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Sum of Soft Topological Spaces

In this paper, we introduce the concept of sum of soft topological spaces using pairwise disjoint soft topological spaces and study its basic properties. Then, we define additive and finitely additive properties which are considered a link between soft topological spaces and their sum. In this regard, we show that the properties of being p-soft Ti, soft paracompactness, soft extremally disconnectedness, and soft continuity are additive. We provide some examples to elucidate that soft compactness and soft separability are finitely additive; however, soft hyperconnected, soft indiscrete, and door soft spaces are not finitely additive. In addition, we prove that soft interior, soft closure, soft limit, and soft boundary points are interchangeable between soft topological spaces and their sum. This helps to obtain some results related to some important generalized soft open sets. Finally, we observe under which conditions a soft topological space represents the sum of some soft topological spaces.


Introduction
In 1999, Molodtsov [1] introduced the concept of soft sets as an innovative approach to deal with uncertainties. He demonstrated that soft sets are a beneficial mathematical method to handle with uncertainty in a parametric manner. Then, Maji et al. have completed two significant works, the first one has showed an application of soft set theory on decision-making problems [2], and the second one has presented the basic concepts between two soft sets such as soft union, intersection, and equality relations [3]. Ali et al. [4] have made some amendments for some results obtained by [3] and have defined new types of soft union and intersection between two soft sets. They have presented these types along with a relative complement of a soft set to keep the De Morgan's laws on a soft setting.
In 2011, Shabir and Naz [5] initiated the concept of soft topological spaces using soft sets that are defined over an initial universe set with a fixed set of parameters. After the inception of soft topology, many authors have investigated soft topological concepts analogously with their counterparts on classical topology. The different types of belong and non-belong relations on soft setting leads to introducing several types of soft axioms in terms of ordinary points [5][6][7][8] and soft points [9,10]. The authors of [11][12][13][14] have corrected some alleged findings on soft separation axioms. The author in [15] has presented and studied soft compactness. Then, the authors in [16] have defined other types of soft compactness depending on the natural belonging of the ordinary points to the covers. The authors in [17] have explored weak types of soft compact spaces, namely almost soft compact and mildly soft compact spaces. Investigation of soft compactness using soft pre-open and soft semi-open sets have been done by [18] and [19], respectively. The authors in [20] have discussed the concepts of

Preliminaries
In what follows, we first recall the main definitions and results which will be used through this work.

Soft Sets
Throughout the paper, Y will be a nonempty set, called an initial universal set, 2 Y its power set, E a nonempty set, called the set of parameters, A, B, C, ... subsets of E. Let us mention that almost all definitions are given for soft sets having a common set of parameters A. Definition 1.
[1] A soft set over Y is an ordered pair (ξ, A) such that A ⊂ E is a set of parameters and ξ is a mapping of A into 2 Y .
Through this work, the collection of all soft sets over Y under a set of parameters A is denoted by SS(Y A ). In addition, we use the different notations (ω, B), (δ, C), (η, D) for soft sets.

Definition 2. [3]
A soft set (ξ, A) over Y is said to be the null soft set if ξ(a) = ∅ for all a ∈ A; and it is said to be the absolute soft set if ξ(a) = Y for all a ∈ A.
The null and absolute soft sets are denoted by ∅ Y and Y, respectively. A soft set (P, A) over Y is called a soft point if there exist a ∈ A and y ∈ Y such that P(a) = {y} and P(b) = ∅, for each b ∈ A \ {a}. A soft point will be shortly denoted by P y a and we say that P y a ∈ (ξ, A), if y ∈ ξ(a) [33]. Definition 3. [34] (ξ, A) is a soft subset of (ω, B), denoted by (ξ, A) ⊆(ω, B), if A is a subset of B, and ξ(a) is a subset of ω(a) for all a ∈ A. The two soft sets are soft equal if each of them is a soft subset of the other. Definition 4. [4] The relative complement of (ξ, A) is a soft set (ξ, A) c = (ξ c , A) such that the map ξ c : A → 2 Y is defined by ξ c (a) = Y \ ξ(a) for each a ∈ A. Definition 5. [3] The union of soft sets (ξ, A) and (ω, B) over Y, denoted by (ξ, A) (ω, B), is the soft set (δ, C), where C = A B and δ : C → 2 Y is a mapping defined by Definition 6. [4] The intersection of soft sets (ξ, A) and (ω, B) over Y, denoted by (ξ, A) (ω, B), is a soft set (δ, C), where C = A B = ∅ and δ : C → 2 Y is a mapping defined by δ(c) = ξ(c) ω(c).
However, in this paper, we need definitions of the union and intersection for an arbitrary family of soft sets over a common universe Y and with a common set of parameters. Definition 7. [3] The union of a family {(ξ i , A) : i ∈ I} of soft sets over the common universe Y, denoted i∈I (ξ i , A), is the soft set (η, A), where, for each a ∈ A, η(a) = i∈I ξ i (a).
The intersection of a family {(ξ i , A); i ∈ I} over the common universe Y, denoted i∈I (ξ i , A), is the soft set (η, A), where, for each a ∈ A, η(a) = i∈I ξ i (a). Definition 8. [33] A soft mapping between SS(Y A ) and SS(Z A ) is a mapping f : Y → Z such that the image of (ξ, A) ∈ SS(Y A ) and preimage of (θ, A) ∈ SS(Z A ) are defined by:
a countable dense soft set [17]. (Recall that a soft set (ξ, A) is countable if ξ(a) is countable for each a ∈ A.) Definition 11. For a subset Z = ∅ of (Y, τ, A), the family τ Z = { Z (ξ, A) : (ξ, A) ∈ τ} is called a soft relative topology on Z and the triple (Z, τ Z , A) is called a soft subspace of (Y, τ, A). Proposition 1. [5] Let (Y, τ, A) be a soft topological space. Then, for each a ∈ A, the family τ a = {ξ(a) : (ξ, A) ∈ τ} defines a topology on Y for each a ∈ A.
We call τ a a parametric topology on Y.
Definition 12. [5,7] Let (ξ, A) be a soft set over Y and y ∈ Y. We write: 1. y (ξ, A) if y ∈ ξ(a) for some a ∈ A; and y (ξ, A) if y ∈ ξ(a) for every a ∈ A.
2. y ∈ (ξ, A) if y ∈ ξ(a) for every a ∈ A; and y ∈ (ξ, A) if y ∈ ξ(a) for some a ∈ A. In particular, y ∈ Y means y ∈ Y.

Definition 14.
[7] A soft set (ξ, A) over Y is called stable provided that there is S ⊆ Y such that ξ(a) = S for each a ∈ A; a soft topological space (Y, τ, A) is called stable provided that all proper non null soft open sets are stable.

Definition 15.
A soft topological space (Y, τ, A) is said to be: 1. soft compact [15] provided that every soft open cover of Y has a finite subcover. 2. soft paracompact [20] if every soft open cover has a soft open, locally finite refinement. 3. soft connected [20] if it can not be expressed as a union of two disjoint soft open sets. 4. soft hyperconnected [30] if every nun-null soft open set is soft dense. 5. soft extremally disconnected [22] if Cl τ (ξ, A) is soft open for every (ξ, A) ∈ τ. Theorem 1. [15,20] A soft closed set in a soft compact (resp. soft paracompact) space is also soft compact (resp. soft paracompact).

Sum of Soft Topological Spaces
In this section, we introduce and study the concept of sum of soft topological spaces. Then, we investigate which properties are additive and finitely additive. Definition 18. A collection of two or more soft sets is said to be pairwise disjoint if the intersection of any two distinct soft sets is the null soft set.
be a family of pairwise disjoint soft topological spaces and Y = i∈I Y i . Then, the collection defines a soft topology on Y with a fixed set of parameters A.
Definition 19. The soft topological space (Y, τ, A) given in the above proposition is said to be the sum of soft topological spaces and is denoted by (⊕ i∈I Y i , τ, A).
Remark 1. The term of sum of soft topological spaces was given in [31] without a condition of pairwise disjointness. Moreover, the authors of [31] did not study the properties of additive, finitely additive and countably additive which represent the main goal of this study. In fact, this definition leads to confusion on how constructing the sum of soft topological spaces and losing some well-known properties of the sum of soft topological spaces as the following example shows: two soft topologies on Y 1 and Y 2 , respectively. The sum of soft topological spaces (Y 1 , τ 1 , A) and (Y 2 , τ 2 , A) does not exist according to Definition 19 because Y 1 Y 2 = ∅. However, the sum of soft topologies τ 1 and τ 2 on Y = Y 1 Y 2 = {1, 2, 3} according to the definition given in [31] is τ = { ∅ Y , Y}. It is clear that Y 1 and Y 2 do not belong to τ and this contradicts the fact that the universal sets Y 1 and Y 2 belong to τ, see Corollary 1. Moreover, (Y, τ, A) is soft connected and this contradicts the fact that the sum of soft topological spaces is soft disconnected, see Corollary 2.
Corollary 1. All soft sets Y i are soft clopen in (⊕ i∈I Y i , τ, A).

Corollary 2.
Every sum of soft topological spaces is soft disconnected.
is a class of pairwise disjoint soft topological spaces and X i is a subspace of Y i for every i ∈ I, then the soft topology of the sum of subspaces {(X i , τ X i , A) : i ∈ I} and the soft topological subspace on i∈I X i of the sum soft topology (⊕ i∈I Y i , τ, A) coincide.

Definition 20.
A property P is said to be: 1. additive if, for any family of soft topological spaces {(Y i , τ i , A) : i ∈ I} with the property P, the sum of this family (⊕ i∈I Y i , τ, A) also has property P. 2. finitely additive (resp., countably additive) if, for any finite (resp., countable) family soft topological spaces with the property P, the sum of this family (⊕ i∈I Y i , τ, A) also has property P.
Theorem 2. The property of being a p-soft T i -space is an additive property for i = 0, 1, 2, 3, 4.

Proof.
We prove the theorem in the case of i = 2. Let y = z ∈ ⊕ i∈I Y i . Then, we have the following two cases: 1. There exists i 0 ∈ I such that y, z ∈ Y i 0 . Since (Y i 0 , τ i 0 , A) is p-soft T 2 , then there exist two disjoint soft open subsets (ξ, A) and (ω, A) of (Y i 0 , τ i 0 , A) such that y ∈ (ξ, A) and z ∈ (ω, A). It follows from Definition 19 that (ξ, A) and (ω, A) are disjoint soft open subsets of (⊕ i∈I Y i , τ, A).
2. There exist i 0 = j 0 ∈ I such that y ∈ Y i 0 and z ∈ Y j 0 . Now, Y i 0 and Y j 0 are soft open subsets of (Y i 0 , τ i 0 , A) and (Y j 0 , τ j 0 , A), respectively. It follows from Definition 19 that Y i 0 and Y j 0 are disjoint soft open subsets of (⊕ i∈I Y i , τ, A).
It follows from the two cases above that (⊕ i∈I Y i , τ, A) is a p-soft T 2 -space. The theorem can be proved similarly in the cases of i = 0, 1.
To prove the theorem in the cases of i = 3 and i = 4, it suffices to prove the p-soft regularity and soft normality, respectively.
First, we prove the p-soft regularity property. Let (η, A) be a soft closed subset of (⊕ i∈I Y i , τ, A) such that y (η, A). It follows from Proposition 3 that Second, we prove the soft normality property. Let (η, A) and (δ, A) be two disjoint soft closed subsets of (⊕ i∈I Y i , τ, A). It follows from Proposition 3 that (η, A) Y i and (δ, Proposition 5. The property of being a discrete soft space is an additive property. is a soft open subset of (⊕ i∈I Y i , τ, A).
The following example shows that soft compactness is not an additive property. Proof. 1. From Corollary 1, (Y i , τ i , A) is a soft closed subspace of (⊕ i∈I Y i , τ, A) for each i ∈ I. It follows from Theorem 1 that (Y i , τ i , A) is soft compact for each i ∈ I. 2. Let (⊕ i∈I Y i , τ, A) be the sum of soft topological spaces. Then, Λ = {Y i : i ∈ I} is a soft open cover of Y = i∈I Y i . It is clear that Λ does not have a finite subcover. This contradicts the fact that (⊕ i∈I Y i , τ, A) is soft compact. Hence, it must be that I is finite. (Y 1 , τ 1 , A) and (Y 2 , τ 2 , A) given in Example 2 are soft hyperconnected. Moreover, they are soft connected. However, the sum of (Y 1 , τ 1 , A) and (Y 2 , τ 2 , A) is neither soft hyperconnected nor soft connected. This means that the properties of soft hyperconnected and soft connected are not finite additive.

Remark 2. It is clear that the soft topological spaces
Similarly to the proof of Proposition 6, we prove the following:  Proof. Necessity: From Corollary 1, (Y i , τ i , A) is a soft closed subspace of (⊕ i∈I Y i , τ, A) for each i ∈ I. It follows from Theorem 1 that (Y i , τ i , A) is soft paracompact for each i ∈ I. Sufficiency: : i ∈ I} be a family of soft mappings. Then, we define a soft mapping f : (⊕ i∈I Y i , τ, A) → (⊕ i∈I Z i , θ, B) as follows: For each soft subsets (ξ, A) and (ω, B) of (⊕ i∈I Y i , τ, A) and (⊕ i∈I Z i , θ, B), respectively, we have: Proof. We merely give a proof for the theorem in the case of soft continuity and one can prove the cases between parentheses similarly. Necessity: Suppose that a soft mapping f : (⊕ i∈I Y i , τ, A) → (⊕ i∈I Z i , θ, B) is soft continuous. Taking an arbitrary soft map f j : (Y j , τ j , A) → (Z j , θ j , B), where j ∈ I. Let (ξ, B) be a soft open subset of (Z j , θ j , B). Then, (ξ, B) is a soft open subset of (⊕ i∈I Z i , θ, B). By assumption, is a soft open subset of (Y j , τ j , A), as required.
is soft continuous for every i ∈ I and let (ω, B) be a soft open subset of (⊕ i∈I Z i , θ, B). Now, (ω, B) Z i is a soft open subset of (Z i , θ i , B) for every i ∈ I. By assumption, ω, B) is a soft open subset of (⊕ i∈I Y i , τ, A), as required.

Corollary 3.
A soft mapping f : (⊕ i∈I Y i , τ, A) → (⊕ i∈I Z i , θ, B) is soft homeomorphism if and only if every soft mapping f i : Theorem 6. Let Int τ i and Int τ (resp. Cl τ i and Cl τ , l τ i and l τ ) be the soft interior (resp. soft closure, soft limit) points of a soft set (η, A) ⊆ Y i in (Y i , τ i , A) and (⊕ i∈I Y i , τ, A), respectively. Then: 2, 3. Following similar above arguments, results 2 and 3 are satisfied.
Lemma 1. Let (η, A) be a soft subset of (⊕ i∈I Y i , τ, A). Then, the collection {(η, A) Y i : i ∈ I} is locally finite.
Sufficiency: The proof follows from the fact that Cl τ (η, Thus, (ω, A) is soft dense. Thus, (⊕ i∈N Y i , τ, A) is soft separable. Hence, the desired result is proved.
The other path of this study is the answer of the following two questions: 1. Under what conditions can a soft topological space represent the sum of soft topological spaces? 2. If a soft topological space represents the sum of soft topological spaces, what is the maximum number of these soft topological spaces?
The following results answer these questions.
Theorem 10. If (Y, τ, A) is stable soft disconnected, then it represents the sum of two soft topological spaces.
Proof. Since (Y, τ, A) is soft disconnected, then it contains at least a proper soft clopen set (ξ, A). Since (Y, τ, A) is stable, then ξ(a) = X ⊆ Y for each a ∈ A. Thus, the two soft subspaces (X, τ X , A) and (X c , τ X c , A) are soft topological spaces such that (Y, τ, A) is their sum. Proof. Straightforward.

Conclusions
The study of soft topological spaces is of great importance because it provides a general frame that consists of parameterized classical topological spaces. The aim of the present work is to study the sum of topological spaces in the soft setting. Our results mainly investigate invariant properties between soft topological spaces and their sum. Thus, we define additive and finitely additive properties. In this regard, we demonstrate some additive properties such as p-soft T i , i = 0, 1, 2, 3, 4, soft paracompactness, soft extremally disconnectedness, and soft continuity. With the help of illustrative examples, we show that the properties of soft compact and soft separable spaces are finitely additive and the properties of soft hyperconnected, soft indiscrete, and door soft spaces are not additive. We made use of interchangeability of soft interior and soft closure operators between soft topological spaces and their sum to obtain some results related to some important generalized soft open sets. We complete this study by investigating the necessary conditions for a soft topological space to represent the sum of some soft topological spaces.
In the upcoming studies, we plan to examine more notions with respect to additive properties such as e-soft separation axioms and w-soft separation axioms. In addition, we are going to introduce and discuss the concept of sum of soft topological spaces on the contexts of ordered soft topological spaces and fuzzy soft topological spaces. Finally, we hope that this work will help the researchers who are interested in soft topology to study additive properties as a new characteristic of the concepts.