Innovative Strategy for Constructing Soft Topology

: To address the complexity of daily problems, soft set theory has emerged as a valuable tool, providing innovative mathematical techniques to manage vast amounts of data and ambiguity. The study of soft topology involves the investigation of various properties of soft sets and functions, as well as the development of new mathematical models and techniques for addressing uncertainty. The main motivation of this paper is to delve deeper into the subject and devise new methodologies to address real-world challenges more effectively and unlock the full potential of soft sets in various applications. In this paper, we present a novel soft topology, which is constructed using soft single points on a nonempty set V in relation to a topology on V . We investigate and study the behaviors and properties associated with this particular type of soft topology. Furthermore, we shed light on the soft separation axioms with this type of soft topology and investigate whether these axioms are inherited from the corresponding ordinary topology or not. Our study is concerned with examining the connection between ordinary topologies and the soft topologies generated that arise from them, with the aim of identifying their interdependencies and potential implications. By studying the connection between soft topologies and their corresponding ordinary topologies, researchers are able to gain a deeper understanding of the properties and behaviors of these structures and develop new modeling approaches for dealing with uncertainty and complexity in data.


Introduction
Experts across various disciplines, including medical science, economics, systems engineering, artificial intelligence, and other fields, regularly encounter the challenge of molding complex systems that involve uncertainty.While classical probability theory, fuzzy set theory [1], and rough set theory [2] are commonly used mathematical tools to address such problems, they may not always yield satisfactory results due to limitations in their parameterizations.In 1999, Molodtsov [3] explored a fresh strategy for dealing with uncertainty, known as soft set theory, which overcomes the shortcomings of other methods.The technique involves the use of soft sets, which are a parametrized family of subsets of a universal set.Unlike other methods, soft set theory does not impose any specific conditions for object illumination and parameters can be selected in various forms, such as sentences, words, numbers, or mappings.As a result, the theory is highly flexible and convenient to apply in practical settings.In addition, Molodtsov applied soft set theory to various areas including the study of function smoothness, game theory, and more, demonstrating its versatility and broad applicability.In [4], Maji et al. examined the role of soft sets in decision-making problems and presented a technique for aggregating soft set parameters to facilitate decision making.In this context, a soft set can be employed to represent each option or alternative under consideration, with its parameters reflecting the various factors that are relevant to the decision, such as financial cost, danger, potential benefit, and other quantitative or qualitative measures.By combining and analyzing these parameters, the proposed method enables decision makers to arrive at informed and effective decisions.In 2003, Maji et al. [5] presented and analyzed the concept of soft operations between soft sets.Yang [6] focused on rectifying certain properties in soft set theory that were introduced in [5].In [7], several researchers have sought to refine some soft operations or develop new ones, in order to enhance their functionality and utility in soft set theory.Recently, in 2019, Hussain et al. [8] defined soft single points and soft real points.They have found intriguing and fundamental characteristics.
The exploration of general topology is a notable and significant area of mathematics that focuses on the application of concepts from set theory and structures employed in the field of topology.In 2011, Shabir and Naz [9] initiated a new area of study within topology called soft topology, which combines the theories of soft sets and topology.They established several fundamental concepts such as soft open (closed), soft neighborhood, soft closure, soft separation axioms, soft regular spaces, and soft normal spaces.In 2012, Hussain and Ahmad [10] furthered the study of soft topology by exploring the properties of soft open (closed), soft neighborhood, and soft closure.In addition, they presented the concepts of the soft interior, soft exterior, and soft boundary.Zarlutana et al. [11] explored and studied the concept of soft mappings and soft continuous mapping between two collections of soft sets.In 2015, Hussain [12] established further characterization of soft continuous mapping.
Numerous research studies have showcased practical applications of soft sets and their topologies across diverse disciplines to address real-world issues.This underscores the importance of studying and investigating these sets and their topologies, providing a strong motivation for us to delve deeper into the subject and devise new methodologies to address real-world challenges more effectively and unlock the full potential of soft sets in various applications.This leads to improved solutions in the face of uncertainty and complexity.
This article aims to establish a cohesive perspective to advance the development of soft topological spaces by building upon the findings of ordinary topology.Our paper presents an innovative methodology to construct a soft topology by utilizing the set V both as an initial universe set and as a set of parameters.Let κ be a topology on V, and let B = { Lv,Q : for all v ∈ V and Q ∈ κ} ∪ Φ be the collection of all soft single points associated with κ, where Q is an open neighborhood of an element v in κ.We proceed to construct a soft topology based on this collection B, which forms a base for this soft topology.We will study the relationship between the soft closure (the soft interior) and the closure (the interior) of any soft set, with respect to soft topology and its corresponding ordinary topology.We will prove that two soft topologies that are generated by soft single points are soft homeomorphic if their corresponding ordinary topologies are homeomorphic.We will define a soft relative topology generated by soft single points with respect to an ordinary relative topology.Furthermore, we will study the soft separation axioms that were introduced in [9] of a soft topology generated by soft single points, with respect to an ordinary topology.We will investigate whether these axioms are inherited from an ordinary topology or not.Investigation methods of producing soft topology from classical topology is an important and fruitful area of research in topology and related fields.Researchers are able to gain a deeper understanding of the properties and behaviors of these structures and develop new tools and techniques for studying these spaces, which can then be applied to other areas of mathematics and science.
The organization of the paper is summarized in the following sections.This section comprises an introduction and two subsections.The first subsection provides a comprehensive literature review for constructing soft topology, while the second subsection points out the related topics of soft sets and their applications.Section 2 provides a review of important definitions and results from soft set theory and soft topological spaces that are relevant to the present study.In Section 3, we introduce the generated soft topology by soft single points on a nonempty set V with respect to any ordinary topology on V. We prove that two soft topologies are soft homeomorphic if their corresponding ordinary topologies are homeomorphic.Also, we examine the soft relative topology generated by soft single points with respect to a relative topology.In Section 4, we investigate the soft separation axioms for this type of soft topology.We verify whether the soft separation axioms of this type of soft topology are inherited from its corresponding ordinary topology or not.In Section 5, we present our conclusion which supports our results and explains the future work.

Literature Review of Studies of Constructing Soft Topology
Let us review the literature that examined the topology and different methods of constructing soft topology.In [13], Milan conducted the study on soft topological space, exploring their connection with a topological space defined on the Cartesian product of two sets through a homeomorphism.In 2019, Terepeta [14] discussed a method of constructing a soft topology from any ordinary topology on a nonempty set V while considering any possible set of parameters.In [15], Al Ghour et al. explored a process of constructing a soft topology by utilizing an indexed family of ordinary topologies on a nonempty set V while considering any possible set of parameters.In 2020, Alcantud [16] thoroughly revisited the standard method for deriving soft topology from ordinary topology, which was introduced in [14].Furthermore, they delved into the concepts of soft separability and soft countability within the framework of this soft topology.In 2022, Azzam et al. [17] conducted an exploration of six soft operators and employed them to construct soft topologies.They inferred that all resulting soft topologies are equivalent and share identical properties with their classical counterparts when considering enriched and extended conditions.Recently, El-Atik and Azzam [18] conducted research on a method of transformation to depict the complex networks of the human brain more simplistically.In addition, they explored a topological model for simplicial complexes and utilized it to represent the brain as a union of simplicial complexes.This approach potentially offers a way to diagnose brain cancer.

Related Topics to Soft Set Theory and Their Applications
In 1965, Zadeh [1] proposed a fuzzy set theory as a way to handle uncertainty in data representation and reasoning.Maji et al. [19] presented the concept of fuzzy soft sets.This concept combines the principles of fuzzy sets and soft sets, providing a more comprehensive framework for handling uncertainty in decision making and data analysis.The application of fuzzy sets and fuzzy soft sets has been studied by many researchers across various domains.
In 1982, Pawlak [2] presented a rough set theory: a mathematical framework that offers a systematic approach to address the challenges of uncertainty and imprecision in data analysis.In 2011, Feng et al. [20] worked to generalize the rough sets model that builds upon soft sets and proposed the concept of soft rough sets.The soft rough set is characterized as a parametrized subset of a set, utilized for determining lower and upper approximations of a subset, diverging from the traditional approach of equivalent classes.This soft rough set has been applied in decision making by many researchers.Several academic experts employed soft rough sets to address medical challenges, like the diagnosis of Chikungunya disease, the diagnosis of Coronavirus disease, and many medical applications.One of our plans is to conduct a comprehensive study and undertake the construction of soft topology on these sets in the future.

Preliminaries
This section provides a summary of relevant definitions and results with soft sets theory and soft topology that will be referenced later in the paper.In the rest of the paper, we use V to denote an initial universe set and P(V) to denote its power set.The symbol ∅ is used to denote the empty set.

Soft Sets Theory
Definition 1. [3] Suppose that D is a set of parameters and L is a mapping from D into P(V).A soft set over V, symbolized by LD , is defined as: The collection of all soft sets over V with respect to the set of parameters D will be denoted by S(V) D .

Definition 2. [5,7]
• A soft set LD over V is said to be a null soft set if L(d) = ∅ for all d ∈ D and is symbolized by ΦD .

•
A soft set LD over V is said to be an absolute soft set if L(d) = V for all d ∈ D and is symbolized by ṼD .

•
The relative complement of a soft set LD is the soft set where L(d) c is the complement of a subset L(d).
Definition 3. Suppose that LD and QE are soft sets over V.
The soft sets LD and QE are soft equal if LD ⊆ QE and QE ⊆ LD .

•
Ref. [7] The restricted intersection of two soft sets LD and QE , symbolized by LD ∩ QE , is the soft set TH , such that H = D ∩ E, and T represents a map from H into P(V) given by T Ref. [5] The union of two soft sets LD and QE , symbolized by LD ∪ QE , is the soft set TH , such that H = D ∪ E, and T represents a map from H into P(V) defined as: Definition 5. [9] Suppose that LD is a soft set over V and v ∈ V. We say that v belongs to the soft set Now, we recall the definition of soft mappings between two collections of soft sets.Definition 6. [11] Suppose that D and E are sets of parameters and V and Z are initial universe sets.We assume S(V) D and S(Z) E are two collections of soft sets and λ : V → Z and µ : D → E are mappings.Then a soft mapping is defined by : 1.
The image of LD ∈ S(V) D is a soft set in S(Z) E such that for all e ∈ E,
The following definitions and results center around the soft single points that were presented in [8].
Definition 7. Suppose that V is a nonempty set.Then for each v ∈ V and M ⊆ V, we have a function called a fixed point function, The set {(v, L(v))} is called a soft single point on V.The function L varies uniquely for each v ∈ V and M ⊆ V.The collection of all soft single points on V is symbolized by S V .

Remark 2. •
The set {(v, ∅)} is considered an empty soft single point for each v ∈ V.

•
If V is a set containing n elements, then the total count of soft single points on V is equal to

Soft Topology
Definition 9. [9] Let κ be a collection of S(V) D .Then, (V, κ) D is called a soft topological space if and only if the following requirements are met: 1.

2.
The union of any number of soft sets in κ belongs to κ.

3.
The restricted intersection of any two soft sets in κ belongs to κ.
Every element of κ is called a soft open set in V.The relative complement of the soft open set is called a soft closed set in V. Proposition 1. [9] Suppose that (V, κ) D is a soft topological space.Then, for all d ∈ D, we have The validity of the converse of the above proposition is generally not true, as stated in reference [9].Definition 10. [9,10] Suppose that (V, κ) D is a soft topological space and LD ∈ S(V) D . 1.
The soft closure of LD is the restricted intersection of all soft closed sets which contain LD and is symbolized by Cl κ ( LD ).

2.
The soft interior of LD is the union of all soft open sets which are contained in LD and is symbolized by Int κ ( LD ).

3.
LD is a soft closed set if and only if LD = Cl κ ( LD ).

3.
LD is a soft open set if and only if Definition 11. [9] Suppose that (V, κ) D is a soft topological space and LD ∈ S(V) D .Then, the closure of LD with respect to the topology κ d is the soft set Cl κ d ( LD ), where Cl Proposition 2. [9] For any soft topological space (V, κ) D and LD ∈ S(V) D , we have Cl κ d ( LD ) ⊆ Cl κ ( LD ).
Definition 12. [12] Suppose that (V, κ1 ) D and (Z, κ2 ) E are two soft topological spaces.Suppose that λ : V → Z and µ : D → E are two mappings.Then, a map Υ λ,µ : S(V) D → S(Z) E is said to be a soft homeomorphism if Υ λ,µ is bijection; Definition 13. [9] Let (V, κ) D be a soft topological space and M be a nonempty subset of V.Then, the soft relative topology on M is defined as: Now, we state the main definitions of soft separation axioms with respect to ordinary points in V, which were introduced in [9].Definition 14. Suppose that (V, κ) D is a soft topological space.1.
(V, κ) D is said to be a soft T 0 -space if for any two distinct points v, w ∈ V, there exists a soft open set LD such that either v ∈ LD and w / ∈ LD or v / ∈ LD and w ∈ LD .

2.
(V, κ) D is said to be a soft T 1 -space if for any two distinct points v, w ∈ V, there exist two soft open sets LD and QD such that v ∈ LD , w / ∈ LD and v / ∈ QD , w ∈ QD .

3.
(V, κ) D is said to be a soft T 2 -space if for any two distinct points v, w ∈ V, there exist two soft open sets LD and QD such that v ∈ LD , w ∈ QD , and LD ∩ QD = ΦD .4.
(V, κ) D is said to be a soft regular space if for any soft closed set TD and v / ∈ TD , there exist two soft open sets LD and QD such that v ∈ LD , TD ⊆ QD , and LD ∩ QD = ΦD .

5.
(V, κ) D is said to be a soft normal space if for any two soft disjoint closed sets RD and TD in κ, there exist two soft open sets LD and QD such that RD ⊆ LD , TD ⊆ QD , and LD ∩ QD = ΦD .Definition 15.Suppose that (V, κ) D is a soft topological space.1.
(V, κ) D is said to be a soft T 3 -space if it is both soft T 1 -space and soft regular space.2.
(V, κ) D is said to be a soft T 4 -space if it is both soft T 1 -space and soft normal space.

A New Approach for Constructing a Soft Topology from an Ordinary Topology
This section is focused on introducing a soft topology that is generated by soft single points on V in relation to any ordinary topology on V. Additionally, we examine several properties pertaining to this particular type of soft topology.Definition 16.Suppose that (V, κ) is a topological space.We define a fixed point function for each v ∈ V and Q ∈ κ, , for all w = v} is called a soft single point on V with respect to κ.The set of all soft single points on V with respect to κ is symbolized by B.
Definition 17. Suppose that (V, κ) is a topological space.Let where Φ = {(v, ∅) : for all v ∈ V}.Then, the soft topology generated by soft single points on V with respect to κ is as follows: and is symbolized by (STGP) (V,κ) .
To clarify our terminology, V will henceforth indicate both the initial universe set and the set of parameters.We will use the notation S(V) to represent the collection of all soft sets over V and any soft set over V will be symbolized by L instead of LV .
The first thing we do is to prove that κ in the above definition meets the soft topology conditions.Theorem 4. Suppose that (V, κ) is a topological space.Then, κ in Definition 17 is a soft topology on V.

Proof. 1.
Since V ∈ κ and V is an open neighborhood for each v ∈ V, Lv,V ∈ B for all v ∈ V.Then, their union belongs to κ.Thus, Ṽ = { ∪ Lv,V : for all v ∈ V} ∈ κ.Also, Φ ∈ κ by Definition 17.

3.
Let Qi ∈ κ.Then, we have for all i, Qi = { ∪ Lv,G : Lv,G ∈ B}.Since κ is the union of an arbitrary number of elements in B, then Hence, κ is a soft topology on V.
Definition 18. Suppose that κ is an (STGP) (V,κ) .Then, each element of κ is called a soft single point open set.Example 3. • Let V be an infinite set and κ be a co-finite topology on V.Then, is a soft base of κ.Thus, κ is called a soft co-finite topology generated by soft single points on V with respect to κ.

•
Let R be the set of real numbers and U be the usual topology on R.Then, is a soft base of Ũ .Thus, Ũ is called a soft usual topology generated by soft single points on R with respect to U .
Proof.Suppose that κ 1 is finer or strictly finer than κ 2 .We assume that B1 and B2 are soft bases of κ1 and κ2 , respectively.Let R ∈ κ2 .Then, Hence, R ∈ κ1 .This proves the theorem.The following lemma provides evidence that De-Morgan's laws apply to all soft sets over V with respect to the set of parameters.The proof can be found in ( [9]).
The restricted intersection of any numbers of soft single point closed sets is a soft single point closed set; 3.
The union of any two soft single point closed sets is a soft single point closed set.
Proof.It is clear by applying Theorem 4 and Lemma 1.
While the converse of Proposition 1 may not hold (as noted in [9]), our subsequent result confirms that the converse does indeed hold in the particular situation we are considering.
Theorem 9. Let (V, κ) be a topological space.Then, κ is an (STGP) (V,κ) if and only if κ v = {L(v) : L ∈ κ} is a topology on V for all v ∈ V.
Proof.Necessity.Suppose that κ is an (STGP) (V,κ) .Therefore, by Proposition 1, κ v is a topology on V for all v ∈ V. Sufficiency.Suppose that κ v = {L(v) : L ∈ κ} is a topology on V for all v ∈ V.By the definition of κ, κ v = {∅ and Q, where Q is an open neighborhood of v in κ} for all v ∈ V.For all v ∈ V, we define the soft sets Lv,Q = {(v, Q): v ∈ Q and (w, ∅) for all w = v}.Therefore, we have the set B = { Lv,Q : for all v ∈ V and Q ∈ κ} ∪ Φ.Thus, by Theorem 4, As Propositions 2 and 3 state, Cl κ d ( L) ⊆ Cl κ ( L) for any L ∈ S(V) D , and the equality holds if and only if Cl κ d ( L) is a soft closed set.However, the converse holds in general in our specific case, as demonstrated by the following theorem.
By the definition of κ, we have for all closed set Now, we introduce the definition of the closure of any L ∈ S(V) with respect to any ordinary topology on V. Definition 20.Let (V, κ) be a topological space.Let L ∈ S(V).Then, the closure of L with respect to κ is defined as: where Cl κ (L(v)) is the closure of L(v) in κ.
In the theorem that follows, we describe the relationship between Cl κ ( L) and Cl κ ( L) for any L ∈ S(V).
Proof.We have by Theorem 11, L ⊆ Cl κ ( L) ⊆ Cl κ ( L).Then, In general, Theorem 11 cannot be reversed, as shown by the following example.Proof.Necessity.Suppose that Cl κ ( L) = R and R is a soft single point closed set.Then, R is a soft single point closed set which contains L. If Cl κ ( L) = Q, then Q is the smallest soft single point closed set which contains L. Thus, Q ⊆ R since Q is the smallest soft single point closed set which contains L. Therefore, Cl κ ( L) ⊆ Cl κ ( L).By Theorem 11, we obtain Cl κ ( L) = Cl κ ( L).
Sufficiency.It is clear.
Now, we proceed to define the interior of any L ∈ S(V) with respect to any ordinary topology on V.
Definition 21.Suppose that (V, κ) is a topological space and L ∈ S(V).Then, the interior of L with respect to κ is defined as: where Int κ (L(v)) is the interior of L(v) in κ.Theorem 14.Let κ be an (STGP) (V,κ) and L ∈ S(V).The interior of L with respect to the topology κ v is the soft set Int κ v ( L), where Int The relationship between Int κ ( L) and Int κ ( L) for any L ∈ S(V) is explained in the following theorem.Theorem 15.Let κ be an (STGP) (V,κ) and L ∈ S(V).Then, Int κ ( L) ⊆ Int κ ( L).
Proof.By Theorem 15, we have Int κ ( L) ⊆ Int κ ( L) ⊆ L.Then, From the following example, it is evident that the converse of Theorem 15 is not true in general.
Sufficiency.It is clear.
Corollary 4. Suppose that κ is an (STGP) (V,κ) and L ∈ S(V).Then Int κ (Int κ ( L)) = Int κ ( L). Suppose that κ 1 is a topology on V and κ 2 is a topology on Z.We assume that κ1 is an (STGP) (V,κ 1 ) and κ2 is an (STGP) (Z,κ 2 ) .We study whether κ 1 and κ 2 being homeomorphic as topological spaces implies that κ1 and κ2 are homeomorphic as soft topological spaces.In our situation, where V and Z are treated as initial universe sets and the sets of parameters, Definition 6 specifies that we should only consider the mapping λ =: µ : V → Z.
Proof.Necessity.Suppose that Υ λ is a soft continuous map.Let Q be a nonempty open set in κ 2 , Lz,Q = {(z, Q) : Q is an open neighborhood of z in κ 2 and L(w) = ∅ for all w = z} ∈ κ2 .

Conclusions
In this paper, we have introduced a new technique for constructing a soft topology on a nonempty set V by using soft single points with respect to an ordinary topology on V.The relationship between the soft closure (the soft interior) and the closure (the interior) of a soft set, under both the soft topology and the corresponding ordinary topology, has been investigated.We have demonstrated that the soft topologies that are generated by soft single points exhibit soft homeomorphism if their corresponding ordinary topologies are homeomorphic.In addition, a soft relative topology that arises using soft single points with respect to an ordinary relative topology has been studied.We have examined the soft separation axioms of this type of soft topology, specifically determining whether these axioms are inherited from the ordinary topology or not.We have observed that this soft topology inherits the T 0 and T 1 axioms from its corresponding ordinary topology, while also being recognized as non-hausdorff space (T 2 space) when the set V has more than one element.Furthermore, the conditions that are necessary for this soft topology to satisfy the requirements of a soft regular space and a soft normal space have been examined.The construction of a soft topology from an ordinary topology provides a means of applying the principles of the soft topology to real-world problems.It allows for the use of soft sets in a way that is consistent with the existing framework of an ordinary topology, making it easier to apply soft topology concepts in practical applications.Additionally, the relationship between the soft topology and the corresponding ordinary topology can be explored, leading to a better understanding of the connections between the two and the properties that are shared or differ between them.This method of constructing a soft topology from an ordinary topology is an important step in making these tools more widely applicable and accessible in a variety of fields.To advance this area of research, we plan to investigate and analyze further properties of this soft topology, like soft compactness, soft connectedness, and others.In addition, we plan to extend the study into other directions, such as geometry and algebraic topology, to gain fresh perspectives and expand the scope of knowledge in this field.

Remark 3 .
The set B in Definition 17 is a soft base of κ since every soft single open set is a union of elements from B.

Remark 5 .Definition 19 .
If κ 1 and κ 2 are incomparable, then κ1 and κ2 are incomparable too.The relative complement of a soft single point open set is called a soft single point closed set.

Proof.
Suppose that Int κ ( L) = R.Then, R is the largest soft single point open set which contained in L. Therefore, R(v) ⊆ L(v) for all v ∈ V, and R(v) is an open set which contained in L(v) in κ v .By the definition of κ, we have for all open set R(v) in κ v , {(v, R(v)) : R(v) is an open neighborhood of v in κ and (w, ∅) for all w = v} is a soft single point open set.Since R is the largest soft single point open set which contained in L, then R(v) is the largest open set which contained in L