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Article

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

by
Arif Mehmood
1,
Mohammed M. Al-Shomrani
2,
Muhammad Asad Zaighum
1 and
Saleem Abdullah
3,*
1
Department of Mathematics & Statistics, Riphah International University, Sector I-14, Islamabad 44000, Pakistan
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2100; https://doi.org/10.3390/math8122100
Submission received: 9 October 2020 / Revised: 5 November 2020 / Accepted: 10 November 2020 / Published: 24 November 2020

Abstract

:
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 i : i = 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 ( S 1   and   S 2   ) space if each coordinate space is soft ( S 1   and   S 2   ) 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).

1. 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] introduced new soft separation structures rests on soft weak open sets, namely, soft b-open sets which are in true sense the generalization of soft open sets.
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 T 3 , soft normal and soft T 4 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., 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. [33] investigated some basic notions of soft topological spaces by using a new soft point concept. Later the authors addressed T i -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.

2. Basic Concept

In this section, we present the vital definitions and results of soft set theory that are needed in this article.
Definition 1.
([1]) Let X be the universal set, and E be the set of expressive parameters. Let P ( X ) supposes the power set of X which contains all possible subsets of X and E be the super set of A . An ordered pair ( f , A ) is called a soft set over X , where f is a mapping given by f : A P ( X ) . This signifies that a soft set over X is a parametrized family of subsets of the universal set X . For e A , f ( e ) is value at particular expressive parameter which engages a particular subset of the universal set and considered as the set of e -approximate element of the soft set ( f , A ) and if e A , then f ( e ) = φ implies meaningless that is ( f , A ) = { f ( e ) : e A ˜ E   , ( f : A P ( X   ) } . The family of all these soft sets is symbolized as S S ( X   ) A .
Definition 2
([1]). For two soft sets ( f , A ) and ( g , B ) over the same universe of discourse X , we say ( f , A )   ˜ ( g , B ) if A B , f ( e )   g ( e ) A .
And if A , f ( e ) ˜ g ( e ) B . Then the two soft sets are said to be soft equal. This is possible only if we are playing with the same expressive parameter set.
Definition 3
([17]). A soft set ( f , A )   S S ( X   ) A is said to be an absolute soft set, denoted by ( X , ˜ A ) if f ( e ) = X ,   e A . Moreover, soft set ( f , A )   S S ( X   ) A is said to be null soft set denoted by ( φ , ˜ A ) if f ( e ) = φ ,   e A .
Definition 4
([17]). A soft set ( f , A )   S S ( X   ) A is said to have complement denoted by ( f , A ) c where, f c : A P ( X ) is a mapping given by f c ( e ) = X f ( e ) ,   e A .
Definition 5
([17]). The difference of two soft sets ( f , A ) and ( g , A ) over the same parameter set and universe of discourse X denoted by ( f , A ) ( g , A ) is the soft set namely, ( H , A ) , where   e A ,   H ( e ) = f ( e ) g ( e )   o r ( g , A ) ( f , A ) .
Definition 6
([1]). Let ( f , A ) and ( g , B ) be two soft sets over two different sets of parameters and the same universe of discourse X . The soft union of ( f , A ) and ( g , B ) is the soft set ( H , C ) such that ( f , A ) ˜ ( g , B )   = ( H , C ) , where C = A B and for all e C
H ( e ) = { f ( e )   if   B A ,   i . e . ,   if   e ( A \ B ) g ( e )   if   A B ,   i . e . ,   if   e ( B \ A ) f ( e ) g ( e )   if   e ( A B )   }
Definition 7
([1]). Let ( f , A ) and ( g , B ) be two soft sets over two different sets of parameters and the same universe of discourse X . The soft intersection of ( f , A ) and ( g , B ) is the soft set ( H , C ) such that ( f , A ) ˜ ( g , B ) = ( H , C ) , where C = A B and H ( e ) = f ( e ) ˜ g ( e )   f o r   e a c h   e C .
Definition 8
([34]). A s o f t   s e t   ( f , A )   S S ( X   ) A is said to be a soft point in ( X ˜ , A ) If there exists x X and e A in such a track that f ( e ) = {   x } subject to f ( e c ) = φ for each e c = A {   e }   . This soft point is signified by   x e where,   x e   : A P ( X   ) following track
(   x e   , A ) = ( {   x } , i f   e = a φ   , i f   e a   f o r   a l l   e A     )
A soft point (   x e   , A ) is said to be housed off in s o f t   s e t ( f , A ) signifying   x e (   x e   , A ) , if x e   ( e ) ˜ f ( e ) , that is {   x } ˜ f ( e ) . Obviously,   x e (   x e   , A )   i f f   (   x e   , A ) ˜ ( f , A ) . In addition, two soft points   x e 1 ,     y e 2 relative to the crisp set X are said to be equal if x   =   y   a n d   e 1   =   e 2 that is bi-equal. This means that if the equality symbol is disturbed in either case, then the equality between   x e 1 and y e 2 will automatically be unbalanced.
Definition 9
([17]). Let τ ˜ be a collection of soft sets over a universe of discourse X with a fixed set of expressive parameters E . Thence τ ˜ ˜ S S ( X ) E is called a soft topology on X if it qualifies the following axioms:
(1) 
( X ˜ , E ) ,   ( φ ˜ , E ) ˜ τ ˜ , Where X ˜ ( e ) = X ,   e E and φ ˜ ( e ) = φ ,   e E
(2) 
Union of any number of soft sets in τ ˜ belongs to τ ˜
(3) 
The intersection of any two or finite number of soft sets in τ ˜ belongs to τ ˜ .
The structure of ordered triple ( X ˜ , τ ˜ , E ) is called a soft topological structure. Any candidate of τ ˜ is said to be a soft open set.
Definition 10
([35]). Suppose ( f , E ) be any soft set of soft topological space ( X , ˜ τ   ˜ , E ) . Then ( f , E ) is said to be a soft semiopen set of ( X , ˜ τ   ˜ , E ) if ( f , E )   ˜ c l ( i n t ( ( f , E )   ) ) and said to be soft semiclosed if ( f , E )   ˜ i n t ( c l ( ( f , E )   ) ) . The set of all soft semiopen sets is denoted by S S O ( X ) E and the set of all soft semiclosed sets is denoted by S S C ( X ) E .

3. 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.
Definition 11.
A quadrable system ( X   , τ ˜ 1 , τ ˜ 2 , E ) is S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X , where τ ˜ 1 and τ ˜ 2 are supposed to be arbitrary soft topologies on the crisp set X and E be the set of parameters.
Example 1.
Suppose X   =   { x 1 , x 2 , x 3 } and E   =   { e 1 , e 2 } . We develop the following soft sets relative to crisp set X ;
{ ( f 1 , E ) = { ( e 1 , { x 1 , x 2 } ) , ( e 2 , { x 2 , x 3 } ) } , ( f 2 , E ) = { ( e 1 , { x 1 } ) , ( e 2 , { x 1 , x 2 } ) } , ( f 3 , E ) = { ( e 1 , { x 2 } ) , ( e 2 , { x 2 } ) } , ( f 4 , E ) = { ( e 1 , { x 1 , x 2 } ) , ( e 2 , { x 3 } ) } , ( g 1 , E ) = { ( e 1 , { x 1 } ) , ( e 2 , { x 1 , x 2 } ) } , ( g 2 , E ) = { ( e 1 , { x 2 , x 3 } ) , ( e 2 , { x 2 , x 3 } ) } , ( g 3 , E ) = { ( e 1 , ) , ( e 2 , { x 2 } ) } . T h e n ( X , τ ˜ 1 , τ ˜ 2 , E ) i s   S B T 𝒮 𝓅 𝒶 𝒸 𝑒 . τ ˜ 1 = { ˜ , X ˇ , ( f 1 , E ) , ( f 2 , E ) , ( f 3 , E ) , ( f 4 , E ) } , τ ˜ 2 = { ˜ , X ˇ , ( g 1 , E ) , ( g 2 , E ) , ( g 3 , E ) }
Definition 12.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X . A soft set ( G , E ) over X is said to be soft semiopen If there exists a soft semiopen set ( G 1 , E ) τ ˜ 1 and soft semiopen set ( G 2 , E ) τ ˜ 2 such that ( G , E ) = ( G 1 , E ) ˜ ( ( G 2 , E ) . A soft set ( G , E ) over X is said to be close in ( X   , τ ˜ 1 , τ ˜ 2 , E ) if its respective complement is soft semiopen in ( X   , τ ˜ 1 , τ ˜ 2 , E ) .
Definition 13.
Suppose ( f , E ) be any soft set of soft bitopological space ( X   , τ ˜ 1 , τ ˜ 2 , E ) . Then ( f , E ) is said to be a soft semiopen set of ( X   , τ ˜ 1 , τ ˜ 2 , E ) if ( f , E ) ˜ τ ˜ 1 c l ( τ ˜ 2 i n t ( ( f , E ) ) ) and said to be soft semiclosed if ( f , E ) ˜ τ ˜ 1 i n t ( τ ˜ 2 c l ( ( f , E ) ) ) . The set of all soft semiopen sets is denoted by S S O S O S B T ( X ) E and the set of all soft semiclosed sets is denoted by S S C S O S B T ( X ) E   .
Definition 14.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 over the crisp sets X 1 and X 2 respectively and f : ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) be a soft mapping. Then f is said to be soft continuous at a soft point x e ˜ S S ( X ) E   i f f for each soft semiopen set ( H , E ) in ( X 2   , F 1 ˜ , F 2 ˜ , E ) containing f ( x e ) there exists a soft semiopen ( G , E ) set in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) containing x e such that f ( ( G , E )   ) ˜ ( H , E ) that is f ( x e )   ˜ f ( G , E )   ˜ ( H , E ) or x e ˜ ( G , E )   ˜ ( H , E ) .
Example 2.
Suppose X = { x 1 , x 2 , x 3 } , Y = { y 1 , y 2 , y 3 } and E = { e 1 , e 2 } . Then τ ˜ 1 = { , ˜ X ˇ , ( F , E ) } ,   τ ˜ 2 = { , ˜ X ˇ , ( G , E ) } two soft topological spaces over X and η 1 ˜ = { , ˜ Y ˇ , ( H , E ) } ,   η 2 ˜ = { , ˜ Y ˇ , ( K , E ) } are two soft topological spaces over Y . Here the soft sets over X and Y are defined as follows;
{ ( F , E ) = { ( e 1 , { x 1 , x 3 }   ) , ( e 2 , { x 2 , x 3 }   ) } ( G , E ) = { ( e 1 , { x 2 , x 3 }   ) , ( e 2 , { x 1 , x 3 }   ) } ( H , E ) = { ( e 1 , { y 1 , y 2 }   ) , ( e 2 , { y 2 , y 3 }   ) } ( K , E ) = { ( e 1 , { y 2 , y 3 }   ) , ( e 2 , { y 1 , y 2 }   ) }
Then ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 1   , η 1 ˜ , η 2 ˜ , E ) are two soft bitopological spaces.
τ ˜ 1 τ ˜ 2 = { , ˜ X , ˇ ( F , E ) , ( G , E ) } , η 1 ˜ η 2 ˜ = { , ˜ Y , ˇ ( H , E ) , ( K , E ) } .
If the mapping f : X Y defined as
{ f ( X e 1 1 ) = Y e 1 1 f ( X e 1 1 ) = Y e 1 1 f ( X e 1 1 ) = Y e 1 1 f ( X e 1 1 ) = Y e 1 1
Then f soft semiopen and soft semiclosed.
Definition 15.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X and ( X   , τ ˜ 1 , τ ˜ 2 , E ) . If every element of ( X   , τ ˜ 1 , τ ˜ 2 , E ) can be written as a soft union of elements of W , then W is called a s o f t   b a s e for the soft bitopology ( X   , τ ˜ 1 , τ ˜ 2 , E )   . Each element of W is called a soft base element.
Example 3.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X . Where the crisp set is X = { x 1 , x 2 , x 3 } and E p a r a m a t e r = { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } be the universal set of parameters and A p a r a m a t e r = { e 1 , e 2 } be the subset of the universal set of parameters
E = { e 1 , e 2 , e 3 , e 4 , e 5 , e 6 } ,   ( X   , τ ˜ 1 , E ) = { φ ˜ , X ˜ , f 1 , f 2 , f 3 , f 4 , f 5 , . . f 18 }
then is a S T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X and ( X   , τ ˜ 2 , E )   = { φ ˜ , X ˜ } is a S T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X We define the soft functions as:
{ f 1 ( e 1 ) = φ ˜ , f 1 ( e 2 ) = { x 1 } , f 2 ( e 1 ) = φ ˜ , f 2 ( e 2 ) = { x 2 } , f 3 ( e 1 ) = φ ˜ , f 3 ( e 2 ) = { x 1 , x 2 } f 4 ( e 1 ) = { x 1 } , f 4 ( e 2 ) = φ ˜ , f 5 ( e 1 ) = { x 1 } , f 5 ( e 2 ) = φ ˜ , f 6 ( e 1 ) = { x 1 , x 2 } , f 6 ( e 2 ) = φ ˜ , f 7 ( e 1 ) = { x 1 } , f 7 ( e 2 ) = { x 2 } , f 8 ( e 1 ) = { x 2 } , f 8 ( e 2 ) = { x 1 } , f 9 ( e 1 ) = { x 2 } , f 9 ( e 2 ) = { x 2 } , f 10 ( e 1 ) = { x 2 } , f 10 ( e 2 ) = { x 1 , x 3 } , f 11 ( e 1 ) = { x 1 , x 2 } , f 11 ( e 2 ) = { x 2 } , f 12 ( e 1 ) = { x 1 , x 3 } , f 12 ( e 2 ) = { x 1 , x 3 } , f 13 ( e 1 ) = X , f 13 ( e 2 ) = { x 1 , x 3 } , f 14 ( e 1 ) = { x 1 , x 3 } , f 14 ( e 2 ) = X , f 15 ( e 1 ) = { x 1 , x 2 } , f 15 ( e 2 ) = { x 2 } , f 16 ( e 1 ) = { x 1 } , f 16 ( e 2 ) = { x 2 } , f 17 ( e 1 ) = { x 1 } , f 17 ( e 2 ) = { x 1 , x 2 } , f 18 ( e 1 ) = { x 1 , x 2 } , f 18 ( e 2 ) = { x 1 , x 2 } . P u t   W = { φ ˜ , X ˜ , f 1 , f 2 , f 4 , f 5 , f 9 , f 14 , f 15 , f 17 } . T h e n   W   i s   s o f t   b a s e   f o r   ( X , τ ˜ 1 , τ ˜ 2 , E ) .
Definition 16.
Let ( X , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X and P ( X   , τ ˜ 1 , τ ˜ 2 , E )   ˜ . Then this soft collection is said to be a soft subbase for ( X   , τ ˜ 1 , τ ˜ 2 , E )   i f the collection of all finite soft intersections of members of P is a soft base for ( X   , τ ˜ 1 , τ ˜ 2 , E )   .
Definition 17.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 over the crisp sets X 1 and X 2 respectively. Let f :   ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) be soft mapping. This soft mapping is said to be a soft homeomorphism if this soft mapping is soft one-one, soft one-two and soft bi-continuous.
Definition 18.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X . This soft space is said to be soft, first-countable if every point of X a soft-countable soft local base.
Definition 19.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X . This soft space is said to be soft second if there exists a soft-countable soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) .
Definition 20.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X and ( f 1 , E ) ,   ( f 2 , E ) be soft subsets of the given space, then the soft sets ( f 1 , E ) and ( f 2 , E ) , are said to be soft separated or soft disconnected if
{ ( 1 ) :   ( f 1 , E ) ( φ , E ) , ( f 2 , E ) ( φ , E )   ( 2 ) :   ( f 1 , E ) ˜ ( f 2 , E ) ¯ = ( φ , E ) , ( f 2 , E ) ˜ ( f 1 , E ) ¯ = ( φ , E ) ( 3 ) :   ( X   , τ ˜ 1 , τ ˜ 2 , E )   = ( f 1 , E ) ˜ ( f 2 , E ) .  
Definition 21.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X is said to soft connected if f is not soft disconnected.
Definition 22.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X is said to be soft locally s-connected at x ˜ X if for every soft s-open set ( f , E ) with respect to ( X   , τ ˜ 1 , τ ˜ 2 , E ) where, ( f , E ) = ( f 1 , E ) ˜ ( f 2 , E ) there exists a soft connected s-open set ( G , E ) with respect to the same space ( X   , τ ˜ 1 , τ ˜ 2 , E ) containing 𝓍 contained in ( f , E ) , where ( G , E ) = ( G 1 , E ) ˜ ( G 2 , E ) with ( G 1 , E ) τ ˜ 1 , ( G 2 , E ) τ ˜ 2 . The space ( X   , τ ˜ 1 , τ ˜ 2 , E ) said to be soft locally connected i f f it is soft locally connected at each of its points.

4. Separation Axioms in Soft Bi-Topological Spaces

In this section, some important definitions of soft structures are introduced in soft bitopological spaces with respect to crisp points under soft semiopen sets.
Definition 23.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X , A E and x , y   X such that x > y   o r   x < y If there exists at least one soft ( X   , τ ˜ 1 , τ ˜ 2 , E ) s-open set ( f 1 , A ) or ( f 2 , A ) such that x ( f 1 , A ) ,   y   ( f 1 , A ) or y   ( f 2 , A )   , x ( f 2 , A ) then ( X   , τ ˜ 1 , τ ˜ 2 , E ) is said be a soft S 0 space.
Definition 24.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X , A E and x , y   X such that x > y   o r   x < y If there exists a soft ( X   , τ ˜ 1 , τ ˜ 2 , E ) s-open sets ( f 1 , A ) and ( f 2 , A ) such that x ( f 1 , A ) ,   y   ( f 1 , A ) and y   ( f 2 , A )   , x ( f 2 , A ) then ( X   , τ ˜ 1 , τ ˜ 2 , E ) is said be a soft S 1 space.
Definition 25.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set X , A E and x , y   X such that x > y   o r   x < y If there exists a soft ( X   , τ ˜ 1 , τ ˜ 2 , E ) s-open sets ( f 1 , A ) and ( f 2 , A ) such that x ( f 1 , A ) ,   y   ( f 2 , A ) with ( f 1 , A ) ˜   ( f 2 , A ) = A . then ( X   , τ ˜ 1 , τ ˜ 2 , E ) is said be a soft S 2 space.
Definition 26.
Let ( X   , τ ˜ 1 , τ ˜ 2 , E ) be a S B T 𝒮 𝓅 𝒶 𝒸 𝑒 relative to the crisp set , A E . Then ( X   , τ ˜ 1 , τ ˜ 2 , E ) is said to be regular if for every soft s-closed subset ( f , A ) and every point x   ( f , A ) , there exists a soft s-open sets ( g , A ) and ( h , A ) such that ( f , A ) ( g , A ) ,   x ( h , A ) and the possibility of ( g , A ) rules out the possibility of ( h , A ) .

5. 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 0 axioms with respect crisp points with almost all possibilities, the product of two soft S 1 axioms with respect crisp points with almost all possibilities and product of two soft S 2 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.
Theorem 1.
If ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 over the crisp sets X 1 and X 2 respectively such that these are two soft S 0   𝓈 𝓅 𝒶 𝒸 𝑒 . Then their product ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S 0   𝓈 𝓅 𝒶 𝒸 𝑒 .
Proof. 
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E )     ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product space. We prove that it is soft S 0   𝓈 𝓅 𝒶 𝒸 𝑒 . Suppose ( x 1 , y 1 )   a n d   ( x 2 , y 2 ) are two distinct soft points in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) then different cases are x 1 > x 2   ,   o r   y 1 > y 2 , x 1   x 2 or y 1   y 2 .
Case ( I ) If x 1 > x 2 ,   then ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) being soft S 0   s 𝓅 𝒶 𝒸 𝑒 , corresponding to this pair of distinct soft points there exists a ( X 2   , F 1 ˜ , F 2 ˜ , E ) soft s -open set ( G , E ) such that x 1 ( G , E ) and x 2 ( G , E )   . Thus, ( G , E )   X 2 is soft ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   ) s -open set catching ( x 1 , y 1 ) but bay passing ( x 2 , y 2 )   .
Case ( I I ) If x 1 x 2 ,   then ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) being soft S 0   s 𝓅 𝒶 𝒸 𝑒 , corresponding to this pair of distinct soft points there exists a ( X 2   , F 1 ˜ , F 2 ˜ , E ) soft s -open set ( G , E ) such that x 1 ( G , E ) and x 2 ( G , E )   . Thus, ( G , E )   X 2 is soft ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E )   )   s -open set catching ( x 1 , y 1 ) but bay passing ( x 2 , y 2 )   .
Case ( I I I ) If y 1 >   y 2 ,   then ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) being soft S 0   s 𝓅 𝒶 𝒸 𝑒 , corresponding to this pair of distinct points there exists a ( X 2   , F 1 ˜ , F 2 ˜ , E ) –soft s -open set ( H , A ) such that y 1 ( H , A ) and y 2 ( H , A )   . Thus, X 1 ( H , A ) is soft ( ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) s -open set arresting ( x 1 , y 1 ) and bypassing ( x 2 , y 2 ) . Thus for any two distinct points of i n   X 1 X 2 there exists a ( X 1 , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) soft s -open set arresting one of them and bypassing the other. Hence ( X 1 , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S   0   𝓈 𝓅 𝒶 𝒸 𝑒 .
Case ( I V ) If y 1   y 2 ,   then ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) being soft S 0   s 𝓅 𝒶 𝒸 𝑒 , corresponding to this pair of distinct points there exists a ( X 2   , F 1 ˜ , F 2 ˜ , E ) soft s -open set ( H , A ) such that y 1 ( H , A ) and y 2 ( H , A )   . Thus, X 1 ( H , A ) is soft ( ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) s -open set arresting ( x 1 , y 1 ) and bypassing ( x 2 , y 2 ) . Thus for any two distinct points of i n   X 1 X 2 there exists a ( X 1 , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) soft s -open set arresting one of them and bypassing the other. Hence ( X 1 , τ ˜ 1 , τ ˜ 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ ,   E ) is soft S   0 space. □
Theorem 2.
If ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 over the crisp sets X 1 and X 2 respectively such that these are two soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 . Then their product ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 .
Proof. 
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 such that they are soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 , respectively. Then we have to show that their product that is ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is also soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 . For this proof, it is sufficient to show that each soft subset of ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) consisting of exactly one soft point is a soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   s -closed set. Let ( x 1 , y 1 )   X 1 X 2 so that x 1 X 1 and y 1 X 2 .
Case ( I ) x 1 > y 1 Now, ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) being soft S 1   S 𝓅 𝒶 𝒸 𝑒 , (   x 1 , A ) c is soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   s closed and therefore, ( y 1 , A ) c is soft ( X 2   , F 1 ˜ , F 2 ˜ , E )   s - o 𝓅 𝑒 𝓃 . Also, ( X 2   , F 1 ˜ , F 2 ˜ , E ) being soft S 1   S 𝓅 𝒶 𝒸 𝑒 . So ( (   x 1 , A ) c   ) ( ( y 1 , A ) c   )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) that is { ( (   x 1 , y 1 ) c , A ) } ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and hence, { ( ( ( x 1 , y 1 ) c , A ) } is soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   s closed. Hence ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S 1   S 𝓅 𝒶 𝒸 𝑒 .
Case ( I I ) x 1 y 1 Now, ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) being soft S 1   S 𝓅 𝒶 𝒸 𝑒 , (   x 1 , A ) c is soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   s closed and therefore, ( y 1 , A ) c is soft ( X 2   , F 1 ˜ , F 2 ˜ , E )   s - o 𝓅 𝑒 𝓃 . Also, ( X 2   , F 1 ˜ , F 2 ˜ , E ) being soft S 1   S 𝓅 𝒶 𝒸 𝑒 . So ( (   x 1 , A ) c   ) ( ( y 1 , A ) c   )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) that is { ( (   x 1 , y 1 ) c , A ) } ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and hence, { ( ( ( x 1 , y 1 ) c , A ) } is soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   s -closed. Hence ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S 1   S 𝓅 𝒶 𝒸 𝑒 . □
Theorem 3.
If ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 over the crisp sets X 1   and X 2 respectively such that these are two soft S -Hausdorff spaces. Then their product ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -Hausdorff space.
Proof. 
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the product space. We prove that it is a soft S Hausdorff space. Suppose ( x 1 , x 2 ) and ( y 1 , y 2 ) are two distinct points in X 1 X 2 .
Case ( I ) When x 1 =   y 1 , then x 2 y 2 . let x 2 > y 2 therefore ( x 1 , x 2 ) ( y 1 , y 2 ) By the soft S Hausdorff space property, given a pair of elements x 2 , y 2 X 2 such that x 2   >   y 2 there are disjoint soft s -open sets such that ( G 2 , A ) , ( H 2 , A ) ˜ ( X 2   , F 1 ˜ , F 2 ˜ , E ) such that x 2 ( G 2 , A ) , that x 2 ( H 2 , A ) . Then X 1   ( G 2 , A ) and ( X 1 X 2 ) ( H 2 , A ) are disjoint soft s - o pen sets in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . For, x 1 ( X c r i s p 1 ) A ,   x 2 ( G 2 , A ) implies that ( x 1 , x 2 ) ( ( X c r i s p 1 ) A     ( G 2 , A ) ) , y 1 X 1 , y 2 ( H 2 , A ) implies that ( y 1 , y 2 )   X 1 ( H 2 , A ) . this whole situation leads us to the conclusion that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is a soft S Hausdorff space.
Case ( I I ) If x 1 y 1 let x 1 > y 1 then too ( x 1 , y 1 ) >   ( x 2 , y 2 ) . By the soft S Hausdorff space property, given a pair of elements ( x 1 , y 1 ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) such that x 2 > y 2 , there are disjoint s -open sets in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) such that ( G 1 , A ) , ( H 1 , A ) ˜ ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) such that x 1 ( G 1 , A ) , that y 1 ( H 1 , A ) . Then ( G 1 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and ( H 1 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) are disjoint soft s -open sets in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . For, x 1 ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ,   x 2 ( G 2 , A ) implies that ( x 1 , x 2 ) ( G 1 , A ) X 2   , y 1 ( H 1 , A ) X 2   , y 2 ( H 2 , A ) implies that ( y 1 , y 2 ) ( H 1 , A ) X 2   . This whole situation leads us to the conclusion that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is a soft S -Hausdorff space. □
Theorem 4.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two second-countable S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 then their product, that is ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is also soft second-countable S B T 𝒮 𝓅 𝒶 𝒸 𝑒 .
Proof. 
To prove ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is second-countable S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 . Our assumption implies that there are countable soft s-bases 1 = { B i : i N } and 2 = { C i = i N } for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) respectively. = ( G 1 , A ) ( G 2 , A ) ; such that ( G 1 , A ) and ( G 2 , A ) are soft s -open such that ( G 1 , A ) { ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) , ( G 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   } is a soft b a s e for soft product topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . If we write C = { B i C j ; i , j N } = B 1 B 2 is soft-countable, this implies that C is soft-countable. By definition of soft base , and ( x 1 , y 1 ) N ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) implies that there exists a soft s -open sets ( G , A ) , ( H , A ) such that ( G , A ) ( H , A ) such that ( x 1 , y 1 ) ( G , A ) ( H , A ) ˜ N implies x 1 ( G , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) , y 1 ( H , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) this implies that there exists B i B 1 , C j B 2 such that x 1 B i ˜ ( G , A ) , y 1 C j ˜ ( H , A ) this implies that ( x 1 , y 1 ) B i C j ˜ ( G , A ) ( H , A ) ˜ N . By definition this proves that C is a soft base for the soft product topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   . In addition, C has been shown to be soft-countable. Hence, ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft second-countable relative to soft s -open set. □
Theorem 5.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp sets X 1 and X 2 respectively. the collection = { ( G 1 , A ) ( G 2 , A ) : ( G 1 , A ) ( τ ˜ 1 τ ˜ 2 ) , ( G 2 , A ) ( F 1 ˜ F 2 ˜ ) } is a s o f t   b a s e for some product soft topologies ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , where ( G 1 , A ) and ( G 2 , A ) are soft s -open sets in their corresponding S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 .
Proof. 
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp sets X 1 and X 2 respectively. Suppose ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product topology.
B = { ( u 1 , A ) ( u 2 , A ) : ( u , A ) ( τ ˜ 1 τ ˜ 2 ) , ( u 2 , A ) ( F 1 ˜ F 2 ˜ ) } where ( u 1 , A ) is soft s -open i n   ( F 1 ˜ F 2 ˜ ) and ( u 2 , A ) is soft-open i n   ( τ ˜ 1 τ ˜ 2 ) . We need to prove B is a s o f t b a se for some soft t o p o l o gy on ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . To prove that { B : B } = ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Clearly, ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   this implies that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   = { B : B } . Next, let ( u 1 , A ) ( u 2 , A ) , ( J 1 , A ) ( J 2 , A ) B where ( J 1 , A ) is soft s -open i n   ( F 1 ˜ F 2 ˜ )   a n d   ( J 2 , A )   i s   s o f t   s -open i n   ( τ ˜ 1 τ ˜ 2 ) and suppose ( 𝓍 1 , 𝓍 2 ) ( ( u 1 , A ) ( u 2 , A ) ) ( ( J 1 , A ) ( J 2 , A ) ) B . To prove that there exists ( J 1 , A ) s o f t   s -open i n   ( F 1 ˜ F 2 ˜ )   a n d   ( J 2 , A )   i s   s o f t   s -open i n   ( τ ˜ 1 τ ˜ 2 ) such that ( 𝒻 1 , A ) ( 𝒻 2 , A ) such that ( 𝓍 1 , 𝓍 2 ) ( 𝒻 1 , A ) ( 𝒻 2 , A ) ˜ ( ( u 1 , A ) ( u 2 , A ) ) ( ( J 1 , A ) ( J 2 , A ) ) that is ( 𝓍 1 , 𝓍 2 ) ( ( u 1 , A ) ( u 2 , A ) ) ( ( J 1 , A ) ( J 2 , A ) ) this implies ( 𝓍 1 , 𝓍 2 )   ( ( u 1 , A ) ( u 2 , A ) ) , ( 𝓍 1 , 𝓍 2 ) ( ( J 1 , A ) ( J 2 , A ) ) This implies that 𝓍 1 ( u 1 , A ) , 𝓍 2 ( u 2 , A ) ,   𝓍 1 ( J 1 , A ) , 𝓍 2 ( J 2 , A ) implies that 𝓍 1 ( ( u 1 , A ) ( J 1 , A ) ) , 𝓍 2 ( ( u 2 , A ) ( J 2 , A ) ) implies that 𝓍 1 ( 𝒻 1 , A ) , 𝓍 2 ( 𝒻 2 , A ) . On taking ( 𝒻 1 , A ) = ( u 1 , A ) ( J 1 , A ) ,   ( 𝒻 2 , A ) = ( u 2 , A ) ( J 2 , A ) this implies ( 𝓍 1 , 𝓍 2 ) ( 𝒻 1 , A ) ( 𝒻 2 , A ) . ( A 1 , A ) ( u 2 , A ) , ( J 1 , A ) ( J 2 , A ) this implies that ( u 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ، ( u 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   , ( J 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ,   ( J 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) this implies that ( u 1 , A ) ( J 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) , ( u 2 , A ) ( J 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   . This implies that ( 𝒻 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ,   ( 𝒻 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) this implies that ( 𝒻 1 , A ) ( 𝒻 2 , A ) . Thus we have shown that there exists ( 𝒻 1 , A ) ( 𝒻 2 , A ) such that ( 𝓍 1 , 𝓍 2 ) ( 𝒻 1 , A ) ( 𝒻 2 , A ) . Remaining to prove that ( 𝒻 1 , A ) ( 𝒻 2 , A ) ˜ ( ( u 1 , A ) A ) ( ( J 1 , A ) ( J 2 , A ) ) . Let ( y 1 , y 2 ) ( 𝒻 1 , A ) ( 𝒻 2 , A ) be arbitrary distinct points.   ( y 1 , y 2 ) ( ( 𝒻 1 , A ) ( 𝒻 2 , A ) ) implies that y 1 ( 𝒻 1 , A ) , y 2 ( 𝒻 2 , A ) implies that y 1 ( ( u 1 , A ) ( J 1 , A ) ) ,   y 2 ( ( u 2 , A ) ( J 2 , A ) ) implies that y 1 ( u 1 , A ) , y 1 ( J 1 , A ) ;   y 2 ( u 2 , A ) , y 2 ( J 2 , A ) this implies that ( y 1 , y 2 ) ( ( u 1 , A ) ( u 2 , A ) ) ,   ( y 1 , y 2 ) ( ( J 1 , A ) ( J 2 , A ) ) this implies that ( y 1 , y 2 ) ( ( u 1 , A ) ( u 2 , A ) ) ( ( J 1 , A ) ( J 2 , A ) ) . This implies ( 𝒻 1 , A ) ( 𝒻 2 , A ) ( ( u 1 , A ) ( u 2 , A ) )   ) ( ( J 1 , A ) ( J 2 , A ) ) . Thus is a base. □
Theorem 6.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp sets X 1 and X 2 respectively. Let 1 and 2 be soft bases for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) respectively. Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product space.
Then B = { B 1 B 2 : B 1 1 , B 2 2   } is a soft base for the product soft topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) relative to soft s -open sets.
Proof. 
C = { ( G 1 , A ) ( G 2 , A ) : ( G 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) , ( G 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   }
Then C is a soft base for the topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . We have to prove that B is a soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . According to the soft base, for any ( 𝕩 ˜ 1 , 𝕩 ˜ 2 ) ( G , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) where ( G , A ) is soft s -open relative ( τ ˜ 1 , τ ˜ 2 ) ( ( F 1 ˜ , F 2 ˜ ) implies there exists a soft ( τ ˜ 1 , τ ˜ 2 )   s -open sets ( G 1 , A ) and ( F 1 ˜ , F 2 ˜ ) soft   s -open set ( G 2 , A ) such that ( G 1 , A ) ( G 2 , A ) C such that ( 𝓍 1 , 𝓍 2 ) ( G 1 , A ) ( G 2 , A ) ˜ ( G , A ) . Again ( 𝓍 1 , 𝓍 2 ) ( G 1 , A ) ( G 2 , A ) C implies that 𝓍 1 ( G 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) , 𝓍 2 ( G 2 , A ) ( F 1 ˜ , F 2 ) ˜ . Applying the definition of a soft base, 𝓍 1 ( G 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) this implies that there exists B 1 1 such that 𝓍 1 B 1 ˜ ( G 1 ,   A ) . ( 1 )
𝓍 2 ( G 2 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) implies that there exists B 2 2 such that 𝓍 2 B 2 ˜ ( G 2 , A ) . ( 2 ) . Now B 1 1 , B 2 2   B this implies that 1 B 2 B . Now maxing (1) and (2) implies that there exists 1 B 2 B such that ( 𝓍 1 , 𝓍 2 ) 1 B 2 ˜ ( G 1 , A ) ˜ ( G 2 , A ) ˜ ( G , A ) or ( 𝓍 1 , 𝓍 2 ) 1 B 2 ˜ ( G , A ) . By definition, this proves that B is a soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) relative to soft s   -open sets. □
Theorem 7.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set   X 1 and X 2 respectively. Suppose L and M be soft subbases for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) respectively. Then the co l l e c t i on A of all soft s u b s ets of the f o rm L ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and M ( X 2   , F 1 ˜ , F 2 ˜ , E ) , is a soft subbas e for the product soft ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   w here L ( , A ) , M ( , A )
Proof. 
In order to prove that A is a soft subbase for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , we have to prove that the soft collection of s -open sets ( G ,   A ) of the finite intersection of soft members of A form a soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Since the intersection of empty soft subcollection of ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and so ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   ( G , A ) . Next, we suppose that { ( L 1 A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , ( L 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , ( L 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( L , r A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , ( L p , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) } { ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M 1 , A ) , ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M 2 , A ) , ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M 3 , A ) , ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M r , A ) , ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M q , A ) } be a non-empty infinite soft subcollection of A . This soft intersection of these elements belong to ( G ,   A ) , by the construction of ( G ,   A ) . This elements of ( G , A ) is { ( ( L , 1 A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) ( ( L , 2 A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) ( ( L , 3 A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) ( ( L , r A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 1   , F 1 ˜ , F 2 ˜ , E ) ) , ( L , p A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) } { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M , 1 A ) ) ( ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M , 2 A ) ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M , 3 A ) ) ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M , R A ) ) , ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( M , q A ) ) } .
[ { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   ( ( M , 1 A ) ) ( M , 2 A ) ) ( ( M , 3 A ) ) ( M , R A ) ) , ( M , q A ) ) }   ( ( L , 1 A ) ( ( L , 2 A ) ( ( L , 3 A ) ) ( ( L , r A ) , ( L , p A ) ) ( ( M , 1 A ) ) ( M , 2 A ) ) ( ( M , 3 A ) ) ( M , R A ) ) , ( M , q A ) ) = n = 1 p [ ( L , r A ) ]     r = 1 q [ ( M , r A ) ] . We supposed that B is a soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) generated by the elements of ( ,   A ) and C is a soft base for ( X 2   , F 1 ˜ , F 2 ˜ , E ) generated by the elements of ( ,   A ) . Since the finite soft intersection of soft subbase form, the soft base for that soft topology. In view of the above statements,   n = 1 p [ ( L , r A ) ] , r = 1 q [ ( M , r A ) ] C . Hence of ( G   , A ) is expressible as ( G   , A ) = { B C : B , C } . Then ( G   , A ) is soft s -open such that ( G   , A ) ( τ ˜ 1 , τ ˜ 2 ) ( F 1 ˜ , F 2 ˜ ) is a soft base for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . But ( G   , A ) is obtained from the finite soft intersection of members of A . It follows that A is a soft subbase for ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   .  □
Theorem 8.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set   X 1   a n d   X 2 respectively. Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the product space. Then, the soft projection maps π 1 and π 2 are soft continuous, and soft s -open.
Proof. 
Suppose ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be a product space of S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Then X = ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   . Define soft maps π 1   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) such that π 1 ( 𝓍 1 , 𝓍 2 ) = 𝓍 1   f o r   a l l   ( 𝓍 1 , 𝓍 2 ) ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , π 2   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) such that π 2 ( 𝓍 1 , 𝓍 2 ) = 𝓍 2 for all   ( 𝓍 1 , 𝓍 2 )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Then π 1 and π 2 both are called soft projection maps on the soft first and second coordinate spaces, respectively.
Step (1) First, we show that π 1 is soft continuous. Let ( G , A ) ˜ ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) be arbitrary soft s -open set. π 1 ( G , A ) = { ( 𝓍 1 , 𝓍 2 ) ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2 , F ˜ 1 , F ˜ 2 , E ) : π 1 ( 𝓍 1 , ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ) ( G , A ) } = { ( 𝓍 1 , 𝓍 2 ) ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2 , F ˜ 1 , F ˜ 2 , E ) : 𝓍 1 ( G , A ) } = { ( 𝓍 1 , 𝓍 2 ) ( X 1 , τ ˜ 1 , τ ˜ 2 , E ) ( X 2 , F 1 ˜ , F 2 ˜ , E ) : 𝓍 1 ( G , A ) } = ( G , A ) ( X 2 , F 1 ˜ , F 2 ˜ , E ) i s soft s -open set in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   . This implies π 1 is soft continuous. The proof runs on similar lines to show that π 2 is soft continuous.
Step (2) To prove that the soft projection maps are soft s -open maps. Let ( G , A ) ˜ ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be an arbitrary soft s-open set. Let 𝓍 1   π 2 [ ( G , A ) ] be arbitrary. This means that there exists ( u ˜ 1 , u ˜ 2 ) ( G , A ) such that π 2 ( u ˜ 1 , u ˜ 2 ) 𝓍 2 this implies that u ˜ 1 = 𝓍 2 because π 2 ( u ˜ 1 , u ˜ 2 ) = u ˜ 2 . Now ,   ( u ˜ 1 , 𝓍 2   ) ( G , A ) . Let B be the base for the soft topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Then by definition of a soft base, ( u ˜ 1 , 𝓍 2   ) ( G , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) implies that there exists a soft   ( τ ˜ 1 , τ ˜ 2 ) s-open set ( U 1 , A ) and soft ( F 1 ˜ , F 2 ˜ )   s-open set ( U 2 , A ) such that   ( U 1 , A ) ( U 2 , A ) such that ( u ˜ 1 , 𝓍 2   ) ( U 1 , A ) ( U 2 , A ) ˜ ( G , A ) implies that π 2 ( ( u ˜ 1 , 𝓍 2   ) , 𝓍 2   ) π 2 ( ( U 1 , A ) ( U 2 , A )   ) ˜ ( G , A ) ˜ π 2 ( ( G , A ) ) . for π 2 ( ( U 1 , A ) ( U 2 , A )   ) = { π 2   ( 𝓍 1 , 𝓍 2 ) : ( 𝓍 1 , 𝓍 2 ) ( ( U 1 , A ) ( U 2 , A )   ) } = { 𝓍 2   : 𝓍 1 ( U 1 , A ) , 𝓍 2   ( U 2 , A ) } = ( U 2 , A ) . This proves that 𝓍 2 is a soft interior point of π 2 ( ( G , A ) ) . But 𝓍 2 is an arbitrary soft point of   π 2 ( ( G , A ) ) . Therefore, every point of π 2 ( ( G , A ) ) is a soft interior point. This proves that π 2 ( ( G , A ) ) is soft s -open in ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) . This proves that the soft map π 2 : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is the soft s -open map. A proof can be written on the same lines to show that π 2 is a soft s -open map. Consequently, projection maps are soft s -open maps. This finishes the proof. □
Theorem 9.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set X 1 and X 2 respectively. ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the product space, π 1   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ,   π 2   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft projections maps on the first and second coordinate spaces, respectively. Let ( 𝓎 1 , T 1 , T 2 , E ) be another soft bitopological space such that : ( 𝓎 1 , T 1 , T 2 , E )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Then is soft continuous π 1 o and π 2 o are soft continuous maps.
Proof. 
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product topological space. Let ( 𝓎 1 , T 1 , T 2 , E ) be another soft bitopological space. Let be the soft base for the soft product topology ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Now π 1 o : ( 𝓎 1 , T 1 , T 2 , E )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ,   π 2 o : ( 𝓎 1 , T 1 , T 2 , E )   ( X 2   , F 1 ˜ , F 2 ˜ , E ) are also soft maps. Let be soft continuous. Now π 1 o and π 2 o are continuous soft maps. Conversely, suppose that π 1 o and π 2 o are continuous soft maps. To show that is continuous. Let ( G , A ) ˜ ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be arbitrary soft s -open set. If we prove that 1 ( ( G , A ) ) is soft s - o 𝓅 𝑒 𝓃 in ( 𝓎 1 , T 1 , T 2 , E ) , the result will follow. Let y ˜ 1 ( ( G , A ) ) be an arbitrary, then ( y ˜ ) ( G , A ) . therefore, ( y ˜ ) is an element of ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and hence it can be taken as ( y ˜ ) = ( 𝓍 1 , 𝓍 2 ) ( G , A ) by definition of the soft base, ( 𝓍 1 , 𝓍 2 ) ( G , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( 𝓎 1 , T 1 , T 2 , E ) there exists a soft ( τ ˜ 1 τ ˜ 2 ) s-open set ( U 1 , A ) and soft ( F 1 ˜ F 2 ˜ ) s - o 𝓅 𝑒 𝓃 set ( U 2 , A ) and such that ( U 1 , A ) ( U 2 , A ) such that 𝓍 1   ( U 1 , A ) ( U 2 , A ) ˜ ( G , A ) implies that
π 1 ( ( 𝓍 1 , 𝓍 2 ) ) π 1 ( ( U 1 , A ) ( U 2 , A ) ) ˜ π 1 ( G , A ) and π 2 ( ( 𝓍 1 , 𝓍 2 ) ) π 2 ( ( U 1 , A ) ( U 2 , A ) ) ˜ π 2 ( G , A ) this implies that 𝓍 1 ( U 1 , A ) ˜ π 1 ( G , A ) and 𝓍 2 ( U 2 , A ) ˜ π 2 ( G , A ) . For π 1 ( ( U 1 , A ) ( U 2 , A ) ) = { π 1 ( ( 𝓍 1 , 𝓍 2 ) ) : ( 𝓍 1 , 𝓍 2 ) ( ( U 1 , A ) ( U 2 , A ) ) } = { 𝓍 1 : 𝓍 1 ( U 1 , A ) , 𝓍 2 ( U 2 , A ) , } = ( U 1 , A ) . Similarly   π 2 ( ( U 1 , A ) ( U 2 , A ) ) = ( U 2 , A ) . ( π 1 o ) ( y ˜ ) =   ( π 1 ( ( y ˜ ) ) = π 1 ( ( 𝓍 1 , 𝓍 2 ) ) = 𝓍 1 . Similarity,   ( π 2 o ) (   y ˜ ) =   ( π 2 ( ( y ˜ ) ) = π 2 ( ( 𝓍 1 , 𝓍 2 ) ) = 𝓍 2 . Thus, ( π 1 o ) (   y ˜ ) =   𝓍 1 , ( π 2 o ) (y) =   𝓍 2 . From the above ( π 1 o ) ( y ˜ )   ( U 1 , A ) ˜ π 1 ( ( G , A ) ) , ( π 2 o ) ( y ˜ )   ( U 2 , A ) ˜ π 2 ( ( G , A ) ) this implies that ( y ˜ ) ( π 1 o ) 1   ( U 1 , A ) and ( y ˜ ) ( π 2 o ) 1   ( U 2 , A ) implies that ( y ˜ ) [ ( π 1 o ) 1   ( U 1 , A ) ( π 2 o ) 1   ( U 2 , A )   ] . Therefore, ( π 1 o ) a n d   ( π 2 o ) are given to be soft continuous and hence ( π 1 o ) 1   ( U 1 , A ) and ( π 2 o ) 1   ( U 2 , A ) are soft s -open in ( 𝓎 1 , T 1 , T 2 , E ) this implies that [ ( π 1 o ) 1   ( U 1 , A ) ] [ ( π 2 o ) 1   ( U 2 , A ) ] is soft s -open in ( 𝓎 1 , T 1 , T 2 , E ) On taking ( π 1 o ) 1   ( U 1 , A ) = ( V 2 , A ) ,   ( π 1 o ) 1   ( U 2 , A ) = ( V 2 , A ) . . We have ( V 1 , A ) ( V 2 , A ) are soft s -open in ( 𝓎 1 , T 1 , T 2 , E ) by the above result,   y ˜ ( V 1 , A ) ( V 2 , A ) = ( V , A ) (say). Any,   𝑣 ˜ ( V , A ) implies that 𝑣 ˜ ( V 1 , A ) and 𝑣 ˜ ( V 2 , A ) implies that 𝑣 ˜ ( π 1 o ) 1   ( U 1 , A ) , , 𝑣 ˜ ( π 2 o ) 1   ( U 2 , A ) implies that ( π 1 o ) ( 𝑣 ˜ ) ( U 1 , A ) , ( π 2 o ) ( 𝑣 ˜ ) ( U 2 , A ) implies that ( π 1 o ) ( 𝑣 ) ( U 1 , A ) ˜ π 1 ( ( G , A ) ) and ( π 2 o ) ( 𝑣 ˜ ) ( U 2 , A ) ˜ π 2 ( ( G , A ) ) implies that 𝑣 ˜ ( π 1 o ) 1 [ π 1 ( ( G , A ) ) ] and 𝑣 ˜ ( π 2 o ) 1 [ π 2 ( ( G , A ) ) ] implies that 𝑣 ˜ f 1 ( G , A ) and 𝑣 ˜ f 1 ( G , A ) . Therefore, any 𝑣 ˜ ( V , A ) implies that 𝑣 ˜ f 1 ( G , A ) . This implies that ( V , A ) ˜ f 1 ( G , A ) . Thus we have shown that any y ˜ f 1 ( G , A ) implies that there exists a soft s -open set ( V , A ) ˜ ( 𝓎 1 , T 1 , T 2 , E ) such that y ˜ ( V , A ) ˜ f 1 ( G , A ) . This implies that y is a soft interior point of f 1 ( G , A ) and hence every point of f 1 ( G , A ) is a soft interior point, showing thereby f 1 ( G , A ) is soft s -open in ( 𝓎 1 , T 1 , T 2 , E ) .  □

6. 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 ( S 1   a n d   S 2 ) space if each coordinate space is soft ( S 1   a n d   S 2 ) 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 (first-countable, second-countable) if each coordinate space is (first-countable, second-countable).
Theorem 10.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set s   X 1 and X 2 respectively. ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product space, then the product space ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected if both ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) are soft S -connected.
Proof. 
Suppose ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be a product space of soft bitopological spaces ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) . If the soft product ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected then we have to prove that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) are soft S -connected. Define soft maps π 1   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) such that π 1 ( 𝓍 1 , 𝓍 2   ) = 𝓍 1 f o r   a l l ( 𝓍 1 , 𝓍 2   ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ,   π 2   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) such that π 2 ( 𝓍 1 , 𝓍 2   ) = 𝓍 2   f o r   a l l   ( 𝓍 1 , 𝓍 2   )   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) then π 1 and π 2 are called soft projection maps. Also π 1   a n d   π 1 Now π 1   : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) is soft continuous and ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected.
This implies that π 1 ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) is soft S -connected that is ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) is soft S-connected. Again π 2 ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) is continuous and ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected. This implies π 2 ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) = ( X 2   , F 1 ˜ , F 2 ˜ , E ) . For what we have done, it follows that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) is soft S -connected sets. Conversely, let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the product space of soft S -connected spaces ( X 1   , τ ˜ 1 , τ ˜ 2 , E )   a n d   ( X 2   , F 1 ˜ , F 2 ˜ , E ) . We have to prove that ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected. Pick any point ( 𝓍 1 , 𝓍 2 ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and consider the soft sets ( 𝓍 1 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( 𝓍 2 , A ) . Define the soft maps : ( 𝓍 1 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) by requiring that : ( ( 𝓍 1 , 𝓍 2 ) ) = 𝓍 2 for all 𝓍 2 ( X 2   , F 1 ˜ , F 2 ˜ , E ) and : ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( 𝓍 2 , A ) ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) by writing : ( ( 𝓍 1 , 𝓍 2 ) ) = 𝕩 ˜ 1 for all 𝓍 1 ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) . Now 1 and 1 are continuous soft maps so that   1 [ ( X 2   , F 1 ˜ , F 2 ˜ , E ) and 1 [ ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ] are soft S -connected sets that are ( 𝓍 1 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) and ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( 𝓍 2 , A ) are soft S -connected sets. Write X c r i s p   𝓍 1 = ( 𝕩 ˜ 1 , A ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) , X c r i s p   𝓍 2 = ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( 𝕩 ˜ 2 , A ) , C 𝓍 1 𝓍 2 = X c r i s p   𝓍 1 X c r i s p   𝓍 2 . Then X c r i s p   𝓍 1 and X c r i s p   𝓍 2 are soft S -connected sets. ( 𝓍 1 , 𝓍 2 ) X 2 X 2 . So that X c r i s p   𝓍 1 X   𝓍 2 φ A . being a finite soft union of soft S -connected sets having non-empty intersection. X c r i s p   𝓍 1 X c r i s p   𝓍 2 is soft S -connected that is, C 𝓍 1 𝓍 2 is soft S -connected. Consider the family of soft S -connected sets { C 𝓍 1 𝓍 2 : ( 𝓍 1 , 𝓍 2 ) X } that is { C 𝓍 1 𝓍 2 : ( 𝓍 1 , 𝓍 2 ) X c r i s p } this implies that ( 𝓍 1 , 𝓍 2 ) X [ X c r i s p   𝓍 1 X c r i s p   𝓍 2 ] = ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . Finally, ( 𝓍 1 , 𝓍 2 ) X [ X c r i s p   𝓍 1 X c r i s p 𝓍 2 ] = ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) . If we select fixed members of the above family, then their soft intersection is not soft null. Hence,   ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) is soft S -connected. □
Theorem 11.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set X 1 and X 2 respectively. ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product space, then the product space × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒   i f f each soft coordinate space { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) } is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 .
Proof. 
Suppose each coordinate space { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) } is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 and let ( ϰ , A   )   : Δ be an element of × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } . Then, ( ϰ , A   )   { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) } for each Δ . Since ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) is soft S 1 space, it follows that { ( ϰ , A   )   } is soft ( τ ˜ 1 , τ ˜ 2 ) ( F 1 ˜ , F 2 ˜ ) s   closed for each Δ . Now, each soft projection mapping π α being soft continuous, it follows that ( π )   1 { ( ϰ , A   )   } is soft ( τ ˜ 1 , τ ˜ 2 ) ( F 1 ˜ , F 2 ˜ ) s   closed is × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } for every Δ . Consequently, ( π )   1 { ( ϰ , A   )   } = ( ϰ , A   )   . So, every singleton soft subspace of × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } is soft ( τ ˜ 1 , τ ˜ 2 ) ( F 1 ˜ , F 2 ˜ ) s closed. Hence, × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 . Conversely, let × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 and let ( × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ }   ) β be an arbitrary soft coordinate space of × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } . Let ( { ( ϰ , A ) } ) β and ( { ( y ˜ , A ) } ) β be any two soft distinct points of ( × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ }   ) β . Choose ( ϰ , A   ) and ( y , A ) in × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } whose β - t h coordinate is ( { ( ϰ , A )   } ) β and ( { ( y , A )   } ) β respectively. Since ( { ( ϰ , A )   } ) β ( { ( y , A )   } ) β we have ( ϰ , A ) > ( y , A ) or ( ϰ , A )   < ( y , A ) or ( ϰ , A ) ( y , A ) or ( ϰ , A )   ( y , A ) But × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } being soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 , corresponding to the soft distinct points ( ϰ , A ) and ( y , A ) of { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } there exists soft s -open sets ( G , A ) and ( H , A ) in × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } such that ( ϰ , A ) ( G , A ) but ( y , A ) ( G , A ) and ( y , A )   ( H , A ) but ( ϰ , A ) ( H , A ) . So, there exists basic soft s -open sets ( I , A ) = × {   ( G , A ) α   : α Δ } and ( J , A ) = × {   ( H , A ) α   : α Δ } such that ( ϰ , A ) ( I , A ) ( G , A ) and ( y ˜ , A )   ( J , A ) ( H , A ) . Clearly, ( y , A )   ( I , A ) and ( ϰ , A ) ( J , A ) . Thus, ( G , A ) β is soft s -open set containing ( ϰ , A ) β but not ( y , A )   β ; and, ( H , A ) β is soft s -open set containing ( y , A )   β but not ( ϰ , A )   β . This shows that ( × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ }   ) β is soft S 1   𝒮 𝓅 𝒶 𝒸 𝑒 . □
Theorem 12.
Let ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) and ( X 2   , F 1 ˜ , F 2 ˜ , E ) be two S B T 𝒮 𝓅 𝒶 𝒸 𝑒 𝓈 such that they are soft S 2   𝓅 𝒶 𝒸 𝑒 𝓈 on the crisp set X 1 and X 2 respectively. ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) be the soft product space, then the product space × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } is soft S 2   𝒮 𝓅 𝒶 𝒸 𝑒   i f f each soft coordinate space { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) } is soft S 2   𝒮 𝓅 𝒶 𝒸 𝑒 .
Proof. 
Suppose each soft coordinate space × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) } is soft S 2   𝒮 𝓅 𝒶 𝒸 𝑒 and let ( ϰ , A ) : Δ and ( y , A ) : Δ be two points of × { ( ( X 1   , τ ˜ 1 , τ ˜ 2 , E ) ( X 2   , F 1 ˜ , F 2 ˜ , E ) ) : Δ } such that ( ϰ , A ) : Δ > ( y , A ) : Δ Then,