Characterization of Soft S-Open Sets in Bi-Soft Topological Structure Concerning Crisp Points

: In this article, a soft s-open set in soft bitopological structures is introduced. With the help of this newly defined soft s-open set, soft separation axioms are regenerated in soft bitopological structures with respect to crisp points. Soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable and soft second-countable, soft connected, soft disconnected and soft locally connected spaces are defined with respect to crisp points under s-open sets in soft bitopological spaces. The product of two soft (S  :  = 1,2) axioms with respect crisp points with almost all possibilities in soft bitopological spaces relative to semiopen sets are introduced. In addition to this, soft (countability, base, subbase, finite intersection property, continuity) are addressed with respect to semiopen sets in soft bitopological spaces. Product of soft first and second coordinate spaces are addressed with respect to semiopen sets in soft bitopological spaces. The characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In addition to this, the product of two soft topological spaces is (   and   ) space if each coordinate space is soft (  and   ) space, product of two sot topological spaces is (S regular and C regular) space if each coordinate space is (S regular and C regular), the product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological spaces is (first-countable, second-countable) if each coordinate space is (first countable, second-countable).


Introduction
The soft set theory initiated by Molodtsov [1] has been demonstrated as an intelligent mathematical tool to deal with problems encompassing uncertainties or inexact data. Old-fashioned tools such as fuzzy sets [2], rough sets [3], vague sets [4], probability theory, etc. cannot be cast-off effectively because one of the root problems with these models is the absence of a sufficient number of expressive parameters to deal with uncertainty. In order to add a reasonable number of expressive parameters, Molodtsov [1] has shown that soft set philosophy has a rich potential to exercise in multifarious fields of mathematics. Maji et al. [5] familiarized comprehensive theoretical construction. Works on soft set philosophy are growing very speedily with all its potentiality and are being cast-off in different areas of mathematics [6][7][8][9][10][11]. In the case of the soft set, the parametrization is done with the assistance of words, sentences, functions, etc. For different characteristics of the decision variables present in soft set theory, different hybridization viz. Fuzzy soft sets [12], rough soft sets [13], intuitionistic fuzzy soft sets [14], vague soft sets [15], neutrosophic soft sets [16], etc. have been introduced with the passage of pieces of time research was in progress in the field of soft set theory.
Pakistani mathematician M. Shabir and M. Naz [17] ushered in the novel concept of soft topological spaces, which are defined relative to an initial universe of discourse with a fixed set of decision variables. In his work, different basic recipes and fundamental results were discussed with respect to crisp points, and counter-examples were also planted to clear the doubts. Soft separation axioms with respect to crisp points were discussed. I. Zorlutune et al. [18] tried his hands to fill in the gap that exists in [17] and studied some new features of soft continuous mapping and gave some innovative categorizations of soft continuous, soft open, soft closed mappings and also soft homeomorphisms topological structures.
In 2012, E. Peyghan, B. Samadi and A. Tayebi [19] filled in the gap that exists in [18] and ushered in some new features of soft sets and manifestly explored the notions of soft connectedness in soft topological spaces with respect to crisp points. The same pieces of work were further scrutinized and modified in 2014, M. Al-Khafaj and M. Mahmood [20] with the introduction of some properties of soft connected spaces and soft locally connected spaces with respect to ordinary points. M. Akdag and A. Ozkan [21] investigated some basic characters of basic results for soft topology relative to soft weak open sets, namely soft semiopen (closed)sets and defined soft s(closure and interior) in STS relative to crisp points. S.A. El-sheikh et al. [22]  S. Hussain and B. Ahmad [23] introduced the concept of soft separation axiom in soft topological spaces in full detail for the first time with respect to soft points. They provided examples for almost all results with respect to soft points. Soft regular, soft , soft normal and soft axioms using soft points are discussed. Khattak et al. [24] introduced the concept of α and soft β separation axioms in soft single point spaces and in soft ordinary spaces with respect to crisp points and soft points. M. Naz et al. [25] introduced the concept of soft bitopological spaces. First, the authors discussed the basic concepts of soft bitopology and then addressed different spaces in soft bitopology with respect to soft open sets. The results are supported by suitable examples.
M. Ittanagi [26] opened the door to pairwise notion of sbts and studied some types of soft separation axioms in a pairwise manner for sbts with respect to crisp points. Research on the same structures attracted the attention of researchers and resulted in T.Y Ozturk, and C.G. Aras [27] giving birth to the concept of soft pairwise continuity, soft pairwise open (closed) mappings, pairwise soft homeomorphism and scrutinized their basic characters in sbts. A. Kandil [28] pointed out the notion of pairwise soft connectedness and disconnectedness under the restriction of the idea of pairwise separated soft sets in sbts. They tried their best to explain the newly defined concepts with the support of examples. All the papers that are related so far to soft bitopological spaces do not have any results about weak separation axioms related to soft points. This big gap was bridged by A.M. Khattak et al. [29] for the first time, characterized soft s-separation axioms in soft bitopological spaces with respect to soft points. Soft regularity and normality were also studied with respect to soft points in soft bitopological spaces. In continuation, A.M. Khattak did not stop his work and resulted in Khattak, and some other researchers [30] studied some basic results and hereditary properties in soft bitopological spaces with respect to soft points. G. Senel [31] introduced the concept of soft bitopological Hausdorff space (SBT Hausdorff space) as an original study. First, the author introduced some new concepts in soft bitopological space such as SBT point, SBT continuous function and SBT homeomorphism. Second, the author defined SBT Hausdorff space. The authors analyzed whether an SBT space is Hausdorff or not by SBT homeomorphism defined from an SBT Hausdorff space to researched SBT space. Last, the author finished their study by defining SBT property and hereditary SBT by SBT homeomorphism and investigate the relations between SBT space and SBT subspace.
T.Y. Öztürk, S. Bayramov [32] introduced the concept of the pointwise topology of soft topological spaces. Finally, the authors investigated the properties of soft mapping spaces and the relationships between some soft mapping spaces.
S. Bayramov, G Aras [33] investigated some basic notions of soft topological spaces by using a new soft point concept. Later the authors addressed -soft space and the relationships between them in detail. Finally, the authors define soft compactness and explore some of its important properties. The above references [31][32][33] became a source of motivation for my research work.
There is a big gap for the researchers to define and investigate new structures, namely soft compactness, soft connectedness, soft bases, soft subbases, soft countability, finite intersection properties, soft coordinate spaces etc. In our study, we introduce some new definitions, which are soft semiopen set, soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable, soft connected, soft disconnected and soft locally connected spaces with respect to crisp points under s-open sets in soft bitopological spaces.
In the present article, we will present the notion of soft bitopological structure relative to the soft semiopen set, which is a generalization of the soft open set. The rest of this work is organized as follows: In the next section, concepts, notations and basic properties of soft sets and soft topology are recalled. In Section 3, we introduced some new definitions, which are necessary for our future sections of this article. In Section 4, some important definitions are introduced in soft bitopological spaces with respect to crisp points under soft semiopen sets. In Section 5, the engagement of section two and section three with some important results with respect to the crisp points under soft semiopen sets are addressed. In Section 6, the characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In the final section, some concluding comments are summarized, and future work is included.

Basic Concept
In this section, we present the vital definitions and results of soft set theory that are needed in this article.

Soft S-Open Sets in Soft Bi-Topological Space
In this section, we introduced some new definitions, which are necessary for our future sections of this article. Soft semiopen set, soft continuity at some certain points, soft bases, soft subbase, soft homeomorphism, soft first-countable and soft second-countable, soft connected, soft disconnected and soft locally connected spaces are defined with respect to crisp points under s-open sets in soft bitopological spaces. All most all the results are supported by examples.

Soft S-Open Sets in Soft Bi-Topological Space
In this section, some important definitions of soft structures are introduced in soft bitopological spaces with respect to crisp points under soft semiopen sets.

Product of Soft Separation Axioms and Soft (First and Second) Coordinate Spaces
In this section, the engagement of section two and section three with some important results with respect to crisp points under soft semiopen sets is addressed. The product of two soft S axioms with respect crisp points with almost all possibilities, the product of two soft S axioms with respect crisp points with almost all possibilities and product of two soft S axioms with respect to crisp points with almost all possibilities in soft bitopological spaces relative to semiopen sets are introduced. In addition to this, soft (countability, base, subbase, finite intersection property, continuity) are addressed with respect to semiopen sets in soft bitopological spaces. Finally, the product of soft first and second coordinate spaces are addressed with respect to semiopen sets in soft bitopological spaces. Proof. Let ( , ̃ , ̃ , E) * , , , E be the soft product space. We prove that it is soft S ℯ .Suppose (x , y ) (x , y ) are two distinct soft points in ( , ̃ , ̃ , E) * , , , E then different cases are x > x , y > y , x ≫ x or y ≫ y . , , , be two ℯ over the crisp sets and respectively such that these are two soft ℯ . Then their product ( , ̃ , ̃ , ) * , , , is soft ℯ .

Attachment of Separation Axioms with Soft Connectedness and Soft Coordinate Spaces
In this section, the characterization of soft separation axioms with soft connectedness is addressed with respect to semiopen sets in soft bitopological spaces. In addition to this, the product of two soft topological spaces is ( ) space if each coordinate space is soft ( )space, a product of two sot topological spaces is (S-regular and S-C regular) space if each coordinate space is (S-regular and S-C regular), a product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological +spaces is (firstcountable, second-countable) if each coordinate space is (first-countable, second-countable). Therefore, ( and )space, product of two sot topological spaces is (S regular and C regular) space if each coordinate space is (S regular and C regular), the product of two soft topological spaces is connected if each coordinate space is soft connected and the product of two soft topological spaces is (first countable, second-countable) if each coordinate space is (first countable, second-countable). All the above results are developed with respect to crisp (ordinary) points of the space. In the future, we will try to generate the above structures with respect to the soft points of the space. In this study, the product was limited to two structures. We will try to study the product of finite structures and infinite structures. In the development of the structures, we will use the nearly soft open sets, soft generalized open sets and soft most generalized open sets. After finishing this, we will try our hands to extend the soft bitopological structures to soft tri-topological structures with respect to crisp points and soft points of the spaces under nearly soft open sets, soft generalized open sets and soft most generalized open sets. When studying something in soft topology, we will be very careful because our domain of soft topology is not so strong. By extending the domain will extend the number of soft topologies. The more the topologies, the more accurate the structures will be because soft topology is actually giving information about soft structures. This article is just the beginning of the investigation of a new kind of structure. Hence, it will be necessary to continue the study and carry out more theoretical research in order to build a general framework for practical applications. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflicts of interest.