Soft Mappings Space

Abstract

Various soft topologies are being introduced on a given function space soft topological spaces. In this paper, soft compact-open topology is defined in functional spaces of soft topological spaces. Further, these functional spaces are studied and interrelations between various functional spaces with soft compact-open topology are established.

1. Introduction

Many practical problems in economics, engineering, environment, social science, medical science, and so forth cannot be dealt with by classical methods, because classical methods have inherent difficulties. The reason for these difficulties may be due to the inadequacy of the theories of parameterization tools. Molodtsov [1] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. Maji et al. [2, 3] research deals with operations over soft set. The algebraic structure of set theories dealing with uncertainties is an important problem. Many researchers heve contributed towards the algebraic structure of soft set theory. Aktaş and Çağman [4] defined soft groups and derived their basic properties. Acar et al. [5] introduced initial concepts of soft rings. Feng et al. [6] defined soft semirings and several related notions to establish a connection between soft sets and semirings. Shabir and Ali [7] studied soft ideals over a semigroup. Qiu-Mei et al. [8] defined soft modules and investigated their basic properties. Gunduz (Aras) and Bayramov [9, 10] introduced fuzzy soft modules and intuitionistic fuzzy soft modules and investigated some basic properties. Ozturk and Bayramov defined chain complexes of soft modules and their soft homology modules. Ozturk et al. introduced the concept of inverse and direct systems in the category of soft modules.

Recently, Shabir and Naz [11] initiated the study of soft topological spaces. Theoretical studies of soft topological spaces have also been by some authors in [12–17]. In the study [18] different soft point concepts from the studies [12–17] were given. In this study, soft point consepts in the study [18] are used.

In the present study, first soft compact set on soft topological spaces is defined and the properties of this soft compact set are investigated. Based on this, soft compact-open topology of soft topological spaces is defined on soft mapping space.

2. Preliminaries

In this section we will introduce necessary definitions and theorems for soft sets. Molodtsov [1] defined the soft set in the following way. Let be an initial universe set and let be a set of parameters. Let denote the power set of and .

Definition 1 (see [1]). A pair is called a soft set over , where is a mapping given by .
In other words, the soft set is a parameterized family of subsets of the set . For , may be considered as the set of -elements of the soft set , or as the set of -approximate elements of the soft set.

Definition 2 (see [3]). For two soft sets and over , is called soft subset of if (1) and (2) for all , and are identical approximations. This relationship is denoted by . Similarly, is called a soft superset of if is a soft subset of . This relationship is denoted by . Two soft sets and over are called soft equal if is a soft subset of and is a soft subset of .

Definition 3 (see [3]). The intersection of two soft sets and over is the soft set , where and for all , . This is denoted by .

Definition 4 (see [3]). The union of two soft sets and over is the soft set, where and, for all , This relationship is denoted by .

Definition 5 (see [3]). A soft set over is said to be a NULL soft set denoted by if for all (null set).

Definition 6 (see [3]). A soft set over is said to be an absolute soft set denoted by if, for all , .

Definition 7 (see [11]). The difference of two soft sets and over , denoted by , is defined as for all .

Definition 8 (see [11]). Let be a nonempty subset of , and then denotes the soft set over for which , for all .
In particular, will be denoted by .

Definition 9 (see [11]). Let be a soft set over and be a nonempty subset of . Then the subsoft set of over , denoted by , is defined as follows:  , for all . In other words .

Definition 10 (see [19]). Let and be two soft sets over and , respectively. The cartesian product is defined by , where
According to this definiton, the soft set is soft set over and its parameter universe is .

Definition 11 (see [11]). Let be the collection of soft set over , and then is said to be a soft topology on if(1) belongs to ;(2)the union of any number of soft sets in belongs to ;(3)the intersection of any two soft sets in belongs to .
The triplet is called a soft topological space over .

Definition 12 (see [11]). Let be a soft topological space over , and then members of are said to be soft open sets in .

Definition 13 (see [11]). Let be a soft topological space over . A soft set over is said to be a soft closed in , if its relative complement belongs to .

Proposition 14 (see [11]). Let be a soft topological space over . Then the collection , for each , defines a topology on .

Definition 15 (see [11]). Let be a soft topological space over and let be a soft set over . Then the soft closure of , denoted by , is the intersection of all soft closed super sets of . Clearly is the smallest soft closed set over which contains .

Definition 16 (see [18]). Let be a soft set over . The soft set is called a soft point, denoted by , if, for the element , and for all (briefly denoted by ).

Definition 17 (see [18]). In the two soft points and over a common universe , we say that the points are different points if or .

Definition 18 (see [18]). Let be a soft topological space over . A soft set in is called a soft neighborhood of the soft point if there exists a soft open set such that .

Definition 19 (see [20]). Let and be two soft topological spaces, and let be a mapping. For each soft neighbourhood of , if there exists a soft neighbourhood of such that , then is said to be soft continuous mapping at .
If is soft continuous mapping for all , then is called soft continuous mapping.

Definition 20 (see [20]). Let and be two soft topological spaces, and let be a mapping. If is a bijection, soft continuous and is a soft continuous mapping, then is said to be soft homeomorphism from to . When a homeomorphism exists between and , we say that is soft homeomorphic to .

Definition 21 (see [13]). Let be a soft topological space over . A subcollection of is said to be a base for if every member of can be expressed as a union of members of .

Definition 22 (see [13]). Let be a soft topological space over . A subcollection of is said to be a subbase for if the family of all finite intersections members of forms a base for .

Definition 23 (see [13]). Let be a family of soft mappings and is a family of soft topological spaces. Then, the topology generated from the subbase is called the soft topology (or initial soft topology) induced by the family of soft mappings .

Definition 24 (see [13]). Let be a family of soft topological spaces. Then, the initial soft topology on generated by the family is called product soft topology on . (Here, is the soft projection mapping from to , .)
The product soft topology is denoted by .

Definition 25 (see [16]). Let be a soft topological space and such that . If there exist soft open sets and such that , , and , then is called a soft Hausdorff space.

Definition 26 (see [13]). Let be a soft topological space. If there exists a soft finite subcovering of every soft open covering of the soft topological space , then this soft space is called soft compact space.

Definition 27 (see [18]). Let be a soft topological space over . If, for every soft point in , there exists a soft neighbothood such that is a soft compact subspace of , then is said to be soft locally compact space.

3. Soft Mappings Space

Let be a family of soft topological spaces over the same parameters set . We define a family of soft sets as follows.

If is a soft set over for each , then is defined by . Let us consider the topological product of a family of soft topological spaces . We take the contraction to the diagonal of each soft set . Since there exists a bijection mapping between the diagonal and the parameters set , then the contractions of soft sets are soft sets over .

Now, let us define the topology on . Let be a projection mapping and let the soft set be for each . Then

The soft topology is generated from , as a soft subbase and this soft topology is denoted by .

Definition 28 (see [21]). The soft topological space is called the product of family of the soft topological spaces .

Now, let the family of soft topological spaces be disjoint; that is, for each . For the soft set over the set , define the soft set by and the soft topology define by

It is clear that is a soft topology.

Definition 29 (see [21]). A soft topological space is called soft topological sum of the family of soft topological spaces and denoted by .

Let and be two soft topological spaces. denoted the all soft continuous mappings from the soft topological space to the soft topological space ; that is,

If and are two soft sets over and , respectively, then we define the soft set over as follows:

Now, let be an any soft point. We define the soft mapping by . This mapping is called an evaluation map. For the soft set over is satisfied. The soft topology that is generated from the soft sets as a subbase is called pointwise soft topology and denoted by .

Definition 30 (see [21]). is called a pointwise soft function space (briefly PISFS).

Now, we construct relationships between some function spaces. Let be a family of pairwise disjoint soft topological spaces, let be a soft topological spaces and , and let be a product and sum of soft topological spaces, respectively. Define such that, for all and for all , , where belongs to unique . We define the mapping by for each . It is clear that the mapping is an inverse of the mapping [21].

Let be a soft topological space, let be a family of soft topological spaces, and let be a family of soft mappings. For each soft point , we define the soft map by . If is any soft mapping, then is satisfied for the family of soft mappings .

Now, let be the family of soft topological spaces, and let be a soft topological space. We define mapping by the rule for all , .

Let the mapping be for each , where is a soft projection mapping. It is clear that the mapping is an inverse of the mapping [21].

Now, let , , and be soft topological spaces and let be a soft mapping. Then the induced map is defined by for soft points and . We define exponential law by using induced maps ; that is, . We define the following mapping: which is an inverse mapping as follows [21]:

Let be a soft topological space and let be a soft set. For , since the conditions(1), (2)are satisfied, then is soft topological space over .

Definition 31. The soft set is called a soft compact set if the soft topological space is a soft compact topological space.

Theorem 32. Let be a soft topological space. Then is a soft compact topological space if and only if there is a soft finite subcovering of every soft open covering of the soft set in the soft topological space .

Proof. Suppose that is soft compact topological set and is a soft open covering of the soft set in the soft topological space . Then
For ever each , since , then the family of soft sets is a soft open covering of the soft topological space and since is a soft compact, then
That is, the family is a soft finite covering of the soft set .
Conversely, suppose the given condition holds. Let the family be a soft open covering of the soft topological space . For each , since , choose a soft set such that . Then the family is a soft open covering of in the soft topological spaces . From the hypothesis,
That is, the soft topological space is a soft compact.

Theorem 33. Every soft closed subset of a soft compact topological space is soft compact.

Proof. Let be a soft compact space, let be a soft closed set, and let the family be a soft open covering of in the soft topological spaces . Then the family of sets is a soft open covering of . Since is a soft compact space, then
From here,
Hence from Theorem 32, is soft compact set.

Theorem 34. Every soft compact subset of a soft Husdorff topological space is soft closed.

Proof. Let be a soft Hausdorff space and let be a soft compact set. We will prove that is soft open, so that is soft closed. Each soft point to any the soft point is not equal. Since is a soft Hausdorff space, for
Then the family is a soft open covering of in the soft topological spaces . Since the soft set is a soft compact set, then can be written. We take the soft neighborhoods of the soft point to the corresponding the soft sets such that . The soft set is a soft open neighborhood of the soft point and
Hence holds. That is, is soft open set and is soft closed set.

Remark 35. The union of finite number of soft compact set on soft topological spaces is soft compact set.

Lemma 36. If is a soft locally compact Hausdorff space, is a soft compact set, and , and then there exists a soft open set such that is a soft compact set.

Proof. Since is a soft Hausdorff space, for each the soft point there exists a soft neighborhood such that . Since is a soft locally compact space, for each the soft point there exists a soft neighborhood such that is soft compact. Hence, for soft neighborhood of the soft point , since , then is a soft compact. Since the soft set is a soft compact, there exists the soft finite subcovering of the soft covering of . From here, for the soft open set , the conditions are satisfied.

Let and be two soft topological spaces and let and be two soft sets over and , respectively. is denoted a soft mappings set, where the condition is satisfied.

Definition 37. Let and be two soft topological spaces. The generated topology from the sets as a soft subbase is called soft compact-open topology. The soft set is briefly denoted by .

Example 38. Let , and . If we give the soft sets and for defined by then the families and are soft topology.
Now, let us give the soft continuous mappings set . consists of the mappings
Since the soft topological space is a finite, in this space each soft set is a soft compact. To give subbase of soft compact-open topology in the space , the function family in which each soft set in the space moves to soft set in should be generated.
If we take the soft set , then
If we take the soft set , then and so forth.

Theorem 39. Let be soft locally compact Hausdorff space; let have the soft compact open topology. Then the soft evaluation map is soft continuous.

Proof. Given a soft point of and a soft open set in about the image soft point , we wish to find a soft open about that maps into . First, using the soft continuity and the fact that is soft locally compact Hausdorff space, we can choose a soft open set about having soft compact closure , such that carries into . Then consider the soft open set in . It is a soft open set containing and .

Lemma 40. Let the soft set be a soft compact set in the soft topological space and let be a soft point in the soft topological space . Let be an arbitrary soft open set in the soft product space containing the soft set . Then the soft opens and can be found such that

Proof. For each the soft point , is satisfied. From the definition of soft topological product space, there exist the soft sets and such that
Then the soft family is a soft open covering of the soft set . Since is a soft compact, the soft set is also soft compact. Then is written. From here, the soft open sets and are satisfied in the condition of lemma.

Theorem 41. Let be a soft locally compact Hausdorff space, and let and be arbitrary two soft topological spaces. Let have the soft compact open topology. Then a map is soft continuous if and only if is soft continuous.

Proof. Suppose that is a soft continuous. Consider the functions where denotes the soft identity function on and denotes the switching map. Since it follows that.
, and being the composition of soft continuous maps is itself soft continuous map.
Conversly suppose that is soft continuous map. We prove that is soft continuous. For any soft point in , let soft set which belongs to subbase of be an arbitrary neighborhood of soft point .
For proving the soft continuity of , we want to show that there exists soft open neighborhood of soft point such that
Since , for each soft point , thus . Since is soft continuous, is soft open set and
Hence by Lemma 40, there exists a soft open neighborhood of in such that
Therefore . Now, for any and ,
Hence as desired.

Theorem 42. Let and be soft locally compact Housdorff spaces. Then for any soft topological spaces , the function is a soft homeomorphism.

Proof. Let us show that soft mapping is a soft homeomorphism. The soft mapping is bijective. Now, let us prove that soft mappings and are soft continuous. For proving the soft continuity of , consider arbitrary soft open neighborhood of soft point for any soft point , where belongs to subbase. Here is a soft compact and is soft open in soft soft compact open topology. Without loss of generality we may assume that soft set , where is soft compact and is soft open set. Thus , and this implies that . Since the soft sets and are soft compact, is also soft compact. Then is a soft open set in . We conclude that ; that is, is a soft open neighborhood of soft point . Now we assert that . For any , .
Since , for each and , is satisfied.
Now for proving the soft continuity of , consider soft open neighborhood of for any soft point . Since is soft compact, is a product of soft compact sets. Since , . Let us consider soft open set on soft space . For any soft points , , . Thus we have ; that is, is a soft open neighborhood of soft point .
Now we want to check that . For any and arbitrary , , hence, is soft continuous. Hence is soft homeomorphism.

Theorem 43. If is a family of soft topological spaces and is an arbitrary soft topological space, then the mapping is a soft homeomerphism on soft compact open topology.

Proof. Consider an arbitrary soft open set which belongs to subbasis of the soft space . Since is soft compact in soft topological sum, , -soft compact. Then since is soft open, is soft continuous.
If which belongs to base in is an arbitrary soft open set, then is soft open mapping. Since is injective, surjective, continuous, and soft open mapping, it is soft homeomorphism.

Theorem 44. Let be a soft topological space and let be a family of soft topological spaces. The mapping is a soft homeomorphism in soft compact open topology.

Proof. Let be an arbitrary soft set which belongs to subbasis in soft function space , where is soft compact and is soft open set. We may take soft set as
Then since is soft open, is soft continuous mapping.
For which belongs to base is an arbitrary soft set, since is soft open set, is soft open mapping. This proves that is a soft homeomorphism.

4. Conclusion

First soft compact set on soft topological spaces is defined and the properties of this soft compact set are investigated. Based on these concepts, soft compact-open topology is defined in functional spaces of soft topological spaces. Some properties of these functional spaces are studied. Proof of theorem of evaluation map, exponential law, and others are provided. Further, these functional spaces are studied and interrelations between various functional spaces soft compact-open topology are established. To extend this work, by using soft compact-open topology on function spaces, one could study different subdisciplines of mathematics, for example, algebraic topology, functional analysis, and so forth.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.