Neutrosophic n-Valued Refined Sets and Topologies

In n-Valued refined logic truth value T can be split into many types of truths: T₁, T₂,…, T_p and I into many types of indeterminacies: I₁, I₂,…, I_r and F into many types of falsities: F₁, F₂,…, F_s, where p, r and s are integers greater than 1, and p+r +s = n. Importance of n-valued refined logic and sets appeared in different applications specially in medical diagnosis. In this paper we post a condition on neutrosophic n-valued refined sets to make them functional to be applied in different mathematical branches. We define and study n-valued refined topological spaces. We defined neutrosophic n-valued refined α-open, β-open, pre-open and semi-open sets and studied their properties.