1. Introduction

Neutrosophy has laid the foundation for a whole family of new mathematical theories generalizing both their classical and fuzzy counterparts, such as a neutrosophic set theory. The fuzzy set was introduced by Zadeh [10] in 1965, where each element had a degree of membership. The intuitionstic fuzzy set (Ifs for short) on a universe X was introduced by K. Atanassov [1, 2, 3] in 1983 as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. After the introduction of the neutrosophic set concept [7, 8, 9]. In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental definitions of generalized neutrosophic set operations, we obtain several properties, and discussed the relationship between generalized neutrosophic sets and others. Finally, we extend the concepts of neutrosophic topological space [9].

2. Terminologies

We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in [7, 8], Atanassov in[1, 2, 3] and Salama [9]. Smarandache introduced the neutrosophic components T, I, F which represent the membership, indeterminacy, and non-membership values respectively, where is nonstandard unit interval.

Definition.[7, 8]

Let T, I,F be real standard or nonstandard subsets of

, with

Sup_T=t_sup, inf_T=t_inf

Sup_I=i_sup, inf_I=i_inf

Sup_F=f_sup, inf_F=f_inf

n-sup=t_sup+i_sup+f_sup

n-inf=t_inf+i_inf+f_inf,

T, I, F are called neutrosophic components

Definition [9]

Let X be a non-empty fixed set. neutrosophic set ( for short) is an object having the form Where and which represent the degree of member ship function (namely), the degree of indeterminacy (namely ), and the degree of non-member ship (namely ) respectively of each element to the set .

Definition [9].

The NSS 0_N and 1_N in as follows:

0_N may be defined as:

1N may be defined as:

3. Generalized Neutrosophic Sets

We shall now consider some possible definitions for basic concepts of the generalized neutrosophic set.

Definition

Let X be a non-empty fixed set. generalized neutrosophic set (G for short) is an object having the form Where and which represent the degree of member ship function (namely), the degree of indeterminacy (namely ), and the degree of non-member ship (namely ) respectively of each element to the set where the functions satisfy the condition .

Remark

A generalized neutrosophic can be identified to an ordered triple in on.X, where the triple functions satisfy the condition

Remark

For the sake of simplicity, we shall use the symbol for the G

Example

Every GIFS a non-empty set is obviously on GNS having the form

Definition

Let a GNSS on , then the complement of the set , for short maybe defined as three kinds of complements

One can define several relations and operations between GNSS as follows:

Definition

Let be a non-empty set, and GNSS and in the form , , then we may consider two possible definitions for subsets

may be defined as

and

and

Proposition

For any generalized neutrosophic set the following are holds

Definition

Let be a non-empty set, and, are GNSS Then

maybe defined as:

may be defined as:

Example.3.2. Let and given by:

Then the family is an GNSS on X.

We can easily generalize the operations of generalized intersection and union in definition 3.4 to arbitrary family of GNSS as follow:

Definition

Let be a arbitrary family of in , then

maybe defined as:

1)

2)

maybe defined as:

1)

2)

Definition

Let and are generalized neutrosophic sets then

may be defined as

Proposition

For all two generalized neutrosophic sets then the following are true

i)

ii)

4. Generalized Neutrosophic Topological Spaces

Here we extend the concepts of and intuitionistic fuzzy topological space [5, 7], and neutrosophic topological Space [ 9] to the case of generalized neutrosophic sets.

Definition

A generalized neutrosophic topology (GNT for short) an a non empty set is a family of generalized neutrosophic subsets in satisfying the following axioms

In this case the pair is called a generalized neutrosophic topological space (G for short) and any neutrosophic set in is known as neuterosophic open set ( for short) in . The elements of are called open generalized neutrosophic sets, A generalized neutrosophic set F is closed if and only if it C (F) is generalized neutrosophic open.

Remark A generalized neutrosophic topological spaces are very natural generalizations of intuitionistic fuzzy topological spaces allow more general functions to be members of intuitionistic fuzzy topology.

Example

Let and

Then the family of G in is generalized neutrosophic topology on

Example

Let be a fuzzy topological space in Changes [4] sense such that is not indiscrete suppose now that then we can construct two G on as follows

Proposition

Let be a GNT on , then we can also construct several GNTSS on in the following way:

a)

b)

Proof a)

and are easy.

Let.Since, we have This similar to (a)

Definition

Let be two generalized neutrosophic topological spaces on . Then is said be contained in (in symbols) if for each. In this case, we also say that is coarser than .

Proposition

Let be a family of on . Then is A generalized neutrosophic topology on .Furthermore, is the coarsest NT on containing all. , s

Proof. Obvious

Definition

The complement of (C (A) for short) of is called a generalized neutrosophic closed set (G for short) in .

Now, we define generalized neutrosophic closure and interior operations in generalized neutrosophic topological spaces:

Definition

Let be G NTS and be a G in .

Then the generalized neutrosophic closer and generalized neutrosophic interior of Aare defined by

G

G.It can be also shown that

It can be also shown that is and is a G in

is in if and only if G.

is Gin if and only if G.

Proposition

For any generalized neutrosophic set in we have

(a) G

(b) G

Proof.

Let and suppose that the family of generalized neutrosophic subsets contained in are indexed by the family if G contained in are indexed by the family. Then we see that and hence. Since and and for each , we obtaining . i.e . Hence follows immediately

This is analogous to (a).

Proposition

Let be a G and be two neutrosophic sets in . Then the following properties hold:

Proof (a), (b) and (e) are obvious (c) follows from (a) and Definitions.