Zadeh [1] discovered the fuzzy set (FS) and proposed the concept of membership value in 1965 and explained the concept of uncertainty using a FS. Atanassov introduced [2] the intuitionistic fuzzy (IF) set in 1989, which generalised FS concepts and suggested the degree of non-membership (DNM) as an independent component. Following the introduction of FSs, various types of research on extensions of FS notions were done. Considering the generalisation of FSs, several researchers applied FS theory to a variety of fields in science & technology. Chang [3] established the development of fuzzy topological space (FTS). Intuitionistic FTS was defined by Coker [4]. Kelly proposed [5] the concept of bitopological spaces in 1963 and also, Mwchahary et al. [6] investigated neutrosophic bitopological spaces. 

Smarandache [7-10] proposed the (NS) neutrosophic set and after that many researchers applied it in science & technology. Broumi et al. [11, 12] investigated relations on interval-valued neutrosophic soft sets, as well as Neutrosophic parametrized soft set theory and decision-making. Sumathi et al. [13, 14] discussed the fuzzy neutrosophic groups and later studied the topological group structure of the NS. Many researchers [15-18] have investigated topological spaces based on NSs. Basumatary et al. [19] studied the neutrosophic bitopological group and also, Basumatary et al. [20] discussed the properties of plithogenic neutrosophic hypersoft. Blizard [21] defined the idea of multisets. Kazanc [22] studied how multi-fuzzy soft sets can be used in hyper modules. Many researchers [23, 24] discussed multiset topology, and also many authors [25-27] investigated fuzzy multisets. Bakbak [28] studied neutrosophic multigroup and its applications. Basumatary et al. [29] investigated the neutrosophic multi topological group and its properties and also Basumatary [30] investigated fuzzy closure regarding to the fuzzy boundary. 

In this paper, the definitions of the neutrosophic multi interior (NMI), neutrosophic multi closure (NMC), the neutrosophic multi exterior (NME), and neutrosophic multi boundary (NMB) and their propositions will be studied with examples.

2. Preliminaires   

Definition 2.1: [10]

Let  be the NS on  is expressed as , where ; , and  represents the degree of membership value,    the degree of indeterminacy, and  is the degree of non- membership value respectively.

Definition 2.2: [9]

A neutrosophic multiset (NMS) is a type of NS in which one or more elements with the same or different neutrosophic components are repeated several times.

Definition 2.3: [29]

The Empty NMS can be expressed as .

Definition 2.4: [29]

The Whole NMS can be expressed as .

Definition 2.5: [29]

Let  be a NMS on   and

 ,

then    is the the complement of .

Definition 2.6: [29]

Let  and 

  be neutrosophic multisets on . Then

(i)      

(ii)     .

Definition 2.7: [10]

Let  and  be a family  of neutrosophic multi subsets of  if it satisfies the following:

(i)      

(ii)      for 

(iii)    .

Then  is neutrosophic multi topological space (NMTS).

3. Results

3.1. Definition

Let  be a NMTS and let  be NMS, then the NMI of  is the union of all neutrosophic multi open sets (NMOS) of  contained in  and is defined as 

 .

3.1. Proposition

Let  be a NMTS, then

(i)      , ;

(ii)     ;

(iii)   .

Proof: 

(i)      Since  and  are open sets, we have

 , .

(ii)     Let  ∈ .

  is an interior point of .

  is a neighbourhood of .

 .

Hence, .

(iii)    Let  ∈ .

Then  is an interior point of , so  is a neighbourhood of .

Since , so  is also a neighbourhood of .

    ∈ .

Thus,  ∈  ∈ .

Hence, .

3.1. Example

Let  and  

Then      is a NMTS.

Let         and

 .

Since     .

Then      and 

Hence,   .

3.2. Proposition

Let  be a NMTS, then

(a)     ;

(b)     .

Proof:

(a)      Since  and , then we have from (iii),

  and  

                  … (1)

Again, let .

Then  and .

Hence,  is an interior point of each of the sets  and .

It gives that  and  are neighbourhoods of , so their intersection  is also a neighbourhood of .

Therefore, .

Thus,     

             … (2)

From (1) and (2), we have

 .

(b)     From Proposition 3.1 (iii), we have 

  and 

 . 

Hence, .

3.2. Definition

Let  be a NMTS and let  be NMS, then the NMC of  is the intersection of all neutrosophic multi closed sets (NMCoS) of  containing in  and is defined as 

 .

3.3. Proposition

Let  be a NMTS, then

(a)     , ;

(b)     ;

(c)      is NMCoS iff ;

(d)     .

Proof: 

(a)      Since  and  are closed sets, we have

 , .

(b)     Since  is the smallest NMCoS containing , so .

(c)      Since  is NMCoS, then  itself is the smallest NMCoS containing  and so, .

Conversely, let . Then   is NMCoS and hence  is also NMCoS.

(d)     From (b), we have . Since , so we have . 

But  is a NMCoS. So,  is a NMCoS containing . Since  is the smallest NMCoS containing , so we have .

Hence, .

3.4. Proposition

Let  be a NMTS, then

(a)     ;

(b)     .

Proof of (a):

Since  and . 

From Proposition 3.3 (d), we have  and 

Hence,  … (1)

Since ,  are NMCoSs, so  is also NMCoS.

Also,      and 

 .

Thus,  is a NMCoS containing . 

Since  is the smallest NMCoS containing .

Therefore,  … (2)

From (1) and (2), we have 

 .

3.2. Example

Let  and  and

 .

Then      is a NMTS.

Let         and

 .

Then     

So,         

  and 

Then     

 .

Hence,   .

Proof of (b):

Since  and 

 .

Hence, .

3.3. Example

Let  and  and

 .

Then      is a NMTS.

Let         and

 .

Then     

So,         

  and 

Therefore, .

 .

Hence,   .

3.3. Definition

Let  be a NMTS and let  be NMS, then the NMB of  is defined as 

 .

i.e., .

3.5. Proposition

Let  be a NMTS and  be NMCoS then .

Proof: Since  is NMCoS.

 .

By definition,    

       .

But the converse may not be true.

For this we cite an example-

3.4. Example

Let  and  and

 .

Then      is a NMTS.

Let        ,

Then      and  

So,        

Thus,    .

Hence,   .

But  is not NMCoS.

3.6. Proposition

Let  be a NMTS and , then

(a)     .

(b)     ;

(c)     ;

(d)     ;

(e)     ;

Proof:

(a)      We know that       … (1)

Replacing  by  in (1), we get

                      [from (1)]

Therefore,                    … (2)

Since                     … (3)

and                      … (4)

Using (4) in (1), we have

 .

Then      and 

  and 

Therefore, 

   [from (3)]                   … (5)

From (2) and (5), we have 

        … (6)

Now,     .

                          … (7)

Using (7) in (6), we have

  .

(b)     From (5), we have

Now,

    [as ]

 .

Hence,   .

(c)      From (5), we have

Now,

 .

Hence,   .

(d)     Since  

We have

                         [as ]

Hence, .

(e)      Since 

Then    

But        

So, 

              [as ]

Hence, .

3.7. Proposition

Let  be a NMTS and , then

(a)    ;

(b)   .

Proof:

(a)        

Hence,   .

(b)        

Hence, .

3.4. Definition

Let  be a NMTS and let  be NMS, then the NME of  is defined as 

 .

3.8. Proposition

Let  be a NMTS and , then

(a)     , ;

(b)     ;

(c)     .

Proof:

(a)      Since         

and            .

(b)     Since         .

Hence,   .

(c)      Since 

             [as ]

Hence,   .

3.5. Example

Let  and  

Then      is a NMTS.

Let         and

 .

Then      and 

 .

Hence,   .

3.9. Proposition

Let  be a NMTS and , then

(i)      ;

(ii)     ;

(iii)   .

Proof of (i)

Since     

 .

Hence,   .

3.6. Example

Let  and 

Then  is a NMTS.

Let         and

 .

Then      and 

 .

Since     .

Then      and 

Hence,   .

Proof of (ii)

Since     

 .

Now,     , we have

       … (1)

But       

 .

From (1), we have .

Proof of (iii)

Since     

 .

Hence,   .

5. Conclusions  

In this study, topological space is studied based on NMS. In this work, the definitions of NMI, NMC, NME, and NMB were observed. With examples, the various NMTS propositions are examined.  Hope this study will help in the further development of NMTS. 

Funding: This research received no external funding. 

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

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