Homeomorphism and Quotient Mappings in Infrasoft Topological Spaces

Abstract

In this paper, we contribute to infrasoft topology which is one of the recent generalizations of soft topology. Firstly, we redefine the concept of soft mappings to be convenient for studying the topological concepts and notions in different soft structures. Then, we introduce the concepts of open, closed, and homeomorphism mappings in the content of infrasoft topology. We establish main properties and investigate the transmission of these concepts between infrasoft topology and its parametric infratopologies. Finally, we define a quotient infrasoft topology and infrasoft quotient mappings and study their main properties with the aid of illustrative examples.

1. Introduction

We face vagueness, ambiguity, and representation of imperfect knowledge in different areas such as economics, engineering, medical science, sociality, and environmental sciences. Mathematicians, engineers, and scientists, particularly those who focus on artificial intelligence, are seeking for approaches to solve the problems that contain vagueness. But they experienced a trouble: how they can formulate uncertain concepts that may not involve mathematically definite results. This means that there is a need for alternative mathematical concepts. Therefore, they have begun to look for different fields of research which leads to initiate several set theories as an alternative to George Cantor’s set theory such as fuzzy set (and its generalizations such as intuitionistic fuzzy set and pythagorean fuzzy set), rough set, multiset, and recently soft set.

Soft set was proposed by Molodtsov [1] as one of the nonstatistical mathematical approaches targeting to deal with ambiguous, undefined, and imprecise meaning. It is characterized by flexibility and fruitful applications compared with other uncertainty theories. Since Molodtsov put forward the concept of soft sets, many scholars have applied it in several research areas such as decision-making problems [2], systems of linear equations [3], computer science [4], engineering [5], and medical sciences [6].

The year 2011 was the beginning point of the interaction between soft set theory and topology. Simultaneously, Shabir and Naz [7] and Çaǧman et al. [8] initiated the concept of soft topology. However, they used different techniques to formulate soft topology. On the one hand, Shabir and Naz formulated soft topology on the collection of soft sets over a universal crisp set and a fixed set of parameters. On the other hand, Çaǧman et al. formulated soft topology on the collection of soft sets over an absolute soft set and different sets of parameters. We conduct this study in the frame of Shabir and Naz’s definition which is more analogous to the classical topology.

Kharal and Ahmad [9] defined soft mappings using two ordinary (crisp) mappings, one of them between the sets of parameters and the other between the universal sets. However, we reformulate this definition using the concept of soft points to be convenient for studying in different soft structures. We classify soft mappings by soft spaces into different families such as continuous, open, closed, and homeomorphism mappings. In addition, soft mappings enable us to classify topological concepts in terms of preservation under specific classes of soft mappings. In [10], the authors presented another point of view to study soft mappings with a medical application. Zorlutuna and Çakir [11] investigated continuity between soft topological spaces. The authors of [12] presented new relations between ordinary points and soft sets to define new types of soft separation axioms. Quite recently, Al-shami and Kočinac [13] investigated the conditions under which some concepts are kept between soft topology and its parametric topologies. Kočinac et al. [14] discussed selection principles in the context of soft sets.

Some generalizations of soft topology were introduced and studied. For example, El-Sheikh and Abd El-Latif [15] established the concept of suprasoft topological spaces by neglecting a finite intersection condition of a soft topology. Thomas and John [16] formulated the concept of soft generalized topological spaces, and Zakari et al. [17] originated the concepts of soft weak structures. Lately, Al-shami et al. [18] have initiated and investigated the concept of infrasoft topology. Then, Al-shami [19] studied infrasoft compactness and its application to fixed point theorem. Also, Al-shami and Abo-Tabl [20] defined the concepts of infrasoft connected and infrasoft locally connected spaces. As a continuation of this work, we conduct this study. We aim to redefine soft mappings and explore some types of them through the infrasoft topology content.

We organize this article as follows. After the introduction section, Section 2 mentions some concepts and notions that clarify the investigations of this paper. In Section 3, we show shortcoming of soft mapping defined in [9] and reformulate it simulating to the definition of (crisp) mappings. In Section 4, we define new types of infrasoft mappings and investigate main properties. Among other things, we prove that these infrasoft mappings are preserved under product of infrasoft topological spaces and investigate the transmission of these concepts between infrasoft topology and its parametric infratopologies. We devote Section 5 to study quotient topology and mappings in the content of infrasoft topological spaces. Finally, we outline the fundamental obtained results and suggest some upcoming works in Section 6.

2. Preliminaries

In this section, we mention the main concepts that make this paper self-contained.

Definition 1. (see [1]). We called a pair a soft set over the universal set with a set of parameters provided that is a mapping from to the power set of . We usually write a soft set as follows and . The symbol denotes the set of all soft sets over with any subset of .
We recall some special types of soft sets.(i)If for each , then is called a null soft set [21].(ii)If for each , then is called an absolute soft set [21].(iii)If and for each , then is called a soft point. It is denoted by . The set of all soft points over with is denoted by . We say that belongs to a soft set , denoted by if [22]. More details concerning soft points were given in [23].

Definition 2. (see [24]). A soft set is called a relative complement of a soft set provided that a mapping is defined by for each .
Since an infrasoft topological space is defined under a constant set of parameters, we will recall the previous definitions and results under a constant set of parameters.

Definition 3. (see [21, 24, 25]). Let and be two soft sets over .(i) is a soft subset of a soft set , symbolized by , if for all .  and are called soft equal if and .(ii)The soft intersection of and , symbolized by , is a soft set such that a mapping is given by for each .(iii)The soft union of and , symbolized by , is a soft set such that a mapping is given by for each .More details concerning soft intersection and union were given in [26].

Definition 4. (see [27]). The Cartesian product of and , which are defined over and , respectively, is a soft set, denoted by , given by for each .

Definition 5. (see [9]). Let and be two crisp mappings. Then, a soft mapping is defined as follows: the image of a soft set in is a soft set in such that , and a mapping is given by

Definition 6. (see [9]). Let be a soft mapping. Then, the preimage of a soft set in is a soft set in such that , and a mapping is given by

Definition 7. (see [9]). We call an injective (resp. a surjective, a bijective) soft mapping if and are injective (resp. surjective, bijective) mappings.

Proposition 1. (see [9]). Consider is a soft mapping. Let and be two soft sets in and and be two soft sets in . Then, we have the following results:(i)If , then (ii)If , then (iii) and the equality relation holds if is injective(vi) and the equality relation holds if is surjective(v)(vi)(vii)(viii)

Definition 8. (see [18]). The collection of soft sets over under a fixed set of parameters is called an infrasoft topology on if it is closed under finite soft intersection as well as it contains .
The triple is called an infrasoft topological space. The term given to each member of is called an infrasoft open set, and the relative complement each member of is called an infrasoft closed set.

Definition 9. (see [18]). Let be a subset of .(i)The intersection of all infrasoft closed subsets of which contains a soft set is called the infrasoft closure points of . It is denoted by .(ii)The union of all infrasoft open subsets of which are contained in a soft set is called the infrasoft interior points of . It is denoted by .

Definition 10. (see [18]). Let be an infrasoft topological space and be a nonempty subset of . A class is called an infrasoft relative topology on , and is called an infrasoft subspace of .

Proposition 2. (see [18]). Suppose that is a family of (crisp) classical infratopologies on . Then,which defines an infrasoft topology on (we called this type of infrasoft topology as an infrasoft topology generated by (crisp) classical infratopologies).

Theorem 1. (see [18]). A soft mapping is infrasoft continuous if and only if the inverse image of each infrasoft open (resp. infra soft closed) set is an infrasoft open (resp. infra soft closed) set.

Definition 11. (see [28]). Let be a mapping. A subfamily of the power set of is said to be a quotient topology over (with respect to ) if is the largest topology that makes continuous.

3. Note on Soft Mappings

In this section, we target to achieve two goals; first, we update Definition 5 of soft mappings to be convenient for studying the concepts of open, closed, and homeomorphism mappings in different soft structures such as soft topology, suprasoft topology, and infrasoft topology. Second, we simplify the formulation of Definition 5 using a soft point as the starting point.

We begin by the following example which helps us to clarify the followed approach to achieve the coveted goals.

Example 1. Let and be two universal sets, and let and be two sets of parameters. Consider a soft mapping from to , where the mappings and are defined as follows:For , we have . Note that, for any infrasoft topology (or any soft structure) on and with the sets of parameters and , a soft mapping will not be soft open (soft closed) because the image of any soft sets under is a soft subset of a soft set . To remove this shortcoming, we make a slight modification for Definition 5 to be appropriate for defining soft open and closed mappings.

Definition 12. The image of under a soft mapping , where and are given by such thatfor each .
If there is no confusion, we simplify the above formulation as follows:The following result is easy, but it will be useful for the investigation.

Proposition 3. (i)The image of each soft point is a soft point(ii)The product of two soft points is a soft point

Definition 13. A relation on is a subset of .
Note that Definition 5 does not give meaning of a soft mapping as a self-contained concept. It only gives the method of calculating the image and preimage of soft sets. So, it is nature to wonder what is the formulation of soft mappings that simulates its counterpart on the (crisp) set theory? It is well known that a soft point represents the soft version of an ordinary point so that we redefine a soft mapping between two classes of soft points as follows.

Definition 14. Let and be two crisp mappings. A soft mapping of into is a relation such that each soft point in is related to one and only one soft point in such thatIn addition, for each .
From the above definition, we note two matters; first, reduce calculation burden and its difficulty that arises from Definition 5. Second, Definition 14 gives a logical explanation (justification) for some soft concepts, for example, it can be easily seen why we determine that is injective or surjective according to its two crisp mappings and .
Now, we prove the following results.

Lemma 1. Let be a soft mapping. Then, .

Proof. Since it can be written a soft set as a soft union of its soft points, we obtain , as required.

Theorem 2. The image of soft sets obtained from Definitions 12 and 14 is identical, i.e., Definitions 12 and 14 are identical.

Proof. Let be a soft set in . Then, . It follows from (i) of Proposition 3 that is a soft point in . According to Definition 12, we obtain . Thus, which represents the image of according to Definition 14. Hence, we obtain the coveted result.

Corollary 1. The preimage of soft sets obtained from Definitions 6 and 14 is identical.

We complete this part by presenting some amendments of some results given in [18]. First, the following result is the correct form of Proposition 7 in [18].

Proposition 4. Let and be subsets of , where and are finite sets. If , then .

Proof. It is clear that . Conversely, let . Then, contains an infrasoft open set such that . Since and are finite sets, we consider as a smallest infrasoft open set containing . Now, we have three cases: Case 1: . Then, . Case 2: . Then, . Case 3: and . Then, and . Since is the smallest infrasoft open set containing , and . But this contradicts assumption . Therefore, the only valid cases are Case 1 and Case 2. Thus, . Hence, the proof is complete.Second, we replace Definition 24 by Theorem 8 (which are given in [18]) to keep the systematic line of defining infrasoft continuity, infrasoft openness, and infrasoft closedness. That is, we reformulate Definition 24 of an infrasoft continuous map as follows.

Definition 15. A soft mapping is said to be infrasoft continuous if the preimage of each infrasoft open set is infrasoft open.

4. Infrasoft Homeomorphism Mappings

In this section, we initiate the concepts of infrasoft open, infrasoft closed, and infrasoft homeomorphism mappings. We show the relationships among them and study some properties. We construct some counterexamples to explain some invalid results.

Definition 16. A soft mapping is said to be infrasoft open (resp. infra soft closed) if the image of each infrasoft open (resp. infra soft closed) set is an infrasoft open (resp. infra soft closed) set.

Proposition 5. If is an infrasoft open mapping, then the following statements hold.(i) for each (ii) for each

Proof. To prove (i), let . Then, there exists a soft point such that . Therefore, there exists an infrasoft open set such that . Obviously, . By hypothesis, is an infrasoft open set; hence, , as required.
One can prove (ii) using a similar technique.

Corollary 2. If is an infrasoft open mapping, then the image of an infrasoft neighborhood of is an infrasoft neighborhood of .

Proposition 6. If is an infrasoft closed mapping, then for each .

Proof. Suppose that . Then, . This means for each , there exists an infrasoft open set containing such that . Therefore, . Now, . By assumption, so that . Since is an infrasoft closed mapping, then . Thus, , as required.
In fact, some characterizations of soft open and closed mappings, which are the counterparts of infrasoft open and infrasoft closed mappings, are losing via the structure of infrasoft topology. The following example illustrates this observation.

Example 2. Consider the following soft sets over under a parameter set defined as follows:Then, is an infrasoft topology on . Consider is the indiscrete soft topology (of course, it will be an infrasoft topology) on . Let a soft mapping be defined as follows:One can check that the two conditions given in Proposition 5 hold. Also, the image of any infrasoft neighborhood of is an infrasoft neighborhood of . Moreover, for each . On the contrary, the image of an infrasoft clopen set is which is neither infrasoft open nor infrasoft closed in ; hence, is neither an infrasoft open mapping nor an infrasoft closed mapping.

Proposition 7. Let be a bijective soft map. Then, is infrasoft open if and only if it is infrasoft closed.

Proof. The proof follows from the fact that a bijective soft map implies that .

Proposition 8. Let be an infrasoft open mapping and be an infrasoft open set in . Then, is an infrasoft open mapping.

Proof. Let be an infrasoft open set in . Then, there is an infrasoft open set in such that . This means that is also an infrasoft open set in . By hypothesis, is an infrasoft open set in ; hence, is an infrasoft open mapping.

Definition 17. The composition of two soft mappings and is a soft mapping such that .

Proposition 9. Let and be two infrasoft mappings. Then, the following statements hold.(i)If and are infrasoft open mappings, then is an infrasoft open mapping(ii)If is an infrasoft open mapping and is a surjective infrasoft continuous mapping, then is an infrasoft open mapping(iii)If is an infrasoft open mapping and is an injective infrasoft continuous mapping, then is an infrasoft open mapping

Proof. (i)Straightforward.(ii)Let be an infrasoft open set in . By hypothesis, is an infrasoft open set in ; therefore, is an infrasoft open set in . Since is surjective, then . Thus, is an infrasoft open mapping.(iii)Let be an infrasoft open set in . By hypothesis, is an infrasoft open set in ; therefore, is an infrasoft open set in . Since is injective, then . Thus, is an infrasoft open mapping.Following similar arguments given in the above proof, one can prove the next result.

Proposition 10. Let and be two infrasoft mappings. Then, the following statements hold.(i)If and are infrasoft closed mappings, then is an infrasoft closed mapping(ii)If is an infrasoft closed mapping and is a surjective infrasoft continuous mapping, then is an infrasoft closed mapping(iii)If is an infrasoft closed mapping and is an injective infrasoft continuous mapping, then is an infrasoft closed mapping

Proposition 11. Let be a family of infrasoft topological spaces. Then, is an infrasoft topology on under a set of parameters .

Proof. It is clear that and are members in . Now, let and are two members in . Since for each , then it follows from the fact that is closed under finite soft intersection. Hence, the proof is complete.
We call given in the proposition above a product of infra soft topologies, and a product of infrasoft spaces.

Definition 18. Let be a family of soft mappings. The product of these soft mappings is given by such that .
If is countable, then we write it as follows:

Theorem 3. The product of infrasoft open mappings is an infrasoft open mapping.

Proof. Let be the product of infrasoft open mappings of the family . Let be an infrasoft open set in . Since where for each , then . By hypothesis, is an infrasoft open set in for each . According to the definition of the product of infrasoft spaces, is an infrasoft open set in . Hence, we obtain the coveted result.

Corollary 3. The product of infrasoft closed mappings is an infrasoft closed mapping provided that is bijective for each .

Theorem 4. The product of infrasoft continuous mappings is an infrasoft continuous mapping.

Proof. Similar to the proof of Theorem 3, one can prove this result.

Theorem 5. If is an infrasoft open (resp. infrasoft closed) mapping such that is injective, then is an infraopen (resp. infraclosed) mapping.

Proof. Let be an infraopen set in . Then, there exists an infrasoft open set in such that . Since is an infrasoft open mapping, then is an infrasoft open set in . Therefore, is infrasoft open in . Since is injective, then . Thus, is an infraopen set in . Hence, we obtain the coveted result.
Following the same arguments, the case between parenthesis can be proved.
Example below explains that the converse of the above theorem fails.

Example 3. Let , and consider the two infrasoft topologies and on , whereTaking as the soft identity mapping, it is clear that is an infraopen mapping and an infraclosed mapping for each . But is neither an infrasoft open mapping nor an infrasoft closed mapping because is not an infrasoft open set in and is not an infrasoft closed set in .
We show under which condition the converse of Theorem 5 is true.

Theorem 6. Let be an infrasoft topology induced from the (crisp) classical infratopologies and be a soft mapping such that is injective. Then, is infrasoft open (resp. infrasoft closed) if and only if is infraopen (resp. infraclosed).

Proof. Necessity: it follows from Theorem 5.
Sufficiency: let be an infrasoft open set in . Since is an infraopen mapping, then is an infraopen set in for each . Now, such that either is an infraopen set in or is an infraopen set in . Since is generated from the crisp infratopologies, then is an infrasoft open set in . Hence, the proof is complete.
Following the same arguments, the case between parenthesis can be proved.

Definition 19. A bijective soft mapping is said to be an infrasoft homeomorphism if it is infrasoft continuous and infrasoft open.
We cancel the proofs of the next two propositions because they are easy.

Proposition 12. Let and be two infrasoft homeomorphism mappings. Then, is an infrasoft homeomorphism mapping.

Proposition 13. If is a bijective soft mapping, then the following statements are equivalent.(i) is an infrasoft homeomorphism(ii) and are infrasoft continuous(iii) is infrasoft closed and infrasoft continuousThe proofs of the following two theorems follow from Theorems 5 and 6, respectively.

Theorem 7. If a soft mapping is infrasoft homeomorphism, then a mapping is infrahomeomorphism.

Theorem 8. Let be an infrasoft topology induced from classical (crisp) infratopologies. Then, a soft mapping is infrasoft homeomorphism if and only if a mapping is infrahomeomorphism.

Definition 20. A soft subset of an infrasoft topological space is called an isolated soft set if there exists a soft point such that .

Proposition 14. If is an infrasoft homeomorphism mapping, then the following statements hold for each .(i)(ii)(iii)

Proof. We prove (i), and one can prove the other two cases similarly.
It follows from (i) of Proposition 5 that . Conversely, let . Then, there exists an infrasoft open set such that . By hypothesis, such that is an infrasoft open set in . Therefore, . This means that , as required.

Definition 21. A property is said to be an infrasoft topological invariant if the property possessed by an infratopological space is also possessed by each an infrasoft homeomorphic to .

Theorem 9. The property of an infrasoft dense set (isolated soft set) is an infrasoft topological invariant.

Proof. Let be an infrasoft homeomorphism mapping, and let an infrasoft dense subset of , i.e., . It follows from (ii) of Proposition 14 that . Therefore, is an infrasoft dense subset of . Hence, the proof is complete.

Theorem 10. The product of two infrasoft homeomorphism mappings is an infrasoft homeomorphism mapping.

Proof. It is clear that the product of bijective soft mappings is bijective. Then, Theorem 3 and Corollary 4.15 finish the coveted result.

Proposition 15. Let be a family of pairwise disjoint infrasoft topological spaces. Then, the family is an infrasoft open set in for each produces an infrasoft topology on with a constant set of parameters .

Proof. It is clear that and are members of . To prove that is closed under finite soft intersections, let and be two members of . Then, and for each . Therefore, for each . Thus, . Hence, is an infrasoft topology on .
We call the infrasoft topological space given in the above proposition a sum of infrasoft topological spaces and is denoted by .

Proposition 16. A soft subset of is infrasoft closed if and only if is an infrasoft closed set in for each .

Proof. is an infra soft closed set in is an infra soft open set in for any is an infra soft closed set in for any .

Definition 22. Let be a family of soft mappings. Then, we define a soft mapping as follows: the image of each and the image of each are given by(1)(2)

Theorem 11. A soft mapping is infrasoft open (resp. infrasoft closed) if and only if all soft mappings are infrasoft open (resp. infrasoft closed).

Proof. We prove the theorem in case of the soft mapping is infra soft open.
Necessity: let be an infrasoft open mapping. Taking an arbitrary soft mapping , where , let be an infrasoft open set in . Then, is an infrasoft open set in . Therefore, is an infrasoft open set in . Since for each , then . Thus, is an infrasoft open set in , as required.
Sufficiency: let be an infrasoft open mapping for each , and let be an infrasoft open set in . Since is an infrasoft open set in for each , then is an infrasoft open set in for each . According to the definition of sum of infrasoft topologies, we obtain is an infrasoft open set in . Now, so that is an infrasoft open set in .
Following similar arguments, one can prove the case between the parenthesis.

Corollary 4. A soft mapping is infrasoft homeomorphism if and only if all soft mappings are infrasoft homeomorphism.

5. Infrasoft Quotient Mappings

In this section, we define the concepts of quotient infrasoft topologies and infrasoft quotient mappings. We establish their main properties and investigate transmission of them to (crisp) mappings defined between parametric infrasoft topological spaces.

Definition 23. Let be a soft mapping. A family is said to be a quotient infrasoft topology over (with respect to ) if is the largest infrasoft topology that makes infrasoft continuous.
Note that .
The following example points out how a quotient infrasoft topology is constructed.

Example 4. Let be an infrasoft topology on with a set of parameters such thatLet be another universal set with a set of parameters , and consider a soft mapping from to , where and are defined as follows:Then, is the quotient infrasoft topology on (with respect to ), whereNote that, for any infrasoft topology on is proper finer than , is not infrasoft continuous.

Theorem 12. Let be an infrasoft continuous mapping. Then, the following statements are equivalent.(i) is a quotient infrasoft topology(ii) is an infrasoft open set in if is an infrasoft open set in (iii) is an infrasoft closed set if is an infrasoft closed set

Proof. (i) (ii): since is infrasoft continuous, then is an infrasoft open set in for each infrasoft open set in . Conversely, let be an infrasoft open set in . Since is a quotient infrasoft topology, then , as required.(ii) (i): suppose that there exists an infrasoft topology that makes infrasoft continuous such that . Now, let . Then, . By hypothesis, . Thus, , as required.(ii) (iii): straightforward.

Corollary 5. If is an infrasoft continuous mapping from onto such that is either infrasoft open or infrasoft closed, then is a quotient infrasoft topology.

Proof. It is clear that is an infrasoft open set in for each infrasoft open set in on one hand. On the other hand, is a surjective; then, , and since is an infrasoft open mapping, then is an infrasoft open set so that is infrasoft open. It follows from (ii) of the above theorem that is a quotient infrasoft topology, as required.
The case of is an infrasoft closed mapping is proved in a similar manner.

Proposition 17. Let be an infrasoft continuous mapping from onto a quotient infrasoft topological space . Then, is infrasoft continuous if is infrasoft continuous.

Proof. Necessity: it follows from the fact that the composition of two infrasoft continuous mappings is an infrasoft continuous mapping.
Sufficiency: suppose that is an infrasoft open set in . Since is infrasoft continuous, then is an infrasoft open set in . Since is a quotient infrasoft topology, it follows from Theorem 12 that is an infrasoft open set in . Hence, we obtain the coveted result.
Recall that a mapping is called a quotient inframapping if it is surjective and is equipped with the quotient infratopology with respect to .

Definition 24. A soft mapping is said to be a quotient infrasoft mapping if is surjective and is equipped with the quotient infrasoft topology with respect to .
In other words, is said to be a quotient infrasoft mapping if is surjective and a subset of is infrasoft open if is infrasoft open.

Proposition 18. If be a quotient infrasoft mapping such that generated from crisp infratopologies, then is a quotient inframapping.

Proof. Firstly, it is clear that is a surjective mapping. To prove that is a quotient infratopology, let be an infraopen set in . Then, there is an infrasoft open set in such that . Therefore, is an infrasoft open set in ; thus, is an infrasoft open set in . Conversely, let be an infrasoft open set in . Then, there is an infrasoft open set in such that . Since generated from crisp infratopologies, we can write for each . Now, , where and for each . By hypothesis, is infrasoft open set in . Thus, is infrasoft open set in . Hence, the proof is complete.
The following example shows that a condition imposed on the infrasoft topology of the domain is indispensable.

Example 5. Let be an infrasoft topology on with a set of parameters , and let be an infrasoft topology on with a set of parameters , whereConsider a soft mapping , where and are defined as follows:It is clear that is a quotient infrasoft mapping. Now, we have and are two parametric infratopologies on , and is a parametric infratopology on . On the contrary, is not a quotient inframapping because is an infrasoft open set in in spite of is not an infrasoft open set in .

6. Conclusion

We study some extensions of soft topology, which are defined by reducing the stipulations of soft topology, for various purposes such as obtaining appropriate models to handle some real-life issues, or building some paradigms that demonstrate the relations among some topological notions and ideas, or keeping certain properties under fewer conditions of those given on soft topology. To this end, we have recently defined a new generalization of soft topology, namely, infrasoft topology.

The principal focus of the article was revising the definition of soft mappings and studying some types of soft mapping in the frame of infrasoft topological structures. The main contributions of this paper are listed as follows.(1)Improve the definition of soft mapping given in [9] using soft points(2)Introduce the concepts of infrasoft open, infrasoft closed, and infrasoft homeomorphism mappings, and study several properties with the help of examples(3)Present the concepts of quotient infrasoft topology and quotient infrasoft mappings and investigate main features

To complete building the infrasoft topological structure, we plan to do the following studies in the frame of infrasoft topological spaces.(1)Define some types of separation axioms and show the relationships among them(2)Explore the concepts of compactness and Lindelöffness and establish main characterizations(3)Initiate the concept of connectedness and research fundamental properties

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.