Soft ω-regular open sets and soft nearly Lindelöfness.

In this paper, we define the notion of "soft ω-regular openness," which lies exactly between "soft regular openness" and "soft ω-openness." Through soft ω-regular open sets, we introduce the notions of soft semi ω-regularity as a weaker form of soft semi regularity and soft almost ω-regularity as a strong form of soft almost regularity. We prove that soft ω-regular open sets of a soft topological space form a soft topology. Also, we prove that soft semi ω-regularity and soft almost ω-regularity are independent notions. In addition, we reveal the relationships between soft topology and classical (parametric) topology. Finally, we characterize soft nearly Lindelöfness and improve several results related to soft nearly Lindelöfness using the concept of soft ω-regular open sets.

Introduction and preliminaries

Many real-life problems have their own uncertainties. Accordingly, it is not possible to deal with these problems by traditional methods. For more than thirty years, fuzzy set theory [1], rough set theory [2], and vague set theory [3] have played an essential role in dealing with these problems. Molodostov [4] states that each of these mathematical structures has its drawbacks. The research via soft sets has entered in most scientific fields. Soft set theory has been applied to solve problems using Riemann integral, Beron's integral, game theory, function smoothness, operations research, measure theory, probability, and decision-making problems [4], [5], [6].

Shabir and Naz [7] initiated soft topology. Since that time, the generalization of topological concepts in soft topology has become the focus of many researchers, such as soft compact and soft Lindelöf [8], [9], [10], [11], [12], soft connected [13], soft paracompact [13], soft extremely disconnected [14], soft separable spaces [15], soft separation axioms [16], [17], [18], soft metric spaces [19], [20], [21], and soft homogeneous spaces [22], [23]. The field of research in soft topology is still active (see for example [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]), and substantial contributions can still be made.

Closure and interior operators were used to give rise to several different new classes of sets, while a few others are the so-called regular sets. Researchers have discovered applications for these regular sets not only in mathematics but even in a variety of fields outside of mathematics [38], [39], [40].

The targets of this paper are to scrutinize the behaviors of soft ω-regular open sets via soft topological spaces, to characterize soft nearly Lindelöfness, to improve several results related to soft nearly Lindelöfness using the concept of soft ω-regular open sets, and to open the door to redefine and investigate some of the soft topological concepts such as soft compactness, soft menger, soft separation axioms, and soft continuity, via soft ω-regular open sets.

The arrangement of this article is as follows:

In Section 2, we define soft ω-regular open sets. We investigate the properties of these soft sets and point out their relationships with some famous generalized soft open sets. We also show the relationships between soft ω-regular open sets in soft topology and their general topological counterparts.

In Section 3, we define two new classes of soft separation axioms: soft semi ω-regular and soft almost ω-regular. We investigate their properties and point out their relationships with some famous soft separation axioms. We also show the relationships between each of them in soft topology and their general topological counterparts.

In Section 4, we characterize soft nearly Lindelöfness and improve several results related to soft nearly Lindelöfness using the concept of soft ω-regular open sets.

In Section 5, we give some conclusions and possible future work.

[4] Let Z be a universal set and E be a set of parameters. A soft set over Z relative to E is a function . The collection of all soft sets over Z relative to E will be denoted by .

Let .

(a) [22] If , then K will be denoted by .

(b) [22] If for all , then K will be denoted by .

(c) [19] If , then K will be denoted by and will be called a soft point of . The set of all soft points of will be denoted by .

For any ,

(a) [41] is said to be the null soft set and will be denoted by .

(b) [41] is said to be the absolute soft set and will be denoted by .

(c) [19] is said to be a countable soft set of if is a countable set for each . The collection of all countable soft sets of will be denoted by .

[41] Let . Then

(a) M is said to be a soft subset of N, denoted by , if for every .

(b) the soft union of M and N is denoted by and defined to be the soft set where for every .

(c) the soft intersection of M and N is denoted by and defined to be the soft set where for every .

(d) the difference of M and N is denoted by and defined to be the soft set where for every .

[42] Let . Then

(a) the soft union of these soft sets is the soft set denoted by and defined by for all .

(b) the soft intersection of these soft sets is the soft set denoted by and defined by for all .

[19] Let and . Then is said to belongs to M () if .

[7] A collection δ ⊆ is said to be a soft topology on Z relative to E if

(a) ,

(b) δ is closed under arbitrary soft union,

(c) δ is closed under finite soft intersection.

If δ is a soft topology on Z relative to E, then the triplet is said to be a soft topological space, the members of δ are called soft open sets in and their soft complements are called soft closed sets in . The family of all soft closed sets in will be denoted by .

From now on, topological space and soft topological space will be denoted by TS and STS, respectively. Let be a TS, be a STS, , and . Then the interior of U in , the closure of U in , the soft interior of K in , and the closure of K in will be denoted by , , , and , respectively.

[43] A subset U of a TS is said to be a regular open set in if . The complement of a regular open set in is said to be a regular closed set.

The family of all soft regular open sets (resp. regular closed sets) of a TS is denoted by (resp. ).

[44] A subset W of a TS is said to be an ω-regular open set in if for any , we find such that and is countable. The complement of an ω-regular open set in is said to be an ω-regular closed set.

The family of all ω-regular open sets (resp. ω-regular closed sets) of a TS is denoted by (resp. ).

A TS is said to be

(a) [45] semi regular if for every and every such that , we find such that .

(b) [46] almost regular if for every and every such that , we find such that .

(c) [44] semi ω-regular if for every and every such that , we find such that .

(d) [44] almost ω-regular if for every and every such that , we find such that .

[47] A soft set S of a STS is said to be a soft regular open set in if . The soft complement of a soft regular open set in is said to be a soft regular closed set.

The collection of all soft regular open sets (resp. soft regular closed sets) of a STS is denoted by (resp. ).

A STS is said to be

(a) [48] soft regular if for every and every such that , we find such that .

(b) [49] soft semi regular if for every and every such that , we find such that .

(c) [50] soft almost regular if for every and every such that , we find such that .

[51] Let be a STS and let . Then

(a) is said to be soft nearly Lindelöf if every soft cover of from has a countable subcover.

(b) K is said to be soft nearly Lindelöf relative to if every soft cover of K from has a countable subcover.

For any STS and any , is a topology on Z. This topology is denoted by .

Let be a TS and E be a set of parameters. Then

(a) is a soft topology on Z relative to E. This soft topology is denoted by .

(b) is a soft topology on Z relative to E. This soft topology will be denoted by .

It is not difficult to check that the two types of soft topology in Theorem 1.15 are extended soft topologies [53].

For any collection of TSs , is a soft topology on Z relative to E. This soft topology will be denoted by .

For any STS , is a soft topology on Z relative to E. This soft topology is denoted by .

A non-empty subset is said to have the countable soft intersection property if for any countable subfamily , we have .

Soft ω-regular open sets

In this section, we define soft ω-regular open sets. We investigate the properties of these soft sets and point out their relationships with some famous generalized soft open sets. We also show the relationships between soft ω-regular open sets in soft topology and their general topological counterparts.

A soft set S of a STS is said to be a soft ω-regular open set in if for any , there exists such that and . The soft complement of a soft ω-regular open set in is said to be soft ω-regular closed set.

The family of all soft ω-regular open sets (resp. soft ω-regular closed sets) of a STS is denoted by (resp. ).

For any STS , .

Straightforward. □

For any STS , .

Let and . Let . Then we have and . Hence, . □

The following example shows that the inclusion in Theorem 2.2 cannot be replaced by equality, in general:

Let , , ρ be the usual topology on Z, and . Since and , then by Theorem 2.3, . On the other hand, .

For any STS , .

Let and let . Then there exists such that and . Therefore, . □

Inclusion in Theorem 2.5 cannot be replaced by equality in general, as the following example clarifies:

Let , , and . Then . If , then there exists such that and . Since , then . Thus, . As a result, which is not correct. Therefore, .

For any STS , is a STS.

(1) We have .

(2) Let and let . Then and . So, there exist such that and . Now,

Since , then , and so . Also, . Hence, .

(3) Let . Let . Choose such that . Then there exists such that and . Now,

Since , then . Hence, . □

Let be a STS and . Then if and only if for every , there exists and such that and .

Necessity. Suppose that and let . Then there exists such that and . Put . Then and

Sufficiency. Suppose that for every , there exists and such that and . Let . Then by assumption, there exists and such that and . So, we have , and hence, . It follows that . □

Let be a STS such that for some , for every . Then

(a) For each , .

(b) For each , .

(c) For each , .

(a) Let . Then by Proposition 7 of [7], . To see that , suppose to the contrary that there exists . Then and . Since , then there exists such that and . Choose such that . Then we have , and hence . Since , then by assumption, . But , which is a contradiction.

(b) Let . Then . So, by (a), . And thus,

But , and

. Therefore, .

(c) Let . Since , then by (b), . Since , then by (a), . Thus, □

Let be a STS such that for some , for every . Then for every , .

Let . Then . So, by Proposition 2.9 (c),

Hence, . □

Let be a STS such that for all and and let . Then if and only if for every .

Necessity. Suppose that and let . Then by Theorem 2.10, for every .

Sufficiency. Suppose that for every . Then for every . So, by Proposition 2.9 (c), for every and hence, . It follows that . □

Let be a TS and E be any set of parameters. Let . Then if and only if .

Let be a STS such that for some , for every . Then for every , .

Let and let . Then and so there exists such that and . So, we have and by Theorem 2.10, . Also, since , then is a countable subset of Z. Therefore, . □

Let be a STS such that for all and , and let . If , then for every .

Suppose that and let . Then by Theorem 2.13, for every . □

Let be a STS such that for all and , and let such that for every . Is it true that ?

The following theorem gives a partial answer for Question 2.15:

Let be a TS and E be any set of parameters. Let . Then if and only if .

Necessity. Suppose that and let . Then and so, there exists such that and is countable. So, we have and by Corollary 2.12, . Moreover, . Hence, .

Sufficiency. Let . Choose . Then by Theorem 2.14, . □

Let be a collection of TSs. Then if and only if for all .

Necessity. Suppose that and let . Let . Then . So, there exists such that and . By Proposition 3.28 of [54], . Moreover, since , then is countable. Therefore, .

Sufficiency. Suppose that for all . Let . Then . So, there exists such that and is countable. Now, we have and by Proposition 3.28 of [54], . Moreover, . Therefore, . □

Let be a TS and E be any set of parameters. Let . Then if and only if for every .

Soft semi ω-regularity and soft almost ω-regularity

In this section, we define two new classes of soft separation axioms: soft semi ω-regular and soft almost ω-regular. We investigate their properties and point out their relationships with some famous soft separation axioms. We also show the relationships between each of them in soft topology and their general topological counterparts.

A STS is said to be soft semi ω-regular if for each and each such that , there exists such that .

Every soft semi regular STS is soft semi ω-regular.

Follows from the definitions and Theorem 2.2. □

Soft semi ω-regularity does not imply soft semi regularity in general:

Let , , and

. It is easy to see that . So, by Theorem 2.3, . Hence, is soft semi ω-regular. On the other hand, since but there is no such that , then is not soft semi regular.

Let be a soft semi regular STS such that for some , for every . Then is semi regular.

Let and such that . Choose such that . Then we have . Since is soft semi regular, then there exists such that . By Theorem 2.10, . Also, we have . It follows that is semi regular. □

If is a soft semi regular STS such that for all and , then is semi regular for every .

Let be a STS such that for all and , and is semi regular for all . Is it true that is soft semi regular?

The following theorem gives a partial answer for Question 3.6:

Let be a TS and E be any set of parameters. Then is soft semi regular if and only if is semi regular.

Necessity. Suppose that is soft semi regular. Choose . Then by Theorem 3.4, is semi regular. But . Hence, is semi regular.

Sufficiency. Suppose that is semi regular. Let and let . Then . Since is semi regular, then there exists such that . Thus, and by Corollary 2.12, . Therefore, is soft semi regular. □

Let be a collection of TSs. Then is soft semi regular if and only if is semi regular for all .

Necessity. Suppose that is soft semi regular and let . Let and such that . Then . Since is soft semi regular, then there exists such that . So, we have . Also, by Proposition 3.28 of [54], . Hence, is semi regular.

Sufficiency. Suppose that is semi regular for all . Let and such that . Then . Since is semi regular, then there exists such that . Thus, we have and by Proposition 3.28 of [54], . It follows that is soft semi regular. □

Let be a TS and E be any set of parameters. Then is soft semi regular if and only if is semi regular.

Let be a soft semi ω-regular STS such that for some , for every . Then is semi ω-regular.

Let and such that . Choose such that . Then we have . Since is soft semi ω-regular, then there exists such that . By Theorem 2.13, . Also, we have . It follows that is semi ω-regular. □

If is a soft semi ω-regular STS such that for all and , then is semi ω-regular for every .

Let be a STS such that for all and , and is semi ω-regular for all . Is it true that is soft semi ω-regular?

The following theorem gives a partial answer for Question 3.12:

Let be a TS and E be any set of parameters. Then is soft semi ω-regular if and only if is semi ω-regular.

Necessity. Suppose that is soft semi ω-regular. Choose . Then by Theorem 3.10, is semi ω-regular. But . Hence, is semi ω-regular.

Sufficiency. Suppose that is semi ω-regular. Let where , and let . Then . Since is semi ω-regular, then there exists such that . So, . Also, by Theorem 2.16, . It follows that is soft semi ω-regular. □

Let be a collection of TSs. Then is soft semi ω-regular if and only if is semi ω-regular for all .

Necessity. Suppose that is soft semi ω-regular and let . Let and such that . Then . Since is soft semi ω-regular, then there exists such that . Thus, we have . Also, by Theorem 2.17, . Hence, is semi ω-regular.

Sufficiency. Suppose is semi ω-regular for all . Let and such that . Then . Since is semi ω-regular, then there exists such that . Thus, we have and by Theorem 2.17, . It follows that is soft semi ω-regular. □

Let be a TS and E be any set of parameters. Then is soft semi ω-regular if and only if is semi ω-regular.

A STS is said to be soft almost ω-regular if for each and each such that , there exists such that .

Every soft almost ω-regular STS is soft almost regular.

Follows from the definitions and Theorem 2.2. □

The following two examples will show that the converse of Theorem 3.17 need not be true, and that soft semi ω-regularity and soft almost ω-regularity are independent concepts:

Let , , and . Then is soft almost regular and soft semi regular. On the other hand, since and with but there is not such that , then is not soft almost ω-regular.

Let be as in Example 2.6. It is not difficult to check that . Then clearly that is soft almost ω-regular. On the other hand, since but there is not such that , then is not soft semi ω-regular.

Every soft semi ω-regular soft almost ω-regular is soft regular.

Let be soft semi ω-regular and soft almost ω-regular. Let and such that . Since is soft semi ω-regular, then there exists such that . Since is soft almost ω-regular, then there exists such that . Therefore, is soft regular. □

Let be a TS and E be any set of parameters. Then is soft almost regular if and only if is almost regular.

Necessity. Suppose that is soft almost regular. Let and such that . By Corollary 2.12, . Pick . Since is soft almost regular and , then there exists such that . Thus, . Hence, is almost regular.

Sufficiency. Suppose that is almost regular. Let and such that . Then by Corollary 2.12, . Since is almost regular and , there exists such that . Thus, we have and . Hence, is soft almost regular. □

Let be a TS and E be any set of parameters. Then is soft almost ω-regular if and only if is almost ω-regular.

Necessity. Suppose that is soft almost ω-regular. Let and such that . By Theorem 2.16, . Pick . Since is soft almost ω-regular and , then there exists such that . Thus, . Hence, is almost ω-regular.

Sufficiency. Suppose that is almost ω-regular. Let and such that . Then by Theorem 2.16, . Since is almost ω-regular and , there exists such that . Thus, we have and . Hence, is soft almost ω-regular. □

Let be a collection of TSs. Then is soft almost regular if and only if is almost regular for all .

Necessity. Suppose that is soft almost regular and let . Let and such that . Then and by Proposition 3.28 of [54], . Since is soft almost regular, then there exists such that . Thus, we have and . Also, by Lemma 4.9 of [55], . Hence, is almost regular.

Sufficiency. Suppose that is almost regular for all . Let and such that . Then by Proposition 3.28 of [54], . Since is almost regular and , then there exists such that . Now, we have and . It follows that is soft almost regular. □

Let be a TS and E be any set of parameters. Then is soft almost regular if and only if is almost regular.

Let be a collection of TSs. Then is soft almost ω-regular if and only if is almost ω-regular for all .

Necessity. Suppose that is soft almost ω-regular and let . Let and such that . Then and by Theorem 2.17, . Since is soft almost ω-regular, then there exists such that . Thus, we have and . Also, by Lemma 4.9 of [55], . Hence, is almost ω-regular.

Sufficiency. Suppose that is almost ω-regular for all . Let and such that . Then by Theorem 2.17, . Since is almost ω-regular and , then there exists such that . Now, we have and . It follows that is soft almost ω-regular. □

Let be a TS and E be any set of parameters. Then is soft almost ω-regular if and only if is almost ω-regular.

We end this section with a characterization of soft almost ω-regularity:

A STS is soft almost ω-regular if and only if for each and each such that , there exists such that .

Necessity. Suppose that is soft almost ω-regular. Let and let such that . Then there exists such that . Thus, . Since , then . Let . Then and

Sufficiency. Obvious. □

Soft nearly Lindelöfness

In this section, we characterize soft nearly Lindelöfness and improve several soft nearly Lindelöfness-related results by employing the concept of soft ω-regular open sets.

We start with the following characterization of soft nearly Lindelöf STSs:

For a STS, the following are equivalent:

(a) is soft nearly Lindelöf.

(b) Every soft cover of from has a countable subcover.

(c) For every with , there exists a countable subfamily such that .

(d) For every with the countable soft intersection property, .

(a) ⟹ (b): Let be a soft cover of from . For each , there exists such that . Since , then there exists such that and . Since is soft nearly Lindelöf and soft cover of from , there exists a countable subset such that is a countable soft cover of . We have

Since H is countable and for each , then . Choose a countable subfamily of such that . Set . Then is a countable subcover of .

(b) ⟹ (c): Let with . Then with . So by (b), there exists a countable subfamily such that . Hence, .

(c) ⟹ (d): Suppose to the contrary that there exists with the countable soft intersection property such that . Then by (c), there exists a countable subfamily such that . Thus, does not have the countable soft intersection property, which is a contradiction.

(d) ⟹ (a): Let be a soft cover of from . Then with . By Theorem 2.2, . So, by (d), does not have the countable soft intersection property. Thus, there exists a countable subset such that . Hence, . It follows that is soft nearly Lindelöf. □

Let be soft nearly Lindelöf STS. Then for each , K is soft nearly Lindelöf relative to .

Let be soft nearly Lindelöf and let . Let such that . Since , then . Thus, for each , there exists such that and . Since is soft nearly Lindelöf, , and , then there exist a countable subfamily and a countable set with for each such that . Since and

then there exists a countable subfamily such that . Hence, is a countable subfamily of and . It follows that K is soft nearly Lindelöf relative to . □

The following corollary is an immediate consequence of Theorem 2.2, Theorem 4.2:

Let be soft nearly Lindelöf space. Then for each , K is soft nearly Lindelöf relative to .

Every soft semi ω-regular soft nearly Lindelöf STS is soft Lindelöf.

Let be soft semi ω-regular and soft nearly Lindelöf. Let be a soft cover of from δ. For each , there exists such that . Since is soft semi ω-regular, then for each , there exists such that . Then is a soft cover of from . Since is soft nearly Lindelöf, then by part (b) of Theorem 4.1, there exists a countable subset such that . Hence, is a countable subfamily such that . It follows that is soft Lindelöf. □

Every soft semi regular soft nearly Lindelöf STS is soft Lindelöf.

Every soft regular soft nearly Lindelöf STS is soft Lindelöf.

Let be soft almost regular and soft nearly Lindelöf space. Then for every such that , there exist such that , , and .

Since is soft almost regular, for each by Theorem 3.4 (ii) of [50] there exists such that and . Since is soft nearly Lindelöf space and is a soft cover of from , then there exists a countable set of soft points such that . It follows that and for all . Analogously there exists a family such that and for all . For every , let and . Let and . Then , , , and . □

Let be soft almost ω-regular and soft nearly Lindelöf space. Then for every such that , there exist such that , , and .

Since is soft almost ω-regular, for each by Theorem 3.27, there exists such that and . Since is soft nearly Lindelöf space and is a soft cover of from , then by Part (b) of Theorem 4.1, there exists a countable set of soft points such that . It follows that and for all . Analogously there exists a family such that and for all . For every , let and . Let and . Then , , , and . □

Conclusion

Soft ω-regular open sets have been introduced. It is proved that the collection of soft ω-regular open sets forms a soft topology that lies strictly between the classes of soft regular open sets and soft ω-open sets. Soft semi ω-regularity and soft almost ω-regularity have been also introduced as two new independent variants of soft regularity via soft ω-regular open sets. In addition, the relationships between soft topology and classical (parametric) topology have been revealed. Finally, nearly Lindelöfness has been characterized, and several results related to soft nearly Lindelöfness have been improved. In future studies, the following topics could be considered: 1) define new classes of soft functions using soft ω-regular open sets; 2) investigate the behavior of soft semi ω-regular and soft almost ω-regular STSs in the context of product STSs; and 3) characterize some types of soft Menger spaces using soft ω-regular open sets.

Declarations

Author contribution statement

Samer Al Ghour: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

No data was used for the research described in the article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.