1. Introduction
In our surroundings, there are various difficulties that involve some uncertainties, which are often seen in many different areas such as medical science, biology, ecology, engineering, social science, economics, and so on. For several years, many philosophers, mathematicians, and computer scientists have been attempting to find adequate instruments to cope with uncertainties. One of these instruments is the soft set (SS), which was first proposed by Molodtsov [
1]. Compared to forgoing strategies, the soft set is adequate to process uncertainties and is free from their inherent limitations; for instance, it does not require the preconditions such as a membership function in the fuzzy sets or an equivalence relation in the rough sets.
The first attempt to establish the main characteristics of the soft set theory was conducted by Maji et al. [
2]. They put forward the basic notions between soft sets such as soft intersection and soft union operators, a complement of a soft set, absolute and null soft sets. Some shortcomings and drawbacks caused by this study were adjusted and by Ali et al. [
3]. Quite recently, it has been proposed that some soft operators and operations are consistent with their analogs in the crisp set theory by Al-shami and El-Shafei [
4]. To address the complicated problems, researchers created novel environments by integrating soft sets with some uncertain tools such as fuzzy and rough sets [
5,
6].
As performed with topology initiated by fuzzy sets, topologists made use of soft sets to introduce the concept of a soft topology (ST). Two approaches were given by Çaǧman et al. [
7] and Shabir and Naz [
8] to define STSs. The difference between them is the way of selecting the set of parameters for each element of the soft topology as a variable or constant. This study will go along with the line of Shabir and Naz, who stipulate the necessity of a constant set of parameters. Afterward, many researchers, scholars and intellectuals popularized the classical principles and concepts of topology in the frame of STs. Min [
9] restated the relationship between the spaces of soft
and soft
. He also showed that the soft closed and open subsets of a soft regular space must be stable; i.e., all components are the same. To preserve almost all properties and characterizations produced by the interaction between separation axioms and other topological concepts as well as the systematic relations of soft
in the frame of STSs, researchers introduced a new type of soft
-spaces by El-Shafei et al. [
10]. Singh and Noorie [
11] made a comparative study among some types of soft separation axioms. The concepts of soft compact and Lindelöf spaces were researched by some authors [
12,
13]. Interesting applications of these abstract concepts “soft separation axioms” and “soft compactness” were the goals of the ground-breaking manuscripts [
14]. Kharal and Ahmad [
15] pursued the idea of functions between soft topological spaces and explored main themes. Next, the notions of soft continuity and homeomorphism were examined in [
16,
17]. It worth noting that Al-shami [
18] pioneered soft functions by adopting a binary relation over the family of soft points, which ease the way of computations and justify some definitions of soft themes such as injective and surjective soft functions. Ameen [
19] investigated some structural soft sets that are preserved under some non-continuous soft mappings.
In 2013, Chen [
20] introduced the idea of soft semi-open sets following a similar technique to that given in the classical topologies. He initiated some topological concepts with respect to this idea and pointed out that they widen their counterparts obtained by the family of soft open sets. In a similar way, the concepts of
-open, soft omega open, and soft semi
-open sets were described by [
21,
22,
23], respectively. Many soft topological principles have been investigated from the view of these generalizations as presented in [
24,
25,
26]. To endorse the importance of generalizations of soft open sets, Al-shami [
27] showed how the nutrition systems of individuals are estimated using soft somewhat open sets. Elsayed [
28] defined a new type of soft set called a soft equivalent set and probed its main features.
It is well known that an ST produces some classical topologies (named “parametric topologies”), so it is natural to look at the relationships between them in terms of interchangeable topological concepts and properties between them. To depict how topological concepts and properties navigate from ST to classical topology and vice versa, Al-shami and Kočinac [
29] fostered the basic conceptions that illustrate this theme, in particular, in the cases of an extended ST. This behavior was discussed for some soft topological notions such as operational characterizations [
30], caliber and chain conditions [
31] and soft separation axioms [
32]. The present work’s contribution lies in this area of ST by inventing a new technique to produce novel generalizations of soft open sets using the classical topologies inspired by the original ST. This technique paves the road to reformulate the topological concepts and establish different classes of soft topologies. In the last, we draw the reader’s attention to the fact that there are various contributions to ST structures such as those defined on specific kinds of STs such as vietoris topology [
33], Menger spaces [
34], maximal topologies and expandable spaces [
35]; also, some extensions of ST such as supra soft topology [
36], infra soft topology [
36] and weak soft structures [
37].
To know the content of this work, we mention its layout in the following. The fundamentals to make this manuscript self-contained are reviewed in
Section 2. We adopt a new approach to study generalizations of soft open sets in
Section 3. We define the concepts of weakly soft semi-open and weakly soft semi-closed subsets using their analog obtained from a parametric topology instead of the soft interior and closure operators. In
Section 4, we introduce the operators of a weakly soft semi-interior, weakly soft semi-closure, weakly soft semi-boundary, and weakly soft semi-limit, and scrutinized their master features. In
Section 5, we familiarize the concept of weakly soft semi-continuity and construct an example that elaborates that some equivalent conditions of soft continuity are invalid for this type of continuity. Finally, we dedicate
Section 6 to summarizing the main contributions and suggest some future work.
2. Preliminaries
We invoke in this segment the basic definitions and results necessitated to comprehend this work.
Definition 1 (
[1])
. Let be a universal set, be the family of all subsets of and be a set of parameters (or attributes). A set-valued function is said to be an SS. For simplicity, we symbol an SS by and write as
and ;
That is, an SS over institutes a parameterized families of subsets of such that we call each a -component of .
Through this work, the family of all SSs over with a set of parameters is designated by .
Definition 2 (
[2,3,38,39])
. Let and be SSs over and . Then,
- (i)
If for all , then is called a complement of . We write .
- (ii)
If for every , then we call an absolute soft set; it is symbolized by . The complement of an absolute soft set is called a null soft set; it is symbolized by ϕ.
- (iii)
If and for all , then we call a soft point and symbolized by . We write if .
- (iv)
If or ∅ for all , then we call a pseudo constant soft set. Note that there are pseudo constant soft sets.
Definition 3 (
[40])
. If and are SSs over such that for each , then is called an SS of . Symbolically, we write .
Definition 4 (see,
[4])
. Let and be SSs over . Then:
- (i)
Soft union: where for all .
- (ii)
Soft intersection: where for all .
- (iii)
Soft difference: where for all .
- (iv)
Soft product: where for all .
The definition of soft functions was adjusted by Al-shami [
18] as follows.
Definition 5 (
[18])
. Let and be crisp functions. A soft function of into is a relation such that each is related to one and only one such that
for all .
In addition, for each .
That is, the image of and pre-image of under a soft function are, respectively, given by:
, and
.
A soft function is described as surjective (injective, bijective) if its two crisp functions satisfy this description.
Proposition 1 (
[15,41])
. Let be a soft function and let and be SSs of and , respectively. Then
- (i)
.
- (ii)
If is injective, then .
- (iii)
.
- (iv)
If is surjective, then
Definition 6 ([
8]).
A subfamily ℑ of is said to be an ST if the following terms are satisfied:- (i)
and ϕ are elements of ℑ.
- (ii)
ℑ is closed under the arbitrary unions.
- (iii)
ℑ is closed under the finite intersections.
We will call the triplet an STS. Each element in ℑ is called soft open, and its complement is called soft closed.
Definition 7 (
[8])
. Let be a soft subset of an STS . Then, the soft interior of , denoted by , and the soft closure of , denoted by , are, respectively, given by
- (i)
The soft union of all soft open sets that are contained in .
- (ii)
The soft intersection of all soft closed sets containing .
Proposition 2 (
[8])
. Let be an STS. Then, institutes a topology on for every . We will call this topology a parametric topology.
Definition 8 (see,
[8])
. Let be an SS of an STS . Then and are respectively defined by and , where and are, respectively, the interior and closure of in .
Definition 9 (
[26])
. An STS is called:
- (i)
Full provided in the case that every non-null soft open set has no empty component.
- (ii)
Hyperconnected provided in the case that the soft intersection of any two non-null soft open sets is non-null.
Definition 10 (
[12,38])
. Let be an STS.
- (i)
If all pseudo-constant SSs are elements of ℑ, then ℑ is called an enriched ST.
- (ii)
ℑ with the property “ iff for each ” is called an extended ST.
A deep investigation on the enriched and extended STs was conducted on [
29]. The corresponding properties between these kinds of STs was one of the important and interesting results obtained in [
29]. Henceforth, this type of soft topology will be called an extended soft topology. Under this soft topology, researchers proved several findings that associated soft topology with its parametric topologies. As a matter of fact, the following result represents a key point in the proof of many findings.
Theorem 1 (
[29])
. An STS is extended if and for any soft subset .
Definition 11. A soft subset of is termed as:
- (i)
Soft α-open [21] provided that . - (ii)
Soft semi-open [20] provided that . - (iii)
Soft β-open [42] provided that . - (iv)
Soft somewhat open set (briefly, soft -open) [26] provided that or .
Definition 12 (
[17,20,21,26,42])
. A soft function is said to be:
- (i)
Soft open if is a soft open set where is soft open.
- (ii)
Soft continuous (resp., soft α-continuous, soft semi-continuous, soft -continuous) if is a soft open (resp., soft α-open, soft semi-open, soft -open) set where is soft open.
- (iii)
Soft bi-continuous (resp., soft homeomorphism) if it is soft open and continuous (resp., bijective and soft bi-continuous).
A topological property is a property of an STS that is preserved under any soft homeomorphism.
Theorem 2 (
[29])
. If is soft continuous, then is continuous for each .
3. On Weakly Soft Semi-Open Sets
Herein, we display the concept of “weakly soft semi-open sets”, which is the main idea of this document. It shows that the family of weakly soft semi-open sets is a proper generalization of soft open sets. We provide some interesting examples to prove that the extended soft topology guarantees that this family is a real superset of the families of soft semi-open sets and a genuine subset of the family of soft -open sets. In addition, we demonstrate some differences between this family and other generalizations of soft open sets such as this family is not closed under soft unions. In the end, we reveal how this family conducts itself with some topological principles such as functions and the product of soft spaces.
Definition 13. An SS of an STS is called a weakly soft semi-open set if it is a null soft set or there is a component of it which is a nonempty semi-open set. That is, for all or for some .
We call a weakly soft semi-closed set if its complement is weakly soft semi-open.
Proposition 3. A subset of an STS is weakly soft semi-closed if or for some .
Proof. : If is a weakly soft semi-closed set, then or for some . Therefore, or for some , as demanded.
: Let be an SS such that or for some . Then, or for some . This implies that is weakly soft semi-open. Hence, is weakly soft semi-closed, as demanded. □
The family of weakly soft semi-open (weakly soft semi-closed) subsets is not closed under soft union or soft intersection as the next example clarifies.
Example 1. Let the universe be the real numbers set and a set of parameters be . Let ℑ be the ST on generated by and . Let be the set of rational numbers and take the next weakly soft semi-open sets: Then:
- (i)
is not a weakly soft semi-open set.
- (ii)
is not a weakly soft semi-open set.
Take the next weakly soft semi-closed sets: Then:
- (i)
is not a weakly soft semi-closed set.
- (ii)
is not a weakly soft semi-closed set.
Proposition 4. Let be a full and hyperconnected STS. Then, the soft intersection of soft semi-open and weakly soft semi-open subsets is weakly soft semi-open.
Proof. Assume that and are, respectively, soft semi-open and weakly soft semi-open sets. Then, there exists a non-null soft open set and such that and is a nonempty semi-open subset of . So, there exists a nonempty open subset of . This means that ℑ contains a non-null soft open set with . Since ℑ is soft hyperconnected and full, we obtain . Therefore, and has a nonempty intersection. It follows from general topology that is a nonempty semi-open subset of . Hence, is a weakly soft semi-open set. □
Corollary 1. Let be a full and hyperconnected STS. Then, the soft intersection of soft open subsets and weakly soft semi-open is weakly soft semi-open.
Remark 1. (i)Every pseudo-constant soft subset is a weakly soft semi-subset because for all or for some .
- (ii)
An SS of with (resp. ) is weakly soft semi-open (resp. weakly soft semi-closed).
The next proposition is straightforward.
Proposition 5. Every soft open set is weakly soft semi-open.
Now, we derive the conditions under which some relationships that associate weakly soft semi-open sets with some generalizations of soft open sets are obtained.
Proposition 6. Every soft α-open (soft semi-open) subset of an extended STS is weakly soft semi-open.
Proof. Let be a non-null soft semi-open set. Then, . Since ℑ is an extended soft topology, we obtain for each . This implies that there is a component of which is a nonempty -open set. As we know that any -open set is semi-open; hence, is weakly soft semi-open.
Following similar arguments, one can prove the case of a soft semi-open set. □
Proposition 7. If is extended, then every weakly soft semi-open set is soft -open.
Proof. Let be a non-null weakly soft semi-open set. Then, there is a component of which is a nonempty semi-open set. So, for some . Since ℑ is extended, we obtain . This completes the proof. □
It cannot be dispensed of the stipulation of “extended soft topology” imposed in Proposition 6 and Proposition 7, as the following example elaborates.
Example 2. Let the universe be and a parameters set be . Take the family ℑ consisting of ϕ, and the following SSs over with
;
;
;
;
;
; and
.
Then, is an STS. Remark that is soft semi-open because . However, it is not a weakly soft semi-open set because . In addition, is a weakly soft semi-open set because . However, it is not a soft -open set because .
The example below shows that Proposition 6 and Proposition 7 are irreversible.
Example 3. Let the universe be and a parameters set be . Take the family ℑ consisting of ϕ, and the following SSs over with
;
;
;
;
;
;
;
;
;
;
;
;
; and
.
Then, is an extended STS. Now, is weakly soft semi-open because is a nonempty semi-open set. However, it is not a soft semi-open set because . In addition, , is soft -open because . However, it is not a weakly soft semi-open set because and is empty.
Proposition 8. The image and pre-image of weakly soft semi-open set under a soft bi-continuous function is weakly soft semi-open.
Proof. To show the case of an image, let be a soft bi-continuous function from an STS to an STS and let be a weakly soft semi-subset of . Suppose that there exists such that is a nonempty semi-open subset and let . According to Theorem 2, it follows from the soft bicontinuity of that is a bi-continuous function. It is well known that a continuity of implies that , and an openness of implies that for each subset U of . This implies that . According to Definition 12, we find that is a nonempty semi-open subset of ; hence, is a weakly soft semi-open subset of an STS . □
Corollary 2. The property of being a weakly soft semi-open set is a topological property.
Proposition 9. The product of two weakly soft semi-open sets is weakly soft semi-open.
Proof. Suppose that and are weakly soft semi-open subsets and let . Then, there are such that and are nonempty semi-open subsets. Now, such that . As we know from the classical topology that the product of two nonempty semi-open subsets is still a nonempty semi-open subset; therefore, is a nonempty semi-open subset. Hence, is a weakly soft semi-open subset. □
4. Weakly Semi-Interior and Weakly Semi-Closure Soft Points
This part is dedicated to producing novel operators for each SS, namely, weakly semi-interior, weakly semi-closure, weakly semi-boundary, and weakly semi-limit soft points. We display their main characteristics and depict the relationships among them. By appropriate examples, we evince that the property that says that the “weakly semi-interior (resp., weakly semi-closure) of SS need not be weakly semi-open (resp., weakly semi-closed) sets” is false in general.
Definition 14. The weakly semi-interior soft points of an SS of an STS , denoted by , is defined as the union of all weakly soft semi-open sets contained in .
By Example 1, we remark that the weakly semi-interior soft points of an SS need not be a weakly semi-open set. That is, does not imply that is a weakly semi-open set.
One can prove the next propositions.
Proposition 10. Let be an SS of an STS and . Then, if there is a weakly soft semi-open set that contains such that .
Proposition 11. Let , be SSs of an STS . Then,
- 1.
.
- 2.
if , then .
Corollary 3. For any two subsets , of an STS , we have the following results:
- 1.
.
- 2.
.
Proof. It automatically comes from the following:
1. and .
2. and . □
In Proposition 11 and Corollary 3, the inclusion relations are proper. To demonstrate that, consider Example 1 and take the following SSs:
, ,
, , and .
We remark on the following properties:
- 1.
.
- 2.
whereas .
- 3.
whereas .
- 4.
whereas .
Definition 15. The weakly semi-closure soft points of an SS of an STS , denoted by , is defined as the intersection of all weakly soft semi-closed sets containing .
By Example 1, we remark that the weakly semi-closure points of an SS need not be a weakly semi-closed set. That is, does not imply that is a weakly semi-closed set.
Proposition 12. Let be an SS of an STS and . Then, if for each weakly soft semi-open set contains .
Proof. Let . Suppose that there is weakly soft semi-open set containing with . Then, . Therefore, . Thus, . This is a contradiction, which means that , as demanded.
Suppose that for each weakly soft semi-open set contains . Let . Then, there is a weakly soft semi-closed set containing with . So, and , which contradicts the assumption. Hence, we obtain the desired result. □
Corollary 4. If such that is a weakly soft semi-open set and is an SS in , then .
Proposition 13. The following properties hold for an SS of an STS .
- (1)
.
- (2)
.
Proof. 1. If , then there is a weakly soft semi-open set with . Therefore, and hence, . Conversely, if , we can follow the previous steps to verify .
2. Following similar approach given in 2. □
The proof of the next proposition is following from Definition 15.
Proposition 14. Let , be SSs of an STS . Then
- 1.
.
- 2.
if , then .
Corollary 5. The following results hold for any subsets , of an STS .
- 1.
.
- 2.
.
Proof. It automatically comes from the following:
1. and .
2. and . □
In Proposition 14 and Corollary 5, the inclusion relations are proper. To demonstrate that, consider Example 1 and take the following SSs:
Let , , and , .
We remark on the following properties:
- 1.
.
- 2.
whereas .
- 3.
whereas .
Definition 16. A soft point is said to be a weakly semi-boundary soft point of an SS of an STS if belongs to the complement of .
All weakly semi-boundary soft points of are called a weakly semi-boundary set and denoted by .
Proposition 15. for every SS of an STS .
Proof. (De Morgan’s law);
(Proposition 13(2)). □
Corollary 6. For every SS of an STS , the following properties hold.
- 1.
.
- 2.
.
- 3.
.
- 4.
.
Proof. 1. Obvious.
- 2.
. By 2 of Proposition 13, we obtain the required relation.
- 3.
.
- 4.
;
;
;
. □
Proposition 16. Let be SSs of an STS ; then, the following properties hold.
- 1.
.
- 2.
.
Proof. By substituting in Formula 3 from Corollary 6, the proof follows. □
Proposition 17. Let be an SS of an STS . Then
- 1.
if .
- 2.
if .
Proof. 1. Suppose that . Then, by (4) from Corollary 6, and hence . Conversely, let . Since and , by (3) from Corollary 6, . Therefore, , as demanded.
- 2.
Assume that . Then, , as demanded. Conversely, if , then by (3) from Corollary 6, and hence , as demanded. □
Corollary 7. Let be an SS of an STS . Then, if .
Definition 17. A soft point is said to be a weakly semi-limit soft point of an SS of an STS if for each weakly soft semi-open set containing .
All weakly semi-limit points of are called a weakly semi-derived set and denoted by .
Proposition 18. Let and be subsets of an STS . If , then .
Proof. Straightforward by Definition 17. □
Corollary 8. Consider and are subsets of an STS . Then:
- 1.
.
- 2.
.
Theorem 3. Let be an SS of an STS ; then, .
Proof. The side is obvious. To prove the other side, let . Then, and . Therefore, there is a weakly soft semi-open containing with . Thus, . Hence, we find that . □
Corollary 9. Let be a weakly soft semi-closed subset of an STS ; then, .
5. Continuity via Weakly Soft Semi-Open Sets
In this section, we tackle the concept of soft continuity via weakly soft semi-open sets. We derive its main characterizations and point out that the loss of the property says that “the weakly semi-interior of an SS is weakly soft semi-open” results to evaporating some descriptions of this type of soft continuity. An elucidative counterexample is supplied.
Definition 18. A soft function is said to be weakly soft semi-continuous if the inverse image of each soft open set is weakly soft semi-open.
It is straightforward to prove the next result, so we omit its proof.
Proposition 19. If is a weakly soft semi-continuous function and is a soft continuous function, then is weakly soft semi-continuous.
The proof of the next proposition follows from Proposition 5.
Proposition 20. Every soft continuous function is weakly soft semi-continuous.
Proposition 21. Let be a soft function such that is extended. Then
- 1.
If is soft α-continuous (soft semi-continuous), then is weakly soft semi-continuous.
- 2.
If is weakly soft semi-continuous, then is soft -continuous.
Proof. It, respectively, follows from Proposition 6 and Proposition 7. □
Proposition 22. A soft function is weakly soft semi-continuous if the inverse image of every soft closed subset is weakly soft semi-closed.
Proof. Necessity: Suppose that is a soft closed subset of . Then, is soft open. Therefore, is weakly soft semi-open. Thus, is a weakly soft semi-closed set.
Following a similar argument, one can prove the sufficient part. □
Theorem 4. If is weakly soft semi-continuous, then the next properties are equivalent.
- 1.
For each soft open subset of , we have .
- 2.
For each soft closed subset of , we have .
- 3.
for each .
- 4.
for each .
- 5.
for each .
Proof. (1→2): Suppose that is a soft closed subset of . Then, is soft open. Therefore, . According to Proposition 13, we obtain .
: For any soft set , we have . Then, .
: It is obvious that for each . By 3, we obtain . Therefore, .
: Let be an arbitrary soft set in . Then, . So that, . Hence, .
: Suppose that is a soft open subset in . By 5, we obtain . However, , so , as demanded. □
To prove that the converse of the above theorem is generally false, the next example is shown.
Example 4. Let and with . Let and be two STs defined on and , respectively, with the same set of parameters , where
;
; and
.
Let be a soft function, where is the identity function and is defined as follows
and .
Now, is not a weakly soft semi-open subset of because . Then, is not weakly soft semi-continuous. On the other hand, , and , which means that the all properties given in Theorem 4 hold true.
Now, we introduce the concepts of weakly soft semi-open, weakly soft semi-closed and weakly soft semi-homeomorphism functions.
Definition 19. A soft function is said to be weakly soft semi-open (resp., weakly soft semi-closed) provided that the image of each soft open (resp., soft closed) set is weakly soft semi-open (resp., weakly soft semi-closed).
Proposition 23. Let be a soft function and be any SS of .
- 1.
If is weakly soft semi-open, then .
- 2.
If is weakly soft semi-closed, then .
Proof. 1. Let be an SS of . Then, is a weakly soft semi-open subset of and so .
- 2.
The proof is similar to that of 1. □
Proposition 24. A bijective soft function is weakly soft semi-open if it is weakly soft semi-closed.
Proof. Necessity: Let be a weakly soft semi-closed subset of . Since is weakly soft semi-open, is weakly soft semi-open. By the bijectiveness of , we obtain . So that, is a weakly soft semi-closed set. Hence, is weakly soft semi-closed. To prove the sufficient, we follow a similar approach. □
Proposition 25. Let be a weakly soft semi-closed function and be a soft closed subset of . Then, is weakly soft semi-closed.
Proof. Suppose that is a soft closed subset of . Then, there is a soft closed subset of with . Since is a soft closed subset of , then is also a soft closed subset of . Since , then is a weakly soft semi-closed set. Thus, is a weakly soft semi-closed. □
Proposition 26. The next three statements hold for soft functions and .
- 1.
If is soft open and is soft j-open such that is extended, then is weakly soft semi-open, where .
- 2.
If is weakly soft semi-open and is soft continuous surjective, then is weakly soft semi-open.
- 3.
If is soft open and is weakly soft semi-continuous injective, then is weakly soft semi-open.
Proof. 1. Without loss of generality, let . Then, consider as a soft open subset of . So, is a soft open subset of . Thus, is a soft -open subset. According to Proposition 6, is a weakly soft semi-open subset. Hence, is weakly soft semi-open.
2. Suppose that is a soft open subset of . Then, is a soft open subset of . Therefore, is a weakly soft semi-open subset of . Since is surjective, then . Thus, is weakly soft semi-open.
3. Let be a soft open subset of . Then, is a soft open subset of . Therefore, is a weakly soft semi-open subset of . Since is injective, . Thus, is weakly soft semi-open. □
We cancel the proof of the next finding because it can be obtained following a similar approach to the above proposition.
Proposition 27. The next three statements hold for soft functions and .
- 1.
If is soft closed and is soft j-closed, then is weakly soft semi-closed, where .
- 2.
If is weakly soft semi-closed and is soft continuous surjective, then is weakly soft semi-closed.
- 3.
If is soft closed and is weakly soft semi-continuous injective, then is weakly soft semi-closed.
Definition 20. A bijective soft function in which is weakly soft semi-continuous and weakly soft semi-open is called a weakly soft semi-homeomorphism.