Generalizations of Some Basic Theorems of Real Analysis

We define for each 𝛼 ∈ [0,1], the nested 𝛼 −level sets for fuzzy sets and the nested 𝛼 −intervals for fuzzy numbers as a generalization of the nested intervals of the Real line (ℝ). Furthermore, we prove two essential theorems: the nested 𝛼 −level sets theorems and 
the intermediate-value theorem for fuzzy numbers. The results generalize existing results in the literature.