Homeomorphism and Quotient Mappings in Infrasoft Topological Spaces

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. -en, 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.


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. is means that there is a need for alternative mathematical concepts. erefore, 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].
e 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. e 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. omas 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.
en, 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.

Preliminaries
In this section, we mention the main concepts that make this paper self-contained.
Definition 1 (see [1]). We called a pair (H, A) a soft set over the universal set X ≠ ∅ with a set of parameters A provided that H is a mapping from A to the power set 2 X of X. We usually write a soft set (H, A) as follows (H, A) � (a, H(a)): a ∈ A { and H(a) ∈ 2 X }. e symbol S(X A ) denotes the set of all soft sets over X with any subset of A.
We recall some special types of soft sets. a . e set of all soft points over X with A is denoted by P(X A ). We say that P x a belongs to a soft set (F, A), denoted by P x a ∈ (F, A) if x ∈ F(a) [22]. More details concerning soft points were given in [23].
Definition 2 (see [24]). A soft set (H c , A) is called a relative complement of a soft set (H, A) provided that a mapping H c : A ⟶ 2 X is defined by H c (a) � X∖H(a) for each a ∈ A.
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.
More details concerning soft intersection and union were given in [26].
Definition 4 (see [27]). e Cartesian product of (H, A) and (F, B), which are defined over X and Y, respectively, is a soft set, denoted by (H × F, A × B), given by ( Definition 5 (see [9]). Let g: X ⟶ Y and φ: A ⟶ B be two crisp mappings. en, a soft mapping g φ : S(X A ) ⟶ S(Y B ) is defined as follows: the image of a soft set Definition 6 (see [9]). Let g φ : S(X A ) ⟶ S(Y B ) be a soft mapping. en, the preimage of a soft set (U, (N) ⊆ A, and a mapping g − 1 (U) is given by (2) Definition 8 (see [18]). e collection τ of soft sets over X under a fixed set of parameters A is called an infrasoft topology on X if it is closed under finite soft intersection as well as it contains Φ. e triple (X, τ, A) is called an infrasoft topological space.
e term given to each member of τ is called an infrasoft open set, and the relative complement each member of τ is called an infrasoft closed set.
Proposition 2 (see [18]). Suppose that Ω � Ω a a∈A is a family of (crisp) classical infratopologies on X. en, which defines an infrasoft topology on X (we called this type of infrasoft topology as an infrasoft topology generated by (crisp) classical infratopologies).
Theorem 1 (see [18]). Definition 11 (see [28]). Let g: X ⟶ Y be a mapping. A subfamily θ of the power set P(Y) of Y is said to be a quotient topology over Y (with respect to g) if θ is the largest topology that makes g continuous.

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.
x, y, z be two universal sets, and let A � a 1 , a 2 , a 3 and B � b 1 , b 2 , b 3 be two sets of parameters. Consider a soft mapping g φ from S(X A ) to S(Y B ), where the mappings g: X ⟶ Y and φ: A ⟶ B are defined as follows: . Note that, for any infrasoft topology (or any soft structure) on X and Y with the sets of parameters A and B, a soft mapping g φ will not be soft open (soft closed) because the image of any soft sets under g φ 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.
If there is no confusion, we simplify the above formulation as follows: e following result is easy, but it will be useful for the investigation.

Proposition 3 (i) e image of each soft point is a soft point (ii) e product of two soft points is a soft point
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.
In addition, 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 g φ is injective or surjective according to its two crisp mappings g and φ.
Now, we prove the following results.
Proof. Since it can be written a soft set as a soft union of its soft points, we obtain φ(a) , as required.
φ(a) which represents the image of (H, M) according to Definition 14. Hence, we obtain the coveted result. 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]. A). Since X and A are finite sets, we consider (G, A) as a smallest infrasoft open set containing P x a . Now, we have three cases: A) and P x a ∈ cl(F, A). But this contradicts assumption cl(G, A)⊓cl(F, A) � Φ. erefore, the only valid cases are Case 1 and Case 2.
Second, we replace Definition 24 by eorem 8 (which are given in [18]) to keep the systematic line of defining infrasoft continuity, infrasoft openness, and infrasoft closedness.
at is, we reformulate Definition 24 of an infrasoft continuous map as follows.

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.
is an infrasoft open mapping, then the following statements hold. (H, A). By hypothesis, g φ (U, A) is an infrasoft open set; hence, g φ (U, A)⊑int(g φ (H, A)), as required.
One can prove (ii) using a similar technique.
□ 4 Journal of Mathematics θ, B) is an infrasoft open mapping, then the image of an infrasoft neighborhood of P x a ∈ X is an infrasoft neighborhood of g φ (P x a ). θ, B) is an infrasoft closed mapping, then cl(g φ (H, A))⊑g φ (cl (H, A)) for each Proof. Suppose that P y b ∉ g φ (cl (H, A)).
Since g φ is an infrasoft closed mapping, then cl(g φ (H, A) (H, A)), as required. □ Example 2. Consider the following soft sets over X � w 1 , w 2 under a parameter set A � a 1 , a 2 defined as follows: en, τ � Φ, X, (U i , A): i � 1, 2 is an infrasoft topology on X. Consider θ is the indiscrete soft topology (of course, it will be an infrasoft topology) on X. Let a soft mapping g φ : (X, θ, A) ⟶ (X, τ, A) be defined as follows: One can check that the two conditions given in Proposition 5 hold. Also, the image of any infrasoft neighborhood of P w a is an infrasoft neighborhood of g φ (P w a ). Moreover, cl(g φ (H, A))⊑g φ (cl(H, A)) for each (H, A) ∈ S(X) A . On the contrary, the image of an infrasoft clopen set X is (a 1 , w 1 ), (a 2 , X) which is neither infrasoft open nor infrasoft closed in τ; hence, g φ is neither an infrasoft open mapping nor an infrasoft closed mapping. Proof. e proof follows from the fact that a bijective soft map g φ implies that g φ (H c , A) � (g φ (H, A)) c . e composition of two soft mappings f ψ : (X, τ, A) ⟶ (Y, θ, B) and .
be two infrasoft mappings. en, the following statements hold.

(i) If g φ and f ψ are infrasoft open mappings, then f ψ ∘ g φ is an infrasoft open mapping (ii) If f ψ ∘ g φ is an infrasoft open mapping and g φ is a surjective infrasoft continuous mapping, then f ψ is an infrasoft open mapping (iii) If f ψ ∘ g φ is an infrasoft open mapping and f ψ is an injective infrasoft continuous mapping, then g φ is an infrasoft open mapping
Proof (i) Straightforward.
Following similar arguments given in the above proof, one can prove the next result.

is an infrasoft closed mapping (ii) If f ψ ∘ g φ is an infrasoft closed mapping and g φ is a surjective infrasoft continuous mapping, then f ψ is an infrasoft closed mapping (iii) If f ψ ∘ g φ is an infrasoft closed mapping and f ψ is an injective infrasoft continuous mapping, then g φ is an infrasoft closed mapping
Journal of Mathematics Proposition 11. Let (X i , τ i , A i ): i ∈ I be a family of infrasoft topological spaces. en, τ � i∈I (U i , A i ): Proof. It is clear that X and Φ are members in τ. Now, let that τ is closed under finite soft intersection. Hence, the proof is complete.
We call τ given in the proposition above a product of infra soft topologies, and (X, τ, A) a product of infrasoft spaces.
i ∈ I be a family of soft mappings. e product of these soft mappings is given by If I is countable, then we write it as follows:

Theorem 3. e product of infrasoft open mappings is an infrasoft open mapping.
Proof. Let g φ : (X, τ, A) ⟶ (Y, θ, B) be the product of infrasoft open mappings of the family g iφ :  e product of infrasoft closed mappings g iφ : i ∈ I is an infrasoft closed mapping provided that g iφ is bijective for each i.   θ φ(a) . Hence, we obtain the coveted result.

us, g(U) is an infraopen set in
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 A � a 1 , a 2 , and consider the two infrasoft topologies τ � Φ, X, (U, A) and θ � Φ, X, Taking g φ : (X, τ, A) ⟶ (X, θ, A) as the soft identity mapping, it is clear that g: (X, τ a ) ⟶ (X, τ φ(a)�a ) is an infraopen mapping and an infraclosed mapping for each a ∈ A. But g φ is neither an infrasoft open mapping nor an infrasoft closed mapping because g φ (U

, A) � (U, A) is not an infrasoft open set in θ and g φ (U c , A) � (U c , A) is not an infrasoft closed set in θ.
We show under which condition the converse of eorem 5 is true.

Theorem 6. Let θ be an infrasoft topology induced from the (crisp) classical infratopologies and g
Proof. Necessity: it follows from eorem 5. Sufficiency: let (H, A) be an infrasoft open set in τ. Since g: (X, τ a ) ⟶ (Y, θ φ(a) ) is an infraopen mapping, then g (H(a)) is an infraopen set in θ φ(a) for each a ∈ A. Now, Since θ is generated from the crisp infratopologies, then g φ (H, A) is an infrasoft open set in θ. Hence, the proof is complete.
Following the same arguments, the case between parenthesis can be proved.

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. en, f ψ ∘ g φ is an infrasoft homeomorphism mapping. g φ : (X, τ, A) ⟶ (Y, θ, B) is a bijective soft mapping, then the following statements are equivalent.
We prove (i), and one can prove the other two cases similarly. It follows from (i) of Proposition 5 that g φ (int(H, A))⊑int(g φ (H, A)). Conversely, let P y b ∈ int(g φ (H, A). en, there exists an infrasoft open set (U, B) such that P y b ∈ (U, B)⊑g φ (H, A). By hypothesis, P x a � g −1 is an infrasoft open set in τ. erefore, P x a ∈ int(H, A). is means that P y b ∈ g φ (int (H, A)), as required. □ Definition 21. A property is said to be an infrasoft topological invariant if the property possessed by an infratopological space (X, τ, A) is also possessed by each an infrasoft homeomorphic to (X, τ, A).

Theorem 9. e property of an infrasoft dense set (isolated soft set) is an infrasoft topological invariant.
Proof. Let g φ : (X, τ, A) ⟶ (Y, θ, B) be an infrasoft homeomorphism mapping, and let (H, A) an infrasoft dense subset of (X, τ, A), i.e., cl(H, A) � X. It follows from (ii) of Proposition 14 that cl(g φ (H, A)) � g φ (cl(H, A) , g φ (H, A) is an infrasoft dense subset of (Y, θ, B). Hence, the proof is complete.  (X i , τ i , A) for each i ∈ I} produces an infrasoft topology on X � ∪ i∈I X i with a constant set of parameters A.
Proof. It is clear that X and Φ are members of τ. To prove that τ is closed under finite soft intersections, let (U 1 , A) and (U 2 , A) be two members of τ. en, Hence, τ is an infrasoft topology on X.
We call the infrasoft topological space given in the above proposition a sum of infrasoft topological spaces and is denoted by (⊕X i , τ, A).   A) is an infrasoft open set in (⊕Y i , θ, A).
Following similar arguments, one can prove the case between the parenthesis. □ Corollary 4. A soft mapping g φ :

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.
is said to be a quotient infrasoft topology over Y (with respect to g φ ) if θ is the largest infrasoft topology that makes g φ infrasoft continuous.
Let Y � x, y, z be another universal set with a set of parameters B � b 1 , b 2 , and consider a soft mapping g φ from (X, τ, A) to P(Y B ), where g: X ⟶ Y and φ: A ⟶ B are defined as follows: Note that, for any infrasoft topology on Y is proper finer than θ, g φ is not infrasoft continuous. Theorem 12. Let g φ : (X, τ, A) ⟶ (Y, θ, B) be an infrasoft continuous mapping. en, the following statements are equivalent.    Y, θ, B). en, Proof. Necessity: it follows from the fact that the composition of two infrasoft continuous mappings is an infrasoft continuous mapping.
Sufficiency: suppose that (U, C) is an infrasoft open set in μ. Since g φ ∘ f ψ is infrasoft continuous, then is an infrasoft open set in τ. Since θ is a quotient infrasoft topology, it follows from eorem 12 that g −1 φ (U, C) is an infrasoft open set in θ. Hence, we obtain the coveted result.
Recall that a mapping g: X ⟶ Y is called a quotient inframapping if it is surjective and Y is equipped with the quotient infratopology with respect to g.
Journal of Mathematics Definition 24. A soft mapping g φ : P(X A ) ⟶ P(Y B ) is said to be a quotient infrasoft mapping if g φ is surjective and Y is equipped with the quotient infrasoft topology with respect to g φ . In other words, g φ : P(X A ) ⟶ P(Y B ) is said to be a quotient infrasoft mapping if g φ is surjective and a subset θ, B) be a quotient infrasoft mapping such that τ generated from crisp infratopologies, then g: (X, τ a ) ⟶ (Y, θ φ(a) ) is a quotient inframapping.
Proof. Firstly, it is clear that g is a surjective mapping. To prove that θ φ(a) is a quotient infratopology, let U be an infraopen set in θ φ Consider a soft mapping g φ : (X, τ, A) ⟶ (Y, θ, B), where g: X ⟶ Y and φ: A ⟶ B are defined as follows: It is clear that g φ is a quotient infrasoft mapping. Now, we have τ a 1 � ∅, X, u } are two parametric infratopologies on X, and θ b 1 � θ b 2 � ∅, Y { } is a parametric infratopology on Y. On the contrary, g: (X, τ a 1 ) ⟶ (Y, θ φ(a 1 )�b 1 ) is not a quotient inframapping

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. e 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. e main contributions of this paper are listed as follows.
(1) Improve the definition of soft mapping given in [9] using soft points 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
e author declares that there are no conflicts of interest.