Fuzzy soft topology considers only membership value. It has nothing to do with the non-membership value. So an extension was needed in this direction. Vague soft topology addresses both membership and non-membership values simultaneously. Sometimes vague soft topology (single structure) is unable to address some complex structures. So an extension to vague soft bi-topology (double structure) was needed in this direction. To make this situation more meaningful, a new concept of vague soft bi-topological space is introduced and its structural characteristics are attempted with a new definition. In this article, new concept of vague soft bi-topological space (VSBTS) is initiated and its structural behaviors are attempted. This approach is based on generalized vague soft open sets, known as vague soft

During the study of the potential applications in traditional and non-classical logic, it is essential to have fuzzy soft sets, vague soft sets and neutrosophical soft sets. Researchers now deal with the complications of modeling uncertain information in the economic, engineering, environmental, sociology, medical, and numerous other fields. Classical methods do not always succeed because there may be different types of uncertainties in those fields. Zadeh [

Maji et al. [

Chen et al. [

Cagman et al. [

Hong et al. [

Selvachandran et al. [

This reference [

In this article, the concept of vague soft topology is initiated and its structural characteristics are attempted with new definitions. The rest of the article is pictured as follows. Originality begins from this

In

In this section, some basic definitions, which are soft sets, soft sub-space, soft equal space, soft difference, soft null set, soft absolute set, soft point, soft union, soft intersection, vague soft topology and vague soft neighborhood are addressed.

The set of all NSS over

1. Vague Soft Subset:

2. Vague Soft Equality:

3. Vague Soft Intersection:

4. Vague Soft Union:

More generally, the Vague Soft intersection and the Vague Soft union of a collection of

5. The VSS defined as

6. Vague Soft Complement:

Clearly, the complements of

(1)

(2)

(3)

Then the order triple

In this section, the notion of vague soft bi-topological space is leaked out. Examples are also reflected to understand the structures. Vague soft

Then

Therefore

Here

The set of all pairwise vague open sets in

The set of all pairwise vague

If

If

1. Since

2. Since

As

Therefore

3. Since

Then

It is clear that

The pairwise vague soft

(

Conversely,

_{1} such that _{2} such that _{1}, _{2}) = _{0}. Which leads to the situation

In section, some results are discussed in two vague soft bi-topological space with respect to vague soft

Let, if possible, no point of

Then for each

Fuzzy soft topology considers only membership value. It has nothing to do with non-membership value. So extension was needed in this direction. The concept of vague soft topology was introduced to address the issue with fuzzy soft topology. Vague soft topology addresses both membership and non-membership values simultaneously. To make this object more meaningful, the conception of vague soft bi-topological structure is ushered in and its structural characteristics are attempted with new definitions. An ample of examples are also given to understand the structures. For the non-validity of some results, counter examples are provided. Pair-wise vague open and pair-wise vague soft closed sets are also addressed with examples in vague soft bi-topological spaces. Vague soft

We wish to express our extra sincere appreciation to the PI of our combined project, Professor Choonkil Park, who has the substance (figure) of a genius: he became the driving force, convincingly guided and encouraged us to be professional and do the right things even when the road got tough. Without his persistent help, the goal of this project would not have been realized.