Topology on Soft Continuous Function Spaces

The concept of soft sets was initiated by Molodtsov. Then, some operations on soft sets were defined by Maji et al. Later on, the concept of soft topological space was introduced. In this paper, we introduce the concept of the pointwise topology of soft topological spaces. Finally, we investigate the properties of soft mapping spaces and the relationships between some soft mapping spaces.


Introduction
Because classical methods can inhere troubles, most practical problems in different scientific fields such as engineering, social science, economics, environment, and medical science have required solutions via technical methods rather than dealing with classical methods.Insufficiency of the theories of parameterization tools may result in these difficulties.The concept of soft set theory was initiated by Molodtsov [1] as a new mathematical tool in order to deal with uncertainties.In addition, the works of Maji et al. [2,3] focus on operations over soft set.It can be said that the algebraic structure of set theories bearing on uncertainties is an important problem.Hence, many researchers have been interested in the algebraic structure of soft set theory, and there are many works on this subject.For example, soft groups and their basic properties were introduced by Aktaş and Ça gman [4].Later, initial concepts of soft rings were brought into attention by U. Acar et al. [5].Then, establishing a connection between soft sets and semirings, F. Feng et al. [6] defined soft semirings and several related notions.Later on, M. Shabir et al. [7] worked on soft ideals over a semigroup.Qiu Mei Sun et al. [8] introduced soft modules and their basic properties.Continuing in this way from specific to more general, fuzzy soft modules and intuitionistic fuzzy soft modules were introduced by Gunduz and Bayramov [9,10], respectively, and they investigated some basic properties of these modules.Recently, chain complexes of soft modules and soft homology modules of them were defined by Ozturk and Bayramov [11], and then the concepts of inverse and direct systems in the category of soft modules were introduced by Ozturk et al. [12].
In the present study, the pointwise topology is defined in soft continuous mapping space, and the properties of soft mapping spaces are investigated.Subsequently, we give some relationships between some soft mappings spaces.

Preliminaries
Here we give necessary definitions and theorems for soft sets that have already been given in the literature.Thus, first of all, we present the definition of the soft set given by Molodtsov [1].Throughout the study, we will assume X to be an initial universe set and E to be a set of parameters.Then, P(X) will denote the power set of X. Definition 1.A pair (F, E) is called a soft set over X, where F is a mapping given by F : E → P(X) [1].In other words, the soft set is a parameterized family of subsets of the set X.For e ∈ E, F(e) may be considered as the set of e−elements of the soft set (F, E), or as the set of e−approximate elements of the soft set.and (G, E) is a soft subset of (F, E).Definition 3. The intersection of two soft sets (F, E) and (G, E) over X is the soft set (H, E), where ∀e ∈ E, H(e) = F(e) ∩ G(e).This is denoted by (F, E) Definition 4. The union of two soft sets (F, E) and (G, E) over X is the soft set, where ∀e ∈ E, H(e) = F(e) ∪ G(e).This relationship is denoted by (F, E) Definition 5. A soft set (F, E) over X is said to be a NULL soft set denoted by Φ if for all e ∈ E, F(e) = ∅(null set) [2].Definition 6.A soft set (F, E) over X is said to be an absolute soft set denoted by ∼ X if for all e ∈ E, F(e) = X [2].Definition 7. The difference (H, E) of two soft sets (F, E) and (G, E) over X , denoted by (F, E)\(G, E), is defined as H(e) = F(e)/G(e) for all e ∈ E [13].Definition 8. Let (F, E) be a soft set over X and Y be a non-empty subset of X.Then, the sub soft set of (F, E) over Y denoted by Y F, E , is defined as follows Y F(e) = Y ∩ F(e), for all e ∈ E. In other words, Definition 9. Let (F, A) and (G, B) be two soft sets over X 1 and X 2 , respectively, and A, B ⊆ E [4].
Definition 10.Let τ be the collection of soft sets over X [13]; then, τ is said to be a soft topology on X if (1) Φ, ∼ X belongs to τ; (2) the union of any number of soft sets in τ belongs to τ; (3) the intersection of any two soft sets in τ belongs to τ.
The triplet (X, τ, E) is called a soft topological space over X. Definition 11.Let (X, τ, E) be a soft topological space over X, then members of τ are said to be soft open sets in X [13].
Definition 12.The complement of a soft set (F, E) is denoted by (F, E) and is defined by (F, E) = (F , E), where F : E → P(X) is a mapping given by F (e) = X − F(e) for all e ∈ E [13].
Definition 13.Let (X, τ, E) be a soft topological space over X.A soft set (F, E) over X is said to be soft closed in X if its relative complement (F, E) belongs to τ [13].Definition 14.Let (X, τ, E) be a soft topological space over X and (F, E) be a soft set over X [13].Then, the soft closure of (F, E) denoted by (F, E) is the intersection of all soft closed super sets of (F, E).Clearly, (F, E) is the smallest soft closed set over X which contains (F, E).Definition 15.Let (F, E) be a soft set over X.The soft set (F, E) is called a soft point, denoted by (x e , E) , if for the element e ∈ E, F(e) = {x}, and F(e ) = ∅ for all e ∈ E − {e} (briefly denoted by x e ) [19].Definition 16.For two soft points x e and y e over a common universe X, we say that the points are different points if x = y or e = e .[19] Definition 17.The soft point x e is said to belong to the soft set (F, E) denoted by x e ∼ ∈ (F, E) if x e (e) ∈ F(e); i.e., x ⊆ F(e) [19].Definition 18.Let (X, τ, E) be a soft topological space over X [19].
Definition 19.Let (X, τ, E) and (Y, τ , E) be two soft topological spaces, and f : (X, τ, E) → (Y, τ , E) be a mapping [20].For each soft neighbourhood (H, E) of ( f (x) e , E) , if there exists a soft neighbourhood (F, E) of x e such that f ((F, E)) ∼ ⊆ (H, E), then f is said to be soft continuous mapping at x e .If f is soft continuous mapping for all x e , then f is called soft continuous mapping.
is a family of soft mappings, and {(Y i , τ i , E i )} i∈∆ is a family of soft topological spaces [14].Then, the topology τ generated from the subbase be a family of soft topological spaces [14].Then, the initial soft topology on X = ∏ i∈∆ X i generated by the family {(p i , q i )} i∈∆ is called product soft topology on X. (here, (p i , q i ) is the soft projection mapping from (X, E) to The product soft topology is denoted by ∏ i∈∆ τ i .

Topology on Soft Continuous Function Spaces
Let {(X s , τ s , E)} s∈S be a family of soft topological spaces over the same parameters set E. We define a family of soft sets ∏ s∈S X s , E as follows; Let (X, τ, E) be a soft topological space, {(Y s , τ s , E)} s∈S be a family of soft topological spaces, and {( f s , 1 E ) : (X, τ, E) → (Y s , τ s , E)} s∈S be a family of soft mappings.For each soft point x e ∼ ∈ (X, τ, E) , we define the soft mapping [21].
is soft continuous for each s ∈ S.
Proof.=⇒ Let f be a soft continuous mapping.Since the soft mappings p s are also continuous, the composite mapping will be continuous.
Y s , E be an any soft base of product topology.
s∈S is a family of soft continuous mappings, then the soft mapping Now, let the family of soft topological spaces {(X s , τ s , E)} s∈S be disjoint; i.e., X s 1 ∩ X s 2 = ∅ for each s 1 = s 2 .For the soft set F : E → ∪ s∈S X s over the set E, define the soft set F| X s : E → X s by F| X s (e) = F(e) ∩ X s , ∀e ∈ E and the soft topology τ define by It is clear that τ is a soft topology.Definition 22.A soft topological space ∪ s∈S X s , τ, E is called the soft topological sum of the family of soft topological spaces {(X s , τ s , E)} s∈S and denoted by ⊕ s∈S (X s , τ s , E).
Let {(X s , τ s , E)} s∈S be a family of soft topological spaces, (Y, τ , E) be a soft topological space, and f s : (X s , τ s , E) → (Y, τ , E) s∈S be a family of soft mappings.We define the soft function Proof.=⇒ Let f be a soft continuous mapping.Since the soft mappings i s are also continuous, the composite mapping will be continuous.
⇐= Let (F, E) ∈ τ be a soft open set.The soft set f −1 (F, E) belongs to the soft topology ⊕ s∈S τ s if and only if the soft set f −1 (F) X s , E belongs to τ s .Since s∈S be a family of soft continuous mappings.We define the  Proof.We should show that τ e = ∏ s∈S (τ s ) e .Let us take any set U from τ e .From the definition of the topology τ e , there exists a soft open set The topological sum can be proven in the same way.
Let (X, τ, E) and Y, τ , E be two soft topological spaces.Y X denotes the all soft continuous mappings from the soft topological space (X, τ, E) to the soft topological space Y, τ , E ; i.e., If (F, E) and (G, E) are two soft sets over X and Y, respectively, then we define the soft set G F , E over Y X as follows: for each e ∈ E. E) be an any soft point.We define the soft mapping e x α : This mapping is called an evaluation map.For the soft set The soft topology that is generated from the soft sets (G x α , E) | (G, E) ∈ τ as a subbase is called a pointwise soft topology and denoted by τ p .Definition 23.Y X , τ p , E is called a pointwise soft function space (briefly PISFS).
Example 5. Let X = {x 1 , x 2 }, Y = {y 1 , y 2 , y 3 } and E = {e 1 , e 2 }.If we give the soft sets F i : E → P(X) for i ∈ I and G : E → P(Y) defined by Then, the soft subbase of soft pointwise topology consists of the following sets.
Theorem 7. If the soft topological space Y, τ , E is a soft T i −space for each i = 0, 1, 2, then the soft space Y X , τ p , E is also a soft T i −space.
Proof.The soft points of the soft topological space Y X , τ p , E denoted by ( f α , E); i. e., if there exists soft open sets (F 1 , E), (F 2 , E) ∈ τ where the condition of the soft T i −space is satisfied.Then, the soft open sets E) are neighbourhoods of soft points f α and g β , respectively, where the conditions of soft T i −space are satisfied for these neighbourhoods.Now, we construct relationships betwen some function spaces.Let {(X s , τ s , E)} s∈S be a family of pairwise disjoint soft topological spaces, Y, τ , E be a soft topological space, and ∏ s∈S Y X s , τ s p , E , ⊕ s∈S (X s , τ s , E) be a product and sum of soft topological spaces, respectively.The soft mapping x α belongs to unique (X s 0 , τ s 0 , E).We define the inverse mapping of ∇ is a soft homeomorphism in the pointwise soft topology.
Proof.To prove the theorem, it is sufficient to show that the mappings ∇ and ∇ −1 are soft continuous.For this, we need to show that the soft set ∇ −1 e −1 x α (F, E) is a soft open set, where each e −1 x α (F, E) belongs to a soft subbase of the soft space belongs to subbase of the soft topological space Y ⊕ s∈S X s , τ p , E , the mapping ∇ −1 is soft continuous.Thus, the mapping is a soft homeomorphism.Now, let {(Y s , τ s , E)} s∈S be the family of soft topological spaces, (X, τ, E) be a soft topological space.We define mapping is a soft homeomorphism in the pointwise soft topology.
Proof.Since ∆ is bijective mapping, to prove the theorem it is sufficient to show that the mappings ∆ and ∆ −1 are soft open.First, we show that the mapping ∆ is soft open.Let us take an arbitrary soft set belongs to the base of the product space ∏ s∈S Y X s , τ s p , E .Since the soft set Similarly, it can be proven that ∆ −1 is soft open mapping.Indeed, for each soft open set Hence this set is soft open and the theorem is proved.Now, let (X, τ, E), Y, τ , E and Z, τ , E be soft topological spaces and f : Z, τ , E × (X, τ, E) → Y, τ , E be a soft mapping.Then, the induced map which is an inverse mapping E as follows Generally, in the pointwise topology for each soft continuous map g, the mapping E −1 (g) need not be soft continuous.Let us give the solution of this problem under some conditions.Theorem 10.Let (X, τ, E), Y, τ , E and Z, τ , E be soft topological spaces and the mapping e : Y Z × X → Z, e( f , z) = f (z) be soft continuous.If there is a pointwise soft topology in the function space Y X and the soft mapping ∧ g : X → Y Z is soft continuous, then the soft mapping E −1 ∧ g : Z × X → Y is also soft continuous.
Proof.By using the mapping we take

Conclusions
In this paper, we introduce the concept of the pointwise topology of soft topological spaces.Finally, we investigate the properties of soft mapping spaces and the relationships between some soft mapping spaces.We hope that the results of this study may help in the investigation of soft normed spaces and in many studies.

Theorem 4 .
Let {(X s , τ s , E)} s∈S be a family of soft topological spaces.Then, ∏ s∈S X s , (τ s ) e = ∏ s∈S (X s , (τ s ) e ) and ⊕ s∈S X s , (τ s ) e = ⊕ s∈S X s , (τ s ) e are satisfied for each e ∈ E.
[21]e there exists a bijection mapping between the diagonal ∆ and the parameters set E, then the restrictions of soft sets are soft sets over E[21].
Let us consider the topological product ∏ s∈S X s , ∏ s∈S τ s , ∏ s∈S E s of a family of soft topological spaces {(X s , τ s , E)} s∈S .We take the restriction to the diagonal ∆ ⊂ ∏ s∈S E s of each soft set ∏ s∈S F : ∏ s∈S E s → P( ∏ s∈S X s ).
s n X s such that the set U = F s 1 (e) × ... × F s n (e) × ∏ .Then, from the definition of the topology (τ s i ) e , there exist soft open sets (F s 1 , E), ..., (F s n , E) such that F s 1 (e) = U s 1 , ..., F s n (e) = U s n .Then, s∈S(τ s ) e The evaluation mapping e x α : Y X , τ p , E → Y, τ , E is a soft continuous mapping for each soft Proposition 6.A soft mapping g : (Z, η, E) → Y X , τ p , E -where (Z, η, E) is a soft topological space-is a soft continuous mapping if and only if the soft mapping e x α • g : τ s p , E is the last soft set, ∇ −1 e −1 x α (F, E) is a soft open set on the product space ∏ s∈S Y X s , τ s p , E .Now, we prove that the mapping ∇ −1 : Y