[Retracted] New Soft Structure: Infra Soft Topological Spaces

Abstract

It is always convenient to find the weakest conditions that preserve some topologically inspired properties. To this end, we introduce the concept of an infra soft topology which is a collection of subsets that extend the concept of soft topology by dispensing with the postulate that the collection is closed under arbitrary unions. We study the basic concepts of infra soft topological spaces such as infra soft open and infra soft closed sets, infra soft interior and infra soft closure operators, and infra soft limit and infra soft boundary points of a soft set. We reveal the main properties of these concepts with the help of some elucidative examples. Then, we present some methods to generate infra soft topologies such as infra soft neighbourhood systems, basis of infra soft topology, and infra soft relative topology. We also investigate how we initiate an infra soft topology from crisp infra topologies. In the end, we explore the concept of continuity between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces.

1. Introduction

This paper is at the junction of two disciplines, namely, infra topology and soft set theory. Their hybridization has produced an interesting structure called infra soft topology which is the framework for our contribution. Let us summarize the antecedents and state-of-the-art of the topic.

In 1999, Molodtsov [1] proposed the concept of soft sets as a new mathematical approach to cope with problems containing uncertainties, and he explained the potentiality of soft sets to handle many problems in different areas. This theory has gained much attention from researchers and scientists because of its diverse applications. It is possible to see a rapid growth in soft sets’ research in the last few years, see, for example, [2, 3].

In 2003, Maji et al. [4] put forward some soft operations such as union and intersection and subset and equality relations between two soft sets. They also defined the null and absolute soft sets as a soft version of the empty and universal crisp sets. Ali et al. [5] showed some shortcoming given in [4], defined certain new operations on soft sets, and explored their main properties. Abbas et al. [6] and Qin and Hong [7] described new types of soft equality, which they applied to introduce new types of algebraic structures. Recently, Al-shami and El-Shafei [8] have defined and discussed new types of operations between soft sets.

In 2011, Shabir and Naz [9] introduced a topological structure on soft setting. They defined the fundamental notions of soft topologies such soft open and closed sets, soft subspaces, and belonging and nonbelonging relations which are used to initiate soft separation axioms. Zorlutuna et al. [10] came up with the idea of soft point which helps to study some properties of soft interior points and soft neighbourhood systems. This concept was independently reformulated by Samanta et al. [11, 12], while Das and Samanta [11] applied the new version of the soft point to study the concept of soft metric spaces and Nazmul and Samanta [12] used it to discuss soft neighbourhood systems and reveal some relations of soft limit points of a soft set. Many scholars analyzed the properties of soft topologies and compared their performance with the case of classical topologies, see, for example, [13–23]. Generalizations of open sets were investigated in soft topologies, see [24, 25]. In [26], we corrected some alleged results concerning soft separation axioms, especially those defined using soft points.

Some generalizations of a soft topology were given by weakening a soft topology’s conditions. For example, in 2014, El-Sheikh and Abd El-Latif [27] established the concept of supra soft topological spaces by neglecting a finite intersection condition of a soft topology. This path, therefore, attracted a lot of researchers who studied essential notions related to supra soft topologies, see, for example, [28, 29]. Thomas and John [30] formulated the concept of soft generalized topological spaces which are defined as a family of soft sets that satisfy an arbitrary union condition of a soft topology, and Zakari et al. [31] originated the concepts of soft weak structures which are defined as a family of soft sets that contain the null soft set . Ittanagi [32] studied the concept of soft bitopological space which can be regarded as a soft topological space when the two soft topologies are identical. Lately, Al-shami et al. [33] have constructed soft topology on ordered setting as an extension of soft topology. Similarly, Al-shami and El-Shafei [34] studied supra soft topology on ordered setting.

We note that many properties of soft topological spaces are still valid on infra soft topological spaces, and initiating examples that show some relationships between certain topological concepts are easier on infra soft topological spaces. Therefore, we aim in this paper to perform an exhaustive analysis of infra soft topological spaces.

This paper is structured as follows: after this introduction, Section 2 addresses some definitions and properties that help the reader to well understand this manuscript. In Section 3, we introduce the concept of infra soft topological spaces and disclose the main properties of infra soft interior, infra soft closure, infra soft limit, and infra soft boundary points of a soft set. In Section 4, we tackle some techniques of generating infra soft topology such as infra soft neighbourhoods and infra soft subspaces. In Section 5, we formulate the concept of infra continuous maps between infra soft topological spaces and determine the conditions under which the continuity is preserved between infra soft topological space and its parametric infra topological spaces. We give some conclusions and make a plan for future works in Section 6.

2. Preliminaries

In this section, we recall the technical concepts that we need in this paper.

The notation refers to the set of subsets of .

Definition 1 (see [1]). A pair is a soft set over a nonempty set provided that is a map from the set of parameters to .
For the sake of brevity and ease, henceforth, a soft set is symbolized by instead of . It is identified as .
The set of all soft sets over under a set of parameters is symbolized by .

Definition 2 (see [11]). A soft set is called finite (resp., countable) if is finite (resp., countable) for each .

Definition 3 (see [35]). The relative complement of a soft set is a soft set , where is a map defined by for each .

Definition 4 (see [4]). A soft set over is said to be the null soft set, symbolized by , if for each . Its relative complement is said to be the absolute soft set, symbolized by .

Definition 5 (see [11, 12]). A soft point over is a soft set such that is a singleton set and is the empty set for each . This soft point will be briefly symbolized by .

Since a soft topological space and its generalizations, which are the theme of this manuscript, are defined under a fixed set of parameters, we will mention the definitions and findings given in the previous studies under a fixed set of parameters.

Definition 6 (see [4]). A soft set is a soft subset of a soft set , symbolized by , if for all .
The soft sets and are called soft equal if each is a soft subset of the other.

Definition 7 (see [5]). The intersection of two soft sets and over , symbolized by , is a soft set , where a map is given by for each .

Definition 8 (see [4]). The union of two soft sets and over , symbolized by , is a soft set , where a map is given by for each .

Definition 9 (see [9, 19]). For a soft set over and , we say that(i) (it reads as totally belongs to ) if for each (ii) (it reads as does not partially belong to ) if for some (iii) (it reads as partially belongs to ) if for some (iv) (it reads as does not totally belong to ) if for each

Soft maps are recalled in the next two definitions with some modifications to be convenient for defining the concepts of soft continuous maps.

Definition 10 (see [36]). A soft mapping between and is a pair , denoted also by , of mappings such that and . Let and be subsets of and , respectively. Then, the image of and preimage of are defined as follows:(i) is a soft set in such that for each .(ii) is a soft set in such that

Remark 1. Henceforth, a soft map implies a map from the universal set to the universal set and a map from the set of parameters to itself.

Definition 11 (see [10]). A soft map is said to be injective (resp. surjective and bijective) if and are injective (resp. surjective and bijective).

Definition 12 (see [37]). An infratopology on is a collection of subsets of , that is, closed under finite intersections and satisfies .

Definition 13 (see [9]). The collection of soft sets over under a fixed set of parameters is called a soft topology on if it satisfies the following axioms:(i) and belong to (ii)The union of an arbitrary family of soft sets in belongs to (iii)The intersection of a finite family of soft sets in belongs to The triple is called a soft topological space. The term given to each member of is a soft open set, and the relative complement of each member of is a soft closed set.

3. Infra Soft Topological Spaces

In this section, we introduce the concept of infra soft topology as a class of soft sets, that is, closed under finite soft intersection and contains the null soft set. It lies between soft topology and soft weak structure, and it is independent of a supra soft topology. We define the main concepts of infra soft topology and reveal their main properties. One of the merits of infra soft topology is that many results of soft topology are still valid on infra soft topology, especially those are related to the soft interior and closure operators. A number of examples are provided to validate the obtained results.

Definition 14. The collection of soft sets over under a fixed set of parameters is said to be an infra soft topology on if it is closed under finite soft intersection and the null soft set is a member of .
The triple is called an infra soft topological space. Every member of is called an infra soft open set, and its relative complement is called an infra soft closed set.

The following examples elucidate the uniqueness of infra soft topology than the other celebrated soft structures.

Example 1. Let be a set of parameters and be the universal set. Then, is an infra soft topology on ; we called an excluding point infra soft topology. On the contrary, and are infra soft open sets, but their union is not an infra soft open set. Therefore, is neither supra soft topology nor generalized soft topology. Hence, it is not a soft topology.

Example 2. Let be a set of parameters and be the universal set. Then, is a supra soft topology on ; we called a particular point supra soft topology. On the contrary, and are supra soft open sets, but their intersection is not a supra soft open set. Therefore, is not an infra soft topology. Moreover, it is not a soft topology.

Remark 2. It can be examined that the intersection of any family of infra soft topologies is always infra soft topology. But the union of two infra soft topologies need not be an infra soft topology. We show this fact in the following example.

Example 3. Consider the soft sets below defined on with a set of parameters :Then, and are two infra soft topologies on . But is not an infra soft topology on because .

The following results present two techniques to originate infra soft topology using soft maps.

Proposition 1. Let be a soft map. If is an infra soft topology on , then a class is an infra soft topology on .

Proof. Since , then . Let . Then, there exist such that and . Therefore, . Since , then . Hence, the proof is complete.

In a similar manner, one can prove the following result.

Proposition 2. Let be a bijective soft map. If is an infra soft topology on , then a class is an infra soft topology on .

Definition 15. We define the infra soft interior points and infra soft closure points of a soft subset of which are, respectively, denoted by and as follows:(i) is the union of all infra soft open sets that are contained in (ii) is the intersection of all infra soft closed sets containing

The following example illustrates that the infra soft interior points and infra soft closure points of a soft set need not be infra soft open and infra soft closed sets, respectively.

Example 4. Let be a set of parameters. Then, { =  is finite} is an infra soft topology on the set of real numbers . It is clear that is not a soft topological space. Let , where is the set of natural numbers. We find that equals to which is neither an infra soft open set nor an infra soft closed set.

Proposition 3. Let be a soft subset of . Then, the following properties hold.(i)If is an infra soft open set, then (ii)If is an infra soft closed set, then

Proof. It immediately follows from Definition 15.

Example 4 shows that the converse of the two properties given in the above proposition need not be true in general. These two properties are an example of some soft topological properties that are losing on infra soft topological spaces.

Proposition 4. Let be a soft subset of . Then, the following properties hold.(i) if and only if there exists an infra soft open set such that (ii) if and only if for every infra soft open set containing

Proof. (i)Straightforward.(ii)Necessity: let . Then, belongs to every infra soft closed set containing . Suppose that there exists an infra soft open set containing such that , so . This is a contradiction. Thus, the necessary part holds.Sufficiency: let . Then, is an infra soft open set containing such that . Thus, the proof is complete.

Proposition 5. Let be a soft subset of . Then,(i)(ii)

Proof. (i) = { is an infra soft open set included in }^c = { is an infra soft closed set including }In a similar manner, we prove (ii).

Proposition 6. Let be a soft subset of . Then,(i)If is an infra soft open set, then (ii)If is an infra soft closed set, then

Proof. (i)Let . Then, and . This means that, for each infra soft open set containing , we have . Since is infra soft topology, then is an infra soft open set containing ; consequently, . Thus, . Hence, .(ii)Let . Then, and . Therefore, there is an infra soft open set such that . Suppose that . Then, there is an infra soft open set such that . Now, we have which is an infra soft open set containing such that and . This means that . But this contradicts . Thus, . Hence, .

Theorem 1. Let be soft subsets of . Then, the following properties hold:(i)(ii)(iii)If , then (iv)(v)

Proof.  The proofs of (i), (ii), and (iii) are obvious.(iv)It follows from (ii) that . Conversely, let . Then, there exists an infra soft open set such that . By (iii), we have . Therefore, . This ends the proof of .(v)It is clear that . Conversely, let . Then, there exist two infra soft open sets such that and . Now, is an infra soft open set containing such that . Therefore, . Thus, . Hence, the proof is complete.

One can prove the following result similarly.

Theorem 2. Let be soft subsets of . Then, the following properties hold:(i)(ii)(iii)If , then (iv)(v)

The following example clarifies that the following two equalities are not always true.(i)(ii)

Example 5. Let be a set of parameters and  = { is finite} be a family of soft sets on the set of real numbers . Then, is an infra soft topological space. Let . Then, . Therefore, . But . Also, let . Then, for each . Therefore, . But .

Proposition 7. If are soft subsets of such that , then .

Proof. It follows from (iii) of Theorem 1 that . To prove that , let . Then, there exists an infra soft open set containing such that . Now, we have three cases:

Case 1. . Then, .

Case 2. . Then, .

Case 3. and . Then, and . This implies that, for each infra soft open set containing , we obtain and . But this contradicts . Therefore, this case is impossible.
Thus, Cases 1 and 2 are only valid, and they imply that . Hence, the proof is complete.

The following result explains two methods of generating crisp infra topologies from an infra soft topology.

Proposition 8. Let be an infra soft topological space. Then,(i) is an infra topological space for each , where (ii) is an infra topological space for each , where

Proof. (i)Since , then . To prove that is closed under finite intersection, let . Then, there exists two infra soft open subsets of such that and . Owing to the fact that , we obtain , as required.Following similar arguments, one can prove (ii).

We called given in the above proposition “a parametric infra topological space.”

The converse of the above proposition need not be true, as shown in the following example.

Example 6. Consider the soft sets below defined on with a set of parameters :Then, is not an infra soft topology on . On the contrary, and are two infra soft topologies on . Also, , , and are three infra soft topologies on .

Definition 16. Let be a soft subset of . Then,(i) is a soft set given by , where is the closure of in for each (ii) is a soft set given by , where is the interior of in for each

Proposition 9. Let be a soft subset of . Then,(i)(ii)

Proof. (i)Let . Then, there exists an infra soft open set containing such that . Now, is an infra open subset of for every such that . Therefore, ; thus, , as required.Following similar arguments to those given in the proof of (i), one can prove (ii).

The converse of properties (i) and (ii) in the above theorem is not always true as explained in the following example.

Example 7. Assume that is the infra soft topological space given in Example 3. Let and . By calculating, we find and . On the contrary, and . Hence, and .

Definition 17. Let be a soft subset of an infra soft topological space . A soft point is said to be an infra soft limit point of if for each infra soft open set containing .
All infra soft limit points of are said to be an infra-derived soft set of and denoted by .

Proposition 10. If and are soft subsets of , then the following properties hold:(i) and (ii)If , then (iii)If , then (iv)

Proof.  The proofs of (i), (ii), and (ii) are obvious.(iv)It follows from (ii) that . Conversely, let . Then, there exist infra soft open sets containing such that and . It is clear that is an infra soft open set containing such that . This means that . Thus, . Hence, the proof is complete.

The next result investigates the role of infra soft limit points in studying the infra soft closure points of a soft set.

Theorem 3. Let be a soft subset of . Then,(i)If is infra soft closed, then .(ii).(iii).

Proof. (i)Let be an infra soft closed set and . Then, . Now, is an infra soft open set such that . Therefore, . Thus, , as required.(ii)Let . Then, and . Therefore, there exists an infra soft open set such that On the contrary, for each , we have . Then, . Therefore, , and this implies that From (1) and (2), we get . Thus, . Hence, , as required.(iii)It follows from (ii) of Proposition 4 that ; also, it is clear that . Then, . On the contrary, let . Then, for every infra soft open set containing . Without loss of generality, suppose that . Then, . Therefore, . Hence, we obtain the desired result.

The converse of property (i) in the above theorem is not always true as explained in the following example.

Example 8. From Example 4, we demonstrate that . This directly means that . But is not an infra soft closed set.

Definition 18. Let be a soft subset of . The infra soft boundary points of , denoted by , are all soft points which belong to the relative complement of .

Proposition 11. Let be a soft subset of . Then,(i)(ii)

Proof. (i) = { and } = { and } = { and }(ii)

Corollary 1. Let be a soft subset of . Then,(i)(ii)(iii)If is infra soft open, then (iv)If is infra soft closed, then (v)If is both infra soft open and infra soft closed, then

4. Methods of Generating Infra Soft Topologies

In this section, we present some methods of generating infra soft topologies different from those given in Propositions 1 and 2. These methods are infra soft neighbourhood systems, infra soft basis, infra soft subspace, and crisp infra topologies. We research these methods with the help of elucidative examples.

4.1. Infra Soft Neighbourhood and Infra Soft Neighbourhood Systems

Definition 19. A soft subset of is said to be an infra soft neighbourhood of a soft point if there exists an infra soft open set such that .

Definition 20. The infra soft neighbourhood system of a soft point , denoted by , is the class of all infra soft neighbourhoods of . In other words,  = { is an infra soft neighbourhood of a soft point }.

Proposition 12. If is an infra soft open subset of , then it is an infra soft neighbourhood of its all soft points.

Proof. Straightforward.

To demonstrate that the converse of the above proposition fails, consider Example 5. It is clear that every infinite infra soft set is an infra soft neighbourhood of its all soft points, but it is not an infra soft open set.

Theorem 4. The infra soft neighbourhood system of a soft point satisfies the following properties. (IN1):  (IN2): if and , then  (IN3): if , then  (IN4): for each , there is an infra soft neighbourhood of a soft point contained in such that for each

Proof.  (IN1): since is an infra soft open set containing every soft point , then it is an infra soft neighbourhood of its all soft points. Therefore, for every . Thus, . (IN2): let . Then, there exists an infra soft open set such that . If , then . Therefore, . (IN3): let . Then, there exist two infra soft open sets such that and . Since is an infra soft topology, then is an infra soft open set. Obviously, . Thus, . (IN4): let . Then, there exists an infra soft open set such that . Putting , we obtain the desired result.

Theorem 5. Let is the class of all families satisfying the four properties given in Theorem 4. Then,  = {: for each } forms an infra soft topology on the universal set . Moreover, it forms a soft topology.

Proof. It is well known that is an infra soft neighbourhood of all its soft points; then, ; also, . Let . For each , we have and . Then, and . It follows from (IN3) that . Thus, . Hence, is an infra soft topology.
To prove that is a soft topology, let for each . Suppose that . Then, there exists such that . Therefore, . Since , then, by (IN2), we obtain ; hence, , as required.

4.2. Basis of Infra Soft Topology

Definition 21. Let be an infra soft topological space. A class is said to be a basis for if the finite soft intersection of members of forms .

It is clear that is a basis for itself.

Proposition 13. Every subclass of containing (or two disjoint infra soft open sets) is a basis for a unique infra soft topology on .

Proof. Let be a family of soft sets generated by . Since the empty soft intersection is the absolute soft set, then . By hypothesis, , so . It follows from the definition of , which is generated by , that is closed under finite soft intersection. Hence, we obtain the desired result.
To prove the uniqueness, let be another infra soft topology generated by . Let . Then, can be expressed as a finite soft intersection of members of . Therefore, . Thus, . Similarly, one can prove that . Hence, , as required.

Example 9. Consider the class defined on with a set of parameters , whereBy taking the finite soft intersection of the members of , we obtain , whereObviously, is an infra soft topology on .

Remark 3. The basis for an infra soft topology need not be unique. In other words, there are more than one basis for an infra soft topology in general.

4.3. Subspace

Definition 22. Let be an infra soft topological space and be a nonempty subset of . A class is called an infra soft relative topology on , and is called an infra soft subspace of .

It can be easily checked that given in the above definition is an infra soft topology on .

Theorem 6. Let be an infra soft subspace of . Then, is an infra soft closed subset of if and only if there exists an infra soft closed subset of such that .

Proof. Necessity: let be an infra soft closed subset of . Then, there exists an infra soft open set in such that . This means that there exists an infra soft open set in such that . Therefore, . Putting ends the proof of the necessary part.
Sufficiency: let such that is an infra soft closed set in . Then, . Since is an infra soft open set in , then is an infra soft open set in . Therefore, is an infra soft closed set in . Hence, the proof is complete.

The proofs of the following two propositions are straightforward, and thus, they are omitted.

Proposition 14. Let be an infra soft open subset of . Then, is an infra soft open subset of if and only if it is an infra soft open subset of .

Proposition 15. Let be an infra soft subspace of . Then, is an infra soft neighbourhood of in if and only if there exists an infra soft neighbourhood of in such that .

Theorem 7. Let be an infra soft subspace of such that . Let , , and be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in , and let , , and be, respectively, the infra soft interior, infra soft closure, and infra soft limit points of a soft set in . Then,(i)(ii)(iii)

Proof. We only prove (i). One can prove the other cases using similar techniques.
Let . Then, there exists an infra soft open subset of such that . Therefore, . Thus, and . Hence, . Conversely, let . Then, there exist two infra soft open subsets of such that and . Since is an infra soft topology, then is an infra soft open set containing such that . Consequently, . Thus, . Hence, the proof is complete.

4.4. Producing Infra Soft Topology from Crisp Infra Topologies

Proposition 16. Suppose that is a family of crisp infra topologies on . Then,which defines an infra soft topology on .

Proof. Let the given assumptions be satisfied. Since for each , then . To prove that is closed under finite soft intersection, let . According to the structure of , we have for each . Since is infra topology, then . This implies that , as required. Hence, is an infra soft topological space.

Definition 23. The infra soft topological space given in the above proposition is called the infra soft topology on generated by .
We write if for each .

Remark 4. The largest infra soft topology whose parametric infra topologies are is .

We explain in the following example how we can apply Proposition 16 to construct an infra soft topology from crisp infra topologies.

Example 10. Let and be two crisp infra topologies on . To produce an infra soft topology from and , we construct infra soft open sets by choosing any set in as an image of , say, . Then, we can choose the image of by four different ways because the number of infra open sets in is four. Therefore, we obtain the following four infra soft sets:We repeat this manner with each set in . Then, we obtain the following infra soft open sets:Hence, is an infra soft topology on .

Proposition 17. Every infra soft topology generated by infra topologies contains all soft sets, in which their e-components are or .

Proof. Since for each , then any soft set defined as is or , which is a member of .

The converse of the above proposition fails as illustrated in the following example.

Example 11. Let and be an infra soft topology on , whereIt is clear that contains all soft sets, in which their e-components are or . But does not generate from crisp infra topologies because and ; however, .

In the following two examples, we show how we can examine whether the infra soft topology is generated by crisp infra topologies or not?

Example 12. Let and be an infra soft topology on , whereThen, and are the parametric (crisp) infra topologies of an infra soft topology . Consider and . Now, and ; however, . Thus, is not generated from crisp infra topologies.

Example 13. Consider the soft sets below defined on with a set of parameters :Then, is an infra soft topology on . It is clear that and are the parametric (crisp) infra topologies of an infra soft topology . Consider and . It can be seen that is generated from the crisp infra topologies and .

5. Continuity between Infra Soft Topological Spaces

In this section, we define the concept of continuity between infra soft topological spaces and then give its equivalent conditions using infra soft open and infra soft closed sets. Also, we discuss losing some equivalent conditions of soft continuity on infra soft topology with the help of an illustrative example. We close this section by studying “transmission” of continuity between an infra soft topological space and its parametric infra topological spaces.

Definition 24. Definition 24A soft map from to is said to be infra soft continuous at a soft point if for each infra soft open set containing , there is an infra soft open set containing such that .

Definition 25. A soft map from to is said to be infra soft continuous if it is infra soft continuous for all .

Theorem 8. A soft map is infra soft continuous if and only if the inverse image of each infra soft open set is an infra soft open set.

Proof. Necessity: let be an infra soft open subset of . Without loss of generality, consider . Then, for each , we have an infra soft open subset of containing such that . Thus, and . Hence, is infra soft open.
Sufficiency: suppose that and is an infra soft open set containing . Then, is an infra soft open set containing such that . Therefore, is infra soft continuous at which we choose arbitrarily; hence, is infra soft continuous.

Corollary 2. A soft map is infra soft continuous if and only if the inverse image of each infra soft closed set is an infra soft closed set.

Proof. Necessity: let be an infra soft closed subset of . Then, is infra soft open. Therefore, is an infra soft open set; hence, is infra soft closed.
Sufficiency: following similar arguments, we prove the sufficient part.

The following properties are equivalent to soft continuity on soft topological and supra soft topological spaces.(i) for each (ii) for each (iii) for each

But they are not equivalent to soft continuity on the infra soft topological spaces.

Definition 26. Let be a soft map and be a nonempty subset of . A soft map from to , which is given by for each , is called the restriction soft map of on .

Theorem 9. If is infra soft continuous, then the restriction soft map is infra soft continuous.

Proof. Let be an infra soft open subset of . Then, . By hypothesis, is an infra soft open subset of ; therefore, is an infra soft open subset of . Hence, is an infra soft continuous map, as required.

It is easy to prove the following two results; thus, their proofs will be omitted.

Proposition 18. If and are infra soft continuous maps, then is an infra soft continuous map.

Proposition 19. Let be a family of infra soft topologies on with a set of parameters . If is infra soft continuous for each , then is infra soft continuous.

Proposition 20. Let be a soft map of an infra soft topological space to . Then, is an infra soft topology on . Moreover, is the strongest infra soft topology on which makes as infra soft continuous.

Proof. Since , then . Let . Then, there exist . Since is infra soft topology, then . Therefore, ; thus, is infra soft topology on . To prove that is the strongest infra soft topology on , it makes as infra soft continuous. Suppose that is an infra soft topology on such that is infra soft continuous. Let ; then, . This implies that ; thus, . Hence, the proof is complete.

We complete this section by investigating the concept of infra soft continuity between the soft map and crisp map.

Theorem 10. If a soft map is infra soft continuous, then a map is infra continuous.

Proof. Let be an infra open subset of . Then, there exists an infra soft open subset of such that . Since is an infra soft continuous map, then is an infra soft open subset of . It follows from Definition 10 that , where ; this implies that in the infra open subset of . Hence, we obtain the desired result.

We explain that the converse of the above theorem fails in the example below.

Example 14. Let and and be two infra soft topologies on , whereConsider and are identity maps. Then, is infra continuous for each . But is not an infra soft continuous map because .

We show under which condition the converse of Theorem 10 holds.

Theorem 11. Let be an infra soft topology generated from the crisp infra topologies. Then, a soft map is infra soft continuous if and only if a map is infra continuous.

Proof. The necessary part is proved in Theorem 10.
To prove the sufficient part, let be an infra soft open subset of . Then, is a soft subset of such that for each . Since is an infra open subset of and a map is infra continuous, then is the infra open subset of . By hypothesis, is generated from the crisp infra topologies, so is an infra soft open subset of , as required.

6. Conclusion

This study has introduced the concept of an infra soft topology as a new structure is weaker than a soft topology. The most important goal of investigating this concept is to keep some soft topological properties under fewer conditions than topology.

We have contributed to improve the knowledge about this area in three aspects. First, we have established the basic concepts of infra soft topological spaces and scrutinized properties. We have noted that most properties of interior and closure operators are valid on infra soft topological spaces, while most of them are losing on other generalizations of soft topology such as supra soft topology. Second, we have proposed some techniques of producing infra soft topologies such as soft maps, soft neighbourhood systems, infra soft basis, infra soft subspace, and crisp infra topologies. In fact, the techniques of soft neighbourhood systems and soft operators initiate soft topology which is due to the identical between their properties on soft topology and infra soft topology. Third, we have introduced and investigated the concept of continuity between infra soft topological spaces. We have described this concept using infra soft open and infra soft closed sets. Moreover, we have showed that some characterizations of continuity on soft topology are losing on the frame of infra soft topology, especially those that are based on the interior and closure operators.

In future works, we plan to formulate the soft topological concepts such as separation axioms, compactness, and connectedness on the frame of infra soft topology. In particular, we shed light on discovering which ones of their properties are still valid on the infra soft topologies.

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.