Continuity and Compactness on Neutrosophic Soft Bitopological Spaces

Ahmed B. AL-Nafee^1*, Jamal K. Obeed², Huda E. Khalid³       

^1*Ministry of Education, College of Open Education, Dept. of Mathematics, Babylon, Iraq

" ahm_math_88@yahoo.com"

² University of Missan, College Of Basic Education, Dept. of Mathematics, Iraq

jamal.k@uokerbala.edu.iq

³Administrative Assistant for the President of Telafer University, Telafer, Iraq

dr.huda-ismael@uotelafer.edu.iq

"" *Correspondence: Ahm_math_88@yahoo.com "

Abstract:

In this manuscript, continuity, compactness and  concepts in neutrosophic soft bitopological space have been defined using star bineutrosophic soft open notion. Theorems and properties concerning to these two notions have been investigated here.

Keyword: Neutrosophic Soft Set, Fuzzy Set,  Spaces, Star Bi  Soft Open, -Open, Continuity, Compactness, Soft Set, Neutrosophic Soft Bitopological Spaces  

1.^. Introduction

The notion of the soft set as a general mathematical tool for coping with objects involving vagueness and uncertainty has been introduced by Molodtsov [2]. F. Smarandache [3] introduced the concept of neutrosophic sets which is a generalization_. of _.Zadeh’s_-fuzzy set_, and Atanassov's_-intuitionistic_. fuzzy set, as a new mathematical tool for dealing with problems involving indeterminacy, inconsistent knowledge, incompleteness.

In 2013, the notion of neutrosophic soft set was introduced by combining the concept of soft set and neutrosophic set [4], and later this concept and its operations have been redefined by [5,6,7]. The concept of neutrosophic soft topological spaces was presented by [8,9]. In 2021, Al-Nafee, et al. [1] extended this the concept to the concept of neutrosophic bitopological space which is defined over two neutrosophic' soft topological spaces and they studied the basic topological concepts of these . The concepts of neutrosophic soft continuous mapping and compactness with their properties and some theorems were investigated by many authors (see [8,10,11]). For more details on these concepts see [12-20].

In this manuscript, the authors introduced the concepts of continuity, compactness and  in neutrosophic soft bitopological spaces through presenting the concepts of N₃(bi)^*-continuous mapping, NSbi-open mapping, NSbi-closed mapping, N₃(bi)^*-compact and N₃(bi)^*-  based on the definition of N₃(bi)^*-open, some of related theorems and properties also have investigated.

2.^. Basic Concepts 

In this section, some fundamental and relevant definitions are recalled as background and to give the reader a deep insight on the basic tools of the upcoming section. 

Definition 2.1 [20]

  Consider P( ) the set of all subsets-of- . A soft. set  on over  is a set valued function from E to P( ). we can rewrite it as a set of ordered pairs,  = {(e, (e)), e }, where E is a set of parameters. 

Definition 2.2 [3]

  The 'set S over G is defined_.as follows: "

                       "   S =  "

where the functions_, I,B,F : G ,  and " - 0 +3.

‘’From philosophical point of view the neutrosophic set takes the value from real standard or non-standard subsets of , "."But in real life application in scientific and engineering problems it is difficult to use a neutrosophic set with value from real standard or non-standard subset of , "."Hence we consider the neutrosophic set which takes the value from the subset of [0, 1]’’".

Definition 2.3 [5]

  Consider  an initial universe set and E a set of parameters. P( ) denotes the set of all the neutrosophic subsets from . A soft set (in abbrev, NSs)  on theinitial universe set  is a set defined by a set valued function H representing a mapping from E to P( ), where H is called approximate function of the neutrosophic soft ^.set . that’s mean,  is a parameterized family for some elements of P( ) which implies to it can be rewritten as a set of ordered pairs, { .    [0,1],  respectively known as Truth-Membership, Indeterminacy-Membership, and Falsity-Membership function of . It is well known that the supremum of each  equal 1 , so the inequality,  is apparent.

Note:

1.              From the def. 2.3, and up to the rest of this paper, the notion N₃ (G) will be represent to the set of all neutrosophic sets over .

2.              For the purpose of abbreviation, the authors will denote to the (neutrosophic soft set) by NSs. Also the (Neutrosophic Soft Point) is denoted by NSp.

Definition 2.4 [4,9]

 Consider:  N₃ (G), where   

 {  

 = {  Then:   

o               = { . 

o               = { .

o                  { .

o                = { .

o                = { .

Definition 2.5 [1]

 Let {  N₃ (G). Then, the complement of  ( ) is defined as: 

.

: Definition 2.6 [12]

  Consider   an initial universe  , and   a  of parameters. Then the NSs  is called NSp, for each   , and is defined as :

Definition 2.7 [9]

 Consider   a . A soft topology on  is a family  N₃(G), which satisfies the following conditions:

1)              , .

2)              If    then .

3)              If , for every  then .

(G,E, ) is called a soft topology space (in abbrev, Nst-space). Each member of  is named as a soft open . A NSs is named as a soft closed set if and only if its complement is soft open set.

The soft interior of N₃ (G)"(( )⁰) is defended as: ,

( )⁰   {  :  is a soft open , ⊑ }.

The soft closure of N₃ (G) (  ) is defended as: ,

"     {  :  is a soft closed , ⊑ .

Definition 2.8 [1]

  Let (G,E, ) and (G,E, ) be two Nst-spaces defined on G. Then (G,E, _, ) or (G, for abbreviation purposes) is called a  soft  space or (in abbrev, BIN-space).

From this definition up to the rest of the paper and for the abbreviation purpose, we give an attention to the readers that  will sometimes be represented as G.

Definition 2.9 [1]

  A subset N₃ (G) of BIN-space  is called a star bi  soft open (abbreviation, N₃(bi)^*-open) over  iff and their complement is a star bi  soft closed (in abbrev, N₃(bi)^*-closed). The set of all N₃(bi)^*-open (N₃(bi)^*-closed) over  is denoted by  ( ), respectively.

Definition 2.10 [1]

  Let  be an BIN-space and N₃(G). Then,

·                (bi)^*-neutrosophic.soft interior_. of  ( ) is defined as: 

   {  :  is an N₃(bi)^*-open set, ⊑ }.

·                (bi)^*-neutrosophic-soft closure of    ( ) is defined as:

   {  :  is an N₃(bi)^*-closed set, ⊑ .

Remark 2.11

  In ref. [1] , the theorem 4.9. and theorem 4.12., the equalities (   ⊓ ), =  are in general not true.

  i.e. ( ⊓ ), ,  the following example has been originated by the authors to demonstrate this remark:

Example 2.12 

  Let G =  and E = . And let  N₃( ) such that,

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1)} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 0,0, 1 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{(1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{(0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { is an Nst-space on G,  = { } is an Nst-space on G.

Consequently (   ⊓ ). Also, .

For more details and background on these concepts (return to the ref. [1]).

3. on soft  Spaces

    The authors have dedicated this portion of the manuscript to introducing the concept of N₃(bi)^*-continuous mapping, NSbi-open mapping, NSbi-closed mapping in  soft  spaces,  they also present a deep investigation into the related theorems and properties.

Definition  3.1

  Consider  , and be two BIN-spaces. A mapping :  is supposed to be -continuous at  iff every -open set over containing , there exists -open set  over  containing  such that  If  is an -continuous for all , then  is called -continuous over .

Theorem 3.2

  Consider  be an BIN-spaces and be a mapping.  So the upcoming conditions are identical: 

(1)             is an -continuous.

(2)            For each -open set  over ,  is an -open set over G. 

(3)            For each -closed set  over ,  is an -closed set over G.

(4)            For each N₃ (G), .

(5)            For each N₃ ( ), .

(6)            For each N₃ ( ), .

Proof: (1) → (2) 

  Let  be an -open set over and   ∈  be an arbitrary  NSp. Then ∈ .

Since  is an -continuous mapping, there exists -open set  over  containing  such that 

This implies that ∈ ,  is an -open set over .

(2) → (1) 

  Consider  be a  NSp  be an -open set over containing .

So  is an -open set over and ( ))   .

(2) → (3) (obvious).

(3) → (4) 

   Let N₃ (G). Since is an -closed set over  

\ is an -closed set over 

Now:

  , .

This implies that,     , .

(4) → (5)

   Let N₃( ) and = .

From (4), we have . 

Then       .

(5) → (6)

 Let N₃ ( ). Substituting  for condition in (5).

Then ⊏ .

It is clear that . Then we have,

⊏ .

 (6) → (2)

   Let  be an -open set over .

Since ⊏ ,

then    is obtained.

This implies that is an -open set over G.

Definition 3.3

  Consider  be an BIN-spaces and be a mapping. Then,

1)              A mapping  is called an NSbi-open if the image  of each -open set  over  is an -open set over . 

2)              A mapping  is called an NSbi-closed if the image  of each -closed set  over  is an - closed set over .

Theorem 3.4

  Let  and  be two BIN-spaces,  be a mapping. Then,  is an NSbi-open mapping iff for each N₃ (G),  is satisfied.

Proof

  Let  be an NSbi-open mapping and N₃ (G).

Then is an -open set and ⊏ .

Since is an NSbi-open mapping, is an -open set over and

⊏ . Thus  ⊏  is obtained.

Conversely

  Let be any -open set over G. Then = . 

From the condition of theorem, we have . 

Then . 

This implies that .That is is an NSbi-open mapping. 

Theorem 3.5

  Consider  and  be two BIN-spaces,  be a mapping. Then,  is an NSbi-closed mapping iff for each N₃ (G), ⊏  is satisfied.

Proof

  Let  be an - closed mapping and N₃ (G).

Since  is an NSbi-closed mapping,  is an -closed set over and

⊏ . Thus ⊏  is obtained.

 Conversely

  Let  be any -closed set over G.

From the condition of the theorem ⊏     . 

This means that  . That is is an NSbi-closed mapping. 

Example 3.6

  Let G = , =  and E = .

And let  N₃( ) such that:

= {(e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >})}.

= {(e, {< ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >})}.

= {(e, {< ^{(1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >})}.

= { } is an Nst-space on G.

= { is an Nst-space on G.

Then,

(G,E, _, ) is an BIN-space,

 { }.

And let  N₃( ) such that:

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >}) }.

= { } is an Nst-space on .

= { is an Nst-space on .

Then  is an BIN-space and  { }.

Now, if is a mapping from (G,E, _, ) , defined as follows:

  ( ) = , ( ) = , ( ) = , .

Then it is easy to prove that,  

⁻¹( ) is an N₃(bi)^*-open set  G, for all N₃(bi)^*-open set  over .

( ) is an N₃ (bi)^*-open set , for all N₃ (bi)^*-open set  over G.

Therefore

 is an N₃(bi)^*-continuous mapping from (G,E, _, ) 

 is an NSbi-open mapping from (G,E, _, ) .

Example 3.7

  Let G = ,  =  and E = .

 And let  N₃( ) such that:

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >}) }.

= { is an Nst-space on G.

= { } is an Nst-space on G.

Then,

(G,E, _, ) is an BIN-space,

 { }.

And let  N₃( ) such that:

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { is an Nst-space on .

= { } is an Nst-space on .

Then  is an BIN-space and  { }

Now, if  is a mapping from (G,E, _, ) , defined as follows:

  ( ) = , ( ) = ( ) = , . 

Note that:

=  is an N₃ (bi)^*-open set , 

=  is an N₃ (bi)^*-open set , 

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >})} is not an N₃ (bi)^*-open set .

And

=  is an N₃ (bi)^*-open set , 

=  is an N₃ (bi)^*-open set , 

=  is an N₃ (bi)^*-open set , 

Therefore,  is an NSbi-open mapping from (G,E, _, ) , but not N₃(bi)^*-continuous. 

Theorem 3.8

  Let ,  and be an BIN-spaces. If  and are an N₃(bi)^*-continuous mappings, then  is an N₃(bi)^*-continuous mapping.

Proof

  Let be any -open set over .

Since  and

 is an N₃(bi)^*-continuous mapping. 

Then  is an -open set over . 

On the other hand, since  is an N₃(bi)^*-continuous mapping.

 is an -open set over . 

That is,  is an -open set over  and 

 is an N₃(bi)^*-continuous mapping.

Theorem 3.9

  Let  be an BIN-spaces, be a bijective map-ping. Then the for next terms are identical:

1.               is an -homeomorphism,

2.                 is an -continuous and NSbi-closed mapping,

3.                 is an -continuous and NSbi-open mapping.

4. Compactness on soft  Spaces

  This part of the manuscript has been devoted to introduce the notion of N₃(bi)^*-compact, N₃(bi)^*-  in neutrosophic soft  spaces, we also investigatedtheir related theorems and properties.

Definition 4.1

  A family NS  of N₃(bi)^*-open subsets of  BIN-space  is called an N₃(bi)^*-open cover of N₃(G) iff holds. If , then NS  is supposed to be an N₃(bi)^*-open cover of . If NS is a finite, then NS is called a finite N₃(bi)^*-open cover of .

Definition 4.2

  A finite subfamily of an N₃(bi)^*-open cover of  is called a finite N₃(bi)^*-subcover of , if it is also an N₃(bi)^*-open cover of .

Definition 4.3

  A BIN-space  is supposed to be an N₃(bi)^*-compact iff every an N₃(bi)^*-open cover of  has a finite N₃(bi)^*-subcover.

Definition 4.4

  A subset of an BIN-space  is called an N₃(bi)^*-compact provided for every family  of N₃(bi)^*-open subsets of such that .

Theorem 4.5

  If is an -continuous mapping from an N₃(bi)^*-compact space  onto an BIN-space . Then  is an N₃(bi)^*-compact.

Theorem 4.6

  An BIN-space is an N₃(bi)^*-compact iff given any family  of N₃(bi)^*-closed subsets of  such that the _.intersection of any finite_. number of the  is _.nonempty.

Theorem 4.7

  Every N₃(bi)^*-closed subset of N₃(bi)^*-compact space is an N₃(bi)^*-compact.

Note 

" The proofs of the theorems (4.5, 4.6, 4.7) are similar to the corresponding theorems in the  soft compact topological spaces"(For more details the  reader can return to the ref. [11]).

Definition 4.8

  An BIN-space (G,E, _, ) is called an N₃(bi)^*-  if and only if for each pair of distinct points ,  of (G,E, _, ), there exists two N₃(bi)^*-open sets ,  Such that Î , ,   . ( See Ex. 4.11, (G,E, _, ) is an N₃(bi)^*-  ).

Theorem 4.9 [11]

  Every  soft compact subset of a  soft  topological space is a  soft closed.

Remark 4.10 

  The above theorem in  soft bitopological spaces (BIN-spaces) is not true, the authors have originated the upcoming example to demonstrate this claim.

Example 4.11

  Consider G =  and E = .

And let  N₃( ) such that:

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{(0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 0, 0, 1 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 1, 1, 0 )} >, < ^{( 0, 0, 1 )} >}) }.

= { (e, {< ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { (e, {< ^{(1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) }.

= { is an Nst-space on G.

= { } is an Nst-space on G.

Then,

 { },

(G,E, _, ) is an BIN-space.

Note that:

(G,E, _, ) is an N₃(bi)^*-  is an N₃(bi)^*-compact, also it is an N₃(bi)^*-compact and so any subset of (G,E, _, ). But = { (e, {< ^{(1, 1, 0 )} >, < ^{( 1, 1, 0 )} >‚ < ^{( 0, 0, 1 )} >, < ^{( 1, 1, 0 )} >}) } is not -closed set over .

5.Conclusion

  Topological concepts are used in building important mathematical concepts in different fields as well as their applications in other sciences. The motive of this research is to expand the topological concepts based on the NSs. In this manuscript, the authors introduced the concept of continuity, compactness and  in  soft  spaces by introducing the concept of N₃(bi)^*-continuous mapping, NSbi-open mapping, NSbi-closed mapping, N₃(bi)^*-compact and N₃(bi)^*-  based on the definition of N₃(bi)^*-open, we also investigated the related theorems and properties of these concepts. We hope that the results of this study will be useful for researchers to present additional new studies on the neutrosophic soft sets.

Acknowledgement: This research is supported by the Neutrosophic Science International Association (NSIA) in both of its headquarter in New Mexico University and its Iraqi branch at University of Telafer, for more details about (NSIA) see the URL http://neutrosophicassociation.org/ .

Reference 

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