Bioperators on soft topological spaces

: To contribute to soft topology, we originate the notion of soft bioperators ˜ γ and ˜ γ (cid:48) . Then, we apply them to analyze soft (˜ γ, ˜ γ (cid:48) )-open sets and study main properties. We also prove that every soft (˜ γ, ˜ γ (cid:48) )-open set is soft open; however, the converse is true only when the soft topological space is soft (˜ γ, ˜ γ (cid:48) )-regular. After that, we deﬁne and study two classes of soft closures namely Cl (˜ γ, ˜ γ (cid:48) ) and ˜ τ (˜ γ, ˜ γ (cid:48) ) - Cl operators, and two classes of soft interior namely Int (˜ γ, ˜ γ (cid:48) ) and ˜ τ (˜ γ, ˜ γ (cid:48) ) - Int operators. Moreover, we introduce the notions of soft (˜ γ, ˜ γ (cid:48) )- g .closed sets and soft (˜ γ, ˜ γ (cid:48) )- T 12 spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.


Introduction
Vagueness and uncertainty occupied the human mind for centuries. In modern society, we face uncertainty and vagueness in different areas such as economics, engineering, medical science, sociality, and environmental sciences. Over the years, mathematicians, engineers, and scientists, particularly those who focus on artificial intelligence are seeking for approaches to solve the problems that contain uncertainty or vagueness. They established many tools for this purpose such as soft sets which are the most popular of all these.
The concept of soft sets was first constructed by Molodtsov [29] in 1999 as a general mathematical tool for dealing with uncertain objects. He successfully applied the soft set theory in several 2. Soft set andγ operator Definition 2.1. [29] Let X be an initial universe and E be a set of parameters. Let P(X) denote the power set of X and A be a non-empty subset of E. A pair (F, A) is called a soft set over X, where if F is a mapping given by F : A → P(X). In other words, a soft set over X is a parameterized family of subsets of the universe X. For a particular e ∈ A, F(e) may be considered the set of e-approximate elements of the soft set (F, A) and if e A, then F(e) = ∅. The family of all these soft sets over the universal set X is denoted by S S (X) A .
We call (F, A) a null soft set, denoted byφ if for all e ∈ A, F(e) = φ, and we call it an absolute soft set, denoted byX if for all e ∈ A, F(e) = X.
Definition 2.2. [16] For two soft sets (F, A) and (G, B) over a common universe X, we say that (F, A) is a soft subset of (G, B) (we write (F, A)⊆(G, B)) if A ⊆ B, and F(e) ⊆ G(e) for all e ∈ A. We also say that these two soft sets are soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A). Definition 2.7. [13] A soft set (P, A)∈ S S (X) A is called a soft point inX, denoted by P x e , if there exist e ∈ A and x ∈ X such that P(e) = {x} and P(e ) = φ for every e ∈ A \ {e}. We write P x e∈ (F, A), if x ∈ F(e).
Definition 2.8. [15,32] Let (F, A)∈ S S (X) A , and let x ∈ X. We say that 1. x∈ (F, A) whenever x ∈ F(e) for all e ∈ A. 2. x˜ (F, A) whenever x ∈ F(e) for some e ∈ A.
Note that x˜ (F, A) if x F(e) for some e ∈ A, and x˜ (F, A) if x F(e) for all e ∈ A. Definition 2.9. [32] Let x ∈ X. Then (x, A) is the soft set over X for which x(e) = {x} for all e ∈ A.
Definition 2.10. [32] Letτ be the collection of soft sets over X. Thenτ is said to be a soft topology on X if it satisfies the following axioms: 1.φ,X belong toτ. 2. The soft union of an arbitrary number of soft sets inτ belongs toτ. 3. The soft intersection of a finite number of soft sets inτ belongs toτ.
The triple (X,τ, A) is said to be a soft topological space (or soft space, in short) over X. Every member ofτ is called a soft open set. The complement of soft open set is called a soft closed set.
The main definitions and results aboutγ operator on the soft topologyτ can be found in [11,21,22]. Now, we will define the softγ-open set with respect to a soft point P x e . Definition 2.13. Let (X,τ, A) be a soft topological space andγ :τ → S S (X) A be an operator onτ. A soft set (F, A) of (X,τ, A) is said to be softγ-open if for each P x e∈ (F, A), there exists (V, A)∈τ with P x ẽ ∈ (V, A) andγ(V, A)⊆ (F, A). Definition 2.17. [22] Let (F, A)∈ S S (X) A and P x e∈X . A soft point P x e∈X is in the softγ-closure of (F, A) ifγ(U, A)∩ (F, A)˜ φ for every (U, A)∈τ with P x e∈ (U, A). The set of all softγ-closure points of (F, A) is called the softγ-closure of (F, A) and it is denoted by Clγ(F, A).  Definition 2.22. [22] A soft space (X,τ, A) is said to be softγ-T 1 2 if every softγ-g.closed set of (X,τ, A) is softγ-closed.
The following example shows that the inverse inclusions of Remark 3.1 do not hold and the converses of Proposition 3.1 are not true in general.  For any soft set (F, A) of (X,τ, A), the following statements are equivalent: Proof.
The following example shows that the soft regularity onγ andγ of Proposition 3.3 cannot be removed in general.  Theorem 3.1. Let (X,τ, A) be a soft topological space. Then the following statements are equivalent: Proof.
Proof. The proof follows from Theorem 3.1 and Remark 3.1.
Proof. The proof follows from Lemma 3.2 and Proposition 2.2.
In the end of this section, we introduce two classes of soft interior via bioperatorsγ andγ called Int (γ,γ ) andτ (γ,γ ) -Int operators.  Let (F, A), (G, A)∈ S S (X) A . Then the following statements are true: The following are equivalent: Proof. Straightforward.   (N, A). The class of all soft (γ,γ )-nbds of P x e is called the soft (γ,γ )-nbd system at P x e and is denoted by (N P x e , A). Theorem 4.5. The soft (γ,γ )-nbd system (N P x e , A) at P x e in a soft space (X,τ, A) has the following properties:  (N, A). Then for each P y e∈ (U, A) such that y x, P y ẽ ∈ Int (γ,γ ) (N, A) and hence (N, A)∈ (N P y e , A).  In this section, we introduce soft (γ,γ )-g.closed sets and soft (γ,γ )-T 1 2 spaces, and study some of their characterizations. Remark 5.1. Every soft (γ,γ )-closed set in (X,τ, A) is soft (γ,γ )-g.closed, but its converse is not true as may be shown from the following example.  Proof. The proof is immediate consequence of Proposition 3.1 (3).
The following example shows that the converse of Proposition 5.5 is not true in general.

Conclusions
Researchers and scientists proposed different approaches to handle problems of uncertainty. Among them, soft set theory has received the attention of the topologists who always seek to generalize and apply the topological notions on different structures.
As a contribution to this area, we have presented and studied the concepts of bioperatorsγ andγ on soft topologyτ, and the notion of soft (γ,γ )-open sets. Then, we have defined two soft closure and two soft interior operators, and elucidated the relationships between them. Finally, we have initiated the concepts of soft (γ,γ )-g.closed sets and soft (γ,γ )-T 1 2 spaces and investigated main properties. It was investigated in [10] the interchangeable property of soft interior and closure operators between soft sets and and their components. In the upcoming work, we will study, by making use of this property, the transmission of the concepts given herein from soft topology to its parametric topology and vise versa. Also, we will investigate this work in the contents of supra soft topology and fuzzy soft topology.

Conflict of interest
The authors declare that they have no competing interest.