Soft version of compact and Lindel¨of spaces using soft somewhere dense sets

: Herein, we applied soft somewhere dense sets to initiate six sorts of soft spaces called almost (nearly, mildly) soft S D -compact and almost (nearly, mildly) soft S D -Lindel¨of spaces. We study the master properties of these spaces and illustrate the relations between them with the help of examples. In addition, we clarify that the six soft spaces are equivalent under a soft S D -partition. Moreover, the relationships between the initiated spaces and enriched soft topological spaces and other well-known spaces such as soft S -connected are indicated.


Introduction and preliminaries
Many researchers developed the theory of soft sets after it was established by Molodtsov [23] in 1999 as a new mathematical method for dealing with problems involving uncertainties. A first attempt was made by Maji et al. [21], in 2003, to formulate soft operators. He defined the null and absolute soft sets, a complement of soft set, and soft intersection and union between two soft sets. Ali et al. [2] made a major contribution, in 2009, through the soft set theory. Some soft operators were redefined such as the complement of a soft set and soft intersection between two soft sets, and new soft operators were initiated between two soft sets such as restricted union, restricted and extended intersection and restricted difference. In this sense, Al-shami and El-Shafei [9] relaxed constraints on a parameters set to implement generalized soft operators. Shabir and Naz [27], in 2011, initiated soft topology and presented some properties of soft separation axioms. The work in [27] was continued by Min [22], he corrected a relationship between soft T 2 and soft T 3 -spaces and analyzed the properties of soft regular spaces. In 2012, the relation between soft sets and fuzzy sets was pointed out by Zorlutuna et al. [29]. Also they introduced the soft point in order to analyze some properties of soft interior points and soft neighborhood systems. Aygünoǧlu and Aygün [14] defined the notion of soft compact spaces and introduced the definition of enriched soft topological spaces which will play a remarkable role in this article. In [19], Hida provided a stronger description for soft compact spaces which defined in [14]. Al-shami et al. [10] investigated new types of covering properties called almost soft compact and approximately soft Lindelöf spaces.
Kharal and Ahmad [20] defined soft mappings and established main properties. Then, Zorlutuna and Ç akir [30] explored the concept of soft continuous mappings. Asaad [13] presented the concept of soft extremally disconnected spaces and revealed main properties. In [28], the authors conducted a comparative study on soft separation axioms. Alcantud [1] discussed the countability axioms in the soft topologies. Recently, Al-shami applied the concepts of compactness and soft separation axioms to economic application [6] and information system [7].
At one and the same year, new form of a soft point was introduced in [15] and [24]. This form helps to simulate the followed manner in classical topology to soft topology and makes it easier to prove many soft topological properties. A comparative study on soft points was conducted in [26]. Al-shami [5] restudied main properties of soft separation axioms; especially, those introduced using the different types of soft points.
In 2017, Al-shami [3] initiated a new class of generalized open sets called somewhere dense sets, and in 2019, he and Noiri [11] applied to introduce new types of soft mappings. Al-shami [4] studied this class in soft topology in 2018. Then, he with co-authors [8,17] exploited to study some concepts in soft setting. This study goes in this path of study by introducing new types of soft compact and Lindelöf spaces, namely almost (nearly, mildly) soft S D-compact and almost (nearly, mildly) soft S D-Lindelöf spaces. To clarify some features of these spaces and the relationships between them, we provided some illustrative examples. In addition, we define soft S D-partition and soft S D-hyperconnected spaces to study some properties that link them with the soft spaces that have been introduced, and we deduce some conclusions that associate these spaces with enriched soft topological spaces. Then we investigate when the six initiated soft spaces have a soft hereditary property. Finally, we elucidate that the soft S D-irresolute mappings keep the six initiated soft spaces. Now, with a fixed parameters set Ω we recall some notions and conclusions that are mentioned in the various previous studies.  (ii) The complement of somewhere dense subset is cs-dense.
(iii) The union of all somewhere dense sets contained in M is called the S -interior and it is denoted by S int(M).
(iv) The intersection of all cs-dense sets containing M is called the S -closure of M and it is denoted by S cl(M). (i) Nearly S D-compact (resp. nearly S D-Lindelöf) if every somewhere dense cover of W has a finite (resp. countable) subcover in which its S -closure covers W.
(ii) Almost S D-compact(resp. almost S D-Lindelöf ) if every somewhere dense cover of W has a finite (resp. countable) subcover which the S -closures of whose members cover W.     Then (W, σ, Ω) is said to be a soft topological space (briefly, soft T S ). Each member in σ is called soft open and its relative complement is called soft closed.  (i) Soft point [15,24] if there are p ∈ Ω and t ∈ W with M(p) = {t} and M(q) = ∅, for each q ∈ Ω−{p}.
A soft point is briefly denoted by P t p . (ii) Pseudo constant [25] if M(p) = W or ∅, for each p ∈ Ω. A family of all pseudo constant soft sets is briefly denoted by CS (W, Ω).
Definition 1.14. [14] If (i) of Definition 1.9 is replaced by the following condition: (M, Ω) ∈ σ, for all (M, Ω) ∈ CS (W, Ω), then a soft topology σ on W is said to be enriched. In this case, the triple (W, σ, Ω) is called an enriched soft T S over W.
Definition 1.21. [17] A soft T S (W, σ, Ω) is soft S -connected iff the only soft somewhere dense and soft cs-dense subsets of (W, σ, Ω) are ∅ and W. In this work, we will use S D-connected instead of S -connected.
is the null soft set or a soft somewhere dense set where (M, Ω) is soft open (resp. soft somewhere dense) set.
, Ω) be a soft map. Then the following properties are equivalent: (ii) The inverse image of every soft closed subset of (Z, σ Z , Ω) is W or soft cs-dense.

Almost soft S D-compact and almost soft S D-Lindelöf spaces
In this section, we will use soft somewhere dense sets to define two new spaces, namely almost soft S D-compact and almost soft S D-Lindelöf spaces, also, we study some of their basic properties.
Definition 2.1. A family of soft somewhere dense subsets of (W, σ, Ω) is called a soft somewhere dense cover (briefly, S D-cover) of W if W is a soft subset of this family.
The following example presents a soft topological space which is almost soft S D-compact but not soft compact. , (q, ∅)}. Now, every soft superset of (E, Ω) is soft somewhere dense, but not soft cs-dense. Also, any soft S D-cover of (R, σ, Ω) is superset of (E, Ω). From Proposition 1.19, we find that soft S -closure of any member of the soft S D-cover is the absolute soft set R. Thus, (R, σ, Ω) is almost soft S D-compact.
Proof. It follows from Definition 2.2. All soft points in N construct a soft S D-cover of (N, σ, Ω). Since every soft subset of (N, σ, Ω) (except the null and absolute soft sets) is both soft somewhere dense and soft cs-dense. Therefore, (N, σ, Ω) is not almost soft S D-compact.
Proof. It is obvious.
Proposition 2.8. If a soft T S (W, σ, Ω) is almost soft S D-compact (resp. almost soft S D-Lindelöf), then every soft cs-dense subset of (W, σ, Ω) is almost soft S D-compact (resp. almost soft S D-Lindelöf).
The case between parentheses can be achieved similarly. : i ∈ I} be a family of soft sets. If i∈Λ S int(E i , Ω) ∅ for any finite (resp. countable) set Λ, then E is said to have the first type of finite (resp. countable) S D-intersection property.
Theorem 2.11. A soft T S (W, σ, Ω) is almost soft S D-compact (resp. almost soft S D-Lindelöf) iff i∈I (E i , Ω) ∅ for every family E = {(E i , Ω) : i ∈ I} of soft cs-dense sets has the first type of finite (resp. countable) S D-intersection property.
Proof. By contrary, suppose that E = {(E i , Ω) : i ∈ I} is a soft cs-dense subsets of (W, σ, Ω) with , which is a contradiction. Conversely, let H = {(H i , Ω) : i ∈ I} be a soft S D-cover of (W, σ, Ω). Then ∅ = i∈I (H c i , Ω) and so by the first type of finite S D-intersection property, we have ∅ = The case of almost soft S D-compact can be achieved similarly .   The case between parentheses can be achieved similarly.  Proof. It is obvious.

Nearly soft S D-compact and nearly soft S D-Lindelöf spaces
In this section, we present generalizations for almost soft S D-compact and almost soft S D-Lindelöf spaces called nearly soft S D-compact and nearly soft S D-Lindelöf spaces. Also, some properties of those generalizations are studied.
Definition 3.1. A soft T S (W, σ, Ω) is said to be nearly soft S D-compact (resp. nearly soft S D-Lindelöf) if for every soft S D-cover H = {(H i , Ω) : i ∈ I} of (W, σ, Ω), there is a finite (resp. countable) set Λ ⊆ I with W = S cl( i∈Λ (H i , Ω)).  The following example shows that the converse of Proposition 3.2 is not true.  with (E 1 , Ω) ⊆S cl( i∈I * 1 (H i , Ω)), . . . , (E n , Ω) ⊆S cl( i∈I * n (H i , Ω)), . . . . Therefore, The case of a nearly soft S D-compact space can be achieved similarly. Proof. It follows from i∈I S cl(H i , Ω) ⊆S cl( i∈I (H i , Ω)), where (H i , Ω) is a soft subset of (W, σ, Ω).  The following example demonstrates the converse of Proposition 3.6 is not true. Then (E, Ω) is soft somewhere dense iff 1 (E, Ω) or 2 (E, Ω). Set E = {(E, Ω) is finite such that there is only one parameter r ∈ Ω with 1 ∈ E(r) or 2 ∈ E(r)}, then E is a soft S D-cover of R. Now, any soft S D-cover of R contains four soft somewhere dense subsets of E which contains a soft open set (E 3 , Ω). Since a soft somewhere dense set (E 3 , Ω) is soft S D-dense, then (R, σ, Ω) is nearly soft S D-compact. However, (R, σ, Ω) is not almost soft S D-Lindelöf since E is a soft S D-cover does not have a countable subcover in which its soft S D-closure of whose members covers R. Ω) : i ∈ I} be a family of soft sets. If S int( i∈Λ (E i , Ω)) ∅ for any finite (resp. countable) set Λ, then E is said to have the second type of finite (resp. countable) S Dintersection property.
Note that if a family E = {(E i , Ω) : i ∈ I} has the second type of finite (resp. countable) S Dintersection property, then it has the first type of finite (resp. countable) S D-intersection property. Proof. We give the proof when (W, σ, Ω) is nearly soft S D-compact. The other case can be made similarly. By contrary, suppose that E = {(E i , Ω) : i ∈ I} is a family of soft cs-dense subsets of (W, σ, Ω) with i∈I (E i , Ω) = ∅. Since (W, σ, Ω) is nearly soft S D-compact and W = i∈I (E c i , Ω), then W = S cl( Ω)), which is a contradiction. Conversely, it follows from Theorem 2.11 and Proposition 3.6.
Proof. Assume that {H i (p) : i ∈ I} is a somewhere dense cover for (W, σ p ). We construct a soft S D-cover {(E i , Ω) : i ∈ I} for (W, σ, Ω) with E i (p) = H i (p) and for each p p, E i (p ) = W. Since (W, σ, Ω) is nearly soft S D-Lindelöf, there is a countable set Λ with W = S cl( i∈Λ (E i , Ω)) ⊆(S cl( E i ), Ω). Therefore, W = S cl( i∈Λ E i (p 1 )) = S cl( i∈Λ H i (p 1 )) and so (W, σ p ) is nearly S D-Lindelöf. The case of a nearly soft S D-compact space can be achieved similarly.
By using Theorem 2.12, we can prove the following theorem. Theorem 3.13. Let Ψ Θ : (W, σ W , Ω) → (Z, σ Z , Ω) be a soft S D-irresolute map. Then the image of a nearly soft S D-compact (resp. nearly soft S D-Lindelöf) set is nearly soft S D-compact (resp. nearly soft S D-Lindelöf).

Mildly soft S D-compact and mildly soft S D-Lindelöf spaces
In this section, we introduce another generalizations of almost soft S D-compact and almost soft S D-Lindelöf spaces, we put some restrictions to become all of the soft spaces that introduced are equivalent.           If we replace the natural numbers set N in Example 2.6 by a set W = {1, 2, 3}, then we obtain (W, σ, Ω) is mildly soft S D-compact. But it is soft S D-disconnected. This shows that the converse of the above proposition fails.
To illustrate that not every nearly soft S D-Lindelöf space is mildly soft S D-Lindelöf, we give the following example.  i ∈ I} is a family of soft S C-subsets of W which has the finite intersection property, so i∈I (H c i , Ω) ∅. Therefore, W i∈I (H i , Ω), which is a contradiction. Hence, (W, σ, Ω) is mildly soft S D-compact. The case between parentheses can be achieved similarly.
Proof. By using a similar technique of the proof of Theorem 2.12, the proposition holds.
The proof of the following two propositions is obvious, so it will be omitted.    (i) (W, σ, Ω) is almost soft S D-Lindelöf (resp. almost soft S D-compact).
(ii) → (iii) Assume that H ={(H i , Ω) : i ∈ I} is an S C-cover of a nearly soft S D-Lindelöf space (W, σ, Ω). Then there is a countable set Λ ⊆ I with W = S cl( i∈Λ (H i , Ω)). Now, since (W, σ, Ω) is a soft S D-partition, then S cl( i∈Λ (H i , Ω)) = i∈Λ (H i , Ω). Hence, (W, σ, Ω) is mildly soft S D-Lindelöf. Then H is an S C-cover of a mildly soft S D-Lindelöf space (W, σ, Ω) , so there is a countable set Λ ⊆ I with W = i∈Λ S cl(H i , Ω). The case between parentheses can be achieved similarly.     The proof of the following proposition is obvious, so it will be omitted.

Conclusions
We define some new types of soft spaces based on soft somewhere dense sets, namely, almost (nearly, mildly) soft S D-compactness and almost (nearly, mildly) soft S D-Lindelöfness. We use examples to illustrate the relationships between these concepts, and we analyze the image of these spaces under soft S D-continuous and soft S D-irresolute mappings.
The different types of soft compact and Lindelöf spaces introduced herein help us to classify soft structures into new different families which help us to model some real-life problems as those given in [7]. Also, we can apply soft somewhere dense sets and these families to initiate new types of approximations and accuracy measures in the content of rough sets models. Finally, the given concepts herein allow us to study many results induced from their interaction with some soft topological notions such as soft Menger and soft connected spaces.