Topological interpretations of fuzzy subsets. A unified approach for fuzzy thresholding algorithms

We show that the classical definition of a fuzzy subset carries additional structures of a topological nature. We look at the concept of a fuzzy subset and its corresponding α-cuts from an alternative point of view: namely, a fuzzy subset may be interpreted as a nested topology on a crisp set of reference, called a universe. Several kinds of fuzzy subsets associated to this interpretation are analyzed. Other topologies induced by fuzzy subsets are considered, paying special attention to their relationship with total preorders defined on the universe. Moreover, this theoretical approach allows us to provide a unified framework for most of the fuzzy thresholding algorithms that can be found in the literature.