Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces

In this paper we introduce definitions of generalized neutrosophic sets. After given the fundamental defin itions 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], intuition istic fuzzy topological space [5, 6], and fuzzy topological space [4] to the case of generalized neutrosophic sets. Possible applicat ion to GIS topology rules are touched upon.


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 defin itions 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].

Definiti on [9].
The NSS 0 N and 1 N in X as follows: 0 N may be defined as: 1 N may be defined as:

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

Definiti on
Let X be a non-empty fixed set. A generalized neutrosophic set (G NS for short) A is an object having the form

Re mark
A generalized neutrosophic For the sake of simp licity, we shall use the symbol , , ,

Example
Every GIFS A a non-empty set X is obviously on GNS having the form defined as three kinds of comp lements One can define several relat ions and operations between GNSS as fo llo ws: Definiti on Let x be a non-empty set, and GNSS A and B in the form , , , then we may consider two possible definit ions for subsets ( ) For any generalized neutrosophic set A the following are holds

Definiti on
Let X be a non-empty set, and may be defined as:

Definiti on
Let A and B are generalized neutrosophic sets then A B may be defined as For all , A B two generalized neutrosophic sets then the following are true i)

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.

Definiti on
A generalized neutrosophic topology (GNT for short) an a non empty set X is a family τ of generalized neutrosophic subsets in X satisfying the following axio ms In this case the pair ( ) ,τ X be a fuzzy topological space in Changes [4] sense such that 0 τ is not indiscrete suppose now that X τ be a GNT on X , then we can also construct several GNTSS on X in the following way : a) . In this case, we also say that 1 τ is coarser than 2 τ . Proposition be a family of NTSS on X . Then , , Then the generalized neutrosophic closer and generalized neutrosophic interio r of Aare defined by G .It can be also shown that It can be also shown that