. This means that $\widetilde{\phi} = (\widetilde{\bigcup}_{\rho = 1}^{n} (H^c_\rho, \mathbf{A}))^c = \widetilde{\bigcap}_{\rho = 1}^{n} (H_\rho, \mathbf{A})$. Hence, $\mathcal{C}$ has has a finite intersection property, as required.

Sufficiency: Consider $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ as a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. Then $\widetilde{\phi} = \widetilde{\bigcap}_{\rho\in I}(H_\rho^c, \mathbf{A})$ and so by the finite $sw$-intersection property of $\mathcal{C}$, we obtain $\widetilde{\phi} = \widetilde{\bigcap}_{\rho = 1}^{n} (H_\rho^c, \mathbf{A})$. Thus, $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n} (H_\rho, \mathbf{A})$, which proves that $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact.

Similarly, it can be proved the case between parentheses.

Lemma 3.10. (i) The inverse image of a soft $sw$-open set under a surjective soft $sw$-continuous mapping is soft $sw$-open.

(ii) The image of a soft $sw$-open set under a soft $sw$-open mapping is soft $sw$-open.

Proof. To prove (i), let $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (X, \mathcal{T}_{X}, \mathbf{A})$ be a soft $sw$-continuous mapping and let $(F, \mathbf{A})$ be a soft $sw$-open subset of $(X, \mathcal{T}_{X}, \mathbf{A})$. Then, there is a non-null soft open subset $(G, \mathbf{A})$ of $(Z, \mathcal{T}_{Z}, \mathbf{A})$ such that $(G, \mathbf{A})\widetilde{\subseteq}(F, \mathbf{A})$. Now, $\Omega^{-1}_\pi(G, \mathbf{A})\widetilde{\subseteq}\Omega^{-1}_\pi(F, \mathbf{A})$. By hypothesis, $\Omega^{-1}_\pi(G, \mathbf{A})$ is a non-null soft open subset of $(E, \mathcal{T}_{E}, \mathbf{A})$. Hence, $\Omega^{-1}_\pi(F, \mathbf{A})$ is soft $sw$-open. Following similar technique, it can be proved (ii).

According to the above lemma, we see that the property of being a soft $sw$-open set is a soft topological property. Also, the concepts of soft $sw$-continuity and soft $sw$-irresolute are identical under a surjective soft mapping.

Proposition 3.11. The surjective soft continuous image of a soft $sw$-compact (resp., soft $sw$-Lindelöf) set is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Proof. Consider $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (X, \mathcal{T}_{X}, \mathbf{A})$ is a soft continuous mapping and let $(F, \mathbf{A})$ be a soft $sw$-Lindelöf subset of $(E, \mathcal{T}_{E}, \mathbf{A})$. Let us consider $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $\Omega_\pi(F, \mathbf{A})$. Obviously, $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho\in I}\Omega_\pi^{-1}(H_\rho, \mathbf{A})$. It follows from Lemma 3.10 that $\Omega_\pi^{-1}(H_\rho, \mathbf{A})$ is a soft $sw$-open set for each $\rho\in I$. According to soft $sw$-Lindelöfness of $(F, \mathbf{A})$, there is a countable set $\delta$ such that $(F, \mathbf{A})\widetilde{\subseteq}\widetilde{\bigcup}_{\rho\in \delta} \Omega_\pi^{-1}(H_\rho, \mathbf{A})$. Now, we obtain $\Omega_\pi(F, \mathbf{A})\widetilde{\subseteq}\widetilde{\bigcup}_{\rho\in \delta}\Omega_\pi(\Omega_\pi^{-1}(H_\rho, \mathbf{A}))\widetilde{\subseteq} (H_\rho, \mathbf{A})$, which means that $\Omega_\pi(F, \mathbf{A})$ is soft $sw$-Lindelöf. It can be prove the case of soft $sw$-compact in a similar way.

One can prove the next two propositions following a similar arguments.

Proposition 3.12. The soft $sw$-continuous image of a soft $sw$-compact (resp., soft $sw$-Lindelöf) set is soft compact (resp., soft Lindelöf).

Proposition 3.13. The soft $sw$-irresolute image of a soft $sw$-compact (resp., soft $sw$-Lindelöf) set is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Now, we explore the relationship between soft topology and its parametric topologies with respect to possessing the properties of $sw$-compactness and $sw$-Lindelöfness. First, we demonstrate that possessing the properties of soft $sw$-compactness and $sw$-Lindelöfness by soft topologies leads to possess these properties by their parametric topologies. In fact, this is one of the unique properties of these types of covering properties which is invalid for the other types of compactness and Lindelöfness produced by soft $\alpha$-open, soft semi-open, soft pre-open and soft $b$-open sets.

Theorem 3.14. Every parametric topological space $(E, \mathcal{T}_{a})$ inspired by a soft $sw$-compact (resp., soft $sw$-Lindelöf) space $(E, \mathcal{T}, \mathbf{A})$ is $sw$-compact (resp., $sw$-Lindelöf) for each $a\in \mathbf{A}$.

Proof. Suppose that $\{H_\rho:\rho\in I\}$ is an $sw$-open cover of a parametric topological space $(E, \mathcal{T}_{a})$. Then, for each $\rho\in I$ there exists a nonempty open subset $G_\rho$ of $\mathcal{T}_{a}$ such that $G_\rho\subseteq H_\rho$. Accordingly, for each $\rho\in I$ there exists a soft open subset $(V_\rho, \mathbf{A})$ of $\mathcal{T}$ such that $V_\rho = G_\rho$. Now, defining a family of soft open sets $(W_\rho, \mathbf{A})$ as follows $W_\rho(a) = H_\rho$ and $W_\rho(a') = E$ for $a'\neq a$. Then, $\{(W_\rho, \mathbf{A}):\rho\in I\}$ forms a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. By hypothesis of soft $sw$-compactness of $(E, \mathcal{T}, \mathbf{A})$, we obtain $\widetilde{E} = \bigcup\limits_{\rho = 1}^{n} (W_\rho, \mathbf{A})$. This implies that $E = \bigcup\limits_{\rho = 1}^{n} W_\rho(a) = \bigcup\limits_{\rho = 1}^{n} H_\rho$. Hence, $(E, \mathcal{T}_{a})$ is $sw$-compact, as required. Following similar argument, one prove the case between parentheses.

We point out by the next example that the converse of Theorem 3.14 is false.

Example 3.15. Let $\mathcal{T} = \{\phi, \widetilde{\mathbb{R}}, (G_1, \mathbf{A}), (G_2, \mathbf{A})\}$ be a soft topology over the set of real numbers $\mathbb{R}$ with a set of parameters $\mathbf{A} = \{a, b\}$, where $(G_1, \mathbf{A}) = \{(a, \mathbb{R}), (b, \emptyset)\}$ and $(G_1, \mathbf{A}) = \{(a, \emptyset), (b, \mathbb{R})\}$. Now, both parametric topologies inspired by $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ are the indiscrete topology, so they are $sw$-compact. On the other hand, a soft$_{TS} \; (\mathbb{R}, \mathcal{T}, \mathbf{A})$ is not soft $sw$-Lindelöf.

It should be noted that Theorem 3.14 does not hold true for compactness and Lindelöfness defined by the other kinds of generalizations of soft open sets as the next example elucidates.

Example 3.16. Let $\mathbf{A} = \{a, b\}$ be a set of parameters and $\mathcal{T}$ be a soft topology over the set of real numbers $\mathbb{R}$ consists of the absolute soft sets and all soft sets $(H, \mathbf{A})$ satisfying that $1\not\in H(a)$. It can be noted that a soft subset of $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is soft open iff it is soft pre-open. This automatically leads to that $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is soft pre-compact. On the other hand, a parametric topology $\mathcal{T}_b$ is the discrete topology, so $(\mathbb{R}, \mathcal{T}_b)$ is not pre-Lindelöf.

Second, we investigate under which conditions the converse of Theorem 3.14 holds true.

Definition 3.17. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be full if all components of every non-null soft open set are nonempty.

Proposition 3.18. Let $(H, \mathbf{A})$ be a soft $sw$-open subset of a full soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then, $H(a)$ is an $sw$-open subset of $(E, \mathcal{T}_a)$ for each $a\in \mathbf{A}$.

Proof. Since $(H, \mathbf{A})$ is a soft $sw$-open set, there exists a soft open set $(G, \mathbf{A})$ such that

By hypothesis, $(E, \mathcal{T}, \mathbf{A})$ is a full soft$_{TS}$, so $G(a)$ is a nonempty open subset of $(E, \mathcal{T}_a)$ for each $a\in \mathbf{A}$. It follows from (3.1) that $G(a)\subseteq H(a)$ for each $a\in \mathbf{A}$. This means that $H(a)$ is an $sw$-open subset of $(E, \mathcal{T}_a)$ for each $a\in \mathbf{A}$. Hence, the proof is complete.

To elucidate that the converse of the above proposition fails, we build the next example.

Example 3.19. Let $\mathcal{T} = \{\phi, \widetilde{E}, (G_\rho, \mathbf{A}):\rho = 1, 2, ..., 6\}$ be a soft topology over the universal set $E = \{x, y, z\}$ with a set of parameters $\mathbf{A} = \{a, b\}$, where

$(G_1, \mathbf{A}) = \{(a, \{x\}), (b, \{x\})\}$;

$(G_2, \mathbf{A}) = \{(a, \{y\}), (b, \{y\})\}$;

$(G_3, \mathbf{A}) = \{(a, \{z\}), (b, \{z\})\}$;

$(G_4, \mathbf{A}) = \{(a, \{x, y\}), (b, \{x, y\})\}$;

$(G_5, \mathbf{A}) = \{(a, \{x, z\}), (b, \{x, z\}\}$;

$(G_6, \mathbf{A}) = \{(a, \{y, z\}), (b, \{y, z\})\}$.

It is clear that $(E, \mathcal{T}, \mathbf{A})$ is a full soft$_{TS}$. For a soft set $(H, \mathbf{A}) = \{(a, \{y\}), (b, \{x\})\}$, $int(H(a))\neq \emptyset$ and $int(H(b))\neq \emptyset$. But $(H, \mathbf{A})$ is not soft $sw$-open.

Remark 3.20. Note that a soft$_{TS}$ given in Example 3.15 is extended but not full, whereas a soft$_{TS}$ given in Example 3.19 is full but not extended. This points out that the concepts of extended soft topology and full soft topology are independent of each other. Moreover, if $\mathcal{T}$ is a non-indiscrete soft topology, then a full soft$_{TS}$ is not extended, and an extended soft$_{TS}$ is not full. This means that the equivalence between them is obtained in the case of $\mathcal{T}$ is indiscrete soft topology.

Theorem 3.21. Let $\mathbf{A}$ be a finite (resp., countable) parameter set and $(E, \mathcal{T}, \mathbf{A})$ be a full soft$_{TS}$. Then, all parametric topological spaces $(E, \mathcal{T}_a)$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is $sw$-compact (resp., $sw$-Lindelöf) if and only if $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf).

Proof. We prove the theorem for the compactness property and the Lindelöf property is proved in a similar way.

Necessity: Assume that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$ and let $\mid\mathbf{A}\mid = m$. Then $E = \bigcup\limits_{\rho\in I}H_\rho(a)$ for each $a\in \mathbf{A}$. According to Proposition 3.18, $H_\rho(a)$ is an $sw$-open set for each $a\in \mathbf{A}$. By hypothesis, $(E, \mathcal{T}_a)$ is $sw$-compact for each $a\in \mathbf{A}$, so we obtain $E = \bigcup\limits _{\rho = 1}^{n_1} H_\rho(a_1)$, $E = \bigcup\limits _{\rho = n_1+1}^{n_2} H_\rho(a_2) , \dots, E = \bigcup\limits _{\rho = n_{m-1}+1}^{n_m} H_\rho(a_m)$. This implies that $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n_m} (H_\rho, \mathbf{A})$. Hence, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact.

Sufficiency: Follows from Theorem 3.14.

To see that the condition of "full soft topology" in the above theorem is not superfluous, see, Example 3.15.

Now, we review some properties and results that are invalid for soft $sw$-compact and soft $sw$-Lindelöf spaces. The property reports that every finite (resp., countable) topological space is $sw$-compact (resp., $sw$-Lindelöf) is not valid via soft topologies. Also, the property says that every compact subset of $T_2$-space is closed does not hold true via soft $sw$-compact spaces. It can be illustrated these facts by the next example.

Example 3.22. Let $(E, \mathcal{T}, \mathbb{R})$ be a soft$_{TS}$, where $E = \{x, y\}$ is the universal set, the set of real numbers $\mathbb{R}$ is a set of parameters and $\mathcal{T}$ is the discrete soft topology. Obviously, every parametric topological space $(E, \mathcal{T}_{r})$ inspired by $(E, \mathcal{T}, \mathbb{R})$ is $sw$-compact. On the other hand, a soft$_{TS} \; (E, \mathcal{T}, \mathbb{R})$ is not soft $sw$-Lindelöf.

Example 3.23. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ given in Example 3.19 is a $tt$-soft $swT_2$-space. Now, $\{(a, \{y\}), (b, \{z\})\}$ is a soft $sw$-compact subset of $(E, \mathcal{T}, \mathbf{A})$ but it is not soft $sw$-closed.

The following properties of soft compactness and Lindelöfness and some of their generalizations in soft topologies are invalid for soft $sw$-compact and $sw$-Lindelöf spaces as it can be seen from Example 3.15 in cases of compactness and Lindelöfness, $\alpha$-compactness and $\alpha$-Lindelöfness, and semi-compactness and semi-Lindelöfness.

(i) Let $\mathbf{A}$ be a finite (resp., countable) set of parameters. If $(E, \mathcal{T}_a)$ is compact (resp., Lindelöf) for each $a\in\mathbf{A}$, then $(E, \mathcal{T}, \mathbf{A})$ is soft compact (resp., soft Lindelöf).

(ii) Let $(E, \mathcal{T}, \mathbf{A})$ be an extended soft$_{TS}$. Then,

● if every parametric topological space $(E, \mathcal{T}_{a})$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is compact (resp., $\alpha$-compact, semi-compact, pre-compact, $b$-compact), then $(E, \mathcal{T}, \mathbf{A})$ is soft compact (resp., soft $\alpha$-compact, soft semi-compact, soft pre-compact, soft $b$-compact).

● if every parametric topological space $(E, \mathcal{T}_{a})$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is Lindelöf (resp., $\alpha$-Lindelöf, semi-Lindelöf, pre-Lindelöf, $b$-Lindelöf), then $(E, \mathcal{T}, \mathbf{A})$ is soft Lindelöf (resp., soft $\alpha$-Lindelöf, soft semi-Lindelöf, soft pre-Lindelöf, soft $b$-Lindelöf).

It can be easily proved the next result that associates the concepts of soft $sw$-compact and soft $sw$-Lindelöf spaces with the cardinality number of the universal set and a set of parameters.

Lemma 3.24. An extended soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf) if and only if the universal set $E$ and set of parameters $\mathbf{A}$ are finite (resp., countable).

We close this subsection by the next remark which gives a unique characteristic of soft $sw$-compact and soft $sw$-Lindelöf spaces.

Remark 3.25. Let $(E, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$ such that $E$ is infinite (resp., uncountable) and $\mathcal{T}$ is a non-indiscrete soft topology. If $\mathcal{T}$ contains a finite (resp., countable) proper non-null soft set, then $(E, \mathcal{T}, \mathbf{A})$ is not soft $sw$-compact (resp., not soft $sw$-Lindelöf).

3.2.   Almost soft $sw$-compact and almost soft $sw$-Lindelöf spaces

Definition 3.26. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) if every soft $sw$-open cover has a finite (resp., countable) subfamily such that the soft $sw$-closure of whose members covers $\widetilde{E}$. That is, for every soft $sw$-open cover $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ there is a finite (resp., countable) subset $\delta$ of $I$ with $\widetilde{E} = \widetilde{\bigcup}_{\rho\in \delta} swcl(H_\rho, \mathbf{A})$.

In what follows, we provide two soft topological spaces, one of them is almost soft $sw$-compact and the other is not almost soft $sw$-Lindelöf.

Example 3.27. In a soft$_{TS} \; (\mathbb{R}, \mathcal{T}, \mathbf{A})$ given in Example 3.4, note that every soft $sw$-open cover of $\widetilde{\mathbb{R}}$ contains a soft $sw$-open set $\{(a, \{1\}), (b, \mathbb{R}\setminus\{1\})\}$. Since $swcl(\{(a, \{1\}), (b, \mathbb{R}\setminus\{1\})\}) = \widetilde{\mathbb{R}}$, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact.

Example 3.28. Let $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$, where $\mathbb{R}$ is the set of real numbers, $\mathbf{A} = \{a, b\}$ and $\mathcal{T} = \{\mathbb{\phi}, \mathbb{R}, \{(a, \{1\}), (b, \phi)\}, \{(a, \phi), (b, \{1\})\}, \{(a, \{1\}), (b, \{1\})\}\}$. Note that the family $\{\{(a, \{1, r\}), (b, \phi)\}, \{(a, \phi), (b, \{1, r\})\}: r\in \mathbb{R}\}\}$ forms a soft $sw$-open cover of $\widetilde{\mathbb{R}}$. Note that every member of this cover is soft $sw$-open and soft $sw$-closed set, so $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is not almost soft $sw$-Lindelöf.

The proof of the following result is easy, so we remove it.

Proposition 3.29. (i) Every almost soft $sw$-compact space is almost soft $sw$-Lindelöf.

(ii) Every soft $sw$-compact (resp., soft $sw$-Lindelöf) space is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

(iii) The family of almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) subsets is closed under finite (resp., countable) union.

If we replace the universal set $\mathbb{R}$ in Example 3.28 by the set of natural numbers $\mathbb{N}$, then we obtain a soft$_{TS}$ which is almost soft $sw$-Lindelöf but not almost soft $sw$-compact. Also, it is illustrated in Example 3.27 that a soft$_{TS}$ given in Example 3.4 is almost soft $sw$-compact but this soft$_{TS}$ is not soft $sw$-Lindelöf. This elucidates that the converse of Proposition 3.29 fails.

Proposition 3.30. If a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf), then every soft $sw$-clopen subset of $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

Proof. Let $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ be a soft $sw$-open cover of a soft $sw$-clopen subset $(F, \mathbf{A})$ of $(E, \mathcal{T}, \mathbf{A})$. Then, $\{(H_\rho, \mathbf{A}):\rho\in I\} \widetilde{\bigcup}(H^c, \mathbf{A})$ is a soft $sw$-open cover of an almost soft $sw$-compact space $(E, \mathcal{T}, \mathbf{A})$. By hypothesis, $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n} swcl(H_\rho, \mathbf{A}) \widetilde{\bigcup}(H^c, \mathbf{A})$, which implies that $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho = 1}^{n}swcl(H_\rho, \mathbf{A})$. This finishes the proof that $(F, \mathbf{A})$ is almost soft $sw$-compact. Similarly, it can be proved the case between parentheses.

Corollary 3.31. The soft intersection of almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) and soft $sw$-clopen sets is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

Definition 3.32. A family of soft sets $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is said to have the first kind of finite (resp., countable) $sw$-intersection property if $\widetilde{\bigcap}_{\rho\in \delta} swint(H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for any finite (resp., countable) subset $\delta$ of $I$.

Theorem 3.33. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) if and only if $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for every family $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ of soft $sw$-closed sets has the first kind of finite (resp., countable) $sw$-intersection property.

Proof. We prove the theorem in case of compactness and one can prove the case of Lindelöfness in a similar way.

Necessity: Let us consider $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ as a family of soft $sw$-closed subsets of an almost soft $sw$-compact $(E, \mathcal{T}, \mathbf{A})$. Suppose that $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A}) = \widetilde{\phi}$. Then, $\widetilde{E} = \widetilde{\bigcup}_{\rho\in I}(H_\rho^c, \mathbf{A})$. Therefore, $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n} swcl(H_\rho^c, \mathbf{A})$. Thus, $\widetilde{\phi} = (\widetilde{\bigcup}_{\rho = 1}^{n}swcl(H^c_\rho, \mathbf{A}))^c = \widetilde{\bigcap}_{\rho = 1}^{n} swint(H_\rho, \mathbf{A})$, as required.

Sufficiency: Suppose that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. Then $\widetilde{\phi} = \widetilde{\bigcap}_{\rho\in I}(H_\rho^c, \mathbf{A})$. According to the first kind of finite $sw$-intersection property, we obtain $\widetilde{\phi} = \widetilde{\bigcap}_{\rho = 1}^{n}swint(H_\rho^c, \mathbf{A})$. This automatically means that $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n}swcl(H_\rho, \mathbf{A})$, which proves that $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact.

Theorem 3.34. A soft $sw$-continuous image of an almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) set is almost soft compact (resp., almost soft Lindelöf).

Proof. Consider $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (Z, \mathcal{T}_{Z}, \mathbf{A})$ is a soft $sw$-continuous mapping and let $(F, \mathbf{A})$ be an almost soft $sw$-Lindelöf subset of $(E, \mathcal{T}, \mathbf{A})$. Suppose that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft open cover of $\Omega_\pi(F, \mathbf{A})$. Obviously, $\Omega_\pi^{-1}(H_\rho, \mathbf{A})$ is a soft $sw$-open set for each $\rho\in I$ such that $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho\in I}\Omega_\pi^{-1}(H_\rho, \mathbf{A})$. By hypothesis of almost soft $sw$-Lindelöfness of $(F, \mathbf{A})$ there is a countable set $\delta$ with $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho\in \delta} swcl(\Omega_\pi^{-1}(H_\rho, \mathbf{A}))$. Now, $\Omega_\pi(F, \mathbf{A})\widetilde{\subseteq}\widetilde{\bigcup}_{\rho\in \delta}\Omega_\pi(swcl(\Omega_\pi^{-1}(H_\rho, \mathbf{A})))$; it follows from (iii) of Theorem 2.22 that $\Omega_\pi(swcl(\Omega_\pi^{-1}(H_\rho, \mathbf{A}))) \; \widetilde{\subseteq}cl(\Omega_\pi(\Omega_\pi^{-1}(H_\rho, \mathbf{A}))) \widetilde{\subseteq}cl(H_\rho, \mathbf{A})$. Hence, $\Omega_\pi (F, \mathbf{A})\widetilde{\subseteq} \; \widetilde{\bigcup}_{\rho\in \delta}cl(H_\rho, \mathbf{A})$ which proves that $\Omega_\pi(F, \mathbf{A})$ is almost soft Lindelöf.

Following similar technique, the case of compactness is proved.

Corollary 3.35. The soft $sw$-irresolute image of an almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) set is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

Definition 3.36. For each soft subset $(H, \mathbf{A})$ of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$, define $(swcl(H), \mathbf{A})$ as $swcl(H)(a) = swcl(H(a))$, where $swcl(H(a))$ is the $sw$-closure of $H(a)$ in a parametric topological space $(E, \mathcal{T}_a)$ for each $a\in \mathbf{A}$.

Lemma 3.37. Let $(H, \mathbf{A})$ be an extended soft subset of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then $swcl(H, \mathbf{A}) = (swcl(H), \mathbf{A})$.

Proof. It is clear that $(H, \mathbf{A})\widetilde{\subseteq} (swcl(H), \mathbf{A})$. Since $\mathcal{T}$ is extended, $(swcl(H), \mathbf{A})$ is a soft $sw$-closed set. As it is well known that $swcl(H, \mathbf{A})$ is the smallest soft $sw$-closed set containing $(H, \mathbf{A})$, so $swcl(H, \mathbf{A})\widetilde{\subseteq}(swcl(H), \mathbf{A})$.

Note that $(E, \mathcal{T}, \mathbf{A})$ given in Example 2.20 is an extended soft$_{TS}$. For a soft set $(H, \mathbf{A}) = \{(a, \{x\}), (b, \{y\})\}$, we find that $swcl(H, \mathbf{A}) = (H, \mathbf{A})$ whereas $(swcl(H), \mathbf{A}) = \{(a, E), (b, \{y\})\}$. This confirms that the converse of the above lemma need not be true in general.

Theorem 3.38. Let $(E, \mathcal{T}, \mathbf{A})$ be extended. Then, every parametric topological space $(E, \mathcal{T}_{a})$ inspired by almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) space $(E, \mathcal{T}, \mathbf{A})$ is almost $sw$-compact (resp., almost $sw$-Lindelöf) for each $a\in \mathbf{A}$.

Proof. Let $\{H_\rho:\rho\in I\}$ be an $sw$-open cover of a parametric topological space $(E, \mathcal{T}_{a})$. Consider a family of soft sets $\{(V_\rho, \mathbf{A}):\rho\in I\}$, where $V_\rho(a) = H_\rho$ and $V_\rho(a') = E$ for $a'\neq a$. Obviously, this family forms a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. By almost soft $sw$-compactness of $(E, \mathcal{T}, \mathbf{A})$, we obtain $\widetilde{E} = \bigcup\limits_{\rho = 1}^{n} swcl(V_\rho, \mathbf{A})$. Since $\mathcal{T}$ is extended, it follows from Lemma 3.37 that $swcl(V_\rho, \mathbf{A})\widetilde{\subseteq}(swcl(V_\rho), \mathbf{A})$. So, $\widetilde{E} = \bigcup\limits_{\rho = 1}^{n}(swcl(V_\rho), \mathbf{A})$. Thus, $E = \bigcup\limits_{\rho = 1}^{n} swcl(V_\rho(a)) = \bigcup\limits_{\rho = 1}^{n} swcl(H_\rho)$ which ends the proof that $(E, \mathcal{T}_{a})$ is almost $sw$-compact. One can prove the case between parentheses following similar technique.

Example 3.22 shows that the converse of Theorem 3.38 is false in general. In the next example, we clarify that the topological condition of extended in the mentioned-above theorem is indispensable.

Example 3.39. Let $\mathbf{A} = \{a, b\}$ be a set of parameters and $\mathcal{T}$ be a soft topology over the set of real numbers $\mathbb{R}$ consists of the null soft sets and all soft sets $(H, \mathbf{A})$ satisfying that $1\in H(a)$. Now, $\{(a, \emptyset), (b, \{2\})\}\not\in \mathcal{T}$ despite $\emptyset\in \mathcal{T}_a$ and $\{2\}\not\in \mathcal{T}_b$, so $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is not extended. Note that every soft $sw$-open cover of $\widetilde{\mathbb{R}}$ contains a soft $sw$-open set containing $\{(a, \{1\}), (b, \emptyset)\}$. Since $swcl(\{(a, \{1\}), (b, \emptyset)\}) = \widetilde{\mathbb{R}}$, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact. On the other hand, a parametric topology $\mathcal{T}_b$ is the discrete topology, so $(\mathbb{R}, \mathcal{T}_b)$ is not almost $sw$-Lindelöf.

Lemma 3.40. Let $(H, \mathbf{A})$ be a soft subset of a full soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then $(swcl(H), \mathbf{A})\widetilde{\subseteq} swcl(H, \mathbf{A})$.

Proof. Let $P^e_a\not\in swcl(H, \mathbf{A})$. Then, we can find a soft $sw$-open set $(G, \mathbf{A})$ containing $P^e_a$ such that $(G, \mathbf{A})\widetilde{\bigcap} (H, \mathbf{A}) = \phi$. This means that $G(a)\cap H(a) = \emptyset$ for each $a\in \mathbf{A}$. Since $\mathcal{T}$ is full, $G(a)$ is an $sw$-open subset of $\mathcal{T}_a$. This implies that $G(a)\cap swcl(H(a)) = \emptyset$. Therefore, $P^e_a\not\in(swcl(H), \mathbf{A})$. Hence, $(swcl(H), \mathbf{A})\widetilde{\subseteq} swcl(H, \mathbf{A})$, as required.

To point out that the converse of the above lemma is generally false, consider a full soft$_{TS}$ displayed in Example 3.19 and take a soft set $(H, \mathbf{A}) = \{(a, \{y\}), (b, \{x\})\}$. Now, $swcl(H, \mathbf{A}) = \{(a, \{x, y\}), (b, \{x, y\})\}$ but $(swcl(H), \mathbf{A}) = \{(a, \{y\}), (b, \{x\})\}$.

Theorem 3.41. Let $\mathbf{A}$ be a finite (resp., countable) parameter set and $(E, \mathcal{T}, \mathbf{A})$ be a full soft$_{TS}$. Then, all parametric topological spaces $(E, \mathcal{T}_a)$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is almost $sw$-compact (resp., almost $sw$-Lindelöf) if and only if $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

Proof. We prove the theorem for the compactness property and the Lindelöf property is proved in a similar way.

Necessity: Assume that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$ and let $\mid\mathbf{A}\mid = m$. Then $E = \bigcup\limits_{\rho\in I}H_\rho(a)$ for each $a\in \mathbf{A}$. According to Proposition 3.18, $H_\rho(a)$ is an $sw$-open set for each $a\in \mathbf{A}$. By hypothesis, $(E, \mathcal{T}_a)$ is almost $sw$-compact for each $a\in \mathbf{A}$, so we obtain $E = \bigcup\limits _{\rho = 1}^{n_1} swcl(H_\rho(a_1))$, $E = \bigcup\limits _{\rho = n_1+1}^{n_2} swcl(H_\rho(a_2)) , \dots, E = \bigcup\limits _{\rho = n_{m-1}+1}^{n_m} swcl(H_\rho(a_m))$. This implies that $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n_m} (swcl(H_\rho), \mathbf{A})$. According to Lemma 3.40, we obtain $(swcl(H), \mathbf{A})\widetilde{\subseteq} swcl(H, \mathbf{A})$, so $\widetilde{E} = \widetilde{\bigcup}_{\rho = 1}^{n_m} swcl(H_\rho, \mathbf{A})$. Hence, $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact.

Sufficiency: Follows from Theorem 3.38.

The next lemma associates these types of spaces with the cardinality numbers of the universal set and a set of parameters.

Lemma 3.42. An extended soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) if and only if the universal set $E$ and set of parameters $\mathbf{A}$ are finite (resp., countable).

Proof. The necessary part follows from the fact that all soft sets which their components are the universal set or empty set are soft $sw$-open and soft $sw$-closed sets. The sufficient part is obvious.

Finally, we draw attention of the readers to that Remark 3.25 does not hold true for almost soft $sw$-compact and almost soft $sw$-Lindelöf spaces as illustrated in the next example.

Example 3.43. Let $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$, where $\mathbb{R}$ is the set of real numbers, $\mathbf{A} = \{a, b\}$ and $\mathcal{T} = \{\mathbb{\phi}, \mathbb{R}, \{(a, \{1\}), (b, \phi)\}\}$. Then, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ satisfies the conditions of a soft$_{TS}$ given in Remark 3.25. On the other hand, note that $\{(a, \{1\}), (b, \phi)$ is a soft subset of a member of any soft $sw$-open cover of $\widetilde{\mathbb{R}}$. Since $swcl(\{(a, \{1\}), (b, \phi)) = \widetilde{\mathbb{R}}$, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact.

3.3.   Weakly soft $sw$-compact and weakly soft $sw$-Lindelöf spaces

Definition 3.44. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) if every soft $sw$-open cover has a finite (resp., countable) subfamily such that its soft $sw$-closure covers $\widetilde{E}$. That is, for every soft $sw$-open cover $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ there is a finite (resp., countable) subset $\delta$ of $I$ with $\widetilde{E} = swcl(\widetilde{\bigcup}_{\rho\in \delta }(H_\rho, \mathbf{A}))$.

In what follows, we provide two soft topological spaces, one of them is weakly soft $sw$-compact and the other is not weakly soft $sw$-compact.

Example 3.45. In a soft$_{TS} \; (\mathbb{R}, \mathcal{T}, \mathbf{A})$ given in Example 3.28, note that every soft $sw$-open cover of $\widetilde{\mathbb{R}}$ contains a soft $sw$-open set $\{(a, \{1\}), (b, \{1\})\}$. Since $swcl(\{(a, \{1\}), (a, \{1\})\}) = \widetilde{\mathbb{R}}$, $(\mathbb{R}, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact.

Example 3.46. Let $\mathcal{T}$ be the soft discrete over the set of real numbers $\mathbb{R}$ with nay set of parameters $\mathbf{A}$. Then, a soft$_{TS} \; (\mathbb{R}, \mathcal{T}, \mathbf{A})$ is not weakly soft $sw$-Lindelöf.

It can be easily prove the next result, so we remove it.

Proposition 3.47. (i) Every weakly soft $sw$-compact is weakly soft $sw$-Lindelöf.

(ii) Every almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) space is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf).

(iii) The family of weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) subsets is closed under finite (resp., countable) union.

If we replace the universal set $\mathbb{R}$ in Example 3.46 by any infinite countable set, then we obtain a soft$_{TS}$ which is weakly soft $sw$-Lindelöf but not weakly soft $sw$-compact. Also, it follows from illustration given in Example 3.45 that a soft$_{TS}$ given in Example 3.28 is weakly soft $sw$-compact but this soft$_{TS}$ is not almost soft $sw$-Lindelöf. Hence, the converse of Proposition 3.47 is incorrect in general.

Definition 3.48. A family of soft sets $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is said to have the second kind of finite (resp., countable) $sw$-intersection property if $swint(\widetilde{\bigcap}_{\rho\in \delta}(H_\rho, \mathbf{A}))\neq\widetilde{\phi}$ for any finite (resp., countable) subset $\delta$ of $I$.

Theorem 3.49. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) if and only if $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for every family $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ of soft $sw$-closed sets has the second kind of finite (resp., countable) $sw$-intersection property.

Proof. We prove the theorem in case of compactness and one can prove the case of Lindelöfness in a similar way.

Necessity: Let us consider $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ as a family of soft $sw$-closed subsets of a weakly soft $sw$-compact $(E, \mathcal{T}, \mathbf{A})$. Suppose that $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A}) = \widetilde{\phi}$. Then, $\widetilde{E} = \widetilde{\bigcup}_{\rho\in I}(H_\rho^c, \mathbf{A})$. Therefore, $\widetilde{E} = swcl(\widetilde{\bigcup}_{\rho = 1}^{n}(H_\rho^c, \mathbf{A}))$. Thus, $\widetilde{\phi} = (swcl(\widetilde{\bigcup}_{\rho = 1}^{n}(H^c_\rho, \mathbf{A})))^c = swint(\widetilde{\bigcap}_{\rho = 1}^{n}(H_\rho, \mathbf{A}))$, as required.

Sufficiency: Suppose that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. Then $\widetilde{\phi} = \widetilde{\bigcap}_{\rho\in I}(H_\rho^c, \mathbf{A})$. According to the second kind of finite $sw$-intersection property, we obtain $\widetilde{\phi} = swint(\widetilde{\bigcap}_{\rho = 1}^{n}(H_\rho^c, \mathbf{A}))$. This automatically means that $\widetilde{E} = swcl(\widetilde{\bigcup}_{\rho = 1}^{n}(H_\rho, \mathbf{A}))$, which proves that $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact.

Theorem 3.50. A soft $sw$-continuous image of a weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) set is weakly soft compact (resp., weakly soft Lindelöf).

Proof. Consider $\Omega_\pi: (E, \mathcal{T}_{E}, \mathbf{A})\rightarrow (Z, \mathcal{T}_{Z}, \mathbf{A})$ is a soft $sw$-continuous mapping and let $(F, \mathbf{A})$ be a weakly soft $sw$-Lindelöf subset of $(E, \mathcal{T}, \mathbf{A})$. Suppose that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft open cover of $\Omega_\pi(F, \mathbf{A})$. Obviously, $\Omega_\pi^{-1}(H_\rho, \mathbf{A})$ is a soft $sw$-open set for each $\rho\in I$ such that $(F, \mathbf{A})\widetilde{\subseteq} \widetilde{\bigcup}_{\rho\in I}\Omega_\pi^{-1}(H_\rho, \mathbf{A})$. By hypothesis of weakly soft $sw$-Lindelöfness of $(F, \mathbf{A})$ there is a countable set $\delta$ with $(F, \mathbf{A})\widetilde{\subseteq} swcl(\widetilde{\bigcup}_{\rho\in \delta} (\Omega_\pi^{-1}(H_\rho, \mathbf{A})))$. Now, $\Omega_\pi(F, \mathbf{A})\widetilde{\subseteq}\Omega_\pi(swcl(\widetilde{\bigcup}_{\rho\in \delta}(\Omega_\pi^{-1}(H_\rho, \mathbf{A})))$; it follows from (iii) of Theorem 2.22 that $\Omega_\pi(swcl(\widetilde{\bigcup}_{\rho\in \delta}\Omega_\pi^{-1}(H_\rho, \mathbf{A}))) \; \widetilde{\subseteq}cl(\widetilde{\bigcup}_{\rho\in \delta}\Omega_\pi(\Omega_\pi^{-1}(H_\rho, \mathbf{A}))) \widetilde{\subseteq}cl(\widetilde{\bigcup}_{\rho\in \delta}(H_\rho, \mathbf{A})$. Hence, $\Omega_\pi (F, \mathbf{A})\widetilde{\subseteq} \; \widetilde{\bigcup}_{\rho\in \delta}cl(H_\rho, \mathbf{A})$ which proves that $\Omega_\pi(F, \mathbf{A})$ is weakly soft Lindelöf.

Following similar technique, the case of compactness is proved.

Corollary 3.51. The soft $sw$-irresolute image of a weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) set is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf).

Theorem 3.52. Let $(E, \mathcal{T}, \mathbf{A})$ be extended. Then, every parametric topological space $(E, \mathcal{T}_{a})$ inspired by weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) space $(E, \mathcal{T}, \mathbf{A})$ is weakly $sw$-compact (resp., weakly $sw$-Lindelöf) for each $a\in \mathbf{A}$.

Proof. Let $\{H_\rho:\rho\in I\}$ be an $sw$-open cover of a parametric topological space $(E, \mathcal{T}_{a})$. Consider a family of soft sets $\{(V_\rho, \mathbf{A}):\rho\in I\}$, where $V_\rho(a) = H_\rho$ and $V_\rho(a') = E$ for $a'\neq a$. Obviously, this family forms a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$. By weakly soft $sw$-compactness of $(E, \mathcal{T}, \mathbf{A})$, we obtain $\widetilde{E} = swcl(\bigcup\limits_{\rho = 1}^{n} (V_\rho, \mathbf{A}))$. Since $\mathcal{T}$ is extended, it follows from Lemma 3.37 that $E = swcl(\bigcup\limits_{\rho = 1}^{n} (V_\rho(a))) = swcl(\bigcup\limits_{\rho = 1}^{n}(H_\rho))$. Hence, $(E, \mathcal{T}_{a})$ is weakly $sw$-compact. One can prove the case between parentheses following similar technique.

Example 3.22 sets forth that the converse of Theorem 3.52 is false, and Example 3.39 demonstrates that the topological condition of extended in the mentioned-above result is indispensable.

Theorem 3.53. Let $\mathbf{A}$ be a finite (resp., countable) parameter set and $(E, \mathcal{T}, \mathbf{A})$ be a full soft$_{TS}$. If all parametric topological spaces $(E, \mathcal{T}_a)$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is weakly $sw$-compact (resp., weakly $sw$-Lindelöf), then $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf).

Proof. We prove the theorem for the compactness property and the Lindelöf property is proved in a similar way. To do this, assume that $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ is a soft $sw$-open cover of $(E, \mathcal{T}, \mathbf{A})$ and let $\mid\mathbf{A}\mid = m$. Then $E = \bigcup\limits_{\rho\in I}H_\rho(a)$ for each $a\in \mathbf{A}$. According to Proposition 3.18, $H_\rho(a)$ is an $sw$-open set for each $a\in \mathbf{A}$. By hypothesis, $(E, \mathcal{T}_a)$ is weakly $sw$-compact for each $a\in \mathbf{A}$, so we obtain $E = swcl(\bigcup\limits _{\rho = 1}^{n_1} (H_\rho(a_1)))$, $E = swcl(\bigcup\limits _{\rho = n_1+1}^{n_2} (H_\rho(a_2))) , \dots, E = swcl(\bigcup\limits _{\rho = n_{m-1}+1}^{n_m} (H_\rho(a_m))$. This implies that $\widetilde{E} = (swcl(\widetilde{\bigcup}_{\rho = 1}^{n_m} (H_\rho)), \mathbf{A})$. According to Lemma 3.40, we obtain $\widetilde{E} = swcl(\widetilde{\bigcup}_{\rho = 1}^{n_m} (H_\rho, \mathbf{A}))$. Hence, $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact.

The next result points out the cardinality numbers of the universal set and a set of parameters in the weakly soft $sw$-compact and weakly soft $sw$-Lindelöf spaces.

Lemma 3.54. An extended soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf) if and only if the universal set $E$ and set of parameters $\mathbf{A}$ are finite (resp., countable).

Proof. The necessary part follows from the fact that all soft sets which their components are the universal set or empty set are soft $sw$-open and soft $sw$-closed sets. The sufficient part is obvious.

3.4.   Mildly soft $sw$-compact and mildly soft $sw$-Lindelöf spaces

Definition 3.55. We call a soft subset of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ a soft $swoc$-set if it is both soft $sw$-open and soft $sw$-closed.

Definition 3.56. We call a family of soft $swoc$-subsets of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ a soft $swoc$-cover of $\widetilde{E}$ if it covers $\widetilde{E}$.

Definition 3.57. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is said to be mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) if every soft $swoc$-cover has a finite (resp., countable) subcover.

In a soft$_{TS}$ displayed in Example 3.4, it can be noted that the only soft $swoc$-sets are absolute and null soft sets. So, this soft$_{TS}$ is weakly soft $sw$-compact. Correspondingly, in a soft$_{TS}$ displayed in Example 3.28, it is illustrated that $\{\{(a, \{1, r\}), (b, \phi)\}, \{(a, \phi), (b, \{1, r\})\}: r\in \mathbb{R}\}\}$ is a soft $swoc$-cover of the universal set which has not a countable subcover. So, this soft$_{TS}$ is not weakly soft $sw$-Lindelöf. These two examples clarify the existence and uniqueness of weakly soft $sw$-compact and weakly soft $sw$-Lindelöf spaces.

Proposition 3.58. (i) Mildly soft $sw$-compact spaces are mildly soft $sw$-Lindelöf.

(ii) Almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf) spaces are mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

(iii) The family of mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) subsets is closed under finite (resp., countable) union.

Proof. Straightforward.

Corollary 3.59. Soft $sw$-compact (resp., soft $sw$-Lindelöf) spaces are mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

A discrete soft$_{TS}$ defined over any infinite countable set with any finite set of parameters is mildly soft $sw$-Lindelöf but not mildly soft $sw$-compact, so the converse of (i) in Proposition 3.58 fails. To show that the converse of (ii) in Proposition 3.58 is also incorrect, we suffice by an example introduced in general topology. In Example 3.3 if we consider a set of parameters any uncountable set, then it can be checked that this soft$_{TS}$ is not soft $sw$-Lindelöf. On the other hand, note that the soft $swoc$-subsets of this soft$_{TS}$ are the absolute and null soft sets, which means that this soft$_{TS}$ is mildly soft $sw$-compact. Hence, the converse of Corollary 3.59 is generally false as well.

One can prove the next results following similar arguments in the proofs of their counterparts given in Subsection 3.1

Proposition 3.60. If a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf), then every soft $swoc$-subset of $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

Corollary 3.61. The intersection of a mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) set and a soft $swoc$-set is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

Theorem 3.62. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) if and only if $\widetilde{\bigcap}_{\rho\in I}(H_\rho, \mathbf{A})\neq\widetilde{\phi}$ for every family $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ of soft $swoc$-sets has a finite (resp., countable) intersection property.

Proposition 3.63. The property of being a mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) set is preserved by a surjective soft continuous mapping.

Proposition 3.64. The soft $sw$-continuous image of a mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) set is mildly soft compact (resp., mildly soft Lindelöf).

Proposition 3.65. The property of being a mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) set is preserved by a soft $sw$-irresolute mapping.

Lemma 3.66. An extended soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) if and only if the universal set $E$ and set of parameters $\mathbf{A}$ are finite (resp., countable).

Theorem 3.67. Every parametric topological space $(E, \mathcal{T}_{a})$ inspired by a mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf) space $(E, \mathcal{T}, \mathbf{A})$ is mildly $sw$-compact (resp., mildly $sw$-Lindelöf) for each $a\in \mathbf{A}$.

Example 3.15 also demonstrates that the converse of Theorem 3.67 is not true in general.

Theorem 3.68. Let $\mathbf{A}$ be a finite (resp., countable) parameter set and $(E, \mathcal{T}, \mathbf{A})$ be a full soft$_{TS}$. Then, all parametric topological spaces $(E, \mathcal{T}_a)$ inspired by $(E, \mathcal{T}, \mathbf{A})$ is mildly $sw$-compact (resp., mildly $sw$-Lindelöf) if and only if $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

Definition 3.69. We call $(E, \mathcal{T}, \mathbf{A})$ a soft $sw$-locally indiscrete soft$_{TS}$ if every soft $sw$-open set is soft $sw$-closed.

Proposition 3.70. The next concepts are identical under a soft $sw$-locally indiscrete soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$.

(i) $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact (resp., soft $sw$-Lindelöf).

(ii) $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact (resp., almost soft $sw$-Lindelöf).

(iii) $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact (resp., weakly soft $sw$-Lindelöf).

(iv) $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact (resp., mildly soft $sw$-Lindelöf).

Proof. We prove the proposition for compactness and the case of Lindelöfness is proved in a similar way.

The proofs of the directions (i) $\rightarrow$ (ii) and (ii) $\rightarrow$ (iii) follow from Propositions 3.5 and 3.29, respectively. To prove that (iii) $\rightarrow$ (iv), let $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ be a soft $swoc$-cover of a weakly soft $sw$-compact space $(E, \mathcal{T}, \mathbf{A})$. Then there exists a finite subset $\delta$ of $I$ such that $\widetilde{E} = swcl(\widetilde{\bigcup}_{\rho\in \delta}(H_\rho, \mathbf{A}))$. Since $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-locally indiscrete, $swcl(\widetilde{\bigcup}_{\rho\in\delta}(H_\rho, \mathbf{A})) = \widetilde{\bigcup}_{\rho\in \delta}(H_\rho, \mathbf{A})$. Thus, $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact. Finally, we prove that (iv) $\rightarrow$ (i). To do this, let $\mathcal{C} = \{(H_\rho, \mathbf{A}):\rho\in I\}$ be a soft $sw$-open cover of a soft $sw$-locally indiscrete soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$. Then $\mathcal{C}$ is also a soft $swoc$-cover. By hypothesis of mildly soft $sw$-compactness, there exists a finite subset $\delta$ of $I$ such that $\widetilde{E} = \widetilde{\bigcup}_{\rho\in \delta}(H_\rho, \mathbf{A})$. Thus, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-compact. Hence, the proof is complete.

4.   Soft $sw$-connected spaces

In this section, we define the concept of soft $sw$-connected spaces and discuss its characterizations induced from soft $sw$-open sets. Then, we describe this concept by disjoint soft open sets, which implies the equivalence between soft $sw$-connected and soft hyperconnected spaces. Also, we explore the main properties of this concept; especially, those related to specific types of soft compactness, soft separation axioms, and soft continuous mappings. Finally, with the aid of counterexamples, we demonstrate that the property of this type of connectedness is not commutative between soft topologies and their parametric topologies. We also prove that possessing this property by a full soft topology implies possessing by its parametric topologies and vice versa.

Definition 4.1. We call the soft subsets $(H, \mathbf{A})$ and $(K, \mathbf{A})$ of a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A}) \; sw$-separated soft sets if $(H, \mathbf{A})\widetilde{\bigcap}swcl(K, \mathbf{A}) = \widetilde{\phi}$ and $swcl(H, \mathbf{A})\widetilde{\bigcap}(K, \mathbf{A}) = \widetilde{\phi}$.

It is clear that a condition of $sw$-separated soft sets is proper stronger than a condition of disjoint.

Definition 4.2. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is called soft $sw$-disconnected if there are two non-null $sw$-separated soft sets $(H, \mathbf{A})$ and $(K, \mathbf{A})$ which their soft union is $\widetilde{E}$. Otherwise, we call $(E, \mathcal{T}, \mathbf{A})$ a soft $sw$-connected soft$_{TS}$.

In this case, we call $(H, \mathbf{A})$ and $(K, \mathbf{A})$ a soft $sw$-disconnection of $\widetilde{E}$.

Proposition 4.3. Every soft disconnected soft$_{TS}$ is soft $sw$-disconnected.

Proof. Follows from the fact that $swcl(H, \mathbf{A})\widetilde{\subseteq} cl(H, \mathbf{A})$ for any soft subset $(H, \mathbf{A})$ of $\widetilde{E}$.

A soft$_{TS}$ displayed in Example 3.28 is soft connected. But it is soft $sw$-disconnected because $\{(a, \widetilde{\mathbb{R}}), (b, \emptyset)\}$ and $\{(a, \emptyset), (b, \widetilde{\mathbb{R}})\}$ are $sw$-separated soft sets in which their soft union is the absolute soft set $\widetilde{\mathbb{R}}$. So, the converse side of Proposition 4.3 is not always true.

Proposition 4.4. The next properties are equivalent.

(i) $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-connected.

(ii) It cannot exist two proper disjoint soft $sw$-closed sets whose soft union is $\widetilde{E}$.

(iii) It cannot exist two proper disjoint soft $sw$-open sets whose soft union is $\widetilde{E}$.

(iv) The absolute and null soft sets are the only soft $sw$-open and soft $sw$-closed.

Proof. $(i)\Rightarrow (ii)$: If we consider there exist two disjoint proper soft $sw$-closed sets $(H, \mathbf{A})$ and $(K, \mathbf{A})$ such that $(H, \mathbf{A})\widetilde{\bigcup}(K, \mathbf{A}) = \widetilde{E}$. Then we obtain $(H, \mathbf{A})$ and $(K, \mathbf{A})$ are $sw$-separated soft sets because $swcl(H, \mathbf{A}) = (H, \mathbf{A})$ and $swcl(K, \mathbf{A}) = (K, \mathbf{A})$. But this contradicts $(i)$. Hence, $(ii)$ hold true.

$(ii)\Rightarrow (iii)$: Obvious.

$(iii)\Rightarrow (iv)$: Let us consider $(H, \mathbf{A})$ is both soft $sw$-open and soft $sw$-closed set. This leads to that $(H^c, \mathbf{A})$ is soft $sw$-open such that $(H, \mathbf{A})\widetilde{\bigcap}(H^c, \mathbf{A}) = \widetilde{\phi}$ and $(H, \mathbf{A})\widetilde{\bigcup}(H^c, \mathbf{A}) = \widetilde{E}$, which contradicts $(iii)$. Hence, $(iv)$ hold true.

$(iv)\Rightarrow (i)$: Suppose, by contrary, that there is a soft set $(H, \mathbf{A})$ which is both soft $sw$-open and soft $sw$-closed. Then $swcl(H, \mathbf{A}) = (H, \mathbf{A})$ and $swcl(H^c, \mathbf{A}) = (H^c, \mathbf{A})$. So, $(H, \mathbf{A})$ and $(H^c, \mathbf{A})$ are two $sw$-separated soft sets such that their union is $\widetilde{E}$, thus $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected. Hence, $(iv)\Rightarrow (i)$ hold true.

Now, we describe the concept of soft $sw$-disconnectedness using soft open sets instead of soft $sw$-open sets.

Theorem 4.5. A soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected if and only if $\mathcal{T}$ contains two non-null disjoint soft open sets.

Proof. To prove the necessary part, let $(E, \mathcal{T}, \mathbf{A})$ be soft $sw$-disconnected. Then, it follows from (iii) of Proposition 4.4 that there exist two proper non-null disjoint soft $sw$-open sets; say, $(H, \mathbf{A})$ and $(K, \mathbf{A})$. Thus, there exist non-null disjoint soft open sets $(G_1, \mathbf{A}), (G_2, \mathbf{A})$ such that $(G_1, \mathbf{A})\widetilde{\subseteq}(H, \mathbf{A})$ and $(G_2, \mathbf{A})\widetilde{\subseteq}(K, \mathbf{A})$, as required. To prove the sufficient part, let $(U, \mathbf{A})$ and $(V, \mathbf{A})$ be two non-null disjoint soft open sets. Then, $(U, \mathbf{A})\widetilde{\subseteq}(V^c, \mathbf{A})$, which means that $(V^c, \mathbf{A})$ is a proper soft $sw$-open set. Accordingly, we obtain $(V, \mathbf{A})$ and $(V^c, \mathbf{A})$ are non-null disjoint soft $sw$-open subsets of $\widetilde{E}$ such that their soft union is $\widetilde{E}$. Hence, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected.

The above theorem informs us that the concepts of soft $sw$-connected and soft hyperconnected spaces are identical.

Corollary 4.6. The next properties are equivalent.

(i) $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-connected.

(ii) There do not exist two non-null soft open sets are disjoint.

(iii) There do not exist two proper soft closed sets such that their union is $\widetilde{E}$.

(iv) Every non-null soft open set is soft dense.

(v) The soft interior of any proper soft closed set is the null soft set.

(vi) Every soft subset is soft dense or soft nowhere dense.

(vii) There do not exist two soft points separated by disjoint soft neighbourhoods.

Proposition 4.7. A non-null proper subset $(K, \mathbf{A})$ of a soft $sw$-connected space $(E, \mathcal{T}, \mathbf{A})$ has a non-null soft $sw$-boundary points.

Proof. Suppose, by contrary, that $swb(K, \mathbf{A}) = \widetilde{\phi}$. Then $swcl(K, \mathbf{A}) = swint(K, \mathbf{A})$, which means that $(K, \mathbf{A})$ is both soft $sw$-open and soft $sw$-closed. But this contradicts the soft $sw$-connectedness of $(E, \mathcal{T}, \mathbf{A})$. Hence, $swb(K, \mathbf{A})\neq\widetilde{\phi}$, as required.

Definition 4.8. A soft subset $(K, \mathbf{A})$ of $(E, \mathcal{T}, \mathbf{A})$ is called a soft $sw$-connected set if there do not exist two non-null $sw$-separated soft sets $(G, \mathbf{A})$ and $(H, \mathbf{A})$ such that their union is $(K, \mathbf{A})$. Otherwise $(K, \mathbf{A})$ is called a soft $sw$-disconnected set.

Lemma 4.9. Let $(H, \mathbf{A})$ and $(G, \mathbf{A})$ be two soft $sw$-disconnection sets of $(E, \mathcal{T}, \mathbf{A})$. If $(K, \mathbf{A})$ is a soft $sw$-connected subset of $(E, \mathcal{T}, \mathbf{A})$, then $(K, \mathbf{A})\widetilde{\subseteq}(H, \mathbf{A})$ or $(K, \mathbf{A})\widetilde{\subseteq}(G, \mathbf{A})$.

Proof. Since $(H, \mathbf{A})$ and $(G, \mathbf{A})$ are soft $sw$-disconnection sets of $(E, \mathcal{T}, \mathbf{A})$, we obtain $(H, \mathbf{A})\widetilde{\bigcup}(G, \mathbf{A}) = \widetilde{E}$ and $[(H, \mathbf{A})\widetilde{\bigcap}swcl(G, \mathbf{A})] \widetilde{\bigcup}[swcl(H, \mathbf{A})\widetilde{\bigcap}(G, \mathbf{A})] = \widetilde{\phi}$. Now, $(K, \mathbf{A}) = [(K, \mathbf{A})\widetilde{\bigcap}(H, \mathbf{A})] \widetilde{\bigcup}[(K, \mathbf{A})\widetilde{\bigcap}(G, \mathbf{A})]$. It is clear that

So, we infer that $(K, \mathbf{A})\widetilde{\bigcap}(H, \mathbf{A})$ and $(K, \mathbf{A})\widetilde{\bigcap}(G, \mathbf{A})$ are soft $sw$-disconnection sets of $(K, \mathbf{A})$. But this contradicts the soft $sw$-connectedness of $(K, \mathbf{A})$. Hence, $(K, \mathbf{A})\widetilde{\bigcap}(H, \mathbf{A}) = \widetilde{\phi}$ or $(K, \mathbf{A})\widetilde{\bigcap}(G, \mathbf{A}) = \widetilde{\phi}$, which means that $(K, \mathbf{A})\widetilde{\subseteq}(H, \mathbf{A})$ or $(K, \mathbf{A})\widetilde{\subseteq}(G, \mathbf{A})$.

Theorem 4.10. Let $(K, \mathbf{A})$ be a soft subset of $(E, \mathcal{T}, \mathbf{A})$ such that for each $P^e_a, P^x_b\in (K, \mathbf{A})$ there is a soft $sw$-connected subset $(F, \mathbf{A})$ of $(K, \mathbf{A})$ containing $P^e_a, P^x_b$. Then $(K, \mathbf{A})$ is soft $sw$-connected.

Proof. Suppose, by contrary, that $(K, \mathbf{A})$ is soft $sw$-disconnected. Then there exist soft $sw$-disconnection sets $(H, \mathbf{A})$ and $(G, \mathbf{A})$ of $(K, \mathbf{A})$. So there are soft points $P^e_a, P^x_b$ such that $P^e_a\in (H, \mathbf{A})$ and $P^x_b\in (G, \mathbf{A})$. By the given, there exists a soft $sw$-connected set $(F, \mathbf{A})$ containing $P^e_a, P^y_e$ such that $(F, \mathbf{A})\widetilde{\subseteq}(K, \mathbf{A}) = (H, \mathbf{A})\widetilde{\bigcup} (G, \mathbf{A})$. By Lemma 4.9, we get $(F, \mathbf{A})\widetilde{\subseteq}(H, \mathbf{A})$ or $(F, \mathbf{A})\widetilde{\subseteq}(G, \mathbf{A})$. Consequentially, $(H, \mathbf{A})\widetilde{\bigcap}(G, \mathbf{A})\neq\widetilde{\phi}$, which contradicts that $(H, \mathbf{A})$ and $(G, \mathbf{A})$ are soft $sw$-disconnection of $(K, \mathbf{A})$. This means that $(K, \mathbf{A})$ is soft $sw$-connected.

Corollary 4.11. If $(K, \mathbf{A})$ is a soft union of soft $sw$-connected sets $(H_\rho, \mathbf{A})$ such that their soft intersection is non-null, then $(K, \mathbf{A})$ is soft $sw$-connected.

Proof. Suppose, by contrary, that $(K, \mathbf{A})$ is soft $sw$-disconnected. Then there are two soft $sw$-disconnection sets $(F, \mathbf{A})$ and $(G, \mathbf{A})$ of $(K, \mathbf{A})$. Since $\widetilde{\bigcap}(H_\rho, \mathbf{A})\neq\widetilde{\phi}$, then there is a soft point $P^e_a$ such that $P^e_a\in (H_\rho, \mathbf{A})$ for each $\rho$. Now, either $P^e_a\in (F, \mathbf{A})$ or $P^e_a\in (G, \mathbf{A})$. Say, $P^e_a\in (F, \mathbf{A})$. Then $[\widetilde{\bigcap}(H_\rho, \mathbf{A})]\widetilde{\bigcap}(F, \mathbf{A}) \neq\widetilde{\phi}$. According to the above theorem we obtain $(H_\rho, \mathbf{A})\widetilde{\subseteq}(F, \mathbf{A})$ for each $\rho$. This means that $(K, \mathbf{A})\widetilde{\subseteq}(F, \mathbf{A})$. But this is a contradiction. Hence, $(K, \mathbf{A})$ is soft $sw$-connected.

Proposition 4.12. Let $\Omega_{\pi}$ be a soft continuous mapping of a soft $sw$-connected space $(E, \mathcal{T}_{E}, \mathbf{A})$ onto a soft$_{TS} \; (X, \mathcal{T}_{X}, \mathbf{A})$. Then $\Omega_{\pi}(\widetilde{E})$ soft $sw$-connected.

Proof. Suppose that $\Omega_{\pi}(\widetilde{E}) = \widetilde{X}$ is soft $sw$-disconnected. Then, by Theorem 4.5 we get $\mathcal{T}_{X}$ contains two non-null disjoint soft open sets $(F, \mathbf{A})$ and $(H, \mathbf{A})$. By hypothesis, $\Omega^{-1}_{\pi}(F, \mathbf{A})$ and $\Omega^{-1}_{\pi}(H, \mathbf{A})$ are disjoint soft open subsets in $\mathcal{T}_{E}$. Since $\Omega_{\pi}$ is surjective, these soft open sets are non-null. Again it follows from Theorem 4.5 that $(E, \mathcal{T}_{E}, \mathbf{A})$ is soft $sw$-disconnected. But this is a contradiction. Hence, $(X, \mathcal{T}_{X}, \mathbf{A})$ is soft $sw$-connected.

Proposition 4.13. (i) Any type of soft $sw$-Hausdorff space is soft $sw$-disconnected.

(ii) Every soft $sw$-connected is almost soft $sw$-compact.

A soft$_{TS}$ displayed in Example 3.19 is almost soft $sw$-compact, but it is soft $sw$-disconnected. The next example provides a soft$_{TS}$ which is soft $sw$-disconnected, but not $pp$-soft $swT_2$.

Example 4.14. Let $\mathcal{T} = \{\widetilde{\phi}, \widetilde{E}, (G_1, \mathbf{A}), (G_2, \mathbf{A})\}$ be a soft topology over the universal set $E = \{x, y, z\}$ with a set of parameters $\mathbf{A} = \{a, b\}$, where $(G_1, \mathbf{A}) = \{(a, \{x\}), (b, \{x\})\}$ and $(G_2, \mathbf{A}) = \{(a, \{y, z\}), (b, \{y, z\})\}$. It is clear that $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected, but it is not $pp$-soft $swT_2$.

Proposition 4.15. Let $(E, \mathcal{T}, \mathbf{A})$ be a soft$_{TS}$ such that $E$ or $\mathbf{A}$ is infinite countable (resp., uncountable). Then every soft $sw$-disconnected is not soft $sw$-compact (resp., soft $sw$-Lindelöf).

The converse of the above proposition need not be true as illustrated by Example 3.4.

Proposition 4.16. The next properties are identical under a soft $sw$-connected space $(E, \mathcal{T}, \mathbf{A})$.

(i) $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-Lindelöf.

(ii) $(E, \mathcal{T}, \mathbf{A})$ is almost soft $sw$-compact.

(iii) $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-Lindelöf.

(iv) $(E, \mathcal{T}, \mathbf{A})$ is weakly soft $sw$-compact.

(v) $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-Lindelöf.

(vi) $(E, \mathcal{T}, \mathbf{A})$ is mildly soft $sw$-compact.

Proof. The proof follows from the fact that $swcl(H, \mathbf{A}) = \widetilde{E}$ for any non-null soft open subset $(H, \mathbf{A})$ of a soft $sw$-connected space $(E, \mathcal{T}, \mathbf{A})$.

In rest of this section, we study the relationships between soft topologies and their parametric topologies with respect to this type of connectedness. A soft$_{TS}$ displayed in Example 3.15 is soft $sw$-disconnected, whereas its parametric topological spaces are $sw$-connected. This means the property of being $sw$-connectedness does not navigate from parametric topologies to soft topology. On the other hand, the property of being soft $sw$-connectedness does not transmit from soft topology to its parametric topologies as we elaborate by the next example.

Example 4.17. Let $E = \{x, y, z\}$ be the universal set and the set of natural numbers $\mathbb{N}$ be a set of parameters. Then, the family $\{\widetilde{\phi}, \widetilde{E}, (H_\rho, \mathbb{N}): H_\rho(n) = E$ for all but finitely many $n\in \mathbb{N}\}$ forms a soft topology over $E$ with $\mathbb{N}$. It can be checked that the families of soft open sets and soft $sw$-open sets are equal. Since we cannot get two disjoint soft open sets, $(E, \mathcal{T}, \mathbb{N})$ is soft $sw$-connected. Whereas all parametric topological spaces inspired by $(E, \mathcal{T}, \mathbb{N})$ is $sw$-disconnected.

Theorem 4.18. Let $\mathbf{A}$ be a finite set of parameters. If all parametric topological spaces inspired by a soft$_{TS} \; (E, \mathcal{T}, \mathbf{A})$ is $sw$-disconnected, then $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected.

Proof. Without loss of generality, take $\mathbf{A} = \{a, b\}$. By hypothesis, $(E, \mathcal{T}_a)$ and $(E, \mathcal{T}_b)$ are $sw$-disconnected. Then, there exist two disjoint nonempty open sets $V, W$ in $\mathcal{T}_a$ and two disjoint nonempty open sets $X, Y$ in $\mathcal{T}_b$. This implies that there exist four non-null soft open subsets $(H_1, \mathbf{A}), (H_2, \mathbf{A}), (H_3, \mathbf{A}), (H_4, \mathbf{A})$ of $(E, \mathcal{T}, \mathbf{A})$ such that $H_1(a) = V, H_2(a) = W$ and $H_3(b) = X, H_4(b) = Y$. Now, we have two cases:

(i) there exist two soft open sets of them are disjoint.

(ii) there does not exist two soft open sets of them are disjoint. Then, $(H_5, \mathbf{A}) = (H_1, \mathbf{A})\widetilde{\bigcap}(H_2, \mathbf{A})$ and $(H_6, \mathbf{A}) = (H_3, \mathbf{A})\widetilde{\bigcap}(H_4, \mathbf{A})$ are non-null soft open sets. Since $H_5(a) = \emptyset$ and $H_6(b) = \emptyset$, $(H_5, \mathbf{A})$ and $(H_6, \mathbf{A})$ are disjoint.

Hence, according to Theorem 4.5, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected.

By Example 3.15, we see that the converse of the above theorem fails.

Theorem 4.19. Let $(E, \mathcal{T}, \mathbf{A})$ be a full soft$_{TS}$. Then, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected if and only if all parametric topological spaces inspired by $(E, \mathcal{T}, \mathbf{A})$ is $sw$-disconnected.

Proof. Necessity: Let $(E, \mathcal{T}, \mathbf{A})$ be soft $sw$-disconnected. Then, according to Theorem 4.5, $\mathcal{T}$ contains two disjoint non-null soft open sets $(F, \mathbf{A})$ and $(H, \mathbf{A})$. Since $(E, \mathcal{T}, \mathbf{A})$ is full, $F(a)$ and $H(a)$ are both nonempty disjoint open subsets of $(E, \mathcal{T}_a)$ for each $a\in\mathbf{A}$. This implies that $(E, \mathcal{T}_a)$ is $sw$-disconnected for each $a\in\mathbf{A}$.

Sufficiency: Let a parametric topological space $(E, \mathcal{T}_a)$ be $sw$-disconnected. Then, there exist two disjoint nonempty open sets $V, W$ in $\mathcal{T}_a$. So, there exist two non-null soft open subsets $(F, \mathbf{A})$ and $(H, \mathbf{A})$ of $(E, \mathcal{T}, \mathbf{A})$ such that $F(a) = V$ and $H(a) = W$. Obviously, $(F, \mathbf{A})\widetilde{\bigcap}(H, \mathbf{A})$ is a soft open set with an empty component; i.e. $(F\bigcap H)(a) = \emptyset$. Since $(E, \mathcal{T}, \mathbf{A})$ is full, we obtain $(F\bigcap H)(a) = \emptyset$ for all $a\in \mathbf{A}$. This means that $(F, \mathbf{A})$ and $(H, \mathbf{A})$ are also disjoint. Hence, $(E, \mathcal{T}, \mathbf{A})$ is soft $sw$-disconnected.

5.   Conclusion remarks and future works

Soft set theory has been widely used in many real-world applications for its tremendous practical value. Typologists have exploited soft sets to define soft topological spaces and studied classical topological concepts via soft frame. As we noted in the published literature the abstract topological concepts in soft settings such as soft compactness ^[8], soft separation axioms ^[12] and generalized soft open sets ^[13] have been successfully applied to address practical issues in the information systems, economic and medical science. Also, the variety of belonging and non-belonging relations between ordinary points and soft sets produces a fruitful environment to display classical topological concepts from different standpoints as illustrated to soft compact spaces and soft separation axioms.

In this manuscript, we have benefited from one of the generalizations of soft open sets called "soft $sw$-open sets" to introduce the concepts of compactness, Lindelöfness and connectedness via the frame of soft topologies. First, we have presented novel types of soft compact and Lindelöf spaces namely, soft $sw$-compact, soft $sw$-Lindelöf, almost soft $sw$-compact, almost soft $sw$-Lindelöf, weakly soft $sw$-compact, weakly soft $sw$-Lindelöf, mildly soft $sw$-compact and mildly soft $sw$-Lindelöf spaces. We have established the main properties of these spaces and provided some interesting examples to show the relationships between them. Some exciting results described the behaviours of these spaces via soft topology and its parametric topologies have been obtained and illustrated with some counterexamples.

Then, we have introduced the concept of soft $sw$-connected spaces and described by soft $sw$-open and $sw$-closed sets. One of the important results that we have proved is the identity between soft $sw$-connected and soft hyperconnected spaces. We have also scrutinized the relationships that associate this concept with some types of soft compactness and soft separation axioms. Moreover, we have elaborated that the property of being $sw$-connectedness is commutative between soft topology and its parametric topological spaces under a condition of full soft topology.

We draw attention to that the concepts introduced herein have some unique characterizations that are invalid for their counterparts defined by the other types of famous generalizations of soft open sets such as soft $\alpha$-open, soft semi-open, soft pre-open, soft $b$-open and soft $\beta$-open sets. For example, the properties of being soft $sw$-compact and soft $sw$-Lindelöf (mildly soft $sw$-compact and mildly soft $sw$-Lindelöf) spaces are transmitted to all their parametric topological spaces; see, Theorems 3.14 and 3.67. But it is imposed a condition of extended soft topology to preserve this characteristic for the other types given in the published literature. Also, some descriptions of the previous types of compactness and Lindelöfness are preserved under a condition of extended soft topology, whereas these descriptions are satisfied for the current types of compactness and Lindelöfness under a condition of full soft topology; see, Theorems 3.21, 3.41, 3.53 and 3.68. With respect to soft $sw$-connectedness, it has been shown that the property of being a soft $sw$-connected space is navigated to all its parametric topological spaces provided that a set of parameters is finite; see, Theorem 4.18. Also, it has been demonstrated that this property is interchangeable between soft topology and its parametric topologies under a condition of full soft topology; see, Theorem 4.19.

Regarding the relationships between the current covering properties and the previous ones, it can be noted that the covering properties produced by soft $\alpha$-open and soft semi-open sets are special cases of their counterparts introduced herein. This fact can be seen from Example 3.15. On the other hand, any indiscrete soft$_{TS}$ defined over an uncountable universal set and a soft$_{TS}$ displayed in Example 3.16 clarify that the current covering properties and those defined by soft pre-open, soft $b$-open and soft $\beta$-open sets are independent of each other.

In future work, we are going to study further topological concepts via the family of soft $sw$-open sets such as paracompactness, local compactness, and local connectedness. Also, we will research the possibility of applying the concepts introduced herein to handle some real-life issues as presented in ^[8]. Moreover, we analysis rough set models via soft frame using the concepts of soft $sw$-open sets as given in the classical frame by ^[10,11].