The concept of  neutrosophy is a new branch of science introduced by F.smarandache [1,2], and has many applications in different fields of science such as topology. As a generalization of the concept of topological spaces, Also A.A. Salama and et al. [3] defined neutrosophic crisp topological spaces and many kinds of neutrosophic crisp open and closed sets in 2014. A.A. Salama and F. Smarandache [4] defined neutrosophic topological spaces and its neutrosophic sets. in 2012.

supra topological space was introduced by A.S. Mashhour et al. [5] in 1983, as a generalization of the concept of topological space.

G.Jayaparthasarathy, et al. generalised this concept and introduced the concept of neutrosophic supra topological space [6] in 2019, by using the neutrosophic sets.

Furthermore, Mony reserchers extended some of the topololgy to bitopology as,  The concept of supra bi-topological spaces which was introduced by R. Gowri, A.K.R. Rajayal [8 ] in 2017. On the other hand, The concept of bi-topological spaces was introduced by Kelly [8] as an extension of topological spaces in 1963. 

Also, R.K.Al-Hamido [10] extended neutrosophic topological spaces to neutrosophic bi-topological spaces in 2019. This concept has been studied in [11].

 Also, The concept of neutrosophic crisp bi-topological spaces was introduced by R.K.Al-Hamido [ 12] as an extension of neutrosophic crisp topological spaces in 2018.

M. Parimala, et al. introduced the concept of neutrosophic minimal structure [14] in 2020, by using the neutrosophic sets.

In this paper, we use the neutrosophic crisp sets to introduce neutrosophic crisp minimal structure. Also, we introduce new neutrosophic crisp minimal open (closed) sets in this neutrosophic crisp minimal structure, and we study some basic properties of this new neutrosophic crispminimal open (closed) sets.

2. Preliminaries

In this part, we recall some basic definitions and properties which are useful in this paper. 

Definition 2.1. [3]

Let  be a fixed set. A neutrosophic crisp set (NCS) U is an object having the form ; are subsets of X.

Definition 2.2. [3] 

A neutrosophic crisp topology (NCT) on a non-empty set  is a family T of neutrosophic crisp subsets in  satisfying the following axioms:

1.     X_N and  Æ_N belong to T.

2.     T is closed under finite intersection. 

3.     T is closed under arbitrary union.

The pair (X,T) is said to be a neutrosophic crisp topological space (NCTS) in X. Moreover,  The elements in T are said to be neutrosophic crispopen sets (NCOS). A neutrosophic crisp set F is closed (NCCS) if and only if its complement is an open neutrosophic crisp set.

Definition 2.3. [13]  

A neutrosophic crisp supra topology (NCT) on a non-empty set  is a family T of neutrosophic crisp supra subsets in  satisfying the following axioms:

1.     X_N and  Æ_N belong to T.

2.     T is closed under arbitrary union.

The pair(X,T) is said to be a neutrosophic crisp supra topological space (NCTS) in X. Moreover,  The elements in T are said to be neutrosophiccrisp supra open sets (NCSOS), A neutrosophic crisp set F is neutrosophic crisp supra closed (NCSCS) if and only if its complement is an neutrosophic crisp supra open.

Definition 2.4. [3]  

    _" Let X be a non-empty fixed set. A neutrosophic_. crisp set (NCS_. for short) B is an object having the form B=<B₁,B₂,B₃> where B₁,B₂ and B₃are subsets of X."

3. Neutrosophic crisp minimal structure

In this section, we introduce the neutrosophic crisp minimal structure. Moreover, we introduce  new types of neutrosophic crisp minimal open (closed) sets in this space, and study their properties, we examine the relationship between them in details.

Definition 3.1.

A neutrosophic crisp minimal structure (NCMS) on a non-empty set  is a family M of neutrosophic crisp subsets in X satisfying the following axioms:

1.     Æ_N belong to M.

2.     X_N belong to M.

The pair(X,M) is said to be a neutrosophic crisp minimal structure (NCMS) in X. Moreover,  The elements in M are said to be neutrosophic crispminimal open sets (NCMOS), A neutrosophic crisp set F is neutrosophic crisp minimal closed (NCMCS) if and only if its complement is an neutrosophic crisp minimal open.

Example 3.2.                

Let X={a,b}, M={Æ_N, X_N, A, B, E}, A={<{a},Æ,Æ>}, B={<{b},Æ,Æ>}, E={<{a},{b},Æ>}neutrosophic sets over X. Then   is neutrosophic crisp minimal structure.

Remark 3.3.

- the family of all neutrosophic crisp minimal open sets is denoted  by ( NCMOS(X) ).

- the family of all neutrosophic crisp minimal closed sets is denoted  by ( NCMCS(X) ).

Example  3.4.  

In Example 3.2.

the neutrosophic minimal crisp open sets is denoted  by are : 

NCMOS(X) ={Æ_N, X_N, A, B, E}.

Remark 3.5.

 Let  be  NCMS then 

the union of two neutrosophic crisp minimal open sets is not necessary neutrosophic crisp minimal open set.

as the following example:

Example 3.6.

Let X={a,b}, M={Æ_N, X_N, A, B, E}, A={<{a},Æ,Æ>}, B={<{b},Æ,{a,b}>}, E={<{a},{b},Æ>}neutrosophic sets over X. Then   is neutrosophic crisp minimal structure.

  are two neutrosophic crisp minimal open sets but ,  is not neutrosophic crisp minimal open set.

Remark 3.7.

Let  be  NCMS then the intersection of two neutrosophic minimal open sets is not necessary neutrosophic minimal open set.

as the following example:

Example 3.8.

In Example 3.6,  are two neutrosophic crisp minimal open sets but ,  is not neutrosophic crisp minimal open set.

Remark 3.9.

 Let  be  NCMS then 

the union of two neutrosophic crisp minimal closed sets is not necessary neutrosophic crisp minimal closed set.

as the following example:

Example 3.10.

Let X={a,b}, M={Æ_N, X_N, A, B, E}, A={<{a},Æ,Æ>}, B={<{b},Æ,{a}>}, E={<{a},{b},Æ>}neutrosophic crisp sets over X. Then   is neutrosophic crisp minimal structure.

  are two neutrosophic crisp minimal closed sets but ,  is not neutrosophic crisp minimal closed set.

Remark 3.11.

Let  be  NCMS then the intersection of two neutrosophic minimal closed sets is not necessary neutrosophic minimal closed set.

as the following example:

Example 3.12.

In Example 3.10

  are two neutrosophic crisp minimal closed sets but , is not neutrosophic crisp minimal closed set.

Remark 3.13.

Every NCS is NCMS but the converse is not true.

Example 3.14. 

In Example 3.6.

  is NCMS is. But   is not NCS.

Remark 3.15.

Every neutrosophic supra topological space is NCMS but the converse is not true.

Example 3.16

In Example 3.6.

  is NCMS is. But   is not a neutrosophic supra topological space.

4. The interior and the closure via neutrosophic crisp Minimal open (closed) sets 

In this section we define the closure and interior Neutrosophic crisp minimal set based on these new varieties of Neutrosophic  crisp minimal open and closed sets. Also we introduce the basic properties of closure and the interior. 

Definition 4.1.

Let  be  NCMS, and A a neutrosophic  crisp minimal set then :

The union of any neutrosophic crisp minimal open sets, contained in A is called neutrosophic crisp minimal interior of A ( NCMint(A) ).

NCMint(A) =È{B  ;BÍA; BÎNMOS(X)}.

Theorem 4.2.

Let  be  NCMS then, A, B are neutrosophic crisp  sets then :

1. NCMint(A) Í A.
2. NCMint(A) is not necessary neutrosophic minimal open set.
3. AÍB  Þ  NCMint(A)  Í  NCMint(B).

Proof :

1. Follow from the defintion of NCMint(A) as a union of any neutrosophic crisp minimal open sets ,contained  in A.
2. Follow from remark 3.8.
3. The Proof is obvious.

Definition 4.3.

Let  be  NCMS then, A is neutrosophic crisp  sets then :

The intersection of any neutrosophic crisp minimal open sets ,containing A is called neutrosophic crisp minimal closure of A ( (NCMcl(A) ).

NCMcl(A)=Ç{B  ;BÊA; BÎNCMCS(X)}

Theorem 4.4.

Let  be  NCMS then, A is neutrosophic crisp  sets then :

1. AÍ NCMcl(A).
2. NCMcl(A) is not necessary neutrosophic crisp minimal closed set.

Proof :

1. Follow from the defintion of NCMcl(A) as a intersection of any neutrosophic crisp minimal closed set containing A.
2. Follow from remark 3.10.

5. Neutrosophic crisp minimal structure subspace:

In this part, we intruduced the neutrosophic crisp minimal structure subspace of neutrosophic crisp minimal structure, and studied some of its properties.

Theorem 5.1: If  is a neutrosophic crisp minimal structure  neutrosophic crisp subset over X,

 , then  is  neutrosophic crisp minimal structure.

proof: 

since Æ_N , then Æ_N  , X_N , then  therefore  is a neutrosophic crisp minimal structure.

Definition5.2: If  is a neutrosophic crisp minimal strucre  neutrosophic crisp subset over X,

 , then  is called a neutrosophic crisp minimal structure subspace of .

Example 5.3:

 Let X={a,b}, M={Æ_N, X_N, A, B, E}, A={<{a},Æ,Æ>}, B={<{b},Æ,Æ>}, E={<{a},{b},Æ>}neutrosophic crisp sets over X. Then   is neutrosophic crisp minimal structure. Let G={<{a},{b},X>},

 .

  is a neutrosophic crisp minimal structure subspace of .

Definition5.4,

Let  is a neutrosophic crisp minimal strucre, then

- Arbitrary union of neutrosophic minimal open sets in  is neutrosophic minimal open. (Union Property)

- Finite intersection of neutrosophic minimal open sets in is neutrosophic minimal open (intersection   Property)

Theorem 5.6.

 let  is a neutrosophic crisp minimal structure, then

1-If the neutrosophic minimal structure space  has the union property, then

the subspace  also has union property.

2-If the neutrosophic minimal structure space  has the intersection property, then the subspace   also has union property.

Proof. Suppose the family of open set in neutrosophic minimal subspace   

then there exist a family of open sets  in neutrosophic minimal structure space

  such that  then ; since the neutrosophic minimal structure space  has the union property then   therefore the neutrosophic minimal structure space  also has the union propert.

-The proof of (2) is similarly to (1).

4. Conclusion  

In this paper, we have defined a new topological space by using neutrosophic crisp sets. This new space called neutrosophic crisp minimal structure space.Then we have introduced new neutrosophic crisp open(closed) sets in neutrosophic crisp minimal structure space. Also we studied some of their basic properties and their relationship with each other. This paper is just a beginning of a new structure and we have studied a few ideas only, it will be necessary to carry out more theoretical research to establish a general framework for the practical application. In the future, using these notions, various classes of mappings on neutrosophic crisp minimal structure space, separation axioms on the neutrosophic crisp minimal structure space, Neutrosophic crisp minimal α-open sets , Neutrosophic crisp minimal β-open sets , Neutrosophic crisp minimal pre-open sets , Neutrosophic crisp minimal semi-open sets and many researchers can be studied.

References

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[2]   F. Smarandache, “Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability, and Statistics” University of New Mexico, Gallup, NM 87301, USA (2002).

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