1. Introduction
Some problems in medicine, engineering, the environment, economics, sociology, and other fields have their own doubts. Therefore, we are unable to deal with these problems by conventional 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. Molodtsov [
4] argues that each of these theories has its own set of problems. These difficulties mainly come from the inadequacy of the parameterization tool for the theories. Research through soft set theory has included almost all branches of science. 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 [
5,
6,
7,
8,
9,
10].
General topology, as one of the main branches of mathematics, is the branch of topology that deals with the basic definitions of set theory and structures used in topology. It is the foundation of most other branches of topology, including algebraic topology, geometric topology, and differential topology. Shabir and Naz [
11] initiated soft topology, which is a new branch of topology that combines soft set theory and topology. Since that time, the generalization of topological concepts in soft topology has become the focus of many researchers, such as soft compact [
12], soft connected [
13], soft paracompact [
13], soft extremely disconnected [
14], soft Menger spaces [
15], soft separable spaces [
16], soft separation axioms [
17,
18,
19], soft metric spaces [
20,
21,
22], soft homogeneous spaces [
23,
24], and soft maps [
25,
26], and substantial contributions can still be made.
Generalizations of soft open sets play an effective role in soft topology through their use to improve on some known results or to open the door to redefine and investigate some of the soft topological concepts such as soft compactness, soft correlation, soft class axioms, soft assignments, etc. Some important generalizations of soft open sets appear in [
27,
28,
29,
30,
31,
32,
33,
34]. Soft
-open sets were defined and investigated by Al Ghour and Hamed [
35], who used them to characterize soft Lindelofness and soft weakly Lindelofness. Then several research papers related to soft
-open sets were introduced [
36,
37]. The authors of [
38] defined and investigated
-open sets as a generalization of open sets, which are a strong form of
-open sets. In the present paper, we extend
-open sets to include soft topological spaces. We investigate this class of soft sets, especially in soft Lindelof and soft anti-locally soft topological spaces. Additionally, via soft
-open sets, we introduce the class of soft
-regular spaces. This research will be of interest to many researchers, especially those who are interested in generalizing soft topological concepts or those who are interested in linking topology and soft topology.
The arrangement of this article is as follows:
In
Section 2, we collect the main definitions and results that will be used in this research.
In
Section 3, we define the class of soft
-open sets. With the help of examples, we show that this class of soft sets forms a soft topology that lies strictly between the soft topology of soft open sets and the soft topology of soft
-open sets, and we give some sufficient conditions for the equality between the three soft topologies. Additionally, we show that soft closed soft
-open sets are soft
-open sets in soft Lindelof soft topological spaces. In addition, we study the correspondence between soft
-open sets in soft topological spaces and
-open sets in topological spaces.
In
Section 4, we study the correspondence between soft anti-locally countable soft topological spaces and anti-locally countable topological spaces. The relationships between the soft
-open sets (respectively, soft regular open sets, soft
-open sets) of a given soft anti-locally countable soft topological space and the soft
-open sets (respectively, soft regular open sets, soft
-open sets) of the soft topological space of soft
-open sets generated by it are then investigated.
In
Section 5, we introduce
-regularity in topological spaces via
-open sets, which is strictly between regularity and
-regularity. We also introduce soft
-regularity in soft topological spaces via soft
-open sets, which is strictly between soft regularity and soft
-regularity. We give several characterizations and relationships regarding
-regularity and soft
-regularity. Moreover, we study the correspondence between soft
-regularity in soft topological spaces and
-regularity in topological spaces.
In
Section 6, we give some conclusions and possible future work.
3. Soft -Open Sets
Definition 8. A soft set M of a STS is called a soft -open set in if for any , we find and K such that and . Soft complements of soft -open sets in are called soft -closed sets in .
The family of all soft -open sets in will be denoted by .
Theorem 5. For any STS , .
Proof. For the inclusion , let and . If , then K and . This shows that . For the inclusion , let and let . Then we find and K such that and . This shows that . □
The following two examples will show that any of the inclusions in Theorem 5 cannot be replaced by equality, in general:
Example 1. Let , , μ be the usual topology on Y, and σ ={}. Then, clearly, . On the other hand, if , then there exist and K such that and , and so . This implies that and so which is impossible. It follows that .
Example 2. If , , and , then .
Theorem 6. If is a soft and soft p-space, then .
Proof. Follows from Theorem 5 of this paper and Theorem 9 of [
35]. □
Theorem 7. For any STS , is a STS.
Proof. According to Theorem 5, , and so , . Let and let . Then we have and . Thus, we find and such that , , , and . Hence, , , and . Therefore, . Let and let . Choose such that . Then there exist and K such that and . Therefore, . □
Theorem 8. If is a STS such that , then .
Proof. Follows from Theorem 5 of this paper and Theorem 4 of [
35]. □
Theorem 9. For any soft locally countable STS , is a discrete STS.
Proof. Let M∈. Choose . Since is soft locally countable, then there exists such that . Thus, we have , and . Therefore, M∈. □
Corollary 1. Let be a STS such that Y is a countable set. Then is a discrete STS.
Theorem 10. Let be a STS and . Then if and only if for each , there exists such that and .
Proof. Necessity. Let and let . Then we find and K∈ such that and . As , then . Thus, .
Sufficiency. Suppose that for each , there exists such that and . Let . Then we find such that and . If , then and . Thus, . □
Theorem 11. Let be a STS. If , then .
Proof. Let and let . Choose such that . As , then . Thus, we find and K such that and . Therefore, we have , , and . □
Corollary 2. Let be a STS. If , then .
As can be shown by the following example, the condition ‘’ is essential in Theorem 11.
Example 3. Let , , , μ be the usual topology on Y, and . Since , then . On the other hand, if , then there exists and such that and . Thus, , and hence . Therefore, W is a countable set and , which is impossible.
Theorem 12. Let be a STS. Then for each , .
Proof. Let . Let and let . Choose such that . Since , then we find and K such that and . So, we have , is countable, and . On the other hand, . Therefore, . □
Corollary 3. Let be a STS. If , then for every .
Proof. Let and let . Then . Thus, by Theorem 12, . □
Lemma 1. Let Y be an initial universe and B be a set of parameters. If is a collection of TSs, then for every and , .
Proof. Let and . Since , then . Since , then . □
Theorem 13. Let Y be an initial universe and B be a set of parameters. If is a collection of TSs, then .
Proof. By Theorem 12 of this paper and Theorem 8 of [
23] we have
for every
. Hence,
. To show that
, by Theorem 6 of [
23], it is sufficient to show that
for all
and
. Let
and
. Let
. Then we have
. Hence, there exist
and a countable subset
such that
and
. Thus, we have
,
and
. Since by Lemma 1,
, then
. Hence,
. □
Corollary 4. If is a TS and B is any set of parameters, .
Proof. Let for all . Then . Thus, by Theorem 13, □
Theorem 14. If soft , then is soft .
Proof. Let such that . As is a soft , then we find such that , , and . Therefore, is a soft . □
The following example will show that the converse of Theorem 14 is false:
Example 4. Let ,, and . Then it is clear that is not a soft . On the other hand, by Corollary 1, isa soft .
Theorem 15. Let be soft Lindelof. If , then .
Proof. Let . By Theorem 10, for every , we find such that and . Set . Then is a soft open cover of M. As and is soft Lindelof, then there exists a countable subcover of . Since and , then . □
Corollary 5. Let be soft second countable. If , then .
Theorem 16. If is a soft Lindelof STS, then .
Proof. Suppose that is soft Lindelof. Let and let . Then we find and such that and . Thus, . Since and by Theorem 5, , then . Hence, . □
Corollary 6. If is a soft second countable STS, then
Theorem 17. Let be STS. If , then for some and .
Proof. Let
. If
, then
such that
and
. If
, then we find
. Since
, then we find
and
such that
and
. Thus,
Let . Then such that . □
Corollary 7. Let be STS. If , then for some and .
Proof. Follows from Theorems 5 and 17. □
4. Soft Anti-Local Countability
From now on, ALC and SALC will denote anti-locally countable and soft anti-locally countable, respectively.
Theorem 18. A STS is SALC if and only if is SALC.
Proof. Follows from Theorems 5 and 13 of [
35]. □
Theorem 19. Let be a STS. If is ALC for every , then is SALC.
Proof. Suppose that is ALC for every . Let . Choose . Then . Since is ALC, then is uncountable. Hence, . It follows that is SALC. □
Now we will give an example that shows that the opposite of Theorem 19 is not true in general.
Example 5. Let , , and where and . Then is SALC. On the other hand, since , then is not ALC.
Theorem 20. Let Y be an initial universe and let B be a set of parameters. Let be a collection of TSs. Then is SALC if and only if is ALC for each .
Proof. Necessity. Suppose that is SALC. Let and let . Then . Since is SALC, then . Since for every , then is uncountable. Hence, is ALC.
Sufficiency. Suppose that
is ALC for each
. Let
. By Theorem 11 of [
23],
for each
. Therefore, by Theorem 19,
is SALC. □
Corollary 8. For any TS and any set of parameters B, is ALC if and only if is SALC.
Proof. For each , set . Then . Thus, by Theorem 20 we get the result. □
Theorem 21. If is SALC and , then .
Proof. Suppose that is SALC and let . Then clearly . We are going to show that . Let . We find N and such that and . Thus, and hence . Since and by Theorem 18, is SALC, then . Therefore, we have with , and hence . □
Corollary 9. If is SALC and is a soft -closed set, then .
As the following example shows, the condition “SALC” is necessary in Theorem 21.
Example 6. Let , and . Then ⊆ and . On the other hand, since Y is countable, then by Corollary 1 is a discrete STS, and hence .
Theorem 22. For any SALC STS , .
Proof. Let
be SALC and let
. Then
. By Theorem 5,
. Thus, we have
On the other hand, since , then by Theorem 21, we have . Thus, . Hence, . □
The inclusion in Theorem 22 cannot be replaced by equality in general, as will be shown in the following example:
Example 7. Consider the STS , where μ is the usual topology on . Since is ALC, then by Corollary 8, is SALC. On the other hand, it is not difficult to check that .
Theorem 23. For any SALC STS , .
Proof. To show , let . Then . Since , then by Theorem 21, , and so . On the other hand, since is soft -closed in , then by Corollary 9, . Therefore, .
To show that , let . Then . Since is soft -closed in , then by Corollary 9, . On the other hand, since , then by Theorem 21, . Therefore, . Hence, . □
In Theorem 23, the condition in ’SALC’ is essential, as the following example shows:
Example 8. Let , . Consider the STS . Then , while .
Theorem 24. For any SALC STS , .
Proof. Let
. Then
. Since
is soft
-closed in
, then by Corollary 9,
. Thus,
. Additionally, since by Theorem 5,
, then
Therefore, . □
Theorem 25. If is SALC and soft , then is soft .
Proof. Let be SALC and soft . Let such that . As is soft , then we find with . Since by Theorem 5, , then . Additionally, by Theorem 21, we have and . Hence, . Therefore, is a soft . □
5. Soft -Regularity
Definition 9. A TS is called -regular if given any point and such that , there exist and such that , , and .
Theorem 26. A TS is -regular if given any and , there exists such that .
Proof. Necessity. Suppose that is soft -regular. Let and . Then and . Since is -regular, then there exist and such that , , and . Since and , then .
Sufficiency. Suppose that whenever and , then there exists such that . Let and . Then there exists such that . Let . Then and . Therefore, is -regular. □
Theorem 27. Locally countable TSs are -regular.
Proof. Let
be locally countable. Let
and
. Since
is locally countable, then
is a discrete TS (see [
38]). Therefore,
. Hence, we have
and
. Therefore, by Theorem 26,
is
-regular. □
Theorem 28. Every regular TS is -regular.
Proof. Let be a regular TS. Let and . Since is regular, then there exists such that . Therefore, is -regular. □
The following is an example of an -regular TS that is not regular:
Example 9. Let and . Then is not regular, while by Theorem 27, is -regular.
Theorem 29. Every ALC -regular TS is regular.
Proof. Let
be ALC and
-regular. Let
and
. Since
is
-regular, then there exists
such that
. Since
is ALC, then
(see [
38]). Therefore,
. Hence,
is regular. □
Theorem 30. Every -regular TS is ω-regular.
Proof. Let be -regular. Let and . Since is -regular, then there exists such that . Since by Theorem 5, , then . Therefore, . Hence, is -regular. □
The following is an example of an -regular TS that is not -regular:
Example 10. Let and . Since is ω-locally indiscrete, then by Corollary 15 of [42], is ω-regular. Claim 1. is not -regular.
Proof of Claim 1. Suppose to the contrary that is -regular. Then by Theorem 26, there exists such that . Therefore, . Thus, and so we find and a countable set such that and . Therefore, we have , a contradiction. □
Definition 10. A STS is called soft -regular if given any and such that , there exist and such that , , and .
Theorem 31. For any STS , the following are equivalent:
(a) is soft -regular;
(b) For any and any such that , there exists such that .
Proof. (a) ⟹ (b): Let and such that . Then and . By (a), we find and such that , and . Therefore, .
(b) ⟹ (a): Let and such that . Then by (b), we find such that . Let . Then and . This shows that is soft -regular. □
Theorem 32. Soft locally countable STSs are soft -regular STSs.
Proof. Follows from Theorems 9 and 31. □
Theorem 33. Soft regular STSs are soft -regular STSs.
Proof. Let be soft regular. Let and such that . We find such that . As , then . Thus, . Therefore, is soft -regular. □
Theorem 34. Soft -regular STSs are soft ω-regular STSs.
Proof. Let be soft -regular. Let and such that . We find such that . As , then . Thus, . Therefore, is soft -regular. □
Theorem 35. If is soft -regular, then is -regular for every .
Proof. Let
. Let
and let
such that
. Pick
such that
. Since
and
is soft
-regular, then we find
such that
, and thus,
. By Theorem 12, we have
. Additionally, by Proposition 7 of [
11], we have
. Therefore, we have
, and
. It follows that
is
-regular. □
Theorem 36. Let be a collection of STSs. Then is soft -regular if and only if is -regular for every .
Proof. Necessity. Let
be soft
-regular and let
. Then by Theorem 36,
is
-regular. Furthermore, by Theorem 8 of [
23],
. Therefore,
is
-regular.
Sufficiency. Let
be
-regular for every
. Let
. Let
and let
. Then
. Since
is
-regular, then there exists
such that
. Thus, we have
with
. Additionally, by Theorem 13, we have
. Hence, by Proposition 2 of [
37],
. This shows that
is soft
-regular. □
Corollary 10. For any TS and any set of parameters B, is soft -regular if and only if is -regular.
Proof. For every , let . Then . The result follows from Theorem 37. □
Now we give an example to show that the converse of Theorem 33 need not be true, in general:
Example 11. Let be as in Example 8 and B be any set of parameters. Then is -regular but not regular. Thus, by Corollary 10 and Corollary 4 of [37], is soft -regular but not soft regular. Now we give an example to show that the converse of Theorem 35 need not be true, in general:
Example 12. Let be as in Example 9 and B be any set of parameters. Then is ω-regular but not -regular. Thus, by Corollary 5 of [37] and Corollary 10, is soft ω-regular but not soft -regular. The following example shows that the implication in Theorem 36 is not reversible, in general:
Example 13. Let be as in Example 7 of [37]. It is proved in [37] that and are regular TSs. Thus, they are -regular TSs. On the other hand, it is proved in [37] that is not soft ω-regular, and so is not soft -regular. Theorem 37. If is a soft -regular STS and , then is soft -regular.
Proof. Let and . Choose such that . As , , and is soft -regular, we find and such that , and . Therefore, , with , and . Moreover, by Corollary 2, . □