Generalized Neutrosophic Set and Generalized Neutrosophic Topological Spaces

A A Salama

doi:10.5923/j.computer.20120207.01

Abstract

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 fu zzy topological space (4) to the case of generalized neutrosophic sets. Possible applicat ion to GIS topology rules are touched upon.

Highlights

Let T, I,F be real standard or nonstandard subsets of ] [ 0−,1+, withNeutrosophy has laid the foundation for a whole family of Sup_T=t_sup, inf_T=t_inf new mathematical theories generalizing both their classicalSup_I=i_sup, inf_I=i_ inf and fuzzy counterparts, such as a neutrosophic set theory.Sup_F=f_sup, inf_F=f_infThe fuzzy set was introduced by Zadeh [10] in 1965, where n -s u p =t_su p +i_ s u p +f_s u p each element had a degree of membership
The fuzzy set was introduced by Zadeh [10] in 1965, where n -s u p =t_su p +i_s u p +f_s u p each element had a degree of membership
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 (GNT1 ) ON,1N ∈τ, ( ) GNT2 G1 G2 ∈τ for any G1,G 2 ∈τ, (GNT3 ) Gi ∈τ ∀{Gi : i ∈ J } ⊆ τ ( ) In this case the pair X,τ is called a generalized neutrosophic topological space (G NTS for short) and any neutrosophic set in τ is known as neuterosophic open set ( NOS for short) in X

Summary

Introduction

Neutrosophy has laid the foundation for a whole family of Sup_T=t_sup, inf_T=t_inf new mathematical theories generalizing both their classical. Sup_I=i_sup, inf_I=i_ inf and fuzzy counterparts, such as a neutrosophic set theory. The fuzzy set was introduced by Zadeh [10] in 1965, where n -s u p =t_su p +i_ s u p +f_s u p each element had a degree of membership. The intuitionstic n-inf=t_ inf+i_inf+f_ inf, fuzzy set (Ifs for short) on a universe X was introduced by K. { } ( ) ( ) ( ) the neutrosophic set concept [7, 8,= 9]. Where introduce definitions of generalized neutrosophic sets. ( ) non-member ship (namely γ A x ) respectively of each element x ∈ X to the set A

Terminologies

Generalized Neutrosophic Sets

A B may be defined as:

Generalized Neutrosophic Topological Spaces

G NTSS on X as follows