A function theoretical view of fuzzy sets: new extension principle

By definition, a fuzzy set is uniquely characterized by its membership function. Mathematically, a membership function is a unit-interval valued function. So a fuzzy set theory can be viewed from such a functional view. In this paper, fuzzy theories are examined for categories of classical sets, topological spaces (with nice conditions, such as compact Hausdorff) and smooth spaces (closed manifold in Euclidean space). The last two are often required in control theory. Taking this view the traditional extension principle is not valid in these categories, because the principle does not produce continuous/smooth membership function. In this paper, a new general extension principle is established.